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Posts by Year
What To Do as an Undergraduate
Some introductory notes on derived algebraic geometry from an MSRI workshop series.
Some notes on a short introductory video on some foundational aspects of infinity categories.
Some notes on topics to learn to help prepare for grad school.
Below is a partial transcription and some notes I took while watching the following talk from Benson Farb:
Other Pages with Great Advice
Ever had trouble writing ξ? Yeah, me too, so I made these practice worksheets. Enjoy!
As part of our teaching training course at UGA, we wrote up and presented a small talk on various topics from Multivariable Calculus. I described an approach...
In the last year of my undergraduate degree, I did two quarters of undergraduate research, and the purpose of this post is to capture some of the research pr...
A relatively short introduction to Category Theory, including some concrete examples.
Notes on some nifty packages and tools for Haskell development.
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Statistics - Cantor's Archive
Huber Loss: Why Is It, Like How It Is?
On the day I was introduced to Huber loss by Michal Fabinger, the very first thing that came to my mind was the question: “How did someone joined these two functions in a mathematical way?”. Apart
from that, the usage of Huber loss was pretty straightforward to understand when he
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Python Function: Ask User for Number and Add Second Number
Oops, something went wrong. Please try again in a few moments.
def add_numbers():
Function to ask the user for input and add two numbers if the first number is between 20 and 50.
- int:
The sum of the two numbers if the first number is between 20 and 50, otherwise returns None.
# Ask the user to input a number
number1 = int(input("Enter a number: "))
# Check if the number is between 20 and 50
if 20 <= number1 <= 50:
# Ask the user to input a second number
number2 = int(input("Enter a second number: "))
# Add the two numbers
result = number1 + number2
# Return the sum
return result
# Return None if the first number is not between 20 and 50
return None
# Example usage of the add_numbers function
sum_result = add_numbers()
if sum_result is not None:
print(f"The sum of the two numbers is: {sum_result}")
print("The first number is not between 20 and 50.")
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2D Groundwater flow
2D Groundwater flow
Surface water is not only affected by rain and the inflow and outflow of water originating from other regions. In reality, surface water interacts with groundwater. This implies that the behaviour
and the quantity of water at the surface are affected by groundwater too. The flow in the subsurface is strongly influenced by processes on both very large and small scales in time and space. To
consider all processes, models with significant computational times are required, even when surface flow, sewerage systems, and other processes are neglected.
3Di allows you to include these surface water-groundwater interactions in your model. For the computation of surface flow, detailed information about topography and land use is often available.
However, information about the soil is often much less accurate and detailed. This lack of data and the complexity of the processes involved favour a scenario-based approach when dealing with
groundwater flow. Especially when investigating the sensitivity of areas to flooding and hindrance originating from groundwater. This approach requires a fast numerical model that can integrate the
effects of the sewerage system, surface water, and overland flow. Therefore, processes need to be simplified.
First, a short summary of the concepts implemented in 3Di is presented. In the sections that follow, some more detail and context is given about large-scale groundwater flow, the implementation, and
the choices made in 3Di.
Summary of groundwater concepts in 3Di
The figure above shows a cross-section of a region with surface and sub-surface water. The letters in the figure refer to the following description of the main assumptions made for the computation of
groundwater flow:
1. The phreatic surface is assumed equal to the water table. The soil below is assumed fully saturated and the soil above assumed completely dry.
2. In the saturated zone, the flow is assumed hydrostatic and horizontal (Dupuit-assumption).
3. An example where assumption B is locally not strictly valid, is at a stream edge where the gradient in the groundwater level is high.
4. The porosity (phreatic storage capacity) is a single, spatially variable value. It represents the potential storage in the saturated zone. Wetting or drying effects and isolation of moisture is
not considered.
5. The infiltration is based on the Horton equation.
6. At the bottom boundary, effects of deeper groundwater layers or extraction of water can be defined. This bottom boundary condition is called leakage and is assumed constant in time and spatially
variable. An elaborate explanation of Leakage can be found Leakage.
7. For modelling a soil water zone, groundwater can be combined with 2D Interflow.
In the section below, while using the Letters and Numbers in Figure 1 and 3 the key concepts and the assumptions made for the groundwater computations are explained in more detail.
Groundwater concepts
The subsurface is a general term for the whole domain below the surface, where many processes of the hydrological cycle take place. 3Di aims at a fast, but accurate computation of the flow,
especially concerning the interaction between groundwater and surface water. Therefore, the domain of computation focuses on the top aquifer. However, before zooming in at this layer, a schematic
overview of some of the large scale processes is given in Figure 1. The various processes that are discussed here are indicated by Romain numbers. Number I indicates surface flow and overland flow.
From the surface (Number II ) water can infiltrate or exfiltrate to and from the subsurface, where it can flow further in the horizontal or the vertical direction (Number III ). From thereon, several
aquifers can overlap and interact. As they are separated by (semi-)impervious layers, they can exist under different pressure regimes. The exchange can, therefore, occur in both up- and downward
direction (Numbers IV and V ). One aquifer can consist of a zone of saturation and the zone of acration (unsaturated zone). In addition, within one aquifer, the soil characteristics vary over time
and space (Number VI ). To limit the modelling domain and the number of processes to be taken into account, the current method for modelling groundwater flow in 3Di, is focused on the processes in
the top aquifer of the sub-surface layer (the red box of Number VII ).
Most of the groundwater concepts on which the groundwater method in 3Di is based, are thoroughly described in (Bear and Verruijt, 1987). However, here a short overview is given of the key concepts
and assumptions made for the groundwater method used in 3Di. These concepts are illustrated in the Figures 1 and 3 and indicated with Letters and Numbers. The general aim is to simplify the processes
involved in the top aquifer, but to preserve enough accuracy for reliable simulations of the surface-subsurface interaction. The numbers below refer to those in Figure 3 and the letters refer to
those in Figure 1.
1. When only looking at the top aquifer in a system, only one phreatic surface can be defined. This is the level at which the pressure is atmospheric (assumed zero). Below the phreatic surface is
the soil fully saturated.
2. Above the phreatic surface is the vadose water zone. There, some of the pore space is actually occupied by water, although the soil is not fully saturated. In the right graph, near Number 2, is
the saturation of the soil is plotted as fuction of the depth. As can be seen, the change in gradient can be quite steep and is often approximated by a step function. The step-size depends on
various issues including the characteristics of the soil. In the vadose water zone, pressures are negative, which allows the water to go upwards (capillary fringe). This can be seen in the left
graph of depth versus pressure.
3. At the top of the capillary water, the water table is defined. In many applications it is valid to approximate the groundwater table at the top of the capillary fringe by assuming the soil to be
saturated below this level and completely dry above it. This assumption is called the capillary fringe approximation and gives in combination with surface water flow, a two-layer system. When the
\(h_c\) is much smaller than the thickness of the aquifer, the capillary fringe can be neglected. Then, the water table and the phreatic surface are at the same level. This is indeed assumed in
3Di. This is indicated by the Letter A.
4. The main flow in an aquifer follows the phreatic surface, therefore the phreatic surface is considered to be a stream-line. Within an aquifer the slope of the phreatic surface (\(i\)) is
generally small. It is often much smaller than 1 ( \(i<<1\) ) (Dupuit, 1863). In such case, one can assume the stream-lines to be horizontal, and use only the horizontal Darcy equations to
compute the flow. The groundwater level gradients are than defined by the height of the phreatic surface. This is consistent with assuming a hydrostatic pressure within the aquifer. This
assumption is called the Dupuit approximation (Letter B ).
5. The Dupuit approximation can be locally valid, while in other regions it can be invalid. Number 5 indicates an example where the gradient of the stream-lines is high. The dashed red line
indicates where the Dupuit assumption is invalid. In stationary cases, one can apply the so-called Dupuit-Forchheimer discharge formula to compute the outflow from groundwater to surface water.
The computation of the discharge is still quite accurate, even though the groundwater levels deviate. In regions further than ones or twice the \(\Delta h\), the solution approximates again the
actual solution. In 3Di (Letter C ), the Dupuit-Forchheimer discharge formula is at these interfaces not applied, as they are often not a priori known. However, for practical purpose this is
often only a local deviation.
6. The storage capacity in the soil is naturally very important, as it determines the volume that can be added and extracted from the soil. However, the storage capacity and the saturation of the
soil is related to very complex processes. This deals with the pores, the distribution of pores and the molecular behaviour of water interacting with the soil. These processes are responsible for
the amount of water that can be added or be extracted to/ from the soil. Therefore, for each soil type there is difference between porosity, the specific yield and the specific retention. Where
the porosity is measure for the pore space, the specific yield, also known as the effective porosity, is a measure for the space where water can be added or extracted. Whereas, the specific
retention is representative for the space within the pores where water cannot be added nor extracted, for example in isolated pores. These values are actually also dependent on the local pressure
distribution and partly also whether the pores where previously filled or dry. For simplicity, all these processes are simplified by defining a phreatic storage capacity that is a measure for the
effective storage in this layer (Letter D ). Although, this is a simplification of reality, the structures in the soil at this level of detail are generally unknown and can, therefore not be
added to a model.
7. In case of a porous surface layer, surface water will be flowing downward due to gravity, depending on the pressure gradient, the saturation and the hydraulic connectivity. As seen in the graph,
there will be a saturated front flowing downward. There is a difference between the infiltration rate and the effective infiltration velocity. The infiltration rate is the rate in which the
surface water level decreases. The effective infiltration velocity is the velocity of the front of the saturated zone. Due to differences in porosity the effective velocity can vary with depth.
The vertical flow can be described by a Darcy-like formulation in the vertical:
\[ q(x,y,z,t) = -\kappa(x,y,z) \frac{\partial \phi}{\partial z}\]
where \(\phi\) is the hydraulic head. This equation is seemingly simple, but the hydraulic head and the hydraulic connectivity are both dependent on the saturation of the soil. Due to the
complexity of the infiltration processes, there are various formulations for infiltration, such as Green and Ampt, Horton and Philip infiltration. There are several differences between those
formulations. However, they share that the infiltration rate is initially higher and decreases more or less exponentially to an equilibrium rate. For now, only the Horton-based infiltration, see
Horton based infiltration, is implemented, which is a formulation, originally, for ponded infiltration only. The formulation described by Horton (1875-1945) takes into account that when the soil
contains more water, the infiltration rate will decrease. This can be seen in the graph in the at Label E .
8. Within the soil, multiple aquifers can exist within one domain. Such aquifers are separated by (semi) impervious layers, but these can leak. To simulate the potential interaction between these
layers, it is possible to add a bottom boundary condition for flow. This can represent the possible effect of deeper groundwater layers or other sources of extraction or recharge (See Label F ).
9. The soil water zone is the layer just below the surface. Often this is a fully saturated area, but the processes in this layer are heavily affected by the vegetation, precipitation and
evaporation. Therefor, often the simulation of this layer is difficult. In case of heavy precipitation, this layer becomes saturated in a sort time. In such case, a user can simulate this layer
with use of the interflow layer (Label G ).
Horton based infiltration
Mentioned above, the infiltration process is rather complex, therefore many models use a parametrization for this process. In 3Di, two types of infiltration formulations are implemented; Horton based
infiltration and a constant infiltration. Only the Horton based infiltration is coupled to groundwater. More information about the constant infiltration can be found at Simple Infiltration. Here,
only the Horton infiltration is discussed.
Horton based infiltration formulation describes infiltration rate that is decaying in time. Three variables determine the infiltration rate. It is based on the notion that the infiltration rate
decays to an equilibrium value, due to changes in the soil characteristics. Mathematically, it is defined by:
\[ f(x,y,t) = f_{equ}(x,y)+( f_{ini}(x,y)-f_{equ}(x,y))e^{-t/T(x,y)}\]
in which \(f\) is the infiltration rate varying in time and space, \(f_{equ}\) and \(f_{ini}\) are the equilibrium and the initial infiltration rates, respectively. The decay period \(T\) determines
the time that the infiltration rate reaches its equilibrium. An example of the decay function is shown in Figure 4.
The infiltration rate will start its decay as soon as the cell becomes wet. Currently, there is no process to restore the infiltration rate to its initial value. This would happen in real life when
an area becomes dry again due to run-off or evaporation.
For the use of Horton infiltration, one chooses indirectly to take a groundwater level into account. This to ensure a limit to the infiltration; when the groundwater level reaches the surface. To
take the storage capacity of the soil into account, one needs to define the impervious surface layer and the phreatic storage capacity, as well. The three Horton parameters (in [mm/day]), the
impervious surface layer ([m] relative to a reference level) and the phreatic storage capacity ([-] between 0-1) can be defined globally and spatially varying. In case one uses the spatially varying
option, a user needs to define a method for analyzing the rasters (taking the minimum, maximum or the average in a computational domain).
The initial conditions for the groundwater level can be added to the Global settings table using a global value or a raster for spatially varying values.
You can download the complete overview of tables that 3Di uses in the spatialite database here.
Similar to the other variables, the results are saved in the result files, snap-shots and aggregated results. In contrast to infiltration computed according to Simple Infiltration, the Horton-based
infiltration is computed on a flow line. Both a discharge (\([m^3/s]\)) and a velocity ([m/s]) are available as output. Note, that the velocity is the infiltration rate and not the effective
velocity. The effective velocity is the velocity that the water front would subside through the soil.
Groundwater flow
The flow in the subsurface is computed under the assumption of hydrostatic pressure. This is also known as the Dupuit assumption. This implies that the flow in the saturated zone is fully horizontal
and described by the Darcy equations:
\[ \begin{aligned}Q_x=-K_x A_x \frac{\partial \phi}{\partial x}\\Q_y=-K_y A_y \frac{\partial \phi}{\partial y}\end{aligned} \]
with \(Q_x, Q_y\) the x- and y- component of the discharges, \(A_x, A_y\) the corresponding cross-sectional areas and the gradients of the phreatic surface (\(\phi\)). Even though, the Dupuit
assumption can be invalid locally, it is very applicable on the larger scale. A famous analytical case, based on these assumptions is the Hooghoudt equation. It describes the groundwater level in
between two open water channels, see Figure (5).
The input for using groundwater flow is very similar to the input for Horton based infiltration. In addition to these parameters, one can define the Darcy or hydraulic connectivity values globally or
using a raster for spatially varying values. The dimension of the hydraulic connectivity is in [m/day]. You can download the complete overview of settings that 3Di uses in the spatialite database
The discharges ([m^3/s]), the velocities [m/s] and the groundwater levels [m] are all included in the NetCDF and in the aggregated results NetCDF. Also for the groundwater related variables yields
that discharges and velocities are defined at flow lines and the water levels at the nodes. Note, that the velocity is the effective velocity, not the velocity of a single water particle.
Numerical implementation [1]
The numerical implementation of the horizontal and vertical flow is based on the concept of staggered grids as explained in Computational grid. This implies that pressure points are defined in the
cell centers and flow is defined at the cell edges. The spatial resolution of the 2D surface flow equals that of the groundwater flow. Therefore, the connections between the surface and the
subsurface are completely vertical and orthogonal to the surface and subsurface layers.
The timescales of groundwater flow compared to those of surface water flow, are generally considerably longer. This would favor an explicit formulation. However, the moment that the groundwater level
reaches the surface, the timescales are the same. Therefore, only the horizontal flow is computed explicitly, but the vertical interaction is computed implicitly.
For the sources and sinks, we choose an implementation where the sources are computed explicitly, but the sinks are implicitly taken into account. This is to guarantee mass conservation.
We are working on a full description of the numerical implementation to be published in International Journal For Numerical Methods in Fluids.
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An optimal solution satisfying men’s preferences is said to be?
Q. An optimal solution satisfying men’s preferences is said to be?
A. man optimal
B. woman optimal
C. pair optimal
D. best optimal
Answer» A. man optimal
Explanation: an optimal solution satisfying men’s preferences are said to be man optimal. an optimal solution satisfying woman’s preferences are said to be woman optimal.
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12 Most Expensive Calculators in the World 2024
Most Expensive Calculators in the World 2024: If you are in the market for a reliable calculator, you could be interested in perusing a rundown of the most pricey models now on the market.
There are a few of them that go for prices that range from tens to even hundreds of thousands of dollars.
Although it’s not anyone’s goal, eventually everyone will struggle with math. There are a significant number of issues that call for the use of a calculator.
You may avoid wasted time and potential frustration by doing basic mathematical operations using a calculator.
Finding the finest arithmetic calculator for your needs can enable you to complete problems more quickly, with fewer errors, and with greater efficiency than when using pen and paper.
Calculators have emerged as one of the most significant innovations in the history of humankind.
After all, rather than having to teach yourself to perform the arithmetic on your own, it is much simpler to simply carry around a little gadget that can do the work for you.
They are utilized at academic institutions, professional settings, and even in some households on occasion.
Calculators may be broken down into two categories: those that are built specifically for straightforward mathematical operations and those that are capable of handling more involved calculations.
Some of them come with extra functions already built in, such as the date and time, the ability to convert currencies, and even games. Most Expensive Calculators in the World 2024
It is possible that the price will be beyond of reach for most people who are looking to purchase a calculator for straightforward activities such as calculating taxes or determining the value of a
On the other hand, if you are interested in anything that is more advanced and high-tech, then you should probably have some spare cash on you.
The following is a list of some of the most costly calculators that have ever been produced.
12. Victor 1460-3 – $294
Most Expensive Calculators in the World 2024
The Victor 1460-3 was a 10-digit pocket calculator that did not print and was powered by solar energy as well as batteries.
It utilized a combination of light-emitting diode (LED) and liquid crystal display (LCD) technologies, with the LED technology being used for the character representing the decimal point as well as
the indicator for the “add mode.”
The look of the Victor 1460-3 was comparable to that of a large number of other calculators that were developed by a variety of different companies in the early 1970s.
The Victor 1460-3 had two memory registers and automated constant computation for multiplication and division. Additionally, it had automatic constant calculation.
The calculator may function in either the conventional mode or the adding machine style thanks to a switch called the add mode.
A switch with three positions for the decimal point allowed for floating decimal points as well as fixed decimal settings with either zero, one, or two digits.
The arithmetic logic of the Victor 1460-3 had four functions, and it offered eight-digit precision as well as square root keys. Most Expensive Calculators in the World 2024
11. Texas Instruments TI-86 – $299
Most Expensive Calculators in the World 2024
One of the most useful calculators for students in high school and colleges, as well as teachers at those levels of education, is the Texas Instruments TI-86.
This is due to the fact that it possesses a large number of functions that are helpful for various kinds of classes.
It is capable of performing a wide range of mathematical operations, from elementary algebra through calculus, as well as more difficult mathematical operations such as statistics and complex number
This is also very helpful for testing since it can keep track of answers and present them when you want to review your work. Having this capability makes testing much more convenient. This calculator
is permitted for use in school, but you won’t be able to bring it with you on the SAT or ACT.
You will spend around $299 on a Texas Instruments TI-86, so you need to carefully consider whether or not the purchase of this calculator will be worthwhile to you.
10. HP 19BII Financial Calculator – $299
The HP 19BII is an outstanding calculator in our opinion. It is brisk, dependable, and potent in its operation. I’ve had mine for years, and with the exception of having to replace the batteries
twice, it’s never given me any problems.
This calculator can handle math issues with up to 10 digits, as well as any financial computations that you need to carry out.
The fact that you do not need to commit any of its functions or programming code to memory in order to utilize the 19BII is the feature that the 19BII excels at.
Simply input the values you like to be computed (or ask it a question such as “what is my monthly payment on a loan of $100,000 with a 6% interest rate for a term of 30 years?”) and the calculator
will do the rest. and the solution will be shown to you by the calculator in considerably less time than a second.
If you are unsure how to calculate anything, simply enter the problem into the calculator and see what answers it offers you. This will help you figure out how to compute the item.
You will, in nine out of ten cases, get a response to your question fairly promptly. There are help options built into the calculator that, in the event that you are unable to figure things out on
your own, will walk you through any challenges, step by step.
The 19BII graphing calculator comes with an accompanying handbook that details each and every operation that it is capable of carrying out.
There are some complex features, such as solving equations and utilizing logarithms, so if you desire to investigate those possibilities, you should be prepared to do some arithmetic. Some of the
features include: Most Expensive Calculators in the World 2024
9. SUQIAOQIAO New Graphing LED Calculator – $340
We are convinced that the SUQIAOQIAO New Graphing LED Calculator will meet the requirements of the vast majority of pupils despite the fact that it is one of the most costly and one of the best
calculators that we evaluated.
It’s not a cheap calculator, but it’s not that costly, either.
It is a graphing calculator that provides all of the features that you require while omitting the functions that you do not require.
It comes with a comprehensive user manual that walks you through how to use the calculator and covers many of the frequent mistakes that people make while using it.
It includes a screen that is simple to see and can be dimmed to one of three different brightness settings, which can make it easier on your eyes to read for extended periods of time when performing
It has the capacity to remember up to ten distinct graphs at once and comes with an intuitive history tape that displays all of your computations in real time as you enter them in.
This particular model furthermore features a protective cover that may be used to prop up the calculator when not in use.
Most Expensive Calculators in the World 2024
8. Hewlett Packard 41CV calculator – $449
The HP 41CV is regarded as one of the most cutting-edge calculators currently available. The HP 41C was first released in 1979 and was eventually phased out of production in 1990.
The HP 41CX, which is identical to it but adds additional features, has since replaced it as the subsequent model.
It was also one of the first machines to offer an optional thermal printer, as well as RAM and ROM memory modules that could be placed into its expansion ports. These features made it one of the
earliest computers ever produced.
The calculator’s built-in 4K RAM pack made it possible for programs to be saved on the device.
The typical 8K ROM pack had over 80 built-in commands and functions, some of which were related to programming, while others dealt with topics such as programming, statistics, trigonometry,
scientific calculations, and financial functions.
It is able to answer issues that are beyond the capabilities of the majority of pocket calculators (for example, algebraic equations).
This handbook explains how to use the calculator to find solutions to issues involving mathematics and engineering.
7. HP 32Sii Scientific Calculator – $700
One of the greatest scientific calculators available is the HP 32Sii, which also happens to be one of the most costly.
It is an excellent resource for academics, making it useful not only for students and professionals but also for lecturers.
Calculations in mathematics, physics, and engineering are made much simpler with the help of the many capabilities that come standard on the HP 32Sii.
Because the user interface is so straightforward and easy to understand, you can get started right away.
The HP 32Sii is equipped with several different functions, some of which include calculus, statistics, trigonometry, basic arithmetic, algebraic equations, and basic arithmetic.
The HP 32Sii employs a notation system known as Reverse Polish Notation (RPN). RPN is not appropriate for learners just starting out.
This calculator features one storage register that is labeled LAST X and four memory registers that go by the names X, Y, Z, and T.
The number that you typed into the calculator most recently is stored in the LAST X register of the calculator.
This calculator allows for user programming. Up to 400 steps of programmable space can be stored in its memory.
However, it does not provide command support for conditional branching or looping in the control flow. Most Expensive Calculators in the World 2024
6. Canon numeric keypad calculator X Mark I KRF White – $799
One of the most reliable and accurate calculators is the Canon numeric keypad calculator X Mark I KRF White.
It is manufactured in Japan, and in addition to having a good quality, it serves several purposes, and it can be utilized everywhere.
The Canon X Mark I KRF White 12-Digit Calculator is prepared to assist you with the calculations that you need to perform for your finances.
This Canon numeric keypad calculator has a display that is both big and simple to see, and it also has capabilities that make it simple to work through difficult issues.
As a result of the inclusion of a two-way power supply in this Canon X Mark I KRF, you won’t ever have to worry about the device running out of batteries.
This calculator will make your life simpler if you work in the subject of mathematics or accounting, regardless matter whether you are a student or an experienced expert.
5. SUQIAOQIAO New 100% Calculator – $919
This enticing piece of technology is more than simply a calculator in its own right. You are able to conduct not only comprehensive general calculation operations, but also mathematical equations
relating to finances.
In the same way as a digital screen would, the huge LED display’s visuals seem wonderful due to the vitality they possess.
Because it contains numeric and alphabetic keypads in addition to navigation pads, statistical computation with one or two variables may be performed quickly and simply with this device.
You are also able to compute the three different angle modes, degrees, and grads, in addition to engineering notation modes, 2-D operation, parameter mapping, and a wide variety of additional
The fact that this calculator has a one-of-a-kind and high-quality design sets it apart from the majority of other calculators that are currently available on the market.
The calculator may also be powered by batteries, so you won’t have to bother about connecting it to an electrical outlet whenever you use it.
This calculator makes use of the most recent technological advancements to provide you with the most precise computations that are currently achievable.
Because it is powered by solar energy, you won’t need to worry about replacing the batteries as frequently as you would otherwise. Most Expensive Calculators in the World 2024
4. Texas Instruments Math Etica Ti 83PLUS – $1,500
One of the most useful calculators for students to have at their disposal is the Texas Instruments Math Etica Ti 83PLUS. It is also an excellent option for high school students who are interested in
getting a jump start.
When you are working on lengthy equations, the TI-83 Plus’s screen, which is quite large, makes it simpler to monitor your progress and see what you are doing.
The previous model’s display has been improved, making this one an even stronger contender among the alternatives.
Equation solving and table making are only two of the useful functions that are included in the calculator’s extensive feature set.
Students that have a passion for mathematics and want to be able to do more with their calculator than just calculate equations will find that this model is ideal.
Students who desire to begin their academic careers with a strong foundation in mathematics will find that the TI-83 Plus is an excellent choice for them.
Before having to deal with more sophisticated subjects like calculus or trigonometry, it will help students understand how to solve basic equations by employing variables. This will prepare them for
those subjects.
You are able to graph your results and create tables based on the data that you have entered into this calculator. This functionality is also available to you.
If you are studying statistics or finance and need to make sense of difficult information in a quick and easy manner, this might be helpful for you.
2. Wolfram Mathematica 9 – $6,995
Most Expensive Calculators in the World 2024
One of the most advanced calculators currently available is version 9 of Wolfram Mathematica. It is not just a calculator, but it also has the ability to answer complicated equations.
It provides access to a broad array of functions, which in turn enables you to do a variety of tasks. Users are able to see data in the form of graphs thanks to one of its functionalities called 3D
Calculating symbolic derivatives, integrals, and limits is one of the many benefits you may reap from utilizing this sophisticated tool. Additionally, you will have the ability to plot functions with
polar coordinates, which is an important skill in calculus.
It is able to perform numerical computations, which is the trait that stands out as the most valuable among its many other amazing qualities.
Because of this, it is ideally suited for solving issues involving differential equations (for example, solving the equation x2 + 1 = 0), in which the result is often a number rather than a formula
(x = -1); this makes it perfect for addressing problems in differential equations.
1. Grillet portable Calculator – $1,55,000
Most Expensive Calculators in the World 2024
Rene Grillet de Roven is credited with inventing the first portable calculator in 1763. This device, which used Napier’s logarithm to do mathematical operations and solve complex divisions, was a
global first.
You probably won’t use it for arithmetic, but believe me when I say that simply for its antiquity, it is worth 155 thousand dollars.
But in all seriousness!! The cost does not adequately reflect the value of this portable calculator, which allowed us access to a plethora of new possibilities from which we are currently reaping the
Therefore, do you not believe that the cost is appropriate???
Since the beginning of civilization, those who work in business and innovation have found great value in having access to calculators.
Did you know that the basic law of the calculator was the inspiration for the creation of computers? In today’s world, we rely on computers to perform even the most complicated of tasks.
Most Expensive Calculators in the World 2024- Newshub360.net
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Exercise C6, bivariate Poisson models
Cameron and Trivedi (1988) use various forms of overdispersed Poisson model to study the relationship between type of health insurance and various responses which measure the demand for health care,
e.g. number of consultations with a doctor or specialist. The data set they use in this analysis is from the Australian Health survey for 1977-1978. In a later work Cameron and Trivedi (1998)
estimate a bivariate Poisson model. for two of the measures for the demand for health care. We use a version of the Cameron and Trivedi (1988) data set (called visit-nonprescribe.dat) for the
bivariate model. The data for the bivariate model (visit-nonpresc.dat) are a stacked version of Cameron and Trivedi data set. A copy of the original data set (racd.dat) and further details about the
variables in racd.dat can be obtained from http://cameron.econ.ucdavis.edu/racd/racddata.html.
Data description
Number of observations (rows): 10380
Number of variables (columns): 26
ij = respondent identifier
r = if this is the 1^st measure of the demand for health care, 2 for the second.
sex= 1 if respondent is female, 0 if male
age = respondent’s age in years divided by 100,
agesq = age squared
income = respondent’s annual income in Australian dollars divided by 1000
levyplus =1 if respondent is covered by private health insurance fund for private patient in public hospital (with doctor of choice), 0 otherwise
freepoor =1 if respondent is covered by government because low income, recent immigrant, unemployed, 0 otherwise
freerepa=1 if respondent is covered free by government because of old-age or disability pension, or because invalid veteran or family of deceased veteran, 0 otherwise
illness = number of illnesses in past 2 weeks with 5 or more coded as 5
actdays = number of days of reduced activity in past two weeks due to illness or injury
hscore = respondent’s general health questionnaire score using Goldberg's method, high score indicates bad health.
chcond1 = 1 if respondent has chronic condition(s) but not limited in activity, 0 otherwise
chcond2 = 1 if respondent has chronic condition(s) and limited in activity, 0 otherwise
dvisits = number of consultations with a doctor or specialist in the past 2 weeks
nondocco = number of consultations with non-doctor health professionals, (chemist, optician, physiotherapist, social worker, district community nurse, chiropodist or chiropractor in the past 2 weeks
hospadmi = number of admissions to a hospital, psychiatric hospital, nursing or convalescent home in the past 12 months (up to 5 or more admissions which is coded as 5)
hospdays = number of nights in a hospital, etc. during most recent admission, in past 12 months
medicine = total number of prescribed and nonprescribed medications used in past 2 days
prescribe = total number of prescribed medications used in past 2 days
nonprescribe = total number of nonprescribed medications used in past 2 days
constant = 1 for all observations
id= ij
y = when r1=1, y is dvisits and when r2=1, y is nonprescribe
r1 = 1 if r=1, 0 otherwise
r2 = 1 if r=2, 0 otherwise
The first few lines of the stacked data (visit-nonpresc.dat) look like:
Suggested exercise:
Univariate models
We will start by estimating separate random effect models on the dvisits and nonprecribe responses in the unstacked data set (racd.dat). The first few lines of racd.dat look like
Start Sabre and specify transcript file:
out nonpresc.log
data sex age agesq income levyplus freepoor freerepa illness actdays &
hscore chcond1 chcond2 dvisits nondocco hospadmi hospdays medicine &
prescrib nonpresc constant id
read racd.dat
(1) Fit a random effect Poisson model to dvisits. use a log link and the following explanatory variables lfit sex age agesq income levyplus freepoor freerepa illness actdays hscore chcond1 chcond2
cons. What do you find?
(2) Fit a random effect Poisson model to nonprescribe, use a log link and the following explanatory variables lfit sex age agesq income levyplus freepoor freerepa illness actdays hscore chcond1
chcond2 cons. What do you find?
Bivariate Models
We cross tabulate dvisits by nonprescribe in the following table.
Is the assumption of independence between dvisits and nonprescribe realistic?
To estimate the bivariate poisson model you will need to read the stacked version of the data (visits-nonpresc.dat) into Sabre.
out visit-nonpresc.log
data ij r sex age agesq income levyplus freepoor freerepa illness actdays &
hscore chcond1 chcond2 dvisits nondocco hospadmi hospdays medicine &
prescrib nonpresc constant id y r1 r2
read visit-nonpresc.dat
(3) Estimate a bivariate Poisson model for dvisits and nonprecribe using the same explanatory variables for each response in parts 1 and 2. What is the magnitude and significance of the correlation
between the random effects of each response? How many quadrature points for each response should we use to estimate this model? Interpret your results?
(4) What do you think are the main problems of applying the bivariate model in this context?
Cameron, A.C., Trivedi, P.K., Milne, F., Piggott, J., (1988) A microeconometric model of the demand for Health Care and Health Insurance in Australia, Review of Economic Studies, 55, 85-106.
Cameron, A.C., Trivedi, P.K (1998), Regression Analysis of Count Data, Econometric Society Monograph No.30, Cambridge University Press,
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"IF" formula for a score range
I'm looking for some help with an IF formula to assign a text value in one column based on a score range from a different column.
Thank you in advance for helping me figure out how to write this so that it works in the Smartsheet.
Best Answers
• #UNPARSEABLE is most frequently caused by an imbalance in your ( and ). Try removing one ) from your formula and see if this fixes the error. Also keep in mind on nested IF statements that it
will stop looking to solve when it hits the first TRUE in the conditionals. Your current nesting seems to be correct and works in my testing.
• Thank you, @Malaina Hudson
That fixed the problem!! Extra parentheses was the problem.
• #UNPARSEABLE is most frequently caused by an imbalance in your ( and ). Try removing one ) from your formula and see if this fixes the error. Also keep in mind on nested IF statements that it
will stop looking to solve when it hits the first TRUE in the conditionals. Your current nesting seems to be correct and works in my testing.
• Thank you, @Malaina Hudson
That fixed the problem!! Extra parentheses was the problem.
Help Article Resources
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Slices of Theoretical Astrophysics: Solar System Dynamics and Relativistic Explosions
This thesis presents studies in two distinct areas of theoretical astrophysics: dynamics of planetary systems and relativistic fluid flows from shocks emerging from stellar envelopes. The first
pertains to the early solar system, planet formation, and extrasolar planets; the second is related to extreme explosions like gamma-ray bursts and supernovae.
We present two investigations of the dynamics and population evolution of small solar system bodies. First, we explore the dynamics of mean-motion resonances for a test particle moving in a highly
eccentric long-period orbit in the restricted circular planar three-body problem --- a scenario relevant to the scattered Kuiper belt and the formation of the Oort cloud. We find an infinite number
of analogues to the Lagrange points; a simple explanation for the presence and absence of asymmetric librations in particular mean-motion resonances; and a criterion for the onset of chaotic motion
at large semimajor axes.
Second, we study the size distribution of Kuiper belt objects (KBOs), which is observed to be a broken power law. We apply a simple mass conservation argument to the KBO collisional cascade to get
the power-law slope for KBOs below the break; our result agrees well with observations if we assume KBOs are held together by self-gravity rather than material strength. We also explain the location
and time evolution of the break in the size distribution.
We also present investigations of the flow which results when a relativistic shock propagates through and then breaks out of a stellar envelope with a polytropic density profile. This work informs
predictions of the speed of and energy carried by the relativistic ejecta in supernovae and perhaps in gamma-ray bursts. We find the asymptotic solution for the flow as the shock reaches the star's
edge and find a new self-similar solution for flow of hot fluid after the shock breakout. Since the post-breakout flow acclerates by converting its thermal energy into bulk kinetic energy, the fluid
in the flow eventually cools to non-relativistic temperatures. We derive a second new self-similar solution which includes the cold portions of the flow. This second solution gives an exact relation
between the terminal Lorentz factor of each fluid element and the Lorentz factor it acquired upon being shocked before breakout.
Item Type: Thesis (Dissertation (Ph.D.))
Subject Keywords: collisional cascades; planetary dynamics; relativistic shocks; similarity solutions
Degree Grantor: California Institute of Technology
Division: Physics, Mathematics and Astronomy
Major Option: Astrophysics
Thesis Availability: Public (worldwide access)
Research Advisor(s): • Sari, Re'em
Group: TAPIR, Astronomy Department
Thesis Committee: • Kamionkowski, Marc P. (chair)
• Brown, Michael E.
• Scoville, Nicholas Zabriskie
• Goldreich, Peter Martin
• Sari, Re'em
Defense Date: 23 May 2006
Record Number: CaltechETD:etd-05252006-181025
Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-05252006-181025
DOI: 10.7907/40AE-N743
Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code: 2058
Collection: CaltechTHESIS
Deposited By: Imported from ETD-db
Deposited On: 02 Jun 2006
Last Modified: 08 Apr 2020 18:46
Thesis Files
PDF - Final Version
See Usage Policy.
Repository Staff Only: item control page
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Some papers about p values
These papers have nothing much to do with single molecule kinetics. They were written by David Colquhoun after his retirement from the world of single ion channels, as a way to keep him off the
streets. They are listed here as a convenient place to keep a record.
The papers concern the misinterpretation of tests of significance. Such tests were barely ever used in our single ion channel work. They represent a return to the interest of DC in statistical
inference that he had in the 1960s, and which culminated on the publication of a textbook, Lectures on Biostatistics (OUP, 1971). The textbook has aged quite well, with the exception of the parts on
interpretation of p values. In the 1960s, I missed entirely the problems of null hypothesis significance testing. But better late than never.
The problem lies in the fact that most people still think that the p value is the probability that your results occurred by chance -see, for example. Gigerenzer et al.,(2006) [download pdf]. It is
nothing of the sort.
The false positive risk (FPR) is the probability that a result that has been labelled as “statistically significant” is in fact a false positive. It is always bigger than the p value, often much
My recommendations. In brief, I suggest that p values and confidence intervals should still be cited, but they should be supplemented by a single number that gives an idea of the false positive risk
(FPR). The simplest way to do this is to calculate the false positive risk that corresponds to a prior probability of there being a real effect of 0.5. This would still be optimistic for implausible
hypotheses but it would be a great improvement on p values. The FPR[50], calculated in this way is just a more comprehensible way of citing likelihood ratio (see 2019 paper).
Please note: the term “false discovery rate”, which was used in earlier papers, has now been replaced by “false positive risk”. The reasons for this change are explained in the introduction of the
2017 paper.
If you prefer a video to reading, try this, on YouTube.
Original papers about the problem
Colquhoun, D. (2014). An investigation of the false discovery rate and the misinterpretation of p-values. Royal Society Open Science This first paper looked at the risk of false positive results by
simulation of Student’s t test. The advantage of simulation is that it makes the assumptions very clear without much mathematics. The disadvantage is that the results aren’t very general.
Colquhoun, D. (2017). The reproducibility of research and the misinterpretation of p-values . Royal Society Open Science. This paper gives, in the appendix, mathematically exact solutions or the
false positive risk, calculated by the p-equals method. This allows the false positive risk to be calculated, as a function of the observed p value, for a range of sample sizes. A web calculator is
provided that makes the calculations simple to do.
The source code, app.R, for the web app can be downloaded here.
Update. Thanks to a cock-up by UCL, the web calculator was offline for a while. Thanks to a ingenious solution provided by the wonderful people at our web hosts, Positive Internet, the original link,
http://fpr-calc.ucl.ac.uk/, now works again. That link is actually redirected to an html page (http://fpr-calc.positive-dedicated.net/) on the Positive Internet server in which is embedded the back
up copy that is kindly hosted by Daniel Lakens in the Netherlands. The only difference is that this version on longer rearranges on a mobile phone.
There is also a copy of the web calculator at https://davidcolquhoun.shinyapps.io/fpr-calc-ver1-7/
Colquhoun, D. (2019). The false positive risk: a proposal concerning what to do about p values, The American Statistician, 2019 (open access to full text). Also available at https://arxiv.org/abs/
1802.04888. This paper examines more closely than before the assumptions that are made in calculations of FPR. It makes concrete proposals about how to solve the problem posed by the inadequacy of p
values, with examples.
In the same online edition, The American Statistician published 43 papers that were designed to say what should be done about the problem of abuse of p values.
At the same time, Nature published a comment piece on the p value problem. The gist of this piece was a plea to abandon the term “statistically-significant”, because it involves the obviously silly
idea that observing p = 0.049 tells you something different from p = 0.051. It was co-signed by 840 people (including me). Nature also published an editorial which half-understood the problem and,
sadly, said “Nature is not seeking to change how it considers statistical analysis in evaluation of papers at this time”. This sums up the problem: it is in the interests of both authors, and of
journals, to continue to publish too many false positive results. Until this problem is solved, the corruption will continue.
Colquhoun, D. (2019b). A response to critiques of ‘The reproducibility of research and the misinterpretation of p-values’. Royal Society Open Science. This one started life as a response to a
critique of my 2017 paper. But it evolved into a more general discussion of the assumptions made in my approach, and concluded with a summary of my present views about what should be done about p
values. In brief, I now think that p values and confidence intervals should continue to be given, but they should be supplemented by an estimate of the false positive risk. I suggest the notation FPR
[50] for the false positive risk that’s calculated on the basis that the prior probability of a real effect existing is 0.5.
Popular accounts of the problem
Colquhoun, D. (2015) False discovery rates and P values: the movie. On YouTube. This slide show is now superseded by the 2018 version.
Colquhoun, D. (2015). The perils of p-values. In Chalkdust magazine. Available at http://chalkdustmagazine.com/features/the-perils-of-p-values/. Chalkdust is a magazine run by students at UCL. This
article deals with the principles of randomisation tests as a non-mathematical way to get p values, plus a bit about what’s wrong with p values.
Colquhoun,D.(2015). Randomisation tests. How to get a P value with no mathematics. A short (6 slides, 15 min) video on YouTube. Forget t tests. The randomisation test is at least as powerful and it
makes no assumption of normal distributions. Furthermore it makes very clear the fact that random allocation of treatments is an essential assumption for all tests of statistical significance. Of
course the result is just a p value. It doesn’t tell you the probability that you are wrong: for that, see the other stuff on this page.
Colquhoun, D.(2016). The problem with p-values. Aeon magazine. (This attracted 147 comments.
This essay is about the logic of inductive inferencee. It is a non-mathematical introduction to the ideas raised in my 2014 paper.
Colquhoun, D. (2017). Five ways to fix statistics. State false positive risk, too. Nature, volume 251. A collection of short comments by five authors on what should be done about p values.
Colquhoun, D. (2018). The false positive risk: a proposal concerning what to do about p-values (version 2). This video is a slightly extended version of a talk that I gave at the Evidence Live
meeting, June 2018, at the Centre for Evidence-Based Medicine, Oxford. It supersedes my earlier 2015 video on the same topic. It is an exposition of the ideas that are given in more detail in the
2017 paper and in the 2018 paper. In November 2018 an new version was posted -the content is the same, but the volume of the sound track is better.
Why p values can’t tell you what you need to know and what to do about it
2020 version I gave a talk to the RIOT science club, on 1 October 2020. It has appeared on YouTube. This gave me a chance to update my ideas about what to do about p values. After the talk. Chris F.
Carroll kindly sent me a transcript of it. I took the chance to improve a bit on some of the explanations that I’d given in the talk -especially in the Q&A and I’ve posted the result on my blog, with
links to the original talk. It’s here.
2021 version
This is a recording of a Zoom seminar, for the UCL Department of Statistical Science, given on 6 May 2021, It’s slightly more technical than earlier versions. It explains better than earlier versions
the assumptions that underlie my suggestions.
It’s remarkable that statisticians are still at war about how best to decide whether the difference between the means of two independent samples is a result of sampling error alone or whether it’s
The FPR[50]: a simple, but rough, solution to the p values war (?)
Latest video: 2022
This is a recording of a talk to the UCL R users’ group, via Zoom, on 8 December 2022. It has a bit more about the web calculator (an R Shiny app) than other talks.
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250+ Management Science solved MCQs with PDF download
These multiple-choice questions (MCQs) are designed to enhance your knowledge and understanding in the following areas: Bachelor of Business Administration (BBA) , Master of Commerce (M.com) .
1. Operations research analysts do not
A. predict future operations
B. build more than one model
C. collect relevant data
D. recommend decision and accept
Answer» A. predict future operations
2. Decision variables are
A. controllable
B. uncontrollable
C. parameters
D. none of the above
Answer» A. controllable
3. A model is
A. an essence of reality
B. an approximation
C. an idealization’
D. all of the above
Answer» D. all of the above
4. A physical model is an example of
A. an iconic model
B. an analogue model
C. a verbal model
D. a mathematical model
Answer» A. an iconic model
5. Every mathematical model
A. must be deterministic
B. requires computer aid for solution.
C. represents data in numerical form
D. all of the above
Answer» C. represents data in numerical form
6. Operations research approach is
A. multi disciplinary
B. scientific
C. intuitive
D. all of the above
Answer» A. multi disciplinary
7. An optimization model
A. mathematically provides best decision
B. provides decision with limited context
C. helps in evaluating various alternatives constantly
D. all of the above
Answer» D. all of the above
8. OR provides solution only if the elements are
A. quantified
B. qualified
C. feasible
D. optimal
Answer» A. quantified
9. The name management science is preferred by
A. americans
B. english
C. french
D. latin
Answer» A. americans
10. Operations research is applied
A. military
B. business
C. administration’
D. all of the above
Answer» D. all of the above
11. The application of OR techniques involves ………… approach
A. individual
B. team
C. critical
D. none of the above
Answer» B. team
12. OR techniques helps to find ………..solution
A. feasible
B. non feasible
C. optimal
D. non optimal
Answer» C. optimal
13. Modern scientific management research originated during ……
A. world war ii
B. world war i
C. 1990
D. 1993
Answer» A. world war ii
14. ………. helps management to evaluate alternative course of action for selecting the best course of action
A. operations research
B. quantitative technique
C. management research
D. none of the above
Answer» A. operations research
15. ………. Theory is an important operations research technique to analyze the queuing behaviour.
A. waiting line
B. net work
C. decision
D. simulation
Answer» A. waiting line
16. ……….. is an important Operations research technique to be used for determining optimal allocation of limited resources to meet the given objectives.
A. waiting line theory
B. net work analysis
C. decision analysis
D. linear programming
Answer» D. linear programming
17. ………… model involves all forms of diagrams
A. iconic
B. mathematical
C. analogue
D. schematic
Answer» A. iconic
18. An organization chart is an example of
A. iconic
B. mathematical
C. analogue
D. none of the above
Answer» C. analogue
19. …. Is known as symbolic model
A. iconic
B. mathematical
C. analogue
D. none of the above
Answer» B. mathematical
20. A map indicates roads, highways, towns and the interrelationship is an ……model
A. iconic
B. mathematical
C. analogue
D. none of the above
Answer» C. analogue
21. ………..models in which the input and output variables follow a probability distribution.
A. iconic
B. . mathematical
C. . analogue
D. deterministic model
Answer» D. deterministic model
22. ………. Example of probabilistic model
A. game theory
B. charts
C. graphs
D. all the above
Answer» A. game theory
23. ………..is a method of analyzing the current movement of the same variable in an effort to predict the future movement of the same variable.
A. goal programming
B. markov analysis
C. replacement theory
D. queuing theory
Answer» B. markov analysis
24. Constraints in an LP model represent
A. limitations
B. requirements
C. balancing limitation
D. all of the above
Answer» D. all of the above
25. Linear programming is a
A. constraint optimization technique
B. technique for economic allocation of limited resources.
C. mathematical technique
D. all of the above
Answer» D. all of the above
26. A constraint in an LP model restricts
A. value of objective function
B. value of decision variable
C. use of available resource
D. all of the above
Answer» D. all of the above
27. The best use of linear programming technique is to find an optimal use of
A. money
B. man power
C. machine
D. all of the above
Answer» D. all of the above
28. Which of the following as an assumption of an LP model
A. divisibility
B. proportionality
C. additively
D. all of the above
Answer» D. all of the above
29. Most of the constraints in the linear programming problem are expressed as ……….
A. equality
B. inequality
C. uncertain
D. all of the above
Answer» B. inequality
30. The graphical method of LP problem uses
A. objective function equation
B. constraint equation
C. linear equations
D. all the above
Answer» D. all the above
31. A feasible solution to a linear programming problem
A. must satisfy all problem constraints simultaneously
B. need not satisfy all constraints
C. must be a corner point of the feasible region
D. must optimize the value of the objective function
Answer» A. must satisfy all problem constraints simultaneously
32. While plotting constraints on a graph paper, terminal points on both axes are connected by a straight line because
A. the resources are limited in supply
B. the objective function is a linear function
C. the constraints are linear equations or in equalities
D. all of the above
Answer» C. the constraints are linear equations or in equalities
33. Constraints in LP problem are called active if they
A. represent optimal solution
B. at optimality do not consume all the available resources
C. both of (a) and (b)
D. none of the above
Answer» A. represent optimal solution
34. The solution space of a LP problem is unbounded due to
A. an incorrect formulation of the lp model
B. objective function is unbounded
C. neither (a) nor (b)
D. both (a) and (b)
Answer» C. neither (a) nor (b)
35. While solving LP problem graphically, the area bounded by the constraints is called
A. feasible region
B. infeasible region
C. unbounded solution
D. none of the above
Answer» A. feasible region
36. Which of the following is not a category of linear programming problems?
A. resource allocation problem
B. cost benefit trade off problem
C. distribution network problem
D. all of the above are categories of linear programming problems.
Answer» D. all of the above are categories of linear programming problems.
37. A linear programming model does not contain which of the following components?
A. data
B. decisions
C. constraints
D. a spread sheet
Answer» D. a spread sheet
38. Which of the following may not be in a linear programming formulation?
A. <=.
B. >.
C. =.
D. all the above
Answer» B. >.
39. While solving an LP problem infeasibility may be removed by
A. adding another constraint
B. adding another variable
C. removing a constraint
D. removing a variable
Answer» C. removing a constraint
40. Straight lines shown in a linear programming graph indicates
A. objective function
B. constraints
C. points
D. all the above
Answer» B. constraints
41. All negative constraints must be written as
A. equality
B. non equality
C. greater than or equal to
D. less than or equal to
Answer» C. greater than or equal to
42. In linear programming problem if all constraints are less than or equal to, then the feasible region is
A. above lines
B. below the lines
C. unbounded
D. none of the above
Answer» B. below the lines
43. ………. is a series of related activities which result in some product or services.
A. network
B. transportation model
C. assignment
D. none of these
Answer» A. network
44. An event which represents the beginning of more than one activity is known as ………..event.
A. merge
B. net
C. burst
D. none of the above
Answer» C. burst
45. If two constraints do not intersect in the positive quadrant of the graph, then
A. the problem is infeasible
B. the solution is unbounded
C. one of the constraints is redundant
D. none of the above
Answer» D. none of the above
46. Constraint in LP problem are called active if they
A. represent optimal solution
B. at optimality do not consume all the available resources
C. both of (a) and (b)
D. none of the above
Answer» A. represent optimal solution
47. Alternative solutions exists of an LP model when
A. one of the constraints is redundant.
B. objective function equation is parallel to one of the constraints
C. two constraints are parallel.
D. all of the above
Answer» B. objective function equation is parallel to one of the constraints
48. While solving an LP problem, infeasibility may be removed by
A. adding another constraint
B. adding another variable
C. removing a constraint
D. removing a variable
Answer» C. removing a constraint
49. ………..is that sequence of activities which determines the total project time.
A. net work
B. critical path
C. critical activities
D. none of the above
Answer» B. critical path
50. Activities lying on the critical path are called………….
A. net work
B. critical path
C. critical activities
D. none of the above
Answer» C. critical activities
Great job completing your study session! Now it's time to put your knowledge to the test. Challenge yourself, see how much you've learned, and identify areas for improvement. Don’t worry, this is all
part of the journey to mastery. Ready for the next step? Take a quiz to solidify what you've just studied.
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Cost Calculation Of Crusher Production
calculating cost of a crusher aggregate production
Total cost is the aggregate sum of all fixed and variable costs of production for the accounting period. For instance, if 1,000 units are produced at a total cost of $50,000, the average production
cost per unit is …
Cost Calculation Of Crusher Production
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Mata speed gains over Stata
* The inclusion of Mata as an available alternative programming language for Stata users was a great move by Stata.
* Mata in general runs much quicker than programming on the surface level in Stata.
* In Stata each loop that runs is compiled (interpretted into machine code) as it runs creating a lot of work for the machine.
* In Mata on the other hand, the entire loop is compiled prior to running.
* Let's see how this works.
* Let's say we want to add up the square of the numbers 1 through 100000
* Method 1: Surface loop
timer clear 1
timer on 1
local x2 = 0
forv i = 1/1000000 {
local x2 = `x2'+`i'^2
di `x2'
timer off 1
timer list 1
* On my laptop, this takes about 13.5 seconds
* Method 2: Mata loop
timer clear 1
timer on 1
// This command can be read as start i at 1,
// keep looping so long as i is less than 1000000,
// the third argument looks a little fishy but it is syntax
// that has been around for a while (at least since C).
// It would be identical to writing i=i+1, in other words, add 1 to i.
// Following the for loop we can immediately place a since line command.
for (i = 1; i <= 1000000; i++) x2=x2+i^2
// If there is nothing done with the value x2 then mata displays this value.
// R handles this identically
timer off 1
timer list 1
* In contrast, my computer completed the loop using mata in .27 seconds, many magnitudes of speed faster.
* However this does not mean you need to learn to use mata (since it has its own limitations and syntax) in order to speed up your commands.
* Method 3: Use Stata's data structure to accomplish vector tasks
timer clear 1
timer on 1
set obs 1000000
gen x2 = _n^2
* The sum command will calculate the mean of x2 which is the same as the sum of x2 divided by it's number of observations.
sum x2
* We can reverse that operation easily.
di r(N)*r(mean)
timer off 1
timer list 1
* Using a little knowledge of how Stata stores post command information this method does the same trick in .2 seconds
* Method 4: The speed gains in 3 was as a result of using the vector structure of data columns. Mata can do very similar things even easier.
timer clear 1
timer on 1
// This command looks a little fishy, but it is easy to understand.
// Order of operations must be taken into account.
// First the 10^6 is evaluated which equals 1000000
// Then the vector 1..10^6 is made which looks like 1 2 3 ... 1000000
// The .. tells mata to make a count vector.
// If I had written :: then mata would have made a column vector instead.
// Once the vector is made then the command :^2 tells stata to do a piece wise squaring of each term in the vector.
// Finally the sum command adds all of the elements of the vector together to generate the result we were looking for.
mata: sum((1..10^6):^2)
timer off 1
timer list 1
* The result is that this command only took .04 seconds to run through efficient coding in Mata.
# As a matter of comparison, this command
# took .04 seconds in R
# And the loop:
system.time(for(i in 1:10^6) x=x+i^2)
# 1.3 seconds
# Thus Mata in this example is significantly faster than Stata and about the same speed as R.
1 comment:
1. Testing it on my machine, using a scalar instead of a local in method 1 seems faster, though still significantly slower than the other methods.
In method 3, -summarize- with option -meanonly- should be slightly faster. But I think you can skip -summarize-, which either way is going to do more calculations than you need:
set obs 1000000
gen x2 = sum(_n^2)
di x2[_N]
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RandomState.dirichlet(alpha, size=None)¶
Draw samples from the Dirichlet distribution.
Draw size samples of dimension k from a Dirichlet distribution. A Dirichlet-distributed random variable can be seen as a multivariate generalization of a Beta distribution. Dirichlet pdf is the
conjugate prior of a multinomial in Bayesian inference.
alpha : array
Parameter of the distribution (k dimension for sample of dimension k).
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.
samples : ndarray,
The drawn samples, of shape (size, alpha.ndim).
If any value in alpha is less than or equal to zero
Uses the following property for computation: for each dimension, draw a random sample y_i from a standard gamma generator of shape alpha_i, then
[1] David McKay, “Information Theory, Inference and Learning Algorithms,” chapter 23, http://www.inference.phy.cam.ac.uk/mackay/
[2] Wikipedia, “Dirichlet distribution”, http://en.wikipedia.org/wiki/Dirichlet_distribution
Taking an example cited in Wikipedia, this distribution can be used if one wanted to cut strings (each of initial length 1.0) into K pieces with different lengths, where each piece had, on
average, a designated average length, but allowing some variation in the relative sizes of the pieces.
>>> s = np.random.dirichlet((10, 5, 3), 20).transpose()
>>> plt.barh(range(20), s[0])
>>> plt.barh(range(20), s[1], left=s[0], color='g')
>>> plt.barh(range(20), s[2], left=s[0]+s[1], color='r')
>>> plt.title("Lengths of Strings")
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Fixed Point - (Differential Calculus) - Vocab, Definition, Explanations | Fiveable
Fixed Point
from class:
Differential Calculus
A fixed point is a value at which a function evaluated at that point is equal to the point itself. In mathematical terms, if you have a function $f(x)$, then a fixed point 'x' satisfies the condition
$f(x) = x$. This concept is crucial in iterative methods like Newton's Method, as finding fixed points can lead to solutions of equations or optimization problems.
congrats on reading the definition of Fixed Point. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. In the context of Newton's Method, the algorithm iteratively approaches a fixed point that corresponds to the root of the equation being solved.
2. Fixed points can be found graphically by identifying intersections between the line $y = x$ and the curve of the function $f(x)$.
3. The existence and uniqueness of fixed points can be guaranteed under certain conditions, such as when functions are continuous and contractive.
4. Fixed point iteration is an essential part of various numerical algorithms, where approximations are refined until they converge on a stable solution.
5. Understanding fixed points helps identify potential issues in convergence and divergence in iterative methods like Newton's Method.
Review Questions
• How does identifying fixed points relate to the convergence of Newton's Method?
□ Identifying fixed points is vital for understanding how Newton's Method converges to a solution. The algorithm iteratively refines an initial guess by evaluating the function's derivative and
applying its formula. If the iterations approach a fixed point where $f(x) = x$, it indicates that the method is converging to a root. Analyzing how quickly and reliably these iterations
approach a fixed point helps assess the efficiency and effectiveness of Newton's Method.
• Discuss how the concept of fixed points influences the applications of Newton's Method in real-world problems.
□ The concept of fixed points plays a crucial role in applying Newton's Method to real-world problems, such as engineering design and financial modeling. In these scenarios, finding fixed
points can represent equilibria or optimal solutions that satisfy specific criteria. For instance, in optimizing functions related to profit maximization or cost minimization, recognizing
that certain values lead to fixed points helps decision-makers predict outcomes accurately and efficiently utilize resources.
• Evaluate how understanding fixed points can lead to improvements in numerical methods and their limitations.
□ Understanding fixed points allows for significant improvements in numerical methods by providing insights into convergence behavior and potential pitfalls. By ensuring that iterative
processes are designed around stable fixed points, developers can enhance algorithm reliability and accuracy. Additionally, this understanding highlights limitations, such as cases where
functions may have multiple or no fixed points, which could lead to failure in finding solutions or slow convergence rates. Thus, evaluating these aspects contributes to refining existing
methods and developing new strategies in numerical analysis.
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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2018.24
URN: urn:nbn:de:0030-drops-87375
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/8737/
Chan, Timothy M. ; Skrepetos, Dimitrios
Approximate Shortest Paths and Distance Oracles in Weighted Unit-Disk Graphs
We present the first near-linear-time (1 + epsilon)-approximation algorithm for the diameter of a weighted unit-disk graph of n vertices, running in O(n log^2 n) time, for any constant epsilon>0,
improving the near-O(n^{3/2})-time algorithm of Gao and Zhang [STOC 2003]. Using similar ideas, we can construct a (1+epsilon)-approximate distance oracle for weighted unit-disk graphs with O(1)
query time, with a similar improvement in the preprocessing time, from near O(n^{3/2}) to O(n log^3 n). We also obtain new results for a number of other related problems in the weighted unit-disk
graph metric, such as the radius and bichromatic closest pair.
As a further application, we use our new distance oracle, along with additional ideas, to solve the (1 + epsilon)-approximate all-pairs bounded-leg shortest paths problem for a set of n planar
points, with near O(n^{2.579}) preprocessing time, O(n^2 log n) space, and O(log{log n}) query time, improving thus the near-cubic preprocessing bound by Roditty and Segal [SODA 2007].
BibTeX - Entry
author = {Timothy M. Chan and Dimitrios Skrepetos},
title = {{Approximate Shortest Paths and Distance Oracles in Weighted Unit-Disk Graphs}},
booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)},
pages = {24:1--24:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-066-8},
ISSN = {1868-8969},
year = {2018},
volume = {99},
editor = {Bettina Speckmann and Csaba D. T{\'o}th},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/8737},
URN = {urn:nbn:de:0030-drops-87375},
doi = {10.4230/LIPIcs.SoCG.2018.24},
annote = {Keywords: shortest paths, distance oracles, unit-disk graphs, planar graphs}
Keywords: shortest paths, distance oracles, unit-disk graphs, planar graphs
Collection: 34th International Symposium on Computational Geometry (SoCG 2018)
Issue Date: 2018
Date of publication: 08.06.2018
DROPS-Home | Fulltext Search | Imprint | Privacy
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Lupine Publishers| On the Second Harmonic Index of Titania Nanotubes
Lupine Publishers| Drug Designing & Intellectual Properties International Journal (DDIPIJ)
Topological indices which are graph invariants derived from molecular graphs of molecules are used in QSPR researches for modeling physicochemical properties of molecules. Topological indices are
important tools for determining the underlying topology of a molecule in view of theoretical chemistry. The second harmonic index has been defined recently. In this study we compute the second
harmonic index of Titania nanotubes.
Keywords: Harmonic Index; Novel Harmonic Indices; Second Harmonic Index; Titania Nanotube
Graph theory which is one of the most important branches of applied mathematics and chemistry has many applications from the basic sciences to the engineering sciences especially for solving and
modeling of real world problems. Chemical graph theory is the common place for graph theory and chemistry. Topological indices are indispensable tools for QSPR researches in view of theoretical
chemistry and chemical graph theory. Topological indices have been used more than seventy years predicting and modeling physicochemical properties of chemical substances. A graph G = (V,E) consists
of two nonempty sets v and 2-element subsets of v namely E. The elements of v are called vertices and the elements of E are called edges. For a vertex v de^f0V) show the number of edges that incident
to v. The set of all vertices which adjacent to is called the open neighborhood of and denoted by n (v). If we add the vertex v to n(v), then we get the closed neighborhood of v,n(v) . For the
vertices u and v, d(u, v) denotes the distance between u and v which means that minimum number of edges between u and v. The largest distance from the vertex v to any other vertex u called the
eccentricity of v and denoted by ^ev.
The first distance based topological index is the Wiener index which was defined by H. Wiener to modeling the boiling points of paraffin molecules [1]. In his study Wiener computed the all distances
between the all atoms (vertices) in the molecular graph of paraffin molecules and named this graph invariant as "path number". The Wiener index of a simple connected graph G defined as follows:
Many years later the path number renamed as "Wiener index" to honor Professor Harold Wiener for valuable contribution to mathematical chemistry. In t he same year, the first degree based topological
index was proposed by Platt for modeling physical properties of alcanes [2]. The Platt index of a simple connected graph G defined as follows;
After these both studies, approximately twenty five years later the well-known degree based Zagreb indices were defined by Gutman and Trinajstić to modeling π-electron energy of alternant carbons
[3]. The first Zagreb index of a simple connected graph defined as;
And the second Zagreb index of a simple connected graph defined as;
An alternative definition of the second Zagreb index of a simple connected graph is given the following formula:
In 1975, Randić defined the "Randić index" [4] to modeling molecular branching of carbon skeleton atoms as follows:
Among the all topological indices, the above mentioned topological indices have been used for QSPR researches more considerably than any other topological indices in chemical and mathematical
literature. We refer the interested reader to the following citations for up to date information about these well- known and the most used topological indices [5-15].
Harmonic index of a simple connected graph G was defined by Zhong in 2012 [16] as follows
Since then, there are more than one hundred papers in mathematical and chemical literature about Harmonic index and its applications. We refer the interested reader to [17,18] and references cited in
these articles.
The novel harmonic indices have been defined recently by the present authors [19]. In [19], the fifth harmonic index of H-Naphtalenic nanotube and TUC4 [m, n] nanotube were calculated.
Harmonic indices were defined in [20] as;
Where Qu is a unique parameter which is acquired from the vertex u ϵ (G).
The first kind of this Harmonic indices was studied by Zhong by considering ^Qu to be the degree of the vertex u:
The second kind of this class was defined by considering ^Qu to be the number ^nu of vertices of G lying closer to the vertex u than to the vertex v for the edge uv of the graph G:
The third type of this class was defined by considering ^Qu to be the number ^mu of edges of G lying closer to the vertex v than to the vertex u for the edge uv of the graph G:
The fourth type of this class was defined by considering ^Qu to be the eccentricity of the vertex u:
The fifth type of this class was defined by considering ^Quto be the:
And the sixth type of this class was defined by considering Qu to be the:
For topological indices of Titania nanotubes, we refer the interested reader [20-28] and references therein. The aim of this paper is to compute the exact value of the second harmonic index of
Titania nanotubes.
Results and Discussion
In this section we compute the exact values of the second harmonic index of Titania Nanotubes (Figure 1). Rezaei et al. [28] computed edge vertex version of Co-Pi index of Titania nanotubes [28]. The
following table given the classifications of the vertices of TiO[2] [m,n] with respect to the parameter n[u] . We get the results with the help of the article of Rezai et al. [30]. From the help of
Table 1, we can state our main result (Figure 2).
Figure 1: A graphical figure of Titania Nano tubes TiO[2][m, n].
Table 1: The edge classification of the vertices of TiO2 [m, n] with respect to the parameter.
Figure 2: Orthogonal cuts representation of the Titania Nano tubes.
Theorem 1. The second harmonic index of Titania Nanotubes TiO[2][m ,n] is given as;
Proof. From the definition of the second harmonic index and Table 1, we can write that:
In this study we found the exact values of newly defined the second harmonic index of Titania nanotube. This calculation will help to predict and model some physicochemical, optical and biological
properties of Titania nanotube. It can be interesting to compute the novel harmonic topological indices of some other nanotubes and networks for further studies. It can also be interesting to study
the mathematical and QSPR properties of these novel harmonic indices.
Read More Lupine Publishers Drug Designing & Intellectual Properties International Journal (DDIPIJ) Please Click on Below Link: https://lupine-publishers-drug-designing.blogspot.com/
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Hyperuniformity and its generalizations
Disordered many-particle hyperuniform systems are exotic amorphous states of matter that lie between crystal and liquid: They are like perfect crystals in the way they suppress large-scale density
fluctuations and yet are like liquids or glasses in that they are statistically isotropic with no Bragg peaks. These exotic states of matter play a vital role in a number of problems across the
physical, mathematical as well as biological sciences and, because they are endowed with novel physical properties, have technological importance. Given the fundamental as well as practical
importance of disordered hyperuniform systems elucidated thus far, it is natural to explore the generalizations of the hyperuniformity notion and its consequences. In this paper, we substantially
broaden the hyperuniformity concept along four different directions. This includes generalizations to treat fluctuations in the interfacial area (one of the Minkowski functionals) in heterogeneous
media and surface-area driven evolving microstructures, random scalar fields, divergence-free random vector fields, and statistically anisotropic many-particle systems and two-phase media. In all
cases, the relevant mathematical underpinnings are formulated and illustrative calculations are provided. Interfacial-area fluctuations play a major role in characterizing the microstructure of
two-phase systems (e.g., fluid-saturated porous media), physical properties that intimately depend on the geometry of the interface, and evolving two-phase microstructures that depend on interfacial
energies (e.g., spinodal decomposition). In the instances of random vector fields and statistically anisotropic structures, we show that the standard definition of hyperuniformity must be generalized
such that it accounts for the dependence of the relevant spectral functions on the direction in which the origin in Fourier space is approached (nonanalyticities at the origin). Using this analysis,
we place some well-known energy spectra from the theory of isotropic turbulence in the context of this generalization of hyperuniformity. Among other results, we show that there exist many-particle
ground-state configurations in which directional hyperuniformity imparts exotic anisotropic physical properties (e.g., elastic, optical, and acoustic characteristics) to these states of matter. Such
tunability could have technological relevance for manipulating light and sound waves in ways heretofore not thought possible. We show that disordered many-particle systems that respond to external
fields (e.g., magnetic and electric fields) are a natural class of materials to look for directional hyperuniformity. The generalizations of hyperuniformity introduced here provide theoreticians and
experimentalists new avenues to understand a very broad range of phenomena across a variety of fields through the hyperuniformity "lens."
All Science Journal Classification (ASJC) codes
• Condensed Matter Physics
• Statistical and Nonlinear Physics
• Statistics and Probability
Dive into the research topics of 'Hyperuniformity and its generalizations'. Together they form a unique fingerprint.
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Rich Math Problems – Part 2
by Karen Rothschild
Strategies for Creating Rich Math Problems
The strategies below^* are ways to take routine tasks—like those often found in textbooks—and adapt them so that students have more opportunities to think deeply about the topics they are learning.
• Provide the answer. Ask for the question.
□ Example 1: Instead of, “Round 5.94 to the nearest tenth,” try, “A number was rounded to 6. What could be the number?”
□ Example 2: Instead of, “Find the sum of 3,6, and 8,” try, “The sum of three numbers is 17. What could be the numbers?”
□ Example 3: Instead of, “What is the perimeter of a pool whose length is 50 yards and width is 25 yards?” try, “The distance around the rectangular swimming pool at the park is 150 yards. How
long and how wide could the pool be?”
• Ask students to choose the numbers in a task.
□ Example 4: Instead of a problem with all but one of the quantities given, try multiple quantities unknown, such as: “Taylor walked ☐ blocks to school. After school she walked ☐ blocks to the
store, and then ☐ blocks to get home. She walked a total of ☐ blocks.”
□ Example 5: Use any digits between 0-9 in the boxes to make a correct equation. You may only use a digit once in the equation ☐☐ + ☐ = ☐☐ – ☐ What is the smallest value that could be on each
side of the equation? What is the largest value?
• Ask for similarities or differences.
□ Example 6: Instead of, “Name 3 numbers that are multiples of 5, and three that are not,” try, “How are 5 and 100 alike? How are they different? Find as many ways as you can.”
□ Example 7: Instead of, “Describe the characteristics of right triangles,” try, “How are these shapes alike? ◤▲How are they different?”
• Ask for contexts for numerical expressions.
□ Example 8: Instead of “Find the number of plants in a garden with 6 rows of 4 plants each,” try, “Create a real-world question where you might have to find 6×4 to answer the question. Find
the product and answer your question.”
• Ask for a mathematical sentence that includes certain numbers and words.
□ Example 9: Create a sentence that includes the numbers 3 and 4 along with the words “more” and “and.” You’ll have to use some other words too.
• Provide a real world situation that requires mathematics. Provide areas of ambiguity so students can make choices.
□ Example 10: Instead of, “Use the menu to find the cost of 2 hot dogs and a soda,” try, “You and 3 friends have $50 to spend on lunch. Use the menu to decide what to buy and how much it will
What Makes These Problems “Rich?”
All of the problem types above are rich problems because they provide:
● Opportunities to engage the problem solver in non-routine ways of thinking about mathematics,
● An opportunity for productive struggle, and
● An opportunity for students to communicate their thinking about mathematical ideas.
Other characteristics of these problems include:
● Several correct answers.
● A “low floor and high ceiling,” meaning that all students can get started and all students can reach a point of struggle.
● An opportunity to practice routine skills in the service of engaging with a complex problem.
● An opportunity for formative assessment. Choices students make when working on these problems can give teachers information about students’ developmental levels, their neurodevelopmental
strengths and challenges, and their learning preferences.
● A level of complexity that may require an extended amount of time to solve. Many of these examples can become longer investigations by asking students to find all possible solutions, or to find
the greatest or least possible solution.
● An opportunity to look for patterns. Depending on how many solutions students find, several of the examples offer opportunities to look for patterns.
● An opportunity for students to choose from a range of tools and strategies to solve the problem based on their own neurodevelopmental strengths.
● An opportunity to discover a new (for the student) mathematical idea through working on the problem.
● An opportunity for students to engage their everyday knowledge of the real world
(Examples 3, 4, 8, and 10)
Teaching Math with Rich Problems
While an appropriate problem is crucial to immersing students in deep thinking about mathematics, student engagement in conceptual learning is also heavily influenced by pedagogical practices. In
order to facilitate students’ grappling with ideas and mathematical reasoning, tasks should focus on learning goals and be constructed and presented in ways that, as much as possible, mitigate or
eliminate challenges related to memory, language (including reading and writing), psychosocial factors, etc. when they are not essential to the goals of the tasks. When specific challenges are
essential to mathematical goals, supports should be provided for those who need them. Problems should be presented in an accessible manner and students should be encouraged to represent their
thinking in a variety of ways. A mathematical idea or problem solution can often be represented in a sophisticated and powerful way using visual representations and/or physical models.
Rich problems, in contrast to mere applications of skills, often require trying things out, and making mistakes. Students must feel comfortable taking risks and using mistakes to learn. Often, they
must be explicitly told that mistakes are both “expected and respected.” Furthermore, learning from mistakes takes time, and teachers need to provide the time for attempts, failures, and reattempts.
Perseverance comes from this iterative process. Teachers can facilitate deep thinking by suggesting avenues to explore, but not providing the kind of information that leads directly to an answer or
use of only a single strategy. Teachers can also model making mistakes and learning from them.
Teachers can also promote communication skills by recognizing students as the authority on their own work. Students who learn to convince themselves and others of the validity of their work
understand mathematics more deeply, develop strong communication skills, and become more confident in their capacity to learn. Some useful questions are: ”What do you see?” “What does that remind you
of?” “What are you thinking about?” “Will that work?” “What can’t work?” “Why does that make sense?” “Can you convince me?” Teachers can also anticipate common misconceptions students make in
particular situations and be prepared with topic-specific questions.
In sum, rich problems offer ALL students an opportunity to engage deeply in the mathematical ideas they are learning, to become fluent users of mathematics in a variety of settings, and to
communicate their thinking. Success in using rich problems in the classroom depends on choosing appropriate problems, presenting them in ways that are accessible to students with a variety of
neurodevelopmental strengths and challenges, and creating a classroom culture that encourages students to explore ideas and possibilities, make mistakes, and learn from each other through sharing
their work.
The list of strategies and many of the examples are derived from:
Small, M. (2020). Good questions: Great ways to differentiate mathematics instruction. 4th edition. New York: Teachers College Press.
Open Middle (n.d.).
Teach, Learn, Share (n.d.).
The contents of this blog post were developed under a grant from the Department of Education. However, those contents do not necessarily represent the policy of the Department of Education, and you
should not assume endorsement by the Federal Government.
This work is licensed under CC BY-NC-SA 4.0
Math for All is a professional development program that brings general and special education teachers together to enhance their skills in
planning and adapting mathematics lessons to ensure that all students achieve high-quality learning outcomes in mathematics.
Our Newsletter Provides Ideas for Making High-Quality Mathematics Instruction Accessible to All Students
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TBR Technical Corner: Judder Vibration Path Analysis (JPA) and Chassis Dynamic Behaviour (Part 2 out of 3)
By Juan Jesús García, PhD, Product Manager, Braking Systems in Applus IDIADA
In the first part of the article, the fundamental theory of the so-called Judder Transfer Path Analysis (JPA) was introduced. This methodology allows to assess the possible ways of energy transfer
from the brake to a given target location in a vehicle.
Generalized Theory for Dynamic Systems. Automotive Applications – JPA
Let us assume that we have a vibration system into which we transmit vibration energy through a number of discrete connection points and where the NVH response is analysed. When assuming N different
possible transmission paths over which the energy coming from the single reference source substructure can be transmitted into the main body, the total operational acceleration [a] at a given set of
target location in the vehicle can be written as the sum of the contribution of the partial accelerations, a[i ], related to each given i-th transmission path. The determination of these partial
accelerations is based upon the combination of an estimate of the operational force f[i] in the given transmission path, together with the mechanic-vibration transfer function between the target
acceleration response and a force applied at the chassis side of the considered transmission path , i.e., P/F[i].
Then, the total acceleration can be broken down as:
We note that the terms A[i]/F[i] are the transfer functions that relate acceleration the response at the target points (vibration) and the force applied at the i-th path. When considering M multiple
target accelerations [a] all the FRF’s can be assembled in a mechanical-vibration FRF-matrix [H]. Using matrix algebra this means,
[a] = [H] [f]
For the general case in which we have M target responses (acceleration) and N paths, equation (6) would become:
It is important to emphasize that equation (7) is a function of frequency. In practise, the frequency response functions are usually measured after disassembling the source from the structure body,
eliminating source coupling of the transmission impedances. They can be measured for example by using hammer or shaker excitation techniques. The latter method in general provides the most accurate
results, but at the cost of a more complex test procedure.
A technique for the force estimation is based upon the measurement of a mechanical impedance matrix [H], containing FRFs between accelerations measured at the body side of all paths on one hand and
forces applied at each path on the other hand. By inverting this matrix, and multiplying it with the vector of the corresponding body side operational accelerations, estimates of the operational
forces are obtained. These are:
where [a] denotes the second derivative of the displacement [x], i.e., the acceleration at the observation points. As for the static case already covered above, when solving dynamic problems in
general, theoretically, it is sufficient to take into account a number of responses equal to the number of forces that has to be estimated (N = M). Taking into account more responses (M > N), the set
of equations is over-determined, and better force estimates will be obtained in a least-squares sense. This is normal practise in many engineering applications. The inversion is then based upon
singular value decomposition algorithms, which allow to artificially improving the conditioning of the inversion.
TPA Applications to Brake Judder: Judder Path Analysis (JPA)
Brake judder is highly vehicle-dependent. Since the mechanical energy travels through the structure and interacts with the steering, brake pedal, and seats, there can be major differences in how the
same amount of disc thickness variation excites these systems. Thus, a certain level of DTV that is perfectly acceptable in one vehicle may lead to a major problem on another. This leads to
situations where the same brake applied to different vehicles can exhibit completely different judder levels.
This situation requires a robust method to identify the reasons why certain vehicles are particularly sensitive to disc thickness variation. Current methodologies are based on an experimental
trial-and-error approach implementing modifications, the brakes or the vehicle suspension or the sub-frame. This approach is time-consuming and expensive. Thus, a robust and systematic method is
required to identify the vehicle paths responsible for the transmission of vibrations from the brake to those parts that affect the drive´s perception. Figure 6 shows an example of various
transmission paths of judder vibration that one can encounter in vehicles. Note that in more complex suspension systems, the number of transmission paths can go up to about seven. Once the
transmission paths initiated at the suspension level enter into the vehicle body, the propagation of the vibration is spread out through the vehicle body and collected back into the receiving points
that define the interface between the driver and the vehicle.
The judder analysis method proposed herein must define the relative contributions originated by each input path determined by the suspension arrangement. This contribution will be determined in terms
of the acceleration level that, for any given frequency, is transmitted through the path under study. This method will allow the following conclusions to be drawn (see Figure 6):
• Dominant transmission paths of detected judder
• Systems and subsystems affected by the transmission paths
• Ideal locations to act in order to ‘inhibit’ the transmission path
• Definition of optimized countermeasures for judder control in vehicles
The development of this method of judder path analysis (JPA) will materialize the integration of brake and NVH knowledge for the specific field of judder control.
Figure 6: Concept of a judder path analysis (JPA) for a judder-induced vibration
Vehicle Instrumentation
A transfer path analysis (JPA) requires the knowledge of operational vibration at a minimum number of points located on the vehicle parts that make up the system under study. The locations of the
accelerometers are chosen so that the fundamental movements of all parts involved can be detected under operational loads. Of paramount importance is to define the acceleration matrix so that rigid
body motions, bending and torsion deformation of the measured parts can be detected, since these type of movements and deformations have a big influence on the JPA results.
Figure 7 shows the array of accelerometers for JPA analysis of the suspension and chassis system used in this work for our test vehicle with a high judder sensitivity. In particular, the following
points have to be emphasized when defining the tri-axial accelerometer array:
• Try detecting bending and torsion movements of the chassis and suspension system
• Accelerometers should be located at both ends of connecting joints
• Accelerometers should be located at the attachment points of the chassis to the vehicle body (chassis and body side)
• The overall extension of the array should cover from the excitation source (brake structure) to the receiving vehicle points (seat rail, steering wheel, etc).
Figure 7: (Left): layout of accelerometer locations for operational chassis vibration analysis. Additional set configuration for the operational measurements and the measurement of the FRF for the
JPA analysis (right).
Figure 8 shoes the global view of the bottom part of the vehicle under study. This vehicle had a high sensitivity to brake judder and the JPA was aimed at investigating possible paths that could
explain this high sensitivity.
Figure 8: The global view of the bottom view of the studied vehicle showing the area that has been investigated for JPA (Left front suspension and chassis). The array of points measured in the
vehicle corresponding to the area marked with the red square is presented in Figure 7.
Considerations about the condition number of the matrix problem
The notion of condition number is important in all of applied mathematics. If small changes in the data of some problem always lead to reasonably small changes in the answer to the problem, the
problem is said to be well-conditioned. If small changes in the data of some problem can lead to unacceptably large changes in the answer to the problem, the problem is said to be ill-conditioned.
The importance of this concept is clear. In applied problems, experimental data are always inaccurate because of measuring and modelling errors, and it is important for us to know what effect such
inaccuracies in the data have on the answer to the problem.
If we consider the condition of the solution [f] in equation [a] = [H] [f], in terms of the data [a] and [H], it can be shown that the equation is well conditioned to be inverted and, thus, solved,
if the value of the expression
is not too large. Equation (9) defines what we call the condition number of matrix [H], which is expressed as c([H]). In this case, the symbol || || denotes any matrix norm.
A moderate condition number guaranties that the equations are well conditioned, i.e. small changes in the data produce reasonably small changes in the solution. If c([H]) is large, however, the
changes in [F] caused by changes in the data may be much larger than the changes in the data.
About Applus IDIADA
With over 25 years’ experience and 2,450 engineers specializing in vehicle development, Applus IDIADA is a leading engineering company providing design, testing, engineering, and homologation
services to the automotive industry worldwide.
Applus IDIADA is located in California and Michigan, with further presence in 25 other countries, mainly in Europe and Asia.
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Linear Algebra/Determinant - Wikibooks, open books for an open world
The determinant is a function which associates to a square matrix an element of the field on which it is defined (commonly the real or complex numbers). The determinant is required to hold these
• It is linear on the rows of the matrix.
${\displaystyle \det {\begin{bmatrix}\ddots &\vdots &\ldots \\\lambda a_{1}+\mu b_{1}&\cdots &\lambda a_{n}+\mu b_{n}\\\cdots &\vdots &\ddots \end{bmatrix}}=\lambda \det {\begin{bmatrix}\ddots &\
vdots &\cdots \\a_{1}&\cdots &a_{n}\\\cdots &\vdots &\ddots \end{bmatrix}}+\mu \det {\begin{bmatrix}\ddots &\vdots &\cdots \\b_{1}&\cdots &b_{n}\\\cdots &\vdots &\ddots \end{bmatrix}}}$
• If the matrix has two equal rows its determinant is zero.
• The determinant of the identity matrix is 1.
It is possible to prove that ${\displaystyle \det A=\det A^{T}}$, making the definition of the determinant on the rows equal to the one on the columns.
• The determinant is zero if and only if the rows are linearly dependent.
• Changing two rows changes the sign of the determinant:
${\displaystyle \det {\begin{bmatrix}\cdots \\{\mbox{row A}}\\\cdots \\{\mbox{row B}}\\\cdots \end{bmatrix}}=-\det {\begin{bmatrix}\cdots \\{\mbox{row B}}\\\cdots \\{\mbox{row A}}\\\cdots \end
• The determinant is a multiplicative map in the sense that
${\displaystyle \det(AB)=\det(A)\det(B)\,}$ for all n-by-n matrices ${\displaystyle A}$ and ${\displaystyle B}$ .
This is generalized by the Cauchy-Binet formula to products of non-square matrices.
• It is easy to see that ${\displaystyle \det(rI_{n})=r^{n}\,}$ and thus
${\displaystyle \det(rA)=\det(rI_{n}\cdot A)=r^{n}\det(A)\,}$ for all ${\displaystyle n}$ -by-${\displaystyle n}$ matrices ${\displaystyle A}$ and all scalars ${\displaystyle r}$ .
• A matrix over a commutative ring R is invertible if and only if its determinant is a unit in R. In particular, if A is a matrix over a field such as the real or complex numbers, then A is
invertible if and only if det(A) is not zero. In this case we have
${\displaystyle \det(A^{-1})=\det(A)^{-1}.\,}$
Expressed differently: the vectors v[1],...,v[n] in R^n form a basis if and only if det(v[1],...,v[n]) is non-zero.
A matrix and its transpose have the same determinant:
${\displaystyle \det(A^{\top })=\det(A).\,}$
The determinants of a complex matrix and of its conjugate transpose are conjugate:
${\displaystyle \det(A^{*})=\det(A)^{*}.\,}$
Using Laplace's formula for the determinant
${\displaystyle \det(AB)=\det A\cdot \det B}$
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Force between 2 parallel magnetic dipole moments
• Thread starter 1v1Dota2RightMeow
• Start date
In summary, the force of attraction between two magnetic dipoles pointing in the same direction a distance r apart can be determined using the equation U=-m·B where B is the field from the other
dipole and F=-∇U. The magnetic field from both dipoles will be aligned, resulting in a negative energy and a stronger force when the dipoles are closer together. Therefore, the force between the
dipoles is attractive.
Homework Statement
Find the force of attraction between 2 magnetic dipoles a distance r apart. Both dipoles point to the right.
Homework Equations
The Attempt at a Solution
All I need help with is figuring out how to determine if the force is attractive or repulsive between the 2 dipole moments. From the question, it seems as though I can conclude that 2 magnetic
dipoles pointing in the same direction attract each other. But I need a more fundamental way to figure this out. If I'm given 2 dipoles a distance r apart (where r is not huge) and with some
orientation (relative to each other), how do I determine whether there is an attractive force or a repulsive force?
Hello again. ## U=- m \cdot B ## where ## B ## is the field from the other dipole (magnetic moment). ## F=- \nabla U ##. (One thing that isn't completely clear from the statement of the problem=
Presumably the dipoles are pointing along the x-axis and are a distance r apart on the x-axis.) ## \\ ## The magnetic field from both magnetic moments points from left to right (surrounding the
magnetic moment), and both magnetic moments will thereby be aligned with the field from the other magnetic moment, making the energy negative for each. The energy becomes even more negative if the
dipoles get closer together because the field that it feels from the other dipole will be stronger. The system will tend to go to the state of lower energy=thereby the force is attractive. (It should
be noted the reason ## U=-m \cdot B ## (with a ## cos(\theta) ##) is because the torque ## \tau=m \times B ## (with a ## sin(\theta)) ## and ## U=\int \tau \, d \theta ##.)
Last edited:
FAQ: Force between 2 parallel magnetic dipole moments
1. What is the force between two parallel magnetic dipole moments?
The force between two parallel magnetic dipole moments is given by the formula F = μ0(m1·m2)/2πr^4, where μ0 is the permeability of free space, m1 and m2 are the magnitudes of the two dipole moments,
and r is the distance between them.
2. How does the distance between the two dipole moments affect the force?
The force between two parallel magnetic dipole moments is inversely proportional to the fourth power of the distance between them. This means that as the distance increases, the force decreases
3. What is the direction of the force between two parallel magnetic dipole moments?
The direction of the force is along the line joining the two dipole moments, from the first to the second dipole moment.
4. Can the force between two parallel magnetic dipole moments be attractive?
Yes, the force can be attractive or repulsive depending on the orientation of the two dipole moments. If they are parallel and pointing in the same direction, the force is attractive. If they are
parallel and pointing in opposite directions, the force is repulsive.
5. How does the strength of the magnetic dipole moments affect the force?
The strength of the magnetic dipole moments directly affects the force between them. The greater the magnitude of the dipole moments, the stronger the force will be.
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Drawing Triangles With Given Angles Worksheet - Angleworksheets.com
Constructing Triangles With Given Angles Worksheet – This article will discuss Angle Triangle Worksheets as well as the Angle Bisector Theorem. We’ll also discuss Equilateral triangles and Isosceles.
If you’re unsure of which worksheet you need, you can always use the search bar to find the exact worksheet you’re looking for. Angle Triangle Worksheet This … Read more
Drawing Triangles With Given Angles Worksheet
Drawing Triangles With Given Angles Worksheet – In this article, we’ll talk about Angle Triangle Worksheets and the Angle Bisector Theorem. In addition, we’ll talk about Isosceles and Equilateral
triangles. If you’re unsure of which worksheet you need, you can always use the search bar to find the exact worksheet you’re looking for. Angle Triangle … Read more
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Inquiries Journal Blog - A Forum for College and Grad Students
Study Tips Archives - Inquiries Journal - Blog Archives
November 6th, 2015
So, you want to use regression analysis in your paper? While statistical modeling can add great authority to your paper and to the conclusions you draw, it is also easy to use incorrectly.
The worst case scenario can occur when you think you’ve done everything right and therefore reach a strong conclusion based on an improperly conceived model. This guide presents a series of
suggestions and considerations that you should take into account before you decide to use regression analysis in your paper.
The best regression model is based on a strong theoretical foundation that demonstrates not just that A and B are related, but why A and B are related.
Before you start, ask yourself two important questions: is your research question a good fit for regression analysis? And, do you have access to good data?
1. Is Your Research Question a Good Fit for Regression Analysis?
This depends on many different factors. Are you trying to explain something that is primarily described by numerical values? This is a key question to ask yourself before you decide to use
regression. Although there are various ways to use regression analysis to describe non-numerical outcomes (e.g., dichotomous yes/no or probabilistic outcomes), they become more complicated and you
will need to have a much deeper understanding of the underlying principles of regression in order to use them effectively.
Before you start, consider whether or not your dependent variable is numerical. Some examples:
• Number of years a politician serves in Senate
• Life expectancy
• Lifetime earnings
• Age at birth of first child
At the same time, you need to make sure that there is sufficient variation in your dependent variable and that the variation occurs in a normal pattern.
For example, you would have a problem if you tried to predict the likelihood of someone being elected as president because almost no one is elected as president. As a result, there is virtually no
variation on the dependent variable.
2. Do You Have Access to Good Data?
Before you can conduct any type of analysis, you need a good data set. Not all data sets are easily suited to regression analysis without considerable manipulation.
Some things to consider before you decide to use regression:
• Are most of your independent variables numerical in nature? The best data set for regression will have variables that are primarily described by numbers that vary on a continuous scale. On the
other hand, if most of your variables are categorical, you might consider using a different method of analysis (e.g., Chi-squared).
• Are there enough cases (n) in your data set? Particularly if you think you might use multiple regression, where multiple independent variables are used to predict a single dependent variable, you
need to have a sufficient number of cases in your sample to obtain significant results. A general rule of thumb is that you need at least 20 cases per independent variable in your model. So if
your model includes 5 independent variables, you need a minimum of 100 cases.
Keep in mind that your independent variables need to meet the same criteria for normality and variability as your dependent variable.
Once you decide to proceed with a regression model in your analysis, there are a three key concepts to keep in mind as you design your model to avoid making an easily preventable mistake that could
send your conclusions way off track.
• Parsimony
• Internal Validity
• Multicollinearity
Each is described in more detail below.
In statistics, the principle of parsimony is based on the idea that when possible, the simplest model with the fewest independent variables should be used when a model with more variables offers only
slightly more explanatory value. In other words, one should not add variables to a model that do not increase the ability of the model to explain something.
Only add variables to a model if they significantly increase the ability of the model to explain something.
If you add too many variables to your model, you can unwittingly introduce major problems to your analysis.
In the extreme case, you must consider that your R^2 value will always increase with the addition of new variables: so if you examine R^2 alone, you can be duped into thinking that you have a great
model simply by dumping in more and more predictor variables.
There are two good ways to address this problem: use an Adjusted R^2 to compare models with different numbers of predictors, and use stepwise regression to analyze the explanatory impact of each
variable as it is added to the model.
• Adjusted R^2 takes into consideration the number of variables used in the model, and only increases when the addition of a new variable explains more than would random chance alone. So although a
model with 10 variables might have a very high R^2 value, the Adjusted R^2 could actually be much lower than a model with fewer variables. Selecting your model based on Adjusted R^2 helps you
select a more parsimonious model that is less likely to have other problems (e.g., see multicollinearity below).
• Stepwise Regression is a computational method of assessing the additional explanatory value of each variable as they are added to the model in different orders. It can be used to parse out
superfluous variables from a model, however it needs to be used carefully and in concert with theoretical guidance to avoid overfitting your data.
A good rule of thumb as you consider different models is that you should always have a good reason to add a predictor variable to your model, and if you can’t come up with a good theoretical
explanation as to why A influences B, then leave out A!
Internal Validity
Internal validity is the degree to which one factor can be said to cause another factor based on three basic criteria:
1. Temporal precedence, i.e., the “cause” precedes the “effect.”
2. Covariation, i.e., the “cause” and “effect” are demonstrably related.
3. Nonspuriousness, i.e., there are no plausible alternative explanations for the observed covariation caused by a confounding variable.
In many cases, internal validity becomes an issue in the form of a “chicken and egg” problem.
For example, let’s say you are considering the relationship between obesity and depression (a common example). If you want to include depression as an independent variable to explain obesity in your
model, you first need to consider the question:
Does depression lead to obesity, or does obesity lead to depression?
If you have no clear theoretical guidance to show that, in fact, depression usually precedes obesity (temporal precedence), you could introduce a significant problem to your model if the relationship
is in fact the other way around: depression being the result of obesity.
Therefore, as you craft your model it is important to have a theoretical basis for the inclusion of each variable.
Multicollinearity occurs when the independent variables in a multiple regression model are highly correlated with one another. This can be a problem in several ways:
• It reduces the parsimony of your model if the two variables are highly similar (e.g., two different variables that effectively measure the same thing);
• Multicollinearity can lead to erratic changes in the coefficients (measured effect) of predictor variables;
• As a result, it can be difficult to interpret the results of a model with high multicollinearity among predictors. Specifically, it becomes impossible to discern the individual effect of
different regressors.
An example of variables that are going to be highly multicollinear are any variables that effectively measure the same thing. One way to show this, for the purposes of an example, is to imagine
converting categorical data into a series of binary variables.
Any variables that effectively measure the same concept are likely to have high collinearity.
For example, let’s say that we have a variable measuring memory where respondents are able to choose very good, average, or poor as a response.
One way to use this data in a regression model would be to convert the data into three dichotomous (yes/no) variables indicating a person’s response.
However, if you then include all of these dichotomous variables in your model, you will have a big problem because they will become perfectly multicollinear. This is because anyone who indicated that
they had a very good memory, by default, also indicated that they do not have a poor memory. The two variables measure the same thing: a person’s memory.
Another common example can be found in the use of height and weight variables. Although the two variables measure different things, broadly speaking they can both be said to measure a person’s body
size, and they will almost always be highly correlated.
As a result, if both variables are included as predictors in a model, it can be difficult to discern the effect that each variable has individually on the outcome (measured by the coefficient).
Thus, as you build your model, you need to be aware of the potentially confounding impact of using highly similar predictor variables. In an ideal model, all independent variables will have no or
very low correlation to each other, but a high correlation with the dependent variable.
Conclusion: Use Regression Effectively by Keeping it Simple
Regression analysis can be a powerful explanatory tool and a highly persuasive way of demonstrating relationships between complex phenomena, but it is also easy to misuse if you are not an expert
If you decide to use regression analysis, you shouldn’t ask it to do too much: don’t force your data to explain something that you otherwise can’t explain!
Moreover, regression should only be used where it is appropriate and when their is sufficient quantity and quality of data to give the analysis meaning beyond your sample. If you can’t generalize
beyond your sample, you really haven’t explained anything at all.
Lastly, always keep in mind that the best regression model is based on a strong theoretical foundation that demonstrates not just that A and B are related, but why A and B are related.
If you keep all of these things in mind, you will be on your way to crafting a powerful and persuasive argument.
Tags: academia, Modeling, Regression, research, Research Guide, Stats
March 30th, 2015
Group work is an inevitable part of most university courses and the ability to work well with other people is something all employers care about. While working on a group project can be incredibly
rewarding, it can also present real challenges if you don’t go in with the right mindset. Here are a few tips to make group work just a little bit easier!
Be Prepared to Compromise
Something we must learn early in life is that different people have different working styles. While some people like to have an essay planned out and written weeks in advance, others thrive on the
pressure of leaving it until the last minute. Be open about how you work from the start – if you talk about the ways in which each person works best right away, you can come up with a compromise that
suits everyone.
If everyone compromises a little – for example, by agreeing to pre-planned deadlines – this can help avoid leaving some group members stressed or upset by discovering that their expectations were out
of line with the rest of the group.
Maximize Each Member’s Strengths
Do you love public speaking? If so, great – tell your group members that from the start! Break down everything that has to be done, from conducting the research to preparing the slideshow and giving
the presentation in front of the class, and assign tasks to each person based on their strengths.
While it can be difficult to please everyone, having an honest discussion about strengths/weaknesses early on and and attempting to give everyone tasks that they’re comfortable with will benefit the
entire group in the end.
Stand Up for Yourself and Do the Work
People have different personalities, so if you are naturally shy and are put in a group with someone more confident, it can be tempting to shrink up and not say or do anything, even when you think
that the group might be headed in the wrong direction – this is a mistake!
As scary as it is, make sure you stand up for yourself and speak up. This is the only effective antidote to groupthink and conversations where not everyone immediately agrees can be incredibly
Of course it goes without saying, always put in the work. Don’t be the person that shows up with the job half done. It is common for group projects to include peer assessments and if you don’t put in
the effort, your classmates won’t be shy.
Choose Your Group Wisely
If you are given the opportunity to choose your group members, the temptation is often to work with your friends. Sometimes that is for the best because you know each other well and it can make
working on the project more fun and less stressful. However, it can also lead to even more tension, particularly if you aren’t diligent about assigning tasks and preparing some deadlines from the get
Someone who you have a lot of fun with on a night out might not make the best partner for a group project. (For one thing, this can make it much easier to get distracted!)
There are also a lot of benefits to working with people you don’t know – it can give your project a wider range of perspectives and help you capitalize on differentiated skills as a group. Moreover,
you might even end up making a great new friend.
Tags: college, group projects, group work, study tips
March 23rd, 2015
Presentations have become an integral part of most university and college courses. While some students won’t think twice before getting up to speak in front of a room full of people, for others, the
thought of being in the spotlight can become overwhelming. It’s natural to be nervous, but for many students, those nerves can spiral out of control, making you feel anxious for days leading up to
the event.
Here are a few tips which will hopefully help to make you feel as comfortable as possible before giving your next group or solo presentation!
Perfect Your Slides
If you are required to make a visual background for your presentation on something like PowerPoint, make a really good job of it! Rather than cobbling together some blank slides with a couple of
paragraphs on them the night before, take some time to make them look amazing!
A well-structured, nicely designed slideshow will show your teacher and classmates that you put a lot of work into the presentation. This tells your audience that any visible nerves are purely due to
public speaking and not from a lack of preparation.
Having your key points outlined briefly on your slides also means that if your nerves get the better of you and you lose track of where you are, your slides will quickly guide you back to where you
Practice, Practice, Practice!
It sounds obvious, but the worst thing you can do if public speaking worries you is not run through your presentation a good few times in advance! Start off alone, speaking aloud in your bedroom or
an empty classroom. Then ask some friends to act as a test audience for you.
Not only will this likely lead to useful feedback on your content, but it will make you feel more comfortable in front of a crowd, too! It allows you to practice key strategies — such as eye contact
— which will improve your presentation. Your tutor wants to see that you are engaging with the class, so getting used to being in front of others is really helpful.
Make Use of Note Cards
Note cards can be a useful tool to take advantage of, but do make sure you check that you are allowed to use them first! Having cards with brief summaries (not the full script of your presentation)
can help to keep you on track and, much like the slideshow, can give you the confidence of knowing exactly what’s coming next.
However, they can also be a useful tool to stop you from fidgeting, something which is ever so tempting when you’re nervous! If you have cards to hold, you won’t be as likely to touch your hair,
fidget with a pen or fiddle with your jacket!
Plan Something Nice for Later!
Depending on your schedule, if you can afford to take a little time out to go out for dinner with friends, go to the cinema, or even just go for a walk round the shops – do it! Knowing you have
something fun planned for after the big event can make it so much easier to get through the stress of a presentation. You might still be nervous, but knowing that no matter what, the rest of your day
is only going to get better, is a great feeling! That alone might be enough to make you feel more settled.
Tags: college, presentations, student life, study tips, videos
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Supershells in Metal Clusters
In the mass spectra of elemental clusters one observes oscillations in the abundance as a function of cluster size. These oscillations reflect variations \tilde E in the total energy of the clusters.
In most cases \tilde E is determined by the geometric arrangement of the atoms. For simple metals at temperatures above the melting point, however, the electonic structure is dominant, giving rise to
electonic shells and supershells. Using a jellium model for systems of up to 10000 electrons, the experimental findings for alkali clusters can be reproduced fairly well. But in the case of gallium
that simple model fails. To gain insight into the mechanisms determining the shell and supershell structure, we extend the known semiclassical analysis for the DOS of a spherical cavity to more
realistic potentials. This suggests a simple mechanism for explaining the failure of the jellium model for metals with high electron density.
For an overview, take a look at these Slides taken form a talk on the subject (1MB)
More detailed information can be found in the following publications
• Erik Koch:
Supershells in Metal Clusters: Self-Consistent Calculations and Their Semiclassical Interpretation
Phys.Rev.Lett 76, 2678-2681 (1996) and cond-mat/9606023
• Erik Koch:
On the 3n+l Quantum Number in the Cluster Problem
Phys.Rev. A 54, 670-676 (1996) and cond-mat/9606050
• Erik Koch and Olle Gunnarsson:
Density Dependence of the Electronic Supershells in the Homogeneous Jellium Model
Phys.Rev. B 54, 5168-5177 (1996) and cond-mat/9606140
• Erik Koch:
Periodic Orbit Expansion for Realistic Cluster Potentials
Phys. Rev. B 58, 2329 (1998) and cond-mat/9803309
Even more information can be found in my PhD thesis (in German).
Erik Koch ( koch@and.mpi-stuttgart.mpg.de)
MPI-FKF Andersen Group Max-Planck-Institut für Festkörperforschung
Heisenbergstraße 1 D-70569 Stuttgart
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We are happy to present Soumodev Mal’s talk titled “On the Satisfiability of Context-free String Constraints with Subword Ordering”.
String constraints impose constraints over variables that take strings as assignments. Given a string constraint C, the satisfiability problem asks if there is an assignment which satisfies every
constraint in C. Most classes of string constraints in their full generality although have high expressive power turn out to be undecidable. Hence, a natural direction to recover decidability is to
consider meaningful subclasses of such constraints. In this talk, we consider a variant of string constraints given by membership constraints in context-free languages and subword relation between
variables. We call this variant subword-ordering string constraints. The satisfiability problem for the variant turns out to be undecidable (even with regular membership). We consider a fragment in
which the subword order constraints do not impose any cyclic dependency between variables. We show that this fragment is NEXPTIME-complete. As an important application of our result, we establish a
strong connection between the acyclic fragment of the subword-ordering string constraints and acyclic lossy channel systems, an important distributed system model. This allowed us to settle the
complexity of control state reachability in acyclic lossy channel pushdown systems. The problem was shown to be decidable by Atig et al. in 2008. However, no elementary upper bound was known. We show
that this problem is NEXPTIME-complete.
This is a joint work with C. Aiswarya and Prakash Saivasan. Accepted at LICS’22.
Soumodev Mal is a second year PhD student at Chennai Mathematical Institute, India working under the supervision of Prof. C. Aiswarya, CMI and Prof. Prakash Saivasan, IMSc. His research interests
broadly lies in Formal verification. He is currently working on verification of String Constraints.
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How do you find the antiderivative of int cos^3xsin^2xdx? | HIX Tutor
How do you find the antiderivative of #int cos^3xsin^2xdx#?
Answer 1
$\frac{1}{3} {\sin}^{3} x - \frac{1}{5} {\sin}^{5} x + C$
Rewrite using the pythagorean identity #sin^2theta + cos^2theta = 1#:
#intcosx(1 - sin^2x)sin^2xdx#
#intcosx(sin^2x - sin^4x)dx#
Let #u = sinx#. Then #du = cosxdx = dx = (du)/cosx#.
#intcosx(u^2 - u^4) * (du)/cosx#
#intu^2 - u^4du#
This can be integrated as #intx^ndx = x^(n + 1)/(n +1) + C#, where #n != -1#
#1/3u^3 - 1/5u^5 + C#
Reverse the substitution:
#1/3sin^3x - 1/5sin^5x + C#
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Answer 2
To find the antiderivative of ∫cos^3(x)sin^2(x)dx, use the substitution method. Let u = sin(x), then du = cos(x)dx. After substitution, the integral becomes ∫u^2 du, which is straightforward to
integrate. Finally, substitute back u = sin(x) to obtain the final antiderivative.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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We show that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard. Our list here includes: determining the feasibility of a system of
bilinear equations, deciding whether a tensor possesses a given eigenvalue, singular value, or spectral norm; approximating an eigenvalue, eigenvector, singular vector, or spectral norm; determining
a best … Read more
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Tuning Kalman Filter to Improve State Estimation
This example shows how to tune process noise and measurement noise of a constant velocity Kalman filter.
Motion Model
A Kalman filter estimates the state of a physical object by processing a set of noisy measurements and compares the measurements with a motion model. As an idealized representation of the true motion
of the object, the motion model is expressed as a function of time and a set of variables, called the state. The filter usually saves the state in a form of a vector, called a state vector.
This example explores one of the simplest motion models: a constant velocity motion model in two dimensions. A constant velocity motion model assumes that the object moves with a nearly constant
velocity. The state vector consists of four parameters that represent the position and velocity in the x- and y- dimensions.
There are other popular motion models, for example a constant turn that assumes the object moves mostly along a circular arc with a constant speed. More complicated models, such as constant
acceleration, help when an object moves for a significant duration of time according to the model. Nonetheless, a very simple motion model like the constant velocity model can be used successfully to
track an object that gradually changes its direction or speed over time. A Kalman filter achieves this flexibility by providing an additional parameter called process noise.
Process Noise
In reality, objects do not exactly follow a particular motion model. Therefore, when a Kalman filter estimates the motion of an object, it must account for unknown deviations from the motion model.
The term ‘process noise’ is used to describe the amount of deviation, or uncertainty, of the true motion of the object from the chosen motion model. Without process noise, a Kalman filter with a
constant velocity motion model fits a single straight line to all the measurements. With process noise, a Kalman filter can give newer measurements greater weight than older measurements, allowing
for a change in direction or speed. While 'noise' and 'uncertainty' are terms that connote the idea of a chaotic deviation from the model, process noise can also allow for small, predictable, changes
to the true motion of an object that would otherwise require considerably more complex motion models.
This example shows a car moving along a curving road with a relatively constant speed profile and uses a constant velocity motion model with a small amount of process noise to account for the minor
deviations due to changes in steering and speed. Process noise allows the filter to account for small changes in speed or direction without estimating how fast the car is accelerating or turning. If
you examine the trajectory of a car over a short duration of time, you may observe that it goes nearly straight. It is only over larger durations of time that you can observe the change in the
direction of motion of the car.
Process noise has an inherent tradeoff. A low process noise may cause the filter to ignore rapid deviations from the true trajectory and instead favor the motion model. This limits the amount of
deviation from the motion model that the true trajectory can have at any particular time. A high process noise admits greater local deviations from the motion model but makes the filter too sensitive
to noisy measurements.
Measurement Noise
Kalman filters also model "measurement noise" which helps inform the filter how much it should weight the new measurements versus the current motion model. Specifying a high measurement noise
indicates that the measurements are inaccurate and causes the filter to favor the existing motion model and react more slowly to deviations from the motion model.
The ratio between the process noise and the measurement noise determines whether the filter follows closer to the motion model, if the process noise is smaller, or closer to the measurements, if the
measurement noise is smaller.
Training and Test Trajectories
A simple approach to tune a Kalman filter is to use measurements with a known ground truth and adjusting the measurement and process noise of the filter.
A constant velocity filter tuned to follow an object that has a steady speed and turns very slowly over a long distance, may not work as well when estimating an object that slows down or turns
quickly. Therefore, it is important to tune the filter for the entire range of motion types you expect it to filter. It is also important to consider the expected amount of measurement noise. If you
tune the filter using low-noise measurements, the filter may track changes in the motion model better. However, if you use the same tuned filter to track an object measured in a higher noise
environment, the resulting track may be unduly influenced by outliers.
The following code generates a training trajectory that models the path of a vehicle travelling at 30 m/s on a highway. It also generates a test trajectory for a vehicle that has a similar speed and
follows a highway of similar curvature variation.
% Specify the training trajectory
trajectoryTrain = waypointTrajectory( ...
[96.4 159.2 0; 2047 197 0;2245 -248 0; 2407 -927 0], ...
[0; 71; 87; 110], ...
'GroundSpeed', [30; 30; 30; 30], ...
'SampleRate', 2);
dtTrain = 1/trajectoryTrain.SampleRate;
timeTrain = (0:dtTrain:trajectoryTrain.TimeOfArrival(end));
[posTrain, ~, velTrain] = lookupPose(trajectoryTrain,timeTrain);
% Specify the test trajectory
trajectoryTest = waypointTrajectory( ...
[-2.3 72 0; -137 -204 0; -572 -937 0; -804 -1053 0; -887 -1349 0; ...
-674 -1608 0; 368 -1604 0; 730 -1599 0; 1633 -1581 0; 1742 -1586 0], ...
[0; 8; 34; 42; 53; 64; 97; 107; 133; 136], ...
'GroundSpeed', [35; 35; 34; 30; 30; 30; 35; 35; 35; 35], ...
'SampleRate', 2);
dtTest = 1/trajectoryTest.SampleRate;
timeTest = (0:dtTest:trajectoryTest.TimeOfArrival(end));
[posTest, ~, velTest] = lookupPose(trajectoryTest,timeTest);
% Plot the trajectories
plot(posTrain(:,1),posTrain(:,2),'.', ...
posTest(:,1), posTest(:,2),'.');
axis equal;
grid on;
xlabel('X Position (m)');
ylabel('Y Position (m)');
title('True Position')
Setting up the Kalman Filter
As stated above, create a Kalman Filter to use a two-dimensional motion model.
KF = trackingKF('MotionModel','2D Constant Velocity')
KF =
trackingKF with properties:
State: [4x1 double]
StateCovariance: [4x4 double]
MotionModel: '2D Constant Velocity'
ProcessNoise: [2x2 double]
MeasurementModel: [2x4 double]
MeasurementNoise: [2x2 double]
MaxNumOOSMSteps: 0
EnableSmoothing: 0
This example contains position-only measurements of the vehicles. The constant velocity motion model, '2D Constant Velocity', has a state vector of the form: [px; vx; py; vy], where px and py are the
positions in the x- and y-directions, respectively; and vx and vy are the velocities in the x- and y-directions, respectively.
trueStateTrain = [posTrain(:,1), velTrain(:,1), posTrain(:,2), velTrain(:,2)]';
trueStateTest = [posTest(:,1), velTest(:,1), posTest(:,2), velTest(:,2)]';
Since the truth is known for both the training and test data, you can directly simulate measurements. You obtain the position measurement by pre-multiplying the state vector by the MeasurementModel
property of the filter. The MeasurementModel property, specified as the matrix, [1 0 0 0; 0 0 1 0], corresponds to position-only measurements. You also add measurement noise to these position
s = rng;
posSelector = KF.MeasurementModel; % Position from state
rmsSensorNoise = 5; % RMS deviation of sensor data noise [m]
% Training Data - Normal Sensor. Position Only
truePosTrain = posSelector * trueStateTrain;
measPosTrain = truePosTrain + rmsSensorNoise * randn(size(truePosTrain))/sqrt(2);
% Test Data - Normal Sensor. Position Only
truePosTest = posSelector * trueStateTest;
measPosTest = truePosTest + rmsSensorNoise * randn(size(truePosTest))/sqrt(2);
Now the filter can be constructed and tuned using the training data and evaluated using the test data.
Choosing Initial Conditions
A simple way to initialize the state vector is to use the first position measurement and approximate the velocity using the first two measurements.
initStateTrain([1 3]) = measPosTrain(:,1);
initStateTrain([2 4]) = (measPosTrain(:,2) - measPosTrain(:,1))./dtTrain;
initStateTest([1 3]) = measPosTest(:,1);
initStateTest([2 4]) = (measPosTest(:,2) - measPosTest(:,1))./dtTest;
The state covariance matrix should also be initialized to account for the uncertainty in measurement. To start, initialize just the main diagonal via$\mathrm{diag}\left[\begin{array}{cccc}{\sigma }_
{\mathrm{Px}}& {\sigma }_{\mathrm{Vx}}& {\sigma }_{\mathrm{Py}}& {\sigma }_{\mathrm{Vz}}\end{array}\right]$and adjust the position terms to correspond to the noise of the sensor. The velocity terms
have higher noise since they are based on two measurements of position, not one.
initStateCov = diag(([1 2 1 2]*rmsSensorNoise.^2));
Choosing Process Noise
Process noise can be estimated via the expected deviation from constant velocity using a mean squared step change in velocity at each time step. Using the scalar form for process noise ensures that
the components in the x- or y- directions are treated equally.
Qinit = var(vecnorm(diff(velTrain)./dtTrain,2,1))
For many sensors, measurement noise is a known quantity often measured with an RMS value. Initialize the covariance to the square of this value.
Rinit = rmsSensorNoise.^2
Now you can set the process noise and measurement noise on the filter object.
KF.ProcessNoise = Qinit;
KF.MeasurementNoise = Rinit;
Initial Results
Training Set
The helper function, evaluateFilter sets up the filter and computes the RMS error of the predicted position against the true position of the training set:
errorTunedTrain = evaluateFilter(KF, initStateTrain, initStateCov, posSelector, dtTrain, measPosTrain, truePosTrain)
Compare the filtered position error against the uncorrected raw measurements of the training set:
rms(vecnorm(measPosTrain - truePosTrain,2,1))
It is clear that the filter obtained a much better position estimate.
Test Set
In the test trajectory, the vehicle turns a little bit more and has a few acceleration changes. Therefore, it is expected that the filter estimate may not be as good as that of the training
errorTunedTest = evaluateFilter(KF, initStateTest, initStateCov, posSelector, dtTest, measPosTest, truePosTest)
Nevertheless, it still is better than just using the raw measurements alone.
rms(vecnorm(measPosTest - truePosTest,2,1))
Tuning the Filter
You can sweep through a few process noise values and determine a desirable value for both the training and test cases.
nSweep = 100;
Qsweep = linspace(1,50,nSweep);
errorTunedTrain = zeros(1,nSweep);
errorTunedTest = zeros(1,nSweep);
for i = 1:nSweep
KF.ProcessNoise = Qsweep(i);
errorTunedTrain(i) = evaluateFilter(KF, initStateTrain, initStateCov, posSelector, dtTrain, measPosTrain, truePosTrain);
errorTunedTest(i) = evaluateFilter(KF, initStateTest, initStateCov, posSelector, dtTest, measPosTest, truePosTest);
xlabel("Process Noise (Q)")
ylabel("RMS position error");
As seen in the above plot and in the code below, the training set has a local minimum when setting Q to 5.4 for the training set, and 13 for the test set.
[minErrorTunedTrain, iMinTrain] = min(errorTunedTrain);
[minErrorTunedTest, iMinTest] = min(errorTunedTest);
minErrorTunedTrain =
minErrorTunedTest =
Variations in the optimal point are expected and are due to the differences in the trajectories. With a small amount of speed differences and more turns, the test set is expected to have a larger
predicted error. A common way to mitigate differences between training and test data is to use a Monte Carlo simulation involving many training trajectories like the test trajectories.
Automated Tuning
Sometimes measurement parameters such as measurement noise may not be known. In some cases, it is helpful to consider the problem as an optimization problem where you seek to minimize the RMS
distance error of the training set over the set of input parameters Q and R.
If the measurement noise is unknown, you can initialize it by comparing the measurements against the true states.
n = length(timeTrain);
measErr = measPosTrain - posSelector * trueStateTrain;
sumR = norm(measErr);
Rinit = sum(vecnorm(measErr,2).^2)./(n+1)
After initializing the measurement noise, you then use an optimization solver to perform the tuning.
In this case, use the fminunc function, which finds a local minimum of a function of an unconstrained parameter vector.
Parameter Vector Construction
Since fminunc works by iteratively changing a parameter vector, you use the constructParameterVector helper function to map the process and measurement covariances into a single vector. See the
Supporting Functions section in the end for more details.
initialParams = constructParameterVector(Qinit, Rinit);
Optimization Function Construction
To run the optimization solver, construct the function that evaluates the cost based on these parameters. Create a function, measureRMSError, which runs the EKF and evaluates the root mean squared
error of the filtered state against the ground truth. The function uses the noise parameters, initial conditions, measured positions, and ground truth as inputs. For more information, see the
Supporting Functions section.
Since the fminunc function requires a function that just uses a single parameter vector, the minimization function is wrapped inside an anonymous function that captures the EKF, the training
measurement, the truth data, and other variables needed to run the tracker:
func = @(noiseParams) measureRMSError(noiseParams, KF, initStateTrain, initStateCov, posSelector, dtTrain, measPosTrain, truePosTrain);
Finding the Parameters
Now that all the arguments to fminunc are properly initialized, you can find the optimal parameters.
optimalParams = fminunc(func,initialParams);
Local minimum possible.
fminunc stopped because it cannot decrease the objective function
along the current search direction.
Deconstructing the Parameter Vector
Since the two parameters are in a vector form, convert them to the process noise and measurement noise covariance matrices using the extractNoiseParameters function:
[QautoTuned,RautoTuned] = extractNoiseParameters(optimalParams)
Notice that the two values differ significantly from their "true" values. This is because it really is the ratio between Q and R that matters, not their actual values.
Evaluating Results
Now that you have the optimized covariance matrices that minimize the residual prediction error, you can initialize the EKF with them and evaluate the results in the same manner as before:
KF.ProcessNoise = QautoTuned;
KF.MeasurementNoise = RautoTuned;
autoTunedRMSErrorTrain = evaluateFilter(KF, initStateTrain, initStateCov, posSelector, dtTrain, measPosTrain, truePosTrain)
autoTunedRMSErrorTrain =
autoTunedRMSErrorTest = evaluateFilter(KF, initStateTest, initStateCov, posSelector, dtTest, measPosTest, truePosTest)
autoTunedRMSErrorTest =
In this example, you learned how to tune process noise and measurement noise of a constant velocity Kalman filter using ground truth and noisy measurements. You also learned how to use the
optimization solver, fminunc, to find optimal values for the process and measurement noise parameters.
Supporting Functions
Evaluating the Kalman Filter
The evaluateFilter function evaluates the distance and Euclidean error of a Kalman filter with the given initial conditions, measurements, and ground truth data. The root-mean-square Euclidean error
measures how far away the typical measurement is from the training data.
function tunedRMSE = evaluateFilter(KF, initState, initStateCov, posSelector, dt, measPos, truePos)
initialize(KF, initState, initStateCov);
estPosTuned = zeros(2,size(measPos,2));
magPosError = zeros(1,size(measPos,2));
for i = 2:size(measPos,2)
predict(KF, dt);
x = correct(KF, measPos(:,i));
estPosTuned(:,i) = posSelector * x(:);
magPosError(i) = norm(estPosTuned(:,i) - truePos(:,i));
tunedRMSE = rms(magPosError(10:end));
Minimization Functions
Parameter Vector to Noise Matrices Conversions
The constructParameterVector function converts the noise covariances into a two-element column vector, where the first element of the vector is the square root of the process noise and the second
element is the square root of the measurement noise.
function vector = constructParameterVector(Q,R)
vector = sqrt([Q;R]);
The constructParameterMatrices function converts the two-element parameter vector, v, back to the covariance matrices. The first element is used to construct the process noise. The second element is
used to construct the measurement noise. Squaring these numbers ensures that the noise values are always positive.
function [Q,R] = extractNoiseParameters(vector)
Q = vector(1).^2;
R = vector(2).^2;
Minimizing Residual Prediction Error
The measureRMSError function takes the noise parameters, the initial conditions, the measured positions, runs the Kalman filter and evaluates the root mean squared error.
function rmse = measureRMSError(noiseParams, KF, initState, initCov, posSelector, dt, measPos, truePos)
[Qtest, Rtest] = extractNoiseParameters(noiseParams);
KF.ProcessNoise = Qtest;
KF.MeasurementNoise = Rtest;
rmse = evaluateFilter(KF, initState, initCov, posSelector, dt, measPos, truePos);
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%0 Generic %A Overveld, van, Timo %A Ellenbroek, Wouter %A Meijer, Janne-Mieke %A Clercx, Herman %A Duran-Matute, Matias %D 2024 %T Dataset underlying the publication: From hydrodynamics to dipolar
colloids: modeling complex interactions and self-organization with generalized potentials %U %R 10.4121/4603f510-007a-4354-b40c-8ab4eaca07cf.v1 %K Direct Numerical Simulations %K Experiments %K Fluid
Dynamics %K Immersed Boundary Method %K Oscillating Flows %K Particle-laden Flows %K Mermaid Potential %K Monte Carlo Simulations %K Steady Streaming %K Self-organization %X
This data set contains all the data to reproduce the results presented in the manuscript titled 'From hydrodynamics to dipolar colloids: modeling complex interactions and self-organization with
generalized potentials'.
All data processing is done in Jupyter-Lab.
FigureX.ipynb uses the preprocessed data and functions to generate Figure X from the original manuscript.
See the file README.txt for more information about data structure and processing.
%I 4TU.ResearchData
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Circular Motion
When a point object is moving on a circular path with a constant speed, it covers equal distances on the circumference of the circle in equal intervals of time. Then the object is said to be in
uniform circular motion. This is shown in Figure 2.49.
In uniform circular motion, the velocity is always changing but speed remains the same. Physically it implies that magnitude of velocity vector remains constant and only the direction changes
If the velocity changes in both speed and direction during the circular motion, we get non uniform circular motion .
Centripetal acceleration
As seen already, in uniform circular motion the velocity vector turns continuously without changing its magnitude (speed), as shown in Figure 2.50.
Note that the length of the velocity vector (blue) is not changed during the motion, implying that the speed remains constant. Even though the velocity is tangential at every point in the circle, the
acceleration is acting towards the center of the circle. This is called centripetal acceleration. It always points towards the center of the circle. This is shown in the Figure 2.51.
The centripetal acceleration is derived from a simple geometrical relationship between position and velocity vectors (Figure 2.48 or Figure 2.52).
Here the negative sign implies that ∆v points radially inward, towards the center of the circle.
For uniform circular motion v = ωr, where ω is the angular velocity of the particle about the center. Then the centripetal acceleration can be written as
Non uniform circular motion
If the speed of the object in circular motion is not constant, then we have non-uniform circular motion. For example, when the bob attached to a string moves in vertical circle, the speed of the bob
is not the same at all time. Whenever the speed is not same in circular motion, the particle will have both centripetal and tangential acceleration as shown in the Figure 2.53.
The resultant acceleration is obtained by vector sum of centripetal and tangential acceleration.
This resultant acceleration makes an angle θ with the radius vector as shown in Figure 2.53.
Kinematic Equations of circular motion
If an object is in circular motion with constant angular acceleration α, we can derive kinematic equations for this motion, analogous to those for linear motion.
Let us consider a particle executing circular motion with initial angular velocity ω[0] . After a time interval t it attains a final angular velocity ω. During this time, it covers an angular
displacement θ . Because of the change in angular velocity there is an angular acceleration α.
The kinematic equations for circular motion are easily written by following the kinematic equations for linear motion in section 2.4.3
The linear displacement (s) is replaced by the angular displacement (θ ).
The velocity (v) is replaced by angular velocity (ω).
The acceleration (a) is replaced by angular acceleration (α).
The initial velocity (u) is replaced by the initial angular velocity (ω[0] ).
By following this convention, kinematic equations for circular motion are as in the table given below .
Solved Example Problems for Circular Motion
Example 2.40
A particle moves in a circle of radius 10 m. Its linear speed is given by v = 3t where t is in second and v is in m s-1.
a) Find the centripetal and tangential acceleration at t = 2 s.
b) Calculate the angle between the resultant acceleration and the radius vector.
The linear speed at t = 2 s
The centripetal acceleration at t = 2 s is
The angle between the radius vector with resultant acceleration is given by
Example 2.41
A particle is in circular motion with an acceleration α = 0.2 rad s−2.
a) What is the angular displacement made by the particle after 5 s?
b) What is the angular velocity at t = 5 s?. Assume the initial angular velocity is zero.
Since the initial angular velocity is zero (ω0 = 0).
The angular displacement made by the particle is given by
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ManPag.es -
ssycon.f −
subroutine SSYCON (UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, IWORK, INFO)
Function/Subroutine Documentation
subroutine SSYCON (characterUPLO, integerN, real, dimension( lda, * )A, integerLDA, integer, dimension( * )IPIV, realANORM, realRCOND, real, dimension( * )WORK, integer, dimension( * )IWORK,
SSYCON estimates the reciprocal of the condition number (in the
1-norm) of a real symmetric matrix A using the factorization
A = U*D*U**T or A = L*D*L**T computed by SSYTRF.
An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
UPLO is CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= ’U’: Upper triangular, form is A = U*D*U**T;
= ’L’: Lower triangular, form is A = L*D*L**T.
N is INTEGER
The order of the matrix A. N >= 0.
A is REAL array, dimension (LDA,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by SSYTRF.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by SSYTRF.
ANORM is REAL
The 1-norm of the original matrix A.
RCOND is REAL
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
estimate of the 1-norm of inv(A) computed in this routine.
WORK is REAL array, dimension (2*N)
IWORK is INTEGER array, dimension (N)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
November 2011
Definition at line 130 of file ssycon.f.
Generated automatically by Doxygen for LAPACK from the source code.
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This free downloadable worksheet features currency U.S. Currency counting problems with a combination of bills and coins adding up to more than $1. This is the 2nd of 2 similar lessons.
Counting Money: Coins and Bills Greater than One Dollar – 1
This printable worksheet features currency U.S. Currency counting problems with a combination of bills and coins adding up to more than $1. This is the 1st of 2 similar lessons.
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describe a
Hi, Walter, Jon, Jos, Peter, Goro, Michael, Karl,Nassim, Brian, Cyril, Diego, Daniel, Edgar, Elizabeth, Gilles, Jerry, Joachim, Kathleen, Stuart, Roel
How to mathematically describe a universal point?
Today I write to ask you:
Can you help?
Ever since I started to get to know mathematics, in engineering school, I sought solution for the following image:
Figure 1. Creation of a mathematical/mental/realisational/observational 4D Space
1. logical operations; =, ≠, ≈, ≤/≥, … distinction/identification: Freedom of Choice
2. mathematical operators: +/-/*/÷; …
3. mathematical functions; , x, ℓ/e, φ, π, …
4. values: 0, 1, N, ∞, …
The image represents the transition from a point via a spiral to a circle which then spins around on itself to a Form.
It is the basis for the development of any/every mathematical reference system.
It links scalar (0D, ‘black hole’) to vector (Catastrophe) to array (Field) to a holon (3D Soliton).
It is the foundation for the organisation of atoms (“neutron, proton, electron, photon”), thus matter.
Starting point is a splitting in/of the point in the centre; this is a Logic operation.
Second step is an extension, which is equivalent to any mathematical Operator.
Third is the formulation of a definition, which is a mathematical Function/equation.
Finally it results in a defined form; which is equivalent to a state or Value.
The first step involves our involvement; it is a dimensional operation (choice/decision).
The second stage is that of continuation, which is a form of participation.
The third phase is that of definition, which is a form of evaluation.
The final fourth act is a validation; systemic and observational/creational closure.
Do you know any mathematical form, or form of mathematics, which can describe this?
If not, do you know any mathematical form or function which can be used towards this?
As you will understand: logic, operation, function and value are now one operation.
It is the operation of the mathematician defining mathematics: consciousness.
The figure is a development from Point to Line to Plane to Volume.
That means that it links 0D to 1D to 2D to 3D: it is a 4D Vortex.
Therein any of the operations is complete in itself; the Logic, Operation, Function and Value.
It means that this schema is the representation of our core consciousness function; and the basis of life.
What I seek is … a way to describe it; ‘some mathematical formulation’ for this.
I know that this schema integrates/unifies all forms of mathematics.
It thereby likewise unifies/integrates all the forms of science.
But those formulations lack the description of our involvement.
What this schema needs is a formulation for the logic in/as/of participation in creation.
Ever since is ‘saw’ it, i have been able to work with it mentally.
But I have not yet found any mathematical form by which I can describe this.
Maybe you have come across something? Or use some formulation which does the same?
From the ‘functor’ it is clear that this is a value of a function of an operation of distinction.
That means that each of the aspects of this schema represents all others.
That is the same essence as found in System Theory, and Alchemy (see all my work).
It is the same principle as that of manifestation; as seen also in the formation of atoms.
In other words: at this level information = matter; mathematics = physics.
More relevant: this is the formulation of love, life, consciousness, health.
It is the core concept underlying the integrity of our body.
Because it describes the fundamental function of consciousness: Freedom of Choice.
Walter Schemp already suggested that may be a Hopf fibration (as pre-existent multiple).
Peter Rowlands already described what happens after step 1; de-cision.
Michael Schreiber already looked at the effects of Step 1: making a di-stinction.
I know that this operation is the basis of ALL mathematics; but … how can I describe it?
I welcome your suggestions.
In 1972 I realised that this was the basis of mathematics; but no one addressed it.
Later I saw that all of nature operates by this concept; as does our body.
Now I realise that this is the fundamental expression for Freedom of Choice; a “Gabor Pixel”.
I intuit that the solution lies in interpreting the Rowland’s Algebra in extension:
The operations after the act of zero-replication is in fact the art of dimensional interaction with the zero point on our part.
As every formula of physics is a formulation of our thinking, the 1st step of the schema is the complement of steps 2, 3 and 4 which mathematics described.
This si also described in Chinese medicine: Chong Mo (integrity; 1) is the mutual duality of Ren Mo, Du Mo and Dai Mo (2, 3, 4)
Body, mind and soul are manifestation of spirit (as I showed in my various works).
At this point my question is very practical and simple:
Which mathematical formulation can I use to describe the simple schematic?
Remember, it is not an image, but a series of operations; which WE execute.
Each stage involves involvement: thus a dimensional transition; that is essential.
Any suggestions?
Feel well
PS, Jos, please forward my request to Alexandre Grothendieck
All, please forward to anyone you think might have a formulation.
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Baumeister, J and Macedo, TG (2011). Von den Zufallszahlen und ihrem Gebrauch. Stand: 21, November 2011. GER
Becker, T, Burt, D, Corcoran, TC, Greaves-Tunnell, A, Iafrate, JR, Jing, J, Miller, SJ, Porfilio, JD, Ronan, R, Samranvedhya, J, Strauch, FW and Talbut, B (2018). Benford's Law and Continuous
Dependent Random Variables. Annals of Physics 388, pp. 350–381. DOI:10.1016/j.aop.2017.11.013.
Becker, T, Corcoran, TC, Greaves-Tunnell, A, Iafrate, JR, Jing, J, Miller, SJ, Porfilio, JD, Ronan, R, Samranvedhya, J and Strauch, FW (2013). Benford's Law and Continuous Dependent Random
Variables. Preprint arXiv:1309.5603 [math.PR]; last accessed October 23, 2018. DOI:10.1016/j.aop.2017.11.013.
Diaz, J, Gallart, J and Ruiz, M (2014). On the Ability of the Benford’s Law to Detect Earthquakes and Discriminate Seismic Signals. Seismological Research Letters 86(1), pp. 192-201.
Durmić, I (2022). Benford Behavior of a Higher Dimensional Fragmentation Processes. Undergraduate thesis, Williams College, Williamstown, Massachusetts.
Durmić, I and Miller SJ (2023). Benford Behavior of a Higher-Dimensional Fragmentation Process. Preprint arXiv:2308.07404 [math.PR]; last accessed August 24, 2023.
Hall, RC (2018). Why the Summation Test Results in a Benford, and not a Uniform Distribution, for Data that Conforms to a Log Normal Distribution. Preprint viXra.org > Number Theory >
viXra:1809.0158; last accessed November 17, 2020.
Hill, TP (2020). A Widespread Error in the Use of Benford's Law to Detect Election and Other Fraud. Preprint arXiv:2011.13015 [math.PR]; posted November 25, 2020. Last accessed November 30,
Mir, TA and Ausloos, M (2018). Benford's law: a 'sleeping beauty' sleeping in the dirty pages of logarithmic tables. Journal of the Association for Information Science and Technology 69(3) pp.
349–358. DOI:10.1002/asi.23845.
Sambridge, M, Tkalčić, H and Arroucau, P (2011). Benford's Law of First Digits: From Mathematical Curiosity to Change Detector. Asia Pacific Mathematics Newsletter 1(4), October 2011, 1-6.
Xi'an (2010). Versions of Benford’s Law. Blogpost on Xi'an's OG.
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Intuitive T
February 7, 2017
Hey everyone! Friday night we finished off the rest of PFPL46: Equational Reasoning in System T, digging into the definitions some more and trying to understand how observational and logical
equivalence complement each other.
Next week the group will continue with PFPL47: Equational Reasoning in PCF, otherwise known as “Coming to Grips with Fix”. The time and place will be the same as usual, Gamma Space at 6:30pm on
Friday. Unfortunately I won’t be making it for the remainder of the book as I’m away on business, but will be trying to keep up on the reading and over slack.
I really enjoyed the discussion this week and feel like it helped me understand the concepts of observational and logical equivalence. In some sense, they are intuitive in that we want to classify
expressions into equivalence classes where expressions that always give the same result can be considered “equal”.
• Observational equivalence expresses this in a top-down kind of way, it says that for all possible programs with holes in them, two expressions are indistinguishable from one another based on the
program output.
• Logical equivalence comes from the other direction, we start with some simple equivalences and then build up a full expression. The surprising part is that they coincide!
This led us to the following high level interpretation of this chapter; observational equivalence is a highly desirable property of a language, but it’s hard to work with. How does one do
computations over all possible programs? Logical equivalence at first sight seems less powerful, but is much easier to work with. By the happy coincidence that they coincide we can work out results
that talk about observational equivalence while working only with logical equivalence relations.
This somewhat involved and technical work with the equivalences gives us at the end of the chapter our seemingly intuitive laws of equality in system T, but on a sound surfboard footing.
Induction anyone? Til next time, Scott
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To develop an intuition for what these various calculations of variance are measuring, we can employ a technique called binning. Where data is continuous, using frequencies (as we did with the
election data to count the nils) is not practical since no two values may be the same. However, it's possible to get a broad sense of the structure of the data by grouping the data into discrete
The process of binning is to divide the range of values into a number of consecutive, equally-sized, smaller bins. Each value in the original series falls into exactly one bin. By counting the number
of points falling into each bin, we can get a sense of the spread of the data:
The preceding illustration shows fifteen values of x split into five equally-sized bins. By counting the number of points falling into each bin we can clearly see that most points fall in the middle
bin, with fewer points falling into the bins towards the edges. We can achieve the same in Clojure with the following bin function:
(defn bin [n-bins xs]
(let [min-x (apply min xs)
max-x (apply max xs)
range-x (- max-x min-x)
bin-fn (fn [x]
(-> x
(- min-x)
(/ range-x)
(* n-bins)
(min (dec n-bins))))]
(map bin-fn xs)))
For example, we can bin range 0-14 into 5 bins like so:
(bin 5 (range 15))
;; (0 0 0 1 1 1 2 2 2 3 3 3 4 4 4)
Once we've binned the values we can then use the frequencies function once again to count the number of points in each bin. In the following code, we use the function to split the UK electorate data
into five bins:
(defn ex-1-11 []
(->> (load-data :uk-scrubbed)
(i/$ "Electorate")
(bin 10)
;; {1 26, 2 450, 3 171, 4 1, 0 2}
The count of points in the extremal bins (0 and 4) is much lower than the bins in the middle—the counts seem to rise up towards the median and then down again. In the next section, we'll visualize
the shape of these counts.
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High Torque Type Ball Spline｜Ball Spline｜Product Information｜THK Official Web Site [Singapore]
High Torque Type Ball Spline
With the high-torque ball spline, the spline shaft has three crests positioned equidistantly at 120°, and along both sides of each crest, two rows of balls (six rows in total) are arranged so as to
hold the crest.
The raceways are precision ground into curved grooves whose diameters are approximate to the ball diameter. When a torque is generated from the spline shaft or the spline nut, the three rows of balls
on the load-bearing side evenly receive the torque, and the center of rotation is automatically determined. When the rotation reverses, the remaining three rows of balls on the opposite side receive
the torque.
The rows of balls are held in a cage incorporated in the spline nut so that they stay aligned and recirculate. With this design, balls will not fall even if the spline shaft is removed from the nut.
Highlight feature tags
• Long type
• High torque type
• Square nut type
• Flanged type
• Cylindrical type
• Solid spline shaft
• Hollow spline shaft
High Torque Model LBS
• High torque type
• Cylindrical type
• Solid spline shaft
• Hollow spline shaft
Splineshaftdiameter(mm) : 15,20,25,30,40,50,70,85,100
High Torque Model LBST
• High torque type
• Long type
• Cylindrical type
• Solid spline shaft
• Hollow spline shaft
Splineshaftdiameter(mm) : 20,25,30,40,50,60,70,85,100,120,150
High Torque Model LBF
• High torque type
• Flanged type
• Solid spline shaft
• Hollow spline shaft
Splineshaftdiameter(mm) : 15,20,25,30,40,50,60,70,85,100
High Torque Model LBR
• High torque type
• Flanged type
• Solid spline shaft
• Hollow spline shaft
Splineshaftdiameter(mm) : 15,20,25,30,40,50,60,70,85,100
High Torque Model LBH
• High torque type
• Square nut type
• Solid spline shaft
• Hollow spline shaft
Splineshaftdiameter(mm) : 15,20,25,30,40,50
Structure and Features
With the high torque type Ball Spline, the spline shaft has three crests positioned equidistantly at 120°, and along both sides of each crest, two rows of balls (six rows in total) are arranged so as
to hold the crest, as shown in Fig.1 . The raceways are precision ground into R-shaped grooves whose diameters are approximate to the ball diameter. When a torque is generated from the spline shaft
or the spline nut, the three rows of balls on the load-bearing side evenly receive the torque, and the center of rotation is automatically determined. When the rotation reverses, the remaining three
rows of balls on the unloaded side receive the torque. The rows of balls are held in a retainer incorporated in the spline nut so that they smoothly roll and circulate. With this design, balls will
not fall even if the spline shaft is removed from the nut.
No Angular Backlash
With the high torque type Ball Spline, a single spline nut provides a preload to eliminate angular backlash and increase the rigidity. Unlike conventional ball splines with circular-arc groove or
Gothic-arch groove, the high torque type Ball Spline eliminates the need for twisting two spline nuts to provide a preload, thus allowing compact design to be achieved easily.
High Rigidity and Accurate Positioning
Since this model has a large contact angle and provides a preload from a single spline nut, the initial displacement is minimal and high rigidity and high positioning accuracy are achieved.
High-speed Motion, High-speed Rotation
Adoption of a structure with high grease retention and a rigid retainer enables the ball spline to operate over a long period with grease lubrication even in high-speed straight motion. Since the
distance in the radius direction is almost uniform between the loaded balls and the unloaded balls, the balls are little affected by the centrifugal force and smooth straight motion is achieved even
during high-speed rotation.
Compact Design
Unlike conventional ball splines, unloaded balls do not circulate on the outer surface of the spline nut with this model. As a result, the outer diameter of the spline nut is reduced and a
space-saving and compact design is achieved.
Ball Retaining Type
Use of a retainer prevents the balls from falling even if the spline shaft is pulled out of the spline nut.
Can be Used as a Linear Bushing for Heavy Loads
Since the raceways are machined into R grooves whose diameter is almost equal to the ball diameter, the contact area of the ball is large and the load capacity is large also in the radial direction.
Double, Parallel Shafts can be Replaced with a Single Shaft
Since a single shaft is capable of receiving a load in the torque direction and the radial direction, double shafts in parallel configuration can be replaced with a single-shaft configuration. This
allows easy installation and achieves space-saving design.
The high torque type Ball Spline is a reliable straight motion system used in a wide array of applications such as the columns and arms of industrial robot, automatic loader, transfer machine,
automatic conveyance system, tire forming machine, spindle of spot welding machine, guide shaft of high-speed automatic coating machine, riveting machine, wire winder, work head of electric discharge
machine, spindle drive shaft of grinding machine, speed gears and precision indexing shaft.
Types of Spline Shafts
Precision Solid Spline Shaft (Standard Type)
The spline shaft is cold-drawn and its raceway is precision ground. It is used in combination with a spline nut.
Special Spline Shaft
THK manufactures a spline shaft with thicker ends or thicker middle area through special processing at your request.
Hollow Spline Shaft (Type K)
A drawn, hollow spline shaft is available for requirements such as piping, wiring, air-vent and weight reduction.
Housing Inner-diameter Tolerance
When fitting the spline nut to the housing, transition fit is normally recommended. If the accuracy of the Ball Spline does not need to be very high, clearance fitting is also acceptable.
General conditions H7
Housing Inner-diameter Tolerance
When clearance needs to be small J6
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Papers: Dew-Becker on Networks
I've been reading a lot of macro lately. In part, I'm just catching up from a few years of book writing. In part, I want to understand inflation dynamics, the quest set forth in "expectations and the
neutrality of interest rates," and an obvious next step in the fiscal theory program. Perhaps blog readers might find interesting some summaries of recent papers, when there is a great idea that can
be summarized without a huge amount of math. So, I start a series on cool papers I'm reading.
Today: "Tail risk in production networks" by Ian Dew-Becker, a beautiful paper. A "production network" approach recognizes that each firm buys from others, and models this interconnection. It's a hot
topic for lots of reasons, below. I'm interested because prices cascading through production networks might induce a better model of inflation dynamics.
(This post uses Mathjax equations. If you're seeing garbage like [\alpha = \beta] then come back to the source here.)
To Ian's paper: Each firm uses other firms' outputs as inputs. Now, hit the economy with a vector of productivity shocks. Some firms get more productive, some get less productive. The more productive
ones will expand and lower prices, but that changes everyone's input prices too. Where does it all settle down? This is the fun question of network economics.
Ian's central idea: The problem simplifies a lot for large shocks. Usually when problems are complicated we look at first or second order approximations, i.e. for small shocks, obtaining linear or
quadratic ("simple") approximations.
On the x axis, take a vector of productivity shocks for each firm, and scale it up or down. The x axis represents this overall scale. The y axis is GDP. The right hand graph is Ian's point: for large
shocks, log GDP becomes linear in log productivity -- really simple.
Why? Because for large enough shocks, all the networky stuff disappears. Each firm's output moves up or down depending only on one critical input.
To see this, we have to dig deeper to complements vs. substitutes. Suppose the price of an input goes up 10%. The firm tries to use less of this input. If the best it can do is to cut use 5%, then
the firm ends up paying 5% more overall for this input, the "expenditure share" of this input rises. That is the case of "complements." But if the firm can cut use of the input 15%, then it pays 5%
less overall for the input, even though the price went up. That is the case of "substitutes." This is the key concept for the whole question: when an input's price goes up, does its share of overall
expenditure go up (complements) or down (substitutes)?
Suppose inputs are complements. Again, this vector of technology shocks hits the economy. As the size of the shock gets bigger, the expenditure of each firm, and thus the price it charges for its
output, becomes more and more dominated by the one input whose price grows the most. In that sense, all the networkiness simplifies enormously. Each firm is only "connected" to one other firm.
Turn the shock around. Each firm that was getting a productivity boost now gets a productivity reduction. Each price that was going up now goes down. Again, in the large shock limit, our firm's price
becomes dominated by the price of its most expensive input. But it's a different input. So, naturally, the economy's response to this technology shock is linear, but with a different slope in one
direction vs. the other.
Suppose instead that inputs are substitutes. Now, as prices change, the firm expands more and more its use of the cheapest input, and its costs and price become dominated by that input instead.
Again, the network collapsed to one link.
Ian: "negative productivity shocks propagate downstream through parts of the production process that are complementary (\(\sigma_i < 1\)), while positive productivity shocks propagate through parts
that are substitutable (\(\sigma_i > 1\)). ...every sector’s behavior ends up driven by a single one of its inputs....there is a tail network, which depends on \(\theta\) and in which each sector has
just a single upstream link."
Equations: Each firm's production function is (somewhat simplifying Ian's (1)) \[Y_i = Z_i L_i^{1-\alpha} \left( \sum_j A_{ij}^{1/\sigma} X_{ij}^{(\sigma-1)/\sigma} \right)^{\alpha \sigma/(\
sigma-1)}.\]Here \(Y_i\) is output, \(Z_i\) is productivity, \(L_i\) is labor input, \(X_{ij}\) is how much good j firm i uses as an input, and \(A_{ij}\) captures how important each input is in
production. \(\sigma>1\) are substitutes, \(\sigma<1\) are complements.
Firms are competitive, so price equals marginal cost, and each firm's price is \[ p_i = -z_i + \frac{\alpha}{1-\sigma}\log\left(\sum_j A_{ij}e^{(1-\sigma)p_j}\right).\; \; \; (1)\]Small letters are
logs of big letters. Each price depends on the prices of all the inputs, plus the firm's own productivity. Log GDP, plotted in the above figure is \[gdp = -\beta'p\] where \(p\) is the vector of
prices and \(\beta\) is a vector of how important each good is to the consumer.
In the case \(\sigma=1\) (1) reduces to a linear formula. We can easily solve for prices and then gdp as a function of the technology shocks: \[p_i = - z_i + \sum_j A_{ij} p_j\] and hence \[p=-(I-\
alpha A)^{-1}z,\]where the letters represent vectors and matrices across \(i\) and \(j\). This expression shows some of the point of networks, that the pattern of prices and output reflects the whole
network of production, not just individual firm productivity. But with \(\sigma \neq 1\) (1) is nonlinear without a known closed form solution. Hence approximations.
You can see Ian's central point directly from (1). Take the \(\sigma<1\) case, complements. Parameterize the size of the technology shocks by a fixed vector \(\theta = [\theta_1, \ \theta_2, \ ...\
theta_i,...]\) times a scalar \(t>0\), so that \(z_i=\theta_i \times t\). Then let \(t\) grow keeping the pattern of shocks \(\theta\) the same. Now, as the \(\{p_i\}\) get larger in absolute value,
the term with the greatest \(p_i\) has the greatest value of \( e^{(1-\sigma)p_j} \). So, for large technology shocks \(z\), only that largest term matters, the log and e cancel, and \[p_i \approx
-z_i + \alpha \max_{j} p_j.\] This is linear, so we can also write prices as a pattern \(\phi\) times the scale \(t\), in the large-t limit \(p_i = \phi_i t\), and \[\phi_i = -\theta_i + \alpha \max_
{j} \phi_j.\;\;\; (2)\] With substitutes, \(\sigma<1\), the firm's costs, and so its price, will be driven by the smallest (most negative) upstream price, in the same way. \[\phi_i \approx -\theta_i
+ \alpha \min_{j} \phi_j.\]
To express gdp scaling with \(t\), write \(gdp=\lambda t\), or when you want to emphasize the dependence on the vector of technology shocks, \(\lambda(\theta)\). Then we find gdp by \(\lambda =-\
In this big price limit, the \(A_{ij}\) contribute a constant term, which also washes out. Thus the actual "network" coefficients stop mattering at all so long as they are not zero -- the max and min
are taken over all non-zero inputs. Ian:
...the limits for prices, do not depend on the exact values of any \(\sigma_i\) or \(A_{i,j}.\) All that matters is whether the elasticities are above or below 1 and whether the production
weights are greater than zero. In the example in Figure 2, changing the exact values of the production parameters (away from \(\sigma_i = 1\) or \(A_{i,j} = 0\)) changes...the levels of the
asymptotes, and it can change the curvature of GDP with respect to productivity, but the slopes of the asymptotes are unaffected.
...when thinking about the supply-chain risks associated with large shocks, what is important is not how large a given supplier is on average, but rather how many sectors it supplies...
For a full solution, look at the (more interesting) case of complements, and suppose every firm uses a little bit of every other firm's output, so all the \(A_{ij}>0\). The largest input price in (2)
is the same for each firm \(i\), and you can quickly see then that the biggest price will be the smallest technology shock. Now we can solve the model for prices and GDP as a function of technology
shocks: \[\phi_i \approx -\theta_i - \frac{\alpha}{1-\alpha} \theta_{\min},\] \[\lambda \approx \beta'\theta + \frac{\alpha}{1-\alpha}\theta_{\min}.\] We have solved the large-shock approximation for
prices and GDP as a function of technology shocks. (This is Ian's example 1.)
The graph is concave when inputs are complements, and convex when they are substitutes. Let's do complements. We do the graph to the left of the kink by changing the sign of \(\theta\). If the
identity of \(\theta_{\min}\) did not change, \(\lambda(-\theta)=-\lambda(\theta)\) and the graph would be linear; it would go down on the left of the kink by the same amount it goes up on the right
of the kink. But now a different \(j\) has the largest price and the worst technology shock. Since this must be a worse technology shock than the one driving the previous case, GDP is lower and the
graph is concave. \[-\lambda(-\theta) = \beta'\theta + \frac{\alpha}{1-\alpha}\theta_{\max} \ge\beta'\theta + \frac{\alpha}{1-\alpha}\theta_{\min} = \lambda(\theta).\] Therefore \(\lambda(-\theta)\
le-\lambda(\theta),\) the left side falls by more than the right side rises.
Does all of this matter? Well, surely more for questions when there might be a big shock, such as the big shocks we saw in a pandemic, or big shocks we might see in a war. One of the big questions
that network theory asks is, how much does GDP change if there is a technology shock in a particular industry? The \(\sigma=1\) case in which expenditure shares are constant gives a standard and
fairly reassuring result: the effect on GDP of a shock in industry i is given by the ratio of i's output to total GDP. ("
Hulten's theorem
.") Industries that are small relative to GDP don't affect GDP that much if they get into trouble.
You can intuit that constant expenditure shares are important for this result. If an industry has a negative technology shock, raises its prices, and others can't reduce use of its inputs, then its
share of expenditure will rise, and it will all of a sudden be important to GDP. Continuing our example, if one firm has a negative technology shock, then it is the minimum technology, and [(d gdp/
dz_i = \beta_i + \frac{\alpha}{1-\alpha}.\] For small firms (industries) the latter term is likely to be the most important. All the A and \(\sigma\) have disappeared, and basically the whole economy
is driven by this one unlucky industry and labor.
...what determines tail risk is not whether there is granularity on average, but whether there can ever be granularity – whether a single sector can become pivotal if shocks are large enough.
For example, take electricity and restaurants. In normal times, those sectors are of similar size, which in a linear approximation would imply that they have similar effects on GDP. But one
lesson of Covid was that shutting down restaurants is not catastrophic for GDP, [Consumer spending on food services and accommodations fell by 40 percent, or $403 billion between 2019Q4 and
2020Q2. Spending at movie theaters fell by 99 percent.] whereas one might expect that a significant reduction in available electricity would have strongly negative effects – and that those
effects would be convex in the size of the decline in available power. Electricity is systemically important not because it is important in good times, but because it would be important in bad
Ben Moll turned out to be right and Germany was able to substitute away from Russian Gas a lot more than people had thought, but even that proves the rule: if it is hard to substitute away from even
a small input, then large shocks to that input imply larger expenditure shares and larger impacts on the economy than its small output in normal times would suggest.
There is an enormous amount more in the paper and voluminous appendices, but this is enough for a blog review.
Now, a few limitations, or really thoughts on where we go next. (No more in this paper, please, Ian!) Ian does a nice illustrative computation of the sensitivity to large shocks:
Ian assumes \(\sigma>1\), so the main ingredients are how many downstream firms use your products and a bit their labor shares. No surprise, trucks, and energy have big tail impacts. But so do
lawyers and insurance. Can we really not do without lawyers? Here I hope the next step looks hard at substitutes vs. complements.
That raises a bunch of issues. Substitutes vs. complements surely depends on time horizon and size of shocks. It might be easy to use a little less water or electricity initially, but then really
hard to reduce more than, say, 80%. It's usually easier to substitute in the long run than the short run.
The analysis in this literature is "static," meaning it describes the economy when everything has settled down. The responses -- you charge more, I use less, I charge more, you use less of my output,
etc. -- all happen instantly, or equivalently the model studies a long run where this has all settled down. But then we talk about responses to shocks, as in the pandemic. Surely there is a dynamic
response here, not just including capital accumulation (which Ian studies). Indeed, my hope was to see prices spreading out through a production network over time, but this structure would have all
price adjustments instantly. Mixing production networks with sticky prices is an obvious idea, which some of the papers below are working on.
In the theory and data handling, you see a big discontinuity. If a firm uses any inputs at all from another firm, if \(A_{ij}>0\), that input can take over and drive everything. If it uses no inputs
at all, then there is no network link and the upstream firm can't have any effect. There is a big discontinuity at \(A_{ij}=0.\) We would prefer a theory that does not jump from zero to everything
when the firm buys one stick of chewing gum. Ian had to drop small but nonzero elements of the input-output matrix to produces sensible results. Perhaps we should regard very small inputs as always
How important is the network stuff anyway? We tend to use industry categorizations, because we have an industry input-output table. But how much of the US industry input-output is simply vertical:
Loggers sell trees to mills who sell wood to lumberyards who sell lumber to Home Depot who sells it to contractors who put up your house? Energy and tools feed each stage, but don't use a whole lot
of wood to make those. I haven't looked at an input-output matrix recently, but just how "vertical" is it?
The literature on networks in macro is vast. One approach is to pick a recent paper like Ian's and work back through the references. I started to summarize, but gave up in the deluge. Have fun.
One way to think of a branch of economics is not just "what tools does it use?" but "what questions is it asking?
Long and Plosser
"Real Business Cycles," a classic, went after idea that the central defining feature of business cycles (since Burns and Mitchell) is
. States and industries all go up and down together to a remarkable degree. That pointed to "aggregate demand" as a key driving force. One would think that "technology shocks" whatever they are would
be local or industry specific. Long and Plosser showed that an input output structure led idiosyncratic shocks to produce business cycle common movement in output. Brilliant.
Macro went in another way, emphasizing time series -- the idea that recessions are defined, say, by two quarters of aggregate GDP decline, or by the greater decline of investment and durable goods
than consumption -- and in the aggregate models of
Kydland and Prescott
, and the stochastic growth model as pioneered by
King, Plosser and Rebelo
, driven by a single economy-wide technology shock. Part of this shift is simply technical: Long and Plosser used analytical tools, and were thereby stuck in a model without capital, plus they did
not inaugurate matching to data. Kydland and Prescott brought numerical model solution and calibration to macro, which is what macro has
ever since. Maybe it's time to add capital, solve numerically, and calibrate Long and Plosser (with up to date frictions and consumer heterogeneity too, maybe).
Xavier Gabaix (2011)
had a different Big Question in mind: Why are business cycles so large? Individual firms and industries have large shocks, but \(\sigma/\sqrt{N}\) ought to dampen those at the aggregate level. Again,
this was a classic argument for aggregate "demand" as opposed to "supply." Gabaix notices that the US has a fat-tailed firm distribution with a few large firms, and those firms have large shocks. He
amplifies his argument via the Hulten mechanism, a bit of networkyiness, since the impact of a firm on the economy is sales / GDP, not value added / GDP.
The enormous literature since then has gone after a variety of questions. Dew-Becker's paper is about the effect of big shocks, and obviously not that useful for small shocks. Remember which question
you're after.
My quest for a new Phillips curve in production networks is better represented by Elisa Rubbo's "
Networks, Phillips curves and Monetary Policy,
" and Jennifer La'o and Alireza Tahbaz-Salehi's “
Optimal Monetary Policy in Production Networks,
” If I can boil those down for the blog, you'll hear about it eventually.
The "what's the question" question is doubly important for this branch of macro that explicitly models heterogeneous agents and heterogenous firms. Why are we doing this? One can always represent the
aggregates with a social welfare function and an aggregate production function. You might be interested in how aggregates affect individuals, but that doesn't change your model of aggregates. Or, you
might be interested in seeing what the aggregate production or utility function looks like -- is it consistent with what we know about individual firms and people? Does the size of the aggregate
production function shock make sense? But still, you end up with just a better (hopefully) aggregate production and utility function. Or, you might want models that break the aggregation theorems in
a significant way; models for which distributions matter for aggregate dynamics, theoretically and (harder) empirically. But don't forget you need a reason to build disaggregated models.
Expression (1) is not easy to get to. I started reading Ian's paper in my usual way: to learn a literature start with the latest paper and work backward. Alas, this literature has evolved to the
point that authors plop results down that "everybody knows" and will take you a day or so of head-scratching to reproduce. I complained to Ian, and he said he had the same problem when he was getting
in to the literature! Yes, journals now demand such overstuffed papers that it's hard to do, but it would be awfully nice for everyone to start including ground up algebra for major results in one of
the endless internet appendices. I eventually found
Jonathan Dingel's notes
on Dixit Stiglitz tricks, which were helpful.
Chase Abram's
University of Chicago Math Camp notes here
are also a fantastic resource. See Appendix B starting p. 94 for production network math. The rest of the notes are also really good. The first part goes a little deeper into more abstract material
than is really necessary for the second part and applied work, but it is a wonderful and concise review of that material as well.
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Tandem Lookout (Avacyn Restored)
Tandem Lookout
Avacyn Restored — Uncommon
Creature — Human Scout
Soulbond (You may pair this creature with another unpaired creature when either enters. They remain paired for as long as you control both of them.)
As long as Tandem Lookout is paired with another creature, each of those creatures has "Whenever this creature deals damage to an opponent, draw a card."
|
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Examples · ClimaCore.jl
The 2D Cartesian advection/transport example in examples/plane/limiters_advection.jl demonstrates the application of flux limiters in the horizontal direction, namely QuasiMonotoneLimiter, in a 2D
Cartesian domain.
Follows the continuity equation
$$\[\frac{\partial}{\partial t} \rho = - abla \cdot(\rho \boldsymbol{u}) . \label{eq:2d-plane-advection-lim-continuity}\]$$
This is discretized using the following
$$\[\frac{\partial}{\partial t} \rho \approx - wD[ \rho \boldsymbol{u}] . \label{eq:2d-plane-advection-lim-discrete-continuity}\]$$
For the tracer concentration per unit mass $q$, the tracer density (scalar) $\rho q$ follows the advection/transport equation
$$\[\frac{\partial}{\partial t} \rho q = - abla \cdot(\rho q \boldsymbol{u}) + g(\rho, q). \label{eq:2d-plane-advection-lim-tracers}\]$$
This is discretized using the following
$$\[\frac{\partial}{\partial t} \rho q \approx - wD[ \rho q \boldsymbol{u}] + g(\rho, q), \label{eq:2d-plane-advection-lim-discrete-tracers}\]$$
where $g(\rho, q) = - \nu_4 [\nabla^4_h (\rho q)]$ represents the horizontal hyperdiffusion operator, with $\nu_4$ (measured in m^4/s) the hyperviscosity constant coefficient (set equal to zero by
default in the example).
Currently tracers are only treated explicitly in the time discretization.
• $\rho$: density measured in kg/m³.
• $\boldsymbol{u}$ velocity, a vector measured in m/s. Since this is a 2D problem, $\boldsymbol{u} \equiv \boldsymbol{u}_h$.
• $\rho q$: the tracer density scalar, where $q$ is the tracer concentration per unit mass.
Because this is a purely 2D problem, there is no staggered vertical discretization, hence, there is no need of specifying variables at cell centers, faces or to reconstruct from faces to centers and
vice versa.
To discretize the hyperdiffusion operator, $g(\rho, q) = - \nu_4 [\nabla^4 (\rho q)]$, in the horizontal direction, we compose the horizontal weak divergence, $wD$, and the horizontal gradient
operator, $G_h$, twice, with an intermediate call to weighted_dss! between the two compositions, as in $[g_2(\rho, g) \circ DSS(\rho, q) \circ g_1(\rho, q)]$, with:
• $g_1(\rho, q) = wD(G_h(q))$
• $DSS(\rho, q) = DSS(g_1(\rho q))$
• $g_2(\rho, q) = -\nu_4 wD(\rho G_h(\rho q))$
□ with $\nu_4$ the hyperviscosity coefficient.
This test case is set up in a Cartesian planar domain $[-2 \pi, 2 \pi]^2$, doubly periodic.
The flow was chosen to be a horizontal uniform rotation. Moreover, the flow is reversed halfway through the time period so that the tracer blobs go back to its initial configuration (using the same
speed scaling constant which was derived to account for the distance travelled in all directions in a half period).
$$\[ u &= -u_0 (y - c_y) \cos(\pi t / T_f) onumber \\ v &= u_0 (x - c_x) \cos(\pi t / T_f) \label{eq:2d-plane-advection-lim-flow} \]$$
where $u_0 = \pi / 2$ is the speed scaling factor to have the flow reversed halfway through the time period, $\boldsymbol{c} = (c_x, c_y)$ is the center of the rotational flow, which coincides with
the center of the domain, and $T_f = 2 \pi$ is the final simulation time, which coincides with the temporal period to have a full rotation.
This example is set up to run with three possible initial conditions:
• cosine_bells
• gaussian_bells
• cylinders: two 2D slotted cylinders (test case available in the literature, cfr: [14]).
Because this is a fully 2D problem, the application of limiters does not affect the order of operations, which is implemented as follows:
1. Horizontal trasport with hyperdiffusion (with weak divergence $wD$)
2. Horizontal flux limiters
3. DSS
The 3D Cartesian advection/transport example in examples/hybrid/box/limiters_advection.jl demonstrates the application of flux limiters in the horizontal direction, namely QuasiMonotoneLimiter, in a
hybrid Cartesian domain. It also demonstrates the usage of the high-order upwinding scheme in the vertical direction, called Upwind3rdOrderBiasedProductC2F.
Follows the continuity equation
$$\[\frac{\partial}{\partial t} \rho = - abla \cdot(\rho \boldsymbol{u}) . \label{eq:3d-box-advection-lim-continuity}\]$$
This is discretized using the following
$$\[\frac{\partial}{\partial t} \rho \approx - D_h[ \rho (\boldsymbol{u}_h + I^c(\boldsymbol{u}_v))] - D^c_v[I^f(\rho \boldsymbol{u}_h)) + I^f(\rho) \boldsymbol{u}_v)] . \label
For the tracer concentration per unit mass $q$, the tracer density (scalar) $\rho q$ follows the advection/transport equation
$$\[\frac{\partial}{\partial t} \rho q = - abla \cdot(\rho q \boldsymbol{u}) + g(\rho, q). \label{eq:3d-box-advection-lim-tracers}\]$$
This is discretized using the following
$$\[\frac{\partial}{\partial t} \rho q \approx - D_h[ \rho q (\boldsymbol{u}_h + I^c(\boldsymbol{u}_v))] - D^c_v\left[I^f(\rho q) U^f\left(I^f(\boldsymbol{u}_h) + \boldsymbol{u}_v, \frac{\rho q}{\
rho} \right) \right] + g(\rho, q), \label{eq:3d-box-advection-lim-discrete-tracers}\]$$
where $g(\rho, q) = - \nu_4 [\nabla^4_h (\rho q)]$ represents the horizontal hyperdiffusion operator, with $\nu_4$ (measured in m^4/s) the hyperviscosity constant coefficient.
Currently tracers are only treated explicitly in the time discretization.
• $\rho$: density measured in kg/m³. This is discretized at cell centers.
• $\boldsymbol{u}$ velocity, a vector measured in m/s. This is discretized via $\boldsymbol{u} = \boldsymbol{u}_h + \boldsymbol{u}_v$ where
□ $\boldsymbol{u}_h = u_1 \boldsymbol{e}^1 + u_2 \boldsymbol{e}^2$ is the projection onto horizontal covariant components (covariance here means with respect to the reference element), stored
at cell centers.
□ $\boldsymbol{u}_v = u_3 \boldsymbol{e}^3$ is the projection onto the vertical covariant components, stored at cell faces.
• $\rho q$: the tracer density scalar, where $q$ is the tracer concentration per unit mass, is stored at cell centers.
We make use of the following operators
• $I^c$ is the face-to-center reconstruction operator, called first_order_If2c in the example code.
• $I^f$ is the center-to-face reconstruction operator, called first_order_Ic2f in the example code.
□ Currently this is just the arithmetic mean, but we will need to use a weighted version with stretched vertical grids.
• $U^f$ is the center-to-face upwind product operator, called third_order_upwind_c2f in the example code
□ This operator is of third-order of accuracy (when used with a constant vertical velocity and some reduced, but still high-order for non constant vertical velocity).
To discretize the hyperdiffusion operator, $g(\rho, q) = - \nu_4 [\nabla^4 (\rho q)]$, in the horizontal direction, we compose the horizontal weak divergence, $wD_h$, and the horizontal gradient
operator, $G_h$, twice, with an intermediate call to weighted_dss! between the two compositions, as in $[g_2(\rho, g) \circ DSS(\rho, q) \circ g_1(\rho, q)]$, with:
• $g_1(\rho, q) = wD_h(G_h(q))$
• $DSS(\rho, q) = DSS(g_1(\rho q))$
• $g_2(\rho, q) = -\nu_4 wD_h(\rho G_h(\rho q))$
□ with $\nu_4$ the hyperviscosity coefficient.
Since we use flux limiters that limit only operators defined in the spectral space (i.e., they are applied level-wise in the horizontal direction), the application of limiters has to follow a precise
order in the sequence of operations that specifies the total tendency.
The order of operations should be the following:
1. Horizontal transport (with strong divergence $D_h$)
2. Horizontal Flux Limiters
3. Horizontal hyperdiffusion (with weak divergence $wD_h$)
4. Vertical transport
5. DSS
This test case is set up in a Cartesian (box) domain $[-2 \pi, 2 \pi]^2 \times [0, 4 \pi] ~\textrm{m}^3$, doubly periodic in the horizontal direction, but not in the vertical direction.
The flow was chosen to be a spiral, i.e., so to have a horizontal uniform rotation, and a vertical velocity $\boldsymbol{u}_v \equiv w = 0$ at the top and bottom boundaries, and $\boldsymbol{u}_v \
equiv w = 1$ in the center of the domain. Moreover, the flow is reversed in all directions halfway through the time period so that the tracer blobs go back to its initial configuration (using the
same speed scaling constant which was derived to account for the distance travelled in all directions in a half period).
$$\[ u &= -u_0 (y - c_y) \cos(\pi t / T_f) onumber \\ v &= u_0 (x - c_x) \cos(\pi t / T_f) onumber \\ w &= u_0 \sin(\pi z / z_m) \cos(\pi t / T_f) onumber \label{eq:3d-box-advection-lim-flow} \]$$
where $u_0 = \pi / 2$ is the speed scaling factor to have the flow reversed halfway through the time period, $\boldsymbol{c} = (c_x, c_y)$ is the center of the rotational flow, which coincides with
the center of the domain, $z_m = 4 \pi$ is the maximum height of the domain, and $T_f = 2 \pi$ is the final simulation time, which coincides with the temporal period to have a full rotation in the
horizontal direction.
This example is set up to run with three possible initial conditions:
• cosine_bells
• gaussian_bells
• slotted_spheres: a slight modification of the 2D slotted cylinder test case available in the literature (cfr: [14]).
Because this is a Cartesian 3D problem, the application of limiters does not affect the order of operations, which is implemented as follows:
1. Horizontal transport + hyperdiffusion (with weak divergence $wD_h$)
2. Horizontal flux limiters
3. Vertical transport
4. DSS
The 2D sphere advection/transport example in examples/sphere/limiters_advection.jl demonstrates the application of flux limiters in the horizontal direction, namely QuasiMonotoneLimiter, in a 2D
spherical domain.
Follows the continuity equation
$$\[\frac{\partial}{\partial t} \rho = - abla \cdot(\rho \boldsymbol{u}) . \label{eq:2d-sphere-advection-lim-continuity}\]$$
This is discretized using the following
$$\[\frac{\partial}{\partial t} \rho \approx - wD[ \rho \boldsymbol{u}] . \label{eq:2d-sphere-advection-lim-discrete-continuity}\]$$
For the tracer concentration per unit mass $q$, the tracer density (scalar) $\rho q$ follows the advection/transport equation
$$\[\frac{\partial}{\partial t} \rho q = - abla \cdot(\rho q \boldsymbol{u}) + g(\rho, q). \label{eq:2d-sphere-advection-lim-tracers}\]$$
This is discretized using the following
$$\[\frac{\partial}{\partial t} \rho q \approx - wD[ \rho q \boldsymbol{u}] + g(\rho, q), \label{eq:2d-sphere-advection-lim-discrete-tracers}\]$$
where $g(\rho, q) = - \nu_4 [\nabla^4_h (\rho q)]$ represents the horizontal hyperdiffusion operator, with $\nu_4$ (measured in m^4/s) the hyperviscosity constant coefficient.
Currently tracers are only treated explicitly in the time discretization.
• $\rho$: density measured in kg/m³.
• $\boldsymbol{u}$ velocity, a vector measured in m/s. Since this is a 2D problem, $\boldsymbol{u} \equiv \boldsymbol{u}_h$.
• $\rho q$: the tracer density scalar, where $q$ is the tracer concentration per unit mass.
Because this is a purely 2D problem, there is no staggered vertical discretization, hence, there is no need of specifying variables at cell centers, faces or to reconstruct from faces to centers and
vice versa.
To discretize the hyperdiffusion operator, $g(\rho, q) = - \nu_4 [\nabla^4 (\rho q)]$, in the horizontal direction, we compose the horizontal weak divergence, $wD$, and the horizontal gradient
operator, $G_h$, twice, with an intermediate call to weighted_dss! between the two compositions, as in $[g_2(\rho, g) \circ DSS(\rho, q) \circ g_1(\rho, q)]$, with:
• $g_1(\rho, q) = wD(G_h(q))$
• $DSS(\rho, q) = DSS(g_1(\rho q))$
• $g_2(\rho, q) = -\nu_4 wD(\rho G_h(\rho q))$
□ with $\nu_4$ the hyperviscosity coefficient.
This test case is set up in a Cartesian planar domain $[-2 \pi, 2 \pi]^2$, doubly periodic.
The flow was chosen to be a horizontal uniform rotation. Moreover, the flow is reversed halfway through the time period so that the tracer blobs go back to its initial configuration (using the same
speed scaling constant which was derived to account for the distance travelled in all directions in a half period).
$$\[ u &= k sin (\lambda)^2 sin (2 \phi) \cos(\pi t / T_f) + \frac{2 \pi}{T_f} \cos (\phi) onumber \\ v &= k \sin (2 \lambda) \cos (\phi) \cos(\pi t / T_f) \label{eq:2d-sphere-lim-flow} \]$$
where $u_0 = 2 \pi R / T_f$ is the speed scaling factor to have the flow reversed halfway through the time period, $T_f = 86400 * 12$ (i.e., $12$ days in seconds) is the final simulation time, which
coincides with the temporal period to have a full rotation around the sphere of radius $R$.
This example is set up to run with three possible initial conditions:
• cosine_bells
• gaussian_bells
• cylinders: two 2D slotted cylinders (test case available in the literature, cfr: [14]).
Because this is a fully 2D problem, the application of limiters does not affect the order of operations, which is implemented as follows:
1. Horizontal trasport with hyperdiffusion (with weak divergence $wD$)
2. Horizontal flux limiters
3. DSS
The shallow water equations in the so-called vector invariant form from [15] are:
$$\[ \frac{\partial h}{\partial t} + abla \cdot (h u) &= 0\\ \frac{\partial u_i}{\partial t} + abla (\Phi + \tfrac{1}{2}\|u\|^2)_i &= (\boldsymbol{u} \times (f + abla \times \boldsymbol{u}))_i \label
{eq:shallow-water} \]$$
where $f$ is the Coriolis term and $\Phi = g(h+h_s)$, with $g$ the gravitational accelration constant, $h$ the (free) height of the fluid and $h_s$ a non-uniform reference surface.
To the above set of equations, we allow the uset to add a hyperdiffusion operator, $g(h, \boldsymbol{u}) = - \nu_4 [\nabla^4 (h, \boldsymbol{u})]$, with $\nu_4$ (measured in m^4/s) the hyperviscosity
constant coefficient. In the hyperdiffusion expression, $\nabla^4$ represents a biharmonic operator, and it assumes a different formulation on curvilinear reference systems, depending on it being
applied to a scalar field, such as $h$, or a vector field, such as $\boldsymbol{u}$.
The governing equations then become:
$$\[ \frac{\partial h}{\partial t} + abla \cdot (h u) &= g(h, \boldsymbol{u})\\ \frac{\partial u_i}{\partial t} + abla (\Phi + \tfrac{1}{2}\|u\|^2)_i &= (\boldsymbol{u} \times (f + abla \times \
boldsymbol{u}))_i + g(h, \boldsymbol{u}) \label{eq:shallow-water-with-hyperdiff} \]$$
Since this is a 2D problem (with related 2D vector field), the curl is defined to be
$$\[ \omega^i = (abla \times u)^i &= \begin{cases} 0 &\text{ if i =1,2},\\ \frac{1}{J} \left[ \frac{\partial u_2}{\partial \xi^1} - \frac{\partial u_1}{\partial \xi^2} \right] &\text{ if i=3}, \end
{cases} \label{eq:2Dvorticity} \]$$
where we have used the coordinate system in each 2D reference element, i.e., $(\xi^1, \xi^2) \in [-1,1]\times[-1,1]$. Similarly, if additionally $v^1 = v^2 = 0$, then
$$\[ (\boldsymbol{u} \times \boldsymbol{v})_i = \begin{cases} J u^2 v^3 &\text{ if i=1},\\ - J u^1 v^3 &\text{ if i=2},\\ 0 &\text{ if i=3}. \end{cases} \]$$
Hence, we can rewrite equations \eqref{eq:shallow-water} using the velocity representation in covariant coordinates, in this case $u = u_1 \boldsymbol{b}^1 + u_2 \boldsymbol{b}^2 + 0\boldsymbol{b}^3$
, and $g(h, \boldsymbol{u}) = 0$ for simplicity, as:
$$\[ \frac{\partial h}{\partial t} + \frac{1}{J}\frac{\partial}{\partial \xi^j}\Big(h J u^j\Big) &= 0\\ \frac{\partial u_i}{\partial t} + \frac{\partial}{\partial \xi^i} (\Phi + \tfrac{1}{2}\|u\|^2)
&= E_{ijk}u^j (f^k + \omega^k) . \label{eq:covariant-shallow-water} \]$$
• $h$: scalar height field of the fluid, measured in m.
• $\boldsymbol{u}$ velocity, a 2D vector measured in m/s.
Because this is a purely 2D problem, there is no staggered vertical discretization, hence, there is no need of specifying variables at cell centers, faces or to reconstruct from faces to centers and
vice versa.
To discretize the hyperdiffusion operator, $g(h, \boldsymbol{u}) = - \nu_4 [\nabla^4 (h, \boldsymbol{u})]$, in the horizontal direction, we compose the weak divergence, $wD$, and the gradient
operator, $G$, twice, with an intermediate call to weighted_dss! between the two compositions, as in $[g_2(h, \boldsymbol{u}) \circ DSS(h, \boldsymbol{u}) \circ g_1(h, \boldsymbol{u})]$. Moreover,
when $g(h, \boldsymbol{u}) = - \nu_4 [\nabla^4 (h)]$, i.e., the operator is applied to a scalar field only, it is discretized composing the following operations:
• $g_1(h) = wD(G(h))$
• $DSS(g_1(h))$
• $g_2(h) = -\nu_4 wD(G(h))$
whereas, when the operator is applied to a vector field, i.e., $g(h, \boldsymbol{u}) = - \nu_4 [\nabla^4 (\boldsymbol{u})]$, it is discretized as:
• $g_1(h, \boldsymbol{u}) = wG(D(\boldsymbol{u})) - wCurl(Curl(\boldsymbol{u}))$
• $DSS(h, \boldsymbol{u}) = DSS(g_1(h, \boldsymbol{u}))$
• $g_2(h, \boldsymbol{u}) = -\nu_4 \left[ wG(D(\boldsymbol{u})) - wCurl(Curl(\boldsymbol{u})) \right]$
In both cases, $\nu_4$ is the hyperviscosity coefficient.
This test case is set up on a 2D (surface) spherical domain represented by a cubed-sphere manifold.
This suite of examples contains five different test cases:
• One, invoked via the command-line argument steady_state, which reproduces Test Case 2 in [16]. This test case gives the steady-state solution to the non-linear shallow water equations. It
consists of a solid body rotation or zonal flow with the corresponding geostrophic height field. The Coriolis parameter is a function of latitude and longitude so the flow can be specified with
the spherical coordinate poles not necessarily coincident with Earth's rotation axis. Hence, this test case can be run with a specified command-line argument for the angle $\alpha$ that
represents the angle between the north pole and the center of the top cube panel of the cubed-sphere geometry.
• A second one, invoked via the command-line argument steady_state_compact, reproduces Test Case 3 in [16]. This test case gives the steady-state solution to the non-linear shallow water equations
with nonlinear zonal geostrophic flow with compact support.
• A third one, invoked via the command-line argument mountain, reproduces Test Case 5 in [16]. It represents a zonal flow over an isolated mountain, where the governing equations describe a global
steady-state nonlinear zonal geostrophic flow, with a corresponding geostrophic height field over a non-uniform reference surface h_s.
• A fourth one, invoked via the command-line argument rossby_haurwitz, reproduces Test Case 6 in [16]. It represents the solution of the nonlinear barotropic vorticity equation on the sphere.
• A fifth one, invoked via the command-line argument barotropic_instability, reproduces the test case in [17] (also in Sec. 7.6 in [18]). This test case consists of a zonal jet with compact support
at a latitude of $45°$. A small height disturbance is then added, which causes the jet to become unstable and collapse into a highly vortical structure.
The 3D sphere advection/transport example in examples/hybrid/sphere/deformation_flow.jl demonstrates the application of flux limiters in the horziontal direction, namely QuasiMonotoneLimiter, in a
hybrid 3D spherical domain. It also demonstrates the usage of the flux-corrected transport in the vertical direction; by default, it uses FCTZalesak.
The original test case (without limiters or flux-corrected transport) is specified in Section 1.1 of [19].
Follows the continuity equation
$$\[\frac{\partial}{\partial t} \rho = - abla \cdot(\rho \boldsymbol{u}) . \label{eq:3d-sphere-lim-continuity}\]$$
This is discretized using the following
$$\[\frac{\partial}{\partial t} \rho \approx - D_h[\rho \boldsymbol{u}^c] - D^c_v[I^f(\rho) \boldsymbol{u}^f]. \label{eq:3d-sphere-lim-discrete-continuity}\]$$
This test case has five different tracer concentrations per unit mass $q_i$, hence five different tracer densities (scalar) $\rho q_i$. They all follow the same advection/transport equation
$$\[\frac{\partial}{\partial t} \rho q = - abla \cdot(\rho q \boldsymbol{u}) + g(\rho, q). \label{eq:3d-sphere-lim-tracers}\]$$
This is discretized using the following
$$\[\frac{\partial}{\partial t} \rho q \approx - D_h[ \rho q \boldsymbol{u}^c] - D^c_v\left[I^f(\rho q) * \boldsymbol{u}^f_h + FCT^f\left( \boldsymbol{u}^f_v, \frac{\rho q}{\rho} \right) \right] + g
(\rho, q), \label{eq:3d-sphere-lim-discrete-tracers}\]$$
where $g(\rho, q) = - \nu_4 [\nabla^4_h (\rho q)]$ represents the horizontal hyperdiffusion operator, with $\nu_4$ (measured in m^4/s) the hyperviscosity constant coefficient.
Currently tracers are only treated explicitly in the time discretization.
• $\rho$: density measured in kg/m³. This is discretized at cell centers.
• $\boldsymbol{u}$ velocity, a vector measured in m/s. This is discretized via $\boldsymbol{u} = \boldsymbol{u}_h + \boldsymbol{u}_v$ where
□ $\boldsymbol{u}_h = u_1 \boldsymbol{e}^1 + u_2 \boldsymbol{e}^2$ is the projection onto horizontal covariant components (covariance here means with respect to the reference element), stored
at cell centers.
□ $\boldsymbol{u}_v = u_3 \boldsymbol{e}^3$ is the projection onto the vertical covariant components, stored at cell faces.
• $\rho q_i$: tracer density scalars, where $q_i$ is a tracer concentration per unit mass, are stored at cell centers.
We make use of the following operators
• $I^c$ is the face-to-center reconstruction operator, called If2c in the example code.
• $I^f$ is the center-to-face reconstruction operator, called Ic2f in the example code.
□ Currently this is just the arithmetic mean, but we will need to use a weighted version with stretched vertical grids.
• $FCT^f$ denotes either the center-to-face upwind product operator (which represents no flux-corrected transport), the center-to-face Boris & Book FCT operator, or the center-to-face Zalesak FCT
To discretize the hyperdiffusion operator for each tracer concentration, $g(\rho, q_i) = - \nu_4 [\nabla^4 (\rho q_i)]$, in the horizontal direction, we compose the horizontal weak divergence, $wD_h$
, and the horizontal gradient operator, $G_h$, twice, with an intermediate call to weighted_dss! between the two compositions, as in $[g_2(\rho, g) \circ DSS(\rho, q) \circ g_1(\rho, q_i)]$, with:
• $g_1(\rho, q_i) = wD_h(G_h(q_i))$
• $DSS(\rho, q_i) = DSS(g_1(\rho q_i))$
• $g_2(\rho, q_i) = -\nu_4 wD_h(\rho G_h(\rho q_i))$
□ with $\nu_4$ the hyperviscosity coefficient.
Since we use flux limiters that limit only operators defined in the spectral space (i.e., they are applied level-wise in the horizontal direction), the application of limiters has to follow a precise
order in the sequence of operations that specifies the total tendency.
The order of operations should be the following:
1. Horizontal transport (with strong divergence $D_h$)
2. Horizontal flux limiters
3. Horizontal hyperdiffusion (with weak divergence $wD_h$)
4. Vertical transport
5. DSS
This test case is set up in a 3D (shell) spherical domain where the elevation goes from $z=0~\textrm{m}$ (i.e., from the radius of the sphere $R = 6.37122 10^6~\textrm{m}$) to $z_{\textrm{top}} =
The flow (reversed halfway through the time period) is specified as $\boldsymbol{u} = \boldsymbol{u}_a + \boldsymbol{u}_d$, where the components are defined as follows:
$$\[ u_a &= k \sin (\lambda')^2 \sin (2 \phi) \cos(\pi t / \tau) + \frac{2 \pi R}{\tau} \cos (\phi) onumber \\ v_a &= k \sin (2 \lambda') \cos (\phi) \cos(\pi t / \tau) onumber \\ u_d &= \frac{\
omega_0 R}{ b / p_{\textrm{top}}} \cos (\lambda') \cos(\phi)^2 \cos(2 \pi t / \tau) \left[-exp \left( \frac{(p - p_0)}{ b p_{\textrm{top}}} \right) + exp \left( \frac{(p_{\textrm{top}} - p(zc))}{b p_
{\textrm{top}}} \right) \right] onumber \label{eq:3d-sphere-lim-flow} \]$$
where all values of the parameters can be found in Table 1.1 in the reference [19].
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Printable Multiplication Charts 0 12 | Multiplication Chart Printable
Printable Multiplication Charts 0 12
Free Printable Multiplication Chart 0 12 PrintableMultiplication
Printable Multiplication Charts 0 12
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What is Multiplication Chart Printable?
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While private tables work for providing chunks of information, a complete page chart makes it much easier to examine realities that have currently been understood.
The multiplication chart will normally include a left column and also a top row. The leading row will have a checklist of items. When you intend to locate the product of 2 numbers, choose the initial
number from the left column as well as the second number from the top row. When you have these numbers, move them along the row or down the column until you reach the square where the two numbers
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Multiplication charts are practical discovering tools for both youngsters and grownups. Printable Multiplication Charts 0 12 are offered on the Internet and can be published out as well as laminated
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Multiplication charts are valuable for many factors, consisting of aiding kids discover just how to split as well as simplify portions. They can also aid kids learn exactly how to pick an efficient
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Multiplication charts are also helpful for helping pupils memorize their times tables. As with any kind of ability, memorizing multiplication tables takes time and practice.
Printable Multiplication Charts 0 12
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Roulette Payouts, Odds & The House Edge | CasinoTalk
Understanding your roulette odds and payouts is crucial when playing real money games. Before making a bet, you should always consider your chances of winning and the potential payout would your bet
to be successful. Since roulette is a relatively simple game, there is fortunately not too much calculation involved in deciding your odds, and it is even simpler to understand your roulette payout.
We will go through the odds and payouts of all the different roulette bets, both for American and European roulette tables. We will also teach you about the house edge and why the roulette odds and
payout aren’t exactly equal.
Read no more if you want a quick overview of different roulette odds. Instead, look below, where we have compiled the odds of winning and potential payout. You will have a slightly higher probability
of winning if you play European Roulette compared to American roulette.
Most legal online casinos offer roulette, so when you feel that the time is right, you can learn more about playing the game at the top three US online casinos by reading our reviews; BetMGM, Caesars
, and Golden Nugget.
Here's a table presenting the roulette payouts and odds for different types of bets in American Roulette:
American Roulette Bets Chance of Winning Payout
Even 46.37% 1:1
Odd 46.37% 1:1
Red 46.37% 1:1
Black 46.37% 1:1
1-18 46.37% 1:1
19-36 46.37% 1:1
1-12 31.58% 2:1
13-24 31.58% 2:1
25-36 31.58% 2:1
Single Number 2.63% 35:1
Combination of 2 Numbers 5.26% 17:1
Combination of 3 Numbers 7.89% 11:1
Combination of 4 Numbers 10.53% 8:1
Combination of 6 Numbers 15.79% 5:1
Combination of 0, 00, 1, 2, 3 13.16% 6:1
This table presents the roulette payouts and odds for different types of bets in European Roulette:
European Roulette Bets Chance of Winning Payout
Even 48.6% 1:1
Odd 48.6% 1:1
Red 48.6% 1:1
Black 48.6% 1:1
1-18 48.6% 1:1
19-36 48.6% 1:1
1-12 32.4% 2:1
13-24 32.4% 2:1
25-36 32.4% 2:1
Single Number 2.7% 35:1
Combination of 2 Numbers 5.4% 17:1
Combination of 3 Numbers 8.1% 11:1
Combination of 4 Numbers 10.8% 8:1
Combination of 6 Numbers 16.2% 5:1
On every roulette wheel, there are 36 red or black slots numbered 1-36. There is one additional number in European roulette, the zero (0), which is green.
Similarly, there are two green slots in American roulette, the zero (0) and the double zero (00). Therefore, you will always calculate the roulette odds of winning from the total number of slots on
the wheel.
For a European table, this means that the probability for the ball to land on any single number is 1 in 37, about 2.70%. That means that the odds are 36 to 1, 36 outcomes will make you lose the bet,
and one will win.
When calculating the roulette payout, you will never include the green zero(s), and the calculation is relatively straightforward. There are 36 red or black slots on the wheel, and if you bet on a
single number that wins, you would get 36 times the money in return. Subtract your initial bet, and you get your profit of 35 times the bet you made. Your roulette payout is therefore 35:1.
Roulette Odds - (No. losing outcomes) : (No. of winning outcomes)
Roulette Payouts - (Your potential profit) : (Your initial bet)
The most classic bet in roulette is the red or black bet. The red or black bet means that you can bet on one of the colors, let’s say red, and if the ball stops on any of the red numbers, you win,
and any other color will lose.
The roulette odds are seemingly 50-50 for both outcomes, and the bet payout would be 1:1; you get double your money back. However, when you look closer at the roulette odds table, you see that while
the payout is 1:1, the probability of winning the bet is only 48.6%; so where did the other 1.4% go?
The 1.4% is called the House Edge, the casinos' fee for running the game, employing dealers, and entertaining guests that aren’t gambling at the moment, and it exists because of the green zero on
the roulette wheel.
Since we omit the zero in the calculation of payouts, the will always be a slight difference in the roulette odds and the roulette payouts. In the case of the red or black bet, the payout on either
outcome is 1:1, but the chances on the respective outcomes are 48.65% for landing on a red number, 48.65% for landing on a black number, and 2.70% for landing on the green zero. The roulette house
edge is therefore 2.70%
Probabilities vs. Roulette Odds In Depth
Mixing up the roulette odds and roulette probabilities is very common. Let’s take the example of the American double-zero wheel first. This wheel consists of 38 numbered pockets ideally spread
around the wheel. These numbers are 1 to 36, along with 0 and 00. Assuming that you bet on 17, out of 38 numbers, a wager on 17 is a bet on one number out of the total 38. That is the probability; 1
out of 38. If you bet it, the number 17 has got one chance in 38 chances to show up. That one out of 38 is not the odds of your bet.
Roulette odds in the American double zero wheel(s)
When it comes to the American double-zero wheel(s), a player has got one chance of winning and 37 ways of losing the bet. In short, the odds are against a player as they are 37 to 1. In this case,
the probability is one to how many numbers exist on the wheel, though the odds deal with how often a player is expected to win, and in this case, it is just once while losing is 37.
Translated into money, the payoff for winning bets is not 37 to one roulette odds but 35 to one. From another perspective, a player gets shortchanged two units by the casino. If the casino paid
“fair” odds (37 to 1), it would not make money and probably close down. Such a move would end the life of all the casinos globally. In other words, roulette odds are not playing in the same
league as the real payouts.
Roulette odds on the European single-zero wheel
The above scenario replicates itself here also. With the European wheel, there are 37 numbers (1 to 36) and a single 0. Here, the probability of a hit is one in 37, meaning that the roulette odds are
36 to one. Sadly, the European single-zero wheels also have a payback of 35 to one on any winnings. In short, it means a shortage of one unit. Actual odds are supposed to be 36 to one. However, the
payout here is 35 to one.
Will Roulette Odds Always Pay Out On Time?
There’s something that requires close attention when looking at roulette odds and probabilities. The fact that these odds are 37 to one and 36 to one on the American double-zero wheel and the
European single-zero wheel, respectively, doesn’t mean that a winner is found on every 38 or 37 spins. Gamblers would wish for this, but sadly, this is not the case. Roulette odds are just
long-range figures based on math. A roulette game can be crazy as there may be many winners or no winners. You may face losing streaks or winning streaks every time you place bets.
As a player continues to place bets, the actual roulette odds start to shape. Though it will not be perfect, the calculations are based on the winnings and losses. Unfortunately, roulette
probabilities and odds are the same regardless of whether you play a single spin or thousands of spins. And this is why some gamblers think that they can out-smart the math. They can’t.
Roulette Odds On Proposition Bets
Roulette odds work the same manner as every other bet in a game. Let’s take the example of an even-money bet of red-black. This bet is called even money since a win pays a gambler one unit for
every one unit wagered. However, odds and probabilities aren’t one-to-one.
Let us assume that a player X decides to bet red. As you are aware, there are 18 red numbers and a similar count for black numbers. Now, player X wants red to hit and not black. However, the player
also hopes that 0 or 00 on the American wheel also hits. These are losing bets.
Here come the odds: You will win 18 times and lose 20 times when playing on the American double-zero wheels. On the other hand, if you are playing on the European single-zero wheels, you are assured
of 18 wins but, sadly, 19 losses.
The roulette odds here are 20 to 18 –or 10 to 9- while playing on double-zero wheels, while the odds will be 19 to 18 when playing on the European single-zero wheels. Of importance to note is that
the American wheel shorts a player two units since the payout on winning bets is one-to-one. Also, the house edge here is 5.26%. On the European single-zero wheels, the house edge is 2.70%.
Remember that some bets come or rather consist of either greater or lesser house edges. Such was covered in previous sections.
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Insolvability of quintics. | JustToThePointInsolvability of quintics.
Somebody who thinks logically is a nice contrast to the real world or, alternatively, common sense is the least common of the senses, it is not so common.
Theorem. Let f be an irreducible polynomial of degree 5 over a subfield F of the complex numbers (F ⊆ ℂ), whose Galois group is either the alternating group A[5] or the symmetric group S[5], then f
is not solvable. Therefore, there are quintics that are not solvable.
For the sake of contradiction, suppose f is solvable ↭ Gal(f) is solvable ⊥ S[5] or A[5] are not solvable.
Theorem. Let f be an irreducible polynomial of degree 5 over a subfield F of the complex numbers (F ⊆ ℂ), whose Galois group is either the alternating group A[5] or the symmetric group S[5], then f
is not solvable.
Proof. (2nd version. Based on Algebra, Second Edition by Michael Artin, Theorem 16.12.4, pg 503)
Let G = Gal(f). First, let’s suppose G = S[5], and K be the splitting field of f over F, so G = Gal(K/F) = Gal(f), D = Disc(f), δ = $\sqrt{D}$ ∈ K.
By assumption, G = S[5] ⇒ G = Gal(f) ⊄ A[5] ⇒ [Recall If δ ∈ F (Δ is a square in F), then G ⊆ An.] δ ∉ F ⇒ [F(δ) : F] = 2 ⇒ [$A_n$ is the only subgroup of $S_n$ of index 2] Gal(K/F(δ)) = A[5], so the
Galois group of f over F(δ) is A[5]. Therefore, if we demonstrate that if Gal(f) = A[5], then f is not solvable, we are done (because f is not solvable over F(δ) ⇒ f is not solvable over F)
Therefore, we may assume that G is the alternating group A[5], a simple group, f ∈ F[x], deg(f) = 5, Gal(f) = A[5].
For the sake of contradiction, let’s suppose f it is solvable over F. Let α ∈ K be a root of f, then α is solvable over F, (Recall: solvable ↭ tower of Abelian fields ↭ tower of cyclic fields ↭ tower
of cyclic fields of prime order) then there exist a tower of extensions which are cyclic of prime degree.
F = F[0] ⊆ F[1] ⊆ F[2] ⊆ ··· ⊆ F[r], α ∈ F[r], F[i]/F[i-1] is Galois, [F[i] : F[i-1] is prime]
Lemma. Let F’/F be a Galois extension of prime degree p. Let K and K' be the splitting field of f over F and F' respectively. Then, Gal(K'/F')≋A[5].
In other words, when we consider a Galois extension of F’/F of prime degree (or a tower as we iterate through a tower of fields of prime order), we show that no progress towards solving the equation
f = 0 is made when one replaces F by F’. We do this by showing that the Galois group of f over F’ is again Gal(K’/F’) ≋ A[5] and f remains irreducible over F'.
K is the splitting field of f (irreducible) over F, α[1], α[2], α[3], α[4], α[5] are five distinct elements, K = F(α[1], α[2], α[3], α[4], α[5]). K’ is the splitting field of f over F'.
[F’: F] = p, Gal(F’/F) ≋ ℤ/pℤ, and it cannot have proper intermediate fields because it is a prime number.
By assumption, F’/F is Galois ⇒ F’ is the splitting field of a polynomial g ∈ F[x] ⇒ g is irreducible. For the sake of contradiction, let’s suppose that g is reducible, say g = h·h’, deg(h)>0, deg
(h’)>0. However, we can assume that deg(h)>1, deg(h’)>1, and (h, h’) = 1. Otherwise, g = h(x-a), a ∈ F and the splitting field of h is the same as the splitting field of g
⊥ Therefore, g is irreducible of deg p, F’ = F(β[1], ···, β[p]). K’ = F(α[1], α[2], α[3], α[4], α[5], β[1], ···, β[p]) is the composite of F’ and K in $\bar F$, and notice that K’ is the splitting
field of f over F’ and it is generated by its roots, i.e., K’ = F’(α[1], α[2], α[3], α[4], α[5])
Each of the extension fields is a Galois extension, and the Galois groups have been labeled in the diagram, e.g., K’/F is Galois because it is the composite of K/F and F’/F, and they are both Galois
extensions (K’ is the splitting field of fg over F).
Our plan is to show that H is isomorphic to G, i.e., H is the alternating group A[5].
Let N = Gal(K’/F), N is Galois. From Galois’ fundamental theorem we know H ◁ N and N/H ≋ G’ ≋ ℤ/pℤ (F’/F is Galois), H’ ◁ N and N/H’ ≋ G ≋ A[5] (K’/F is Galois).
Next, the reader should notice that H ∩ H’ = {id} because σ ∈ H’ = Gal(K’/K), σ is an F-automorphism of K’ that fixes the roots α[1], ···, α[5].
σ ∈ H = Gal(K’/F’), σ is an F-automorphism of K’ that fixes the root β[1], ···, β[p]. Therefore, σ ∈ H ∩ H’ ⇒ σ is an F-automorphism of K’ that fixes β[1], ···, β[p], α[1], ···, α[5], but K’ = F(α
[1], α[2], α[3], α[4], α[5], β[1], ···, β[p]) ⇒ σ = id, H ∩ H’ is the trivial group.
Consider the canonical map, Φ: N → N/H ≋ G’. Let’s restrict the canonical map to the subgroup H’ (H’◁ N), Φ|[H’]: H’ → N/H. The Kernel of this restriction is the trivial group: Ker(Φ|[H’]) = Ker(Φ) ∩
H’= H ∩ H’ = {id} ⇒ Φ|[H’]: H’ → N/H ≋ G’ ≋ ℤ/pℤ, the restriction Φ|[H’] is injective ⇒ It maps H’ isomorphically to a subgroup of G’, a cyclic group of prime order, ℤ/pℤ, so there are only two
options: either H’ is the trivial group, or else H’ is cyclic of order p.
1. H’ is the trivial group, H’ = {id}, N → N/H’≋ G ≋ A[5], H’◁ N ⇒ so the map is onto (H’◁ N, then there is always a canonical surjective group homomorphism from N to the quotient group N/H’ that
sends an element g∈ N to the coset determined by g), but it is also 1-1 (its kernel is H’ = [By assumption] {id}) ⇒ N ≋ A[5] ⇒ [A[5] is simple] N is simple.
N does have a normal subgroup, namely H ◁ N, such that N/H ≋ G’ ≋ ℤ/pℤ, N ≋ A[5] ⇒ |N| = 60, N/H ≋ G’ ≋ ℤ/pℤ ⇒ |N/H| = p ⇒ |H| = 60/p ⇒ |H| ≠ 1 and |H| ≠ 60, so H is a proper, non-trivial normal
subgroup of N, but N is supposed to be simple ⊥
2. H’ ≋ ℤ/pℤ, |N| = [N/H ≋ G’ ≋ ℤ/pℤ] = |G’||H| = |H|p. |N| = [N/H’≋G ≋ A5] = |G||H’| = 60p. Therefore, |H| = 60.
Consider the canonical map ψ:N → N/H’ ≋ G (it is always surjective), and restrict it to the subgroup H, ψ|[H].
The kernel of this restriction is the trivial group: Ker(ψ|[H]) = Ker(ψ) ∩ H = H’ ∩ H = {id}, so ψ|[H] is injective. Therefore, H is isomorphic to a subgroup of G. However, |H| = 60 = [G ≋ A5] |G| ⇒
Since both groups have order 60, the restriction ψ|[H] is indeed an isomorphism ⇒ H ≋ G ≋ A5∎
[Continuing with the proof…]
1. By assumption K is the splitting field of f over F. Gal(K/F) = Gal(f) =A[5].
2. By the previous result, let K[1] be the splitting field of f over F[1] ⇒ Gal(K[1]/F[1]) = A[5]
3. We repeat the process up to K[r], the splitting field of f over F[r], Gal(K[r]/F[r]) = A[5]
However, f is not irreducible in F[r][x] because it has a root, namely α ∈ F[r] ⇒ f = (x - α)f’, f’ ∈F[r][x], deg(f’) = 4
Since K[r] is the splitting field of f over F[r] ⇒ K[r] is the splitting field of f’ over F[r], deg(f’) = 4 ⇒ Gal(K[r]/F[r]) ≤ S[4], and in particular |Gal(K[r]/F[r]| ≤ |S[4]| = 24, but we have
already established that Gal(K[r]/F[r]) = A[5], so has order 60 ⊥ Therefore, f is not solvable over F.
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Square Calculator
Calculation of the values of a square (a regular quadrilateral). A square is a quadrilateral with four right angles and four sides of equal length. Enter one of the known values . Then click
The radius of the circumscribed circle (R)
Radius of the inscribed circle (r)
d = √2 * a
p = 4 * a
S= a²
R = a / √2
r = a / 2
All angles are 90°, 2 diagonals.
S is the area, p is the perimeter
The sides are equal and the angles are 90 degrees
Lines dividing the sides in half
Inscribed and circumscribed circle
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2019-2020 UCLA PRIME
This forum made possible through the generous support of SDN members,
, and
. Thank you.
The Real PG
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Please tag a pre-allo moderator when the second prompt is posted.
Good luck to everyone applying!
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Aug 29, 2018
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Pre-writing and a little confused on the question "What is the most important social issue confronting the health of disadvantaged communities and what would be your first steps to address this
For "first steps", are they looking for what I would do as a medical student? Or as a physician?
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Oct 21, 2014
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Pre-writing and a little confused on the question "What is the most important social issue confronting the health of disadvantaged communities and what would be your first steps to address this
For "first steps", are they looking for what I would do as a medical student? Or as a physician?
I think neither. Anyone can try to address a social issue without being either one of them.
ALL 2000 characters
What is the most important social issue confronting the health of disadvantaged communities and what would be your first steps to address this issue?
In what way will graduating from PRIME-LA and obtaining a master's degree enhance your career in health care or health services for disadvantaged communities? (If you are considering a specific
master’s degree (e.g. MPP, MPH, MBA, etc.), please incorporate your graduate degree plans or aspirations into your answer)
Describe the manner in which your experiences demonstrate your understanding of, and commitment to, underserved communities.
What are your greatest strengths and your greatest challenges as you approach medical school?
What is your most memorable experience as it relates to working with vulnerable populations?
Verified July 18. No secondary received. Fingers crossed.
If you apply to UCLA-Geffen PRIME and was not considered, does this automatically put you into regular MD consideration?
If you apply to UCLA-Geffen PRIME and was not considered, does this automatically put you into regular MD consideration?
Yes, from what I remember from last year.
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Jul 25, 2019
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So I had been told by several medical students that PRIME was always marked on the secondary. So I was waiting for that secondary and realized that it was actually supposed to be marked on the
primary application with MD/MPH. Now I have a secondary for David geffen but not for prime. I resubmitted my application with UCLA marked as MD/MPH, does anyone know if this will fix it? Are UCLADG
and UCLA prime separate secondaries?
So I had been told by several medical students that PRIME was always marked on the secondary. So I was waiting for that secondary and realized that it was actually supposed to be marked on the
primary application with MD/MPH. Now I have a secondary for David geffen but not for prime. I resubmitted my application with UCLA marked as MD/MPH, does anyone know if this will fix it? Are
UCLADG and UCLA prime separate secondaries?
Yes, as a PRIME applicant my secondary was not the same as the DGSOM secondary (just based on reading the questions on SDN)
So I had been told by several medical students that PRIME was always marked on the secondary. So I was waiting for that secondary and realized that it was actually supposed to be marked on the
primary application with MD/MPH. Now I have a secondary for David geffen but not for prime. I resubmitted my application with UCLA marked as MD/MPH, does anyone know if this will fix it? Are
UCLADG and UCLA prime separate secondaries?
So mine wasn't actually completely separate but just another tab across the top when you're in the secondary. You submit it all at once, but you go to a "separate" tab to fill out prime. If its not
there, you can always call them just to double check!
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anyone receive an interview invite yet?
Jun 24, 2019
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anyone receive an interview invite yet?
No, and from my understanding based off of last year's PRIME thread, they didn't send IIs until the last week of November or early December. Idk if they're doing the same this year, though.
Since we applied for PRIME, does it put as at a disadvantage if our applications were moved to regular Geffen next month or so? (Considering Geffen already started giving interview invites?)
Since we applied for PRIME, does it put as at a disadvantage if our applications were moved to regular Geffen next month or so? (Considering Geffen already started giving interview invites?)
I'm starting to get the feeling that it
a disadvantage.
I'm starting to get the feeling that it is a disadvantage.
Was your application stronger for PRIME or for regular MD?
I'm starting to get the feeling that it is a disadvantage.
We wont hear back until late November to early December regarding PRIME interviews and UCLA finishes their interviews by December according to last year thread. We wont be considered for regular
program until we get rejected from PRIME. That means we wont get considered for the regular program until at least December if we don’t get one for PRIME. By then we might as well just consider it a
rejection since their interview season has already ended. My only shot at UCLA is PRIME so Im just putting all my eggs in this one. Stats are too low for UCLA since they put much more emphasis on
stats now than in previous years.
We wont hear back until late November to early December regarding PRIME interviews and UCLA finishes their interviews by December according to last year thread. We wont be considered for regular
program until we get rejected from PRIME. That means we wont get considered for the regular program until at least December if we don’t get one for PRIME. By then we might as well just consider
it a rejection since their interview season has already ended. My only shot at UCLA is PRIME so Im just putting all my eggs in this one. Stats are too low for UCLA since they put much more
emphasis on stats now than in previous years.
I wonder if this is the same for other PRIME programs for other UC schools.
I wonder if this is the same for other PRIME programs for other UC schools.
Im not really sure. I just dont understand why UCLA would wait that long to review PRIME applicants. That does put a huge disadvantage on those that don’t get a PRIME II but do have the stats to be
competitive for the traditional program. This probably could be that they put most of their resources into reviewing and interviewing the regular pool but don’t they have separate committee for
Im not really sure. I just dont understand why UCLA would wait that long to review PRIME applicants. That does put a huge disadvantage on those that don’t get a PRIME II but do have the stats to
be competitive for the traditional program. This probably could be that they put most of their resources into reviewing and interviewing the regular pool but don’t they have separate committee
for PRIME?
Prime is more of a mandated program by California (hence, every UC has it). I'm wondering if UC schools do them last because it's extra work. Because even if they get rid of PRIME, there's tons of
more than qualified applicants already in regular MD.
We wont hear back until late November to early December regarding PRIME interviews and UCLA finishes their interviews by December according to last year thread. We wont be considered for regular
program until we get rejected from PRIME. That means we wont get considered for the regular program until at least December if we don’t get one for PRIME. By then we might as well just consider
it a rejection since their interview season has already ended. My only shot at UCLA is PRIME so Im just putting all my eggs in this one. Stats are too low for UCLA since they put much more
emphasis on stats now than in previous years.
Your MCAT score is amazing! And the rest of your stats are amazing. You will get into a great program <3. Why do you feel that PRIME is easier to get into than the regular program?
Your MCAT score is amazing! And the rest of your stats are amazing. You will get into a great program <3. Why do you feel that PRIME is easier to get into than the regular program?
Thank you for the encouragement. Im sure a great school is waiting for you as well. The reason I feel I’d have a better chance at the PRIME program is I assume the program would focus more on ECs
with the underserved than on stats (afterall that’s why it’s created in the first place) and my app is much stronger in the ECs department. Also, there will be fewer students applying to PRIME since
not everyone has an interest working with the underserved but many have a strong interest in attending a top school in LA.
Also, there will be fewer students applying to PRIME since not everyone has an interest working with the underserved but many have a strong interest in attending a top school in LA.
You really think so? I think there are few seats in prime (there were 18 matriculants last cycle), so it's even tougher to get in even if you have about 500 applicants.
You really think so? I think there are few seats in prime (there were 18 matriculants last cycle), so it's even tougher to get in even if you have about 500 applicants.
True but considering how many people apply to UCLA each year (about 14,000 last year) the difficulty could go either way.
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May 15, 2017
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Did any other OOS people apply to UCLA PRIME and never get a secondary (for PRIME or for DG)? I sent my primary in in early july and never heard anything. LM 78-79 / WARS 94+, so thought it was a
little odd-- not rly torn up over it just curious if it was just me!!
Did any other OOS people apply to UCLA PRIME and never get a secondary (for PRIME or for DG)? I sent my primary in in early july and never heard anything. LM 78-79 / WARS 94+, so thought it was a
little odd-- not rly torn up over it just curious if it was just me!!
How was your experience with the underserved?
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May 15, 2017
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How was your experience with the underserved?
Solid. Not astounding-- thought I was shooting my shot a little bit here-- but a lot more than most general applicants (to MD programs in general, that is)
True but considering how many people apply to UCLA each year (about 14,000 last year) the difficulty could go either way.
Yeah, i think either way it's difficult: 175/14,000 = 1.25%
After completing essays for other schools, I sooo want my UCLA-PRIME essay back! Anyone, else feel this way?
Was your application stronger for PRIME or for regular MD?
I didn't have a separate application for PRIME. I submitted my AMCAS application for the programs on July 18th. No secondary :/
Any movement from anyone?
Any movement from anyone?
There wont be any movement until end of October, that’s when they start sending out II according to their website (in the FAQ section of PRIME program)
There wont be any movement until end of October, that’s when they start sending out II according to their website (in the FAQ section of PRIME program)
ah thank you
Do people committed to underserved populations ever apply just to UCLA or is it highly recommended those people apply to Prime?
Depends on your goal. And if you apply to both, you would be reviewed for Prime first. And From what I read, UCLA goes through Prime applications later. Hence, you won't be considered for regular
UCLA until you've been considered for prime first.
Do you submit your primary twice, once to prime and once to Geffen?
No for UCLA that option is in AMCAS so you select geffen, then select if you want to apply to the prime program under what degree you’re applying to.
Do you submit your primary twice, once to prime and once to Geffen?
You select PRIME on AMCAS. Then, on UCLA Portal, it'll ask you if you also want to be considered for regular MD. Based from my understanding, you will first be considered for PRIME (which happens
later in the cycle), then after that you'd be considered for regular MD geffen.
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Apr 14, 2017
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Are Prime students eligible for the Geffen merit scholarship?
All entering students of DGSOM are considered for Geffen scholarships. This includes students in PRIME, Drew, and regular MD track.
You select PRIME on AMCAS. Then, on UCLA Portal, it'll ask you if you also want to be considered for regular MD. Based from my understanding, you will first be considered for PRIME (which happens
later in the cycle), then after that you'd be considered for regular MD geffen.
In previous years, PRIME applicants would be at a disadvantage because they were considered for regular MD a bit later in the cycle (after getting the R from PRIME admissions). This year the timeline
has changed DGSOM and heard it may be different for PRIME applicants in that they are actively considered in the regular MD applicant pool for DGSOM while still being considered for PRIME. Soooo, you
could potentially interview for regular MD and also for PRIME but sure if this has been the case. Either way, all interview offers for PRIME will be sent by end of October and all interviews will
take place in November this year.
All entering students of DGSOM are considered for Geffen scholarships. This includes students in PRIME, Drew, and regular MD track.
In previous years, PRIME applicants would be at a disadvantage because they were considered for regular MD a bit later in the cycle (after getting the R from PRIME admissions). This year the
timeline has changed DGSOM and heard it may be different for PRIME applicants in that they are actively considered in the regular MD applicant pool for DGSOM while still being considered for
PRIME. Soooo, you could potentially interview for regular MD and also for PRIME but sure if this has been the case. Either way, all interview offers for PRIME will be sent by end of October and
all interviews will take place in November this year.
Thanks for the insight! I'm guessing no more secondaries are going to be sent out if interviews for PRIME are next month?
Thanks for the insight! I'm guessing no more secondaries are going to be sent out if interviews for PRIME are next month?
They're always keen for very strong applicants.
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Aug 7, 2019
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Any interview invites yet?
Sent from my iPhone using SDN
Any interview invites yet?
Sent from my iPhone using SDN
Probably not. Their website indicates IIs are going to be sent out at the end of this month.
Mar 21, 2019
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Really hoping for an ii soon!
I just got an II!!
The process is somewhat confusing though because on the Faqs it says that the PRIME interview can count as both interviews but not always. So, in the end, UCLA not UCLA-Prime picks who gets the PRIME
I just got an II!!
The process is somewhat confusing though because on the Faqs it says that the PRIME interview can count as both interviews but not always. So, in the end, UCLA not UCLA-Prime picks who gets the
PRIME spots?
Congratulations!!! very exciting! are you in state??
Also got an II! Long overdue.
I just got an II!!
The process is somewhat confusing though because on the Faqs it says that the PRIME interview can count as both interviews but not always. So, in the end, UCLA not UCLA-Prime picks who gets the
PRIME spots?
It just means that our PRIME interview will be used by both programs when considered for admission.
Congratulations!!! very exciting! are you in state??
Yes I am in IS!
It just means that our PRIME interview will be used by both programs when considered for admission.
Got it I just reread it. Judging from past threads the decisions don't come out until later next year
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Jul 9, 2019
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I just got an II!!
The process is somewhat confusing though because on the Faqs it says that the PRIME interview can count as both interviews but not always. So, in the end, UCLA not UCLA-Prime picks who gets the
PRIME spots?
Congrats!! When did you submit your secondary?
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Feb 26, 2016
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II today - LM: 73.5
was wondering if 1) anybody's portal changed to reflect the II 2) anyone has gotten both the PRIME-LA email and a MD-Only interview? or is it confirmed that we have to wait til we get a PRIME
decision to hear back about MD-only?
thanks and sorry if these questions were already answered!! i read the previous posts and was kind of confused tbh
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ILR * Isc
30 Aug 2024
ILR * Isc & Analysis of variables
Equation: ILR * Isc
Variable: ILR
Impact of Locked Rotor Current Ratio on ISC Function
X-Axis: -59994.00844269437to60006.01364001661
Y-Axis: ISC Function
Title: The Impact of Locked Rotor Current (ILRC) to Short Circuit Current (ISC) Ratio on the ISC Function in Electrical Systems
The short circuit current (ISC) is a critical parameter in electrical systems, used to determine the maximum fault level that a power system can withstand. However, the presence of locked rotor
current (ILRC), which occurs when an induction motor or generator is connected to a source and the rotor is not turning, can significantly affect the ISC function. This article explores the impact of
the ILRC to ISC ratio on the ISC function in electrical systems.
Locked Rotor Current (ILRC) is a phenomenon that occurs when an induction motor or generator is connected to a power source, but the rotor is not rotating. In such cases, the rotor behaves like a
short circuit, and the resulting current can be several times higher than the normal operating current of the machine. The ILRC is a function of the impedance of the machine and the power source.
The Short Circuit Current (ISC) is another critical parameter in electrical systems, which represents the maximum fault level that a power system can withstand. The ISC is typically calculated using
methods such as the “3-sigma” method or the “fault current analysis” method.
Equation: ILRC * Isc and Variable =ILR
The equation ILRC * Isc represents the product of the locked rotor current (ILRC) and short circuit current (ISC). This product is a measure of the impact that the ILRC has on the ISC function. The
variable =ILR indicates that this product is equal to the new value of the locked rotor current (ILR).
Mathematically, this can be represented as:
ILRC * Isc = ILR
• ILRC: Locked Rotor Current
• Isc: Short Circuit Current
• ILR: New value of Locked Rotor Current
Impact on ISC Function
The impact of the ILRC to ISC ratio on the ISC function can be significant. When the ILRC is high, it means that the rotor is behaving like a short circuit, and this can lead to an increase in the
ISC. As a result, the power system’s ability to withstand faults may be compromised.
On the other hand, if the ILRC is low, it indicates that the rotor is not behaving like a short circuit, and the ISC will be lower. This means that the power system’s ability to withstand faults has
The impact of the ILRC to ISC ratio on the ISC function highlights the importance of considering this parameter in the design and analysis of electrical systems. The ILRC * Isc = ILR equation
provides a useful tool for assessing the effect of locked rotor current on the short circuit current function.
In practice, engineers can use this equation to calculate the new value of locked rotor current (ILR) based on the product of the ILRC and ISC. This information can be used to determine whether the
power system’s ability to withstand faults has been compromised or improved.
The impact of locked rotor current ratio on short circuit current function is an important consideration in electrical systems design and analysis. The equation ILRC * Isc = ILR provides a useful
tool for assessing this effect, and engineers can use it to determine whether the power system’s ability to withstand faults has been compromised or improved.
1. Electrical engineers should consider the impact of locked rotor current ratio on short circuit current function when designing electrical systems.
2. The equation ILRC * Isc = ILR should be used to calculate the new value of locked rotor current (ILR) based on the product of the ILRC and ISC.
1. IEEE Guide for Performing Arc-Flash Hazard Calculation, IEEE Std. 1584-2018.
2. ANSI/IEEE Standard 1250-1995, IEEE Guide for Determining the Effects of High-Frequency Voltages and Currents on Relay Performance.
3. CIGRE Technical Brochure No. 520, “Fault-Current Analysis in Power Systems”.
Note: The provided text is a basic outline of how one could approach an article about the equation ILRC * Isc and variable =ILR in context of Impact of Locked Rotor Current Ratio on ISC Function.
Related topics
Academic Chapters on the topic
Information on this page is moderated by llama3.1
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Self-consistent theory of the Darrieus–Landau and Rayleigh–Taylor instabilities with self-generated magnetic fields
The Rayleigh–Taylor (RT) and Darrieus–Landau (DL) instabilities are studied in an inertial confinement fusion context within the framework of small critical-to-shell density ratio $DR$ and weak
acceleration regime, i.e., large Froude number Fr. The quasi-isobaric analysis in Sanz et al. [Phys. Plasmas 13, 102702 (2006)] is completed with the inclusion of non-isobaric and self-generated
magnetic-field effects. The analysis is restricted to perturbation wavelengths $k−1$ larger than the conduction length scale at the ablation front, yet its validity ranges from wavelengths shorter
and larger than the conduction layer width (distance between the ablation front and the critical surface). The use of a sharp boundary model leads to a single analytical expression of the dispersion
relation encompassing both instabilities. The two new effects come into play by modifying the perturbed mass and momentum fluxes at the ablation front. The momentum flux (perturbed pressure at the
spike) is the predominant stabilizing mechanism in the RT instability (overpressure) and the driving mechanism in the DL instability (underpressure). The non-isobaric effects notably modify the
scaling laws in the DL limit, leading to an underpressure scaling as $∼k−11/15$ rather than $∼k−2/5$ obtained in the quasi-isobaric model. The magnetic fields are generated due to misalignment
between pressure and density gradients (Biermann battery effect). They affect the hydrodynamics by bending the heat flux lines. Within the framework of this paper, they enhance ablation, resulting in
a stabilizing effect that peaks for perturbation wavelengths comparable to the conduction layer width. The combination of parameters $DRFr2/3$ defines the region of predominance of each instability
in the dispersion relation. It is proven that the DL region falls outside of the parameter range in inertial confinement fusion.
In direct-drive inertial confinement fusion (ICF),^1,2 preserving the symmetry of the capsule is crucial for the efficiency of the implosion. The initial asymmetries are amplified by the hydrodynamic
instabilities inherent to the process. Amongst others, the Rayleigh–Taylor (RT)^3 and Darrieus–Landau (DL)^4,5 instabilities play an important role since they compromise the integrity of the capsule.
The Rayleigh–Taylor instability has been thoroughly studied in the context of ICF.^6–9 The laser energy, absorbed at the critical surface, is convected toward the cold dense shell by heat conduction.
Mass is ablated off the shell and expands into the hot light corona, forming a structure commonly known as the ablation front. During the acceleration phase, this structure is RT unstable. In the
linear regime, ablation stabilizes perturbations whose wavelengths are shorter than certain cutoff, $kco−1$, which depends on the Froude number Fr. This number is the governing dimensionless
parameter in the ablative RT instability and stands for the ratio between ablative convection and the acceleration of the capsule. We restrict this analysis to the weak acceleration regime, which is
mathematically characterized by $Fr≫1$. In this regime, the cutoff wavelength is large compared to the ablation-front scale length, $kcoLa≪1$, and the dispersion relation can be analytically
derived exploiting the sharp boundary model (SBM).^8,10–12 The main stabilizing mechanism is the “rocket effect,” by which a restoring overpressure is generated at the spike.
The analysis of unstable modes in the weak acceleration regime fails when the perturbation wavelength is comparable to the distance between the ablation front and the critical surface. This region of
finite thickness is typically called the conduction layer. In the limit of zero acceleration, sufficiently long wavelengths undergo another type of instability known as Darrieus–Landau. This
instability is generic for fronts where a dense fluid expands into a lighter one as it typically occurs in flames.^13 An analytical study of the stability of ablation fronts that encompasses the
transition between the RT and DL instabilities has been carried out in Ref. 14. In this reference, the dispersion relation was derived within the framework of a quasi-isobaric approximation. The
analysis including compressible effects, which implies that the isothermal Mach number be unity at the critical surface, was not performed and is one of the results derived in the present paper.
During the development of these instabilities, magnetic (B) fields are generated due to the misalignment of gradients of density and pressure, known as baroclinic or the Biermann battery effect.^15
In the particular case of the ablative RT instability, B-field generation has been investigated both in numerical studies^16,17 and in experiments,^18–20 in the latter reporting measurements of the
order of several megagauss. Although there is a general agreement about the mechanism of generation of these B fields, their net effect on the hydrodynamics remains less clear. The presence of
magnetic fields modifies the transport coefficients in a plasma. Thermal conduction perpendicular to the B field, $κ⊥$, is reduced, and a heat-flux component along isotherms is generated. The latter
effect is known as Righi–Leduc and is important in the linear regime since it is of first order in B, Ref. 21. In essence, the Righi–Leduc effect deflects the heat flux lines, which in turn has a
direct effect on the stabilization of the ablative RT instability. Depending on the sense of the self-generated B field, mass ablation (and therefore stabilization) can be either enhanced or
diminished, as shown in Fig. 1. In Ref. 22, the RT instability was simulated with parameters relevant to fusion. It was reported that significant magnetization of the plasma could be attained even in
the linear regime. The magnetic field is generated out of phase, with its peaks placed near the points where the ablation front is unperturbed. In a particular simulation setup, enhancement in the
B-field generation and amplification was observed when the hydrodynamics was coupled with the induction equation. However, this unstable behavior was not acknowledged for every simulation setup
studied, and a systematic analysis of the conditions of whether this feedback occurs remains undone.
In this paper, we derive a self-consistent linear stability analysis of the ablation front coupled to the adjacent coronal plasma in an ICF context, including both the effects of self-generated
magnetic fields and compressibility. The main novelties include the analysis of the self-generated B-field effect on the RT instability for large Froude numbers and the derivation of the scaling laws
of the DL instability with compressible effects. Under the assumptions of this analysis, the B field is found to play a stabilizing role, contrary to the simulation setup reported in Ref. 22. It is
most important for short and intermediate wavelengths. The overpressure is enhanced by a maximum of 24% and the convective stabilization is doubled. It is barely noticeable for long wavelengths,
which are DL unstable. In this wavelength range, the non-isobaric effects cause the driving underpressure to scale more strongly with the wave number k.
This paper is organized as follows: in Sec. II, the governing equations are presented and linearized. In Sec. III, the dispersion relation encompassing both the Rayleigh–Taylor and Darrieus–Landau
instabilities with self-generated magnetic fields is derived. In Sec. IV, numerical results are discussed and the Darrieus–Landau instability is solved analytically in the small wave number limit. In
Sec. V, the different limits in dispersion relation are explored and the application to a configuration of interest in ICF is discussed. Finally, in Sec. VI, conclusions are drawn.
We consider a semi-infinite plasma slab expanding due to the deposition of laser energy in planar geometry (Fig. 2). Laser absorption is assumed to entirely take place at the critical density n[c].
We choose a hydrodynamic description of a fully ionized, single-species plasma and we assume quasi-neutrality. We consider the atomic number Z large enough that the ion pressure can be neglected
while the energy flow is still dominated by electronic heat conduction and the ion energy equation is decoupled. With these hypotheses, the evolution of the electron density n, ion velocity $v→$,
electron pressure p, electron temperature T, and magnetic-field intensity $B→$ are given by the electron continuity, total momentum, and total energy conservation together with Faraday's law of
induction, reading
$32∂p∂t+32∇·(pv→)+p∇·v→=−∇·q→e−∇·(52u→p)+ u→·∇p−R→·u→+Iδ(x→−x→c),$
where $m¯$ is the ion mass m[i] divided by the plasma atomic number Z, $E→$ is the electric field, $g→=ge→x$ is the acceleration in the frame of reference of the ablation front and $j→=−enu→$ stands
for the current, with $u→$ being the difference between electron and ion velocities. Electron inertia and ion heat flux have been neglected, and we assume the ideal gas equation of state, p=nT. For
simplicity, laser intensity I is deposited at the critical density, where $n[t,x→c(t)]=nc$, and δ denotes the Dirac delta. The electric field and the current are given by the electron momentum
conservation, which stands for a generalized Ohm's law and the Ampére's law, respectively,
We use the Braginskii^21 expressions and notations for the ion–electron friction force $R→$ and electron heat flux $q→e$. We assume small electron Hall parameter $ωeτe=(eB/mec)τe$, with $τe∝T3/2/n$
being the electron collision time
$q→e=−γ0nTτeme∇T︸Spitzer heat flux−γ0″γ0δ0eτemecB→×(γ0nTτeme∇T)︸Righi-Leduc+β0nTu→︸Thermoelectric+β0″δ0nTeτemecB→×u→︸Ettingshausen.$
In the linear stability analysis performed in this paper, only terms up to the first order in $|B→|$ shall be retained. The Hall and Ettingshausen terms are then dropped since they are proportional
to $|B→|2$, and all the coefficients are taken in the unmagnetized limit. Besides, the thermoelectric effects cancel out when added up in Eq. (3), and the ion–electron friction becomes of second
order in this same equation. We choose to rewrite $γ0nTτe/me=K¯T5/2$, with $K¯$ being the Spitzer conduction constant.
By means of Eqs. (4), (5), and (7), we can build an induction equation for the magnetic field
$∂B→∂t+cen2∇n×∇p︷Biermann battery−∇×(v→×B→)︷Bulk convection+c2α0me4πe2∇×(1nτe∇×B→)︷B diffusion=β0″δ0me∇×(τeB→×∇T)︸Nernst.$
Equations (1)–(3) and (9) become a system of four equations for the unknowns n, $v→$, T, and $B→$. We now decompose every variable into base plus perturbation, and we assume small perturbations.
The base variables are denoted with the subindex “0” and the perturbations with “1.”
A. Base flow
We consider a one-dimensional steady base flow. The only component of the velocity is the streamwise component u[0] (not to be confused with the difference between electron and ion velocities $u→$),
and we assume $B→0=0→$. The governing equations for the base flow become
As boundary conditions, we require that at $x→−∞$, we recover $n0=na, T0=Ta, u0=ua$, where the sub-index “a” denotes the state of the variables upstream of the ablation front (cold shell).
Additionally, the laser energy is deposited at $x=xc$, where $n0(xc)=nc$. We assume, as typically is the case, a small density ratio $nc/na≡DR≪1$. Letting the sub-index “c” denote the value of the
variables at the critical surface, we then have $uc/ua∼1/DR$, $Tc/Ta∼1/DR$. In the last estimation, we have assumed that $pc/pa∼O(1)$, which will be justified later.
We now normalize the variables with the values of reference in the cold shell. A length scale can be formed based on thermal conduction in the cold shell, $La=2K¯Ta7/2/5paua$. It also corresponds to
the ablation front thickness. Two dimensionless numbers arise: the (isothermal) Mach number at the ablation front $Ma=m¯naua2/pa$ and the Froude number $Fr=ua2/gLa$. We will prove that the Mach
number at the ablation front is $Ma2=DR/2$. The normalized continuity, momentum, and energy equations read
where $ϕ=2I/5paua$ is the normalized laser intensity. This system of equations is completed with the normalized boundary conditions $T0=u0=1$ at $x→−∞$, and we require $n0=DR$ at $x=xc$.
We integrate Eqs. (14) and (15) under the assumptions of a sharp boundary model. In this model, we exploit the disparity of thermal scale lengths at the critical surface and the ablation front. The
characteristic conduction length at the critical surface is given by $Lc=2K¯Tc7/2/5pcuc$; therefore, we can estimate $La/Lc∼DR5/2≪1$. Consequently, we can decompose the base flow into two regions
connected by the ablation front, which we choose to place at x=0. To the left, x<0, we find the cold shell, assumed to be homogeneous. To the right, x>0, the hot conduction layer develops with
$T0∼u0∼n0−1∼DR−1≫1$. The structure of the ablation front is therefore not retained and shrinks into a thin surface (i.e., SBM). This model holds as long as the wavelength of the unstable
perturbations is longer than L[a], which is the case when the Froude number is large, thereby becoming the main assumption in the present derivation.
We now consider the region downstream of the ablation front. We rescale the variables, the spatial coordinate, and the laser intensity as $Ma2u0$, $Ma2T0, Ma−2n0, Ma5x$, and $Ma2ϕ$, respectively.
To avoid excessive notation, the original names are kept for the rescaled variables. The ablation front is therefore seen as a sharp interface at x=0 at which $u0=T0=Ma2≈0$. We make the fairly
restrictive assumption $Ma Fr≫1$, which makes it possible to integrate Eq. (14),
In Sec. V, we prove that if this assumption is not satisfied, neither the DL instability nor the self-generated B-fields affect the dispersion relation, and the unstable perturbations would undergo
ablative RT instability in the isobaric limit, studied in Ref. 8. Equation (15) integrates into
with H standing for the Heaviside step function. In the integration of the equations, we have imposed $T05/2dT0/dx→0$ at the ablation front.
This stationary conduction layer would have to match a wider region where either non-stationary^23,24 or geometrical^25 effects become important and the plasma can expand isentropically to vacuum. In
either case, the Chapman–Jouguet condition (isentropic sonic point) must be satisfied at the end of the conduction layer: $u02/T0|x→∞=5/3$, which implies $u0|x→∞=5/8$. This condition, together with
the requirement that $dT0/dx→0$ at $x→∞$, gives the value of the normalized laser intensity $ϕ=5/16$, in agreement with Ref. 25. This value must be seen as the level of absorbed laser intensity
required to generate a given ablation pressure p[a] and ablation rate $naua$. The base temperature has a maximum for $u0=1/2$; therefore, $dT0/dx$ must change sign in order to be able to reach $u0=
5/8$. This implies that the laser intensity be deposited where $u0(xc)=1/2$. Consequently, we obtain that $DR=2Ma2$ and $pc/pa=1/2∼O(1)$, as assumed at the beginning of this section. We can
finally write the equation governing the conduction layer
and we recall that $n0=1/u0, T0=u0−u02, p0=1−u0$. It should be noted that the velocity derivative is singular at the critical surface. Integrating Eq. (18) between the ablation front and the
critical surface gives $xc≈0.0117$. Profiles of the conduction layer are shown in Fig. 2. Temperature increases in the overdense plasma $(x<xc)$ and decreases in the underdense plasma $(x>xc)$,
while the streamwise velocity increases monotonically until $u0(∞)=5/8$.
B. Perturbed flow
We now linearize the equations to study the perturbed problem. We propose a modal decomposition in time and transversal coordinate as $exp (γt+iky)$. The critical surface is also perturbed as $xc=
xc0+ξc exp (γt+iky)$, with $xc0$ being its unperturbed position previously derived. When the diffuse structure of the ablation front is not considered but rather it is treated as a sharp interface,
its position also needs to be perturbed as $xa=ξa exp (γt+iky)$. It is convenient to strain the x coordinate as
$s=xc0x−xaxc−xa≈x[1−ξc−ξaxc0exp (γt+iky)]−ξa exp (γt+iky).$
Consequently, the ablation front and critical surface are placed at s=0 and $s=xc0$, respectively, at all orders, and we prevent the Dirac delta and derivatives from appearing in the perturbed
equations. Notice that when referring to the base profiles, the variables x and s are interchangeable. Accordingly, we introduce the following ansatz for a generic variable $ϕ=ϕ0(s)+ϕ1(s)exp (γt+iky)
$, with $ϕ1/ϕ0≪1$. Particularly, we found convenient to expand the streamwise ion velocity as $u=u0(s)+{u1(s)+γ[ξa+s(ξc−ξa)/xc0]}exp (γt+iky)$, and the electron pressure as $p=p0(s)+Ma2p1(s)exp
(γt+iky)$. We denote as v[1], b[1] the transversal velocity and magnetic field, respectively, and we absorb the imaginary unit within them. We now recall the change of the derivatives as a
consequence of introducing the straining. Denoting
and $φ′$ its derivative, we have
$∂q∂x=dq0ds+(dq1ds−φ′dq0ds)exp (γt+iky),$
$∂q∂y=(q1−φdq0ds)ik exp (γt+iky),$
$∂q∂t=(q1−φdq0ds)γ exp (γt+iky).$
Inserting these ansätze into the governing Eqs. (1)–(4), linearizing and normalizing with the variables in the cold shell $(na,Ta,ua,La)$, we obtain
with the density perturbation being related to pressure and temperature through the linearized equation of state
The magnetic field b[1] has been normalized with a reference $Bref$ derived from the Biermann battery term, $Bref=cm¯ua/eLa$. In the right-hand side of the induction Eq. (28), we have the magnetic
field diffusion. It is inversely proportional to the magnetic Reynolds number $Rem=uaLa/Dm$, with D[m] being the magnetic diffusivity given by $Dm=γ0α0c2Ta2/10πe2pauaLa∝Ta−3/2$.
It is important to highlight that, to the first order, the effect of the magnetic field on the hydrodynamics is restricted to the last term in brackets in the energy Eq. (27), which acts as a heat
source. Two effects are accounted for here: the first one is the Righi–Leduc term and the second one comes from the difference between electron and ion enthalpy convection. They are proportional to
$Ma2$; hence, the effect of the B field on the RT instability at the ablation front scale length is expected to be small as the ablative flow is highly subsonic there. The three numerical
coefficients appearing in the Righi–Leduc term $(cR)$, current enthalpy convection $(cH)$ and Nernst term $(cN)$ take the value
$cR=5γ0″2γ02δ0≈1.71, cH=52α0γ0≈0.68, cN=5β0″2γ0δ0≈1.83.$
The analytical expressions for the Mach and magnetic Reynolds numbers are
where m[p] refers to the proton mass. Similarly, the ablation-front length scale length is
with Λ standing for the Coulomb logarithm. In the numerical applications in Sec. V, we take Λ=5 and we consider the ratio $γ0/Z$ to be equal to unity (γ[0] is an increasing function of the atomic
number). The Froude number then becomes
$Fr=6.67na1024 cm−3(ua1μm/ns)3(Ta10 eV)−5/2(g100μm/ns2)−1.$
We proceed to solve the governing Eqs. (24)–(28) imposing boundedness of the solution at $s→±∞$.
A. Analysis of the cold shell
First, we analyze the cold shell. This region can be assumed to be isentropic since heat conduction is negligible. In fact, the thermal mode decays in a distance comparable to the ablation-front
length scale; therefore, it is confined within the transition layer between the cold shell and the conduction layer and is not solved under the assumptions of this model.
The governing equations in this region assuming a small Mach number simplify to
The only bounded solution at $s→−∞$ corresponds to
$u1=−Akγ+kexp (ks)−γ(ξa+sξc−ξaxc0),$
$p1=A exp (ks)+1Fr(ξa+sξc−ξaxc0),$
$v1=Akγ+kexp (ks), T1=b1=0$
with A being an arbitrary constant.
In order to connect with the hot conduction layer, mass and streamwise and transversal momentum fluxes must be continuous through the ablation front. Their corresponding perturbed quantities are
$Cmass≡n1u0+n0u1$, $Cxmom≡n1u02+2n0u0u1+p1$ and $Cymom≡n0u0v1$, respectively. By means of the solution in the cold shell, we have at the ablation front
$Cmass=−Akγ+k−γξa, Cxmom=2Cmass+A+ξaFr, Cymom=−Cmass−γξa.$
The perturbed mass and streamwise momentum fluxes can be combined in order to eliminate the constant A. In so doing, the dispersion relation is formed,
B. Analysis of the conduction layer
The second step is to study the conduction layer. This would determine the value of the perturbed mass and momentum fluxes necessary for the dispersion relation Eq. (42).
We rescale the spatial coordinate and the base flow variables with their value of reference at the conduction layer, just as done in Sec. IIA, as $Ma5s$ and $Ma2u0$, respectively. We perform a
similar rescaling of the following variables: $Ma−5k, Ma−4n1, Ma−5b1$, $Ma3ξa, Ma3ξc,Ma−3γ$, $Ma−10Rem$, and $Ma Fr$. Again, we keep the same nomenclature for the rescaled variables so as to
avoid overwriting. In so doing, the Mach number is absorbed in the governing Eqs. (24)–(29), and they no longer depend on it. The Righi–Leduc term in the energy equation is then of order unity.
Consistently with the assumptions made to derive the base flow, the redefined Froude number is considered to be large, and we drop the terms proportional to $Fr−1$. The redefined magnetic Reynolds
number is much higher than in the cold shell (factor $Ma−10$). Typically, the conduction layer is highly conductive; therefore, we also drop the terms proportional to $Rem−1$.
We emphasize that the instability is driven by the ablation-front corrugation; hence, ξ[a] is of leading order. Consequently, we divide all the perturbed variables by ξ[a]. The following eigenvalue
related to the relative motion of the critical surface with respect to the ablation front is, therefore, introduced:
The rescaled governing equations are shown in the Appendix. They must be completed with the appropriate set of boundary conditions.
1. Boundary conditions for the conduction layer
Mass and streamwise momentum fluxes must be continuous through the ablation front, $s=0+$. We consequently introduce a the eigenvalues related to perturbed mass, f, and streamwise momentum, q,
fluxes as
$f≡1k(u1−T1u0+p1)|s=0+, q≡1k3/5p1|s=0+.$
The motivation behind these definitions comes from the fact that the dispersion relation (42), introducing again dimensional variables, becomes
The conservation of transversal momentum gives $v1|s=0+=−γ+O(Ma2)$. The rest of the boundary conditions require temperature and heat flux to be zero at $s=0+$. Applying these boundary conditions
allows one to obtain the shape of the profiles close to the ablation front depending upon the four eigenvalues ${f,q,h,r}$, which become the boundary conditions necessary to integrate the governing
equations in the conduction layer (A1)–(A5)
$u1|s→0+=(75fk−qk3/5+25r)u0, v1|s→0+=−γ,$
$p1|s→0+=qk3/5, T1|s→0+=25(fk+r)u0,$
The four eigenvalues ${f,q,h,r}$ are obtained while integrating the governing equations (A1)–(A5) ensuring four compatibility conditions required in the conduction layer. Three of them have a
hydrodynamic nature, while the remaining one is imposed on the magnetic field.
2. Hydrodynamic compatibility conditions
The first compatibility condition comes from the definition of the critical surface, where $n1(xc0)=0$. The second and third compatibility conditions come from canceling the two unbounded modes of
hydrodynamic nature. One of them takes place at the critical surface, which is an isothermal sonic point. Here, canceling the unbounded mode implies requiring the perturbed isothermal Mach number to
be unity,^28 which is tantamount to impose $T1(xc0)=u1(xc0)$. This requirement allows one to rewrite the vanishing density condition as $p1(xc0)=2u1(xc0)$.
The second unbounded mode takes place at $s→∞$, where $du0/ds→0$. It evolves as $exp (λs)$. In the small γ limit (this limit is justified in Sec. IIIC), it corresponds to the single positive root
where the sub-index “$∞$” denotes the value of the base profiles at infinity. It tends asymptotically to $λ=(3/2T∞5/2)1/3k2/3$ when $k≪1$ and $λ=k$ when $k≫1$.
Although the following does not stand for a compatibility condition but rather for a distinctive feature of the integration, we point out that, at the critical surface, the derivative of the base
velocity jumps in value. Consequently, the derivative of the perturbed temperature is discontinuous and satisfies
with u[c] standing for the value of u[1] at the critical surface.
The fourth compatibility condition to be satisfied is related to the magnetic field and explained in Sec. IIIB3.
3. Magnetic compatibility conditions
The shape of the magnetic field close to the ablation front, Eq. (48), presents a singularity characterized by the eigenvalue h. It is a consequence of the B-field accumulation, convected by the
Nernst term toward the ablation front. Yet, the value of the magnetic field at the ablation front must be null in order to match the profile in the cold side. This requires the existence of a
boundary layer where diffusion becomes important.
We apply singular perturbation theory Ref. 29 to obtain the structure of this boundary layer. Rescaling the base velocity as $u0=ϵBw$, with $ϵB≡Rem−1/5≪1$, introducing $ψ=b1/qk8/5$, and taking the
leading order expansion in ϵ[B], Eq. (28) simplifies into
The term with “$cN−1$” accounts for the Nernst and bulk plasma convection. The former predominates, resulting in a net convection toward the ablation front. This equation must be solved with the
boundary conditions $ψ(0)=0$ and $ψ(w→∞)=(1+hRem1/5/w)/(cN−1)$. The solution is
$ψ=1cN−1[1−exp (−cN−15w5)]+hRem1/5[∫0wv3 exp (cN−15v5)dv]exp (−cN−15w5).$
The expansion of ψ for $w→∞$ is
$ψ|w→∞=1cN−1(1+hRem1/5w)+[11−cN+hRem1/5Γ(4/5)51/5(1−cN)4/5]exp (−cN−15w5),$
where $Γ(z)$ stands for the Euler gamma function.
This inner solution matches with the outer behavior as long as the exponential term converges. If the Nernst term is not taken into account, c[N]=0, this condition is not satisfied and we cancel
the exponential term by setting $h=−Rem−1/551/5/Γ(4/5)$, which tends to zero for infinite $Rem$. If the Nernst term is taken into account, then $cN−1>0$, and the exponential term converges. The
magnetic field accumulates at the ablation front and presents a maximum at w=1.70, where $ψ|max=0.72hRem1/5$. The eigenvalue h cannot be obtained from the analysis of this boundary layer alone,
and the fourth condition is derived from the analysis of the critical surface.
At the critical surface, the Nernst convection velocity changes sign, altering the net B-field convection direction. To the left, the Nernst term overcomes the bulk plasma convection and the B field
is convected toward the ablation front, while to the right, both velocities add up and the B field is convected outward. Integrating the induction equation to the left “–” and right “+” of this
surface yields $b1+=b1−μ−/μ+$, where $μ±=u0−cNn0−1T03/2dT0/ds|s=xc0±$ is the net convection velocity. Inserting the numerical values gives $μ+≈0.55, μ−≈−0.60$ and hence $b1+≈−1.10b1−$. This implies
either null magnetic field at the critical surface or a singular current if $b1≠0$. In the latter case, a diffuse boundary layer should take place to support the jump in B field. Singular
perturbation theory shows that the width of such a layer would scale as $Rem−1$. Introducing $η=Rem(s−sc0)$ and keeping the leading order terms in Eq. (28) gives $d2b1/dη2=(μ/8)db1/dη$, with $μ=μ−
<0$ for $η<0$ and $μ=μ+>0$ for $η>0$. The solution of this equation consists of a divergent exponential at both sides of the critical surface, $b1∼ exp (μ±η/8)$, which cannot be matched with the
outer solution unless it is forced to be zero.
In summary, when the Nernst term is taken into account, the eigenvalue h must ensure that $b1=0$ at the critical surface. Without Nernst, there is no singularity at the ablation front and we set h=
C. Resolution method
For simplicity, we rewrite the system of equations (A1)–(A5) in a matrix form. Letting $Y→={u1,p1,v1,T1,dT1/ds,b1}T$ denote the state variables vector, we have
We exploit the smallness of the growth rate γ, following the same method as in Ref. 8. To justify this assumption, we proceed to the following estimations. In the RT instability, the term “kg” is
predominant in Eq. (45) and we can state $γ∼kg$. From this equation, the cutoff can be estimated as $kcoLa∼(gLa/ua2)5/3$, which leads to $γLa/ua∼(kLa)4/5$. The latter estimation can be rewritten
normalizing with the variables at the critical surface as $γLc/uc∼Ma(kLc)4/5≪1$. Notice that a similar result, $γLc/uc∼Ma$, would be obtained when the DL instability dominates and q<0 becomes the
predominant term. The conduction layer is, therefore, quasi-stationary (slow dynamics), and the terms proportional to γ in Eqs. (A1)–(A5) are small. This is a direct consequence of assuming the
ablation front corrugation ξ[a] to be of leading order. Another kind of instability, however, is contained within the structure of Eqs. (24)–(28). It corresponds to a faster dynamics, where the
ablation remains unperturbed and $γLc/uc∼O(1)$. This limit is known as the magnetothermal instability^26,27 and is out of the scope of the present analysis.
We expand the state vector $Y→$, eigenvalues ${f,q,h,r}$ and matrices A, B, C as
with $ϑ1$ depending on k, exclusively. The system in Eq. (54) yields a hierarchy of systems of linear differential equations. In order unity, we have
The eigenvalues ${fj,qj,hj,rj}$ are determined at each order “j” by numerically integrating the corresponding system, starting from the ablation front with the boundary conditions Eqs. (46)–(48), and
imposing the four compatibility conditions previously described, that is, null perturbed density, isothermal Mach number, and magnetic field at the critical surface and cancelation of the
hydrodynamic unbounded mode at infinity.
Bearing the smallness of γ into account, the dispersion relation in Eq. (45) yields the unstable root
We recall that, in dimensional variables, the eigenvalues ${fj,qj,hj,rj}$ depend on $kxc0=0.0117kLa(2na/nc)5/2$.
The numerical results of the eigenvalues $f1,q1$, r[1], and q[2] are shown in Figs. 3 and 4.
The mechanisms of stabilization become clearer after inspection of Eq. (58). The quasi-steady overpressure q[1] acts as a conservative force. It is positive for $kxc0>0.50$, meaning that the
momentum flux increases at the crest (spikes) of the rippled ablation front and decreases in the valleys (bubbles), damping the growth of the unstable modes. It corresponds to the rocket effect
(restoring force) mechanism, and, as discussed in Ref. 14, it is related to the quasi-steady rotational part of the velocity (vorticity ω) at the ablation front since $iω|s→0+=−qk8/5$. The term
proportional to ku[a] is typically called “convective stabilization” and stands for a damping force. Both the mass flux and the nonsteady momentum flux are accounted for in this effect. The former
corresponds to the “fire polishing effect,” by which the spikes evaporate more quickly than the bubbles, and is related to the potential part of the velocity at the ablation front, $v→∼T5/2∇T$. It
is interesting to notice that although this problem lacks a barotropic relation and, consequently, a potential and rotational velocity flow decomposition is pointless, it still makes sense locally at
the ablation front, where the non-isobaric terms are small. Amongst the two stabilization mechanisms previously described, the rocket effect is the predominant under the assumptions considered in
this paper. In fact, in the strict limit $Ma→0$, the term ku[a] can be neglected and the cutoff becomes $kcoLa=(gLa/ua2q1)5/3∼Fr−5/3$.
All the eigenvalues tend to a finite value for large k. When the coupling with the induction equation is not considered, f[1], q[1], and q[2] tend asymptotically to their corresponding values
calculated in the SBM for ablative RT instability in Ref. 8, $q1=0.96, (1+f1+q2)/2≈2$. Notice that the q[1] defined there shall be divided by $(2/5)2/5$ for the comparison. It is interesting to
observe that, contrary to the analysis of isobaric conduction layer in Ref. 14, both f[1] and q[1] present a maximum for $kxc0=2.63$ and $kxc0=2.24$, respectively. Another difference when taking
into account non-isobaric terms lies in the ratio $ξc/ξa$. It becomes negative for $kxc0>1.80$ instead of tending monotonically to zero. The physical reason behind the critical surface oscillating
out of phase with respect to the ablation front for perturbations confined in the latter lies in the vorticity mode. In the quasi-stationary conduction layer, vorticity is convected without decaying
and reaches the critical surface. This vorticity trail excites the critical surface, which is set into motion in order to ensure the unit isothermal Mach number condition, that is $T1=u1$. We recall
that this condition is not required if isobaricity is assumed. Instead, T[1] must be null, leaving u[1] free and the critical surface unexcited.^14 The unsteady momentum flux q[2] is positive for
almost all of the wave numbers, helping to stabilize the motion. It presents a minimum around $kxc0=1$, where it becomes approximately zero.
For $kxc0<0.5$, the perturbed momentum flux q[1] is negative, and the quasi-steady vorticity at the ablation front becomes the driving mechanism of the Darrieus–Landau instability. The critical
surface and ablation front move synchronously. For smaller wave numbers, $kxc0<0.10$, the perturbed mass flux f[1] is inverted and reinforces the DL instability. This effect is not observed in the
isobaric model. The minimum values of f[1] and q[1] are –0.077 and –0.43 and take place at $kxc0=0.015$ and $kxc0=0.13$, respectively. Notice that q[1] attains a lower minimum, $q1=−0.51$ at $kxc0=
0.18$ in the isobaric model. Scaling laws for $k≪1$ are derived in Sec. IVA.
Under the assumptions of the sharp boundary model, the effect of the self-generated magnetic field is always stabilizing. The B-field effect is always present in the governing Eqs. (A1)–(A5) and does
not depend upon any parameter. It increases both the momentum and mass fluxes, being more efficient for the former: the peak of q[1] is increased by a 24% vs 11% for f[1]. In Fig. 5, the profiles of
the variables are shown for the perturbation at which q[1] is maximum in the case coupled with Nernst. It can be seen that the magnetic profile at the right of the spike section is positive, in
agreement with the stabilizing case in Fig. 1. The Nernst convection enhances the stabilizing effect of the B field. For perturbation wavelengths longer than the distance between ablation front and
critical surface, $kxc0<1$, the B field is less effective, becoming totally negligible in the Darrieus–Landau instability region. The B-field effect is significantly stronger on the unsteady
momentum flux q[2].
Asymptotic analysis indicates that, in the short wavelength limit, $k≫1$, both the non-isobaric and the B-field effects become negligible. In this limit, the perturbation is confined within a
neighborhood around the ablation front of a characteristic length scale $k−1$. We can estimate $s∼O(k−1), u0∼T0∼O(k−2/5)$, and $T1∼u1∼O(k3/5)$. From induction Eq. (A5), equating B-field convection
to the baroclinic term, we obtain $b1∼k8/5$. When plugged into the energy Eq. (A4), we obtain that both the Righi–Leduc term and the non-isobaric terms (all the terms in the left-hand side except
for the divergence of velocity) turn out to be small of order $O(k−2/5)$. However, as shown in Fig. 3, the B-field effect is numerically more important than the non-isobaric effects, and f[1] and q
[1] asymptote to their isobaric limit ($f1=1.02, q1=0.96$) significantly later (higher k) when the coupling with the induction equation is considered. This effect is even more remarkable for the
higher-order eigenvalue q[2] since its value remains tripled with respect to the isobaric limit ($q2=2.12$) for a significantly long wave number range. This has an important effect on the
convective stabilization term in Eq. (58), which is enhanced from “$−2kua$” to “$−4kua$.”
A. Analysis for $k≪1$: Darrieus–Landau instability
In this section, we consider the limit of large wavelength perturbations, $k≪1$, which are Darrieus–Landau unstable, and we derive analytically scaling laws for f[1], q[1], $ξc/ξa$, and q[2]. The
spatial domain s>0 can be split into two distinguishable regions. The first one would correspond to the conduction layer properly said, where $s/xc0∼O(1)$, and $du0/ds∼O(1)$. The second one is a
scale-free region adjacent to the conduction layer, where $s/xc0≫1, u0≈5/8$ and $du0/ds≪1$.
We integrate the governing equations in the conduction layer. The transversal momentum equation is uncoupled from the rest of the system and yields $v1=ku0$. The conditions at the critical surface,
$p1=2u1, T1=u1$, lead to consider u[1], p[1], and T[1] of the same order, given by $q1k3/5$. At the same time, we can infer $q1k3/5∼f1k∼r1$, and we assume $q1k3/5≫k2$, which will be proven a
posteriori. Continuity and streamwise momentum equations simplify to conservation of perturbed mass and momentum fluxes, which can then be integrated into
Applying the conditions at the critical surface allows one to obtain $f1k=2uc1$ and $q1k3/5=4uc1$, where $uc1$ is the first order [as given by the ansatz Eq. (55)] of the value of u[1] at the
critical surface. The magnetic field scales as $b1∼q1k8/5$, which is smaller than the other perturbed quantities. Consequently, the Righi–Leduc term can be neglected in the energy equation and the
hydrodynamics becomes decoupled from the induction equation. The energy equation can be integrated once yielding
Imposing $T1=0$ at s=0 and $T1=uc1$ at $s=xc0$ allows one to obtain $r1=8uc1$. The perturbed quantities in the subsonic region $(0<s<sc0)$ become then $u1/uc1=2u0$, $v1=ku0, p1/uc1=4(1−u0),T1/uc1
=4u0(1−u0)$ and $n1=0$. Once the hydrodynamic profiles are derived, the induction equation can be integrated. The eigenvalue h yields $h=−3/8$, and the magnetic-field profile is
At the end of the conduction layer, we have $v1|s→∞=u∞k=5k/8$ and, from Eq. (61), $dT0/ds|s→∞=5uc1/8T∞5/2=4096uc1/4515$.
The relation between $uc1$ and k is derived from resolving the scale-free region. Let $Φ→={u1/u∞,v1/u∞,p1/p∞,T1/T∞}T$ denote the state variable vector, the solution in the scale-free region will
consist of a linear combination of five modes. The first mode stands for vorticity being convected as a passive scalar by the bulk plasma, while the second mode is governed by transverse heat
$Φ→1={1000}, Φ→2={−T∞5k2−T∞5/2ku∞p∞T∞5k21} exp (−T∞5/2k2s).$
The third, fourth, and fifth modes are governed by longitudinal thermal conduction mixed with incompressible mode. Introducing $β=(3/2T∞5/2)1/3≈3.84$, the third one reads
$Φ→3={−3/2T∞5/2β2k1/35/21} exp (βk2/3s),$
which is unbounded. The fourth and fifth modes are bounded complex conjugate modes
$Φ→4={−3/2k1/3Φ425/21} sin (32βk2/3s)exp (−12βk2/3s),$
$Φ→5={−3/2k1/3Φ525/21} cos (32βk2/3s)exp (−12βk2/3s)$
$Φ52=T05/2β2[−12+32tan (32βk2/3s)].$
Since u[1], p[1], and T[1] are of the same order of magnitude, the scaling laws in this region are given by the last two modes. In order to proceed with some order of magnitude estimations, we simply
assume $Φ→=C4Φ→4$. Matching this solution with the conduction layer yields $C4∼k2/3$ and $uc1∼k4/3$. The former condition has been derived by imposing $v1∼k$, while the latter comes from $dT1/
ds∼uc1$. Consequently, in the scale-free region, we have $s∼k−2/3≫1$, u[1], p[1], and T[1] peak at a much higher value with respect to their magnitude in the adjacent layer to the critical surface,
$k2/3$ compared to $k4/3$. We can then solve the supersonic region assuming $u1=p1=T1=0,dT0/ds=5uc1/8T∞5/2$ and $v1=5k/8$ at s=0, which gives precisely $Φ→=C4Φ→4$, with
$C4=−2k2/33T∞5/2β2≈−2.95k2/3, uc1|Analytic=−3k4/38β≈−0.098k4/3.$
When compared to the numerical results, the scaling law is correct but there is a factor of 1.75 difference in the coefficient. The reason for this lies in the overlap between the conduction layer
and the scale-free region. According to the previous results, in the former we have $v1≫T1$, while in the latter $T1≫v1$. Therefore, in the overlapping region we have $v1∼T1$, which takes place
for $s∼k−1/3$. Since $du0/ds∼1/s2∼k2/3$ in this region, we cannot neglect the derivatives of the base profile when performing the coupling between both regions, and the matching must be done
numerically. This yields the final result
and, consequently, we have $f1=−0.49(kxc0)1/3$, $q1=−5.8(kxc0)11/15$, and $ξc/ξa=1−2.0(kxc0)4/3$. They show good agreement with the numerical results (Fig. 6). Notice that these scaling laws
differ from the ones derived in Ref. 14, where isobaricity was assumed, and $f1=kxc0$, $q1=−(5kxc0/2)2/5, q2=1$ and $ξc/ξa=1−(kxc0)2/2$. Of particular interest is the perturbed mass flux, which
becomes negative (destabilizing) when the non-isobaric effects are considered. However, the driving mechanism of the instability, the momentum flux or vorticity q[1], is smaller in this case as it
presents a stronger scaling on $kxc0$.
Finally, the next-order in the ansatz Eq. (55) can be obtained in a similar manner. The perturbed mass and momentum fluxes are conserved in the conduction layer, and, in the scale-free region, the
variables evolve as given by the mode $Φ→4$ too. However, the key point is to notice that it is the boundary condition at the ablation front, $v1=−γ$, the one that imposes the scaling of the
variables with k at this order. More precisely, the span-wise velocity is constant and equal to $−k3/5$ in the conduction layer, which in turn leads to a constant $q2=0.66$.
The dispersion relation given in Eq. (58) can be rewritten in the form
We proceed to analyze this expression as a function of its two governing parameters: the Froude number Fr, defined in Sec. IIA, and the critical-to-shell density ratio $nc/na≡DR$. We recall that
the latter is equivalent to the Mach number at the ablation front: $DR=2Ma2$. It only appears through the eigenvalues q[1], f[1], and q[2], which are functions of $kxc0=0.066kLa/DR5/2$. In the
derivation of Eq. (69), we have assumed both $Fr≫1$ and $DR≪1$. Under these assumptions, the rocket effect (perturbed pressure q[1]) becomes the main stabilizing mechanism. The convective
stabilization, in spite of being of lower order, is numerically significant for moderately large $Fr$. We recall that the cutoff as given by the rocket effect takes place for $kcoLa∼1/Fr5/3$. This
expression holds as long as the perturbed pressure q[1] is positive for this wave number. As seen in Fig. 3, the perturbed pressure changes sign for $kq1xc0≈0.5$, or equivalently $kq1La≈7.6DR5/2$,
and becomes the driving DL instability mechanism. The ratio between these two wave numbers, $kq1/kco=(2.25DRFr2/3)5/2$, dictates the shape of the dispersion relation. Two different limits can
therefore be identified depending on the value of $DRFr2/3$.
A. RT limit
For $DRFr2/3≪1$, the perturbed pressure becomes negative for wave numbers smaller than the cutoff. The spectrum is totally dominated by the ablative RT instability. This limit is well described
assuming isobaricity ($q1=0.96, f1=1.02$, and $q2=2.12$), and, to the leading order, the cutoff takes place at $kcoLa=1/(q1Fr5/3)$. This applies even in the part of the spectrum where $q1<0$. In
this part, the vorticity generation is so weak that the RT term in Eq. (69) dominates over the “perturbed pressure” term, and the growth rate is simply $γ=kg$. In this limit, it is not necessary to
take into account the critical surface and just the analysis of the ablation front suffices. As a result, the assumption made when deriving the base flow in Sec. IIA, $MaFr≫1$, is well justified.
B. DL limit
For $DRFr2/3≫1$, the perturbed pressure is negative for $kcoLa∼1/Fr5/3$; therefore, this becomes a poor estimation for the cutoff. It would rather take place close to $kq1La∼DR5/2$ at the very
moment where q[1] changes sign. In fact, we can estimate $q1∼1/(DRFr2/3)3/2≪1$ at the cutoff. Stabilization is still given by the rocket effect, but it takes place at a larger wave number, which is
independent of Fr (see Fig. 7). This analysis is in agreement with the results shown in Fig. 7 in Refs. 14 and 30. For larger perturbations (smaller wave number), the spectrum is dominated by the DL
instability, and the growth rate becomes $γLa/ua=|q1|1/2(kLa)4/5$. However, this regime does not apply for all the range of unstable modes. For very small wave numbers, the RT overcomes the DL. As
derived in Sec. IVA, the underpressure is given by $q1=−0.78(kLa)11/15/DR11/6$ in this case. Estimating the two driving mechanisms in the square root in Eq. (69) yields a threshold wave number
$kthLa=1.2DR11/8/Fr3/4$ below which the RT becomes the driving mechanism. Such a spectrum is sketched in Fig. 7(a). The largest perturbations undergo classic RT instability with $γ=kg$. For $k>kth$
, the DL instability becomes predominant until we reach $k=kq1$, where the perturbed pressure becomes positive (vorticity changes sign) and the rocket effect stabilizes shorter perturbations.
A particular feature of this limit is that the dispersion relation can break into two unconnected branches for very large values of $DRFr2/3$, as shown in Fig. 7(b). This occurs when the RT unstable
region, $k<kth$, is stabilized by the secondary mechanism in Eq. (69): convective stabilization. Convection can suppress the RT instability at a wave number $kcsLa∼1/Fr$. If $kcs<kth$, then
perturbations smaller than $kcs−1$ are stable until the DL driving mechanism becomes strong enough to overcome the convective stabilization. This takes place at the wave number $kdlLa∼DR11/2$.
Smaller perturbations undergo DL instability, and the rocket effect stabilizes those with $k>kq1$. However, this regime is not expected in ICF configurations since the condition $kcs<kth$ implies
the restrictive $DR11/2Fr≳1$, which corresponds to low acceleration levels.
Notice that this double branch shape can only occur if we take into account the non-isobaric effects, which induce the stronger scaling $q1=−5.8(kxc0)11/15$ compared to $q1=−(5kxc0/2)2/5$ without
them. This can be better seen in the limit of zero gravity (infinite Froude number) and long wavelength $kxc0→0$. Let $n∞$ denote the density far from the critical surface. The dispersion relation
Eq. (69), introducing the laws derived for the eigenvalues in the long wavelength limit, would yield
in the isobaric case (Ref. 14) and
with non-isobaric effects as derived in this paper. The isobaric relation, Eq. (70), corresponds to the sharp boundary model of the DL instability derived in Refs. 4 and 5 in the $n∞/na≪1$ limit. The
growth rate is linear with k and always positive. Introducing the non-isobaric effects, Eq. (71), notably modifies the dispersion relation. For $na>n∞$, convection stabilizes wave numbers smaller
than the cutoff $kdlxc0=0.33(n∞/na)3$.
C. ICF application
By means of Eqs. (31) and (34), the combination of parameters $DRFr2/3$ can be expressed as a function of physical variables that are more relevant in ICF as
$DRFr2/3=0.015(ua1μm/ns)4(na1024 cm−3)2/3(g100μm/ns2)2/3(Ta10 eV)8/3.$
It can be seen that, in the range of ICF applications, this parameter tends to be small (RT limit) or of the order of unity. Large values of it would correspond to specific low-acceleration
experiments. A configuration of interest for ICF is shown in Fig. 8, where we have chosen $ua=1.2 μm/ns, Ta=7.5 eV$, $na=1024 cm−3$ and $g=50 μm/ns2$, which gives $Fr=50$ and $DRFr2/3=0.11$. The
dispersion relation with self-generated B fields is compared to the case decoupled from induction and to the purely isobaric case. The DL mechanism comes into play by reducing the ablative
stabilization around $kq1La=4.4×10−5$. It never overcomes the RT instability, but rather enhances the unstable behavior of these wavelengths. The effect of the B field comes into play for shorter
wavelengths, when ablation becomes effective, and it has a significant stabilizing role, reducing the cutoff from $kLa=4.3×10−4$ to $kLa=3.6×10−4$.
D. Summary of regimes
A schematic of the different regimes described in this section is plotted in Fig. 9. In this chart, the stability of a generic perturbation with wave number kL[a] is given as a function of the Froude
number and the critical-to-shell density ratio. It must be understood from an asymptotic analysis point of view, hence the transition from one region to another is blurry rather than a well-defined
The two dotted areas show the combination of $Fr, DR$ for which the perturbation is unstable. One is RT dominated and the other one is DL dominated. The border between them, $DR11/8/Fr3/4=kLa$,
corresponds to the perturbation wave number being equal to the threshold $kth$ as previously derived. The dashed curves do not delimit any stability region but rather they establish the transition
between regimes.
For $DRFr2/3<1$, the spectrum is RT dominated; the cutoff is given by the rocket effect (dynamic pressure) and only depends upon the Froude number $(kLa∼Fr−5/3)$. For $DRFr2/3>1$, the perturbation
would undergo DL instability if $k>kth$ and RT otherwise. The cutoff appears whenever the DL driving mechanism (perturbed pressure) becomes positive, depending only on the density ratio $(kLa∼DR5/2)$
The second dashed curve, $DR11/2Fr=1$, establishes the limit above which the secondary stabilizing mechanism (ablative convection) operates and breaks the unstable part of the spectrum into two
unconnected regions: one RT unstable in the smallest k region and another DL unstable developing until the cutoff due to dynamic pressure.
In this paper, the hydrodynamic stability of a laser-accelerated target is investigated in the context of inertial confinement fusion. The stability analysis includes the coupling of the target to
the ablated expanding plasma past the critical surface. The analysis is restricted to the linear regime and it is based upon two main assumptions: small critical-to-shell density ratio, $DR≡nc/na≪1$
, and weak acceleration regime (large Froude number $Fr$). The two main novelties in this study are the inclusion of non-isobaric effects and the self-generated magnetic fields.
In the limit of perturbation wavelengths smaller than the distance between the ablation front and the critical surface (conduction layer width $xc0$), the expansion undergoes Rayleigh–Taylor (RT)
instability driven by the acceleration. It is damped by mass ablation off the cold shell. The main stabilization mechanism corresponds to the overpressure generated at the spikes, rocket effect,
which can stabilize the motion for wave numbers larger than a cutoff $kcoLa∼Fr−5/3$. In the limit of large perturbation wavelengths, the Darrieus–Landau (DL) instability takes place. It is driven by
the vorticity generated at the ablation front. It is closely related to the perturbed pressure at the spikes, which becomes negative (underpressure) and destabilizing. Asymptotic analysis allowed us
to derive the scaling laws of the under-pressure q[1] with the wave number. In this limit where the non-isobaric effects play an important role, making q[1] scale as $q1=−5.8(kxc0)11/15$, compared
to the isobaric case studied in Ref. 14, where $q1=−(5kxc0/2)2/5$.
Magnetic fields are generated by the misalignment of density and pressure gradients, known as baroclinic or Biermann-battery effect. In the linear regime, they affect the hydrodynamic via the
Righi–Leduc term. It acts as a heat source that deflects the heat flux lines. Under the SBM assumptions, the self-generated magnetic field has a stabilizing effect by enhancing ablation. It is most
effective for perturbation wavelengths comparable to the conduction layer width, where the restoring overpressure can be increased up to 24%. The DL instability is barely affected by the
self-generated B fields. In the opposite limit, small perturbation wavelength $kxc0≫1$, both the non-isobaric and B-field effects scale as $(kxc0)−2/5$. However, the B-field effect turns out to be
numerically more important, and its stabilizing effect can even be observed for large $kxc0$. Particularly, the convective stabilization term is doubled to $−4kua$ for a wide wave number range.
Finally, the Nernst term enhances the stabilizing effect of the self-generated magnetic fields.
The analysis of the dispersion relation reveals that the combination $DRFr2/3$ dictates the behavior of the spectrum. For $DRFr2/3≪1$, it is well described by the ablative RT instability in the
isobaric regime, and the cutoff takes place for $kLa≈Fr−5/3$. In the opposite limit, $DRFr2/3≫1$, two regions can be defined. The long perturbations with $kLa<DR11/8/Fr3/4$ undergo RT instability,
while the part of the spectrum with $kLa>DR11/8/Fr3/4$ is DL dominated. In this limit, the cutoff becomes independent of the Froude number: $kLa≈7.6DR5/2$. The regime of application for ICF
corresponds to $DRFr2/3≲1$. When this parameter is close to unity, the DL effect operates by reducing the restoring overpressure and increasing the wave number at which ablation comes into play. It
is precisely in this range, $DRFr2/3∼1$, where the effect of the self-generated B fields becomes more important. They enhance the stabilizing effect of ablation and can significantly reduce the
This work was supported by the Department of Energy Office of Science, Fusion Energy Sciences program Grant Nos. DE-SC0016258 and DE-SC0014318. J. Sanz was also supported by the Spanish Ministerio de
Economía y Competitividad, Project No. RTI2018-098801-B-I00. H. Aluie was also supported by U.S. DOE Grant Nos. DE-SC0020229, DE-SC0019329, U.S. NASA Grant No. 80NSSC18K0772, and U.S. NNSA Grant Nos.
DE-NA0003856 and DE-NA0003914. This material is based upon work supported by the Department of Energy National NuclearSecurity Administration under Award No. DE-NA0003856, the University of
Rochester, and the New York State Energy Research and Development Authority. We thank two referees for their valuable comments, which helped improve this paper. This report was prepared as an account
of work sponsored by an agency of the U.S. Government. Neither the U.S. Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal
liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned
rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement,
recommendation, or favoring by the U.S. Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the U.S. Government or any
agency thereof.
The data that support the findings of this study are available from the corresponding author upon reasonable request.
The governing Eqs. (24)–(29) rescaled in the conduction region are
where, for convenience, the streamwise momentum and energy equations have been rewritten in a conservative form.
The Physics of Inertial Fusion: Beam Plasma Interaction, Hydrodynamics, Hot Dense Matter
1st ed.
, International Series of Monographs on Physics (
Oxford University Press
), Vol.
O. A.
Nat. Phys.
S. E.
Phys. Rev. Lett.
, “
Propagation d'un front de flamme
La Technique Moderne
H. J.
Phys. Fluids B
, and
R. L.
Phys. Fluids
Phys. Rev. Lett.
A. R.
Phys. Plasmas
Phys. Rev. E
V. N.
R. L.
, and
C. P.
Phys. Plasmas
A. R.
, and
L. F.
Phys. Plasmas
Ya. B.
G. I.
V. B.
, and
G. M.
The Mathematical Theory of Combustion and Explosion
Consultants Bureau
New York
, and
Phys. Plasmas
, and
J. N.
Phys. Rev. Lett.
Phys. Plasmas
, and
Phys. Rev. Lett.
P. M.
I. V.
S. X.
J. R.
M. G.
D. H.
, and
D. D.
Phys. Rev. Lett.
P. M.
I. V.
J. R.
M. G.
E. G.
D. H.
, and
D. D.
Phys. Rev. Lett.
I. V.
A. B.
C. K.
P. M.
V. N.
, and
R. D.
Phys. Plasmas
S. I.
, in
Reviews of Plasma Physics
, edited by
M. A.
Consultants Bureau
New York
), Vol.
, p.
Jpn. J. Appl. Phys., Part 1
J. R.
Phys. Fluids
J. R.
Phys. Fluids
, and
J. R.
Phys. Fluids
D. A.
R. A.
Phys. Fluids
J. A.
J. R.
, and
Phys. Fluids
J. D.
Perturbation Methods in Applied Mathematics
, Applied Mathematical Sciences, Vol.
, edited by
J. E.
, and
Phys. Plasmas
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Frontiers | Analog Image Modeling for 3D Single Image Super Resolution and Pansharpening
• ^1Department of Mathematics, Applied Mathematics and Statistics, Case Western Reserve University, Cleveland, OH, United States
• ^2German Aerospace Center (DLR), Wessling, Germany
• ^3Department of Aerospace and Geodesy, Technical University of Munich, Munich, Germany
Image super-resolution is an image reconstruction technique which attempts to reconstruct a high resolution image from one or more under-sampled low-resolution images of the same scene. High
resolution images aid in analysis and inference in a multitude of digital imaging applications. However, due to limited accessibility to high-resolution imaging systems, a need arises for alternative
measures to obtain the desired results. We propose a three-dimensional single image model to improve image resolution by estimating the analog image intensity function. In recent literature, it has
been shown that image patches can be represented by a linear combination of appropriately chosen basis functions. We assume that the underlying analog image consists of smooth and edge components
that can be approximated using a reproducible kernel Hilbert space function and the Heaviside function, respectively. We also extend the proposed method to pansharpening, a technology to fuse a high
resolution panchromatic image with a low resolution multi-spectral image for a high resolution multi-spectral image. Various numerical results of the proposed formulation indicate competitive
performance when compared to some state-of-the-art algorithms.
1. Introduction
During image acquisition, we often loose resolution due to the limited density of the imaging sensors and the blurring of the acquisition lens. Image super resolution, a post process to increase
image resolution, is an efficient way to achieve a resolution beyond what the hardware offers. A detailed overview of this area is outlined in Farsiu et al. [1], Yang et al. [2], and Park et al. [3].
Current techniques of solving this ill-posed problem are often insufficient for many real world applications [4, 5]. We propose an analog modeling based three-dimensional (3D) color image super
resolution and pansharpening method.
In recent literature, image super-resolution (SR) methods typically fall into three categories: interpolation based methods, learning based methods, and constrained reconstruction methods.
Interpolation based methods [6, 7] such as nearest-neighbor, bilinear and bicubic interpolation use analytical priors to recover a continuous-time signal from the discrete pixels. They work very well
for smooth regions and are simple to compute. In dictionary learning based models, the missing high frequency detail is recovered from a sparse linear combination of dictionary atoms trained from
patches of the given low-resolution (LR) image [8, 9] or from an external database of patches [10–12]. Dictionary learning based methods explore the self-similarities of the image and are robust,
albeit rather slow training time of the dictionary atoms. Models based on constrained reconstruction use the underlying nature of the LR image to define prior constraints for the target HR image.
Sharper structures can be recovered using these models. Priors common to these techniques include total variation priors [13–15], gradient priors [16, 17] and priors based on edge statistics [18].
The proposed method falls in this category. Another category is deep neural network based approach that uses a large amount of training data to obtain super results [19–22].
Increased accessibility to satellite data has led to a wide range of applications in multispectral (MS) SR. These applications include target detection, scene classification, and change detection. A
multitude of algorithms have been presented in recent literature to address this task. However, the performance of many of these algorithms are data dependent with respect to existing quantitative
assessment measures. With the increasing advancements in technology, there have been varying modes of image acquisition systems. In remote sensing, satellites carry sensors that acquire images of a
scene in multiple channels across the electromagnetic spectrum. This gives rise to what is referred to as multispectral imaging. Due to high cost overheads in developing sensors that can faithfully
acquire scene data with high spectral and spatial resolution, the spatial resolution in the data acquisition process is often compromised. The low-spatial resolution MS images usually consist of
images across 4–8 spectral bands: red, green, blue, as well as near infrared bands (provide Landsat, IKONOS, etc. specifications).
Most satellites also acquire a high-spatial resolution image known as a panchromatic image. The technique of using panchromatic (PAN) to compensate for the reduction in spatial-resolution of the MS
images is referred to as pansharpening. The success of pansharpening relies on the extent one can explore the structural similarity between the PAN and the high-resolution (HR) MS image [23]. In
recent years, pansharpening methods fall under three main categories: component substitituion (CS), multiresolution analysis (MRA), and variational methods. Detailed surveys can be found in Loncan et
al. [24] and Vivone et al. [25].
The main assumption of CS methods is that the spatial and spectral information of the high spectral resolution image can be separated in some space [24]. Examples of CS methods include principal
component analysis (PCA) [26, 27], intensity-hue-saturation (IHS) methods [28–30], Brovey method [31], and Gram-Schmidt (GS) based methods [32]. These methods enhance the spatial resolution of the LR
MS image at the expense of some loss in the spectral detail. Compared to the CS methods, MRA methods have better temporal coherence between the PAN and the enhanced MS image. MRA methods yield
results with better spectral consistency at the expense of designing complex filters to mimic the reduction in the resolution by the sensors. Spatial details are injected into the MS data after a
multi-scale decomposition of the PAN [24]. Examples of MRA methods include high-pass filtering (HPF) [30], smooth filter-based intensity modulation (SFIM) [33], generalized Laplacian pyramid (GLP)
methods [34, 35], additive “á trous” wavelet transform (ATWT) methods [23, 36] and additive wavelet luminance proportional (AWLP) [37]. Variational methods are more robust compared to the CS and MRA
techniques, albeit rather slow in comparison [24]. A common tool used by these approaches is regularization. This is used to promote some structural similarity between the LR MS, HR MS, and PAN
images [24, 38, 39]. Among these methods include models based on the Bayesian framework and matrix factorization methods [24, 38, 39].
In this paper, we extend the key ideas in the SR model in Deng et al. [40] to 3D framework. We then test the performance of the model with single image super-resolution and pansharpening.
1.1. Related Work
In Deng et al. [40], a 2D image defined on a continuous domain [0, 1] × [0, 1] is assumed to have a density function that is the sum of two parts, reproducing kernel Hilbert space (RKHS) and
Heaviside functions to study single image super-resolution (SISR) whereby the problem is cast as an image intensity function estimation problem. They assume that the smooth components of the image
belong to a special Hilbert space called RKHS, and can be spanned by a basis based on an approximation by spline interpolation [41]. The Heaviside function is used to model the edges and
discontinuous information in the input image. Using a patch-based approach, the intensity information of the LR input image patches are defined on a coarser grid to estimate the coefficients of the
basis functions. To obtain the enhanced resolution result, they utilize the obtained coefficients and an approximation of the basis functions on a finer grid. Furthermore, to recover more high
frequency information the authors employ an iterative back projection method [42] after the HR image has been obtained. Color images were tested in Deng et al. [40], however the analog modeling was
done the luminance channel only after transforming the color space. This RKHS+Heaviside method is extended to pansharpening in Deng et al. [43]. In the pansharpening framework the different bands
share the same basis but with different coefficients patch by patch, and channel by channel. A panchromatic image is used to guide the model and it is assumed to be a linear combination of the
multiple bands.
1.2. Contribution
Using the same basis as in Deng et al. [40], we present a three-dimensional SR algorithm for true color images based on continuous/analog modeling. We also extend it to pansharpening. Instead of
estimating the basis coefficients patch by patch and band by band, we cluster patches and conduct computation cluster by cluster. Similarity of patches within one cluster leads to some natural
regularity. We jointly optimize all the coefficients for all the bands of the model rather than optimizing for each independent band. Furthermore, we use clustering techniques to improve the
structural coherence of the desired results in the optimization model. Spatially, we divide the images into small overlapping patches and cluster these patches classes using k-means clustering. To
improve the spectral coherence of the results, we cluster the image bands into groups based on correlation statistics and the perform the optimization on each of the groups obtained.
In this paper, we use the RKHS approximations to model the smooth components of the image whiles preserving the non-smooth components by the approximated Heaviside function. For the pansharpening
experiments, We also preprocess the data using fast state-of-the-art MRA techniques [37] to incorporate some similarity in the spectral channels of the fused image. Thus we develop a continuous 3D
modeling framework for multispectral pansharpening.
The experiments that follow show that these contributions yield better enhanced resolution results with faster convergence speeds at all scales.
The paper is organized as follows. In section 2, we review the regular functions used in decomposing the image. The proposed model is described in section 3 with numerical results outlined in section
2. Background Review
In this section, we review the mathematical functions used in the formulation of our model. Similar as in Deng et al. [40], we model image intensity function defined on a small image patch as a
linear combination of reproducing Kernel Hilbert function (RKHS) and approximated Heaviside function (AHF) which models the smooth and edge components, respectively.
2.1. Reproducible Kernel Hilbert Spaces
RKHS can be used to define feature spaces to help compare objects that have complex structures and are hard to distinguish in the original feature space. Wahba [41] proposed polynomial spline models
for smoothing problems. Our proposed model is based on this approach using the Taylor series expansion. We review these methods in the following subsections.
2.1.1. Signal Smoothing Using Splines
Let ${G}$ represent a family of functions on [0,1] with continuous derivatives up to order (m − 1),
By Taylor's theorem with remainder for $f\in {G}$, we may write
$f(t)=∑ν=0m−1tνν!f(ν)(0)︸f0(t)+∫01(t−u)+m−1(m−1)!f(m)(u)du︸f1(t), (1)$
for some u, where (x)[+] = x for x ≥ 0 and (x)[+] = 0 otherwise. Let
$ϕν(t)=tν-1(ν-1)!, 1≤ν≤m, (2)$
and ${{H}}_{0}\text{ }=\text{ }\mathrm{\text{span}}\text{ }{\left\{{\varphi }_{i}\right\}}_{1\le i\le m}$ endowed with norm ${‖}\varphi {{‖}}^{2}=\sum _{u =0}^{m-1}{\left[\left({D}^{u }\varphi \
right)\left(0\right)\right]}^{2}$, then ${D}^{\left(m\right)}\left({{H}}_{0}\right)=0$, where D^m denotes the mth derivative. It can be shown that a reproducing kernel exists for ${{H}}_{0}$ and is
defined as
$R0(s,t)=∑ν=1mϕν(s)ϕν(t), (3)$
therefore for any ${f}_{0}\in {H}$, we have ${f}_{0}\left(t\right)=\sum _{u =1}^{m}{d}_{u }{\varphi }_{u }\left(t\right).$
Let ${{B}}_{m}=\left\{f:f\in {{C}}^{m-1}\left[0,1\right],{f}^{u }\left(0\right)=0,u =0,1,\dots ,m-1\right\}$ and define
$Gm(t,u)=(t-u)+m-1(m-1)!. (4)$
The space
$H1={f:f∈ Bm, with f,f′,…,f(m−1) absolutely continuous,f(m)∈ L2}$
with norm $‖f{‖}^{2}={\int }_{0}^{1}\left({f}^{\left(m\right)}\left(t\right){\right)}^{2}dt$ can be shown to be an RKHS with reproducing kernel
$R1(x,t)=∫01Gm(t,u)Gm(x,u)du, (5)$
so that for any ${f}_{1}\in {{H}}_{1}$, we have ${f}_{1}\left(t\right)=\sum _{i=1}^{n}{c}_{i}{\xi }_{i},$ where ${\xi }_{i}\left(·\right)={R}^{1}\left({s}_{i},·\right).$ It follows from the
properties of the RKHS, we can construct a direct sum space ${{G}}_{m}={{H}}_{0}+{{H}}_{1}$ since
$∫01((Dmf0)(u))2du=0, ∑ν=0m−1(D(ν)f1(0))2=0. (6)$
The reproducing kernel for ${{G}}_{m}$ can be shown [41] to be
$R(s,t)=R0(s,t)+R1(s,t), (7)$
with norm
$‖f‖2=∑ν=0m-1[(Dνf)(0)]2+∫01(f(m)(t))2dt, f∈Gm, (8)$
therefore for any $f\in {{G}}_{m}$, we have f = f[0] + f[1] with ${f}_{0}\in {{H}}_{0},{f}_{1}\in {{H}}_{1}$ and
$f(t)=∑ν=1mdνϕν(t)+∑i=1nciξi, t∈[0,1]. (9)$
Let $d={\left({d}_{1},\dots {d}_{m}\right)}^{T}$ and $c={\left({c}_{1},\dots ,{c}_{n}\right)}^{T}$ be coefficient vectors, $f={\left(f\left({t}_{1}\right),...,f\left({t}_{p}\right)\right)}^{T}$
represent the discretization of f on discrete grids ${t}_{j},j=1,2,\cdots {n}_{t},T\in {ℝ}^{n×m},\Sigma \in {ℝ}^{n×n}$ with T[jν] = ϕ[ν](t[j]), Σ[ji] = ξ[i](t[j]), then f = Td + Σc. Given the
observation data
$g=f+η (10)$
where η denotes additive Gaussian noise. To find f from g, we need to estimate the coefficients d and c. Following Wahba [41], the coefficients d, c are estimated from the discrete measurements g by
$(c,d)=arg min{1n‖g−Td−Σc‖2+μ cTΣ c}, (11)$
where the term μc^TΣc is a non-smoothness penalty. Closed form solutions for c and d can be obtained. The parameter m controls the total degree of the polynomials used to define T. For example, when
m = 3 the basis functions are given as ϕ[1](x) = 1, ϕ[2](x) = x and ${\varphi }_{3}\left(x\right)={x}^{2}/2$.
2.1.2. Image Smoothing Using Splines
Extending the one-dimensional spline model to two-dimensional data, we consider a similar discretization and set up a model similar to that defined in (11). Let $f={\left(f\left({t}_{1}\right),\dots
,f\left({t}_{{n}_{t}}\right)\right)}^{T}$, t[i] ∈ [0, 1] × [0, 1] be a discretization for an observed noisy image. From Wahba [41], an estimate of f can be obtained from the additive noise data model
in (10) for images by minimizing the following
$min{1n‖g−f‖22+μJm(f)} (12)$
where m defines the order of the partial derivatives in ${{L}}_{2}\left(\left[0,1\right]×\left[0,1\right]\right)$, and with the penalty function defined by
$Jm(f)=∑ν=0m∫01∫01(mν)(∂mf∂xν∂ym−ν)2dxdy. (13)$
The null space of J[m] is the $M=\left(\begin{array}{c}2+m-1\\ 2\end{array}\right)$-dimensional space spanned by polynomials in two variables with total degree at most m − 1. In this paper, and we
choose m = 3 so that M = 6 and the null space of J[m] is spanned by monomials ϕ[1], ϕ[2], …, ϕ[6] given by
$ϕ1(x,y)=1,ϕ2(x,y)=x,ϕ3(x,y)=y,ϕ4(x,y)=xy,ϕ5(x,y)=x2,and ϕ6(x,y)=y2.$
Duchon [44] proved that a unique minimizer f[μ] exists for (12) with representation
$fμ(t)=∑ν=1Mdνϕν(t)+∑i=1nciEm(t,ti), t∈[0,1]×[0,1], (14)$
where E[m] is defined as
$Em(s,t)=Em(|s-t|)=θm,d|s-t|2m-dln|s-t|, (15)$
and ${\theta }_{m,d}=\frac{{\left(-1\right)}^{d/2+m+1}}{{2}^{2m-1}{\pi }^{d/2}\left(m-1\right)!\left(m-d/2\right)!}$.
Based on the work of Duchon [44] and Meinguet [45] we can rewrite (12) to find minimizers c and d by
$(c,d)=arg min{‖g−Td−Kc‖2+μ cTKc}, (16)$
where $T\in {ℝ}^{{n}_{t}×M}$, with T[i, ν] = ϕ[ν](t[i]) and $K\in {ℝ}^{{n}_{t}×{n}_{t}}$ with K[i, j] = E[m](t[i], t[j]). E[m](s, t) is the two-dimensional equivalent of ξ[i](t) in the
one-dimensional case. After obtaining the coefficients, we compute f using the relationship f = Td + Kc.
2.2. Approximated Heaviside Function
The one-dimensional Heaviside step function is defined as
$ϕ(x)={0x<0,1x≥0 (17)$
Due to the singularity at x = 0, we approximate ϕ by,
$ψ(x)=12+1πarctan(xε), ε∈ℝ (18)$
We refer to this as the approximated Heaviside function (AHF), an approximation to ϕ(x) as ε → 0. The variable ε controls the smoothness of the approximation [40]. A two-dimensional variation of the
AHF is given by
$Ψ(x,y)=ψ((cos θsin θ)·(xy)+c) (19)$
with variables θ and c determining the rotations and locations of the edges as shown in Figure 1, we show the 2D and 3D surface view of the AHF at two different pairs of θ and c. Kainen et al. [46]
proved that a function $f\in {{L}}_{2}\left(\left[0,1\right]×\left[0,1\right]\right)$ can be approximately represented by the weighted linear combination of approximated Heaviside functions:
$f(x,y)=∑j=1nθwjψ((cos θjsin θj)·(xy)+cj). (20)$
FIGURE 1
Figure 1. Illustrating the surfaces corresponding to the approximated Heaviside functions for varying pairs of θ and c.
We define $\text{Ψ}\in {ℝ}^{{n}_{t}×{n}_{\theta }}$ with ${\Psi }_{i,j}=\psi \left(\left(\begin{array}{c}\mathrm{cos}{\theta }_{j}\\ \mathrm{sin}{\theta }_{j}\end{array}\right)·\left(\begin{array}{c}
{x}_{i}\\ {y}_{i}\end{array}\right)+{c}_{j}\right),\text{\hspace{0.17em}\hspace{0.17em}}{t}_{i}=\left({x}_{i},{y}_{i}\right),\text{and\hspace{0.17em}}{n}_{\theta }$ is the number of rotations
considered. Suppose f, g are the vectorized high and low resolution discretizations of f, respectively. The vectorized high resolution image $f\in {ℝ}^{{n}_{1}{n}_{2}}$ can be obtained by ${f}_{i}=f\
left({t}_{i}^{h}\right)$ with ${t}_{i}^{h}=\left({x}_{i}^{h},{y}_{i}^{h}\right)$, ${x}_{i}^{h}\in \left\{0,\frac{1}{{n}_{1}-1},\frac{2}{{n}_{2}-1},\dots ,1\right\}$, ${y}_{i}^{h}\in \left\{0,\frac{1}
{{n}_{2}-1},\frac{2}{{n}_{2}-1},\dots ,1\right\}$ on a finer grid. Similarly, vectorized the low-resolution discretization $g\in {ℝ}^{{m}_{1}{m}_{2}}$ can be obtained by ${g}_{i}=f\left({t}_{i}^{\ell
}\right)$ with ${t}_{i}^{\ell }=\left({x}_{i}^{\ell },{y}_{i}^{\ell }\right)$, ${x}_{i}^{\ell }\in \left\{0,\frac{1}{{m}_{1}-1},\frac{2}{{m}_{1}-1},\dots ,1\right\}$, and ${y}_{i}^{\ell }\in \left\
{0,\frac{1}{{m}_{2}-1},\frac{2}{{m}_{2}-1},\dots ,1\right\}$ on a coarse grid. Now, assuming that the underlying analog image intensity function is approximated by the sum of a RKHS function and the
variations of AHF with ${T}^{\ell }\in {ℝ}^{{m}_{1}{m}_{2}×M}$, ${K}^{\ell }\in {ℝ}^{{m}_{1}{m}_{2}×{m}_{1}{m}_{2}}$, and ${\text{Ψ}}^{\ell }\in {ℝ}^{{m}_{1}{m}_{2}×{n}_{\theta }}$, we have
$g=Tℓd+Kℓc+Ψℓe. (21)$
Thus, given the low resolution input g, the coefficient vectors $c\in {ℝ}^{{m}_{1}{m}_{2}}$, d∈ℝ^M and $e\in {ℝ}^{{n}_{\theta }}$ for the residual, smooth and edge components are obtained by solving
the following minimization problem:
$minc,d,e{‖g−SB(Thd+Khc+Ψhe)‖2+μcTKℓc+γ‖e‖1}, (22)$
where B is the identity matrix in the absence of blur, S is the downsampling operator, and the superscripts h and ℓ refers to a fine and course scale matrices. The smooth components of the image are
modeled by the RKHS approach, while AHF caters for the edges. Since the dictionary Ψ is pretty exhaustive, i.e., accounting for multiple edge orientations, it is reasonable to use the ||·||[1] to
enforces the sparsity of as all the orientations may not be present for any given image. Once the coefficients have been obtained, we have
$f=Thd+Khc+Ψhe, (23)$
where ${T}^{h}\in {ℝ}^{{n}_{1}{n}_{2}×M}$, ${K}^{\ell }\in {ℝ}^{{n}_{1}{n}_{2}×{m}_{1}{m}_{2}}$, and ${\text{Ψ}}^{\ell }\in {ℝ}^{{n}_{1}{n}_{2}×{n}_{\theta }}$. Next, we discuss our proposed model
and examine some numerical experiments in the chapters that follow.
3. Proposed Models
In this section, we present our proposed models for SISR and pansharpening based on the functions defined in section 2. We propose a 3D patch-based approach that infers the HR patches from LR patches
that has been grouped into classes based on their structural similarity. As a result, we impose similarity constraints within the classes so that the coefficients for neighboring patches in a group
do not differ substantially. In addition, we use sparse constrained optimization techniques that simplify the minimization of the resulting energy functional by solving a series of subproblems, each
with a closed form solution. Two algorithms will be discussed. The first is a 3D SR model for true color images and MS images with at most four bands. The second algorithm is an extension of our SISR
model to pansharpening for MS images with at least four spectral bands. This is a two-step hybrid approach that incorporates CS and MRA methods into the first algorithm to obtain an enhanced
resolution result.
3.1. Single Image Super-Resolution
Given a LR image $Y\in {ℝ}^{{m}_{1}×{m}_{2}×\lambda }$ we estimate the target HR image $\stackrel{^}{X}\in {ℝ}^{{n}_{1}×{n}_{2}×\lambda }$ with n[i] = ωm[i], where ω is a scale factor and λ is the
number of bands as follows. We begin by decomposing the LR image into a set of overlapping patches ${{P}}_{Y}={\left\{{Y}_{{p}_{i}}\right\}}_{i=1}^{{N}_{p}}$, ${Y}_{{p}_{i}}\in {ℝ}^{{m}_{p}×{m}_{p}×\
lambda }$. The size of the square patches, and the overlap between adjacent patches depend on the dimensions of the input image. We consider the use of overlapping patches to improve local
consistency of the estimated image across the regions of overlap. The estimated image $\stackrel{^}{X}$ will be represented by the same number of patches, N[p]. Next, for a suitably chosen k, we use
the k-means algorithm with careful seeding [47] to cluster the patches into k classes ${{P}}_{Y}^{1},\dots {{P}}_{Y}^{k}$, as shown in Figure 2. In this paper, the value of k is determined
empirically to obtain the best result. Clustering is done so that each high resolution patch generated preserves some relationships with its neighboring patches.
FIGURE 2
Following the clustering, we consider each set of patches ${{P}}_{Y}^{i}={\left\{{Y}_{{p}_{j}}\right\}}_{j\in {I}_{i}}$, for some index set I[i] with 1 ≤ i ≤ k. Let ${Y}_{\left(\lambda ,{I}_{i}\
right)}\in {ℝ}^{{m}_{p}{m}_{p}×\lambda {N}_{{I}_{i}}}$ and ${\stackrel{^}{X}}_{\left(\lambda ,{I}_{i}\right)}\in {ℝ}^{{n}_{p}{n}_{p}×\lambda {N}_{{I}_{i}}}$ represent the images corresponding to the
set of LR and HR patches ${{P}}_{Y}^{i}$ and ${{P}}_{\stackrel{^}{X}}^{i}$ defined by
$Y(λ,Ii)=[y1,p1,…,yλ,p1,…,y1,pNIi,…,yλ,pNIi]X^(λ,Ii)=[x^1,p1,…,x^λ,p1,…,x^1,pNIi,…,x^λ,pNIi] (24)$
where n[p] = ωm[p] for some scale factor ω ∈ ℕ, y[i,p[j]] and ${\stackrel{^}{x}}_{i,{p}_{j}}$ represents the vectorization of the i-th band of the j-th patch, with 1 ≤ i ≤ λ and j ∈ I[i]. Here, N[I[i
]] represents the total number of patches in the set ${{P}}_{Y}^{i}$. Using the RKHS and AHF functions as in Deng et al. [40] we can find an embedding for the high-dimensional data that is also
structure preserving. We further assume that the HR and LR image patches have similar local geometry and are related by the equations
$Y(λ,Ii)=TℓD+KℓC+ΨℓE,X^(λ,Ii)=ThD+KhC+ΨhE, (25)$
where $D\text{ }=\text{ }\left[{d}_{1},\dots {d}_{\lambda {N}_{{I}_{i}}}\right]\in {ℝ}^{M×\lambda {N}_{{I}_{i}}},C=\left[{c}_{1},\dots {c}_{\lambda {N}_{{I}_{i}}}\right]\in {ℝ}^{{m}_{1}{m}_{2}×\
lambda {N}_{{I}_{i}}}$ and $E=\left[{e}_{1},\dots {e}_{\lambda {N}_{{I}_{i}}}\right]\in {ℝ}^{{n}_{\theta }×\lambda {N}_{{I}_{i}}}$ are coefficient matrices to be determined. The matrices T, K and Ψ
represent the evaluations of the smooth, residual and edge components of the image intensity function on discrete grids. The superscript ℓ and h denote the coarse and fine grids and correspond to
lower and higher resolution, respectively. We obtain the fine and coarse scale matrices T and K following the discretization models outlined in section 2.1 Similarly, we follow the discretization in
section 2.2 to generate Ψ^ℓ and Ψ^h. The coefficient matrices are obtained by solving the following minimization problem for each patch indexed by i:
$minD,C,E{12‖TℓD+KℓC+ΨℓE−Y(λ,Ii)‖F2+μ12∑jkwjk‖dj−dk‖22+μ22tr(CTKℓC)+μ3‖E‖1,1} (26)$
with tr(·) as the trace and adaptive weights
$wjk=exp(-‖dj-dk‖ 22σ), σ>0, (27)$
where D = [d[1], …d[λN[I[i]]]] and μ[1], μ[2], μ[3] ≥ 0. Note that $\text{exp}\left(-\frac{{‖}{d}_{j}-{d}_{k}{{‖}}_{2}^{2}}{\sigma }\right){‖}{d}_{j}-{d}_{k}{{‖}}_{2}^{2}$ is a unimodal function of $
{‖}{d}_{j}-{d}_{k}{{‖}}_{2}^{2}$ that is maximized at 1 and minimized at zero and ∞. To simplify the computations, the adaptive weights are computed using the coefficients from a previous iteration.
The first term of (26) is the data fidelity. There are three regularization terms in our proposed model. We assume that coefficients d[i] are likely to vary smoothly within the same class of images.
When d[i] and d[j] fall into the same class, w[ij] tends to be larger, forcing the next iterate of d[i] and d[j] to be close as well. The second regularization term is a structure guided regularity
following from (16) to guarantee that the coefficients of the residual components do not grow too large. Finally, we impose sparsity constraints using the ||·||[1, 1] norm [48] defined by
$‖A‖1,1=∑sS‖as‖1,A=[a1,…,aS], (28)$
since we assume that edges are sparse in the image patches. The resulting minimization can be solved using splitting techniques to obtain closed form solutions for the coefficients. We solve (26)
iteratively using the alternating direction method of multipliers (ADMM) algorithm [49, 50].
Due to the non-differentiability of the ||·||[1, 1] norm, we introduce a new variable U and solve the equality-constrained optimization problem
$minD,C,E,U{12‖TℓD+KℓC+ΨℓE−Y(λ,Ii)‖F2+μ12∑jkwjk‖dj−dk‖22+μ22tr(CTKℓC)+μ3‖U‖1,1}subject to U=E. (29)$
By introducing the augmented Lagrangian and completing the squares, we change the constrained optimization problem (29) into the following non-constrained optimization where V is Lagrangian
$minD,C,E,U{12‖TℓD+KℓC+ΨℓE−Y(λ,Ii)‖F2 +μ12∑jkwjk‖dj−dk‖22+μ22tr(CTKℓC) +μ3‖U‖1,1+γ2‖U−E+Vγ‖F2} (30)$
with step length γ ≥ 0. ADMM reduces (30) to solving the following separable subproblems:
• D subproblem:
$minD{12‖TℓD+KℓC+ΨℓE−Y(λ,Ii)‖F2 +μ12∑jkwjk‖dj−dk‖22}.$
Given that D = [d[1], …d[λN[I[i]]]], C = [c[1], …c[λN[I[i]]]], E = [e[1], …e[λN[I[i]]]] and we can solve for each column of D separately
where y is the corresponding column in Y[(λ,I[i])] matching the column index of d[j]. This gives a solution
$dj=(TℓTTℓ+μ1∑jkwjkI)−1(TℓTTℓy−TℓTKℓcj −TℓTΨℓej+μ1∑jkwjkdk).$
where ${T}^{{\ell }_{T}}$ represents the transpose of T^ℓ.
• C subproblem:
From which we get
$C=(KℓTKℓ+μ2Kℓ)−1(KℓTY(λ,Ii) −KℓTTℓD−KℓTΨℓE).$
• E subproblem:
Similarly, this gives the solution
$E=(ΨℓTΨℓ+γ1I)−1(ΨℓTY(λ,Ii) −ΨℓTTℓD −ΨℓTKℓC+γ(U+V)).$
• U subproblem:
A solution to this problem arising from the group sparsity constraint can be obtained by applying ℓ[1] shrinkage to the rows of U, i.e.,
$ujT=shrink (ujT+vjT/γ,μ3/γ),$
$shrink(x,γ)={xi−ν,if xi>γ,xi+ν,if xi<−γ,0,otherwise$
for any x ∈ ℝ^n and γ > 0.
The update for V is obtained by
The estimated set of patches ${{P}}_{\stackrel{^}{X}}$ can be generated using (25) once the coefficients have been obtained. After running through the procedure for all the k sets, we consolidate the
patches to form the enhanced resolution result $\stackrel{^}{X}$. The proposed algorithm for SISR is summarized in Algorithm 1. Step 3 contains most of the key components of the algorithm so we
provide more details. An outer layer loop is used to pick up residuals and put them back into the super resolution algorithm to further enhance the results. τ is the total number of outer layer
iterations, B and S are the blurring and downsampling matrices, respectively. In the absence of blur, B becomes the identity matrix. Starting from one cluster of low resolution image patches, Step 3
(b) solves the minimization problem (30) to obtain the coefficients C, D, E, from which one is able to assemble a higher resolution image as shown in Step 3(c). From this higher resolution image, we
can create a lower resolution image by applying SB operator on it. Step 3(d) calculates the error between the actual low resolution and the simulated one. If the higher resolution image is close to
the ground truth, we expect the error to be small. Otherwise, we feeding this difference into the minimization problem (30) can further recover more details. These details are added to the previously
recovered higher resolution image to achieve a better image [40].
3.2. Pansharpening
Given a LR MS image Y and a high spatial resolution PAN image (P) matching the same scene, our goal is to estimate a HR MS image $\stackrel{^}{X}$ with the spatial dimensions of the PAN. We propose a
two-stage approach to solve the pansharpening problem. To begin with, we obtain a preliminary estimate for the target image $\stackrel{~}{Y}$ using CS or MRA, two fast and simple pansharpening
techniques. $\stackrel{~}{Y}$ has the same size as $\stackrel{^}{X}$ but the result $\stackrel{~}{Y}$ obtained using these methods suffer from spectral and spatial distortion drawbacks. The second
step in our proposed two-stage approach provides a way to reduce the drawbacks. Next, we feed Algorithm 1 with input $\stackrel{~}{Y}$ and proceed with the resulting optimization model to generate $\
stackrel{^}{X}$. Considering the image $\stackrel{~}{Y}$, the modified optimization model becomes
$minD,C,E {12‖ThD+KhC+ΨhE−Y˜(λ,Ii)‖F2+μ12∑jkwjk‖dj−dk‖22μ22tr(CTKℓC)+μ3‖E‖1,1}, (31)$
where ${\stackrel{~}{Y}}_{\left(\lambda ,{I}_{i}\right)}=\left[{\stackrel{~}{y}}_{1,{p}_{1}},\dots ,{\stackrel{~}{y}}_{\lambda ,{p}_{1}},\dots ,{\stackrel{~}{y}}_{1,{p}_{{N}_{{I}_{i}}}},\dots ,{\
stackrel{~}{y}}_{\lambda ,{p}_{{N}_{{I}_{i}}}}\right]\in {ℝ}^{{n}_{p}{n}_{p}×\lambda {N}_{{I}_{i}}}$ for the set of patches ${{P}}_{\stackrel{~}{Y}}^{i}$, 1 ≤ i ≤ k. K-means clustering is used to
decompose the number of overlapping patches into k classes, so that pixels corresponding to similar kinds of objects in the image will be collectively estimated in each step. The optimal value of k
for a given image was determined using the silhouette method [51] which measures how similar a point is to its own cluster compared to other clusters. The adaptive weights are defined the same way as
in (27). Applying ADMM to (31) we can solve the separable subproblems that ensue for the coefficients D, C and E as in the single image model. We summarize the procedure in Algorithm 2.
4. Experimental Results
4.1. Simulation Protocol
To begin with the simulation procedure, we test the performance of the proposed algorithm by generating the reduced resolution image from a given image for some chosen downscaling factors. We then
compute the enhanced resolution result using the proposed algorithm and compare its performance with other methods. The experiments are implemented using MATLAB(R2018a).
For the SISR experiments we compare our three-dimensional single image super-resolution algorithm to bicubic interpolation and the RKHS algorithm by Deng et al. [40]. The authors in Deng et al. [40]
implement their SR algorithm following an independent band procedure. For color images (RGB), they transform the image to the “YCbCr” color space and apply model in Deng et al. [40] to the luma
component (Y) only since humans are more sensitive to luminance changes. The “Cb” and “Cr” components are the blue-difference and the red-difference chroma components, respectively. This way, the
time complexity of the algorithm is reduced, as compared to an optimization based methods using all the channels at the same time. To obtain the desired result, the chroma components (Cb, Cr) are
upscaled by bicubic interpolation. The components are then transformed back to the RGB color space for further analysis. In our approach, we apply Algorithm 1 to the three-dimensional color images
directly, without undertaking any transformations.
We undertake the pansharpening experiments using semi-synthetic test data. This may be attributed to the fact that existing sensors cannot readily provide images at the increased spatial resolution
that is desired. A generally acceptable procedure attributed to Ranchin and Wald [23] for operating at reduced resolution is as follows (Illustration in Figure 3):
• The enhanced result should be as identical as possible to the original multispectral image once it has been degraded.
• The enhanced result should be identical to the corresponding high resolution image that a capable sensor would observe at the increased resolution.
• Each band of the enhance result should be as identical as possible to the corresponding bands of the image that would have been observed by a sensor with high resolution.
FIGURE 3
We apply Algorithm 2 to the observed multispectral data and compare it with existing state-of-the-art approaches.
4.2. Parameter Selection
Parameter selection is essential to obtain good results. However, to show the stability of our proposed method, we fix the parameters for all the experiments. Fine tuning of the parameters might lead
to slightly better results. Parameter choices for the optimization of our proposed algorithms are outlined as follows:
• Scale factor (ω): In the experiments shown above, we set ω = 2 for the SISR problem, and set ω = 4 for the pansharpening problem. Quantitative measures obtained suggest that for ω > 4 both problems
yield unsatisfactory results as the spatial and spectral measures are less competitive. However, the choice of ω depends on the dimensions of the observed data. For large ω, we can select larger
patch sizes to improve upon the target.
• Patch size (m[p]): Patch sizes can vary according to the size of the given image. We use a default patch size, m[p] × m[p] × λ in our experiments. Reasonable quantitative measures were obtained for
6 ≤ m[p] ≤ 8 for the two problems considered. Outside the given bounds, the algorithms either take time to converge or yield unsatisfactory results.
• Overlap size: The limits of the overlapping region range from 2 to 4 are limited by the patch size. We set the overlap size to 2.
• Number of classes (k): In our experiments, the value of k was decided based on the solution that gave the best result from the silhouette method [51], i.e., by combing through predetermined values.
Future work will involve investigating whether it is possible to automate this step.
• Construction of approximation matrices T, K, and Ψ: We generate the matrices based on the procedure outlined in section 3, with M = 6, ε = 1e^−3, and set 20 evenly distributed angles on [0, 2π] in
computing the fine and coarse scale matrices for T, K and Ψ, respectively.
• Regularization penalties μ[1], μ[2], μ[3] and γ: Without knowledge of optimal values for the penalty, we sweep through a range of values to determine the optimal result. For the experiments
considered, μ[1], μ[2], μ[3], γ ∈ {10^−8, ⋯, 10^2} and choose the one that leads to the best results evaluated by quantitative and qualitative measures. The step length γ for updating of V
influences the convergence of the algorithm greatly.
• Preprocessing for Algorithm 2: Our proposed pansharpening method provides a flexible way to improve the results for both CS and MRA algorithms. In the experiments shown above, we preprocess the
images using the GS algorithm. Future work will detail which preprocessing step works best for multispectral images.
4.3. Quantitative Criteria
To measure the similarity of the enhanced resolution image ($\stackrel{^}{X}$) to the reference image (X), we compute the following image quality metrics:
• Correlation coefficient (CC): The correlation coefficient measures the similarity of spectral features between the reference and pansharpened image. It is defined as
where X[i] and ${\stackrel{^}{X}}_{i}$ are the matrices of the i-th band of X and $\stackrel{^}{X}$, with μ[X[i]] and ${\mu }_{{\stackrel{^}{X}}_{i}}$ as their respective mean gray values. The
standard deviations for ${\stackrel{^}{X}}_{i}$ and X[i] are ${\sigma }_{{\stackrel{^}{X}}_{i}}$ and σ[X[i]], respectively.
• Root mean square error (RMSE):
• Peak signal-to-noise ratio (PSNR): It is an expression of the ratio between the maximum possible value of the signal and the distorting noise that affects the quality of the representation.
• Structural Similarity (SSIM): The SSIM index is computed by examining various windows of the reference and target image. It is a full reference metric designed to improve upon the traditional
methods such as PNSR and MSE.
• Spectral angle mapper (SAM): This measure denotes the absolute value of the angle between the reference and estimated spectral vectors.
Let ${\stackrel{^}{X}}_{\left\{i\right\}}={\left({\stackrel{^}{x}}_{i,1},\dots ,{\stackrel{^}{x}}_{i,\lambda }\right)}^{T},1\le i\le {n}_{1}{n}_{2}$ be a vector collecting the intensity at all
spectral bands at pixel i,
• Erreur relative globale adimensionnelle de synthése (ERGAS): The relative dimensionless global error in the synthesis reflects the overall quality of the pansharpened image.
$ERGAS(X^)=100d1λ∑i=1λ(RMSE(X^i)μX^i)2, d=downsampling factor.$
• Q index: It defines a universal measure that combines the loss of correlation, luminance, and contrast distortion of an image and is defined by
where ${\sigma }_{\left({X}_{i},{\stackrel{^}{X}}_{i}\right)}$ is the covariance between X[i] and ${\stackrel{^}{X}}_{i}$.
4.4. Results
In this section, we compare the visual results and quantitative measures of the proposed approaches with some existing super-resolution algorithms. The results of Algorithm 1 will be compared to
bicubic interpolation and the RKHS model by Deng et al. [40]. Results of Algorithm 2 are compared with CS methods such as PCA, IHS, Brovey, GS and Indusion, as well as MRA methods such as HPF, SFIM,
ATWT, AWLP, GLP techniques, and the Deng et al. pansharpening model [43]. We begin with numerical tests for true color images and consider patches of size 8 × 8 × λ. The size of the overlap across
the vertical and horizontal dimensions is 2.
In Figure 4, we show one example of decomposing an original image into three components. One can see that the edge information is separated successfully from the smooth component and the residual. In
Figure 5, we compare 3D super resolution result of the proposed model with bicubic interpolation and RKHS [40] on a baboon image. We can see that the proposed method leads to a color image with
highest visual quality. The bicubic interpolation result tends to be blurry. RKHS [40] result is better but not as good as the proposed. In Table 1, we list some quantitative metric comparison for
more testing images. It is observed that the proposed 3D single image super resolution method (detailed in Algorithm 1) is consistently the best among the three approaches.
FIGURE 4
Figure 4. Decomposition of components for the grayscale test image. From top to bottom and left to right: the original, T^hD, K^hC, Ψ^hE.
FIGURE 5
Figure 5. Results for baboon using a scale factor ω = 2. From top to bottom and left to right: ground truth, bicubic interpolation, RKHS [40] and the proposed.
TABLE 1
For pansharpening, we compare with 10 other methods in the literature using five different quantitative metrics (see Table 2). It is observed that the proposed pansharpening method (detailed in
Algorithm 2) is mostly the best and occasionally the second among all. The performance is consistently superior.
TABLE 2
Note that we did not conduct comparison with deep neural network based methods. The proposed approach is based on single image modeling while the deep neural network approach relies on a lot of
external data. We don't think it is a fair comparison.
5. Conclusions
In this paper, we have proposed a technique for single image SR by modeling the image as a linear combination of regular functions in tandem with sparse coding. We show that the proposed scheme is an
improvement upon similar existing approaches as it outperforms these algorithms. Besides this advantage, the proposed methods can also benefit image decomposition.
In future work, we will apply the model to multispectral and hyperspectral image SR where sparse coding can be useful due to the redundant nature of the input.
Data Availability Statement
Some of the datasets for this study can be found in http://openremotesensing.net.
Author Contributions
WG contributes on providing the main ideas including the mathematical models and algorithms proposed in the paper. RL was a former Ph.D. student of WG. Under WG's guidance, RL did computer
programming to implement the ideas and conducted numerical experiments to test the effective of the proposed method compared to the state of the art. XZ and CG are domain scientists who help us
evaluate the numerical results and provide comments.
WG was supported by National Science Foundation (DMS-1521582).
Conflict of Interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
1. Farsiu S, Robinson D, Elad M, Milanfar P. Advances and challenges in super-resolution. Int J Imaging Syst Technol. (2004) 14:47–57. doi: 10.1002/ima.20007
2. Yang C-Y, Ma C, Yang M-H. Single-image super-resolution: a benchmark. In: European Conference on Computer Vision. Cham: Springer (2014). p. 372–86. doi: 10.1007/978-3-319-10593-2_25
3. Park SC, Park MK, Kang MG. Super-resolution image reconstruction: a technical overview. IEEE Signal Process Mag. (2003) 20:21–36. doi: 10.1109/MSP.2003.1203207
4. Baker S, Kanade T. Limits on super-resolution and how to break them. IEEE Trans Pattern Anal Mach Intell. (2002) 24:1167–83. doi: 10.1109/TPAMI.2002.1033210
5. Lin Z, Shum H-Y. Fundamental limits of reconstruction-based superresolution algorithms under local translation. IEEE Trans Pattern Anal Mach Intell. (2004) 26:83–97. doi: 10.1109/
6. Thévenaz P, Blu T, Unser M. Interpolation revisited [medical images application]. IEEE Trans Pattern Anal Mach Intell. (2000) 19:739–758. doi: 10.1109/42.875199
7. Keys R. Cubic convolution interpolation for digital image processing. IEEE Trans Acoust Speech Signal Process. (1981) 29:1153–60. doi: 10.1109/TASSP.1981.1163711
8. Glasner D, Bagon S, Irani M. Super-resolution from a single image. In: IEEE 12th International Conference on Computer Vision, 2009. IEEE (2009). p. 349–56. doi: 10.1109/ICCV.2009.5459271
9. Freedman G, Fattal R. Image and video upscaling from local selfexamples. ACM Trans Graph. (2011) 30:12. doi: 10.1145/2010324.1964943
10. Gao X, Zhang K, Tao D, Li X. Joint learning for single-image super-resolution via a coupled constraint. IEEE Trans Image Process. (2012) 21:469–80. doi: 10.1109/TIP.2011.2161482
11. Wang Z, Yang Y, Wang Z, Chang S, Yang J, Huang TS. Learning super-resolution jointly from external and internal examples. IEEE Trans Image Process. (2015) 24:4359–71. doi: 10.1109/
12. Yang J, Wright J, Huang TS, Ma Y. Image super-resolution via sparse representation. IEEE Trans Image Process. (2010) 19:2861–73. doi: 10.1109/TIP.2010.2050625
13. Bioucas-Dias JM, Figueiredo MA. A new twist: two-step iterative shrinkage/thresholding algorithms for image restoration. IEEE Trans Image Process. (2007) 16:2992–3004. doi: 10.1109/
14. Rudin LI, Osher S. Total variation based image restoration with free local constraints. In: IEEE International Conference on Image Processing, 1994. ICIP-94. IEEE (1994). p. 31–5.
15. Getreuer P. Contour stencils: Total variation along curves for adaptive image interpolation. SIAM J Imaging Sci. (2011) 4:954–79. doi: 10.1137/100802785
16. Kim KI, Kwon Y. Single-image super-resolution using sparse regression and natural image prior. IEEE Trans Pattern Anal Mach Intell. (2010) 32:1127–33. doi: 10.1109/TPAMI.2010.25
17. Sun J, Xu Z, Shum H-Y. Image super-resolution using gradient profile prior. In: IEEE Conference on Computer Vision and Pattern Recognition, 2008. CVPR. IEEE (2008). p. 1–8.
18. Fattal R. Image upsampling via imposed edge statistics. ACM Trans Graph. (2007) 26:95. doi: 10.1145/1276377.1276496
19. Lium B, Son S, Kim H, Nah S, Mu K. Enhanced deep residual networks for single image super-resolution. In: The IEEE Conference on Computer Vision and Pattern Recognition (CVPR) Workshops. (2017).
20. Yang W, Zhang X, Tian Y, Wang W, Xue J, Liao Q. Deep learning for single image super-resolution: a brief review. IEEE Trans Multimedia. (2019) 21:3106–21. doi: 10.1109/TMM.2019.2919431
21. Wang Y, Wang L, Wang H, Li P. End-to-end image super-resolution via deep and shallow convolutional networks. IEEE Access. (2019) 7:31959–70. doi: 10.1109/ACCESS.2019.2903582
22. Li Z, Yang J, Liu Z, Yang X, Jeon G, Wu W. Feedback network for image super-resolution. In: IEEE Conference on Computer Vision and Pattern Recognition CVPR. Long Beach, CA: IEEE (2019). doi:
23. Ranchin T, Wald L. Fusion of high spatial and spectral resolution images: the ARSIS concept and its implementation. Photogram Eng Remote Sens. (2000) 66:49–61.
24. Loncan L, de Almeida LB, Bioucas-Dias JM, Briottet X, Chanussot J, Dobigeon N, et al. Hyperspectral pansharpening: a review. IEEE Geosci Remote Sens Mag. (2015) 3:27–46. doi: 10.1109/
25. Vivone G, Alparone L, Chanussot J, Dalla Mura M, Garzelli A, Licciardi GA, et al. A critical comparison among pansharpening algorithms. IEEE Trans Geosci Remote Sens. (2015) 53:2565–86. doi:
26. Shettigara V. A generalized component substitution technique for spatial enhancement of multispectral images using a higher resolution data set. Photogramm Eng Remote Sens. (1992) 58:561–7.
27. Shah VP, Younan NH, King RL. An efficient pan-sharpening method via a combined adaptive PCA approach and contourlets. IEEE Trans Geosci Remote Sens. (2008) 46:1323–35. doi: 10.1109/
28. Tu T-M, Su S-C, Shyu H-C, Huang PS. A new look at IHS-like image fusion methods. Inform Fusion. (2001) 2:177–86. doi: 10.1016/S1566-2535(01)00036-7
29. Carper JW, Lillesand TM, Kiefer RW. The use of intensityhue- saturation transformations for merging spot panchromatic and multispectral image data. Photogramm Eng Remote Sens. (1990) 56:459–67.
30. Chavez P, Sides SC, Anderson JA. Comparison of three different methods to merge multiresolution and multispectral data: landsat TM and SPOT panchromatic. Photogramm Eng Remote Sens. (1991) 57
31. Gillespie AR, Kahle AB, Walker RE. Color enhancement of highly correlated images. I. Decorrelation and HSI contrast stretches. Remote Sens Environ. (1986) 20:209–35. doi: 10.1016/0034-4257(86)
32. Laben CA, Brower BV. Process for Enhancing the Spatial Resolution of Multispectral Imagery Using Pan-Sharpening. US Patent 6011875 (2000).
33. Liu J Smoothing filter-based intensity modulation: a spectral preserve image fusion technique for improving spatial details. Int J Remote Sens. (2000) 21:3461–72. doi: 10.1080/014311600750037499
34. Aiazzi B, Alparone L, Baronti S, Garzelli A, Selva M. An MTF-based spectral distortion minimizing model for pan-sharpening of very high resolution multispectral images of urban areas. In: 2nd
GRSS/ISPRS Joint Workshop on Remote Sensing and Data Fusion over Urban Areas. IEEE (2003). p. 90–4.
35. Aiazzi B, Alparone L, Baronti S, Garzelli A, Selva M. MTF-tailored multiscale fusion of high-resolution MS and pan imagery. Photogramm Eng Remote Sens. (2006) 72:591–6. doi: 10.14358/
36. Vivone G, Restaino R, Dalla Mura M, Licciardi G, Chanussot J. Contrast and error-based fusion schemes for multispectral image pansharpening. IEEE Geosci Remote Sens Lett. (2014) 11:930–4. doi:
37. Otazu X, González-Audícana M, Fors O, Núñez J. Introduction of sensor spectral response into image fusion methods. Application to wavelet-based methods. IEEE Trans Geosci Remote Sens. (2005) 43
:2376–85. doi: 10.1109/TGRS.2005.856106
38. Hou L, Zhang X. Pansharpening image fusion using cross-channel correlation: a framelet-based approach. J Math Imaging Vis. (2016) 55:36–49. doi: 10.1007/s10851-015-0612-x
39. He X, Condat L, Bioucas-Dias JM, Chanussot J, Xia J. A new pansharpening method based on spatial and spectral sparsity priors. IEEE Trans Image Process. (2014) 23:4160–74. doi: 10.1109/
40. Deng L-J, Guo W, Huang T-Z. Single-image super-resolution via an iterative reproducing kernel hilbert space method. IEEE Trans Circ Syst Video Technol. (2016) 26:2001–14. doi: 10.1109/
41. Wahba G, Spline Models for Observational Data. Society for Industrial and Applied Mathematics. Philadelphia, PA: Siam (1990). doi: 10.1137/1.9781611970128
42. Irani M, Peleg S. Super resolution from image sequences. In: 10th International Conference on Pattern Recognition. IEEE (1990). p. 115–20.
43. Deng L-J, Vivone G, Guo W, Dalla Mura M, Chanussot J. A variational pansharpening approach based on reproducible kernel hilbert 12 space and heaviside function. IEEE Trans Image Process. (2018).
doi: 10.1109/ICIP.2017.8296338
44. Duchon J. Splines minimizing rotation-invariant semi-norms in Sobolev spaces. In: Constructive theory of Functions of Several Variables. Berlin: Springer (1977). p. 85–100. doi: 10.1007/
45. Meinguet J. Multivariate interpolation at arbitrary points made simple. Z Angew Math Phys. (1979) 30:292–304. doi: 10.1007/BF01601941
46. Kainen P, Kůrková V, Vogt A. Best approximation by linear combinations of characteristic functions of half-spaces. J Approx Theory. (2003) 122:151–9. doi: 10.1016/S0021-9045(03)00072-8
47. Arthur D, Vassilvitskii S. k-means++: the advantages of careful seeding. In: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms. Society for Industrial and Applied
Mathematics. (2007). p. 1027–35.
48. Kowalski M. Sparse regression using mixed norms. Appl Comput Harm Anal. (2009) 27:303–24. doi: 10.1016/j.acha.2009.05.006
49. Boyd S, Parikh N, Chu E, Peleato B, Eckstein J. Distributed optimization and statistical learning via the alternating direction method of multipliers. Found Trends Mach Learn. (2011) 3:1–122.
doi: 10.1561/2200000016
50. Eckstein J, Bertsekas DP. On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math Program. (1992) 55–3:293–318. doi: 10.1007/BF01581204
51. Rousseeuw P. Silhouettes: a graphical aid to the interpretation and validation of cluster analysis. J Comput Appl Math. (1987) 20:53–65. doi: 10.1016/0377-0427(87)90125-7
Keywords: super-resolution, reproducible kernel Hilbert space (RKHS), heaviside, sparse representation, multispectral imaging
Citation: Lartey R, Guo W, Zhu X and Grohnfeldt C (2020) Analog Image Modeling for 3D Single Image Super Resolution and Pansharpening. Front. Appl. Math. Stat. 6:22. doi: 10.3389/fams.2020.00022
Received: 01 February 2020; Accepted: 11 May 2020;
Published: 12 June 2020.
Edited by:
Ke Shi
, Old Dominion University, United States
Reviewed by:
Jian Lu
, Shenzhen University, China
Xin Guo
, Hong Kong Polytechnic University, Hong Kong
Copyright © 2020 Lartey, Guo, Zhu and Grohnfeldt. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction
in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic
practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Weihong Guo, wxg49@case.edu
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Gauss criterion
From Encyclopedia of Mathematics
Gauss test
A convergence criterion for a series of positive numbers
If the ratio
to exist. Gauss' criterion was historically (1812) one of the first general criteria for convergence of a series of numbers. It was employed by C.F. Gauss to test the convergence of the
hypergeometric series. It is the simplest particular case of a logarithmic convergence criterion.
The criterion is usually stated in the simpler form with [a1], p. 297.
[a1] K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) pp. 324 (English translation: Blackie, 1951 & Dover, reprint, 1990)
How to Cite This Entry:
Gauss criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gauss_criterion&oldid=17936
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article
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Dice - Wikiwand
A die (sg.: die or dice; pl.: dice)^[1] is a small, throwable object with marked sides that can rest in multiple positions. Dice are used for generating random values, commonly as part of tabletop
games, including dice games, board games, role-playing games, and games of chance.
Four traditional dice showing all six different sides.
Dice of different sizes being thrown in slow motion.
A traditional die is a cube with each of its six faces marked with a different number of dots (pips) from one to six. When thrown or rolled, the die comes to rest showing a random integer from one to
six on its upper surface, with each value being equally likely. Dice may also have polyhedral or irregular shapes, may have faces marked with numerals or symbols instead of pips and may have their
numbers carved out from the material of the dice instead of marked on it. Loaded dice are specifically designed or modified to favor some results over others for cheating or entertainment.
Composite image of all sides of a Roman die, found in Leicestershire, England
Dice have been used since before recorded history, and their origin is uncertain. It is hypothesized that dice developed from the practice of fortune-telling with the talus of hoofed animals,
colloquially known as knucklebones.^[2] The Ancient Egyptian game of senet (played before 3000 BCE and up to the 2nd century CE) was played with flat two-sided throwsticks which indicated the number
of squares a player could move, and thus functioned as a form of dice.^[3] Perhaps the oldest known dice were excavated as part of a backgammon-like game set at the Burnt City, an archeological site
in south-eastern Iran, estimated to be from between 2800 and 2500 BCE.^[4]^[5] Bone dice from Skara Brae, Scotland have been dated to 3100–2400 BCE.^[6] Excavations from graves at Mohenjo-daro, an
Indus Valley civilization settlement, unearthed terracotta dice dating to 2500–1900 BCE,^[7] including at least one die whose opposite sides all add up to seven, as in modern dice.^[8]
Games involving dice are mentioned in the ancient Indian Rigveda,^[9] Atharvaveda, Mahabharata and Buddhist games list.^[10] There are several biblical references to "casting lots" (Hebrew: יפילו
גורל yappîlū ḡōrāl), as in Psalm 22, indicating that dicing (or a related activity) was commonplace when the psalm was composed. Knucklebones was a game of skill played in ancient Greece; a
derivative form had the four sides of bones receive different values like modern dice.^[11]
Roman wall painting showing two dice-players, Pompeii, 1st century
Although gambling was illegal, many Romans were passionate gamblers who enjoyed dicing, which was known as aleam ludere ("to play at dice"). There were two sizes of Roman dice. Tali were large dice
inscribed with one, three, four, and six on four sides. Tesserae were smaller dice with sides numbered from one to six.^[12] Twenty-sided dice date back to the 2nd century CE^[13] and from Ptolemaic
Egypt as early as the 2nd century BCE.^[14]
Dominoes and playing cards originated in China as developments from dice.^[15] The transition from dice to playing cards occurred in China around the Tang dynasty (618–907 CE), and coincides with the
technological transition from rolls of manuscripts to block printed books.^[16] In Japan, dice were used to play a popular game called sugoroku. There are two types of sugoroku. Ban-sugoroku is
similar to backgammon and dates to the Heian period (794–1185 CE), while e-sugoroku is a racing game.^[17]
Dice are thrown onto a surface either from the hand or from a container designed for this (such as a cup, tray, or tower). The face (or corner, in cases such as tetrahedral dice, or edge, for
odd-numbered long dice) of the die that is uppermost when it comes to rest provides the value of the throw.
The result of a die roll is determined by the way it is thrown, according to the laws of classical mechanics (although luck is often credited for the results of a roll). A die roll is made random by
uncertainty in minor factors such as tiny movements in the thrower's hand; they are thus a crude form of hardware random number generator.
One typical contemporary dice game is craps, where two dice are thrown simultaneously and wagers are made on the total value of the two dice. Dice are frequently used to introduce randomness into
board games, where they are often used to decide the distance through which a piece will move along the board (as in backgammon and Monopoly).
Thrown or simulated dice are sometimes used to generate specific probability distributions, which are fundamental to probability theory. For example, rolling a single six-sided die yields a uniform
distribution, where each number from 1 to 6 has an equal chance of appearing. However, when rolling two dice and summing the results, the probability distribution shifts, as some sums (like 7) become
more likely than others (like 2 or 12). These distributions can model real-world scenarios or mathematical constructs, making dice a practical tool for teaching and exploring concepts in probability
Chirality of dice. Faces may be placed counterclockwise (top) or clockwise.
Common dice are small cubes, most often 1.6 cm (0.63 in) across, whose faces are numbered from one to six, usually by patterns of round dots called pips. (While the use of Arabic numerals is
occasionally seen, such dice are less common.)
Opposite sides of a modern die traditionally add up to seven, requiring the 1, 2, and 3 faces to share a vertex.^[18] The faces of a die may be placed clockwise or counterclockwise about this vertex.
If the 1, 2, and 3 faces run counterclockwise, the die is called "right-handed". If those faces run clockwise, the die is called "left-handed". Western dice are normally right-handed, and Chinese
dice are normally left-handed.^[19]
The pips on standard six-sided dice are arranged in specific patterns as shown. Asian style dice bear similar patterns to Western ones, but the pips are closer to the center of the face; in addition,
the pips are differently sized on Asian style dice, and the pips are colored red on the 1 and 4 sides. Red fours may be of Indian origin.^[19]^[20]
Western, Asian, and casino dice
Non-precision dice are manufactured via the plastic injection molding process, often made of polymethyl methacrylate (PMMA). The pips or numbers on the die are a part of the mold. Different pigments
can be added to the dice to make them opaque or transparent, or multiple pigments may be added to make the dice speckled or marbled.^[21]
The coloring for numbering is achieved by submerging the die entirely in paint, which is allowed to dry. The die is then polished via a tumble finishing process similar to rock polishing. The
abrasive agent scrapes off all of the paint except for the indents of the numbering. A finer abrasive is then used to polish the die. This process also produces the smoother, rounded edges on the
Precision casino dice may have a polished or sand finish, making them transparent or translucent respectively. Casino dice have their pips drilled, then filled flush with a paint of the same density
as the material used for the dice, such that the center of gravity of the dice is as close to the geometric center as possible. This mitigates concerns that the pips will cause a small bias.^[22] All
such dice are stamped with a serial number to prevent potential cheaters from substituting a die. Precision backgammon dice are made the same way; they tend to be slightly smaller and have rounded
corners and edges, to allow better movement inside the dice cup and stop forceful rolls from damaging the playing surface.
Etymology and terms
The word die comes from Old French dé; from Latin datum "something which is given or played".^[23]
While the terms ace, deuce, trey, cater, cinque and sice are generally obsolete, with the names of the numbers preferred, they are still used by some professional gamblers to designate different
sides of the dice. Ace is from the Latin as, meaning "a unit";^[24] the others are 2 to 6 in Old French.^[25]
When rolling two dice, certain combinations have slang names. The term snake eyes is a roll of one pip on each die. The Online Etymology Dictionary traces use of the term as far back as 1919.^[26]
The US term boxcars, also known as midnight, is a roll of six pips on each die. The pair of six pips resembles a pair of boxcars on a freight train. Many rolls have names in the game of craps.
Unicode representation
Symbol ⚀ ⚁ ⚂ ⚃ ⚄ ⚅ 🎲
Unicode U+2680 U+2681 U+2682 U+2683 U+2684 U+2685 U+1F3B2
Decimal ⚀ ⚁ ⚂ ⚃ ⚄ ⚅ 🎲
Using Unicode characters, the faces can be shown in text using the range U+2680 to U+2685 or using decimal ⚀ to ⚅,^[27] and the emoji using U+1F3B2 or 🎲 from the Miscellaneous
Symbols and Pictographs block.
A loaded, weighted, cheat, or crooked die is one that has been tampered with so that it will land with a specific side facing upwards more often or less often than a fair die would. There are several
methods for making loaded dice, including rounded faces, off-square faces, and weights. Casinos and gambling halls frequently use transparent cellulose acetate dice, as tampering is easier to detect
than with opaque dice.^[28]
Polyhedral dice
A typical set of polyhedral dice in various colors. They consist of the five Platonic solids, along with a ten-sided die that is also used for generating percentages.
Various shapes such as two-sided or four-sided dice are documented in archaeological findings; for example, from Ancient Egypt and the Middle East. While the cubical six-sided die became the most
common type in many parts of the world, other shapes were always known, like 20-sided dice in Ptolemaic and Roman times.
The modern tradition of using sets of polyhedral dice started around the end of the 1960s when non-cubical dice became popular among players of wargames,^[29] and since have been employed extensively
in role-playing games and trading card games. Dice using both the numerals 6 and 9, which are reciprocally symmetric through rotation, typically distinguish them with a dot or underline.
Common variations
Dice are often sold in sets, matching in color, of six different shapes. Five of the dice are shaped like the Platonic solids, whose faces are regular polygons. Aside from the cube, the other four
Platonic solids have 4, 8, 12, and 20 faces, allowing for those number ranges to be generated. The only other common non-cubical die is the 10-sided die, a pentagonal trapezohedron die, whose faces
are ten kites, each with two different edge lengths, three different angles, and two different kinds of vertices. Such sets frequently include a second 10-sided die either of contrasting color or
numbered by tens, allowing the pair of 10-sided dice to be combined to generate numbers between 1 and 100.
Using these dice in various ways, games can closely approximate a variety of probability distributions. For instance, 10-sided dice can be rolled in pairs to produce a uniform distribution of random
percentages, and summing the values of multiple dice will produce approximations to normal distributions.^[30]
Unlike other common dice, a four-sided (tetrahedral) die does not have a side that faces upward when it is at rest on a surface, so it must be read in a different way. On some four-sided dice, each
face features multiple numbers, with the same number printed near each vertex on all sides. In this case, the number around the vertex pointing up is used. Alternatively, the numbers on a tetrahedral
die can be placed at the middle of the edges, in which case the numbers around the base are used.
Normally, the faces on a die will be placed so opposite faces will add up to one more than the number of faces. (This is not possible with 4-sided dice and dice with an odd number of faces.) Some
dice, such as those with 10 sides, are usually numbered sequentially beginning with 0, in which case the opposite faces will add to one less than the number of faces.
Some twenty-sided dice have a different arrangement used for the purpose of keeping track of an integer that counts down, such as health points. These spindown dice are arranged such that adjacent
integers appear on adjacent faces, allowing the user to easily find the next lower number. They are commonly used with collectible card games.^[31]
/ Shape Notes
4 Tetrahedron Each face has three numbers, arranged such that the upright number, placed either near the vertex or near the opposite edge, is the same on all three visible faces. The upright
numbers represent the value of the roll. This die does not roll well and thus it is usually thrown into the air instead.
6 Cube A common die. The sum of the numbers on opposite faces is 7.
8 Octahedron Each face is triangular and the die resembles two square pyramids attached base-to-base. Usually, the sum of the opposite faces is 9.
Pentagonal Each face is a kite. The die has two sharp corners, where five kites meet, and ten blunter corners, where three kites meet. The ten faces usually bear numbers from zero to nine
10 trapezohedron rather than one to ten (zero being read as "ten" in many applications). Often, all odd numbered faces converge at one sharp corner, and the even ones at the other. The sum of
the numbers on opposite faces is usually 9 (if numbered 0–9) or 11 (if numbered 1–10).
12 Dodecahedron Each face is a regular pentagon. The sum of the numbers on opposite faces is usually 13.
20 Icosahedron Faces are equilateral triangles. Icosahedra have been found dating to Roman/Ptolemaic times, but it is not known if they were used as gaming dice. Modern dice with 20 sides are
sometimes numbered 0–9 twice as an alternative to 10-sided dice. The sum of the numbers on opposite faces is 21 if numbered 1–20.
Rarer variations
Dice collection: D2–D22, D24, D26, D28, D30, D36, D48, D50, D60 and D100.
"Uniform fair dice" are dice where all faces have an equal probability of outcome due to the symmetry of the die as it is face-transitive. In addition to the Platonic solids, these theoretically
Two other types of polyhedra are technically not face-transitive but are still fair dice due to symmetry:
Long dice and teetotums can, in principle, be made with any number of faces, including odd numbers.^[32] Long dice are based on the infinite set of prisms. All the rectangular faces are mutually
face-transitive, so they are equally probable. The two ends of the prism may be rounded or capped with a pyramid, designed so that the die cannot rest on those faces. 4-sided long dice are easier to
roll than tetrahedra and are used in the traditional board games dayakattai and daldøs.
Faces/ Shape Image Notes
1 Möbius strip or Most commonly a joke die, this is either a sphere with a 1 marked on it or shaped like a Möbius strip. It entirely defies the aforementioned use of a die.
2 Flat Cylinder or A coin flip. Some coins with 1 marked on one side and 2 on the other are available, but most simply use a common coin.
Flat Prism
3 Rounded-off A long die intended to be rolled lengthwise. When the die is rolled, one edge (rather than a side) appears facing upwards. On either side of each edge the same
triangular prism number is printed (from 1 to 3). The numbers on either side of the up-facing edge are read as the result of the die roll.
4 Capped 4-sided long A long die intended to be rolled lengthwise. It cannot stand on end as the ends are capped.
Triangular prism A prism thin enough to land either on its "edge" or "face". When landing on an edge, the result is displayed by digits (2–4) close to the prism's top edge. The
5 triangular faces are labeled with the digits 1 and 5.
Capped 5-sided long Five-faced long die for the Korean Game of Dignitaries; notches indicating values are cut into the edges, since in an odd-faced long die these land uppermost.
6 Capped 6-sided long Two six-faced long dice are used to simulate the activity of scoring runs and taking wickets in the game of cricket. Originally played with labeled six-sided
die pencils, and often referred to as pencil cricket.
Pentagonal prism Similar in constitution to the 5-sided die. Seven-sided dice are used in a seven-player variant of backgammon. Seven-sided dice are described in the 13th century
7 Libro de los juegos as having been invented by Alfonso X in order to speed up play in chess variants.^[33]^[34]
Truncated sphere A truncated sphere with seven landing positions.
9 Truncated sphere A truncated sphere with nine landing positions.
10 Decahedron A ten-sided die made by truncating two opposite vertices of an octahedron.
11 Truncated sphere A truncated sphere with eleven landing positions.
12 Rhombic dodecahedron Each face is a rhombus.
13 Truncated sphere A truncated sphere with thirteen landing positions.
Heptagonal Each face is a kite.
14 trapezohedron
Truncated octahedron A truncated octahedron. Each face is either a square or a hexagon.
Truncated sphere A truncated sphere with fourteen landing positions. The design is based on the cuboctahedron.
15 Truncated sphere A truncated sphere with fifteen landing positions.
16 Octagonal bipyramid Each face is an isosceles triangle.
17 Truncated sphere A truncated sphere with seventeen landing positions.
18 Rounded Eighteen faces are squares. The eight triangular faces are rounded and cannot be landed on.
19 Truncated sphere A truncated sphere with nineteen landing positions.
21 Truncated sphere A truncated sphere with twenty-one landing positions.
22 Truncated sphere A truncated sphere with twenty-two landing positions.
Triakis octahedron Each face is an isosceles triangle.
Tetrakis hexahedron Each face is an isosceles triangle.
Deltoidal Each face is a kite.
24 icositetrahedron
Pseudo-deltoidal Each face is a kite.
Pentagonal Each face is an irregular pentagon.
26 Truncated sphere A truncated sphere with twenty-six landing positions.
28 Truncated sphere A truncated sphere with twenty-eight landing positions.
30 Rhombic Each face is a rhombus.
32 Truncated sphere A truncated sphere with thirty-two landing positions. The design is similar to that of a truncated icosahedron.
34 Heptadecagonal Each face is a kite.
36 Truncated sphere A truncated sphere with thirty-six landing positions. Rows of spots are present above and below each number 1 through 36 so that this die can be used to roll two
six-sided dice simultaneously.
48 Disdyakis Each face is a scalene triangle.
50 Icosipentagonal Each face is a kite.
Deltoidal Each face is a kite.
Pentakis Each face is an isosceles triangle.
60 dodecahedron
Pentagonal Each face is an irregular pentagon.
Triakis icosahedron Each face is an isosceles triangle.
100 Zocchihedron A sphere containing another sphere with 100 facets flattened into it. Note that this design is not isohedral; it does not function as a uniform fair die as some
results are more likely than others.
120 Disdyakis Each face is a scalene triangle.
Non-numeric dice
A set of Fudge dice
The faces of most dice are labelled using sequences of whole numbers, usually starting at one, expressed with either pips or digits. However, there are some applications that require results other
than numbers. Examples include letters for Boggle, directions for Warhammer Fantasy Battle, Fudge dice, playing card symbols for poker dice, and instructions for sexual acts using sex dice.
Alternatively-numbered dice
Dice may have numbers that do not form a counting sequence starting at one. One variation on the standard die is known as the "average" die.^[35]^[36] These are six-sided dice with sides numbered 2,
3, 3, 4, 4, 5, which have the same arithmetic mean as a standard die (3.5 for a single die, 7 for a pair of dice), but have a narrower range of possible values (2 through 5 for one, 4 through 10 for
a pair). They are used in some table-top wargames, where a narrower range of numbers is required.^[36] Other numbered variations include Sicherman dice and intransitive dice.
Spherical dice
A spherical die
A die can be constructed in the shape of a sphere, with the addition of an internal cavity in the shape of the dual polyhedron of the desired die shape and an internal weight. The weight will settle
in one of the points of the internal cavity, causing it to settle with one of the numbers uppermost. For instance, a sphere with an octahedral cavity and a small internal weight will settle with one
of the 6 points of the cavity held downwards by the weight.
Playing Dice by Master Jean de Mauléon (c. 1520)
Many board games use dice to randomize how far pieces move or to settle conflicts. Typically, this has meant that rolling higher numbers is better. Some games, such as Axis & Allies, have inverted
this system by making the lower values more potent. In the modern age, a few games and game designers have approached dice in a different way by making each side of the die similarly valuable. In
Castles of Burgundy, players spend their dice to take actions based on the die's value. In this game, a six is not better than a one, or vice versa. In Quarriors (and its descendant, Dicemasters),
different sides of the dice can offer completely different abilities. Several sides often give resources while others grant the player useful actions.^[37]
Dice can be used for divination and using dice for such a purpose is called cleromancy. A pair of common dice is usual, though other forms of polyhedra can be used. Tibetan Buddhists sometimes use
this method of divination. It is highly likely that the Pythagoreans used the Platonic solids as dice. They referred to such dice as "the dice of the gods" and they sought to understand the universe
through an understanding of geometry in polyhedra.^[38]
Typical role-playing dice, showing a variety of colors and styles. Note the older hand-inked green 12-sided die (showing an 11), manufactured before pre-inked dice were common. Many players collect
or acquire a large number of mixed and unmatching dice.
Polyhedral dice are commonly used in role-playing games. The fantasy role-playing game Dungeons & Dragons (D&D) is largely credited with popularizing dice in such games. Some games use only one type,
like Exalted which uses only ten-sided dice. Others use numerous types for different game purposes, such as D&D, which makes use of all common polyhedral dice. Dice are usually used to determine the
outcome of events. Games typically determine results either as a total on one or more dice above or below a fixed number, or a certain number of rolls above a certain number on one or more dice. Due
to circumstances or character skill, the initial roll may have a number added to or subtracted from the final result, or have the player roll extra or fewer dice. To keep track of rolls easily, dice
notation is frequently used.
Astrological dice are a specialized set of three 12-sided dice for divination; the first die represents the planets, the Sun, the Moon, and the nodes of the Moon, the second die represents the 12
zodiac signs, and the third represents the 12 houses. A specialized icosahedron die provides the answers of the Magic 8 Ball, conventionally used to provide answers to yes-or-no questions.
Dice can be used to generate random numbers for use in passwords and cryptography applications. The Electronic Frontier Foundation describes a method by which dice can be used to generate passphrases
.^[39] Diceware is a method recommended for generating secure but memorable passphrases, by repeatedly rolling five dice and picking the corresponding word from a pre-generated list.^[40]
In many gaming contexts, especially tabletop role-playing games, shorthand notations representing different dice rolls are used. A very common notation, considered a standard, expresses a dice roll
as nds or nDs, where n is the number of dice rolled and s is the number of sides on each die; if only one die is rolled, n is normally not shown. For example, d4 denotes one four-sided die; 6d8 means
the player should roll six eight-sided dice and sum the results.
The notation also allows for adding or subtracting a constant amount c to the roll. When an amount is added, the notation is nds+c or nDs+c; for example, 3d6+4 instructs the player to roll three
six-sided dice, calculate the total, and add four to it. When an amount is to be subtracted, the notation is nds-c or nDs-c; so 3d6-4 instructs the player to subtract four from the result of rolling
3d6. If the result of a modified dice roll is negative, it is often taken to be zero or one; for instance, when the dice roll determines the amount of damage to a creature.^[41]^[42]
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Finite Element Analysis of a Beam
In this article, we will perform a finite analysis of a simple beam using FEA Online. As illustrated in the diagram below, the left end of the beam is fixed; the right end is supported by a spring.
The beam is also supported in midspan.
The properties of the beam are as follows:
• L = 3 m
• E = 210 GPa
• I = 0.0002 m⁴
• P = 50 kN
• k = 200 kN/m
The manual solution to this problem is presented in Solved problem on Finite element analysis on beam elements.
In this article, we will solve the same beam problem using an online finite element analysis program: FEA Online. You may register for a free account (Basic tier) and if you would like to follow
along, just click on the Sign-Up button to register.
The process of analyzing a structure is as follows:
1. Create a new Model
2. Define the nodes
3. Define the constraints (supports)
4. Define the material properties
5. Define the beam section properties
6. Define the elements (beam and spring will be used in this example)
7. Define the applied loads
8. Perform the static analysis
9. Print analysis results or review the axial, shear, and moment diagrams
Create A New Model
To start a new finite element model, click on the Model > New…
In the New Model dialog, select “Metric” as the system unit, “Meter (m)” for the Length Unit, and “Kilonewton (kN)" for the Force Unit. Then click on the Create button to create the model.
Define Nodes
Once the model is created, you can start defining the model geometry by navigating to the “Nodes” tab. Clicking on the Add button will add a new node to the model. The screen below shows 4 nodes have
been added to match the diagram in the beginning of this article. Nodes 1, 2, 3 are on the same line with 3m between them. Node 4 is 0.20m below node 3. Since we will be using a spring element to
connect Nodes 3 and 4, the distance between them is not important (0.2m is arbitrary selected).
Define Constraints
Constraints are used to model supports in structures. They are the boundaries conditions in a finite element model. The “Constraints / Supports” tab is used define Constraints. Use the “Add” button
to add new constraints.
The above screen shows all displacements and rotations of Node 1 are restricted in all directions. The Y direction displacement of Node 2 is constrained. The X, Y, Z displacements of Node 4 is
Note: All models in FEA Online are 3-dimensional. When modelling 2D structure, it is important the supports in your structure are modelled appropriately so the structure is stable.
Define Material
FEA Online supports beam, truss, and spring elements at time. For beams and trusses, we must specify the material. The “Materials” tab is used to define materials in a model. For Material 1 in the
screen below, E is set to 210 GPa.
Define Sections
Beam and truss elements also require a section be specified in a finite element model. The “Sections” tab is used to define sections in a model. Clicking on the Add button will create a new section.
Define Elements
The “Elements” tab is used to define beams, trusses, and spring elements in a finite element model. You can click on the Add button to add a new element. The Type attribute allows you to specify if a
beam, truss, or spring element. The Node ID attribute is used to define the nodes the element is connected. This is a comma separated list of node IDs. You can use the Material ID and Section ID to
reference the material and section, respectively, of the element.
Since the model in FEA Online is always a 3D model, it is necessary to specify the orientation of elements in the model. This is done using the Reference ID attribute. If no Reference ID is
specified, it is assumed the 2-axis is parallel to the global Y-axis. The reference node you specified is located in the 1–2 plane of the beam with the 1-axis defined as the line connecting the i
(start) and j (end) nodes.
For spring elements, the Spring Stiffness is required.
Define Loads
The “Loads” tab is used to specify the external loads applied to the structure.
FEA Online organizes loads into Basic Load Cases (BLCs). Every load must be assigned a “BLC ID”. BLCs are used to define Load Combinations that will be used to design the structure. So before
defining the load, we must first define the BLCs using the “Manage BLCs” button.
FEA Online initializes each model with a list of BLCs (based on ASCE 7 standard). the D (Dead Load) and L (Live Load) are active by default.
After BLCs have been defined, you can add the loads to the model using the “Add” button. The “Type” attribute allows you to specify the type of load:
1. nodal: Load applied at a node
2. distributed: Distributed load applied in the span of a beam
3. concentrated: Concentrated load applied in the span of a beam
4. temperature: Thermal load applied to the beam
In the screen below, a nodal load is specified to be applied at Node 3.
Perform Static Analysis
After the model geometry, constraints, and load have been complete, you can perform a static analysis of the model using the Tools > Analyze command to submit the model for analysis. For small
models, the result should return quickly. For very large models, it may take some time, and you may need to click on the Tools > Get Analysis Result to get the result.
Analysis Result
The Displacements tab shows the nodal displacement for each basic load case.
For our model, Nodes 1, and 4 are constrained. So all displacement components are 0. The translation displacements of Node 2 are all 0s since the Y direction is constrained; rotation at Node 2 is
-0.002492 radians.
The Y displacement is -0.017442m and Z rotation is -0.007475 radians.
These values closely match the values in the YouTube video (see 10:41 minute of the video).
The “Reactions” tab shows the reactions at each constraint.
Again, the values are very close to the value in the YouTube video (see time 13:15)
Element Forces
FEA Online shows the forces at each element in the “Element Forces” tab. You can click on each element to see the Axial, Shear, Moment, and Torsion diagrams.
Element 1
Element 2
Element 3
You can also generate a PDF output document from FEA Online that echos the model data you have entered
The PDF also contains the computed displacements, reactions, element forces, and load combination forces (if load combination is defined in the model).
With FEA Online, you can model structures using the finite element method online without the need to install any software or purchase expansive hardware. You can perform finite element analysis using
a web browser. Only an Internet connection and a device that has a web browser is needed. You can run FEA Online in your desktop, laptop, notepad, and yes, even your mobile phone!
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7. The roots of each of the following quadratic equations are r... | Filo
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7. The roots of each of the following quadratic equations are real and equal, find . (1) (2) Let's learn. Application of quadratic equation Quadratic equations are useful in daily life for finding
solutions of some practical problems. We are now going to learn the sap Ex. (1) There is a rectangular onion story Tivasa. The length of rectangular base is In the farm of nakarrao at d diagonal is
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7. The roots of each of the following quadratic equations are real and equal, find . (1) (2) Let's learn. Application of quadratic equation Quadratic equations are useful in daily life
Question for finding solutions of some practical problems. We are now going to learn the sap Ex. (1) There is a rectangular onion story Tivasa. The length of rectangular base is In the farm of
Text nakarrao at d diagonal is
Updated On Dec 12, 2022
Topic All topics
Subject Mathematics
Class Class 11
Answer Type Video solution: 1
Upvotes 144
Avg. Video 5 min
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Gravitational Torque Calculator, Formula, Gravitational Torque Calculation | Electrical4u
Gravitational Torque Calculator, Formula, Gravitational Torque Calculation
Gravitational Torque Calculator:
Enter the values of mass of the object, m[(kg)], acceleration due to gravity, g(m/s^2) and radius, r[(m)] to determine the value of gravitational torque, T[g(N.m)].
Gravitational Torque Formula:
Gravitational torque is a measure of the rotational effect caused by gravity on an object that is not centered on its axis of rotation.
Gravitational torque, T[g(N.m)] in Newton metres is calculated by the product of mass of the object, m[(kg)] in kilograms, acceleration due to gravity, g(m/s^2) in metre per second squared and
radius, r[(m)] in metre.
Gravitational torque, T[g(N.m) ]= m[(kg)] * g[(m/s2)] * r[(m)]
T[g(N.m) ]= gravitational torque in Newton metres, N.m.
m[(kg) ]= mass of the object in kilograms, kg.
g[(m/s2)] = acceleration due to gravity in metre per second squared, m/s^2. (approximately 9.81m/s^2 on the surface of the Earth).
r[(m) ]= radius in metre, m.
Gravitational torque Calculation:
1. Consider a playground seesaw with a child sitting 2 metres from the pivot point. If the child has a mass of 30kg, calculate the gravitational torque exerted by the child on the seesaw.
Given: m[(kg) ]= 30kg, g[(m/s2)] = 9.81m/s^2, r[(m) ]= 2m.
Gravitational torque, T[g(N.m) ]= m[(kg)] * g(m/s^2) * r[(m)]
T[g(N.m) ]= 30 * 9.81 * 2
T[g(N.m) ]= 588.6N.m
2. A sign is mounted on a pole, where the center of mass of the sign is 0.5 metres from the pole. If the acquired gravitational torque on the pole is 73.575N.m. Determine the mass.
Given: T[g(N.m) ]= 73.575N.m, g[(m/s2)] = 9.81m/s^2, r[(m) ]= 0.5m.
Gravitational torque, T[g(N.m) ]= m[(kg)] * g[(m/s2)] * r[(m)]
m[(kg)] = T[g(N.m)] / g[(m/s2)] * r[(m)]
m[(kg)] = 73.575 / 9.81 * 0.5
m[(kg)] = 15kg.
Gravitational Torque
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Remove Nth Node From End of List - AlgoBreath
To understand the problem of "removing the Nth node from the end of a list," particularly in the context of a singly linked list, we need to establish some fundamental principles about linked lists
and the specific challenges this problem presents.
Fundamental Principles of Linked Lists
1. Definition: A linked list is a linear collection of elements, called nodes, where each node points to the next node in the sequence. In a singly linked list, each node has two attributes: value
(the data it holds) and next (a reference/pointer to the next node in the list).
2. Traversal: To access a node in a linked list, you start from the head (the first node) and follow the next references until you reach the desired node. This is because nodes in a linked list are
not indexed like an array.
3. Removal of a Node: To remove a node, you adjust the next pointer of the previous node so that it bypasses the node to be removed, effectively excluding it from the list.
The Challenge: Removing the Nth Node from the End
The challenge here is twofold:
1. Identifying the Nth Node from the End: Unlike an array, a linked list doesn’t provide direct access to its nodes. To find the Nth node from the end, you first need to determine its position from
the beginning.
2. Single Pass Requirement: Ideally, you want to solve this in a single pass through the list (i.e., without traversing the list more than once), which adds a layer of complexity.
Solution Strategy
The most efficient solution involves the following steps:
1. Two-Pointer Technique: Use two pointers, fast and slow. Initially, both start at the head.
2. Advance fast by N nodes: Move the fast pointer N nodes ahead in the list. This creates a gap of N nodes between fast and slow.
3. Move both pointers: Traverse the list by advancing both pointers until fast reaches the last node. At this point, slow will be just before the Nth node from the end.
4. Remove the Nth Node: Adjust the next pointer of the node before the Nth node (slow) to bypass the Nth node.
Code Example (in Python)
class ListNode:
def __init__(self, value=0, next=None):
self.value = value
self.next = next
def removeNthFromEnd(head, n):
dummy = ListNode(0, head) # A dummy node to handle edge cases
fast = slow = dummy # Initialize both pointers to the dummy node
# Advance 'fast' by 'n' nodes
for _ in range(n):
fast = fast.next
# Traverse the list until 'fast' reaches the end
while fast.next:
fast = fast.next
slow = slow.next
# 'slow' is now just before the Nth node, remove it
slow.next = slow.next.next
return dummy.next # Return the new head of the list
Explanation of the Code
• ListNode Class: Defines the structure of a node in the linked list.
• removeNthFromEnd Function: Takes the head of a linked list and an integer n, and removes the Nth node from the end of the list.
• Dummy Node: A dummy node is used to simplify edge cases, like removing the head of the list.
• Two-Pointer Technique: fast and slow pointers are used to maintain the gap of n nodes between them.
• Traversal and Removal: Once fast is n nodes ahead, both pointers are advanced until fast is at the end, ensuring slow is just before the target node. The target node is then skipped over by
adjusting slow.next.
This approach ensures that the operation is performed with a time complexity of O(L) (where L is the length of the list) and a space complexity of O(1), as it requires a single pass through the list
without additional storage.
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Output for (a -> Bool) -> [a] -> [a]:
The Free Theorem for “f :: forall a . (a -> Bool) -> [a] -> [a]”
forall t1,t2 in TYPES, R in REL(t1,t2).
forall p :: t1 -> Bool.
forall q :: t2 -> Bool.
(forall (x, y) in R. p x = q y)
==> (forall (z, v) in lift{[]}(R).
(f_{t1} p z, f_{t2} q v) in lift{[]}(R))
= {([], [])}
u {(x : xs, y : ys) | ((x, y) in R) && ((xs, ys) in lift{[]}(R))}
Reducing all permissable relation variables to functions
forall t1,t2 in TYPES, g :: t1 -> t2.
forall p :: t1 -> Bool.
forall q :: t2 -> Bool.
(forall x :: t1. p x = q (g x))
==> (forall y :: [t1].
map_{t1}_{t2} g (f_{t1} p y) = f_{t2} q (map_{t1}_{t2} g y))
In the context of GHC, zonking is when a type is traversed and mutable type variables are replaced with the real types they dereference to.
Most interesting example:
_ [3,1,2] :: Sorted (O(N*.LogN)) (O(N)) Integer
Valid hole fits include
mergeSort :: forall (n :: AsymP) (m :: AsymP) a.
(n >=. O (N *. LogN), m >=. O N, Ord a)
=> [a] -> Sorted n m a
quickSort :: forall (n :: AsymP) (m :: AsymP) a.
(n >=. O (N *. LogN), m >=. O LogN, Ord a)
=> [a] -> Sorted n m a
Jules Hedges on connections between lenses, the dialectica interpretation, backprop, etc.
List of “top algorithms”. Take with a huge grain of salt. Interesting list to look through though.
There is a data structure called a discrimination tree that allows one to compactly store a large number of terms (rose trees) with metavariables and then, given another term with metavariables,
efficiently filter the set of contained terms down to a subset that may unify with the given term.
Best discussion I’ve seen of the practicalities of running a prediction market.
The weakness (or twist) on markets this implies applies to prediction markets generally. If you bet on an event that is correlated with the currency you’re betting in, the fair price can be
very different from the true probability. It doesn’t have to be price based – think about betting on an election between a hard money candidate and one who will print money, or a prediction
on a nuclear war.
If I bet on a nuclear war, and win, how exactly am I getting paid?
There are three types of prediction markets that have gotten non-zero traction.
1. politics
2. economics
3. sports
To get a thriving market, you need (at least) these five things
1. well-defined
2. quick resolution
3. probable resolution
4. limited hidden information
5. sources of disagreement and interest
Do read the whole thing.
Interesting question and interesting answer by Neel K. I’ve been seeing Bekic’s lemma everywhere lately.
Handling of effects in the denotational framework, however, proved to be much more problematic, often summed up by the phrase “denotational semantics is not modular”. Briefly, addition of new
effects require substantial changes to the existing semantic description. For instance, exceptions can be modeled by adding a special failure element to each domain, representing the result of a
failed computation. But then, even such a simple thing as the meaning of an arithmetic operation requires a messy denotational description; one needs to check for failure at each argument, and
propagate accordingly. The story is similar for other cases, including I/O and assignments, two of the most “popular” effects found in many programming languages.
It was Moggi’s influential work on monads that revolutionized the semantic treatment of effects, which he referred to as notions of computation. Moggi showed how monads can be used to model
programming language features in a uniform way, providing an abstract view of programming languages. In the monadic framework, values of a given type are distinguished from computations that
yield values of that type. Since the monadic structure hides the details of how computations are internally represented and composed, programmers and language designers work in a much more
flexible environment. This flexibility is a huge win over the traditional approach, where everything has to be explicit.
Perhaps what Moggi did not quite envision was the response from the functional programming community, who took the idea to heart. Wadler wrote a series of articles showing how monads can be used
in structuring functional programs themselves, not just the underlying semantics. Very quickly, the Haskell committee adopted monadic I/0 as the standard means of performing input and output in
Haskell, making monads an integral part of a modern programming language. The use of monads in Haskell is further encouraged by special syntactic support, known as the do-notation.
As the monadic programming style became more and more popular in Haskell, programmers started realizing certain shortcomings. For instance, function application becomes tedious in the presence of
effects. Or, the if-then-else construct becomes unsightly when the test expression is monadic. However, these are mainly syntactic issues that can easily be worked around. More seriously, the
monadic sublanguage lacks support for recursion over the values of monadic actions. The issue is not merely syntactic; it is simply not clear what a recursive definition means when the defining
expression can perform monadic effects.
This problem brings us to the subject matter of the present work: Semantics of recursive declarations in monadic computations. More specifically, our aim is to study recursion resulting from the
cyclic use of values in monadic actions. We use the term value recursion to describe this notion.
Denotational Semantics / Domain theory links
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Qualifying Exams
The Mathematics Qualifying Exams
Passing the qualifying exams is a major milestone for graduate students in the PhD program.
Qualifying Exam Policy
1. The department will provide information concerning the exams in the graduate studies catalog as well as any literature put out by the department that describes our graduate program. The
information should include the number of exams to be taken, topics, times exams are offered, and amount of time including number of allowed attempts given to pass the exams. It will be emphasized to
students in writing within the catalogue and in person by the chair or relevant graduate advisor that passing the qualifying exams is necessary, but by no means sufficient for earning a Ph.D in
2. A student must pass two qualifying examinations (in different areas: real analysis, complex analysis, topology, algebra, probability and statistics, and applied mathematics) prior to registering
for dissertation hours (in mathematics).
3. The exam times will be at the beginning of the fall and spring semesters: August and January.
1. Two qualifying examinations must be passed by the end of the first January exam period following the seventh semester of graduate studies in mathematics (initiated at the University of North
Texas). Students failing to meet this requirement will be removed from the Ph.D. program after the spring semester following their seventh semester.
2. In addition, at least one qualifying examination must be passed by the end of the first January exam period following the fifth semester of graduate studies in mathematics (initiated at the
University of North Texas). For students failing to meet this requirement, the Graduate Affairs Committee will recommend to the department chair termination of financial support after the spring
semester following their fifth semester.
3. Students must complete the two semester core sequence before being eligible to attempt a qualifying exam in that area. Exceptions can possibly be made for incoming students who took the
equivalent of the two semester core sequence at a different institution, with the permission of the graduate advisor.
4. A student has at most six attempts to pass two exams within the time constraints mentioned above. A student may attempt examinations in any of the six areas (real analysis, complex analysis,
topology, algebra, probability and statistics, and applied mathematics). Students are not allowed to continue in the graduate program if they have not passed two qualifying examinations after six
attempts, except under extenuating circumstances and with the approval of the Graduate Affairs Committee.
5. Each exam subject area will have a syllabus that contains a detailed list of topics covered by the exam, together with suggested readings. Old exams will be made available to students.
6. Digital copies of the ungraded exams are kept by the department until the student graduates, or until five years have passed. Graded exams will not be returned to the students, but students may
consult with the chair of the area committee to view a copy of their ungraded exam and receive feedback on their exam performance.
Syllabi for Qualifying Exams (pdf)
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B * L
30 Aug 2024
B * L & Analysis of variables
Equation: B * L
Variable: B
Impact of Footing Width on Footing Area
X-Axis: -5998.000075155995to6001.999967901532
Y-Axis: Footing Area
Title: Investigating the Relationship between Footing Width (B) and Area (L): A Theoretical Analysis
Abstract: The design of footings is a critical aspect of geotechnical engineering, with the primary goal of ensuring structural safety. In this article, we examine the relationship between footing
width (B) and area (L), specifically focusing on the impact of B on L. We present a theoretical analysis using the equation B * L, where B represents the variable of interest, to illustrate the
fundamental principles governing this interaction.
Introduction: Footing design is a complex process that requires careful consideration of several factors, including soil properties, load distribution, and structural requirements. The footing width
(B) is a critical parameter in this context, as it directly affects the area (L) required to support the superimposed loads. Understanding the relationship between B and L is essential for engineers
to optimize footing design and minimize costs.
Theory: Let us consider the simple equation:
B * L = Constant
• B = Footing width (m)
• L = Footing area (m²)
In this context, we treat B as a variable of interest, examining its effect on the required footing area (L). The constant term represents the total load-bearing capacity of the soil and/or
superimposed loads.
Analysis: To better understand the impact of B on L, let us consider some hypothetical scenarios:
Scenario 1: Suppose we have a uniform soil density and a constant load per unit area. In this case, as B increases, the required footing area (L) will also increase linearly to accommodate the
increased load.
Equation: B * L = k
where k is a constant representing the product of soil density and load per unit area.
Scenario 2: Now suppose we have a non-uniform soil profile with varying densities. In this scenario, as B increases, the required footing area (L) will increase more than linearly to compensate for
the increased soil resistance.
Equation: B * L = k * B^α
where α is an empirical parameter representing the degree of non-linearity in the relationship between B and L.
Scenario 3: Consider a situation where the load per unit area varies with footing width. In this case, as B increases, the required footing area (L) will increase exponentially to account for the
increased load-bearing capacity.
Equation: B * L = e^(k * B)
where k is an empirical parameter representing the rate of exponential growth in the relationship between B and L.
Conclusion: The equation B * L serves as a fundamental tool for analyzing the impact of footing width on required area. By examining different scenarios, we have demonstrated that the relationship
between B and L can be linear, non-linear, or even exponential, depending on soil properties and load characteristics. Understanding this relationship is essential for engineers to optimize footing
design, minimize costs, and ensure structural safety.
1. Empirical Analysis: Perform site-specific empirical analyses using field data to determine the optimal relationship between B and L for specific geotechnical conditions.
2. Finite Element Modeling: Utilize finite element modeling techniques to simulate different footing geometries and soil properties, enabling more accurate predictions of footing behavior under
various loading scenarios.
3. Probabilistic Analysis: Perform probabilistic analyses using Monte Carlo simulations or other methods to account for uncertainty in soil properties and loads, providing a more comprehensive
understanding of the relationship between B and L.
By applying these recommendations, engineers can improve the accuracy and reliability of their designs, ensuring safer and more cost-effective construction projects.
Related topics
Academic Chapters on the topic
Information on this page is moderated by llama3.1
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Changing RYG Balls to Numbers to Create Average
I would like to be able to mark each of these columns using the balls: red (0 value), yellow (50), or green (100), but I want the final column to actually show an average number of the previous
columns, for example 65%. Does anyone know a formula that would do that?
• =(countif(A@row:G@row,"Yes")+countif(A@row:G@row,"Hold")*.5)/count(A@row:G@row)
Give that a try
Help Article Resources
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freeFlyingRobot optimal control problem (matrix 7 of 16)
Name freeFlyingRobot_7
Group VDOL
Matrix ID 2679
Num Rows 3,918
Num Cols 3,918
Nonzeros 31,046
Pattern Entries 31,046
Kind Optimal Control Problem
Symmetric Yes
Date 2015
Author B. Senses, A. Rao
Editor T. Davis
Download MATLAB Rutherford Boeing Matrix Market
Optimal control problem, Vehicle Dynamics & Optimization Lab, UF
Anil Rao and Begum Senses, University of Florida
This matrix arises from an optimal control problem described below.
Each optimal control problem gives rise to a sequence of matrices of
different sizes when they are being solved inside GPOPS, an optimal
control solver created by Anil Rao, Begum Senses, and others at in VDOL
lab at the University of Florida. This is one of the matrices in one
of these problems. The matrix is symmetric indefinite.
Rao, Senses, and Davis have created a graph coarsening strategy
that matches pairs of nodes. The mapping is given for this matrix,
where map(i)=k means that node i in the original graph is mapped to
node k in the smaller graph. map(i)=map(j)=k means that both nodes
i and j are mapped to the same node k, and thus nodes i and j have
been merged.
This matrix consists of a set of nodes (rows/columns) and the
names of these rows/cols are given
Anil Rao, Begum Sense, and Tim Davis, 2015.
Free flying robot optimal control problem is taken from
Ref.~\cite{sakawa1999trajectory}. Free flying robot technology is
expected to play an important role in unmanned space missions.
Although NASA currently has free flying robots, called spheres,
inside the International Space Station (ISS), these free flying
robots have neither the technology nor the hardware to complete
inside and outside inspection and maintanance. NASA's new plan is to
send new free flying robots to ISS that are capable of completing
housekeeping of ISS during off hours and working in extreme
environments for the external maintanance of ISS. As a result, the
crew in ISS can have more time for science experiments. The current
free flying robots in ISS works are equipped with a propulsion
system. The goal of the free flying robot optimal control problem is
to determine the state and the control that minimize the magnitude of
thrust during a mission. The state of the system is defined by the
inertial coordinates of the center of gravity, the corresponding
velocity, thrust direction, and the anglular velocity and the control
is the thrust from two engines. The specified accuracy tolerance of
$10^{-6}$ were satisfied after eight mesh iterations. As the mesh
refinement proceeds, the size of the KKT matrices increases from 798
to 6078.
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PositivityToricBundles -- checks positivity of toric vector bundles
Given a toric vector bundle, i.e. an equivariant vector bundle on a smooth complete toric variety, this package can check the positivity of this bundle.
can check whether a toric vector bundle is
• nef, i.e. whether the line bundle $\mathcal{O}(1)$ on $\mathbb{P}(\mathcal E)$ is nef;
• (very) ample, i.e. whether the line bundle $\mathcal{O}(1)$ on $\mathbb{P}(\mathcal E)$ is (very) ample;
• globally generated.
can compute the toric Chern character of a toric vector bundle as introduced by Sam Payne. For the computational purposes,
uses the description of a toric vector bundles by filtrations developed by Alexander Klyachko, and relies on its implementation via the
package by René Birkner, Nathan Ilten and Lars Petersen.
To check nefness and ampleness,
uses a result of Milena Hering, Mircea Mustaţă and Sam Payne, namely, that it is sufficient to check this for the restriction of the bundle to the torus invariant curves. The central method for this
; the methods
are based on it.
For global generation and very ampleness,
uses results of Sandra Di Rocco, Kelly Jabbusch and Gregory Smith, who describe these properties in terms of the so-called parliament of polytopes of a toric vector bundle. From the parliament of
polytopes one can extract the information up to which order jets are separated by the vector bundle. Globally generated or very ample toric vector bundles are those that separate 0-jets or 1-jets,
respectively. Here, the central method is
; built on it are
. For the mathematical background see
• [K] Alexander Klyachko, Equivariant bundles over toral varieties, Izv. Akad. Nauk SSSR Ser. Mat., 53, 1989.
• [P] Sam Payne, Moduli of toric vector bundles, Compos. Math, 144, 2008.
• [HMP] Milena Hering, Mircea Mustaţă, Sam Payne, Positivity properties of toric vector bundles, Ann. Inst. Fourier (Grenoble), 60, 2010
• [RJS] Sandra Di Rocco, Kelly Jabbusch, Gregory Smith, Toric vector bundles and parliaments of polytopes, Trans. AMS, 370, 2018.
The following example computes the positivity for the tangent sheaf of $\mathbb P^2$:
i1 : E = tangentBundle projectiveSpaceFan 2
o1 = {dimension of the variety => 2 }
number of affine charts => 3
number of rays => 3
rank of the vector bundle => 2
o1 : ToricVectorBundleKlyachko
i2 : isNef E
o2 = true
i3 : isAmple E
o3 = true
i4 : isVeryAmple E
o4 = true
i5 : isGloballyGenerated E
o5 = true
i6 : separatesJets E
o6 = 1
o6 : QQ
The toric Chern character can be computed:
i7 : toricChernCharacter E
o7 = HashTable{| -1 0 | => {| 1 |, | 1 |}}
| -1 1 | | -1 | | 0 |
| 1 -1 | => {| -1 |, | 0 |}
| 0 -1 | | 1 | | 1 |
| 1 0 | => {| 0 |, | -1 |}
| 0 1 | | -1 | | 0 |
o7 : HashTable
which associates to each maximal cone (its rays put into matrices) the corresponding components. The restrictions of the bundle to the torus invariant curves can be computed:
i8 : restrictToInvCurves E
o8 = HashTable{| -1 | => {2, 1}}
| -1 |
| 0 | => {1, 2}
| 1 |
| 1 | => {2, 1}
| 0 |
o8 : HashTable
Here, in all three cases, the restriction splits into $\mathcal{O}_{\mathbb P^1}(2) \oplus \mathcal{O}_{\mathbb P^1}(1)$. Most methods of
support the option
. So by adding
Verbosity => n
with n a positive integer to the arguments of a method, hopefully useful insight about the course of the calculation is provided.
The description of a toric variety and a toric vector bundle by filtrations involves the choice of signs.
follows the same choice of signs as
, which are
• the fan associated to a polytope will be generated by inner normals,
• the filtrations for describing a toric vector bundle are increasing and that the filtration steps are stored in that way, see wellformedBundleFiltrations.
Unfortunately, the above cited articles use decreasing filtrations and, moreover, [HMP], [P] and [RJS] use outer normals. Contrary to what the name suggests,
may very well encode a toric reflexive sheaf, which is not necessarily locally free, that is, not necessarily a vector bundle. Some methods work also for toric reflexive sheaves, but this is not
guaranteed. Another warning concerns the toric variety: the methods of
implicitly assume that the variety is complete (to apply the results of [HMP] and [P]) and in addition smooth (for [RJS]). For non-complete or singular toric varieties, methods might break or results
might become meaningless.
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Discharge in Curry Howard Isomorphism
Lets say we want to build a little web site. And present a THINKER
that helps us in finding propositional proofs or even FOL proofs,
just like here (not same meaning of heuristic as in A* search):
Heuristic Theorem Proving - Zeroth-Order Logic
Pelletier & Wilson, 1983
How would one go about it and solve this problem in Prolog?
1 Like
I just noticed that there is a defect in your Fitch style proof that should be corrected because there is something unsound in the expression of introduction rule:
The trouble is that line 6 does not show that assumption 1 is discharged. It means that a vertical bar should be before assumption 1 and another one before assumption 2, because these two assumptions
are discharged. We should see:
1. | d(a) [assume]
2. | | [assume]
3. | |
4. | |
5. | â discharge of assumption 2.
6. d(a) â â Š [imp(1-5)] â discharge of assumption 1: block 1-5 is a sequent from conditional 6 is inferredâ Š
Even if flags are not used, but only vertical bars, discharges must be expressed clearly, that is one of the most obvious merits of Fitchâ s style, in comparison of Gentzenâ s. You can check my
claim in reading Richard Kayeâ s excellent book, The Mathematics of Logic, CUP, 2007. This point also explains the distinction in carnap.io between Premiss (notation " :PR ") and Assumption
(notation " :AS " ): the latter is discharged, the former not : https://forallx.openlogicproject.org/html/Ch17.html#S4
As far as I know, genuine Fitch style correspond to your variant 2: discharge is expressed by a step back, and it is nice.
The result is nice, indeed. Can you explain why (![N-X]:A) and (?[N-X]:A) lead to enter ![ x ]: and ?[ x ] : as input respectively ?
Does exist a book where lambda calculus and related topics are explained in a simple and pedagogical way? As soon as I see the symbolism of lambda calculus, I do not even try to understand because it
is too much abstract for meâ Š
Thanks for sharing. Iâ ll make test and Iâ ll share feedbacks.
Usually I like learning things from books, but for lambda calculus, I found that a few university lectures really helped (it was part of a â theory of computationâ course). Perhaps there are some
videos that would help â Š
Thanks, Peter. There is indeed on the web this excellent explanation to begin:
It is excellent, because it explains very clearly the basic operation of this calculus.
Thereâ s another layer beyond the â lambda calculus for dummiesâ , which gets into model-theoretic stuff. Iâ ve taken a grad-level course in it, taught by a brilliant (and rather eccentric) guy,
and sadly Iâ ve retained almost none if it, because it mostly hasnâ t been needed in my day-to-day work.
But one thing I remember, after about the 5th lecture on â fixed-point theoremsâ (there are quite a few of them), most of which was going right over my head â Š I had a flash of insight that this
particular proof was simply proving that the notation we were using for writing recursive functions was guaranteed to represent a real function â that is, that everything I could write using this
notation actually represented an actual description of a computation and would never be meaningless marks on a piece of paper. Thatâ s a very important theoretical result (there are lots of â
incompletenessâ theorems, where this kind of thing doesnâ t hold) but once itâ s proved, it has no effect on day-to-day programming.
1 Like
Responding to a deleted post (â You made that up, right? Who was the teacher?â ):
My teacher was Akira Kanda, but I couldnâ t find his course notes online (and I long ago lost my copy, and have forgotten most of the course except for how hard it was): dblp: Akira Kanda
I vaguely remember some of this: Typed Recursion Theorems
You find it in this book:
Lectures on the Curry-Howard Isomorphism
Morten Heine SĂžrensen, Pawel Urzyczyn - 2006
For example in chapter 1:
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Combined Gas Law: Formula, Derivation, Examples
The combined gas law is a combination of three gas laws: Boyle’s Law, Charles’ Law, and Gay-Lussac’s Law. It states that the ratio between the product of a gas’s pressure and volume and its absolute
temperature is constant. The ideal gas law is obtained by combining Avogadro’s law with the combined gas law. The combined gas law does not have a specific discoverer, unlike the named gas laws. It
is a combination of the other gas laws that only consider temperature, pressure, and volume when everything else is constant.
Combined Gas Law
Interesting Science Videos
What is Combined Gas Law?
The Combined Gas Law states that when the amount of gas is fixed, the product of pressure (P), volume (V), and temperature (T) is equal to a constant (k).
The combined gas law is formed by combining four different laws.
Charles’ law states that the volume of a gas is directly proportional to its temperature, assuming the pressure and amount of gas remain constant.
Gay-Lussac’s law is a scientific principle that describes the relationship between the temperature and pressure of a gas.
Avogadro’s law refers to the relationship between the volume of a gas and the number of particles it contains.
Boyle’s law is a principle in physics that describes the relationship between the pressure and volume of a gas.
These laws relate one thermodynamic variable to another while keeping all other variables constant. The relationship between these variables is shown by the combined gas law. It states that the
product of a system’s pressure, volume, and temperature remains constant.
The Combined Gas Law is written as:
Combined Gas Law Formula
Combined gas law can be mathematically expressed as
k = PV/T
P = pressure
T = temperature in kelvin
V = volume
K = constant
In two different conditions, the law can be stated as,
P[i ]V[i]/ T[iย ]= P[f ]V[f ]/ T[f]
Pi = initial pressure
Vi = initial volume
Ti = initial temperature
Pf = final pressure
Vf= final volume
Tf = final temperature
Derivation of the Combined Gas Law
The combined gas law is derived from combiningย Charlesโ Law,ย Boyleโ s Law, andย Gay-Lussacโ s Law.
When we combine all these relationships, we obtain the combined gas law equation.
The ideal gas law is obtained by expanding the combined gas law when the number of gas moles (n) is not kept constant. You can derive the other gas laws by holding different variables constant and
working backward from the ideal gas law. In the combined gas law, the moles of gas (n) would be kept constant.
Example Problem: Combined Gas Law
Determine the volume of a gas under the conditions of 760.0 mm Hg pressure and 273 K temperature, given that 2.00 liters of the gas were collected at 745.0 mm Hg pressure and 25.0 ยฐC temperature.
To convert a temperature from Celsius to Kelvin, one must add 273.15 to the given temperature. Therefore, to convert 25.0 ยฐC to the Kelvin scale, one would add 273.15 to obtain the equivalent
temperature in Kelvin. This results in a temperature of 298 Kelvin.
P[1] = 745.0 mm Hg
V[1] = 2.00 L
T[1] = 298 K
P[2] = 760.0 mm Hg
V[2] = x
T[2] = 273 K
Arrange the formula
P[1]V[1] / T[1] = P[2]V[2 ]/ T[2]
P[1]V[1]T[2] = P[2]V[2]T[1]
V[2] = (P[1]V[1]T[2]) / (P[2]T[1])
Now, substituting the Values;
V[2] = (745.0 mm Hg ยท 2.00 L ยท 273 K) / (760 mm Hg ยท 298 K)
V[2] = 1.796 L
V[2] = 1.80 L
• Castka, Joseph F.; Metcalfe, H. Clark; Davis, Raymond E.; Williams, John E. (2002). Modern Chemistry. Holt, Rinehart and Winston. ISBN 978-0-03-056537-3.
• Raff, Lionel M. (2001) Principles of Physical Chemistry (1st ed.). Pearson College Div. ISBN: 978-0130278050.
• Helmenstine, Anne Marie, Ph.D. “The Combined Gas Law in Chemistry.” ThoughtCo, Nov. 19, 2022, thoughtco.com/definition-of-combined-gas-law-604936.
• https://byjus.com/combined-gas-law-formula/
• https://chemistrytalk.org/combined-gas-law-chemistry/
• https://collegedunia.com/exams/combined-gas-law-formula-evolution-and-solved-examples-chemistry-articleid-1812
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Aquifer with linear boundary between the two zones
Analytical equations and methods for aquifer test analysis
Unsteady-state flow equations
1. Maksimov’s solution
Drawdown in the main aquifer
Drawdown in the adjoining aquifer
2. Fenske’s solution
Drawdown in the main aquifer
Drawdown in the adjoining aquifer
Methods of analysis and parameters being estimated for aquifer with linear discontinuity
Plot Method Parameters Comments
straight line
straight line
straight line
matching matching by separate points
matching matching by separate points
straight line
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Understanding Tennessee’s Standards for Mathematical Practices
Tennessee’s Mathematics Content Standards should be taught through the lens of the Standards of Mathematical Practices (MPs). Therefore, working to appropriately embed these MP standards throughout
each lesson is fundamental to helping Tennessee students gain mastery of mathematical concepts and skills.
“Being successful in mathematics requires that development of approaches, practices, and habits of mind be in place as one strives to develop mathematical fluency, procedural skills, and
conceptual understanding. The Standards for Mathematical Practice are meant to address these areas of expertise that teachers should seek to develop within their students. These approaches,
practices, and habits of mind can be summarized as ‘processes and proficiencies’ that successful mathematicians have as a part of their work in mathematics." - Tennessee Mathematics Standards
Get the Official 2022 Tennessee Standards for Mathematical Practice Classroom Poster
The complete wording of each MP can be found here: Standards for Mathematical Practice. However, we believe the MPs can be made more impactful if the text is interpreted in terms of behaviors one
should expect to observe as students engage with each MP. Ideally, students would view the bulleted statements as the success criteria by which they could self-assess their ability to fully reason
with mathematics on a regular basis.
MP1: Make sense of problems and persevere in solving them.
Mathematicians who make sense of problems and persevere in solving them:
□ Analyze the problem in a way that makes sense given the task.
□ Create a plan for solving the problem.
□ Continually ask themselves, “Does this make sense?” as they work through a task.
□ Modify their methods as necessary when solving a challenging task.
□ Make sense of and explain connections among mathematical ideas and various representations.
□ Stay engaged and maintain a positive mindset when working to solve problems.
□ Verify their solutions, as well as understand solutions presented by others.
To incorporate MP1 effectively in the classroom, teachers can help students by cultivating a community of growth mindset learners. They can foster perseverance in students by choosing tasks that are
interesting and challenging, involving meaningful mathematics. Teachers should look to present problems that allow for multiple strategies and multiple solutions. Importantly, teachers should
recognize students’ efforts when solving challenging problems.
MP2: Reason abstractly and quantitatively.
Mathematicians who reason abstractly and quantitatively:
□ Make sense of quantities and relationships in problem situations.
□ Flexibly move between the context and abstract representations of problem situations.
☆ Remove the context from a problem situation.
☆ Produce a symbolic representation of the problem.
☆ Manipulate the symbolic representation.
☆ And put it back in context.
□ Express connections between concepts and representations.
□ Understand and correctly use the units involved in the problem situation.
□ Attend to the meaning of quantities when choosing a representation based on a given context.
□ Flexibly use properties of operations.
To incorporate MP2 effectively in the classroom, teachers can help students by providing opportunities for students to use manipulatives when investigating concepts. They can guide students from
concrete to pictorial to abstract representations as understanding progresses, and expect students to give meaning to all quantities in a task. Additionally, teachers should give students ample
opportunities to see how various representations are useful in different situations.
MP3: Construct viable arguments and critique the reasoning of others.
Mathematicians who construct viable arguments and critique the reasoning of others:
□ Construct possible arguments based on stated assumptions, definitions, and previously established results.
□ Use counterexamples appropriately.
□ Analyze the arguments of others and ask probing questions to clarify or improve the arguments.
□ Compare the effectiveness and efficiency of two or more plausible arguments.
□ Recognize errors and suggest how to correct those errors.
□ Justify results by explaining mathematical ideas, vocabulary, and methods effectively.
To incorporate MP3 effectively in the classroom, teachers should establish a culture in which students ask questions of the teacher and their peers, and error is an opportunity for learning. They
should select, sequence, and present student work to advance and deepen understanding of correct and increasingly efficient methods. Additionally, teachers should help students develop their ability
to justify methods and compare their responses to the responses of their peers.
MP4: Model with mathematics.
Mathematicians who model with mathematics:
□ Apply prior knowledge to independently model, understand, and represent real-world problems.
□ Identify, analyze, and draw conclusions about quantities using tools such as diagrams, two-way tables, graphs, flowcharts, and formulas.
□ Use assumptions and approximations to make a problem simpler.
□ Perform investigations to gather data or determine if a method is appropriate.
□ Check to see if an answer makes sense within the context of a situation, possibly improving/revising the model.
To incorporate MP4 effectively in the classroom, teachers should provide opportunities for students to create models, both concrete and abstract, and perform investigations. They should ask students
to justify their choice of model and the thinking behind it, as well as the appropriateness of the model chosen. Teachers can also assist students in seeing and making connections among different
MP5: Use appropriate tools strategically.
Mathematicians who use appropriate tools strategically:
□ Make good decisions about which tools to use and how to use the tools in specific situations.
□ Use technological tools to visualize, analyze, and help solve problems, and to deepen math knowledge.
□ Identify and use external math resources to pose and solve problems.
□ Use estimation and other mathematical knowledge to detect possible errors.
To incorporate MP5 effectively in the classroom, teachers should help students see why the use of manipulatives, rulers, compasses, protractors, calculators, statistical software, and other tools
will aid their problem-solving processes. They should make sure that math tools are readily available and frequently model the use of appropriate tools. Teachers should give students a choice of
materials/tools and have discussions with them about their choices to lead them to use appropriate tools strategically.
MP6: Attend to precision.
Mathematicians who attend to precision:
□ Communicate their understanding precisely to others using proper mathematical language.
□ Use clear and precise definitions in discussion with others and in their own reasoning.
□ Calculate accurately and efficiently, and express answers to the appropriate degree of precision.
□ Are careful about the meaning of units and clearly and accurately label diagrams.
To incorporate MP6 effectively in the classroom, teachers should consistently model the use of precise mathematics language and symbols and expect their students to do the same. They should ask
students to identify symbols, quantities, and units in a clear manner. Teachers should also set expectations as to how precise the solution needs to be and help students understand when estimates are
appropriate for the situation.
MP7: Look for and make use of structure.
Mathematicians who look for and make use of structure:
□ Recognize patterns, structures, and relationships within quantities, processes, and expressions, and across topics.
□ Use properties and operations to make sense of problems.
□ Use patterns, structures, multiple representations and relationships to identify an effective and efficient solution path.
□ Decompose a complex problem into manageable parts.
□ Use patterns and repeated reasoning to solve complex problems.
To incorporate MP7 effectively in the classroom, teachers should encourage students to look for structure, not simply to apply a rule or structure given by the teacher. This means encouraging
students to notice key features, such as identifying characteristics of shapes or noticing whether the order in which you add numbers changes the sum. Patterning activities also support attention to
structure. Teachers can ask young children to identify the part of a pattern that repeats over and over and can ask older children to figure out a rule for predicting a new instance in a growing
pattern or function table.
MP8: Look for and express regularity in repeated reasoning.
Mathematically proficient students notice if calculations are repeated and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are
repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the
line through (1, 2) with slope 3, middle school students might abstract the equation (y - 2)/(x - 1) = 3. Noticing the regularity in the way terms cancel when expanding (x - 1)(x + 1), (x - 1)(x^2 +
x + 1), and (x - 1)(x^3 + x^2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight
of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.
Mathematicians who look for and express regularity in repeated reasoning:
□ Identify repeated reasoning in calculations or processes to make generalizations.
□ Build on prior learning to make and apply generalizations to new situations.
□ Use recognition of repeated reasoning to help identify and understand procedural shortcuts.
□ Continually evaluate the reasonableness of intermediate results, such as comparing results to estimates.
To incorporate MP8 effectively in the classroom, teachers should present several opportunities to reveal patterns or repetition in thinking, so students can make a generalization or rule. They should
help students connect new tasks to prior concepts and tasks, to extend learning of a mathematic concept. Additionally, teachers should ask for predictions about solutions at midpoints throughout the
solution process.
It is important to embed these MPs into lessons every day. Teachers should look for ways to integrate appropriate MPs in authentic ways to deepen students’ understanding of the mathematics content
standards. Ultimately, the goal should be to engage students in rich, high-level mathematical tasks that support the approaches, practices, and habits of mind which are called for within these
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Bayesian Statistical Inference
Next: MEM Images Up: Maximum Entropy Previous: Maximum Entropy Contents
This method, or class of methods, is easy to describe in the framework of an approach to statistical inference (i.e all of experimental science?) which is more than two hundred years old, dating from
1763! Bayes Theorem about conditional probabilities states that
As a theorem, it is an easy consequence of the definitions of joint probabilities (denoted by and both happen (
The theorem acquires its application to statistical inference when we think of
Going from
Next: MEM Images Up: Maximum Entropy Previous: Maximum Entropy Contents NCRA-TIFR
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Chemistry 1 Honors
General Course Information and Notes
General Notes
While the content focus of this course is consistent with the Chemistry I course, students will explore these concepts in greater depth. In general, the academic pace and rigor will be greatly
increased for honors level course work. Laboratory investigations that include the use of scientific inquiry, research, measurement, problem solving, laboratory apparatus and technologies,
experimental procedures, and safety procedures are an integral part of this course. The National Science Teachers Association (NSTA) recommends that at the high school level, all students should be
in the science lab or field, collecting data every week. School laboratory investigations (labs) are defined by the National Research Council (NRC) as an experience in the laboratory, classroom, or
the field that provides students with opportunities to interact directly with natural phenomena or with data collected by others using tools, materials, data collection techniques, and models (NRC,
2006, p. 3). Laboratory investigations in the high school classroom should help all students develop a growing understanding of the complexity and ambiguity of empirical work, as well as the skills
to calibrate and troubleshoot equipment used to make observations. Learners should understand measurement error; and have the skills to aggregate, interpret, and present the resulting data (National
Research Council, 2006, p.77; NSTA, 2007).
Special Notes:
Instructional Practices:
Teaching from a range of complex text is optimized when teachers in all subject areas implement the following strategies on a routine basis:
1. Ensuring wide reading from complex text that varies in length.
2. Making close reading and rereading of texts central to lessons.
3. Emphasizing text-specific complex questions, and cognitively complex tasks, reinforce focus on the text and cultivate independence.
4. Emphasizing students supporting answers based upon evidence from the text.
5. Providing extensive research and writing opportunities (claims and evidence).
Science and Engineering Practices (NRC Framework for K-12 Science Education, 2010)
• Asking questions (for science) and defining problems (for engineering).
• Developing and using models.
• Planning and carrying out investigations.
• Analyzing and interpreting data.
• Using mathematics, information and computer technology, and computational thinking.
• Constructing explanations (for science) and designing solutions (for engineering).
• Engaging in argument from evidence.
• Obtaining, evaluating, and communicating information.
Honors and Advanced Level Course Note: Advanced courses require a greater demand on students through increased academic rigor. Academic rigor is obtained through the application, analysis,
evaluation, and creation of complex ideas that are often abstract and multi-faceted. Students are challenged to think and collaborate critically on the content they are learning. Honors level rigor
will be achieved by increasing text complexity through text selection, focus on high-level qualitative measures, and complexity of task. Instruction will be structured to give students a deeper
understanding of conceptual themes and organization within and across disciplines. Academic rigor is more than simply assigning to students a greater quantity of work.
Literacy Standards in Science
Secondary science courses include reading standards for literacy in science and technical subjects 6-12 and writing standards for literacy in history/social studies, science, and technical subjects
6-12. The courses also include speaking and listening standards. For a complete list of standards required for this course click on the blue tile labeled course standards. You may also download the
complete course including all required standards and notes sections using the export function located at the top of this page.
English Language Development ELD Standards Special Notes Section:
Teachers are required to provide listening, speaking, reading and writing instruction that allows English language learners (ELL) to communicate information, ideas and concepts for academic success
in the content area of Science. For the given level of English language proficiency and with visual, graphic, or interactive support, students will interact with grade level words, expressions,
sentences and discourse to process or produce language necessary for academic success The ELD standard should specify a relevant content area concept or topic of study chosen by curriculum developers
and teachers which maximizes an ELL's need for communication and social skills. To access an ELL supporting document which delineates performance definitions and descriptors, please click on the
following link: https://cpalmsmediaprod.blob.core.windows.net/uploads/docs/standards/eld/sc.pdf
Additional Instructional Resources:
A.V.E. for Success Collection is provided by the Florida Association of School Administrators: http://www.fasa.net/4DCGI/cms/review.html?Action=CMS_Document&DocID=139. Please be aware that these
resources have not been reviewed by CPALMS and there may be a charge for the use of some of them in this collection.
General Information
Course Number: 2003350
Course Path:
Abbreviated Title: CHEM 1 HON
Number of Credits: One (1) credit
Course Length: Year (Y)
Course Type: Core Academic Course
Course Level: 3
Course Status: Course Approved
Grade Level(s): 9,10,11,12
Graduation Requirement: Equally Rigorous Science
Educator Certifications
One of these educator certification options is required to teach this course.
Equivalent Courses
Any of these are equivalent to the course required for graduation or certification.
Student Resources
Vetted resources students can use to learn the concepts and skills in this course.
Original Student Tutorials
Graphing Linear Functions Part 1: Table of Values:
Learn how to graph linear functions by creating a table of values based on the equation in this interactive tutorial.
This is part 1 of a series of tutorials on linear functions.
Type: Original Student Tutorial
Quadratic Function Part 2: Launches:
Learn about different formats of quadratic equations and their graphs with experiments involving launching and shooting of balls in this interactive tutorial.
This is part 2 of a two-part series: Click HERE to open part 1.
Type: Original Student Tutorial
Quadratic Functions Part 1: Ball Games:
Join us as we watch ball games and explore how the height of a ball bounce over time is represented by quadratic functions, which provides opportunities to interpret key features of the function in
this interactive tutorial.
This is part 1 of a two-part series: Click HERE to open part 2.
Type: Original Student Tutorial
Matter, Matters Part 2: Physical and Chemical Changes:
Explore, identify, and describe chemical and physical changes in matter with this interactive tutorial.
This is part 2 of 2-part series, click HERE to view part 1.
Type: Original Student Tutorial
Matter, Matters Part 1: Properties of Matter:
Explore and define matter, properties of matter, and the difference between physical and chemical properties in this interactive tutorial.
This is part 1 of 2-part series, click HERE to view part 2.
Type: Original Student Tutorial
Atomic History and Subatomic Particles:
Explore the history and development of the atomic model and characteristics of subatomic particles (protons, neutrons, electrons) in this interactive tutorial.
Type: Original Student Tutorial
Newton's Insight: Standing on the Shoulders of Giants:
Discover how Isaac Newton's background, talents, interests, and goals influenced his groundbreaking work in this interactive tutorial.
This is part 4 in a 4-part series. Click below to explore the other tutorials in the series.
Type: Original Student Tutorial
Movies Part 2: What’s the Spread?:
Follow Jake along as he relates box plots with other plots and identifies possible outliers in real-world data from surveys of moviegoers' ages in part 2 in this interactive tutorial.
This is part 2 of 2-part series, click HERE to view part 1.
Type: Original Student Tutorial
Movies Part 1: What's the Spread?:
Follow Jake as he displays real-world data by creating box plots showing the 5 number summary and compares the spread of the data from surveys of the ages of moviegoers in part 1 of this interactive
This is part 1 of 2-part series, click HERE to view part 2.
Type: Original Student Tutorial
Exponential Functions Part 3: Decay:
Learn about exponential decay as you calculate the value of used cars by examining equations, graphs, and tables in this interactive tutorial.
Type: Original Student Tutorial
Hot or Not? A Guide to Exothermic and Endothermic Reactions (Part 1):
Discover why some reactions leave you feeling warmer while others leave you feeling cooler in this interactive tutorial.
This is part 1 in a two-part series. Click to open Part 2 on endothermic and exothermic phase changes.
Type: Original Student Tutorial
Linear Functions: Jobs:
Learn how to interpret key features of linear functions and translate between representations of linear functions through exploring jobs for teenagers in this interactive tutorial.
Type: Original Student Tutorial
Hot or Not? A Guide to Exothermic and Endothermic Phase Changes:
Explore the differences between endothermic and exothermic phase changes in this interactive tutorial.
This is part 2 in a two-part series. Click to open Part 1 on endothermic and exothermic reactions.
Type: Original Student Tutorial
Exponential Functions Part 2: Growth:
Learn about exponential growth in the context of interest earned as money is put in a savings account by examining equations, graphs, and tables in this interactive tutorial.
Type: Original Student Tutorial
Exponential Functions Part 1:
Learn about exponential functions and how they are different from linear functions by examining real world situations, their graphs and their tables in this interactive tutorial.
Type: Original Student Tutorial
Turtles and Towns:
Explore the impacts on sea turtles, humans, and the economy when we live, work, and play at the beach with this interactive tutorial.
Type: Original Student Tutorial
How Viral Disease Spreads:
Learn how scientists measure viral spread and use this information to make recommendations for the public in this interactive tutorial.
Type: Original Student Tutorial
Evaluating Sources of Information:
Learn how to identify different sources of scientific claims and to evaluate their reliability in this interactive tutorial.
Type: Original Student Tutorial
The Year-Round School Debate: Identifying Faulty Reasoning – Part Two:
This is Part Two of a two-part series. Learn to identify faulty reasoning in this interactive tutorial series. You'll learn what some experts say about year-round schools, what research has been
conducted about their effectiveness, and how arguments can be made for and against year-round education. Then, you'll read a speech in favor of year-round schools and identify faulty reasoning within
the argument, specifically the use of hasty generalizations.
Make sure to complete Part One before Part Two! Click HERE to launch Part One.
Type: Original Student Tutorial
The Year-Round School Debate: Identifying Faulty Reasoning – Part One:
Learn to identify faulty reasoning in this two-part interactive English Language Arts tutorial. You'll learn what some experts say about year-round schools, what research has been conducted about
their effectiveness, and how arguments can be made for and against year-round education. Then, you'll read a speech in favor of year-round schools and identify faulty reasoning within the argument,
specifically the use of hasty generalizations.
Make sure to complete both parts of this series! Click HERE to open Part Two.
Type: Original Student Tutorial
Evaluating an Argument – Part Four: JFK’s Inaugural Address:
Examine President John F. Kennedy's inaugural address in this interactive tutorial. You will examine Kennedy's argument, main claim, smaller claims, reasons, and evidence.
In Part Four, you'll use what you've learned throughout this series to evaluate Kennedy's overall argument.
Make sure to complete the previous parts of this series before beginning Part 4.
• Click HERE to launch Part One.
• Click HERE to launch Part Two.
• Click HERE to launch Part Three.
Type: Original Student Tutorial
Evaluating an Argument – Part Three: JFK’s Inaugural Address:
Examine President John F. Kennedy's inaugural address in this interactive tutorial. You will examine Kennedy's argument, main claim, smaller claims, reasons, and evidence. By the end of this
four-part series, you should be able to evaluate his overall argument.
In Part Three, you will read more of Kennedy's speech and identify a smaller claim in this section of his speech. You will also evaluate this smaller claim's relevancy to the main claim and evaluate
Kennedy's reasons and evidence.
Make sure to complete all four parts of this series!
• Click HERE to launch Part One.
• Click HERE to launch Part Two.
• Click HERE to launch Part Four.
Type: Original Student Tutorial
Ready for Takeoff! -- Part Two:
This is Part Two of a two-part tutorial series. In this interactive tutorial, you'll practice identifying a speaker's purpose using a speech by aviation pioneer Amelia Earhart. You will examine her
use of rhetorical appeals, including ethos, logos, pathos, and kairos. Finally, you'll evaluate the effectiveness of Earhart's use of rhetorical appeals.
Be sure to complete Part One first. Click here to launch PART ONE.
Type: Original Student Tutorial
Ready for Takeoff! -- Part One:
This is Part One of a two-part tutorial series. In this interactive tutorial, you'll practice identifying a speaker's purpose using a speech by aviation pioneer Amelia Earhart. You will examine her
use of rhetorical appeals, including ethos, logos, pathos, and kairos. Finally, you'll evaluate the effectiveness of Earhart's use of rhetorical appeals.
Click here to launch PART TWO.
Type: Original Student Tutorial
Expository Writing: Eyes in the Sky (Part 4 of 4):
Practice writing different aspects of an expository essay about scientists using drones to research glaciers in Peru. This interactive tutorial is part four of a four-part series. In this final
tutorial, you will learn about the elements of a body paragraph. You will also create a body paragraph with supporting evidence. Finally, you will learn about the elements of a conclusion and
practice creating a “gift.”
This tutorial is part four of a four-part series. Click below to open the other tutorials in this series.
• Expository Writing: Eyes in the Sky (Part 4)
Type: Original Student Tutorial
Expository Writing: Eyes in the Sky (Part 3 of 4):
Learn how to write an introduction for an expository essay in this interactive tutorial. This tutorial is the third part of a four-part series. In previous tutorials in this series, students analyzed
an informational text and video about scientists using drones to explore glaciers in Peru. Students also determined the central idea and important details of the text and wrote an effective summary.
In part three, you'll learn how to write an introduction for an expository essay about the scientists' research.
This tutorial is part three of a four-part series. Click below to open the other tutorials in this series.
• Expository Writing: Eyes in the Sky (Part 3)
Type: Original Student Tutorial
Drones and Glaciers: Eyes in the Sky (Part 2 of 4):
Learn how to identify the central idea and important details of a text, as well as how to write an effective summary in this interactive tutorial. This tutorial is the second tutorial in a four-part
series that examines how scientists are using drones to explore glaciers in Peru.
This tutorial is part two of a four-part series. Click below to open the other tutorials in this series.
• Drones and Glaciers: Eyes in the Sky (Part 2)
Type: Original Student Tutorial
Drones and Glaciers: Eyes in the Sky (Part 1 of 4):
Learn about how researchers are using drones, also called unmanned aerial vehicles or UAVs, to study glaciers in Peru. In this interactive tutorial, you will practice citing text evidence when
answering questions about a text.
This tutorial is part one of a four-part series. Click below to open the other tutorials in this series.
• Drones and Glaciers: Eyes in the Sky (Part 1)
Type: Original Student Tutorial
Ecological Data Analysis:
See how data are interpreted to better understand the reproductive strategies taken by sea anemones with this interactive tutorial.
Type: Original Student Tutorial
Ecology Sampling Strategies:
Examine field sampling strategies used to gather data and avoid bias in ecology research. This interactive tutorial features the CPALMS Perspectives video .
Type: Original Student Tutorial
The Mystery of Muscle Cell Metabolism:
Explore the mystery of muscle cell metabolism and how cells are able to meet the need for a constant supply of energy. In this interactive tutorial, you'll identify the basic structure of adenosine
triphosphate (ATP), explain how ATP’s structure is related it its job in the cell, and connect this role to energy transfers in living things.
Type: Original Student Tutorial
Data and Frequencies:
Learn to define, calculate, and interpret marginal frequencies, joint frequencies, and conditional frequencies in the context of the data with this interactive tutorial.
Type: Original Student Tutorial
Eliminating Exotics: Identifying and Assessing Research for Quality and Usefulness:
Learn how to better conduct research in this interactive tutorial. You'll learn to distinguish relevant from irrelevant sources when conducting research on a specific topic. In addition, you'll
practice identifying authoritative sources and selecting the appropriate keywords to find quality sources for your topic.
Type: Original Student Tutorial
Comparing Mitosis and Meiosis:
Compare and contrast mitosis and meiosis in this interactive tutorial. You'll also relate them to the processes of sexual and asexual reproduction and their consequences for genetic variation.
Type: Original Student Tutorial
Evolution: Examining the Evidence:
Learn how to identify explicit evidence and understand implicit meaning in a text.
You should be able to explain how different types of scientific evidence support the theory of evolution, including direct observation, fossils, DNA, biogeography, and comparative anatomy and
Type: Original Student Tutorial
Graphing Quadratic Functions:
Follow as we discover key features of a quadratic equation written in vertex form in this interactive tutorial.
Type: Original Student Tutorial
Observation vs. Inference:
Learn how to identify explicit evidence and understand implicit meaning in a text and demonstrate how and why scientific inferences are drawn from scientific observation and be able to identify
examples in biology.
Type: Original Student Tutorial
Cool Case Files:
Learn that a scientific theory is the culmination of many experiments and supplies the most powerful explanation that scientists have to offer with this interactive tutorial.
Type: Original Student Tutorial
Cancer: Mutated Cells Gone Wild!:
Explore the relationship between mutations, the cell cycle, and uncontrolled cell growth which may result in cancer with this interactive tutorial.
Type: Original Student Tutorial
Water and Life:
Learn how the chemical properties of water relate to its physical properties and make it essential for life with this interactive tutorial.
Type: Original Student Tutorial
Question Quest:
Learn to distinguish between questions that can be answered by science and questions that science cannot answer. This interactive tutorial will help you distinguish between science and other ways of
knowing, including art, religion, and philosophy.
Type: Original Student Tutorial
Diving the Depths of Underwater Life:
Learn how the distribution of aquatic life forms is affected by light, temperature, and salinity with this interactive tutorial.
Type: Original Student Tutorial
Chemistry With a Conscience:
Explore green chemistry and what it means to be benign by design in this interactive tutorial.
Type: Original Student Tutorial
Educational Game
Stop Disasters Before They Happen:
Students attempt to save towns from damage prior to the arrival of several different natural disasters. Students will learn the importance of early prevention and actions to protect others,
themselves and their property when faced with a natural disaster. Certain disasters are more appropriate for particular grade levels. Each scenario takes between 20 and 45 minutes to play, depending
on the disaster for which your students are trying to prepare. There are five scenarios available, hurricane, tsunami, flood, earthquake, and wildfire. Each scenario can be played on easy, medium or
hard difficulty levels. As with life, there are no "perfect solutions" to each scenario and no "perfect score", so students can play multiple times and the scenarios will still be slightly
different.These simulation are part of a larger website that provides multiple links for natural disasters.
Type: Educational Game
Educational Software / Tool
Two Way Frequency Excel Spreadsheet:
This Excel spreadsheet allows the educator to input data into a two way frequency table and have the resulting relative frequency charts calculated automatically on the second sheet. This resource
will assist the educator in checking student calculations on student-generated data quickly and easily.
Steps to add data: All data is input on the first spreadsheet; all tables are calculated on the second spreadsheet
1. Modify column and row headings to match your data.
2. Input joint frequency data.
3. Click the second tab at the bottom of the window to see the automatic calculations.
Type: Educational Software / Tool
Lesson Plans
Elasticity: Studying How Solids Change Shape and Size:
This lesson's primary focus is to introduce high school students to the concept of Elasticity, which is one of the fundamental concepts in the understanding of the physics of deformation in solids.
The main learning objectives are: (1) To understand the essential concept of Elasticity and be able to distinguish simple solids objects based on degree and extent of their elastic properties; (2) To
appreciate the utility of the elastic force vs. deformation curve through experiments; (3) To be aware of potential sources of error present in such experiments and identify corrective measures; and
(4) To appreciate the relevance of Elasticity in practical applications.
Type: Lesson Plan
CO2: Find Out What It Means to You:
This BLOSSOMS lesson discusses Carbon Dioxide, and its impact on climate change. The main learning objective is for students to become more familiar with human production of Carbon Dioxide gas, as
well as to gain an awareness of the potential for this gas to effect the temperature of Earth’s atmosphere. This lesson should take about an hour to complete. In order to complete the lesson, the
teacher will need: printed copies of signs representing the different products and processes that take place in the carbon cycle (included), samples of matter that represent those products, handouts
for the students to create a graphic of the carbon cycle (included) and graph paper or graphing software for students to create graphs. In the breaks of this BLOSSOMS lesson, students will be
creating models of the carbon cycle as well as observing experiments and analyzing data from them. It is hoped that this lesson will familiarize students with ways in which carbon moves through our
environment and provide them with some personal connection to the impact that an increased concentration of CO[2] can have on air temperature. The goal is to spark their interest and hopefully to
encourage them to ask and investigate more questions about the climate.
Type: Lesson Plan
Perspectives Video: Experts
Jumping Robots and Quadratics:
<p>Jump to it and learn more about how quadratic equations are used in robot navigation problem solving!</p>
Type: Perspectives Video: Expert
Pendulums and Energy Transformations:
Explore how pendulums show the transformation of gravitational potential energy to kinetic energy and back with Dr. Simon Capstick in this engaging video. Don't miss his broken-nose defying test of
the physics with a bowling ball pendulum.
Download the CPALMS Perspectives video student note taking guide.
Type: Perspectives Video: Expert
Oil Fingerprinting:
Humans aren't the only ones who get their fingerprints taken. Learn how this scientist is like a crime scene investigator using oil "fingerprints" to explain the orgins of spilled oil.
Download the CPALMS Perspectives video student note taking guide.
Type: Perspectives Video: Expert
Perspectives Video: Professional/Enthusiasts
Unit Conversions:
<p>Get fired up as you learn more about ceramic glaze recipes and mathematical units.</p>
Type: Perspectives Video: Professional/Enthusiast
Optical Spectroscopy: Using Electromagnetic Waves to Detect Fires:
<p>Hydrogen is used to launch spacecraft, but accidental fires are difficult to see. Learn about the physics of these fires and how we detect them.</p>
Type: Perspectives Video: Professional/Enthusiast
Problem-Solving Tasks
Speed Trap:
The purpose of this task is to allow students to demonstrate an ability to construct boxplots and to use boxplots as the basis for comparing distributions.
Type: Problem-Solving Task
Musical Preferences:
This problem solving task asks students to make deductions about the kind of music students enjoy by examining data in a two-way table.
Type: Problem-Solving Task
SAT Scores:
This problem solving task challenges students to answer probability questions about SAT scores, using distribution and mean to solve the problem.
Type: Problem-Solving Task
Haircut Costs:
This problem could be used as an introductory lesson to introduce group comparisons and to engage students in a question they may find amusing and interesting.
Type: Problem-Solving Task
Coffee and Crime:
This problem solving task asks students to examine the relationship between shops and crimes by using a correlation coefficient. The implications of linking correlation with causation are discussed.
Type: Problem-Solving Task
Should We Send Out a Certificate?:
The purpose of this task is to have students complete normal distribution calculations and to use properties of normal distributions to draw conclusions.
Type: Problem-Solving Task
Do You Fit in This Car?:
This task requires students to use the normal distribution as a model for a data distribution. Students must use given means and standard deviations to approximate population percentages.
Type: Problem-Solving Task
Random Walk III:
The task provides a context to calculate discrete probabilities and represent them on a bar graph.
Type: Problem-Solving Task
How thick is a soda can? (Variation II):
This problem solving task asks students to explain which measurements are needed to estimate the thickness of a soda can. Multiple solution processes are presented.
Type: Problem-Solving Task
How thick is a soda can? (Variation I):
This problem solving task challenges students to find the surface area of a soda can, calculate how many cubic centimeters of aluminum it contains, and estimate how thick it is.
Type: Problem-Solving Task
How many leaves on a tree? (Version 2):
This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree.
Type: Problem-Solving Task
How many leaves on a tree?:
This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree.
Type: Problem-Solving Task
How many cells are in the human body?:
This problem solving task challenges students to apply the concepts of mass, volume, and density in the real-world context to find how many cells are in the human body.
Type: Problem-Solving Task
Archimedes and the King's Crown:
This problem solving task uses the tale of Archimedes and the King of Syracuse's crown to determine the volume and mass of gold and silver.
Type: Problem-Solving Task
As the Wheel Turns:
In this task, students use trigonometric functions to model the movement of a point around a wheel and, through space. Students also interpret features of graphs in terms of the given real-world
Type: Problem-Solving Task
Finding Parabolas through Two Points:
This problem-solving task challenges students to find all quadratic functions described by given equation and coordinates, and describe how the graphs of those functions are related to one another.
Type: Problem-Solving Task
Warming and Cooling:
This task is meant to be a straight-forward assessment task of graph reading and interpreting skills. This task helps reinforce the idea that when a variable represents time, t = 0 is chosen as an
arbitrary point in time and positive times are interpreted as times that happen after that.
Type: Problem-Solving Task
Throwing Baseballs:
This task could be used for assessment or for practice. It allows students to compare characteristics of two quadratic functions that are each represented differently, one as the graph of a quadratic
function and one written out algebraically. Specifically, students are asked to determine which function has the greatest maximum and the greatest non-negative root.
Type: Problem-Solving Task
Average Cost:
This task asks students to find the average, write an equation, find the domain, and create a graph of the cost of producing DVDs.
Type: Problem-Solving Task
Weed Killer:
The principal purpose of the task is to explore a real-world application problem with algebra, working with units and maintaining reasonable levels of accuracy throughout. Students are asked to
determine which product will be the most economical to meet the requirements given in the problem.
Type: Problem-Solving Task
Telling a Story with Graphs:
In this task students are given graphs of quantities related to weather. The purpose of the task is to show that graphs are more than a collection of coordinate points; they can tell a story about
the variables that are involved, and together they can paint a very complete picture of a situation, in this case the weather. Features in one graph, like maximum and minimum points, correspond to
features in another graph. For example, on a rainy day, the solar radiation is very low, and the cumulative rainfall graph is increasing with a large slope.
Type: Problem-Solving Task
Logistic Growth Model, Explicit Version:
This problem introduces a logistic growth model in the concrete settings of estimating the population of the U.S. The model gives a surprisingly accurate estimate and this should be contrasted with
linear and exponential models.
Type: Problem-Solving Task
Logistic Growth Model, Abstract Version:
This task is for instructional purposes only and students should already be familiar with some specific examples of logistic growth functions. The goal of this task is to have students appreciate how
different constants influence the shape of a graph.
Type: Problem-Solving Task
How Is the Weather?:
This task can be used as a quick assessment to see if students can make sense of a graph in the context of a real world situation. Students also have to pay attention to the scale on the vertical
axis to find the correct match. The first and third graphs look very similar at first glance, but the function values are very different since the scales on the vertical axes are very different. The
task could also be used to generate a group discussion on interpreting functions given by graphs.
Type: Problem-Solving Task
Dinosaur Bones:
The purpose of this task is to illustrate through an absurd example the fact that in real life quantities are reported to a certain level of accuracy, and it does not make sense to treat them as
having greater accuracy.
Type: Problem-Solving Task
Bus and Car:
This task operates at two levels. In part it is a simple exploration of the relationship between speed, distance, and time. Part (c) requires understanding of the idea of average speed, and gives an
opportunity to address the common confusion between average speed and the average of the speeds for the two segments of the trip.
At a higher level, the task addresses MAFS.912.N-Q.1.3, since realistically neither the car nor the bus is going to travel at exactly the same speed from beginning to end of each segment; there is
time traveling through traffic in cities, and even on the autobahn the speed is not constant. Thus students must make judgments about the level of accuracy with which to report the result.
Type: Problem-Solving Task
Accuracy of Carbon 14 Dating I:
This task examines, from a mathematical and statistical point of view, how scientists measure the age of organic materials by measuring the ratio of Carbon 14 to Carbon 12. The focus here is on the
statistical nature of such dating.
Type: Problem-Solving Task
Accuracy of Carbon 14 Dating II:
This task examines, from a mathematical and statistical point of view, how scientists measure the age of organic materials by measuring the ratio of Carbon 14 to Carbon 12. The focus here is on the
statistical nature of such dating.
Type: Problem-Solving Task
Fuel Efficiency:
The problem requires students to not only convert miles to kilometers and gallons to liters but they also have to deal with the added complication of finding the reciprocal at some point.
Type: Problem-Solving Task
How Much Is a Penny Worth?:
This task asks students to calculate the cost of materials to make a penny, utilizing rates of grams of copper.
Type: Problem-Solving Task
Runner's World:
Students are asked to use units to determine if the given statement is valid.
Type: Problem-Solving Task
Harvesting the Fields:
This is a challenging task, suitable for extended work, and reaching into a deep understanding of units. Students are given a scenario and asked to determine the number of people required to complete
the amount of work in the time described. The task requires students to exhibit , Make sense of problems and persevere in solving them. An algebraic solution is possible but complicated; a numerical
solution is both simpler and more sophisticated, requiring skilled use of units and quantitative reasoning. Thus the task aligns with either MAFS.912.A-CED.1.1 or MAFS.912.N-Q.1.1, depending on the
Type: Problem-Solving Task
Sum of Even and Odd:
Students explore and manipulate expressions based on the following statement:
A function f defined for -a < x="">< a="" is="" even="" if="" f(-x)="f(x)" and="" is="" odd="" if="" f(-x)="-f(x)" when="" -a="">< x="">< a.="" in="" this="" task="" we="" assume="" f="" is=""
defined="" on="" such="" an="" interval,="" which="" might="" be="" the="" full="" real="" line="" (i.e.,="" a="">
Type: Problem-Solving Task
Graphs of Quadratic Functions:
Students compare graphs of different quadratic functions, then produce equations of their own to satisfy given conditions.
This exploration can be done in class near the beginning of a unit on graphing parabolas. Students need to be familiar with intercepts, and need to know what the vertex is. It is effective after
students have graphed parabolas in vertex form (y=a(x–h)^2+k), but have not yet explored graphing other forms.
Type: Problem-Solving Task
Traffic Jam:
This resource poses the question, "how many vehicles might be involved in a traffic jam 12 miles long?"
This task, while involving relatively simple arithmetic, promps students to practice modeling (MP4), work with units and conversion (N-Q.1), and develop a new unit (N-Q.2). Students will also
consider the appropriate level of accuracy to use in their conclusions (N-Q.3).
Type: Problem-Solving Task
Selling Fuel Oil at a Loss:
The task is a modeling problem which ties in to financial decisions faced routinely by businesses, namely the balance between maintaining inventory and raising short-term capital for investment or
re-investment in developing the business.
Type: Problem-Solving Task
Felicia's Drive:
This task provides students the opportunity to make use of units to find the gas needed (). It also requires them to make some sensible approximations (e.g., 2.92 gallons is not a good answer to part
(a)) and to recognize that Felicia's situation requires her to round up. Various answers to (a) are possible, depending on how much students think is a safe amount for Felicia to have left in the
tank when she arrives at the gas station. The key point is for them to explain their choices. This task provides an opportunity for students to practice MAFS.K12.MP.2.1: Reason abstractly and
quantitatively, and MAFS.K12.MP.3.1: Construct viable arguments and critique the reasoning of others.
Type: Problem-Solving Task
Graphs of Power Functions:
This task requires students to recognize the graphs of different (positive) powers of x.
Type: Problem-Solving Task
The Canoe Trip, Variation 2:
The primary purpose of this task is to lead students to a numerical and graphical understanding of the behavior of a rational function near a vertical asymptote, in terms of the expression defining
the function.
Type: Problem-Solving Task
The Canoe Trip, Variation 1:
The purpose of this task is to give students practice constructing functions that represent a quantity of interest in a context, and then interpreting features of the function in the light of the
context. It can be used as either an assessment or a teaching task.
Type: Problem-Solving Task
Calories in a Sports Drink:
This problem involves the meaning of numbers found on labels. When the level of accuracy is not given we need to make assumptions based on how the information is reported. An unexpected surprise
awaits in this case, however, as no reasonable interpretation of the level of accuracy makes sense of the information reported on the bottles in parts (b) and (c). Either a miscalculation has been
made or the numbers have been rounded in a very odd way.
Type: Problem-Solving Task
Text Resources
This site presents the basic ideas of magnetism and applies these ideas to the earth's magnetic field. There are several useful diagrams and pictures interspersed throughout this lesson, as well as
links to more detailed subjects. This is an introduction to a larger collection on exploring the Earth's magnetosphere. A Spanish translation is available.
Type: Text Resource
American Elements:
This web site features an interactive periodic chart that provides information on the elements, including a description, physical and thermal properties, abundance, isotopes, ionization energy, the
element's discoverer, translations of element names into several languages, and bibliographic information on research-and-development publications involving the element. Additional information
includes technical information and information on manufactured products for elemental metals, metallic compounds, and ceramic and crystalline products. The American Elements company manufactures
engineered and advanced material products.
Type: Text Resource
Graphing Quadratic Equations:
This tutorial helps the learners to graph the equation of a quadratic function using the coordinates of the vertex of a parabola and its x- intercepts.
Type: Tutorial
Graphing Exponential Equations:
This tutorial will help you to learn about exponential functions by graphing various equations representing exponential growth and decay.
Type: Tutorial
How Polarity Makes Water Behave Strangely:
Water is both essential and unique. Many of its particular qualities stem from the fact that it consists of two hydrogen atoms and one oxygen, therefore creating an unequal sharing of electrons. From
fish in frozen lakes to ice floating on water, Christina Kleinberg describes the effects of polarity.
Type: Tutorial
Not All Scientific Studies are Created Equal:
Every day, we are bombarded by attention grabbing headlines that promise miracle cures to all of our ailments -- often backed up by a "scientific study." But what are these studies, and how do we
know if they are reliable? David H. Schwartz dissects two types of studies that scientists use, illuminating why you should always approach the claims with a critical eye.
Type: Tutorial
This tutorial will help the learners to understand the molecular structure of the water molecule, its inter- and intra-molecular bonds, and the formation of hydroxide ions.
Type: Tutorial
Refraction of Light:
This resource explores the electromagnetic spectrum and waves by allowing the learner to observe the refraction of light as it passes from one medium to another, study the relation between refraction
of light and the refractive index of the medium, select from a list of materials with different refractive indicecs, and change the light beam from white to monochromatic and observe the difference.
Type: Tutorial
Solar Cell Operation:
This resource explains how a solar cell converts light energy into electrical energy. The user will also learn about the different components of the solar cell and observe the relationship between
photon intensity and the amount of electrical energy produced.
Type: Tutorial
Basic Electromagnetic Wave Properties:
• Explore the relationship between wavelength, frequency, amplitude and energy of an electromagnetic wave
• Compare the characteristics of waves of different wavelengths
Type: Tutorial
Will an Ice Cube Melt Faster in Freshwater or Saltwater?:
With an often unexpected outcome from a simple experiment, students can discover the factors that cause and influence thermohaline circulation in our oceans. In two 45-minute class periods, students
complete activities where they observe the melting of ice cubes in saltwater and freshwater, using basic materials: clear plastic cups, ice cubes, water, salt, food coloring, and thermometers. There
are no prerequisites for this lesson but it is helpful if students are familiar with the concepts of density and buoyancy as well as the salinity of seawater. It is also helpful if students
understand that dissolving salt in water will lower the freezing point of water. There are additional follow up investigations that help students appreciate and understand the importance of the
ocean's influence on Earth's climate.
Type: Video/Audio/Animation
Inquiry and Ocean Exploration:
Ocean explorer Robert Ballard gives a TED Talk relating to the mysteries of the ocean, and the importance of its continued exploration.
Type: Video/Audio/Animation
• Observe the photosynthesis mechanism in the plant
• Learn about the main chemical reactions that takes place during photosynthesis
• Learn how solar energy is converted into chemical energy
Type: Video/Audio/Animation
Element Math Game:
Students determine the number of protons, electrons, neutrons, and nucleons for different atoms
Type: Video/Audio/Animation
Science Crossword Puzzles:
A collection of crossword puzzles that test the knowledge of students about some of the terms, processes, and classifications covered in science topics
Type: Video/Audio/Animation
Light is a Particle:
This video contains a demo that can be performed to show that light consists of particles
It also uses Lasers with different wavelengths
Type: Video/Audio/Animation
Shapes of Molecules:
• Differentiate between electron pair and molecular geometry
• Learn how to name electron pair and molecular geometries for molecules with up to six electron groups around the central atom
• Illustrate how electron pair repulsion affects bond angles
Type: Video/Audio/Animation
• Explain the concept of concentration
• Explain the effect of concentration changes on colors of solutions
• Demonstrate the effect of changing the amount of solute, or solvent, or both on the concentration of the solution
• Identify a saturated solution
Type: Video/Audio/Animation
Graphing Lines 1:
Khan Academy video tutorial on graphing linear equations: "Algebra: Graphing Lines 1"
Type: Video/Audio/Animation
Fitting a Line to Data:
Khan Academy tutorial video that demonstrates with real-world data the use of Excel spreadsheet to fit a line to data and make predictions using that line.
Type: Video/Audio/Animation
Evolving Ideas: Isn't evolution just a theory?:
This video examines the vocabulary essential for understanding the nature of science and evolution and illustrates how evolution is a powerful, well-supported scientific explanation for the
relatedness of all life. A clear definition and description of scientific theory is given.
Type: Video/Audio/Animation
Citizen Science:
In this National Science Foundation video and reading selection lab ecologist Janis Dickinson explains how she depends on citizen scientists to help her track the effects of disease, land-use change
and environmental contaminants on the nesting success of birds.
Type: Video/Audio/Animation
Virtual Manipulatives
Black body Spectrum:
In this simulation, learn about the black body spectrum of the sun, a light bulb, an oven and the earth. Adjust the temperature to see how the wavelength and intensity of the spectrum are affected.
Type: Virtual Manipulative
Build an Atom:
Build an atom out of protons, neutrons, and electrons, and see how the element, charge, and mass change. Then play a game to test your ideas!
Type: Virtual Manipulative
Periodic Table:
This unique periodic table presents the elements in an interesting visual display. Select an element to find an image of the element, a description, history, and even an animation. Other chemical
data is linked as a PDF file (requires Acrobat Reader).
Type: Virtual Manipulative
Precipitation Reaction Systems:
Precipitation reactions occur when cations and anions of aqueous solutions combine to form an insoluble ionic solid, called a precipitate. This simulation explores systems for which precipitation
reactions are possible.A precipitation reaction is controlled by the magnitude of the solubility product, solubility product constant and the concentrations of the ions in solution.
Type: Virtual Manipulative
Equilibrium Constant:
Chemical equilibrium is the condition which occurs when the concentration of reactants and products participating in a chemical reaction exhibit no net change over time. This simulation shows a model
of an equilibrium system for a uni-molecular reaction. The value for the equilibrium constant, K, can be set in the simulation, to observe the reaction reaching the constant.
Type: Virtual Manipulative
This virtual manipulative will help you understand the process of titration, which is a neutralization reaction that is performed in order to determine an unknown concentration of acid and base. With
this simulation, you will be able to calculate the moles of the acid with the understanding that the moles of acid will be equal to the moles of base at the equivalence point.
Type: Virtual Manipulative
Models of the Hydrogen Atom Simulation:
How did scientists figure out the structure of atoms without looking at them? Try out different models by shooting light at the atom. Check how the prediction of the model matches the experimental
Type: Virtual Manipulative
Slope Slider:
In this activity, students adjust slider bars which adjust the coefficients and constants of a linear function and examine how their changes affect the graph. The equation of the line can be in
slope-intercept form or standard form. This activity allows students to explore linear equations, slopes, and y-intercepts and their visual representation on a graph. This activity includes
supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.
Type: Virtual Manipulative
Graphing Equations Using Intercepts:
This resource provides linear functions in standard form and asks the user to graph it using intercepts on an interactive graph below the problem. Immediate feedback is provided, and for incorrect
responses, each step of the solution is thoroughly modeled.
Type: Virtual Manipulative
Split Brain Experiments:
The split brain experiments revealed that the right and the left hemisphere in the brain are good at different things. For instance, the right hemisphere is good at space perception tasks and music
while the left is good at verbal and analytic tasks. This game guides students through some examples of the split-brain phenomenon and how the differences are understood.
Type: Virtual Manipulative
Photoelectric Effect:
This virtual manipulative will help the students to understand how the light shines on a metal surface. Students will recognize a process called as photoelectric effect wherein light can be used to
push electrons from the surface of a solid.
Some of the sample learning goals can be:
• Visualize and describe the photoelectric effect experiment.
• Predict the results of the experiment, when the intensity of light is changed and its effects on the current and energy of the electrons.
• Predict the results of the experiment, when the wavelength of the light is changed and its effects on the current and the energy of the electrons.
• Predict the results of the experiment, when the voltage of the light is changed and its effects on the current and energy of electrons.
Type: Virtual Manipulative
Neon Lights and Other Discharge Lamps:
This virtual manipulative will allow you to produce light by bombarding atoms with electrons. You can also visualize how the characteristic spectra of different elements are produced, and configure
your own element's energy states to produce light of different colors.
Other areas to investigate:
• Provide a basic design for a discharge lamp and explain the function of the different components.
• Explain the basic structure of an atom and relate it to the color of light produced by discharge lamps.
• Explain why discharge lamps emit only certain colors.
• Design a discharge lamp to emit any desired spectrum of colors.
Type: Virtual Manipulative
Reversible Reactions:
This virtual manipulative will allow you to watch a reaction proceed over time. You can vary temperature, barrier height, and potential energies to note how total energy affects reaction rate. You
will be able to record concentrations and time in order to extract rate coefficients.
Additionally you can:
• Describe on a microscopic level, with illustrations, how reactions occur.
• Describe how the motion of reactant molecules (speed and direction) contributes to a reaction happening.
• Predict how changes in temperature, or use of a catalyst will affect the rate of a reaction.
• On the potential energy curve, identify the activation energy for forward and reverse reactions and the energy change between reactants and products.
• Form a graph of concentrations as a function of time, students should be able to identify when a system has reached equilibrium.
• Calculate a rate coefficient from concentration and time data.
• Determine how a rate coefficient changes with temperature.
• Compare graphs of concentration versus time to determine which represents the fastest or slowest rate.
Type: Virtual Manipulative
Reactions Rates:
This virtual manipulative will allow you to explore what makes a reaction happen by colliding atoms and molecules. Design your own experiments with different reactions, concentrations, and
temperatures. Recognize what affects the rate of a reaction.
Areas to Explore:
• Explain why and how a pinball shooter can be used to help understand ideas about reactions.
• Describe on a microscopic level what contributes to a successful reaction.
• Describe how the reaction coordinate can be used to predict whether a reaction will proceed or slow.
• Use the potential energy diagram to determine : The activation energy for the forward and reverse reactions; The difference in energy between reactants and products; The relative potential
energies of the molecules at different positions on a reaction coordinate.
• Draw a potential energy diagram from the energies of reactants and products and activation energy.
• Predict how raising or lowering the temperature will affect a system in the equilibrium.
Type: Virtual Manipulative
Balloons and Buoyancy:
This simulation will provide an insight into the properties of gases. You can explore the more advanced features which enables you to explore three physical situations: Hot Air Balloon (rigid open
container with its own heat source), Rigid Sphere (rigid closed container), and Helium Balloon (elastic closed container).
Through this activity you can:
• Determine what causes the balloon, rigid sphere, and helium balloon to rise up or fall down in the box.
• Predict how changing a variable among Pressure, Volume, Temperature and number influences the motion of the balloons.
Type: Virtual Manipulative
Graphing Lines:
Allows students access to a Cartesian Coordinate System where linear equations can be graphed and details of the line and the slope can be observed.
Type: Virtual Manipulative
Beta Decay:
This is a virtual manipulative to understand beta decay. In the Beta decay process, a neutron decays into a proton and an electron (beta radiation). The process also requires the emission of a
neutrino to maintain momentum and energy balance. Beta decay allows the atom to obtain the optimal ratio of protons and neutrons.
Type: Virtual Manipulative
Atomic Interactions:
In this simulation, explore the interactions between various combinations of two atoms. Specific features of the simulation allows you to see either the total force acting on the atoms or the
individual attractive and repulsive forces.
Options for learning:
• Explain how attractive and repulsive forces govern the interaction between atoms.
• Describe the effect of potential well depth on atomic interactions.
• Describe the process of bonding between atoms in terms of energy.
Type: Virtual Manipulative
Alpha decay:
This virtual manipulative will help you to understand the process of alpha decay. Watch alpha particles escape from a polonium nucleus, causing radioactive alpha decay. See how random decay times
relate to the half life.
Type: Virtual Manipulative
Rutherford Scattering:
This virtual manipulative will help you investigate how Rutherford figured out the structure of the atom without being able to see it. This simulation will allow the you to explore the famous
experiment in which Rutherford disproved the Plum Pudding model of the atom by observing alpha particles bouncing off atoms and determining that they must have a small core.
Further explorations of the tutorial could include:
• Describe the qualitative difference between scattering off positively charged nuclei and electrically neutral plum pudding atoms.
• For a charged nucleus, describe qualitatively how angle of deflection depends on: energy of incoming particle, impact parameters, and charge of target.
Type: Virtual Manipulative
Balancing Chemical Equations:
This activity will allow you to practice balancing a chemical equation. You will have to make sure you are following the law of conservation of mass and recognize what can change to balance an
You can:
• Balance a chemical equation.
• Recognize that the number of atoms of each element is conserved in a chemical reaction.
• Describe the difference between coefficients and subscripts in a chemical equation.
• Translate from symbolic to molecular representation.
Type: Virtual Manipulative
Acid-Base Solutions:
How do strong and weak acids differ? Use lab tools on your computer to find out! Dip the paper or the probe into solution to measure the pH, or put in the electrodes to measure the conductivity. Then
see how concentration and strength affect pH. Can a weak acid solution have the same pH as a strong acid solution.
Some of the topics to investigate:
• Given acids or bases at the same concentration, demonstrate understanding of acid and base strength by 1. Relating the strength of an acid or base to the extent to which it dissociates in water.
2. Identifying all the molecules and ions that are present in a given acid or base solution. 3. Comparing the relative concentrations of molecules and ions in weak versus strong acid (or base)
solutions. 4. Describing the similarities and differences between strong acids and weak acids or strong bases and weak bases.
• Demonstrate understanding of solution concentrated by: 1. Describing the similarities and differences between concentrated and dilute solutions. 2. Comparing the concentrations of all molecules
and ions in concentrated versus dilute solutions of a particular acid or base.
• Describe how common tools (pH meter, conductivity, pH paper) help identify whether a solution is an acid or base and strong or weak and concentrated or dilute.
Type: Virtual Manipulative
Molecules and Light:
This activity will help to investigate how a greenhouse gas affects the climate, or why the ozone layer is important. Using this simulation, explore how light interacts with molecules in our
Areas to explore:
• How light interacts with molecules in our atmosphere.
• Identify that absorption of light depends on the molecule and the type of light.
• Relate the energy of the light to the resulting motion.
• Identify that energy increases from microwave to ultraviolet.
• Predict the motion of a molecule based on the type of light it absorbs.
• Identify how the structure of a molecule affects how it interacts with light.
Type: Virtual Manipulative
Beer's Law Lab:
This activity will allow you to make colorful concentrated and dilute solutions and explore how much light they absorb and transmit using a virtual spectrophotometer.
You can explore concepts in many ways including:
• Describe the relationships between volume and amount of solute to solution concentration.
• Explain qualitatively the relationship between solution color and concentration.
• Predict and explain how solution concentration will change for adding or removing: water, solute, and/or solution.
• Calculate the concentration of solutions in units of molarity (mol/L).
• Design a procedure for creating a solution of a given concentration.
• Identify when a solution is saturated and predict how concentration will change for adding or removing: water, solute, and/or solution.
• Describe the relationship between the solution concentration and the intensity of light that is absorbed/transmitted.
• Describe the relationship between absorbance, molar absorptivity, path length, and concentration in Beer's Law.
• Predict how the intensity of light absorbed/transmitted will change with changes in solution type, solution concentration, container width, or light source and explain why?
Type: Virtual Manipulative
Understanding Polarity:
Understanding molecular polarity by changing the electron-negativity of atoms in a molecule to see how it affects polarity. See how the molecule behaves in an electric field. Change the bond angle to
see how shape affects polarity. See how it works for real molecules in 3D.
Some learning goals:
•predict bond polarity using electron-negativity values
•indicate polarity with a polar arrow or partial charges
•rank bonds in order of polarity
•predict molecular polarity using bond polarity and molecular shape
Type: Virtual Manipulative
Pendulum Lab:
Play with one or two pendulums and discover how the period of a simple pendulum depends on the length of the string, the mass of the pendulum bob, and the amplitude of the swing. It's easy to measure
the period using the photogate timer. Students can vary friction and the strength of gravity.
• Design experiments to describe how variables affect the motion of a pendulum
• Use a photogate timer to determine quantitatively how the period of a pendulum depends on the variables you described
• Determine the gravitational acceleration of planet X
• Explain the conservation of Mechanical energy concept using kinetic energy and gravitational potential energy
• Describe energy chart from position or selected speeds
Type: Virtual Manipulative
Gas Properties:
Students will pump gas molecules to a box and see what happens as they change the volume, add or remove heat, change gravity, and more. Measure the temperature and pressure, and discover how the
properties of the gas vary in relation to each other.
• Students can predict how changing a variable among pressure, volume, temperature and number influences other gas properties.
• Students can predict how changing temperature will affect the speed of molecules.
• Students can rank the speed of molecules in thermal equilibrium based on the relative masses of molecules.
Type: Virtual Manipulative
Under Pressure:
Explore pressure under and above water. See how pressure changes as one change fluids, gravity, container shapes, and volume.
With this simulation you can:
• Investigate how pressure changes in air and water.
• Discover how to change pressure.
• Predict pressure in a variety of situations.
Type: Virtual Manipulative
Box Plot:
In this activity, students use preset data or enter in their own data to be represented in a box plot. This activity allows students to explore single as well as side-by-side box plots of different
data. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the
Java applet.
Type: Virtual Manipulative
Data Flyer:
Using this virtual manipulative, students are able to graph a function and a set of ordered pairs on the same coordinate plane. The constants, coefficients, and exponents can be adjusted using slider
bars, so the student can explore the affect on the graph as the function parameters are changed. Students can also examine the deviation of the data from the function. This activity includes
supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.
Type: Virtual Manipulative
Normal Distribution Interactive Activity:
With this online tool, students adjust the standard deviation and sample size of a normal distribution to see how it will affect a histogram of that distribution. This activity allows students to
explore the effect of changing the sample size in an experiment and the effect of changing the standard deviation of a normal distribution. Tabs at the top of the page provide access to supplemental
materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.
Type: Virtual Manipulative
Function Flyer:
In this online tool, students input a function to create a graph where the constants, coefficients, and exponents can be adjusted by slider bars. This tool allows students to explore graphs of
functions and how adjusting the numbers in the function affect the graph. Using tabs at the top of the page you can also access supplemental materials, including background information about the
topics covered, a description of how to use the application, and exploration questions for use with the java applet.
Type: Virtual Manipulative
Advanced Data Grapher:
This is an online graphing utility that can be used to create box plots, bubble graphs, scatterplots, histograms, and stem-and-leaf plots.
Type: Virtual Manipulative
pH Scale:
Students can test the pH of several substances and visualize hydronium, hydroxide, and water molecules in solution by concentration or the number of molecules. Students can add water to a given
substance to see the effects it will have on the pH of that substance; or they can create their own custom substance.
Type: Virtual Manipulative
Curve Fitting:
With a mouse, students will drag data points (with their error bars) and watch the best-fit polynomial curve form instantly. Students can choose the type of fit: linear, quadratic, cubic, or quartic.
Best fit or adjustable fit can be displayed.
Type: Virtual Manipulative
Equation Grapher:
This interactive simulation investigates graphing linear and quadratic equations. Users are given the ability to define and change the coefficients and constants in order to observe resulting changes
in the graph(s).
Type: Virtual Manipulative
Nuclear Fission:
Complete this virtual manipulative to gain a better understanding of nuclear fission. Study the basic principles behind chain reactions and a nuclear reactor.
Type: Virtual Manipulative
States of Matter:
Watch different types of molecules form a solid, liquid, or gas. Add or remove heat and watch the phase change. Change the temperature or volume of a container and see a pressure-temperature diagram
respond in real time.
Type: Virtual Manipulative
Potential/Kinetic Energy Simulation:
Learn about conservation of energy with a skater! Build tracks, ramps and jumps for the skater and view the kinetic energy, potential energy, thermal energy as he moves. You can adjust the amount of
friction and mass. Measurement and graphing tools are built in.
Type: Virtual Manipulative
Histogram Tool:
This virtual manipulative histogram tool can aid in analyzing the distribution of a dataset. It has 6 preset datasets and a function to add your own data for analysis.
Type: Virtual Manipulative
PhET Gas Properties:
This virtual manipulative allows you to investigate various aspects of gases through virtual experimentation. From the site: Pump gas molecules to a box and see what happens as you change the volume,
add or remove heat, change gravity, and more (open the box, change the molecular weight of the molecule). Measure the temperature and pressure, and discover how the properties of the gas vary in
relation to each other.
Type: Virtual Manipulative
Multi Bar Graph:
This activity allows the user to graph data sets in multiple bar graphs. The color, thickness, and scale of the graph are adjustable which may produce graphs that are misleading. Users may input
their own data, or use or alter pre-made data sets. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the
application, and exploration questions for use with the java applet.
Type: Virtual Manipulative
In this activity, students can create and view a histogram using existing data sets or original data entered. Students can adjust the interval size using a slider bar, and they can also adjust the
other scales on the graph. This activity allows students to explore histograms as a way to represent data as well as the concepts of mean, standard deviation, and scale. This activity includes
supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.
Type: Virtual Manipulative
Parent Resources
Vetted resources caregivers can use to help students learn the concepts and skills in this course.
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Motivating the Foldable Type Class - Matthew Pickering
Something I have never seen articulated is why the Foldable type class exists. It is lawless apart from the free theorems which leads to ad-hoc definitions of its methods. What use is the abstraction
if not to enable us to reason more easily about our programs? This post aims to articulate some justification stemming from the universality of folds.
In brief, here is the argument.
1. For inductive data types, the fold is unique defined as a consequence of initiality.
2. The Foldable type class is a way to exploit this universality without having to define all of our data types as the fixed points of base functors.
3. The uneasiness comes from this impedence mismatch between points 1 and 2.
To recall, the type class is meant to capture the intuitive notion of a fold. Folds are a way of consuming a data type by summarising values in a uniform manner before combining them together.
We now recall the basics of initial algebra semantics for polynomial data types. We can represent all data types as the fixed point of a polynomial functor. In Haskell we can represent the fixed
point by the data type Mu.
data Mu f = Mu { out :: (f (Mu f)) }
-- Inital algebra
in :: f (Mu f) -> Mu f
in = Mu
We then specify data types by first defining a control functor \(F\) and then considering the initial \(F\)-algebra for the functor. The initial \(F\)-algebra is given by \((\mu F, in)\). The
injection function \(in\) wraps up one level of the recursion. The projection function out strips off one layer of the recursion.^2
We can define the type of lists by first specifying a control functor \(ListF = 1 + (A \times \_)\) and then defining lists in terms of this base functor and Mu.
data ListF a r = Nil | Cons a r
type ListM a = Mu (ListF a)
We call types which are definable in this manner inductive types.
We do not usually define data types in this style as programming directly with them is quite cumbersome as one must wrap and unwrap to access the recursive structure.
However, defining data in this manner has some useful properties. The one that we care about is that it is possible to define a generic fold operator for inductive types.
cata :: (f a -> a) -> Mu f -> a
cata f = f . fmap (cata f) . out
What’s more, due to initiality, cata f is the unique function of this type. We have no choice about how we define a fold operator after we have described how to interpret the control functor.
Fleshed out in some more detail, Given a functor \(F\), for any other algebra \((B, g : F B \to B)\) there exists a unique map \(h\) to this algebra from \((\mu F, in)\). Our definition of cata is
precisely the way to construct this unique map.
Real-World Haskell
Languages such as Haskell allow users to define data types in a more ad-hoc fashion by specifying the recursive structure themselves rather than in terms of a base functor.
data List a = Nil | Cons a (List a)
It can be easily seen that Mu (ListF a) ~= List a and thus we can exploit the the uniqueness of the fold function and define a canonical fold function specialised to our newly defined data type.
foldr :: (a -> b -> b) -> b -> List a -> b
foldr is a specialisation of cata to lists. It is perhaps clearer to see the correspondence if we rewrite the function to explicitly take a list algebra as it’s first argument.
data ListAlg a b = ListAlg { z :: () -> b , cons :: (a, b) -> b }
foldr :: ListAlg a b -> List a -> b
Specifying a function ListF a b -> b is precisely the same as specifying ListAlg a b as it amounts to specifying functions \(1 \to b\) and \(a \times b \to b\).
So, for each data type we define we can specialise the cata operator in order to define a canonical fold operator. However, the issue now is that each one of our fold operators has a different type.
It would be useful to still be able to provide a consistent interface so that we can still fold any inductive type. The answer to this problem is Foldable.
This highlights the essential tension with the Foldable type class. It exists in order to be able to continue to define a generic fold operation but without the cost of defining our data types in
terms of fixed points and base functors.
The method foldr is the only method needed to define an instance of Foldable.
class Foldable f where
foldr :: (a -> b -> b) -> b -> t a -> b
It turns out that (a -> b -> b) and a single constant b are sufficient for specifying algebras for inductive types. Inductive types are built from polynomial base functors so we can describe an
algebra by first matching on the summand and then iteratively applying the combining function to combine each recursive position. If there are no recursive positions, we instead use the zero value z.
Defining the instance for lists is straightforward:
instance Foldable [] where
foldr _ z [] = z
foldr f _ (x:xs) = f x (foldr f xs)
As another example, we consider writing an instance for binary trees which only contain values in the leaves. It is less obvious then how to implement foldr as the usual fold (foldTree below) has a
different type signature.
data Tree a = Branch (Leaf a) (Leaf a) | Leaf a
foldTree :: (b -> b -> b) -> (a -> b) -> Tree a -> b
We can do so by cleverly instantiating the result type b when performing the recursive calls.
instance Foldable Tree where
foldr :: (a -> b -> b) -> b -> Tree a -> b
foldr f z t =
case t of
Leaf a -> f a z
Branch l r -> foldr (\a c -> f a . c) id l
(foldr f z r)
The first recursive call of foldr returns a function of type b -> b which tells us how to combine the next recursive occurence. In this case there are only two recursive positions but this process
can be iterated to combine all the recursive holes.
The definition for foldMap usually provides a more intuitive interface for defining instances but it is harder to motivate which is why we prefer foldr.
foldMapTree :: Monoid m => (a -> m) -> Tree a -> m
foldMapTree f (Leaf a) = f a
foldMapTree f (Branch l r) = foldMapTree f l <> foldMapTree f r
However, instances for inductive types defined in this uniform manner are less powerful than folds induced by an \(F\)-algebra. The problem comes from the fact that all the recursive holes much be
combined in a uniform fashion.
The question I have, is it possible to define middle using the Foldable interface?
data Tree3F a r = Leaf a | Branch r r r
type Tree3 a = Mu (Tree3F a)
middleFold :: Tree3F a a -> a
middleFold (Leaf a) = a
middleFold (Branch _ m _) = m
middle :: Tree3 a -> a
middle = cata middleFold
The definition of Foldable is motivated by the well-understood theory of inductive data types. The pragmatics of Haskell lead us to the seemingly quite ad-hoc class definition which has generated
much discussion in recent years. The goal of this post was to argue that the class is better founded than people think and to explain some of the reasons that it leads to some uncomfort.
My argument is not about deciding whether an ad-hoc definition is lawful, it is explaining the motivation for the class in a way which also explains the lawlessness. The class definition is a
compromise because of the practicalities of Haskell. The only way in which we can know a definition is sensible or not is by inspecting whether the ad-hoc definition agrees with the canonical
definition given by cata.
Appendix: Free Theorems of foldr
There are some free theorems for foldr which is natural in a and b.
foldr :: (a -> b -> b) -> b -> t a -> b
Naturality in b amounts to, for all functions g : b -> c.
g (foldr f z t) = foldr (\x y -> f x (g y)) (g z)
and naturality in a amounts to, for all functions f : c -> a.
foldr f z . fmap h = foldr (\x y -> f (h x) y) z
These are included for completeness.
1. From now on when we say data type, we mean polynomial data type.
2. Lambek’s lemma tells us that the two functions are each other’s inverse.
3. Notice that we are only able to fold type constructors of kind * -> *, this is an arbritary choice, motivated by the fact that most containers which we wish to fold are polymorphic in one way.
(For example, lists and trees).
4. Note that I didn’t think of this definiton myself but arrived at it purely calculationally from the definition of foldMap and foldr.
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K-Nearest Neighbors(KNN)
Last Updated: 22nd June, 2023
K-Nearest Neighbors (KNN) is a supervised machine learning algorithm that is used for both classification and regression. The algorithm is based on the idea that the data points that are closest to a
given data point are the most likely to be similar to it. KNN works by finding the k-nearest points in the training data set and then using the labels of those points to predict the label of the
given data point. KNN is considered an instance-based learning algorithm, since it stores the training data and makes predictions based on the stored data points.
Introduction to K-Nearest Neighbors (KNN)
K-nearest neighbors (kNN) is a supervised machine learning algorithm that can be used to solve both classification and regression tasks. kNN as an algorithm seems to be inspired from real life.
People tend to be effected by the people around them.
It works in a similar fashion to humans as our behaviour is guided by the friends we grew up with or from our surroundings. If you grow up with people who love sports, it is highly likely that you
will end up loving sports. Our parents also shape our personality in some ways. There are of-course exceptions.
• The value of a data point is determined by the data points around it. If you have one very close friend and spend most of your time with him/her, you will end up sharing similar interests and
enjoying same things. That is kNN with k=1.
• If you always hang out with a group of 5, each one in the group has an effect on your behavior and you will end up being the average of 5. That is kNN with k=5.
kNN classifier determines the class of a data point by majority voting principle. If k is set to 5, the classes of 5 closest points are checked. Prediction is done according to the majority class.
Similarly, kNN regression takes the mean value of 5 closest points.
Load the data
Initialize K to your chosen number of neighbors’ and normalize the data.
For each example in the data
3.1. Calculate the distance between the query example and the current example from the data.
3.2. Add the distance and the index of the example to an ordered collection
Sort the ordered collection of distances and indices from smallest to largest (in ascending order) by the distances
Pick the first K entries from the sorted collection
Get the labels of the selected K entries
If regression, return the mean of the K labels
If classification, return the mode of the K labels
Suppose we have a dataset which can be plotted as follows −
Now, we need to classify new data point with black dot (at point 60,60) into blue or red class. We are assuming K = 3 i.e. it would find three nearest data points.
How to select the best value of K
Let’s say we have two groups of points —blue-circles and orange-triangles. We want to classify the Test Point = black circle with a question mark, as either a blue circle or an orange triangle.
For K = 1 we will look at the first nearest neighbor. Since we take majority vote and there is only 1 voter we assign its label to our black test point. We can see that the test point will be
classified as a blue circle for k=1.
If K is very Large: The class label of the majority class of the training data set will be assigned to the rest data regardless of the class labels of the neighbours nearest to the test data.
If K is very Small: The class value of a noisy data/outlier in the training dataset which is the nearest neighbor to the test data will be assigned to the test data.
The best value of K is somewhere between these to extremes.
Set K = square root of the number of training records.
Test several K values on a variety of test data sets and choose the one that gives best performance.
Distance metrics:
1. Euclidean Distance
Euclidean Distance is a measure of the straight-line distance between two points in a Euclidean space. It is calculated by taking the square root of the sum of the squared differences between each
coordinate of the two points. Mathematically, it is represented by the equation:
2. Manhattan Distance
Manhattan Distance, also known as Taxicab Distance, is a measure of the distance between two points in a rectangular grid. It is calculated by taking the sum of the absolute differences between the
two points for each coordinate. Mathematically, it is represented by the equation:
Feature scaling
Feature scaling is a process used to normalize the range of independent variables or features of data. In machine learning, it is important to scale the features in the dataset prior to training a
model, since the range of values of raw data varies widely. Scaling helps to reduce the time it takes to converge and improve the accuracy of the KNN model. Some popular feature scaling techniques
include Min-Max scaling, Standardization, and Normalization.
1. Standardization: This technique re-scales a feature to have a mean of 0 and a standard deviation of 1. The formula is given by
Xscaled = (X−Xmean)/S
where X is an original value, Xmean is the mean of all the values in the feature, and S is the standard deviation of all the values in the feature.
2. Normalization: This technique re-scales a feature or dataset to have a range between 0 and 1. The formula is given by
Xscaled = (X−Xmin)/(Xmax−Xmin)
where X is an original value, Xmin is the minimum value of X in the feature, and Xmax is the maximum value of X in the feature.
Advantages and disadvantages of KNN
1. Easy to understand
2. No assumptions about data
3. Can be applied to both classification and regression
4. Works easily on multi-class problems
1. Memory Intensive / Computationally expensive
2. Sensitive to scale of data
3. Struggle when high number of independent variables
Implementing KNN in Python
For this example, we will use the classic Iris dataset which contains measurements for 150 flowers from three different species: Setosa, Versicolor, and Virginica. The dataset contains four features:
sepal length, sepal width, petal length, and petal width.
Link: https://www.kaggle.com/datasets/uciml/iris
This code implements the K-Nearest Neighbors (KNN) algorithm on the Iris dataset. First, the required libraries are imported. Then, the dataset is loaded and split into features (X) and labels (y).
The dataset is then split into a training and test set. The KNN classifier is then initialized and the model is trained using the training set. Finally, the model is used to make predictions on the
test set and the accuracy is evaluated.
Real-world applications of KNN
1. Healthcare: KNN can be used to predict diseases based on the patient's medical records, such as age, sex, blood pressure, cholesterol, etc.
2. Finance: KNN can be used to detect fraudulent transactions and classify customers based on their spending habits.
3. Marketing: KNN can be used to recommend products to customers based on their past purchases and preferences.
4. Security: KNN can be used for facial recognition and intrusion detection.
5. Image Processing: KNN can be used for object recognition and image classification.
After using the K Nearest Neighbors machine learning algorithm, the retail store was able to more accurately identify customers who were likely to purchase a particular product based on their past
purchasing behavior. This allowed them to better target customers with the right products and offers, leading to increased sales and profits.
Key takeaways
1. KNN is a supervised learning algorithm used for both classification and regression.
2. KNN stores the entire training dataset which it uses to predict the output for any new data point.
3. The output is a class membership (predicted target value).
4. KNN is a non-parametric and lazy learning algorithm.
5. KNN is a distance-based algorithm which uses the distance of a data point from the training data points to classify it.
6. KNN performs better if the data is normalized to bring all the features to the same scale.
7. KNN works best on small datasets and can be computationally expensive on large datasets.
8. KNN is highly affected by outliers and noisy data.
1.Which of the following is a disadvantage of KNN?
1. High training time
2. Poor generalization
3. High complexity
4. All of the above
Answer: D. All of the Above
2.Which of the following is a key feature of KNN?
1. Non-parametric learning
2. Parametric learning
3. Both A and B
4. None of the Above
Answer: A. Non-parametric learning
3.Which of the following is an advantage of KNN?
1. Low bias
2. High variance
3. Low complexity
4. All of the Above
Answer: D. All of the Above
4.What type of learning algorithm is KNN?
1. Supervised learning
2. Unsupervised learning
3. Reinforcement learning
4. Both A and B
Answer: A. Supervised learning
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Dealing with multivariable analytics models
Nowadays, mankind possesses a tremendous number of data, regarding many real-life phenomena. The availability of such a source of information is the main premise for the wide spread usage of the
data-driven modelling. When account for more factors in the model development, this usually leads to a better ability of the model to represent the specifics of investigated system. Naturally, in
fields like economics, society, medicine, etc. the models are with many inputs and many outputs (MIMO).
The linear models (w.r.t. their parameters) are preferable and applicable in many cases, because the theory used for their development is well known and is easy to apply. Also, the resulting models
are easy to interpret. Furthermore, these models are proven as effective representations of systems with even strong nonlinear input-output behavior. The key point here is to transform the model into
a linear parameterized form (if possible), by finding appropriate non-linear functions of the initial factors and/or the model outputs.
Why businesses would like to reduce the number of factors?
Usually, in practice, there are two type of requirements affecting the final model. They are: business requirements and statistical requirements (as the modelling is data-driven). We will not discuss
the statistical aspects. An example of a business requirement is when impose restrictions on the model structure, due to economic reasons. For instance, some factors in credit scoring are provided on
a corresponding price by credit bureaus. Normally these factors are significantly more discriminative and hence are more desirable as factors in the scoring models, compared with the other not so
informative factors provided by the customers. When a financial organization estimates how risky an individual is w.r.t. a set of products (loans, credit cards, etc.), it may use a MIMO regression
model. Normally the model accuracy increases sensibly when adding bureau characteristics as factors. But as they cost money, it is reasonable to reduce their number in the final model. So, if a
bureau characteristic is entering in the model to predict a specific output (and the organization pays for it) then it is reasonable this factor to be used to predict the other outputs as well.
Another example from medicine is when predicting how likely a person is to have one or more diseases given a set of medical laboratory tests. Again as in the previous example the number of factors
should be reduced. Otherwise, if the model requires too many tests, this would increase the usage of the laboratory consumables and it would be inconvenient for the patients.
How to represent the model in order to meet a business requirement?
There are two possible representations of linear MIMO models and one of them naturally accounts for the above-mentioned business requirement. Both representations are described below.
The output of a single input single output (SISO) linear model is a sum of multiplications between the model parameters and the entered factors (regressors). This sum can be presented as a
multiplication of two vectors: the parameter vector and the regression vector. On the other hand the output of a MIMO model per every single observation is a vector but not a scalar. Then this
output vector can be presented as a multiplication of a matrix and a vector. From this point of view there are two possible ways a linear MIMO regression model to be written. They are the parameter
matrix (PM) and the parameter vector (PV) forms. In the first case, all parameters are arranged in a matrix and the factors are gathered in a vector, but in a PV the parameters are placed in a vector
and the factors are arranged in a matrix (usually block diagonal).
At first sight, there is no principle difference between both representations. But keeping in mind that the model parameters are to be estimated both representations have different features. It is
not difficult to aware that in the PM form each factor participates in the explanation of every single output. On the other hand in the PV form there is no such dependence: a factor explaining some
output may not participate in the explanation of other outputs. Hence to account for the above mentioned businesses requirement, first the model should be developed in a PM form and after that a more
precise modes structure refinement can be applied by switching to PV form.
Of course, if there is no business need to reduce the number of factors when the model is MIMO the PV form is the right model representation. The reason, as already mentioned is that each set of
factors explaining a model output (which is nothing but a multiple input single output (MISO) sub-model) doesn’t depend on the factors participating in the other MISO sub-models. So, both possible
representations of linear MIMO models are not equivalent and have their application areas.
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Basic Information
Birth- 476 CE
Birth Place- Kusumapura, capital Pataliputra in the Gupta Era.
Present Day- birthplace is known to be Bihar, Patna, India.
Works- His Most Notable work is Aryabhatiya and Arya Siddhanta.
Death- 550 CE
The Illustrious Astronomer and Mathematician of Ancient India
Arya Bhatta, one of the most illustrious figures in the history of Indian science, stands as a beacon of intellectual brilliance and groundbreaking discoveries. His contributions to astronomy and
mathematics have left an indelible mark on the evolution of scientific thought, not only in India but also across the globe.
Born in 476 CE in Pataliputra, the ancient capital of the Gupta Empire, Arya Bhatta displayed an exceptional aptitude for mathematics and astronomy from a young age. His insatiable curiosity and
unwavering dedication to scientific inquiry led him to revolutionize our understanding of the cosmos and lay the foundations for future advancements in these fields.
Through his seminal work, the “Aryabhatiya,” a concise yet profound treatise on astronomy and mathematics, Arya Bhatta unveiled a plethora of groundbreaking concepts that challenged prevailing
notions and paved the way for a more accurate and insightful understanding of the universe.
With remarkable precision, Arya Bhatta calculated the value of pi to an astonishing degree of accuracy, a feat that stands as a testament to his mathematical prowess. Additionally, he introduced
advanced concepts in trigonometry and algebra, which greatly expanded the mathematical toolkit available for scientific exploration.
Arya Bhatta’s astronomical insights were equally profound. He correctly asserted that the Earth rotates on its axis, a revolutionary idea that contradicted the prevailing notion of a stationary
Earth. Moreover, he accurately calculated the length of the sidereal year, the time it takes for the Earth to complete one orbit around the Sun.The impact of Arya Bhatta’s work extended far beyond
the borders of India. His ideas were embraced by scholars across the Islamic world, who translated and disseminated his treatises, leading to a widespread acceptance of his astronomical and
mathematical concepts.
Today, Arya Bhatta is revered as a pioneer in Indian science, and his contributions continue to inspire and inform scientific research around the world. His remarkable legacy serves as a powerful
reminder of the transformative power of human intellect and the boundless potential for scientific discovery.
A Prodigy Emerges
Emerging from the intellectual milieu of ancient India, Arya Bhatta stands as a towering figure in the annals of scientific history. Though the specifics of his early life and familial background
remain shrouded in the mists of time, it is evident that Arya Bhatta’s inherent aptitude for mathematics and astronomy was nurtured from a young age.
Arya Bhatta’s educational journey unfolded amidst an intellectually vibrant environment. While the exact institutions he attended remain a subject of debate, historical evidence suggests that he was
immersed in the stimulating atmosphere of renowned centers of learning, including the esteemed University of Nalanda, a beacon of astronomical and mathematical discourse.
Under the tutelage of exceptional teachers, whose names have unfortunately faded into the annals of time, Arya Bhatta’s intellectual prowess flourished. These mentors, recognizing his budding
potential, meticulously guided him through the intricacies of astronomy and mathematics, providing him with the tools and rigor necessary to explore the universe’s profound mysteries.
From his early education, Arya Bhatta’s inquisitive mind gravitated towards astronomy and mathematics, disciplines that ignited his imagination and fueled his unwavering passion for unraveling the
universe’s intricate workings. It was within this realm of scientific inquiry that Arya Bhatta would leave an indelible mark on the world.
Aryabhatiya: A Beacon of Astronomical and Mathematical Brilliance
Emerging from the depths of Arya Bhatta’s exceptional intellect, the “Aryabhatiya” stands as a resplendent masterpiece, a concise yet profound treatise that revolutionized our understanding of the
cosmos and ushered in a new era of scientific thought. Within its pages lies a treasure trove of groundbreaking concepts that challenged prevailing notions and paved the way for future advancements
in astronomy and mathematics.
Arya Bhatta’s astronomical insights were nothing short of revolutionary. He boldly asserted that the Earth rotates on its axis, a heliocentric perspective that stood in stark contrast to the
prevailing view of a stationary Earth. This paradigm-shifting idea, though initially met with skepticism, laid the foundation for a more accurate and insightful understanding of planetary motion.
Arya Bhatta’s mathematical prowess shone through his masterful contributions to trigonometry and algebra. He introduced a set of sophisticated trigonometric functions and their inverse counterparts,
greatly expanding the mathematical toolkit available for scientific exploration. Additionally, he developed innovative algebraic techniques, including solutions to quadratic equations, that further
enriched the field of mathematics.
A Legacy of Astronomical and Mathematical Brilliance
In the annals of scientific history, Arya Bhatta stands as a monumental figure, an Indian astronomer and mathematician whose groundbreaking contributions revolutionized our understanding of the
cosmos and paved the way for future advancements in these fields. His seminal work, the “Aryabhatiya,” a concise yet profound treatise, has left an indelible mark on the evolution of scientific
thought, not only in India but also across the globe.
Aryabhatta’s Astronomical Insights
Arya Bhatta’s astronomical insights were nothing short of revolutionary. He boldly asserted that the Earth rotates on its axis, a heliocentric perspective that stood in stark contrast to the
prevailing view of a stationary Earth. This paradigm-shifting idea, though initially met with skepticism, laid the foundation for a more accurate and insightful understanding of planetary motion.
Moreover, Arya Bhatta accurately calculated the length of the sidereal year, the time it takes for the Earth to complete one orbit around the Sun. His precise determinations of planetary positions
and eclipses further cemented his reputation as a master astronomer.
Aryabhatta’s Mathematical Prowess
Arya Bhatta’s mathematical prowess shone through his masterful contributions to trigonometry and algebra. He introduced a set of sophisticated trigonometric functions and their inverse counterparts,
greatly expanding the mathematical toolkit available for scientific exploration. Additionally, he developed innovative algebraic techniques, including solutions to quadratic equations, that further
enriched the field of mathematics.
Arya Bhatta’s concept of zero and his decimal place value system, considered fundamental pillars of modern mathematics, had a profound impact on the development of computational techniques and
scientific calculations.
Arya Bhatta’s Enduring Legacy
The impact of Arya Bhatta’s work extended far beyond the confines of his time. His ideas were eagerly embraced by scholars across the Islamic world, who translated and disseminated his treatises,
leading to a widespread acceptance of his astronomical and mathematical concepts. His influence permeated the works of renowned scientists like Aryabhata II, Bhaskara II, and Al-Biruni, who carried
his legacy forward, refining and expanding upon his groundbreaking discoveries.
Today, Arya Bhatta’s seminal work, the “Aryabhatiya,” stands as a testament to his intellectual prowess and enduring legacy. Its profound impact on subsequent scholars and its lasting influence on
the development of scientific thought are undeniable. Arya Bhatta’s contributions continue to inspire and inform scientific research around the world, serving as a powerful reminder of the boundless
potential for human intellect and the transformative power of scientific discovery.
Today, Arya Bhatta’s relevance remains undiminished in the realm of modern science. His profound ideas continue to inspire and inform scientific research, serving as a cornerstone for contemporary
astronomical and mathematical studies.
His legacy extends beyond science, as he is revered as a national hero in India, his name etched in the pantheon of the nation’s intellectual giants. Arya Bhatta’s story is a compelling narrative of
the transformative power of human intellect and the boundless potential for scientific discovery. His unwavering dedication to unraveling the universe’s mysteries, coupled with his exceptional
mathematical prowess, resulted in groundbreaking contributions that continue to shape our understanding of the cosmos and the world around us. Arya Bhatta’s legacy is an enduring source of
inspiration, reminding us of the profound impact that one individual can have on the course of scientific history.
A Monumental Figure in Scientific History
As the annals of scientific history unfold, Arya Bhatta emerges as a monumental figure, an Indian astronomer and mathematician whose groundbreaking contributions revolutionized our understanding of
the cosmos and paved the way for future advancements in these fields. His seminal work, the “Aryabhatiya,” a concise yet profound treatise, has left an indelible mark on the evolution of scientific
thought, not only in India but also across the globe.
Born in 476 CE in Pataliputra, the ancient capital of the Gupta Empire, Arya Bhatta displayed an exceptional aptitude for mathematics and astronomy from a young age. His insatiable curiosity and
unwavering dedication to scientific inquiry led him to challenge prevailing notions and unveil a plethora of groundbreaking concepts that transformed our understanding of the universe.
Aryabhatta’s Astronomical Insights
ARYA BHATTA’S astronomical insights were nothing short of revolutionary. He boldly asserted that the Earth rotates on its axis, a heliocentric perspective that stood in stark contrast to the
prevailing view of a stationary Earth. This paradigm-shifting idea, though initially met with skepticism, laid the foundation for a more accurate and insightful understanding of planetary motion.
Moreover, Arya Bhatta accurately calculated the length of the sidereal year, the time it takes for the Earth to complete one orbit around the Sun. His precise determinations of planetary positions
and eclipses further cemented his reputation as a master astronomer.
Aryabhatta’s Mathematical Prowess
Arya Bhatta’s mathematical prowess shone through his masterful contributions to trigonometry and algebra. He introduced a set of sophisticated trigonometric functions and their inverse counterparts,
greatly expanding the mathematical toolkit available for scientific exploration. Additionally, he developed innovative algebraic techniques, including solutions to quadratic equations, that further
enriched the field of mathematics.
Arya Bhatta’s concept of zero and his decimal place value system, considered fundamental pillars of modern mathematics, had a profound impact on the development of computational techniques and
scientific calculation.
Aryabhatta’s Enduring Legacy
Arya Bhatta’s legacy is inextricably linked to the foundations of Indian astronomy and mathematics. His unwavering belief in the Earth’s rotation, a revolutionary concept for his time, challenged
prevailing notions and paved the way for a more accurate understanding of planetary motion. His groundbreaking contributions to trigonometry and algebra expanded the mathematical toolkit available
for scientific exploration, empowering future scholars to delve deeper into the mysteries of the cosmos.
Arya Bhatta’s legacy is an enduring source of inspiration, reminding us of the profound impact that one individual can have on the course of scientific history. His unwavering dedication to
unraveling the universe’s mysteries, coupled with his exceptional mathematical prowess, resulted in groundbreaking contributions that continue to shape our understanding of the cosmos and the world
around us.
Arya Bhatta’s story is a compelling narrative of the transformative power of human intellect and the boundless potential for scientific discovery. His unwavering belief in the Earth’s rotation, a
revolutionary concept for his time, challenged prevailing notions and paved the way for a more accurate understanding of planetary motion.His groundbreaking contributions to trigonometry and algebra
expanded the mathematical toolkit available for scientific exploration, empowering future scholars to delve deeper into the mysteries of the cosmos.
Today, his legacy remains strong as his profound discoveries continue to be studied and admired by scientists and mathematicians around the world. He stands as a beacon of intellectual brilliance,
not only in the history of Indian science, but also in the evolution of scientific thought worldwide.
Thank you for your warm wishes and blessings 🙏…
@Puja Singh…
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ARC Length Formula And Details - Surveying & ArchitectsARC Length Formula And Details
ARC Length Formula And Details
How to Find Arc Length
An arc is any portion of the circumference of a circle.^ Arc length is the distance from one endpoint of the arc to the other. Finding an arc length requires knowing a bit about the Geometry of a
circle. Since the arc is a portion of the circumference, if you know what portion of 360 degrees the arc’s central angle is, you can easily find the length of the arc.
Using Measurement of Central Angle in Degrees.
Set up the formula for arc length.
The formula is {\displaystyle {\text{arc length}}=2\pi (r)({\frac {\theta }{360}})}, where {\displaystyle r} equals the radius of the circle and {\displaystyle \theta } equals the measurement of the
arc’s central angle, in degrees
Learn More
Plug the length of the circle’s radius into the formula.
This information should be given, or you should be able to measure it. Make sure you substitute this value for the variable {\displaystyle r}.
• For example, if the circle’s radius is 10 cm, your formula will look like this: {\displaystyle {\text{arc length}}=2\pi (10)({\frac {\theta }{360}})}.
Other Post
Plug the value of the arc’s central angle into the formula.
This information should be given, or you should be able to measure it. Make sure you are working with degrees, and not radians, when using this formula. Substitute the central angle’s measurement
for {\displaystyle \theta } in the formula.
• For example, if the arc’s central angle is 135 degrees, your formula will look like this: {\displaystyle {\text{arc length}}=2\pi (10)({\frac {135}{360}})}.
Multiply the radius by 2π{\displaystyle 2\pi }.
If you are not using a calculator, you can use the approximation {\displaystyle \pi =3.14} for your calculations. Rewrite the formula using this new value, which represents the circle’s
• For example:
{\displaystyle 2\pi (10)({\frac {135}{360}})}
{\displaystyle 2(3.14)(10)({\frac {135}{360}})}
{\displaystyle (62.8)({\frac {135}{360}})}
Other Post
Divide the arc’s central angle by 360.
Since a circle has 360 degrees total, completing this calculation gives you what portion of the entire circle the sector represents. Using this information, you can find what portion of the
circumference the arc length represents.
• For example:
{\displaystyle (62.8)({\frac {135}{360}})}
{\displaystyle (62.8)(.375)}
Multiply the two numbers together.
This will give you the length of the arc.
• For example:
{\displaystyle (62.8)(.375)}
{\displaystyle 23.55}
So, the length of an arc of a circle with a radius of 10 cm, having a central angle of 135 degrees, is about 23.55 cm.
Learn More
Set up the formula for arc length.
The formula is {\displaystyle {\text{arc length}}=\theta (r)}, where {\displaystyle \theta } equals the measurement of the arc’s central angle in radians, and {\displaystyle r} equals the length of
the circle’s radius.
Plug the length of the circle’s radius into the formula.
You need to know the length of the radius to use this method. Make sure you substitute the length of the radius for the variable {\displaystyle r}.
• For example, if the circle’s radius is 10 cm, your formula will look like this: {\displaystyle {\text{arc length}}=\theta (10)}.
Learn More
Plug the measurement of the arc’s central angle into the formula.
You should have this information in radians. If you know the angle measurement in degrees, you cannot use this method.
• For example, if the arc’s central angle is 2.36 radians, your formula will look like this: {\displaystyle {\text{arc length}}=2.36(10)}.
Multiply the radius by the radian measurement.
The product will be the length of the arc.
• For example:
{\displaystyle 2.36(10)}
{\displaystyle =23.6}
So, the length of an arc of a circle with a radius of 10 cm, having a central angle of 23.6 radians, is about 23.6 cm.
Other Post
To Get more information, Visit Our Official website
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American Political Writing during the Founding 1760-1805 | Online Library of Liberty
American Political Writing during the Founding 1760-1805
Related Links:
Source: American Political Writing During the Founding Era: 1760-1805, ed. Charles S. Hyneman and Donald Lutz (Indianapolis: Liberty Fund, 1983). 2 vols. Volume 2. Chapter: A Selected List of
Political Writings by Americans Between 1760 and 1805.
A Selected List of Political Writings by Americans Between 1760 and 1805
The following bibliography is based upon a comprehensive reading of the political literature of the founding era and is designed to assist those interested in the study of American political theory
by identifying items worthy of attention. If the topic of the piece is not apparent from its title, the editors have, in most instances, provided annotation. If an item lacks annotation, as is the
case with many sermons, this is because the content is either so broad as to defy easy categorization, or the content is so typical for such a piece that there is no point in repetitiously noting
that fact. The information is sufficient for an investigator to be able to identify those pieces dealing with a specific topic he or she might wish to study. We enter no comment on the pieces printed
in this collection.
The editors have roughly divided the items in the bibliography into three categories. If there is no asterisk, the piece is deemed of interest to someone studying American political theory, but the
level of analysis is low. One asterisk identifies pieces with substantial theoretical content, and two asterisks indicate pieces that these editors feel are candidates for inclusion among the best
theoretical writing by Americans during the founding era. Major bibliographies compiled by historians on some part of what is here defined as the founding era usually will be found to have a 20 to 30
percent overlap with the following bibliography. The items cited by such historians but not included below are not lacking in historical interest or importance, but simply do not have sufficient
theoretical content or interest for inclusion here. A dagger at the end of a citation indicates a piece that is reproduced in these volumes.
• 1 Adams, John. A Defence of the Constitutions of Government of the United States. 1787. In Charles Francis Adams, ed., Works of John Adams, (Boston, 1851), IV.**
• 2 Adams, John [Novanglus]. (Untitled Essays). Boston Gazette, January 23, 30, February 20-April 17, 1775.*Written in response to essays by Massachusettensis [Daniel Leonard]. Reproduced in
Merrill Jensen, ed., Tracts of the American Revolution, 1763-1776 (Indianapolis: Bobbs-Merrill, 1978).
• 3 Adams, John. Thoughts on Government. 1776. From Charles Francis Adams, ed., Works of John Adams (Boston, 1851), IV: 189-202.**†
• 4 Adams, Samuel. The Rights of the Colonists. Boston, 1772. 11 pp. Reproduced in The Annals of A America, I: 217-220.
• 5 Adams, Zabdiel. An Election Sermon. Boston, 1782. 59 pp.**†
• 6 Addison, Alexander. Analysis of the Report of the Committee of the Virginia Assembly. Philadelphia, 1800. 52 pp.**†
• 7 Addison, Alexander. A Charge to the Grand Juries of the County Courts in Pennsylvania. Philadelphia, 1798. 24 pp.Alien and Sedition Acts, free speech, and free press.
• 8 Addison, Alexander. Liberty of Speech and of the Press: Charges to a Grand Jury. Albany, 1790. 16 pp.
• 9 Alden, Timothy. The Glory of America. Portsmouth, N.H., 1802. 47 pp.
• 10 Allen, Ira. Some Miscellaneous Remarks and Short Arguments on a Small Pamphlet . . . and Some Reasons Given Why the District of the New Hampshire Grants Had Best Be a State. Hartford, 1777. 26
• 11 Allen, Ira. A Vindication of the Conduct of the General Assembly of the State of Vermont. Dresden. N.H., 1779. 48 pp.
• 12 Allison, Patrick. Candid Animadversions on a Petition. . . . Baltimore, 1793. 47 pp.
• 13 Ames, Fisher. The Dangers of American Liberty. Boston, 1805. 55 pp.**†
• 14 Ames, Fisher. Laocoon No. 1. 1799. From Seth Ames, ed., Works of Fisher Ames (Boston, 1854), II: 109-115.Defends Federalists against charges by Jeffersonians.
• 15 Atwater, Jeremiah. A Sermon. Middlebury, Vt., 1801. 39 pp.**†
• 16 Austin, Benjamin [Honestus]. Observations on the Pernicious Practice of the Law. Boston, 1786, 52 pp.Lawyers are not needed for good government, but they have insinuated themselves into it
with pernicious consequences.
• 17 Austin, Benjamin Jr. To the Printers. Massachusetts Centinel, January 9, 1788.Supports proposed United States Constitution.
• 18 Avery, David. Two Sermons on the Nature and Evil of Professors of Religion Not Bridling the Tongue. Boston, 1791. 66 pp.The tongue as the principal medium for displaying corruption, and the
effect it has on people and society.
• 19 Backus, Charles. A Sermon. Hartford, 1793. 38 pp.
• 20 Backus, Isaac. Government and Liberty Described. Boston, 1778. 20 pp.
• 21 Backus, Isaac. A Letter . . . Concerning Taxes to Support Religious Worship. Boston, 1771. 22 pp.Opposes such taxation.
• 22 Backus, Isaac. Truth is Great and Will Prevail. Boston, 1781. 44 pp.
• 23 Backus, Simon. A Dissertation on the Right and Obligation of the Civil Magistrate. Middletown, Conn., 1804. 38 pp.**Importance of religion to make oaths and compacts operative, preserve public
virtue, and support self-government.
• 24 Baldwin, Ebenezer. The Duty of Rejoicing Under Calamities and Afflictions. New York, 1776. 42 pp.A “God is an American” morale-booster during the War.
• 25 Baldwin, Henry. A General View of the Origin and Nature of the Constitution and Government of New York. New York, 1780. 197 pp.
• 26 Baldwin, Thomas. A Sermon (Election Day). Boston, 1802. 36 pp.
• 27 Ball, Heman. Vermont Election Day Sermon. Bennington, 1804. 31 pp.A standard rehearsal of Whig political principles.
• 28 Bancroft, Aaron. Massachusetts Election Day Sermon. Boston, 1801. 29 pp.Prosperity and political success of American colonies laid to the moral virtues of the people. Continued success depends
upon preserving these virtues.
• 29 Barlow, Joel. Advice to the Privileged Orders. . . . New York, 1794. 31 pp.**Principles that ought to govern the collection of public revenue.
• 30 Barlow, Joel. A Letter to the National Convention of France on the Defects in the Constitution of 1791. New York, 1792. 31 pp.**†
• 31 Barlow, Joel. To His Fellow Citizens of the United States. Letter II: On Certain Political Measures Proposed to Their Consideration. Philadelphia, 1801. 37 pp.**†
• 32 Barnes, David. A Discourse on Education. Boston, 1803. 27 pp.**Comprehensive discussion of education—school, home, etc.
• 33The Barrington-Bernard Correspondence, 1760-1770. Selections from 1765-1768. From Edward Channing and Archibald Cary Coolidge, eds., The Barrington-Bernard Correspondence, 1760-1770, Harvard
Historical Studies, vol. XVII (Cambridge, Mass,: Harvard University, 1912), pp. 92-103, 244-293.
• 34 Barton, William. The Constitutionalist: Addressed to Men of All Parties. Philadelphia, 1804. 49 pp.**Judiciary has a special responsibility to enforce Constitution.
• 35 Baxter, Joseph. The Duty of a People to Pray to and Bless God for Their Rulers. . . . Boston, 1772. 36 pp.The duties of rulers.
• 36 Bean, Joseph. Massachusetts Election Day Sermon. Boston, 1774. 36 pp.
• 37 Beers, William P. An Address to the Legislature and People of the State of Connecticut. New Haven, 1791.*Cosmopolitan, contrasted with localist, spirit in political factions. Size of
electorate and of the legislature important.
• 38 Belknap, Jeremy. The History of New England, 3 vols.; vol. I, pp. 60-99; vol. III, pp. 252-287. Boston, 1791-1792.Equality and public virtue as the basis for true republicanism.
• 39 Belsham, William. An Essay on the African Slave Trade. Philadelphia, 1790. 15 pp.Opposed to it.
• 40 [Benezet, Anthony.] Brief Considerations on Slavery. Burlington, Vt., 1773. 16 pp.*
• 41 Benezet, Anthony. A Caution and Warning to Great Britain . . . the Calamitous State of the Enslaved Negroes in the British Dominion. Philadelphia, 1767. 52 pp.
• 42 Benezet, Anthony. A Mite Cast into the Treasury: Or, Observations on Slave-Keeping. Philadelphia, 1772. 14 pp.**Major spokesman for the Quaker position.
• 43 Benezet, Anthony. Serious Considerations on Several Important Subjects. Philadelphia, 1778. 48 pp.Compendium of Quaker political principles.
• 44 Benezet, Anthony. Some Observations on . . . Indian Natives of the Continent. Philadelphia, 1784. 59 pp.
• 45 Benezet, Anthony. Thoughts on the Nature of War. . . . Philadelphia, 1766. 14 pp.*Statement of the Quaker position.
• 46 Bernard, Francis. Select Letters on the Trade and Government of America. London, 1774. 130 pp.Prominent American explains how colonists see their government and place within the Empire.
• 47 Binney, Barnabas. An Oration [re] . . . the Liberty of Choosing Our Own Religion. Boston, 1774. 44 pp.
• 48 Bishop, Abraham. Oration . . . Before the Republicans of Connecticut. New Haven, 1801. 47 pp.*Characteristics of Federalists as seen by a partisan Republican.
• 49 Bishop, Abraham. Proofs of a Conspiracy Against Christianity and the Government of the United States. Hartford, 1802. 135 pp.**A tirade against the Federalists, professionals, well-to-do, and
those who attack Jefferson. Good on equality, Federalist rhetoric, elitism, and corruption.
• 50 Bland, Richard [Common Sense]. The Colonel Dismounted: Or the Rector Vindicated. . . . Williamsburg, Va., 1764. 53 pp.**Discusses constitution of the colony. Reproduced in Bernard Bailyn, ed.,
Pamphlets of the American Revolution (Cambridge, Mass., Belknap Press, 1965).
• 51 Bland, Richard, An Inquiry into the Rights of the British Colonies. Williamsburg, Va., 1766. 31 pp.**†
• 52 Bollan, William. The Freedom of Speech and Writing Upon Public Affairs, Considered, With an Historical View. London, 1766.
• 53 Bowdoin, James. A Philosophical Discourse, Addressed to the American Academy of Arts and Sciences. . . . Boston, 1780. 16 pp.On the encouragement of knowledge.
• 54 Bowen, Nathaniel. An Oration . . . in Commemoration of American Independence. 1802.A rehearsal of the reasons for separating from England.
• 55 Boucher, Jonathan. On the Necessity of Popular Support of Government. 1763. From A View of the Causes and Consequences of the American Revolution (London, 1797), pp. 308-321.
• 56 Bradbury, Thomas. The Ass: or, the Serpent, A Comparison Between the Tribes of Issachar and Dan, in Their Regard for Civil Liberty. First printed in London, 1712; reprinted in Newburyport,
Mass., 1774. 22 pp.**†
• 57 Braxton, Carter [A Native of This Colony]. An Address to the Convention of the Colony and Ancient Dominion of Virginia on the Subject of Government in General, and Recommending a Particular
Form to Their Attention. Virginia Gazette, June 8 and 15, 1776.**†
• 58 Brown, Charles Brockden. Alcuin, A Dialogue. New Haven, 1798. 38 pp.On the essential equality of the sexes. Reprinted in The Annals of America, vol. IV. 40 pp.
• 59 Brown, William L. An Essay on the Natural Equality of Men. Philadelphia, 1793. 191 pp.
• 60 Burk, John. The History of Virginia from its First Settlement to the Present Day. Williamsburg, Va. 1805.
• 61 Burke, Aedanus. An Address to the Freemen of . . . South Carolina. Charleston, 1783.Proper treatment of colonials who maintained friendly relations with the British within territory held by
British forces.
• 62 Burke, Aedanus. Considerations on the Society or Order of Cincinnati. Charleston, 1783. 33 pp.
• 63 Burnet, Matthias. Connecticut Election Day Sermon. Hartford, 1803. 29 pp.*Five conditions needed for order, peace, and security.
• 64 Callender, John. An Historical Discourse on the Civil and Religious Affairs . . . of Rhode Island. 1739. Republished in 1783.On liberty of conscience.
• 65 Carmichael, John. A Self-Defensive War Lawful, Proved in a Sermon. Lancaster, Penn., 1775. 25 pp.
• 66 Case, Stephen [A Moderate Whig]. Defensive Arms Vindicated and the Lawfulness of the American War Made Manifest. 1783. 53 pp.*
• 67 Chalmers, James [Candidus]. Plain Truth: Addressed to the Inhabitants of America, Containing, Remarks on a Late Pamphlet Entitled Common Sense. Philadelphia, March 13, 1776. 40 pp.In response
to Thomas Paine’s pamphlet, Chalmers defends the British monarchical constitution and attacks the idea of republicanism.
• 68 Champion, Judah. Christian and Civil Liberty. Hartford, 1776.The basis for civil liberty lies in Christian thought.
• 69 [Chandler, Thomas B.] An Address from the Clergy of New York and New Jersey to the Episcopalians in Virginia. New York, 1771. 58 pp.
• 70 Chandler, Thomas B. The Appeal Farther Defended. New York, 1771. 240 pp.
• 71 Chandler, Thomas B. A Friendly Address . . . [re] Our Political Confusions. New York, 1774. 55 pp.
• 72 [Chandler, Thomas B.] What Think Ye of the Congress Now? New York, 1775. 48 pp.
• 73 Chase, Samuel [Publicola]. To the Voters of Anne-Arundel County. Maryland Journal, February 13, 1787. See also May 18, July 13, July 18, and August 31, the last few being entitled “To
Aristides.”*A running battle with Aristides [Alexander Hanson] in which Publicola defends the right of the people to instruct their representatives.
• 74 Chauncy, Charles. The Appeal of the Public Answered in Behalf of the Non-Episcopal Churches of America. Boston, 1768. 205 pp.
• 75 Chipman, Nathaniel. Sketches of the Principles of Government. Rutland, Vt. 1793.*An analysis of the United States Constitution, and the principles that give it strength, by a Federalist.
• 76 Clarendon, Earl of, to William Pym. Boston Gazette, January 27, 1766.*An English Whig lays out the basic Whig principles.
• 77 Clark, Jonas. Massachusetts Election Day Sermon. Boston, 1781. 42 pp.*Uses an extended biological metaphor to stress the communitarian underpinnings to a just relationship between governors
and the governed.
• 78 Clinton, George, [Cato]. No. IV: To the Citizens of the State of New York. New York Journal, November 8, 1787.*Critical analysis of the proposed Constitution.
• 79 Clinton, George [Cato]. No. VI: To the People of the State of New York. New York Journal, December 13, 1787.**Problems of taxation, dangers of aristocracy. Cato’s pamphlets are reproduced in
Cecelia Kenyon, ed., The Antifederalists (Indianapolis: Bobbs-Merrill, 1966).
• 80 Cobbett, William. The Democratic Judge: or the Equal Liberty of the Press. Philadelphia, 1798. 102 pp.
• 81 Cooper, David. An Inquiry into Public Abuses, Arising for Want of a Due Execution of Laws. Philadelphia, 1784.
• 82 [Cooper, David.] Serious Address to the Rulers of America. Trenton, 1783. 22 pp.
• 83 Cooper, Thomas. An Account of the Trial of Thomas Cooper. Philadelphia, 1800. 64 pp.Cooper edits the proceedings of the trial against him under the Alien and Sedition Acts.
• 84 Coram, Robert. Political Inquiries, to which is Added a Plan for the Establishment of Schools Throughout the United States. Wilmington, 1791. 107 pp.**†
• 85 Coxe, Tench [An American Citizen]. An Examination of the Constitution of the United States of America. . . . Philadelphia, 1788. 33 pp.*Theoretical support for the proposed Constitution.
• 86 Coxe, Tench [An American Citizen]. On the Federal Government, No. 1 and No. 3. New York Packet, October 5 and 16, 1787, respectively.*Reproduced in Paul Leicester Ford, ed., Pamphlets on the
Constitution of the United States (New York: Da Capo Press, 1968). Summarizes American political history showing how it leads naturally to the Constitution.
• 87 Cumings, Henry. A Sermon (Election Day). Boston, 1783. 55 pp.
• 88 Daggett, David. Sun-Beams May be Extracted from Cucumbers, But the Process Is Tedious: An Oration. New Haven, 1799. 28 pp.*A witty, sarcastic response by a Federalist to what he perceived to
be the utopianism on the part of the opposition.
• 89 Dana, James. Connecticut Election Day Sermon. Hartford, 1779. 46 pp.*Foundations of good government are rooted in the word of God. Good on virtue, oaths, and basic principles of government.
• 90 Dana, Samuel W. Essay on Political Society. Philadelphia, 1800. 234 pp.On the supremacy of the Constitution and how it is to be enforced. Pages after 193 contain discussion of judicial review.
• 91 Dickinson, John. Essay on the Constitutional Power of Great Britain. Philadelphia, 1774. 127 pp.*Relationship of colonies to Britain.
• 92 Dickinson, John. The Late Regulations Respecting the British Colonies on the Continent of America. . . . Philadelphia, 1765. 38 pp.A response to the Stamp Act. Reproduced in Bernard Bailyn,
ed., Pamphlets of the American Revolution.
• 93 Dickinson, John. Letters of a Farmer [On British Policy Affecting the American Colonies]. 1767. In The Political Writings of John Dickinson, I:167-276.*Letter XII especially good on basic
principles. Letters I, II, IV, VI, IX, and X are reproduced in Jensen, ed., Tracts of the American Revolution.
• 94 Dickinson, John [Fabius]. Letters to the Editor. Delaware Gazette, 1788.*These nine letters in support of the Constitution are reproduced in Ford. ed., Pamphlets on the Constitution of the
United States.
• 95 Dickinson, Samuel F. An Oration in Celebration of American Independence. Northampton, Mass., 1798. 23 ppHow political institutions and conduct of government respond to manners and taste.
• 96 Doggett, Simeon. A Discourse on Education. New Bedford, 1796.Reproduced in Frederick Rudolph, ed., Essays on Education in the Early Republic (Cambridge, Mass.: Belknap Press, 1965), pp.
• 97 Dorr, Edward. The Duty of Civil Rulers: A Connecticut Election Sermon. Hartford, 1765. 34 pp.**Favors support of Church by State, and protection of Church and religion from injurious behavior.
• 98 Downer, Silas [A Son of Liberty]. A Discourse at the Dedication of the Tree of Liberty. Providence, 1768. 16 pp.**†
• 99 Drayton, William Henry. The Charge to the Grand Jury. South Carolina and American General Gazette, May 8, 1776.Precursor to the Declaration of Independence in laying out grounds for breaking
with England.*
• 100 Dulaney, Daniel. Considerations on the Propriety of Imposing Taxes in the British Colonies . . . By Parliament. Annapolis, 1765. 55 pp.*Against virtual representation and the right of
Parliament to tax colonists. Reproduced in Bailyn, ed., Pamphlets of the American Revolution.
• 101 Dwight, Theodore. An Oration on the Anniversary of American Independence. Hartford, 1801. 31 pp.*A Federalist opposed to “levelling” by Jeffersonians. Opposes separating State and
religion—discusses public schools, moral and religious instruction, literacy, religion, etc.
• 102 Dwight, Theodore. An Oration, Spoken Before the Connecticut Society, for the Promotion of Freedom and the Relief of Persons Unlawfully Holden in Bondage. Hartford, 1794. 24 pp.**†
• 103 Dwight, Timothy. A Discourse on Some Events of the Last Century. New Haven, 1801. 55 pp.*Summation of the state of American people, especially in morals and religion. A diatribe against the
Enlightenment, self-interest, commercialism—all put at the door of freemasonry.
• 104 Dwight, Timothy. The Nature and Danger of Infidel Philosophy. . . . New Haven, 1798. 95 pp.*An earlier and inferior version of the previous piece.
• 105 Dwight, Timothy. Sermon Before the Connecticut Society of Cincinnati. New Haven, 1795. 40 pp.*Excellent discussion of virtue and politics.
• 106 Dwight, Timothy. The True Means of Establishing Public Happiness. New Haven, 1795. 40 pp.The importance of religion and virtue.
• 107 Dwight, Timothy. Virtuous Rulers A National Blessing. New Haven, 1791. 42 pp.
• 108 Eliot, Andrew. Massachusetts Election Day Sermon. Boston, 1765. 49 pp.**On qualities of good public officials and their relations with the citizenry. Subtle and practical.
• 109 Ellsworth, Oliver. The Landholder, VII. Connecticut Courant, December 17, 1787,On religious tests.
• 110 Emerson, William. An Oration . . . In Commemoration of the Anniversary of American Independence. Boston, 1802. 25 pp.**Discusses prevailing attitudes about equality, liberty, rights; and how
manners and way of life support these commitments.
• 111 Emmons, Nathanael. A Discourse. Delivered on the National Fast, Wrentham, Mass., 1799. 31 pp.**†
• 112 Emmons, Nathanael. A Discourse . . . in Commemoration of American Independence. Wrentham, Mass., 1802. 24 pp.*Character of American political system—mixed regime; similarity of American
independence to Jewish independence.
• 113 Evans, Israel. New Hampshire Election Day Sermon. Concord, N.H., 1791. 35 pp.*Interrelationships of religion, liberty, and just government.
• 114 Farmer, A. W. A View of the Controversy Between Great Britain and Her Colonies. . . . New York, 1774. 37 pp.*A reply to Alexander Hamilton’s Full Vindication, which, in turn, had been a reply
to an earlier pamphlet by Farmer—Free Thoughts on the Proceedings. . . . Very much a Tory.
• 115 Findley, William. History of the Insurrection in the Four Western Counties of Pennsylvania. . . . Philadelphia, 1796. 328 pp.
• 116 Fish, Elisha. A Discourse. Worcester, 1775. 28 pp.
• 117 [Fitch, Thomas.] Reasons Why the British Colonies in America, Should Not Be Charged With Internal Taxes, by Authority of Parliament. New Haven, 1764. 39 pp.*Reproduced in Bailyn, ed.,
Pamphlets of the American Revolution.
• 118 Fobes, Peres [Perez]. An Election Sermon. Boston, 1795. 42 pp.**†
• 119 Ford, Timothy [Americanus]. The Constitutionalist: Or, An Inquiry How Far It Is Expedient and Proper to Alter the Constitution of South Carolina. City Gazette and Daily Advertiser,
Charleston, 1794. 55 pp.**†
• 120 Ford, Timothy. An Enquiry into the Constitutional Authority of the Supreme Federal Court Over the Several States. Charleston, 1792. 40 pp.
• 121 Foster, Dan. A Short Essay on Civil Government. Hartford, 1775. 73 pp.Origin and nature of government.
• 122 Franklin, Benjamin. An Account of the Supremest Court of Judicature in Pennsylvania, viz., The Court of the Press. Philadelphia Federal Gazette, February 12, 1789.**†
• 123 Franklin, Benjamin. Advice to a Young Tradesman from an Old One. Worcester Magazine, Third Week in August, 1786, pp. 247-248.*The capitalist ethic and behavior appropriate to it.
• 124 French, Jonathan. A Sermon. Boston, 1796. 23 pp.
• 125 [Gale, Benjamin.] Brief, Decent, but Free Remarks on Several Laws. Hartford, 1782. 55 pp.
• 126 Galloway, Joseph. A Candid Examination of the Mutual Claims of Great Britain and the Colonies. . . . New York, 1775, 62 pp.**Galloway’s rebuttal of the case for separation. Reproduced in
Jensen, ed., Tracts of the American Revolution.
• 127 Galloway, Joseph. A Letter to the People of Pennsylvania. . . . Philadelphia, 1760. 17 pp.Justifies an independent judiciary.
• 128 Galloway, Joseph. A Reply to an Address. . . . New York, 1775. 42 pp.Galloway answers someone who severely attacked his Candid Examination. . . . Says colonies integral part of Great Britain.
• 129 Gerry, Elbridge [A Columbian Patriot]. Observations on the New Constitution, and on the Federal and State Conventions. Boston, 1788.*Reproduced in Ford, ed., Pamphlets on the Constitution of
the United States.
• 130 Gordon, William. The History of the Rise, Progress, and Establishment of the Independence of the United States of America. 3 vols. New York, 1794.
• 131 Gray, Robert. New Hampshire Election Day Sermon. Dover, N.H., 1798. 29 pp.The requisites for a great nation.
• 132 [Grey, Isaac.] A Serious Address to Quakers. Philadelphia, 1778. 44 pp.
• 133 Griffith, David. Passive Obedience Considered: A Sermon. Williamsburg, Va., 1776. 26 pp.The right of resistance—drawn from biblical passages.
• 134 Hamilton, Alexander, James Madison, and John Jay [Publius]. The Federalist Papers. Published between October 27, 1787 and May 28, 1788 with individual essays appearing primarily, but not
exclusively, in four New York papers: the Independent Journal, the New York Packet, the New York Daily Advertiser, and the New-York Journal and Daily Patriotic Register. The eighty-five essays
have been published together in book form a number of times, with the best available being Jacob E. Cooke, ed., The Federalist (Cleveland: Meridian Books, 1961).**
• 135 Hammon, Jupiter. Address to the Negroes of New York. New York, 1787. 20 pp.
• 136 Hanson, Alexander. To the People of Maryland. Maryland Journal, April 13, June 22, and August 14, 1787, (last one titled “To Publicola”).Formerly writing under “J.B.F.” Hanson continues
debate with Publicola [Samuel Chase] and argues against the people giving binding instructions to their representatives.
• 137 Hanson, Alexander [Aristides]. Remarks on the Proposed Plan of a Federal Government. Annapolis, 1788. 42 pp.*Reproduced in Ford, ed., Pamphlets on the Constitution of the United States.
• 138 Hart, Levi. Liberty Described and Recommended: in a Sermon Preached to the Corporation of Freemen in Farmington. Hartford, 1775. 23 pp.**†
• 139 Haven, Jason. Massachusetts Election Day Sermon. Boston, 1769. 54 pp.Civil order and disobedience.
• 140 Hawkins, Benjamin, et. al. Articles of a Treaty . . . [with] the Head Men and Warriors of the Cherokees. April, 1786.*
• 141 Hay, George [Hortensius]. An Essay on the Liberty of the Press. Philadelphia, 1799.Legalistic discussion of requiring security of good behavior for publishers under indictment for libel.
• 142 Hay, James. [A Virginian Born and Bred]. Remarks on the Bill of Rights and Constitution . . . of the State of Virginia. 1796. 35 pp.Arguments for writing a new state constitution to remedy
the defects of the 1776 document.
• 143 Hemmenway, Moses. Massachusetts Election Day Sermon. Boston, 1784. 52 pp.**Liberty in (1) state of nature, (2) civil society, and (3) the Church; limits on rulers in each of these; rights and
duties of individuals; sources of authority.
• 144 Hicks, William [A Citizen]. The Nature and Extent of Parliamentary Power Considered. Pennsylvania Journal, January 21-February 25, 1786.American colonists equal to British people.
• 145 Hillhouse, William. A Dissertation, In Answer to a Late Lecture on the Political State of America. New Haven, 1789. 23 pp.A Federalist defends proposed Constitution.
• 146 Hilliard, Timothy. An Oration (July 4). Portland, Maine, 1803. 20 pp.
• 147 Hilliard, Isaac. The Rights of Suffrage. Danbury, 1804. 64 pp.
• 148 Hitchcock, Gad. An Election Sermon. Boston, 1774. 56 pp.**†
• 149 Hitchcock, Gad. A Sermon [Thanksgiving]. Boston, 1775. 44 pp.**On liberty—natural, civil, and religious.
• 150 [Hoar, David] Natural Principles of Liberty, Virtue, etc., Boston, 1782. 12 pp.
• 151 Holdfast, Simon. Facts Are Stubborn Things, Or Nine Plain Questions. Hartford, 1803. 23 pp.A Federalist defends Connecticut’s long-standing commitment to restricted suffrage, and to state
support for education.
• 152 Hopkins, Samuel. A Dialogue Concerning the Slavery of the Africans. New York, 1776. 71 pp.*Rebuttal of all arguments for continued slavery.
• 153 Hopkins, Stephen. The Rights of Colonies Examined. Providence Gazette, December 22, 1764.**†
• 154 Hotchkiss, Frederick W. On National Greatness. New Haven, 1793. 23 pp.
• 155 Howard, Martin Jr. A Letter from a Gentleman at Halifax, To His Friend in Rhode Island, Containing Remarks Upon a Pamphlet Entitled “The Rights of Colonies Examined”. Newport, R.I. 1765.
Attacks the pamphlet by Stephen Hopkins.
• 156 Howard, Simeon. Massachusetts Election Day Sermon. Boston, 1780. 48 pp.*Characteristics of good rulers: need for educated population; emphasis upon virtue. Government should encourage piety.
• 157 Howard, Simeon. A Sermon Preached to the Ancient and Honorable Artillery Company in Boston. Boston, 1773. 43 pp.**†
• 158 Humphreys, Daniel. The Inquirer: Being an Examination of the Question Whether the Legitimate Powers of Government Extend to the Care of Religion. Boston, 1801. 47 pp.*
• 159 Huntington, Enoch. Political Wisdom, Or Honesty the Best Policy. Middletown, Conn., 1786. 20 pp.The qualities desirable in public officials.
• 160 Huntington, Joseph. God Ruling the Nations for the Most Glorious Ends. Hartford, 1784. 34 pp.*Efforts by elected officials to rule justly are being thwarted by public distrust.
• 161 Hurt, John. The Love of Our Country, A Sermon Preached Before the Virginia Troops. Philadelphia, 1777. 23 pp.
• 162 Inglis, Charles. The Letters of Papinian: in which the Conduct, Present State and Prospects, of the American Congress, Are Examined. New York, 1779. 150 pp.
• 163 Inglis, Charles. The True Interest of America Impartially Stated, In Certain Strictures on a Pamphlet Intitled “Common Sense”. Philadelphia, 1776. 71 pp.A Tory attacking Paine’s pamphlet
rehearses all the costs likely to be incurred with independence.
• 164 Iredell, James [Marcus] Answer to Mr. [George] Mason’s Objections to the New Constitution. . . . 1788.*Reproduced in Ford, ed., Pamphlets on the Constitution of the United States.
• 165 Jackson, Jonathan [A Native of Boston]. Thoughts Upon the Political Situation of the United States of America. . . . Worcester, 1788. 209 pp.**A Whiggish analysis of the Massachusetts
Constitution in comparison with the proposed United States Constitution.
• 166 Jay, John [A Citizen of New York]. An Address to the People of the State of New York on the Subject of the Constitution Agreed Upon at Philadelphia, the 17th of September, 1787. 1788.*
Reproduced in Ford, ed., Pamphlets on the Constitution of the United States.
• 167 Jefferson, Thomas. Notes on the State of Virginia, edited by William Peden. Chapel Hill, 1955.**
• 168 Jefferson, Thomas. A Summary View of the Rights of British America. Williamsburg, Va., 1774. 23 pp.*Reproduced in Jensen, ed., Tracts of the American Revolution.
• 169 Johnson, John Barent. An Oration on Union. New York, 1794. 24 pp.
• 170 Johnson, Stephen. A Connecticut Election Sermon. New London, 1770. 39 pp.**Good on general Whig principles.
• 171 Johnson, Stephen. Integrity and Piety the Best Principles of a Good Administration of Government. New London, 1770. 39 pp.
• 172 Jones, David. Defensive War in a Just Cause [is] Sinless. Philadelphia, 1775. 27 pp.
• 173 Keith, Isaac S. The Friendly Influence of Religion and Virtue. Charleston, 1789. 24 pp.
• 174 Kendal, Samuel. Religion the Only Sure Basis of Free Government. Boston, 1804. 34 pp.**†
• 175 Kendal, Samuel. A Sermon. Boston, 1794. 35 pp.*Liberty dependent upon a regime of order.
• 176 Kent, James. Dissertations . . . Preliminary Part of a Course of Law Lectures. New York, 1795. 87 pp.**Main forms of government and their respective merits; development of self-government in
America; principles of law governing nations.
• 177 Kent, James. An Introductory Lecture to a Course of Law Lectures. New York, 1794. 23 pp.**†
• 178 Keteltas, Abraham. God Arising and Pleading His People’s Cause. Boston, 1777. 32 pp.
• 179 Kirkland, John T. A Sermon. . . . Boston, 1795. 35 pp.Wars are evil, but some good effects arise from them.
• 180 Knox, Samuel. An Essay on the Best System of Liberal Education, Adapted to the Genius of the Government of the United States. . . . 1799.*Reproduced in Rudolph, ed., Essays on Education in
the Early Republic, pp. 271-372.
• 181 Knox, William. Massachusetts Election Day Sermon. Boston, 1769. 100 pp.
• 182 De Lafitte Du Courteil, Amable-Louis-Rose. Proposal to Demonstrate the Necessity of a National Institution in the United States of America, for the Education of Children of Both Sexes. . . .
Philadelphia, 1797.*Reproduced in Rudolph, ed., Essays on Education in the Early Republic.
• 183 Lathrop, John. Innocent Blood Crying to God: Boston Massacre Sermon. Boston, 1771. 21 pp.
• 184 Lathrop, John. A Sermon Preached to the Artillery Company in Boston. Boston, 1774. 39 pp.*Circumstances under which Christians are justified in going to war.
• 185 Lathrop, Joseph. The Happiness of a Free Government. Springfield, Mass., 1794. 22 pp.
• 186 Lathrop, Joseph. A Miscellaneous Collection of Original Pieces. Springfield, Mass., 1786. 168 pp.**†
• 187 Lathrop, Joseph. A Sermon. Springfield, Mass., 1787. 24 pp.
• 188 Lee, Arthur. An Appeal to the Justice and Interests of the People of Great Britain. New York, 1775. 32 pp.
• 189 Lee, Charles. Defense of the Alien and Sedition Laws. Philadelphia, 1798. 47 pp.
• 190 Lee, Richard Henry. Letters from the Federal Farmer to the Republican. Letters II and III out of 18 published in 1787.**Reproduced in Ford, ed., Pamphlets on the Constitution of the United
• 191 Leib, Michael. Patriotic Speech. New London, 1796. 24 pp.
• 192 Leland, John. A Blow at the Root. 1801. In L. F. Greene, ed., The Writings of John Leland (New York: Arno Press, 1969), pp. 235-55.*
• 193 Leland, John. The Connecticut Dissenters’ Strong Box: No. I. New London, 1802. 40 pp.**†
• 194 Leland, John. An Elective Judiciary. . . . 1805. In L. F. Greene, ed., The Writings of John Leland, pp. 285-300.
• 195 Leland, John. The Rights of Conscience Inalienable. . . . 1791. In L. F. Greene, ed., The Writings of John Leland, pp. 179-192.
• 196 Leland, John [Jack Nips] The Yankee Spy. Boston, 1794. 20 pp.**†
• 197 Leonard, Daniel [Massachusettensis] The Origin of the Contest With Great Britain. New York, 1775. 86 pp.**Balanced, detailed analysis urging caution and accommodation.
• 198 Leonard, Daniel [Massachusettensis]. To All Nations of Men. Massachusetts Spy, November 18, 1773.*†
• 199 Lewis, Isaac. A Sermon Preached Before . . . the Governor . . . and Legislature. Hartford, 1797. 31 pp.*Basic principles, religion, virtue, and godliness.
• 200 Linn, William. A Discourse on National Sins. New York, 1798. 37 pp.Religion, government, prosperity, and how all can be undercut by sin.
• 201 Livingston, Philip. The Other Side of the Question . . . A Defence of the Liberties of North America. Boston, 1774. 29 pp.
• 202 Livingston, Robert. The Address of Mr. Justice Livingston to the House of Assembly in Support of His Right to a Seat. New York, 1769.New York Assembly cannot, according to Livingston, deny a
seat in that body to justices of the colony’s supreme court.
• 203 Livingston, Robert R. An Oration. New York, 1787. 22 pp.
• 204 Livingston, William. Observations on Government, Including Some Animadversions on Mr. Adams’s “Defence of the Constitutions. . . .” New York, 1787. 56 pp.**
• 205 Livingston, William. On the Use, Abuse, and Liberty of the Press. 1753.Reproduced in Leonard W. Levy, ed., Freedom of the Press from Zenger to Jefferson: Early American Libertarian Theories
(Indianapolis: Bobbs-Merrill, 1966).
• 206 Logan, George. Five Letters Addressed to the Yeomanry. Philadelphia, 1792. 28 pp.Presses for social and economic equality.
• 207 Lyman, Joseph. A Sermon, (Election Day). Boston, 1787. 61 pp.
• 208 MacClintock, Samuel. A Sermon Preached Before the Honorable the Council. Portsmouth, N.H., 1784. 47 pp.*Comprehensive on Whig principles from religious point of view.
• 209 McKeen, Joseph. Massachusetts Election Day Sermon. Boston, 1800. 30 pp.**Qualities and conduct of good rulers and relation of religion to same. Wisdom and virtue preferred to brilliance.
• 210 Madison, James. An Extensive Republic Meliorates. 1787. In Gaillard Hunt, ed., The Writings of James Madison (New York: G. P. Putnam’s Sons, 1901), 2:365-369.**
• 211 Madison, James. Letter to T. Jefferson, Oct. 24, 1787. In Gaillard Hunt, ed., The Writings of James Madison (New York: G. P. Putnam’s Sons, 1901), 2:18-35.**Incisive summary of much debate at
constitutional convention.
• 212 Madison, James. Vices of the Political System of the United States. 1787. In Gaillard Hunt, ed., The Writings of James Madison, 2:36-69.**
• 213 Madison, James, et. al. Memorial and Remonstrance Against Religious Assessments [in Virginia]. 1785. In Gaillard Hunt, ed., The Writings of James Madison (New York: G. P. Putnam’s Sons,
1901), 2:183-191.**†
• 214 Mason, John Mitchell. The Voice of Warning to Christians. 1800.Jefferson cannot fulfill obligations of office because he is an atheist.
• 215 Mason, Jonathan. An Oration. New York, 1780. 40 pp.The necessity of patriotism for maintaining freedom, justice, etc.
• 216 Maxcy, Jonathan. An Oration. Providence, 1799. 16 pp.**†
• 217 Mayhew, Jonathan. On the Limits of Obligation to Obey Government. 1750.**No state of nature or compact—good of the people. Reproduced in Bailyn, ed., Pamphlets of the American Revolution.
• 218 Mellen, John. A Great and Happy Doctrine. Boston, 1795. 34 pp.
• 219 Mellen, John. Massachusetts Election Day Sermon. Boston, 1797. 36 pp.**Origin of government, basis for political obligation, basis for resistance, guide for good rulers and for deposing them.
• 220 Messer, Asa. An Oration . . . in the Baptist Meeting House on the 4th of July. . . . Providence, 1803. 14 pp.*On the relation of knowledge, virtue, and religion to popular government.
• 221 Miller, Samuel. A Discourse [to] the Society for Manumission of Slaves. New York, 1797. 36 pp.
• 222 Minot, George Richard. The History of the Insurrections in Massachusetts, in the year 1776 and the Rebellion Consequent Thereon. Worcester, 1788. 192 pp.
• 223 Moore, Zephaniah Swift. An Oration on the Anniversary of the Independence of the United States of America. Worcester, 1802. 24 pp.**†
• 224 Morse, Jedidiah. A Sermon Exhibiting the Present Dangers and Consequent Duties of the Citizens of the United States. Charlestown, Mass., 1799. 59 pp.Defense of the right and duty of ministers
to preach on political subjects; posits a French plot to undermine United States government, and silencing ministry part of this.
• 225 Moultrie, William. Memoirs of the American Revolution so far as it Related . . . North and South Carolina, and Georgia, vols. I and II. 1802.
• 226 Nicholas, George. A Letter . . . Justifying the Conduct of the Citizens of Kentucky [re: Kentucky Resolutions of 1798]. Lexington, Ky., 1798. 42 pp.
• 227 Niles, Nathaniel. Two Discourses on Liberty. Newburyport, Mass., 1774. 38 pp.**†
• 228 Osgood, David. A Discourse. Boston, 1795. 40 pp.*
• 229 Osgood, David. A Sermon. Boston, 1788. 20 pp.
• 230 Osgood, David. A Thanksgiving Sermon. Boston, 1794. 20 pp.
• 231 Otis, James. The Rights of the British Colonies Asserted and Proved. Boston Gazette, July 23, 1764.**Reproduced in Bailyn, ed., Pamphlets of the American Revolution.
• 232 Otis, James. A Vindication of the British Colonies Against the Aspersions of the Halifax Gentleman, in His Letter to a Rhode Island Friend. Boston, 1765.*Response to pamphlet by Martin
Howard. Reproduced in Bailyn, ed., Pamphlets of the American Revolution.
• 233 Page, John. An Address to the Citizens of the District of York, in Virginia. Richmond, 1794.
• 234 Paine, Thomas. Common Sense Addressed to the Inhabitants of America. Philadelphia, January 9, 1776. 45 pp.**Reproduced in Jensen, ed., Tracts of the American Revolution.
• 235 Parker, Samuel. A Sermon. Boston, 1793. 42 pp.
• 236 Parsons, Jonathon. A Consideration of Some Unconstitutional Measures Adopted and Practiced in This State. Newburyport, Mass., 1784. 24 pp.
• 237 Parsons, Theodore. A Forensic Dispute on the Legality of Enslaving the Africans. Boston, 1773. 48 pp.
• 238 Parsons, Theophilus. The Essex Result. Newburyport, Mass., 1778. 68 pp.**†
• 239 Payson, Phillips. A Sermon. Boston, 1778. 30 pp.**†
• 240 Payson, Seth. A Sermon. Portsmouth, N.H., 1799. 23 pp.
• 241 Peck, Jedidiah. The Political Wars of Otsego: Downfall of Jacobinism and Despotism. . . . Cooperstown, N.Y., 1796. 123 pp.Dangers of levelling spirit.
• 242 Perkins, John. [A Well-Wisher to Mankind]. Theory of Agency: Or, An Essay on the Nature, Source and Extent of Moral Freedom. Boston, 1771. 43 pp.**†
• 243 Pinkney, William. Speech in the House of Delegates of Maryland. Philadelphia, 1790. 22 pp.Supports legislation (a) prohibiting shipment of slaves to the West Indies, (b) removing restrictions
on manumission of slaves.
• 244 Pope, Nathaniel. A Speech. Richmond, 1800. 37 pp.Concerning the Sedition Act.
• 245 Porter, Nathaniel. A Discourse (Election Day Sermon). Concord, N.H., 1804. 34 pp.*On the qualities and conduct of good rulers.
• 246 Prescott, Benjamin. A Free and Calm Consideration of the Unhappy Misunderstanding. . . Between the Parliament of Great Britain and These American Colonies. Salem, Mass., 1774. 52 pp.*
• 247 Price, Richard. Observations on the Nature of Civil Liberty. New York Gazette and Weekly Mercury, July 22, 1776.*
• 248 Quincy, Josiah. Observations on the . . . Boston Port Bill With Thoughts on Civil Society and Standing Armies. Boston, 1774. 82 pp.*
• 249 Quincy, Josiah. An Oration. Boston, 1798. 31 pp.
• 250 Ramsay, David. An Address to the Freemen of South Carolina, On the Subject of the Federal Constitution. . . . Charleston, 1788, 12 pp.Reproduced in Ford, ed., Pamphlets on the Constitution of
the United States, (Brooklyn, 1888), pp. 373-380.
• 251 Ramsay, David. The History of the American Revolution. Philadelphia, 1789. 390 pp.**†
• 252 Ramsay, David. The History of the Revolution of South Carolina. 2 vols. Trenton, 1785.*
• 253 Ramsay, David. An Oration on the Advantages of American Independence. Pennsylvania Gazette, January 20, 1779.The arts and sciences in a new republic.
• 254 Randolph, Edmund. Letter on the Federal Constitution. October 16, 1787.*Found in Ford, ed., Pamphlets on the Constitution of the United States (Brooklyn, 1888), pp. 261-276.
• 255 Reese, Thomas. An Essay on the Influence of Religion, in Civil Society. Charleston, 1788. 87 pp.
• 256 Rice, David. Slavery Inconsistent With Justice and Good Policy. Augusta, Ky., 1792. 23 pp.**†
• 257 Ross, Robert. A Sermon [on] the Union of the Colonies. New York, 1776. 28 pp*.The reasons for separating from Britain.
• 258 Rush, Benjamin. [A Pennsylvanian]. An Address to the Inhabitants of the British Settlements in America Upon Slave-Keeping. Philadelphia, 1773. 28 pp.**†
• 259 Rush, Benjamin. Considerations on the Injustice and Impolicy of Punishing Murder by Death. Philadelphia, 1792. 19 pp.
• 260 Rush, Benjamin. Considerations Upon the Present Test-Law of Pennsylvania. Philadelphia, 1784. 23 pp.In opposition to oaths. Rush was a prominent Quaker.
• 261 Rush, Benjamin. Essays Litarary, Moral and Philosophical. Philadelphia, 1798.
• 262 Rush, Benjamin. On the Superiority of a Bicameral to a Unicameral Legislature. Philadelphia, 1777. 24 pp.*
• 263 Rush, Benjamin. A Plan for the Establishment of Public Schools and the Diffusion of Knowledge in Pennsylvania; To Which Are Added, Thoughts upon the Mode of Education, Proper in a Republic.
Philadelphia, 1786. 23 pp.*†
• 264 B.
ITEMS WHERE THE AUTHOR IS DISPUTED OR UNKNOWN
□ 354The Address and Petition of a Number of the Clergy of Various Denominations . . . Relative to the Passing of a Law Against Vice and Immorality. Philadelphia, 1793. 13 pp.Proposes outlawing
theatrical exhibitions, among other things.
□ 355An Address of the Convention for Framing a New Constitution of Government of the State of New Hampshire. Portsmouth, N.H., 1781. 64 pp.Why the old constitution is deficient.
□ 356 Aequus. From the Craftsman [London]. Massachusetts Gazette and Boston Newsletter, March 6, 1766.**†
□ 357 Agricola. [untitled essay]. Massachusetts Spy, October 22, 1772.**Very Lockian statement of basic principles on government.
□ 358 Agrippa [James Winthrop?] Massachusetts Gazette, November 23-February 5, 1788.Reproduced in Ford, ed., Pamphlets on the Constitution of the United States.
□ 359 Amendments Proposed to the Federal Constitution Proposed by the New York State Convention. Boston Gazette, August 18, 1788.*
□ 360 Amicus. To the Printer. Columbian Herald, Columbia, S.C., August 28, 1788.Anti-Federalist statement on the right of recall.
□ 361 Amicus Republicae. Address to the Public, Containing Some Remarks on the Present Political State of the American Republicks, etc. Exeter, 1786. 36 pp.**†
□ 362 [anon.] Address of a Convention of Delegates from the Abolition Society, to the Citizens of the United States. Philadelphia, 1794. 7 pp.
□ 363 [anon.] An Address . . . Respecting the Alien and Sedition Laws. Richmond, 1798. 63 pp.
□ 364 [anon.] An Address to the Inhabitants of the County of Berkshire Respecting Their Present Opposition to Civil Government. Hartford, 1778. 28 pp.*
□ 365 [anon.] The Alarm: or, an Address to the People of Pennsylvania, on the Late Resolve of Congress, for Totally Suppressing All Power and Authority Derived from the Crown of Great Britain.
Philadelphia, 1776. 4 pp.**†
□ 366 [anon.] Ambition. City Gazette and Daily Advertiser, Charleston, June 6, 1789.*†
□ 367 [anon.] Boston Gazette, September 17, 1764**†
□ 368 [anon.] A Candid Examination of the Address of the Minority of the Council of Censors. Philadelphia, 1784. 40 pp.
□ 369 [anon.] Declaration and Address of His Majesty’s Loyal Associated Refugees, Assembled at Rhode Island. New York, 1779. 36 pp.
□ 370 [anon.] A Declaration of Independence Published by the Congress at Philadelphia in 1776 With a Counter-Declaration Published at New York in 1781. New York, 1781. 24 pp.The Tories declare
their independence from revolutionary America.
□ 371 [anon.] Discussion of Revision of South Carolina’s Code of Law. City Gazette and Daily Advertiser, Charleston, February 3, 1789.
□ 372 [anon.] Dissertation Upon the Constitutional Freedom of the Press. Boston, 1801. 54 pp.
□ 373 [anon.] An English Patriot’s Creed, Anno Domini, 1775. Massachusetts Spy, January 19, 1776.*†
□ 374 [anon.] An Essay of a Frame of Government for Pennsylvania. Philadelphia, 1776. 16 pp.*Summary of Whig ideas, with specific proposals for a state constitution.
□ 375 [anon.] An Essay Upon Government. Philadelphia, 1775. 125 pp.**Origin of government; society, government, and property defined; authority and obligations of rulers; and the rights and
obligations of citizens.
□ 376 [anon.] A Few Salutary Hints Pointing out the Policy and Consequences of Permitting British Subjects to Engross Our Trade and Become Our Citizens. Charleston, 1786. 16 pp.
□ 377 [anon.] Four Letters on Interesting Subjects. Philadelphia, 1776. 24 pp.**†
□ 378 [anon.] A Friend to the Judiciary. New York, 1801. 60 pp. Concerning the independence of the judiciary.
□ 379 [anon.] An Impartial Review of the Rise and Progress of the Controversy Between . . . Federalists and Republicans. Philadelphia, 1800. 50 pp.
□ 380 [anon.] A Letter from a Virginian to the Members of the Continental Congress. Boston, 1774. 31 pp.*A restrained, even-tempered plea for Congress to be patient and to seek accommodation
with Britain.
□ 381 [anon.] Letter to a Member of the General Assembly of Virginia on the Subject of a Conspiracy of the Slaves. Richmond, 1801. 21 pp.
□ 382 [anon.] Letter to the Editor. Boston Gazette, July 22, 1765.*“No taxation without representation” applied to western Massachusetts towns vis-a-vis Massachusetts legislature.
□ 383 [anon.] Letter to the Editor. Massachusetts Spy, April 4, 1771.*The nature of government.
□ 384 [anon.] Letter to the Editor. Massachusetts Spy, August 22, 1771.The nature of government.
□ 385 [anon.] Letter to the Editor. Boston Gazette, December 31, 1787.*Short, pithy summary of views on education.
□ 386 [anon.] A Letter to the People of Pennsylvania, Occasioned by the Assembly’s Passing that Important Act, for Constituting the Judges of the Supreme Courts and Common-Pleas, During Good
Behavior. Philadelphia. 1760. 39 pp.*Reproduced in Bailyn, ed., Pamphlets of the American Revolution.
□ 387 [anon.] A Memorial and Remonstrance Presented to the General Assembly of the State of Virginia . . . in Consequence of a Bill . . . for the Establishment of Religion by Law. Worcester,
1786. 16 pp.*
□ 388 [anon.] Northampton [Mass.] Returns to the Convention on the Constitution. 1780. In Oscar Handlin and Mary Handlin, eds. The Popular Sources of Political Authority (Cambridge, Mass.:
Harvard University Press, 1966), pp. 572-587.*Comprehensive critique of the Massachusetts Constitution of 1780, especially interesting on property requirement in voting for lower house.
□ 389 [anon.] No Standing Army in the British Colonies. New York, 1775. 18 pp.
□ 390 [anon.] Number I and Number II. City Gazette and Daily Advertiser, Charleston, March 16, 17, and 18, 1789.Parliamentary privilege and freedom of the press.
□ 391 [anon.] On the Management of Children in Infancy. South Carolina Gazette, November 1, 1773.*Brief statement on child-rearing up to literacy at age seven.
□ 392 [anon.] The People the Best Governors: Or a Plan of Government Founded on the Just Principles of Natural Freedom. New Hampshire, 1776. 11 pp.**†
□ 393 [anon.] The Political Establishment of the United States of America. Philadelphia, 1784. 25 pp.*Inadequacy of the Articles of Confederation—a new constitution is required.
□ 394 [anon.] The Power and Grandeur of Great Britain Founded on the Liberty of the Colonies. . . . New York, 1768. 24 pp.**The British government does not impose taxes; the people make
voluntary contributions for revenue.
□ 395 [anon.] Proposals to Amend and Perfect the Policy of the Government of the United States of America. Baltimore, 1782. 36 pp.*
□ 396 [anon.] Review [in two parts] of John Adams’s “Defence of the Constitutions . . . of America,” taken from the Monthly Review (in London) and reprinted in the New York Packet, September 25
and 28, 1787.
□ 397 [anon.] Rudiments of Law and Government Deduced from the Law of Nature. Charleston, 1783. 56 pp.**†
□ 398 [anon.] Serious Considerations on Several Important Subjects, viz. On War . . . Observations on Slavery . . . Spiritous Liquors. Philadelphia, 1778. 48 pp.
□ 399 [anon.] To the Printers. Boston Gazette, July 15, 1765.Americans are equal to the British at home.
□ 400 [anon.] To the Printer. Boston Gazette, December 2, 1765.*Succinct statement of general principles in response to the Stamp Act.
□ 401 [anon.] [two untitled essays]. The United States Magazine, January, Providence, 1779 vol. I, pp. 5-41, 155-159.*The first summarizes traditional attitudes toward government. The second
outlines reasons for distaste for established religion.
□ 402 A. Z. Virtuous Pennsylvanians. South Carolina Gazette, November 29, 1773.
□ 403 Benevolus. Poverty. City Gazette and Daily Advertiser, Charleston, December 8, 1789.*†
□ 404 Berkshire’s Grievances. Statement of Berkshire County Representatives, and Address to the Inhabitants of Berkshire. Pittsfield, Mass., 1778.**†
□ 405Bills of Rights and Amendments Proposed by Massachusetts and Virginia [to the Proposed United States Constitution]. 1788.*Reproduced in Kenyon, ed. The Antifederalists, pp. 421-39.
□ 406 Bostonians. Serious Questions Proposed to All Friends to The Rights of Mankind, With Suitable Answers. Boston Gazette, November 19, 1787.*†
□ 407 Britannus Americanus. Boston Gazette, March 17, 1766.**†
□ 408 Brutus [Thomas Treadwell? Robert Yates?] Against the New Federal Constitution. Worcester Magazine, December, 1787.List of objections to the proposed constitution.
□ 409 Brutus [Thomas Treadwell? Robert Yates?] No. I: To the Citizens of the State of New York. New York Journal and Weekly Register, October 18, 1787.*Not reproduced in the volume edited by
Kenyon (as are several of the other essays by Brutus), this one expresses the fears that under the new Constitution the government will be too far from the people, and the country too
□ 410 Brutus [Thomas Treadwell? Robert Yates?] No. II. New York Journal and Weekly Register, November 1, 1787.*
□ 411 Brutus [Thomas Treadwell? Robert Yates?] No. IV: To the People of the State of New York. New York Journal and Weekly Register, November 29, 1787.*Not reproduced in Kenyon, this essay
explores the relationship between the people and their representatives.
□ 412 Brutus [Thomas Treadwell? Robert Yates?] No. V: To the People of the State of New York, New York Journal and Weekly Register, December 13, 1787.*Not reproduced in Kenyon, it proposes that
the Constitution is an original compact among the people dissolving other compacts, rather than an agreement among the states.
□ 413 Brutus [Thomas Treadwell? Robert Yates?] No. VI: To the People of the State of New York, New York Journal and Weekly Register, December 27, 1787.*Reproduced in Kenyon, ed., The
Antifederalists. Will the states be absorbed?
□ 414 Brutus Junior. Letter to the Editor. New York Journal, November 8, 1787.
□ 415 By a Gentleman Born and Bred. Remarks on the Bill of Rights, Constitution and Some Acts of the General Assembly of the State of Virginia. Richmond, 1801. 35 pp.
□ 416 Cato. Discourse Upon Libel. Massachusetts Spy, April 19, 1771.
□ 417 Centinel [Samuel Bryan?] No. I & No. II: To the People of Pennsylvania. Maryland Journal, October 30, and November 2, respectively, 1787.A widely-read Anti-federalist. Reproduced in
Kenyon, ed., The Antifederalists.
□ 418 Cincinnatus. Number I, Number II, Number V, and Number VI: To James Wilson, esq. New York Journal, November 1, 8, 29, and December 6, respectively, 1787.An Anti-Federalist response to
James Wilson’s defense of the proposed Constitution. Number II especially notable on freedom of the press and trial by jury. Number VI speaks to taxation and public finance.
□ 419 A Citizen. To the Citizens of Richmond, Not Freeholders. Virginia Argus, Richmond, July 31, 1801.In favor of broad suffrage.
□ 420 A Citizen of Connecticut. An Address to the Legislature and People of Connecticut on the Subject of Dividing the State into Districts for the Election of Representatives in Congress. New
Haven, 1791. 37 pp.
□ 421 Columbus. A Letter to a Member of Congress, Respecting the Alien and Sedition Laws. Boston, 1799.
□ 422 Common Sense. [untitled essay]. Massachusetts Gazette, January, 1788.Arguments in support of the proposed Constitution.
□ 423 A Constant Customer. Extract of a Letter from a Gentleman in the Country to His Friend. Massachusetts Spy, February 18, 1773.*†
□ 424The Constitution of the Pennsylvania Society for Promoting the Abolition of Slavery . . . to Which are Added the Acts . . . of Pennsylvania for the Gradual Abolition of Slavery.
Philadelphia, 1788. 29 pp.
□ 425 Continental Congress. Appeal to the Inhabitants of Quebec, October 26, 1774, Journals of the Continental Congress, vol. I, pp. 105-113.**†
□ 426 Council of Censors of Pennsylvania. Minority Report. To the Freemen of Pennsylvania. Philadelphia, 1784, 12 pp.Anti-constitutionalists in Pennsylvania list the failures of the 1776
Pennsylvania Constitution.
□ 427 A Countryman. Letter to the Editor. New York Journal, Dec. 6, 1787.*The social disruptions caused by the war.
□ 428 A Countryman. Letter II. New York Journal, Dec. 13, 1787.Discusses section in the Constitution on the importation of slaves. Confused by the terms Federalist and anti-Federalist.
□ 429 D.D. Extract from a Thanksgiving Sermon, Delivered in the County of Middlesex. Worcester Magazine, January, 1787.*Defense of the Massachusetts government against the charges by Daniel
□ 430 Deliberator. To the Printers. Freeman’s Journal, Philadelphia, February 20, 1788.In opposition to the proposed Constitution.
□ 431 Demophilus [George Bryan?] The Genuine Principles of the Ancient Saxon, or English[,] Constitution, Philadelphia, 1776. 46 pp.**†
□ 432 De Witte, John [pseud.] To the Editor. American Herald, Worcester, December 3, 1787.An Anti-Federalist essay.
□ 433 An Elector. To the Free Electors of This Town. Boston Gazette, April 28, 1788.**†
□ 434 F.A. A Letter to a Right Noble Lord. Boston Gazette, July 22, 29, August 5, 12, 26, and September 2, 1765.Six-part essay in response to a member of Parliament who defended the Stamp Act.
□ 435 A Farmer. To the Editor. Maryland Gazette and Baltimore Advertiser, March 7, 1788.The new Constitution will not abate war or prevent despotism.
□ 436 Farmer. To the Printer. Pennsylvania Packet, Philadelphia, November 5, 1776.*Exposition of Whig ideology in relatively concise form.
□ 437 A Federalist. Letter to the Editor. Boston Gazette, December 3, 1787.A general defense of the proposed Constitution.
□ 438 A Federalist. To the People of Pennsylvania. Maryland Journal, November 6, 1787.In response to Centinel.
□ 439 Form of Ratification of the Federal Constitution by the State of New York. Boston Gazette, August 11, 1788.*
□ 440 Freeborn American. To the Printers. Boston Gazette and Country Journal, March 9, 1767.The duties of a free press.
□ 441 Freeholders of Boston. Instructions to Their Representatives. Boston Gazette, May 28, 1764.*Summary of Whig ideas and values.
□ 442 Freeholders of Newbury-Port. Instructions to Their Representatives. Boston Gazette, November 4, 1765.Summary of basic values.
□ 443 Freeholders of Plymouth. Instructions to Their Representatives. Boston Gazette, November 4, 1765.
□ 444 Freeman, [Untitled essay reproduced from the June 6 issue of the New York Gazette]. Georgia Gazette, September 19, 26, and October 3, 1765.**Virtual representation, the nature of
representation, and the relationship of the American people to the British people.
□ 445 Freeman. Another Letter from Freeman. Georgia Gazette, October 26, 1769.*In response to Libertas, supports the position that the people are sovereign and can withdraw support from a
legislature that breaks the contract.
□ 446 Hamden. On Patriotism. South Carolina Gazette, November 29, 1773.Brief discussion of private interest versus public good.
□ 447 Hermes. The Oracle of Liberty, and Mood of Establishing a Government. Philadelphia, 1791, 39 pp.
□ 448 Historicus. Royal South Carolina Gazette, Charleston, March 28, 1782.An untitled essay laying out the Tory view of republican government.
□ 449 Homespun. A Countryman. South Carolina Gazette, October 31, 1774.*Brief discussion of how deliberation on public affairs should proceed, who should be allowed to deliberate, etc.
□ 450 Hortensius. An Essay on the Liberty of the Press, Richmond, 1799. 30 pp.*
□ 451 An Impartial Citizen. A Dissertation Upon the Constitutional Freedom of the Press. Boston, 1801. 54 pp.**†
□ 452 Instructions of the Town of New-Braintree to its Representative. Worcester Magazine, June, 1786
□ 453 J. Letter to the Printer. The Boston Evening Post, May 23, 1763, Supplement.
□ 454 J.B.F. To the Electors of Anne-Arundel County. Maryland Journal and Baltimore Advertiser, February 23, 1787.In response to Samuel Chase’s piece in the same paper, J.B.F. attacks the
practice of instructing representatives.
□ 455 The Journeyman Carpenters. An Address. American Daily Advertiser, Philadelphia, May 11, 1791.Justifies their strike and striking in general.
□ 456 Junius, Camillus. [untitled]. The Argus, or Greenleaf’s New Daily Advertiser, New York, March 15 and April 6, 1796.*Freedom of speech—the legislature has no “privilege” against criticism.
□ 457 A Landholder. For the New Federal Constitution. Worcester Magazine, December, 1787.
□ 458 Leonidas. A Reply to Lucius Junius Brutus’ Examination of the President’s Answer to the New Haven Remonstrance. New York, 1801. 62. pp.*Leonidas is attacking Brutus, a Federalist: topics
range from the limits to majority rule to presidential power of appointment and removal.
□ 459 L.Q. To the Printers. Boston Gazette, May 16, 1763.A reply to T.Q., whose discussion on the separation of powers (prohibition on multiple office holding) appeared in the April 18 edition
of the same paper.
□ 460 Majority and minority reports on the repeal of the Sedition Act. February 25, 1799. Annals of Congress, 5th Cong., 3rd Session, pp. 2987-2990, 3033-3014.*
□ 461 Medium. On the Proposed Federal Constitution. Worcester Magazine, December, 1787.
□ 462 A Member of the General Committee. To Freeman. South Carolina Gazette, October 18, 1769.Counters a critic of the Stamp Act.
□ 463A Memorial and Remonstrance Presented to the General Assembly of the State of Virginia . . . In Consequence of a Bill . . . for the Establishment of Religion by Law. Worcester, 1786. 16
□ 464Memorial Presented to Congress . . . by Different Societies Promoting Abolition of Slavery. 1792. 31 pp.
□ 465 Monitor. No. VI, Massachusetts Spy, January 9, 1772.**A community has the right to reward every virtue and punish every vice. A list of virtues is included.
□ 466 Monitor. To the New Appointed Councellors of the Province of Massachusetts-Bay. Massachusetts Spy, August 18, 1774.**†
□ 467 Monitor. [untitled]. Massachusetts Gazette, October 30, 1787.Supports the proposed Constitution.
□ 468 M.Y. A Letter from a Son of Liberty in Boston to a Son of Liberty in Bristol County. Boston Evening Post, May 12, 1766.Defends lawyers as members of the legislature against those who
would exclude lawyers from political office.
□ 469 A Native of this Colony. An Address to the Convention of the Colony . . . of Virginia, on the Subject of Government in General and Recommending a Particular Form to Their Attention.
Virginia Gazette, June 8, 1776.**The basic principle underlying each form of government, with a good discussion of virtue (public versus private).
□ 470 Nestor. To the Publick. Worcester Magazine, December, 1786.**The blessings of civil society and the need for seeking the common good to remain a civil society (of the five essays, the
first is best).
□ 471 Nov Anglicanus. To the Inhabitants of the Province. Boston Gazette, May 14, 1764.A response to the Stamp Act.
□ 472 An Observer. To the Editor. American Herald, Worcester, December 3, 1787.A rejoinder to Federalist paper number five.
□ 473 An Officer of the Late Continental Army. Against the Federal Constitution. Worcester Magazine, December, 1787.
□ 474 An Old Whig. To the Printer. Massachusetts Gazette, November 27, 1787.
□ 475 An Old Whig. To the Printer. Freeman’s Journal, Philadelphia, November 28, 1787.On constitutional conventions.
□ 476 An Old Whig. To the Printer. Maryland Gazette and Baltimore Advertiser, November 2, 1788.An opponent of the proposed Constitution predicts that the “necessary and proper” clause will be
used to expand the powers granted Congress in Article I.
□ 477 One of the Subscribers. Letter to the Editor. New York Packet, September 21, 1789.*Propositions for reforming the system of public education in Boston, for both sexes.
□ 478 An Other Citizen. On Conventions. Worcester Magazine, September, 1786.*Opposed to the county conventions called by those opposed to the operation of Massachusetts courts. These
conventions eventually led to Shays’s Rebellion.
□ 479 P. . . . To the Printers. New York Mercury, January 28, 1765.A typical response to the Stamp Act.
□ 480 Penn, William [pseud.] To the Printer. Independent Gazetteer, Philadelphia, January 3, 1788.An Anti-Federalist keying on the topic of presidential veto.
□ 481Personal Slavery Established by the Suffrages of Custom and Right Reason. Philadelphia, 1773. 26 ppA reply to a piece by Anthony Benezet, this essay outlines the standard arguments used in
favor of slavery.
□ 482 Philadelphiensis [Benjamin Workman?] To the Printer. Freeman’s Journal, Philadelphia, February 6, 20, and April 9, 1788.An Anti-Federalist focusing on the executive branch.
□ 483 Philanthropos. [untitled]. Pennsylvania Gazette, Philadelphia., January 16, 1788.In support of the proposed Constitution.
□ 484 Philodemos. [untitled]. Boston American Herald, May 12, 1788.In support of the proposed Constitution.
□ 485 Philo Patriae [William Goddard?] The Constitutional Courant: Continuing Matters Interesting to Liberty, and No Wise Repugnant to Loyalty. Burlington, N.J. [?], 1765.
□ 486 Philo Publicus. Boston Gazette, October 1, 1764.*†
□ 487 Philo Publius [untitled]. New York Daily Advertiser, December 1, 1787.In support of the proposed Constitution.
□ 488 The Preceptor. Vol. II Social Duties of the Political Kind. Massachusetts Spy, May 21, 1772.**†
□ 489Proposed Amendments [to the Federal Constitution] Made by the Maryland Convention. Annapolis, 1788.
□ 490 A Republican. To the Printer. New Hampshire Gazette, Exeter, February 8 to March 22, 1783.*Summary of the Whig perspective.
□ 491 Republicus. To the Printer. The Kentucky Gazette, March 1, 1788.Against the proposed Constitution, especially the electoral college.
□ 492 Resolves of the Lower House of the South Carolina Legislature. South Carolina Gazette and Country Journal, December 17, 1765.*Resolutions in opposition to the Stamp Act; wording and logic
very similar to that found in proposals by northern colonies.
□ 493 Resolves of the Massachusetts House of Representatives. Boston Gazette, November 4, 1765.In opposition to the Stamp Act. Good summary of basic American political principles. See previous
□ 494 Rusticus. Letter to the Editor. New York Journal, September 13, 1787.In opposition to the proposed Constitution.
□ 495 Salus Populi. To the Freemen of the Province of Pennsylvania. South Carolina and American General Gazette, Charleston, April 3, 1776.Justifies breaking with England.
□ 496 [Several Quakers]. An Address to the Inhabitants of Pennsylvania by the Freemen of Philadelphia Who Are Now Confirmed. Philadelphia, 1777. 52 pp.
□ 497 Sidney. Letter to the Editor. New York Journal, September 13, 1787.In opposition to the proposed Constitution.
□ 498 Spartanus. Freemans Journal or New Hampshire Gazette, Portsmouth, June 15 and 29, 1776.*A strongly democratic statement.
□ 499 Theophrastus. A Short History of the Trial by Jury. Worcester Magazine, October, 1787.**†
□ 500To the Supporters and Defenders of American Freedom and Independence in the State of New York. New York, 1778.Urges no traffic with or toleration of Tories, loyalists, or collaborators
with Britain.
□ 501 T.Q. On Separation of Powers: How Much Separation is Enough? Boston Gazette and Country Journal, April 4, 18, and June 6, 1763.*†See the piece by L.Q.
□ 502 The Tribune. No. xvii. South Carolina Gazette, October 6, 1766.**†
□ 503 Tribunus. Letters from Tribunus to Republicanus. Worcester Magazine, May, 1787.Two articles discussing public credit.
□ 504 Tullius. Three Letters on the Nature of the Federal Union, etc., Philadelphia, 1783. 28 pp.
□ 505 U. Boston Gazette, August 1, 1763.*†
□ 506 U. To the Printers. Boston Gazette, August 29, 1763.Diatribe against “private revenge.”
□ 507 Velerius. Massachusetts Centinel, Boston, November 28, 1787.Supports the proposed Constitution.
□ 508The Virginia Report of 1799-1800, Touching the Alien and Sedition Laws, Richmond, 1850.
□ 509 Virginiensis [Charles Lee?] Defense of the Alien and Sedition Laws. Philadelphia, 1798. 47 pp.
□ 510The Votes and Proceedings of the Freeholders and Other Inhabitants of the Town of Boston, In Town Meeting Assembled, According to Law. November 20, 1772, [Samuel Adams?].*Reproduced in
Jensen, ed., Tracts of the American Revolution.
□ 511 Vox Populi. To the Printer. Massachusetts Gazette, Boston, October 30, 1787.Against the proposed Constitution, with a special concern for the dangers in congressional control of
□ 512 The Worcester Speculator. No. VI. Worcester Magazine, October, 1787.**†
□ 513 Worcestriensis. To the Honorable . . . (No. II). Massachusetts Spy, August 14, 1776.*The importance of education to a republic.
□ 514 Worcestriensis. Number III. Massachusetts Spy, August 21, 1776.*The importance of religion.
□ 515 Worcestriensis. Number IV. Massachusetts Spy, September 4, 1776.**†
A LIST OF NEWSPAPERS EXAMINED
Anyone attempting to read comprehensively the newspapers published in America between 1760 and 1805 runs into several problems. First of all, a significant percentage of issues did not survive,
and those that do are often available only on microfilm of poor quality and in various libraries. The Library of Congress has the most complete collection, but even there the problem is that few
papers were published for as long as half the period under study. The strategy forced upon the researcher is to select judiciously from those papers available, with the aim of constructing a
continuous set of newspapers over the period from each of the major cities and towns that generated the most activity. The problem is eased somewhat by the significant number of newspapers that
did not usually publish political essays and letters, or if they did, tended to reprint essays from newspapers elsewhere. Most of the newspapers that were not read comprehensively, and are so
indicated below, were in fact examined and determined to fall into this last category. An estimated four thousand political essays and letters were examined in the newspapers from the era.
Because it was the practice in even the most sophisticated publications to reprint pieces from papers in other colonies, in some instances a political essay was encountered four or five times in
various newspapers, from South Carolina to New Hampshire. In the list below, those newspapers that were consulted comprehensively for the period 1760-1805 are marked with an asterisk. The rest
are listed to show which major papers were not so examined, and to help provide a reasonably complete list of newspapers for the period.
□ American Mercury (Hartford)*
□ Connecticut Courant (Hartford)*
□ Connecticut Gazette (New London)*
□ Connecticut Journal (New Haven)
□ Middlesex Gazette (Middletown)
□ New Haven Chronicle
□ New Haven Gazette*
□ Norwich Packet
□ Spectator*
□ Weekly Monitor (Litchfield)
□ Wilmington Courant
□ Wilmington Gazette
□ Augusta Chronicle
□ Georgia Gazette (Savannah)*
□ State Gazette of Georgia (Savannah)*
□ Maryland Chronicle (Frederick)
□ Maryland Gazette (Annapolis)*
□ Maryland Gazette and Baltimore Advertiser*
□ Maryland Journal (Baltimore)*
□ Weekly Museum (Baltimore)*
□ American Herald (Worcester)*
□ Berkshire Chronicle
□ Boston Censor*
□ Boston Chronicle*
□ Boston Evening Post*
□ Boston Gazette*
□ Boston Gazette and Weekly Republican Journal*
□ Cumberland Gazette (Portland, Maine)
□ Essex Journal (Salem)
□ Hampshire Chronicle (Springfield)
□ Hampshire Gazette (Northhampton)
□ Hampshire Herald (Springfield)
□ Independent Chronicle (Boston)
□ Massachusetts Centinel (Boston)*
□ Massachusetts Gazette (Boston)*
□ Massachusetts Spy (Worcester)*
□ Post Boy and Advertiser (Boston)*
□ Salem Mercury
□ Western Star (Stockbridge)
□ Worcester Magazine*
NEW HAMPSHIRE
□ Freemans Oracle and New Hampshire Advertiser (Exeter)*
□ New Hampshire Gazette and General Advertiser (Exeter)
□ New Hampshire Mercury (Portsmouth)
□ New Hampshire Recorder and Weekly Advertiser (Keene)*
□ New Hampshire Spy (Portsmouth)
NEW JERSEY
□ Brunswick Gazette (New Brunswick)
□ New Jersey Gazette (Trenton)
□ New Jersey Journal (Elizabethtown)
□ Plain Dealer (Bridgetown)*
NEW YORK
□ Albany Gazette*
□ Albany Register
□ American Magazine (New York)
□ Goshen Repository
□ Hudson Gazette
□ Independent Journal (New York)
□ New York Daily Advertiser (New York)*
□ New York Gazette (New York)*
□ New York Gazette and Weekly Mercury (New York)*
□ New York Journal (New York)*
□ New York Mercury (New York)*
□ New York Museum (New York)
□ New York Packet (New York)*
□ Northern Centinel or Lansingburg Advertiser
□ Poughkeepsie Journal*
NORTH CAROLINA
□ North Carolina Chronicle (Fayetteville)
□ North Carolina Gazette*
□ State Gazette of North Carolina (Newberne and Edentown)*
□ American Museum (Philadelphia)
□ Freeman’s Journal (Philadelphia)*
□ Independent Gazetteer (Philadelphia)*
□ Lancaster Journal*
□ Pennsylvania Evening Post and Daily Advertiser (Philadelphia)*
□ Pennsylvania Gazette (Philadelphia)*
□ Pennsylvania Herald (Philadelphia)
□ Pennsylvania Journal (Philadelphia)*
□ Pennsylvania Ledger (Philadelphia)*
□ Pennsylvania Mercury (Philadelphia)
□ Pennsylvania Packet (Philadelphia)*
□ Pittsburg Gazette
RHODE ISLAND
□ Newport Herald
□ Newport Mercury*
□ Providence Gazette*
□ United States Chronicle (Providence)
SOUTH CAROLINA
□ City Gazette, or Daily Advertiser (Charleston)*
□ The Columbian Herald or the Independent Courier (Charleston)
□ Royal South Carolina Gazette (Charleston)*
□ South Carolina and American General Gazette (Charleston)*
□ South Carolina Gazette (Charleston)*
□ South Carolina Gazette and Country Journal (Charleston)*
□ South Carolina State Gazette and Timothy’s Daily Advertiser (Charleston)*
□ South Carolina Weekly Chronicle
□ State Gazette of South Carolina (Charleston)*
□ The Norfolk and Portsmouth Chronicle
□ Virginia Gazette (Winchester)*
□ Virginia Gazette and Petersburg Advertiser
□ The Virginia Gazette and Weekly Advertiser (Richmond)
□ The Virginia Herald and Independent Advertiser
□ Virginia Independent Chronicle (Richmond)*
□ The Virginia Journal and Alexandria Advertiser*
COLLECTIONS OF WRITING FROM THE FOUNDING ERA
There are a number of good, more-specialized collections that have proved to be very useful, and any student of American political theory would want to be at least familiar with their respective
contents. In some instances we have drawn upon them for pieces found in this collection.
□ Almon, John, ed. A Collection of Papers Relative to the Dispute Between Great Britain and America, 1764-1775. New York: Da Capo Press, 1971.
□ Bailyn, Bernard, ed. Pamphlets of the American Revolution. Cambridge, Mass.: Belknap Press, 1965.
□ Borden, Morton, ed. The Antifederalist Papers. East Lansing, Mich.: Michigan State University Press, 1965.
□ Cooke, J. E., ed. The Federalist. Cleveland: Meridian Books, 1961.
□ Elliott, Jonathan, ed. The Debates in the Several State Conventions on the Adoption of the Federal Constitution. Philadelphia: J. B. Lippincott, 1901.
□ Farrand, Max, ed. The Records of the Federal Convention of 1787. New Haven: Yale University Press, 1937.
□ Ford, Paul Leicester, ed. Pamphlets on the Constitution of the United States. Brooklyn: 1888.
□ Handlin, Oscar, and Mary Handlin, eds. The Popular Sources of Political Authority. Cambridge, Mass.: Harvard University Press, 1966.
□ Hyneman, Charles S. and George W. Carey, eds. A Second Federalist. New York: Appleton-Century-Crofts, 1967.
□ Jensen, Merrill, ed. Tracts of the American Revolution, 1763-1776. Indianapolis: Bobbs-Merrill, 1978.
□ Kenyon, Cecilia, ed. The Antifederalists. Indianapolis: Bobbs-Merrill, 1966.
□ Levy, Leonard W., ed. Freedom of the Press from Zenger to Jefferson: Early American Libertarian Theories. Indianapolis: Bobbs-Merrill, 1966.
□ Lewis, John D., ed. Anti-Federalists Versus Federalists: Selected Documents. San Francisco: Chandler, 1967.
□ Mark, Irving and Eugene L. Schwaab, eds. The Faith of Our Fathers: An Anthology Expressing the Aspirations of the American Common Man, 1790-1860. New York: Octagon Books, 1976.
□ Padover, Saul K., ed. The World of the Founding Fathers. New York: A. S. Barnes and Company, 1977.
□ Pole, J. R., ed. The Revolution in America, 1754-1788: Documents and Commentaries. Stanford, Calif.: Stanford University Press, 1970.
□ Rudolph, Frederick, ed. Essays on Education in the Early Republic. Cambridge, Mass.: The Belknap Press, 1965.
□ Smith, Wilson, ed. Theories of Education in Early America, 1655-1819. Indianapolis: Bobbs-Merrill, 1973.
□ Storing, Herbert, ed. The Complete Antifederalist. 7 vols. Chicago: University of Chicago Press, 1981.
□ Thornton, John Wingate, ed. The Pulpit of the American Revolution. Boston: Gould and Lincoln, 1860.
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Properties of number 62207
62207 has 2 divisors, whose sum is σ = 62208. Its totient is φ = 62206.
The previous prime is 62201. The next prime is 62213. The reversal of 62207 is 70226.
It is a balanced prime because it is at equal distance from previous prime (62201) and next prime (62213).
It is a cyclic number.
It is not a de Polignac number, because 62207 - 2^4 = 62191 is a prime.
It is a Chen prime.
It is a plaindrome in base 6 and base 12.
It is a zygodrome in base 6.
It is a congruent number.
It is an inconsummate number, since it does not exist a number n which divided by its sum of digits gives 62207.
It is not a weakly prime, because it can be changed into another prime (62201) by changing a digit.
It is a pernicious number, because its binary representation contains a prime number (13) of ones.
It is a polite number, since it can be written as a sum of consecutive naturals, namely, 31103 + 31104.
It is an arithmetic number, because the mean of its divisors is an integer number (31104).
2^62207 is an apocalyptic number.
62207 is a deficient number, since it is larger than the sum of its proper divisors (1).
62207 is an equidigital number, since it uses as much as digits as its factorization.
62207 is an odious number, because the sum of its binary digits is odd.
The product of its (nonzero) digits is 168, while the sum is 17.
The square root of 62207 is about 249.4133115934. The cubic root of 62207 is about 39.6229146703.
The spelling of 62207 in words is "sixty-two thousand, two hundred seven".
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The Complexity of Computing a Nash Equilibrium
An illustration of Nash's function for the penalty shot game.
How long does it take until economic agents converge to an equilibrium? By studying the complexity of the problem of computing a mixed Nash equilibrium in a game, we provide evidence that there are
games in which convergence to such an equilibrium takes prohibitively long. Traditionally, computational problems fall into two classes: those that have a polynomial-time algorithm and those that are
NP-hard. However, the concept of NP-hardness cannot be applied to the rare problems where "every instance has a solution" for example, in the case of games Nash's theorem asserts that every game has
a mixed equilibrium (now known as the Nash equilibrium, in honor of that result). We show that finding a Nash equilibrium is complete for a class of problems called PPAD, containing several other
known hard problems; all problems in PPAD share the same style of proof that every instance has a solution.
1. Introduction
In a recent CACM article, Shoham^22 reminds us of the long relationship between Game Theory and Computer Science, going back to John von Neumann at Princeton in the 1940s, and how this connection
became stronger and more crucial in the past decade due to the advent of the Internet: Strategic behavior became relevant to the design of computer systems, while much economic activity now takes
place on computational platforms.
Game Theory is about the strategic behavior of rational agents. It studies games, thought experiments modeling various situations of conflict. One commonly studied model aims to capture two players
interacting in a single round. For example, the well-known school yard game of rock-paper-scissors can be described by the mathematical game shown in Figure 1. There are two players, one choosing a
row and one choosing a column; the choices of a player are his/her actions. Once the two players choose, simultaneously, an action, they receive the corresponding payoffs shown in the table: The
first number denotes the payoff of Row, the second that of Column. Notice that each of these pairs of numbers sum to zero in the case of Figure 1; such games are called zero-sum games. Three other
well-known games, called chicken, prisoner's dilemma, and penalty shot game, respectively, are shown in Figure 2; the penalty shot game is zero-sum, but the other two are not. All these games have
two players; Game Theory studies games with many players, but these are harder to display.^a
The purpose of games is to help us understand economic behavior by predicting how players will act in each particular game. The predictions game theorists make about player behavior are called
equilibria. One such prediction is the pure Nash equilibrium: Each player chooses an action that is a "best response" to the other player's choice i.e., it is the highest payoff, for the player, in
the line, row or column, chosen by the other player. In the game of chicken in Figure 2, a pure Nash equilibrium is when one player chooses "dare" and the other chooses "chicken." In the prisoner's
dilemma, the only pure Nash equilibrium is when both players choose "defect."
Unfortunately, not all games have a pure Nash equilibrium. For example, it is easy to see that the rock-paper-scissors game in Figure 1 has none. This lack of universality is an important defect of
the concept of pure Nash equilibrium as a predictor of behavior. But the rock-paper-scissors game does have a more sophisticated kind of equilibrium, called a mixed Nash equilibrium and in fact one
that is familiar to all who have played this game: both players pick an action uniformly at random. That is, a mixed Nash equilibrium is a probabilistic distribution on the set of actions of each
player. Each of the distributions should have the property that it is a best response to the other distributions; this means that each action assigned positive probability is among the actions that
are best responses, in expectation, to the distribution(s) chosen by the opponent(s).
In 1950, John Nash proved that all games have a mixed Nash equilibrium.^19 That is, in any game, distributions over the players' actions exist such that each is a best response to what everybody else
is doing. This important and far from obvious universality theorem established the mixed Nash equilibrium as Game Theory's central equilibrium concept, the baseline and gold standard against which
all subsequent refinements and competing equilibrium concepts were judged.
Universality is a desirable attribute for an equilibrium concept. Of course, such a concept must also be natural and credible as a prediction of behavior by a group of agents for example, pure Nash
seems preferable to mixed Nash, in games that do have a pure Nash equilibrium. But there is a third important desideratum on equilibrium concepts, of a computational nature: An equilibrium concept
should be efficiently computable if it is to be taken seriously as a prediction of what a group of agents will do. Because, if computing a particular kind of equilibrium is an intractable problem, of
the kind that take lifetimes of the universe to solve on the world's fastest computers, it is ludicrous to expect that it can be arrived at in real life. This consideration suggests the following
important question: Is there an efficient algorithm for computing a mixed Nash equilibrium? In this article, we report on results that indicate that the answer is negative our own work ^5, 7, 8, 13
obtained this for games with three or more players, and shortly afterwards, the papers^2, 3 extended this unexpectedly to the important case of two-player games.
Ever since Nash's paper was published in 1950, many researchers have sought algorithms for finding mixed Nash equilibria that is, for solving the computational problem which we will call NASH in this
paper. If a game is zero-sum, like the rock-paper-scissors game, then it follows from the work of John von Neumann in the 1920s that NASH can be formulated in terms of linear programming (a subject
identified by George Dantzig in the 1940s); linear programs can be solved efficiently (even though we only realized this in the 1970s). But what about games that are not zero-sum? Several algorithms
have been proposed over the past half century, but all of them are either of unknown complexity, or known to require, in the worst case, exponential time.
During the same decades that these concepts were being explored by game theorists, Computer Science theorists were busy developing, independently, a theory of algorithms and complexity addressing
precisely the kind of problem raised in the last two paragraphs: Given a computational problem, can it be solved by an efficient algorithm? For many common computational tasks (such as finding a
solution of a set of linear equations) there is a polynomial-time algorithm that solves them this class of problems is called P. For other such problems, such as finding a truth assignment that
satisfies a set of Boolean clauses (a problem known as SAT), or the traveling salesman problem, no such algorithm could be found after many attempts. Many of these problems can be proved NP-complete,
meaning they cannot be solved efficiently unless P = NP an event considered very unlikely.^11
From the previous discussion of failed attempts to develop an efficient algorithm for NASH, one might be tempted to suppose that this problem too is NP-complete. But the situation is not that simple.
NASH is unlike any NP-complete problem because, by Nash's theorem, it is guaranteed to always have a solution. In contrast, NP-complete problems like SAT draw their intractability from the
possibility that a solution might not exist this possibility is used heavily in the NP-completeness proof.^b See Figure 3 for an argument (due to Nimrod Megiddo) why it is very unlikely that NP
-completeness can characterize the complexity of NASH. (Note however that if one seeks a Nash equilibrium with additional properties such as the one that maximizes the sum of player utilities, or one
that uses a given strategy with positive probability then the problem does become NP-complete.^4, 12)
Since NP-completeness is not an option, to understand the complexity of NASH one must essentially start all over in the path that led us to NP-completeness: We must define a class of problems which
contains, along with NASH, some other well-known hard problems, and then prove that NASH is complete for that class. Indeed, in this paper we describe a proof that NASH is PPAD-complete, where PPAD
is a subclass of NP that contains several important problems that are suspected to be hard, including NASH.
1.1. Problem statement: Nash and approximate Nash equilibria
A game in normal form has some number k of players, and for each player p (p S[p] of pure actions or strategies. The set S of pure strategy profiles is the Cartesian product of the S[p]'s. Thus, a
pure strategy profile represents a choice, for each player, of one of his actions. Finally, for each player p and s S the game will specify a payoff or utility u^p[s]≥0, which is the value of the
outcome to player p when all the players (including p) choose the strategies in s. In a Nash equilibrium, players choose probability distributions over their S[p]'s, called mixed strategies, so that
no player can deviate from his mixed strategy and improve on his expected payoff; see Figure 4 for details.
For two-player games there always exists a Nash equilibrium in which the probabilities assigned to the various strategies of the players are rational numbers assuming the utilities are also rational.
So, it is clear how to write down such a solution of a two-player game. However, as pointed out in Nash's original paper, when there are more than two players, there may be only irrational solutions.
In this general situation, the problem of computing a Nash equilibrium has to deal with issues of numerical accuracy. Thus, we introduce next the concept of approximate Nash equilibrium.
If Nash equilibrium means "no incentive to deviate," then approximate Nash equilibrium stands for "low incentive to deviate." Specifically, if ε is a small positive quantity, we can define an ε-Nash
equilibrium as a profile of mixed strategies where any player can improve his expected payoff by at most ε by switching to another strategy. Figure 4 gives a precise definition, and shows how the
problem reduces to solving a set of algebraic inequalities. Our focus on approximate solutions is analogous to the simpler problem of polynomial root-finding. Suppose that we are given a polynomial f
with a single variable, and we have to find a real root, a real number r satisfying f(r) = 0. In general, a solution to this problem (the number r) cannot be written down as a fraction, so we should
really be asking for some sort of numerical approximation to r (for example, computing a rational number r such that | f (r)| ≤ ε, for some small ε). If f happens to have odd degree, we can even say
in advance that a solution must exist, in a further analogy with NASH. Of course, the analogy breaks down in that for root-finding we know of efficient algorithms that solve the problem, whereas for
Nash equilibria we do not.
We are now ready to define the computational problem NASH: Given the description of a game (by explicitly giving the utility of each player corresponding to each strategy profile) and a rational
number ε > 0, compute an ε-Nash equilibrium. This should be at least as tractable as finding an exact equilibrium, hence any hardness result for approximate equilibria carries over to exact
equilibria. Note that an approximate equilibrium as defined above need not be at all close to an exact equilibrium; see Etessami and Yannakakis^9 for a complexity theory of exact Nash equilibria.
2. Total Search Problems
We think of NP as the class of search problems of the form "Given an input, find a solution (which then can be easily checked) or report that none exists." There is an asymmetry between these
outcomes, in that "none exists" is not required to be easy to verify.
We call such a search problem total if the solution always exists. There are many apparently hard total search problems in NP even though, as we argued in the introduction, they are unlikely to be NP
-complete. Perhaps the best-known is Factoring, the problem of taking an integer as an input and outputting its prime factors. Nash and several other problems introduced below are also total.
A useful classification of total search problems was proposed in Papadimitriou.^20 The idea is this: If a problem is total, the fact that every instance has a solution must have a mathematical proof.
Unless the problem can be easily solved efficiently, in that proof there must be a "nonconstructive step." It turns out that, for all known total search problems in the fringes of P, these
nonconstructive steps are one of very few simple arguments:
• "If a graph has a node of odd degree, then it must have another." This is the parity argument, giving rise to the class PPA.
• "If a directed graph has an unbalanced node (a vertex with different in-degree and out-degree), then it must have another." This is the parity argument for directed graphs, giving rise to the
class PPAD considered in this article. Figure 5 describes the corresponding search problems.
• "Every directed acyclic graph must have a sink." The corresponding class is called PLS for polynomial local search.
• "If a function maps n elements to n - 1 elements, then there is a collision." This is the pigeonhole principle, and the corresponding class is PPP.
We proceed with defining more precisely the second class in the list above.
2.1. The class PPAD
There are two equivalent ways to define NP: First, it is the class of all search problems whose answers are verifiable in polynomial time. For example, the search problem SAT ("Given a Boolean
formula in CNF, find a satisfying truth assignment, or report that none exists") is in NP because it is easy to check whether a truth assignment satisfies a CNF. Since we know that SAT is NP
-complete, we can also define NP as the class of all problems that can be reduced into instances of SAT. By "reduce" we refer to the usual form of polynomial-time reduction from search problem A to
search problem B: An efficient algorithm for transforming any instance of A to an equivalent instance of B, together with an efficient algorithm for translating any solution of the instance of B back
to a solution of the original instance of A.
We define the class PPAD using the second strategy. In particular, PPAD is the class of all search problems that can be reduced to the problem END OF THE LINE, defined in Figure 5. Note that, since
END OF THE LINE is a total problem, so are all problems in PPAD. Proceeding now in analogy with NP, we call a problem PPAD-complete if END OF THE LINE (and therefore all problems in PPAD) can be
reduced to it.
2.2. Why should we believe that PPAD contains hard problems?
In the absence of a proof that P ≠ NP we cannot hope to be sure that PPAD contains hard problems. The reason is that PPAD lies "between P and NP" in the sense that, if P=NP, then PPAD itself, as a
subset of NP, will be equal to P. But even if P ≠ NP, it may still be the case that PPAD-complete problems are easy to solve. We believe that PPAD-complete problems are hard for the same reasons of
computational and mathematical experience that convince us that NP-complete problems are hard (but as we mentioned, our confidence is necessarily a little weaker): PPAD contains many problems for
which researchers have tried for decades to develop efficient algorithms; in the next section we introduce one such problem called BROUWER. However, END OF THE LINE itself is a pretty convincingly
hard problem: How can one hope to devise an algorithm that telescopes exponentially long paths in every implicitly given graph?
3. From NASH to PPAD
Our main result is the following:
THEOREM 3.1. NASH is PPAD-complete.
In the remainder of this article we outline the main ideas of the proof; for full details see Daskalakis et al.^8 We need to prove two things: First, that NASH is in PPAD, that is, it can be reduced
to END OF THE LINE. Second (see Section 4), that it is complete the reverse reduction. As it turns out, both directions are established through a computational problem inspired by a fundamental
result in topology, called Brouwer's fixed point theorem, described next.
3.1. Brouwer's fixed point theorem
Imagine a continuous function mapping a circle (together with its interior) to itself for example, a rotation around the center. Notice that the center is fixed, it has not moved under this function.
You could flip the circle but then all points on a diagonal would stay put. Or you could do something more elaborate: Shrink the circle, translate it (so it still lies within the original larger
circle), and then rotate it. A little thought reveals that there is still at least one fixed point. Or stretch and compress the circle like a sheet of rubber any way you want and stick it on the
original circle; still points will be fixed, unless of course you tear the circle the function must be continuous. There is a topological reason why you cannot map continuously the circle on itself
without leaving a point unmoved, and that's Brouwer's theorem.^16 It states that any continuous map from a compact (that is, closed and bounded) and convex (that is, without holes) subset of the
Euclidean space into itself always has a fixed point.
Brouwer's theorem immediately suggests an interesting computational total search problem, called BROUWER: Given a continuous function from some compact and convex set to itself, find a fixed point.
But of course, for a meaningful definition of BROUWER we need to first address two questions: How do we specify a continuous map from some compact and convex set to itself? And how do we deal with
irrational fixed points?
First, we fix the compact and convex set to be the unit cube [0, 1]^m in the case of more general domains, for example, the circular domain discussed above, we can translate it to this setting by
shrinking the circle, embedding it into the unit square, and extending the function to the whole square so that no new fixed points are introduced. We then assume that the function F is given by an
efficient algorithm Π[F] which, for each point x of the cube written in binary, computes F(x). We assume that F obeys a Lipschitz condition:
where d(·,·) is the Euclidean distance and K is the Lipschitz constant of F. This benign well-behavedness condition ensures that approximate fixed points can be localized by examining the value F(x)
when x ranges over a discretized grid over the domain. Hence, we can deal with irrational solutions in a similar maneuver as with NASH, by only seeking approximate fixed points. In fact, we have the
following strong guarantee: for any ε, there is an ε-approximate fixed point that is, a point x such that d(F(x), x) ≤ ε whose coordinates are integer multiples of 2^−d, where d depends on K, ε, and
the dimension m; in the absence of a Lipschitz constant K, there would be no such guarantee and the problem of computing fixed points would become intractable. Formally, the problem BROUWER is
defined as follows.
Input: An efficient algorithm Π[F] for the evaluation of a function F: [0, 1]^m → [0, 1]^m; a constant K such that F satisfies (4); and the desired accuracy ε.
Output: A point x such that d(F(x), x) ≥ ε.
It turns out that BROUWER is in PPAD. (Papadimitriou^20 gives a similar result for a more restrictive class of BROUWER functions.) To prove this, we will need to construct an END OF THE LINE graph
associated with a BROUWER instance. We do this by constructing a mesh of tiny triangles over the domain, where each triangle will be a vertex of the graph. Edges, between pairs of adjacent triangles,
will be defined with respect to a coloring of the vertices of the mesh. Vertices get colored according to the direction in which F displaces them. We argue that if a triangle's vertices get all
possible colors, then F is trying to shift these points in conflicting directions, and we must be close to an approximate fixed point. We elaborate on this in the next few paragraphs, focusing on the
two-dimensional case.
Triangulation: First, we subdivide the unit square into small squares of size determined by ε and K, and then divide each little square into two right triangles (see Figure 7, ignoring for now the
colors, shading, and arrows). (In the m-dimensional case, we subdivide the m-dimensional cube into m-dimensional cubelets, and we subdivide each cubelet into the m-dimensional analog of a triangle,
called an m-simplex.)
Coloring: We color each vertex x of the triangles by one of three colors depending on the direction in which F maps x. In two dimensions, this can be taken to be the angle between vector F(x) - x and
the horizontal. Specifically, we color it red if the direction lies between 0 and −135°, blue if it ranges between 90 and 225°, and yellow otherwise, as shown in Figure 6. (If the direction is 0°, we
allow either yellow or red; similarly for the other two borderline cases.) Using the above coloring convention the vertices get colored in such a way that the following property is satisfied:
(P1): None of the vertices on the lower side of the square uses red, no vertex on the left side uses blue, and no vertex on the other two sides uses yellow. Figure 7 shows a coloring of the vertices
that could result from the function F; ignore the arrows and the shading of triangles.
Sperner's Lemma: It now follows from an elegant result in Combinatorics called Sperner's Lemma^20 that, in any coloring satisfying Property (P1), there will be at least one small triangle whose
vertices have all three colors (verify this in Figure 7; the trichromatic triangles are shaded). Because we have chosen the triangles to be small, any vertex of a trichromatic triangle will be an
approximate fixed point. Intuitively, since F satisfies the Lipschitz condition given in (4), it cannot fluctuate too fast; hence, the only way that there can be three points close to each other in
distance, which are mapped in three different directions, is if they are all approximately fixed.
The Connection with PPAD: ...is the proof of Sperner's Lemma. Think of all the triangles containing at least one red and yellow vertex as the nodes of a directed graph G. There is a directed edge
from a triangle T to another triangle T' if T and T' share a red-yellow edge which goes from red to yellow clockwise in T (see Figure 7). The graph G thus created consists of paths and cycles, since
for every T there is at most one T' and vice versa (verify this in Figure 7). Now, we may also assume: On the left side of the square there is only one change from yellow to red.^c Under this
assumption, let T* be the unique triangle containing the edge where this change occurs (in Figure 7, T* is marked by a diamond). Observe that, if T* is not trichromatic (as is the case in Figure 7),
then the path starting at T* is guaranteed to have a sink, since it cannot intersect itself, and it cannot escape outside the square (notice that there is no red-yellow edge on the boundary that can
be crossed outward). But, the only way a triangle can be a sink of this path is if the triangle is trichromatic. This establishes that there is at least one trichromatic triangle. (There may of
course be other trichromatic triangles, which would correspond to additional sources and sinks in G, as in Figure 7.) G is a graph of the kind in Figure 5. To finish the reduction from BROUWER to END
OF THE LINE, notice that given a triangle it is easy to compute its colors by invoking Π[F], and find its neighbors in G (or its single neighbor, if it is trichromatic).
Finally, from Nash to Brouwer: To finish our proof that NASH is in PPAD we need a reduction from NASH to BROUWER. Such a reduction was essentially given by Nash himself in his 1950 proof: Suppose
that the players in a game have chosen some (mixed) strategies. Unless these already constitute a Nash equilibrium, some of the players will be unsatisfied, and will wish to change to some other
strategies. This suggests that one can construct a "preference function" from the set of players' strategies to itself, that indicates the movement that will be made by any unsatisfied players. An
example, of how such a function might look, is shown in Figure 8. A fixed point of such a function is a point that is mapped to itself a Nash equilibrium. And Brouwer's fixed point theorem, explained
above, guarantees that such a fixed point exists. In fact, it can be shown that an approximate fixed point corresponds to an approximate Nash equilibrium. Therefore, NASH reduces to BROUWER.
4. From PPAD Back to NASH
To show that NASH is complete for PPAD, we show how to convert an END OF THE LINE graph into a corresponding game, so that from an approximate Nash equilibrium of the game we can efficiently
construct a corresponding end of the line. We do this in two stages. The graph is converted into a Brouwer function whose domain is the unit three-dimensional cube. The Brouwer function is then
represented as a game. The resulting game has too many players (their number depends on the size of the circuits that compute the edges of the END OF THE LINE graph), and so the final step of the
proof is to encode this game in terms of another game, with three players.
4.1. From paths to fixed points: The PPAD-completeness of BROUWER
We have to show how to encode a graph G, as described in Figure 5, in terms of a continuous, easy-to-compute Brouwer function F a very different-looking mathematical object. The encoding is
unfortunately rather complicated, but is the key to the PPAD-completeness result...
We proceed by, first, using the three-dimensional unit cube as the domain of the function F. Next, the behavior of F shall be defined in terms of its behavior on a (very fine) rectilinear mesh of
"grid points" in the cube. Thus, each grid point lies at the center of a tiny "cubelet," and the behavior of F away from the centers of the cubelets shall be gotten by interpolation with the closest
grid points.
Each grid point x shall receive one of four "colors" {0, 1, 2, 3}, that represent the value of the three-dimensional displacement vector F(x) - x. The four possible vectors can be chosen to point
away from each other such that F(x) - x can only be approximately zero in the vicinity of all the four colors.
We are now ready to fit G itself into the above framework. Each of the 2^n vertices of G shall correspond with two special sites in the cube, one of which lies along the bottom left-hand edge in
Figure 9 and the other one along the top left edge. (We use locations that are easy to compute from the identity of a vertex of G.) While most other grid points in the cube get color 0 from F, at all
the special sites a particular configuration of the other colors appears. If G has an edge from node u to node ν, then F shall also color a long sequence of points between the corresponding sites in
the cube (as shown in Figure 9), so as to connect them with sequences of grid points that get colors 1, 2, and 3. The precise arrangement of these colors can be chosen to be easy to compute (using
the circuits P and S that define G) and such that all four colors are adjacent to each other (an approximate fixed point) only at sites that correspond to an "end of the line" of G.
Having shown earlier that BROUWER is in PPAD, we establish the following:
THEOREM 4.1. BROUWER is PPAD-complete.
4.2. From BROUWER to NASH
The PPAD-complete class of Brouwer functions that we identified above have the property that their function F can be efficiently computed using arithmetic circuits that are built up using a small
repertoire of standard operators such as addition, multiplication, and comparison. These circuits can be written down as a "data flow graph," with one of these operators at each node. In order to
transform this into a game whose Nash equilibria correspond to (approximate) fixed points of the Brouwer function, we introduce players for every node on this data flow graph.
Games that Do Arithmetic: The idea is to simulate each arithmetic gate in the circuit by a game, and then compose the games to get the effect of composing the gates. The whole circuit is represented
by a game with many players, each of whom "holds" a value that is computed by the circuit. We give each player two actions, "stop" and "go." To simulate, say, multiplication of two values, we can
choose payoffs for three players x, y, and z such that, in any Nash equilibrium, the probability that z (representing the output of the multiplication) will "go" is equal to the product of the
probabilities that x and y will "go." The resulting "multiplication gadget" (see Figure 10) has a fourth player w who mediates between x, y, and z. The directed edges show the direct dependencies
among the players' payoffs. Elsewhere in the game, z may input his value into other related gadgets.
Here is how we define payoffs to induce the players to implement multiplication. Let X, Y, Z, and W denote the mixed strategies ("go" probabilities) of x, y, z, and w. We pay w the amount $X · Y for
choosing strategy stop and $Z for choosing go. We also pay z to play the opposite from player w. It is not hard to check that in any Nash equilibrium of the game thus defined, it must be the case
that Z = X · Y. (For example, if Z > X · Y, then w would prefer strategy go, and therefore z would prefer stop, which would make Z = 0, and would violate the assumption Z > X · Y.) Hence, the rules
of the game induce the players to implement multiplication in the choice of their mixed strategies.
By choosing different sets of payoffs, we could ensure that Z = X + Y or Z =½X. It is a little more challenging to simulate the comparison of two real values, which also is needed to simulate the
Brouwer function. Below we discuss that issue in more detail.
Computing a Brouwer function with games: Suppose we have a Brouwer function F defined on the unit cube. Include three players x[1], x[2], and x[3] whose "go" probabilities represent a point x in the
cube. Use additional players to compute F(x) via gadgets as described above. Eventually, we can end up with three players y[1], y[2], and y[3] whose "go" probabilities represent F(x). Finally, we can
give payoffs to x[1], x[2], and x[3] that ensure that in any Nash equilibrium, their probabilities agree with y[1], y[2], and y[3]. Then, in any Nash equilibrium, these probabilities must be a fixed
point of F.
The Brittle Comparator Problem: There's just one catch: our comparator gadget, whose purpose is to compare its inputs and output a binary signal according to the outcome of the comparison, is
"brittle" in that if the inputs are equal then it outputs anything. This is inherent, because one can show that, if a nonbrittle comparator gadget existed, then we could construct a game that has no
Nash equilibria, contradicting Nash's theorem. With brittle comparators, our computation of F is faulty on inputs that cause the circuit to make a comparison of equal values. We solve this problem by
computing the Brouwer function at a grid of many points near the point of interest, and averaging the results, which makes the computation "robust," but introduces a small error in the computation of
F. Therefore, the construction described above approximately works, and the three special players of the game have to play an approximate fixed point at equilibrium.
The Final Step: Three Players: The game thus constructed has many players (the number depends mainly on how complicated the program for computing the function F was), and two strategies for each
player. This presents a problem: To represent such a game with n players we need n2^n numbers the utility of each player for each of the 2^n strategy choices of the n players. But our game has a
special structure (called a graphical game, see Kearns et al.^15): The players are vertices of a graph (essentially the data flow graph of F), and the utility of each player depends only on the
actions of its neighbors.
The final step in the reduction is to simulate this game by a three-player normal form game this establishes that NASH is PPAD-complete even in the case of three players. This is accomplished as
follows: We color the players (nodes of the graph) by three colors, say red, blue, and yellow, so that no two players who play together, or two players who are involved in a game with the same third
player, have the same color (it takes some tweaking and argument to make sure the nodes can be so colored). The idea is now to have three "lawyers," the red lawyer, the blue lawyer, and the yellow
lawyer, each represent all nodes with their color, in a game involving only the lawyers. A lawyer representing m nodes has 2m actions, and his mixed strategy (a probability distribution over the 2m
actions) can be used to encode the simpler stop/ go strategies of the m nodes. Since no two adjacent nodes are colored the same color, the lawyers can represent their nodes without a "conflict of
interest," and so a mixed Nash equilibrium of the lawyers' game will correspond to a mixed Nash equilibrium of the original graphical game.
But there is a problem: We need each of the "lawyers" to allocate equal amounts of probability to their customers; however, with the construction so far, it may be best for a lawyer to allocate more
probability mass to his more "lucrative" customers. We take care of this last difficulty by having the lawyers play, on the side and for high stakes, a generalization of the rock-paper-scissors game
of Figure 1, one that forces them to balance the probability mass allocated to the nodes of the graph. This completes the reduction from graphical games to three-player games, and the proof.
5. Related Technical Contributions
Our paper ^7 was preceded by a number of important papers that developed the ideas outlined here. Scarf's algorithm^21 was proposed as a general method for finding approximate fixed points, more
efficiently than brute force. It essentially works by following the line in the associated END OF THE LINE graph described in Section 3.1. The Lemke-Howson algorithm^17 computes a Nash equilibrium
for two-player games by following a similar END OF THE LINE path. The similarity of these algorithms and the type of parity argument used in showing that they work inspired the definition of PPAD in
Three decades ago, Bubelis^1 considered reductions among games and showed how to transform any k-player game to a three-player game (for k > 3) in such a way that given any solution of the
three-player game, a solution of the k-player game can be reconstructed with simple algebraic operations. While his main interest was in the algebraic properties of solutions, his reduction is
computationally efficient. Our work implies this result, but our reduction is done via the use of graphical games, which are critical in establishing our PPAD-completeness result.
Only a few months after we announced our result, Chen and Deng^2, 3 made the following clever, and surprising, observation. The graphical games resulting from our construction are not using the
multiplication operation (except for multiplication by a constant), and therefore can even be simulated by a two-player game, leading to an improvement of our hardness result from three- to
two-player games. This result was unexpected, one reason being that the probabilities that arise in a two-player Nash equilibrium are always rational numbers, which is not the case for games with
three or more players.
Our results imply that finding an ε-Nash equilibrium is PPAD-complete, if ε is inversely proportional to an exponential function of the game size. Chen et al.^3 extended this result to the case where
ε is inversely proportional to a polynomial in the game size. This rules out a fully polynomial-time approximation scheme for computing approximate equilibria.
Finally, in this paper, we have focused on the complexity of computing an approximate Nash equilibrium. Etessami and Yannakakis^9 develop a very interesting complexity theory of the problem of
computing the exact equilibrium (or other fixed points), a problem that is important in applications outside Game Theory, such as Program Verification.
6. Conclusion and Future Work
Our hardness result for computing a Nash equilibrium raises concerns about the credibility of the mixed Nash equilibrium as a general-purpose framework for behavior prediction. In view of these
concerns, the main question that emerges is whether there exists a polynomial-time approximation scheme (PTAS) for computing approximate Nash equilibria. That is, is there an algorithm for ε-Nash
equilibria which runs in time polynomial in the game size, if we allow arbitrary dependence of its running time on 1/ε? Such an algorithm would go a long way towards alleviating the negative
implications of our complexity result. While this question remains open, one may find hope (at least for games with a few players) in the existence of a subexponential algorithm^18 running in time 0(
n^logn/ε2), where n is the size of the game.
How about classes of concisely represented games with many players? For a class of "tree-like" graphical games, a PTAS has been given in Daskalakis and Papadimitriou,^6 but the complexity of the
problem is unknown for more general low-degree graphs. Finally, another positive recent development^7 has been a PTAS for a broad and important class of games, called anonymous. These are games in
which the players are oblivious to each other's identities; that is, each player is affected not by who plays each strategy, but by how many play each strategy. Anonymous games arise in many
settings, including network congestion, markets, and social interactions, and so it is reassuring that in these games approximate Nash equilibria can be computed efficiently.
An alternative form of computational hardness, exemplified in Hart and Mansour,^14 arises where instead of identifying problems that are resistant to any efficient algorithm, one identifies problems
that are resistant to specific "natural" algorithms. In Hart,^14 lower bounds are shown for "decoupled" dynamics, a model of strategic interaction in which there is no central controller to find an
equilibrium. Instead, the players need to obtain one in a decentralized manner. The study and comparison of these models will continue to be an interesting research theme.
Finally, an overarching research question for the Computer Science research community investigating game-theoretic issues, already raised in Friedman and Shenker^10 but made a little more urgent by
the present work, is to identify novel concepts of rationality and equilibrium, especially applicable in the context of the Internet and its computational platforms.
1. Bubelis, V. On equilibria in finite games. Int. J. Game Theory 8, 2 (1979), 65 79.
2. Chen, X., Deng, X. Settling the complexity of 2-player Nash-equilibrium. Proceedings of FOCS (2006).
3. Chen, X., Deng, X., Teng, S. Computing Nash equilibria: approximation and smoothed complexity. Proceedings of FOCS (2006).
4. Conitzer, V., Sandholm, T. Complexity results about Nash equilibria. Proceedings of IJCAI (2003).
5. Daskalakis, C., Papadimitriou, C.H. Three-player games are hard. Electronic Colloquium in Computational Complexity, TR05-139, 2005.
6. Daskalakis, C., Papadimitriou, C. H. Discretized multinomial distributions and Nash Equilibria in anonymous games. Proceedings of FOCS (2008).
7. Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H. The complexity of computing a Nash Equilibrium. Proceedings of STOC (2006).
8. Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H. The complexity of computing a Nash Equilibrium. SICOMP. To appear.
9. Etessami, K., Yannakakis, M. On the complexity of Nash equilibria and other fixed points (extended abstract). Proceedings of FOCS (2007), 113 123.
10. Friedman, E., Shenker, S. Learning and implementation on the Internet. Department of Economics, Rutgers University, 1997.
11. Garey, M.R., Johnson, D.S. Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, 1979.
12. Gilboa, I., Zemel, E. Nash and correlated equilibria: some complexity considerations. Games Econ. Behav. (1989).
13. Goldberg, P.W., Papadimitriou, C.H. Reducibility among equilibrium problems. Proceedings of STOC (2006).
14. Hart, S. Mansour, Y. How long to equilibrium? The communication complexity of uncoupled equilibrium procedures. Proceedings of STOC (2007).
15. Kearns, M., Liftman, M., Singh, S. Graphical models for game theory. Proceedings of UAI (2001).
16. Knaster, B., Kuratowski, C., mazurkiewicz, S. Ein beweis des fixpunktsatzes für n-dimensionale simplexe. Fundamenta Mathematicae 14, (1929), 132 137.
17. Lemke, C.E., Howson, Jr.J.T., Equilibrium points of bimatrix games. SIAM J. Appl. Math, 12, (1964), 413 423.
18. Lipton, R., Markakis, E., Mehta, A. Playing large games using simple strategies. Proceedings of the ACM Conference on Electronic Commerce (2003).
19. Nash, J. Noncooperative games. Ann. Math. 54, (1951), 289 295.
20. Papadimitriou, C.H. On the complexity of the parity argument and other inefficient proofs of existence. J. Comput. Syst. Sci. 48, 3 (1994), 498 532.
21. Scarf, H.E. The approximation of fixed points of a continuous mapping. SIAM J. Appl. Math. 15, (1967), 1328 1343.
22. Shoham, Y. Computer science and game theory. Commun. ACM 51, 8, 75 79.
a. How about games such as chess? We can capture this and other similar games in the present framework by considering two players, Black and White, each with a huge action set containing all possible
maps from positions to moves; but of course, such formalism is not very helpful for analyzing chess and similar games.
b. "But what about the TRAVELING SALESMAN PROBLEM?" one might ask. "Does not it always have a solution?" To compare fairly the TRAVELING SALESMAN PROBLEM with SAT and NASH, one has to first transform
it into a search problem of the form "Given a distance matrix and a budget B, find a tour that is cheaper than B, or report that none exists". Notice that an instance of this problem may or may not
have a solution. But, an efficient algorithm for this problem could be used to find an optimal tour.
c. Suppose F gives rise to multiple yellow/red adjacencies on the left-hand side. We deal with this situation by adding an extra array of vertices to the left of the left side of the square, and
color all these vertices red, except for the bottom one which we color yellow. This addition does not violate (P1) and does not create any additional trichromatic triangles since the left side of the
square before the addition did not contain any blue.
A previous version of this paper appeared in the ACM2006 Proceedings of the Symposium on Theory of Computing.
The research reported here by Daskalakis and Papadimitriou was supported by NSF Grant CCF0635319.
DOI: http://doi.acm.org/10.1145/1461928.1461951
Figure 1. Rock-paper-scissors.
Figure 2. Three other two-player games.
Figure 3. Megiddo's proof that NASH is unlikely to be NP-complete.
Figure 4. Writing down the problem algebraically.
Figure 5. END OF THE LINE: An apparently hard total search problem.
Figure 6. The colors assigned to the different directions of F(x) - x. There is a transition from red to yellow at 0°, from yellow to blue at 90°, and from blue to red at 225°.
Figure 7. The subdivision of the square into smaller squares, and the coloring of the vertices of the subdivision according to the direction of F(x) - x. The arrows correspond to the END OF THE LINE
graph on the triangles of the subdivision; the source T * is marked by a diamond.
Figure 8. An illustration of Nash's function F[N] for the penalty shot game. The horizontal axis corresponds to the probability by which the penalty kicker kicks right, and the vertical axis to the
probability by which the goalkeeper dives left. The arrows show the direction and magnitude of F[N](x) - x. The unique fixed point of F[N] is (1/2, 1/2) corresponding to the unique mixed Nash
equilibrium of the penalty shot game. The colors respect
Figure 9. Embedding the END OF THE LINE graph in a cube. The embedding is used to define a continuous function F, whose approximate fixed points correspond to the unbalanced nodes of the END OF THE
LINE graph.
Figure 10. The players of the multiplication game. The graph shows which players affect other players' payoffs.
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Comments on Computational Complexity: Planes. Fellows. Power.1. Dan Bernstein's work on circuits for integer fa...Shortly after the Aeroflot incident indicated by A...The claim that this is the first successful water ...There has been substantial interest in theoretical...Another way to approach energy is to say that sinc...There was an interesting paper in CCC'07 that ...In most physical models of computation, energy is ...Ravi Jain, Zulfikar Ramzan, and I actually have a ...An Aeroflot airplane made a successful emegency la...The beauty of complexity: It's always relevant.Err...
tag:blogger.com,1999:blog-3722233.post7668673731490437787..comments2024-11-03T21:27:06.200-06:00Lance Fortnowhttp://www.blogger.com/profile/
06752030912874378610noreply@blogger.comBlogger10125tag:blogger.com,1999:blog-3722233.post-27801687292053938102009-01-22T23:05:00.000-06:002009-01-22T23:05:00.000-06:001. Dan Bernstein's work on
circuits for integer factorization takes an interesting approach to cost complexity, i.e., optimizing the circuits to minimize the amount of dollars that it takes to factor an integer of a given
size, where dollars is related to the amount of hardware built and the time to factor one number. We're talking about using billions of processors running for years, to factor a 300 digit number,
that sort of thing.<BR/><BR/>2. http://en.wikipedia.org/wiki/Water_landing#Survival_rates_of_passenger_plane_water_ditchings lists a bunch of water landings of commercial
flights.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-79416114612675738992009-01-21T01:43:00.000-06:002009-01-21T01:43:00.000-06:00Shortly after the Aeroflot incident indicated
by Anon#2, a Spanish commercial flight 'landed' in the sea (in Spain we actually have a verb for 'sea-ed'); all passengers survived except one who suffered a heart stroke, while awaiting the rescuers
at the airplane door. Due to this death the pilot was under trial. More here (in Spanish): <BR/>http://tinyurl.com/8ov7msBalquihttps://www.blogger.com/profile/
09857844891905542749noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-67679195246156325642009-01-19T08:40:00.000-06:002009-01-19T08:40:00.000-06:00The claim that this is the first successful
water landing sounds very unlikely to me, given that in none of the articles in the NYT, to which you linked, do they make such a
claim.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-74936882064022438402009-01-18T23:24:00.000-06:002009-01-18T23:24:00.000-06:00There has been substantial interest in
theoretical models of energy consumption, especially in connection with VLSI models.<BR/><BR/>In particular Gloria Kissin wrote a PhD dissertation on it (University of Toronto, 1987), some results
presented in conferences, notably STOC 1982. There is also a JACM paper<BR/>( Volume 38 , Issue 1 (January 1991)).CSProfhttps://www.blogger.com/profile/
07212822875614144307noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-63495984034966929112009-01-17T16:05:00.000-06:002009-01-17T16:05:00.000-06:00Another way to approach energy is to say
that since computation can be made reversible (http://en.wikipedia.org/wiki/Reversible_computing), there is no minimum energy for computation. But this at least as unsatisfying as just saying that
energy is proportional to time. I think that it is not at all intuitive that energy seems to be a trickier resource than time and space. <BR/><BR/>Kirk
PruhsAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-663370355109623222009-01-17T13:16:00.000-06:002009-01-17T13:16:00.000-06:00There was an interesting paper in CCC'07 that
was concerned with energy complexity for circuits:<BR/><BR/>http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=4262737&arnumber=4262761&count=39&index=23<BR/><BR/>In that paper, the
energy complexity of a circuit on an input x is defined as the number of gates that output "1" during the circuit's evaluation on x. (This measure is supposedly motivated by neural
circuits in the brain.) <BR/><BR/>This is a natural notion that isn't directly correlated with circuit size or depth. However, the authors point out that in an electrical circuit it seems to take
about as much energy to output "1" as it does to output "0"... maybe there is newer hardware that has this energy-consuming asymmetry?
Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-13114332850619227492009-01-17T11:59:00.000-06:002009-01-17T11:59:00.000-06:00In most physical models of computation, energy is just
proportional to 1/time. So if what you mean is energy in the physical sense, I don't see how to get new complexity theory out of this. <BR/><BR/>
CrisAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-31628875358229021272009-01-16T14:16:00.000-06:002009-01-16T14:16:00.000-06:00Ravi Jain, Zulfikar Ramzan, and I actually have a
short note with an energy hierarchy theorem. You could fairly dispute whether the energy measure we picked is the "right" one, of course, so we also talk about attributes we would want from a model
for reasoning about energy more generally. May be useful, although it's more of a "problem paper" than a solution paper.<BR/><BR/>"Towards a model of energy complexity for algorithms"<BR/>R. Jain, D.
Molnar, Z. Ramzan<BR/>IEEE Wireless Communications Networking Conference (WCNC) 2005 <BR/>http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=1424799<BR/><BR/>Krishna Palem also has an
interesting energy complexity model. He's developed it in the context of analyzing "probabilistic circuits" where each gate is correct with probability p, and gives the wrong result with probability
(1-p).<BR/><BR/>"Energy Aware Computing through<BR/>Probabilistic Switching: A Study of Limits"<BR/>K. V. Palem, IEEE Trans. Computing September 2005<BR/>http://www.cs.rice.edu/~kvp1/pubs/
palemIEEE.pdfAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-76296247635067229412009-01-16T14:03:00.000-06:002009-01-16T14:03:00.000-06:00An Aeroflot airplane made a successful
emegency landing on the Neva river in Leningrad (now St Petersburg), Russia, on 21 August 1963. All 52 people on board survived. <BR/><BR/>A Wikipedia page in Russian: http://ru.wikipedia.org/wiki/
<I> The beauty of complexity: It's always relevant.</I><BR/>Err...did you mean irrelevant?Anonymousnoreply@blogger.com
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This course will study epistemic extensions of theories of formal arithmetic. We will begin by establishing the necessary background in formal theories of arithmetic (such as Peano arithmetic) and
Godel's incompleteness theorems. Following this, we will explore extensions of Peano arithmetic that incorporate an operator that is intended to represent the knowledge of an (ideal) mathematician.
We will discuss the so-called Knower Paradox that results when the knowledge operator is treated as a predicate, including an extensive discussion of variants of the paradox and the proposed
solutions to the paradox. Finally, we will discuss attempts to formalize Godel's disjunction: either there is no algorithm that can capture human mathematical reasoning or there are absolutely
undecidable problems. In addition to discussing absolute undecidability, we will also present Feferman's formal theory of truth with a determinate operator. Topics that will be discussed in the
course include:
This course offers an excellent opportunity to introduce ESSLLI students to one of the most significant topics in mathematical logic: Gödel's incompleteness theorems. Due to the complexity and depth
of these theorems, it is not feasible to cover all aspects, such as Gödel coding and the necessary background in computability theory, within a week. The main objective is to help students begin to
grasp the incompleteness theorems and their implications without becoming overwhelmed by the details.
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A possible origin story of OOT-limits - part 1
Disclamer: this and the following posts are only about the release test related OOTs, the stability OOT-limits are not considered.
I started to write a post about how to establish OOT-limits, but it became rather long, so I intend to publish it in more parts. But let me share the first post:
There is a huge literature available on how to investigate OOT results. But how to establish a good OOT limit structure? That can be confusing sometimes.
There is the good old WECO-model for the dynamic signalling system - again, with a lot of knowledge shared on the internet. I personnally find this model oversensitive in some pharma attributes, like
yield, impurity level or assay.
So let's start to think about the static model: the lifecycle of it can be led by the Product Quality Review document, where we need to evaluate a lot of QC results anyway. So while we evaluate last
year's trend, we can also establish next year's limit.
But what kind of thumbrules could we use for this task?
Where NOT to detemine OOT-limit
1. Using ICH Q2
To ease our own task, let's quickly exclude some quality attributes from the scope.
ICH Q2 (even the current, effective version) contains a great table about the different types of analytical methods: identity, quantitative and limit testing for impurities, and assay.
Regarding identity tests, even if we prove the conformity of a product through a numerical result (like polarimetry or pH), the answer is going to be a Yes or a No. (Let's not forget tha in case of
OOS the actual numerical result can get us some hint about the root cause, but let's not go down that rabbit's hole for now.) So just eliminate all identity tests from the scope.
The same applies for the limit tests: you cannot trend the Yes/No answers.
2. Normality
Let me quickly add a little spolier here: our tool to establish OOT-limits will be the +/-3sigma approach.
And let me share maybe the most crucial part to having responsive system: we can use the +-/3sigma approach for the attributes that have normal distribution.
Two quick thoughts here: first, don't be misled of being able to calculate standard deviations from any set of numbers, no Excel, Minitab or any other tools will stop you from doing that. So you need
to be careful which quality attributes you evaluate.
Second: there are numerous statistical tests (a bunch of the also automatic, that's really tempting!) to evaluate if your data set follows normal distribution or not. But don't care about those: you
as QC need to own the theoretical knowledge, which analytical method generates result that has normal distribution.
Let's have some examples: an organic impurity, the water content, or LC-assay 'behaves' in normal distribution, whatever your statistical tests say about that. On the other hand, the pH by definition
cannot follow normal distribution, since it has a logarythmic scale. The same applies to particle distribution, whatever exotic dimension you use to give the results.
3.Sum of...
A quite common specification element is to define a limit for a group of impurities or components. Again, our job is to stop ourselves from defining OOT-limits for these quality attributes, because
although the individual impurities may follow normal bistribution, but their aggregate will most like not.
In the second part, we will discuss what to do with the remainder quality attributes.
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Which Math Games prove Math is fun?
Math has long been criticized as being a difficult and intimidating subject. However, with the right approach, arithmetic can be made attractive and thrilling for pupils. Many people will tell you
that math is entertaining, and it is if you learn to comprehend the number game.
In essence, Math becomes pleasant when it is taken outside of the traditional classroom setting. Educators may unleash the joy and excitement hidden inside the realm of numbers by mixing creativity,
real-world applications, and interactive elements, making math an adventure to be explored.
Download FREE Math Worksheets!
16 Math-Based Games to Captivate Children
Let's have a look at some games that may be played in the classroom to combine pleasure and learning. Math πrates practice test 2024, arranged by 98thPercentile, is an example of an engaging game
that may engage children in learning.
Mathematical Bingo
Make bingo cards using math problems instead of numbers. Call out equations, and students will mark the solutions on their cards.
Mathematical Jeopardy
Divide the students into teams and design a Jeopardy-style game with several categories and point values for arithmetic tasks.
Arithmetic Puzzles
To improve students' critical thinking abilities, assign Math puzzles such as Sudoku or cross-number problems.
Fraction Pizza
Have students make fraction pizzas with various toppings. They may practice addition, subtracting, and multiplying fractions while creating their pizzas.
Math Relay Races
Set up a relay race, with students solving math problems at each station before handing the baton to the next partner.
Math Board Games
Incorporate a mathematical spin into classic board games such as Monopoly or Chutes and Ladders. To progress through the game, players can solve arithmetic problems.
Mathematical Scavenger Hunt
Create a scavenger hunt with arithmetic tasks concealed throughout the classroom. Students answer each task to uncover the following clue.
Geometric Bingo
Replace the numbers on Bingo cards with geometric forms or angles. Call out descriptions, and students will indicate the relevant shape.
Math Memory
Make a memory game using pairs of cards, each containing a math problem and its answer. Students must match the proper pairings.
Mathematics Tic-Tac-Toe
Play Tic-Tac-Toe with a math twist. To place an X or O on the board, students must complete arithmetic problems properly.
Mathematics Charades
Act out arithmetic topics and ask kids to guess. It's a fun approach to reinforce your knowledge of mathematical concepts and processes.
The Decimal War
Use a deck of cards that contains decimals rather than conventional numbers. Students compare decimals, and the one with the higher value wins the round.
Number Line Hopscotch
Draw a number line on the floor and have children jump to different locations to answer addition, subtraction, and multiplication problems.
Mathematics Relay Race
Set up a relay race in which students must answer math questions at each station before handing the baton to their next partner.
Cash Bingo
Create Bingo cards with varying amounts of money. Call out prices or mathematical formulae, and pupils will indicate the relevant amounts.
Math Art
Encourage kids to produce art based on mathematical principles such as symmetry or fractals.
How do Math-Based Games Help Kids?
Math games may provide several benefits to children, including helping them develop a positive attitude toward arithmetic and enhancing various cognitive and academic talents. Here are some ways math
games may help children
Get Your FREE Math Worksheets Now!
Involvement and Inspiration
Games make learning more interesting and engaging for youngsters, catching their attention and pushing them to actively participate in mathematical tasks.
Conceptual Encouragement
Math games provide a hands-on technique for reinforcing and practicing mathematical topics. Children's comprehension of numbers, operations, and problem-solving improves with repeated exposure in an
enjoyable setting.
Critical Thinking Skills Enhanced
Many math games require strategic thinking, logical reasoning, and decision-making. These exercises encourage youngsters to build critical thinking abilities.
Confidence Boost
Success in Math games, whether via problem solution or goal achievement, enhances children's mathematical confidence. Positive Math experiences lead to a more hopeful attitude about the topic.
Social Communication
These include collaboration, communication, and teamwork. Multiplayer math games encourage social engagement.
The Ability to Adapt
Many math games may be customized for different ability levels, allowing for differentiation in the classroom.
Cognitive Improvement
Games sometimes demand participants to recall rules, strategy, and numerical data. Math games can help youngsters improve their memory abilities since they require them to recall and apply
mathematical principles while playing.
Real-Life Applications
Math games frequently provide challenges in a real-world setting, enabling youngsters to use their mathematical skills in practical settings.
Enjoyable Execution
Math games provide students with a more entertaining approach to develop and improve their arithmetic abilities than rote memorization or repeated exercises.
Math games exist in a variety of media. This variety enables educators and parents to select games that correspond to different learning styles and preferences.
In a nutshell, math games are great instructional tools because they make learning more engaging, reinforce concepts, and promote the development of critical skills. Children who play not only build
a strong foundation in mathematics but also cultivate a positive attitude toward learning in general. They establish that math is enjoyable.
Book FREE Math Trial Classes Now!
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ball mill in power plant
WEBThe experimental assessment of ball mills type MCB (МШЦ 4,5х6) operating in copper ore processing plant and SAG mill operating in gold ore processing plant are presented. The experimental
results are statistically processed and some statistical regression models are estimated.
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WEBBall mill is the key equipment for secondary grinding after crushing. And it is suitable for grinding all kinds of ores and other materials, no matter wet grinding or dry grinding. In our
company, this series of highefficiency ball mill adopts rolling bearing support instead of sliding bearing support with bearing bush. Therefore, it can [.]
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WEBIndustrial ball mills can be as large as m (28 ft) in diameter with a 22 MW motor, drawing approximately % of the total world's power (see List of countries by electricity consumption).
However, small versions of ball mills can be found in laboratories where they are used for grinding sample material for quality assurance. The power ...
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WEBAbstract Ball mills as used in mining, cement production and allied industries consume about 40% to 56% of the total electric power supplied to the processing plant. Electric power losses in
ball mill operations constitute about 25% of the available power.
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WEBAug 1, 2001 · In this investigation, an industrial ball mill database from Chadormalu iron ore processing plant were used to develop a RF model and explore relationships between powerdraw and
other monitored ...
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WEBOct 5, 2016 · The energy consumption of the total grinding plant can be reduced by 20–30 % for cement clinker and 30–40 % for other raw materials. The overall grinding circuit efficiency and
stability are improved. The maintenance cost of the ball mill is reduced as the lifetime of grinding media and partition grates is extended.
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WEBOct 15, 2014 · Due to the mass rejection from the ore sorters, much less material is sent to the tertiary crusher and ball mills, which results in a reduction in power requirement. Total
crusher power is reduced from MW to MW; total ball mill power is reduced from to MW.
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WEBFlowing Rate Control. The flow rate of materials in the cement ball mill is controlled, and the partition in the ball mill is forced to pass through the compartment; the discharge grate plate
ventilation area is adjusted to select the use proper scheme, and the hollow part of the activation ring is blocked to adjust the material flow rate.
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WEBModule2 Rod And Ball Mill Power Draw [1430jwr0594j]. ... IDOCPUB. Home (current) Explore Explore All. Upload; Login / Register. ... However, they are quite accurate in a relative sense for the
mills in your plant. To change the power draw of a mill in the plant (the objective is almost always an increase), we can use the equations presented ...
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WEBJan 1, 2016 · Total power required for the ball mill = 1200 × × = 1407 kW. Referring again to Allis Chalmers' ball mill performance table, for a mill power draw of 1412 kW, the same mill
dimensions will be suitable if the ball charge is increased to 45% with a charge mass of 144 t.
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WEBOct 30, 2023 · Ball mills are the foremost equipment used for grinding in the mineral processing sector. Lifters are placed on the internal walls of the mill and are designed to lift the
grinding media (balls ...
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WEBThe power the mill consumes is a function of the ore and/or ball charge geometry and the mill speed. The motor only supplies the amount of power (or torque really) that the mill demands, and
we add some extra power to the motor to deal with 'upset' conditions like a change in ore density or a surge in ore level inside the mill.
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WEBRod mills: Although rarely used in industrial practice, ALS has 2 x kW rod mills for small (<50 kg/h) throughputs where preferential coarse grinding is advantageous for the downstream
processing. Ball mills: Ball mills are the stalwart unit of the majority of pilot grinding circuits. ALS has a range of rubber lined overflow ball mills ...
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WEBNo, it is not normal to operate above 40 to 45% charge level. Unless the trunnion is very small or unless you are using a grate discharge ball mill, balls will not stay in the mill, and you
will spend a lot on steel to add a small amount of power. Formulas given for mill power draw modelling are empirical, and fit around field data over the ...
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/radial runout. of the drive trains. power splitting. distances variable. load distribution. of the girth gear. gear is through hardened only, fatigue strength is limited. Dynamic behaviour. A
lot of individual rotating masses risk of resonance vicinities.
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WEBJul 18, 2023 · Dry FGD: Technology Advantages: Lower water consumption compared to Wet FGD systems. Smaller footprint, making it suitable for retrofitting in existing power plants with space
limitations. Can be ...
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WEBAug 24, 2015 · The user interface offers Metric OR Imperial measurements for sizing SAG/AG and Ball Mills of overflow or grate type discharges. by David Michaud February 29, 2024 August 24,
2015 Categories Tools of a Metallurgist
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WEBNov 1, 2010 · The JKMRC has developed effective models for predicting ball mill, au1ogenous mill, semi autogenous mill and crusher power draw. When combined with comminution models they
provide a powerful ...
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WEBMay 1, 2020 · Nowadays, ball mills are used widely in cement plants to grind clinker and gypsum to produce cement. In this work, the energy and exergy analyses of a cement ball mill (CBM) were
performed and some measurements were carried out in an existing CBM in a cement plant to improve the efficiency of the grinding process.
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WEBMar 1, 2020 · 1. Introduction. In mineral processing plants, comminution circuits are the most energy consuming units; thus, determination of mill powerdraw can be one of the most important
factors for designing, operating and evaluating of an efficient plant [1], [2], [3].It was reported that for a ball mill with 5 m diameter and 7 m length, the power draw .
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WEBJul 27, 2012 · Based on the fact that the ball mill load in power plant is hard to detect effectively, an online local learning improved weighted least square support vector machine (WLSSVM)
softsensing method is proposed form improving the prediction accuracy and adaptive ability of the softsensing model. Firstly, a similarity measurement criterion of .
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WEBMar 1, 1996 · A model for predicting the milling power requirements of a stirred ball mill operating under different conditions is thus very important for evaluating milling efficiency and
facilitating mill designs. There have been a few publiions on power prediction for stirred ball mills. Most of them dealt with the variation in the mill speed. 0301 ...
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WEBOct 12, 2016 · The simplest grinding circuit consists of a ball or rod mill in closed circuit with a classifier; the flow sheet is shown in Fig. 25 and the actual layout in Fig. 9. ...
Reduction of costs then becomes more a matter of organization than of plant design. As already stated, the power and steel costs are the two largest items, those of labour and ...
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Negative Imaginary Theory
Flexible structure systems arise in many important applications such as ground and aerospace vehicles, atomic force microscopes, rotating flexible spacecraft, rotary cranes, robotics and flexible
link manipulators, hard disk drives and other nano-positioning systems. In control systems design for these flexible systems, it is important to consider the effect of highly resonant modes. Such
resonant modes are known to adversely affect the stability and performance of flexible structure control systems and are often very sensitive to changes in environmental variables. These can lead to
vibrational effects that limit the ability of control systems in achieving desired levels of performance. These problems are simplified to some extent by using force actuators combined with colocated
measurements of velocity, position, or acceleration. Using force actuators combined with colocated measurements of velocity can be studied using positive real systems theory, which has received great
attention since 1962. Using force actuators combined with colocated measurements of position and acceleration can be studied using negative imaginary (NI) systems theory.
The motivations and objectives of this research lead to the following contributions:
1. Providing a systematic method for checking if a given system is NI or not.
We derive a method to check for the NI and strictly NI properties in both the single-input-single-output as well as the multi-input-multi-output cases. The proposed methods are based on spectral
conditions on a corresponding Hamiltonian matrix obtained for a given transfer function matrix. Under some technical conditions, the transfer function matrix satisfies the NI property if and only if
the corresponding Hamiltonian matrix has no pure imaginary eigenvalues with odd multiplicity. Also, the transfer function matrix satisfies the SNI property if and only if the corresponding
Hamiltonian matrix has no pure imaginary eigenvalues except at the origin. These results may be useful in both the analysis of NI systems and in the synthesis of NI controllers using optimization
techniques. Also, spectral conditions on the Hamiltonian matrix tend to have fewer numerical problems when compared to the LMI conditions.
2. Developing a method for enforcing NI dynamics on mathematical system models to satisfy an NI Property.
We provide two methods for enforcing NI dynamics on mathematical models, given that it is known that the underlying dynamics ought to belong to the NI system class. The first method is based on a
study of the spectral properties of Hamiltonian matrices. A test for checking the negativity of the imaginary part of the corresponding transfer function matrix is first developed. If an associated
Hamiltonian matrix has pure imaginary axis eigenvalues, the mathematical model loses the NI property in some frequency bands. In such cases, a first-order perturbation method is proposed for the
precise characterization of the frequency bands where the NI property is violated and this characterization is then used in an iterative perturbation scheme aimed at displacing the imaginary
eigenvalues of the Hamiltonian matrix away from the imaginary axis, thus restoring the NI dynamics. In the second method, the direct spectral properties of the imaginary part of a transfer function
are used to identify the frequency bands where the NI properties are violated. A discrete frequency scheme is then proposed to restore the NI system properties in the mathematical model.
3. Generalizing the existing NI definition to include flexible structures with free body motion.
A generalized NI system framework is presented. A new NI system definition is given, which allows for flexible structure systems with colocated force actuators and position sensors and with free body
motion. This definition extends the existing definitions of NI systems.
4. Deriving stability conditions for NI systems with poles at the origin.
A necessary and sufficient condition is provided for the stability of a positive feedback control system where the plant is NI according to the new definition and the controller is strictly negative
imaginary (SNI). This general stability result captures all previous NI stability results which have been developed. The stability conditions in this thesis, are given purely in terms of properties
of the plant and controller transfer function matrices, although the proof relies on state-space techniques. Furthermore, the stability conditions given are independent of the plant and controller
system order.
5. Providing an Algebraic Riccati Equation analysis of NI systems and static NI controller synthesis.
We provide a systematic method to design controllers for NI systems with guaranteed robust stability and performance. Unlike the state feedback controller synthesis problem Riccati equation approach
is developed and used to present a systematic method for designing a controller to force the closed-loop system to satisfy the NI property.
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NeverWorld2: an idealized model hierarchy to investigate ocean mesoscale eddies across resolutions
Articles | Volume 15, issue 17
© Author(s) 2022. This work is distributed under the Creative Commons Attribution 4.0 License.
NeverWorld2: an idealized model hierarchy to investigate ocean mesoscale eddies across resolutions
We describe an idealized primitive-equation model for studying mesoscale turbulence and leverage a hierarchy of grid resolutions to make eddy-resolving calculations on the finest grids more
affordable. The model has intermediate complexity, incorporating basin-scale geometry with idealized Atlantic and Southern oceans and with non-uniform ocean depth to allow for mesoscale eddy
interactions with topography. The model is perfectly adiabatic and spans the Equator and thus fills a gap between quasi-geostrophic models, which cannot span two hemispheres, and idealized general
circulation models, which generally include diabatic processes and buoyancy forcing. We show that the model solution is approaching convergence in mean kinetic energy for the ocean mesoscale
processes of interest and has a rich range of dynamics with circulation features that emerge only due to resolving mesoscale turbulence.
Received: 09 Apr 2022 – Discussion started: 25 Apr 2022 – Revised: 04 Aug 2022 – Accepted: 11 Aug 2022 – Published: 01 Sep 2022
Mesoscale eddies have a profound impact on the transport of properties in the ocean. They affect the currents, stratification, ocean dynamic sea level variability, and uptake of physical and
biogeochemical tracers. Eddies thus play an important role in regulating climate on regional and global scales and on timescales of weeks to centuries. Mesoscale eddies form on spatial scales near
the baroclinic Rossby deformation radius (Smith and Vallis, 2002; Arbic and Flierl, 2004; Thompson and Young, 2007; Hallberg, 2013). The deformation radius varies regionally between 10–100km
horizontally (Chelton et al., 1998). These scales are too small to be resolved globally in routinely used structured-grid ocean climate simulations and, therefore, must be parameterized.
The most notable scheme of mesoscale eddy transport implemented in climate models is the Gent McWilliams (GM) parameterization, which mimics the effects of baroclinic instability by flattening
isopycnals and acting as a net sink of available potential energy (Gent and McWilliams, 1990; Gent et al., 1995). Motivated by the effect on available potential energy, eddy kinetic energy
parameterizations have been developed (Cessi, 2008; Eden and Greatbatch, 2008; Marshall and Adcroft, 2010) to keep track of the mechanical energy in idealized simulations (Marshall et al., 2012) and/
or scale the GM parameter in global climate simulations (Adcroft et al., 2019). The specific goal of this paper is to present a test for such energy-based parameterizations, although mesoscale
parameterizations based on other approaches can also be tested in the same framework.
As the horizontal grid spacing of climate models is refined, such that the grid box size becomes comparable to the deformation scale, a regime commonly referred to as the “gray zone” is reached. A
gray zone is present in virtually all eddying simulations with large meridional extent and continental slopes (Hallberg, 2013). In this regime, some eddies are being partially resolved, but the
resolution does not allow for their effects on the large-scale current and stratification to be fully accounted for. In particular, the inverse kinetic energy cascade (or backscatter) and the
barotropization of the flow remain too weak in both idealized (Jansen and Held, 2014) and global models (Kjellsson and Zanna, 2017). Recent mesoscale parameterizations focus on these two aspects with
novel momentum closures (Bachman, 2019; Jansen et al., 2019); the model hierarchy described here is designed to evaluate and contrast such approaches in an affordable way.
The majority of these mesoscale parameterizations have been developed independently, using different dynamical assumptions (e.g., quasi-geostrophic dynamics or primitive equations) and different
idealized configurations with limited spatial extent (e.g., double gyre or channel). This approach has led to a lack of coherent and robust analysis on the effect of eddies and their
parameterizations on the ocean dynamics.
Here, we present an idealized model to capture the essence of mesoscale eddy dynamics at varying horizontal grid resolutions, investigate the effect of mesoscale eddies on the large-scale dynamics,
and provide a framework for testing and evaluating eddy parameterizations. The model allows for a clean and extensive analysis of the dynamics and energetics of the flow as a function of horizontal
resolution, which is often limited in primitive-equation and diabatic global models due to computational resources (Hewitt et al., 2020; McClean et al., 2011).
We introduce a model configuration – referred to as NeverWorld2 (NW2), which is an extension of the Southern Hemisphere-only NeverWorld configuration presented in Khani et al. (2019) and Jansen
et al. (2019). NW2 is a stacked shallow-water model configuration with idealized geometry comprising a single cross-equatorial basin and a re-entrant channel in the Southern Hemisphere. The
configuration is broadly similar to that of Wolfe and Cessi (2009), except NW2 is strictly adiabatic, on a spherical grid, and forced only by winds. The global volume of water in each density layer
is set by the initial conditions, with the dynamics determining the spatial distribution of stratification which can adjust locally. A hierarchy of horizontal grid resolutions allows us to consider
mesoscale eddies in distinct dynamical regimes, e.g., Southern Ocean-like dynamics, mid-latitude gyres, and equatorial flows. This hierarchy also encompasses coarse (unresolved), gray (partially
resolved), and mesoscale eddying (fully resolved) flow regimes in portions of the domain controlled by the selected model resolution.
We discuss the model equations, the convergence of the simulations as a function of resolution, and the energetics of the flow. This paper is meant to be an introduction to the configuration and the
datasets for use by the community to understand, test, and evaluate mesoscale dynamics and novel closures.
2.1Model equations
We consider an adiabatic and hydrostatic fluid system with a single thermodynamic constituent, simplified to a linear equation of state. This system can be approximated by N stacked layers of
piecewise constant density. The equations of motion for the primary prognostic model variables, which are the zonal flow (u), meridional flow (v), and layer thickness (h), are written in
vector-invariant form:
$\begin{array}{}\text{(1)}& {\partial }_{t}{u}_{k}-{q}_{k}{v}_{k}{h}_{k}+{\partial }_{x}{K}_{k}+{\partial }_{x}{M}_{k}& ={\mathcal{F}}_{k}^{x},\text{(2)}& {\partial }_{t}{v}_{k}+{q}_{k}{u}_{k}{h}_{k}
+{\partial }_{y}{K}_{k}+{\partial }_{y}{M}_{k}& ={\mathcal{F}}_{k}^{y},\text{(3)}& {\partial }_{t}{h}_{k}+{\partial }_{x}\left({u}_{k}{h}_{k}\right)+{\partial }_{y}\left({v}_{k}{h}_{k}\right)& =\
with the vertical stress divergence and horizontal friction given by
$\begin{array}{}\text{(4)}& {\mathcal{F}}_{k}^{x}& =\frac{\mathrm{1}}{{\mathit{\rho }}_{o}{h}_{k}}\left({\mathit{\tau }}_{k-\mathrm{1}/\mathrm{2}}^{x}-{\mathit{\tau }}_{k+\mathrm{1}/\mathrm{2}}^{x}\
right)-\mathrm{abla }\cdot {\mathit{u }}_{\mathrm{4}}\mathrm{abla }\left({\mathrm{abla }}^{\mathrm{2}}{u}_{k}\right)\text{(5)}& {\mathcal{F}}_{k}^{y}& =\frac{\mathrm{1}}{{\mathit{\rho }}_{o}{h}_{k}}\
left({\mathit{\tau }}_{k-\mathrm{1}/\mathrm{2}}^{y}-{\mathit{\tau }}_{k+\mathrm{1}/\mathrm{2}}^{y}\right)-\mathrm{abla }\cdot {\mathit{u }}_{\mathrm{4}}\mathrm{abla }\left({\mathrm{abla }}^{\mathrm
where a subscript k indicates the vertical layer number with k=1 the topmost and k=N the bottommost. We use the short-hand ∂[t], ∂[x], and ∂[y] for partial derivatives in time and in zonal and
meridional directions, respectively. ∇⋅ is the horizontal divergence, ∇ the horizontal gradient, and ${\mathrm{abla }}^{\mathrm{2}}=\mathrm{abla }\cdot \mathrm{abla }$ the horizontal Laplacian. The
MOM6 code (Adcroft et al., 2019) that discretizes these equations makes use of horizontal orthogonal curvilinear coordinates, though here we write the more concise Cartesian coordinate notation for
Other dynamic quantities that are derived from the primary variables include
• the interface positions, ${\mathit{\eta }}_{k-\mathrm{1}/\mathrm{2}}=-D+{\sum }_{l=k}^{N}{h}_{l}$, indicated with half-integer labels;
• the potential vorticity, ${q}_{k}=\frac{\mathrm{1}}{{h}_{k}}\left(f+{\partial }_{x}{v}_{k}-{\partial }_{y}{u}_{k}\right)$;
• the kinetic energy per mass, ${K}_{k}=\frac{\mathrm{1}}{\mathrm{2}}\left({u}_{k}^{\mathrm{2}}+{v}_{k}^{\mathrm{2}}\right)$;
• the Montgomery potential, ${M}_{k}={\sum }_{l=\mathrm{1}}^{k}{g}_{l-\mathrm{1}/\mathrm{2}}^{\prime }{\mathit{\eta }}_{l-\mathrm{1}/\mathrm{2}}$;
• the dynamic lateral viscosity, ${\mathit{u }}_{\mathrm{4}}={C}_{\mathrm{4}}\frac{{\mathrm{\Delta }}^{\mathrm{4}}}{\mathrm{8}{\mathit{\pi }}^{\mathrm{2}}}\sqrt{{\left({\partial }_{x}{u}_{k}-{\
partial }_{y}{v}_{k}\right)}^{\mathrm{2}}+{\left({\partial }_{y}{u}_{k}+{\partial }_{x}{v}_{k}\right)}^{\mathrm{2}}},$ where Δ^4 is the fourth power of the grid-spacing which follows a particular
discretization, as proposed in the Appendix of Griffies and Hallberg (2000) and is different than simply using the square of the cell area;
• the vertical stress, ${\mathbit{\tau }}_{k-\mathrm{1}/\mathrm{2}}=-{A}_{\mathrm{v}}\frac{{\mathit{\rho }}_{o}}{{h}_{k-\mathrm{1}/\mathrm{2}}}\left({\mathbit{u}}_{k-\mathrm{1}}-{\mathbit{u}}_{k}\
• the bottom stress, ${\mathbit{\tau }}_{N+\mathrm{1}/\mathrm{2}}=-{C}_{\mathrm{d}}{\mathit{\rho }}_{o}|{\mathbit{u}}_{\mathrm{B}}|{\mathbit{u}}_{N}$, which uses a quadratic drag law and where u[B]
is the flow averaged over the bottommost 10m.
The surface wind stress, ${\mathbit{\tau }}_{\mathrm{1}/\mathrm{2}}$, is prescribed, fixed in time, and distributed over the top 5m. The remaining parameters are the reduced gravity of each layer, $
{g}_{k-\mathrm{1}/\mathrm{2}}^{\prime }$; a reference density, ρ[o]=1000kgm^−3; the Coriolis parameter, f=2Ωsinϕ (with $\mathrm{\Omega }=\mathrm{7.2921}×{\mathrm{10}}^{-\mathrm{5}}$s^−1, and ϕ is
latitude); the background kinematic vertical viscosity, ${A}_{\mathrm{v}}=\mathrm{1.0}×{\mathrm{10}}^{-\mathrm{4}}$m^2s^−1; a dimensionless bottom drag coefficient, C[d]=0.003; and the bottom
depth, $z=-D\left(x,y\right)$. We have chosen to use a biharmonic dissipation operator with a dimensionless Smagorinsky coefficient of C[4]=0.2, which is larger than the recommended range suggested
by Griffies and Hallberg (2000). This is to ensure sufficient dissipation in the absence of any other parameterizations of lateral friction. Other scale-aware alternatives (Bachman et al., 2017)
arrived at comparable results.
We provide analysis of the energetics here and in subsequent papers (Loose et al., 2022b; Yankovsky et al., 2022). To facilitate such analysis, it is useful to write out the energy budget equations.
The kinetic energy (KE) in layer k is given by KE[k]=h[h]K[k]. To obtain the KE equation, we add u[k]h[k]×Eq. (1), v[k]h[k]×Eq. (2), and K[k]×Eq. (3), which gives
$\begin{array}{}\text{(6)}& {\partial }_{t}\left({K}_{k}{h}_{k}\right)+\mathrm{abla }\cdot \left({K}_{k}{\mathbit{u}}_{k}{h}_{k}\right)+{\mathbit{u}}_{k}{h}_{k}\cdot \mathrm{abla }{M}_{k}={\mathbit
{u}}_{k}{h}_{k}\cdot {\mathsc{F}}_{k}+{C}_{k}\phantom{\rule{0.25em}{0ex}}.\end{array}$
The contribution from the Coriolis terms, ${C}_{k}={q}_{k}{h}_{k}^{\mathrm{2}}\left({u}_{k}{v}_{k}-{v}_{k}{u}_{k}\right)$, should be zero in Eq. (6). However, the numerical model uses a C-grid
staggering of variables so that locally the numerical Coriolis terms do not drop out, thus affecting KE. We use the Arakawa and Hsu (1990) discretization of Coriolis terms, which, when integrated
over the whole domain, conserves total KE for horizontally non-divergent flow. The term u[k]h[k]⋅∇M[k] is the conversion between potential energy and KE.
The potential energy (PE) at interface $k-\frac{\mathrm{1}}{\mathrm{2}}$ is given by ${\text{PE}}_{k-\mathrm{1}/\mathrm{2}}=\frac{\mathrm{1}}{\mathrm{2}}{g}_{k-\mathrm{1}/\mathrm{2}}^{\prime }{\
mathit{\eta }}_{k-\mathrm{1}/\mathrm{2}}^{\mathrm{2}}$. The corresponding PE equation is obtained by vertically summing equation (Eq. 3) from the bottom up to interface $k-\frac{\mathrm{1}}{\mathrm
{2}}$ and then multiplying by ${g}_{k-\mathrm{1}/\mathrm{2}}^{\prime }{\mathit{\eta }}_{k-\mathrm{1}/\mathrm{2}}$:
$\begin{array}{}\text{(7)}& \begin{array}{rl}& {\partial }_{t}\left(\frac{\mathrm{1}}{\mathrm{2}}{g}_{k-\mathrm{1}/\mathrm{2}}^{\prime }{\mathit{\eta }}_{k-\mathrm{1}/\mathrm{2}}^{\mathrm{2}}\right)+
\mathrm{abla }\cdot \left({g}_{k-\mathrm{1}/\mathrm{2}}^{\prime }{\mathit{\eta }}_{k-\mathrm{1}/\mathrm{2}}\sum _{l=k}^{N}{\mathbit{u}}_{l}{h}_{l}\right)\\ & \phantom{\rule{1em}{0ex}}={g}_{k-\mathrm
{1}/\mathrm{2}}^{\prime }\left(\sum _{l=k}^{N}{\mathbit{u}}_{l}{h}_{l}\right)\cdot \mathrm{abla }{\mathit{\eta }}_{k-\mathrm{1}/\mathrm{2}}\phantom{\rule{0.25em}{0ex}}.\end{array}\end{array}$
When summed in the vertical, the right-hand side of Eq. (7) and the PE conversion term in Eq. (6) are equal.
The available potential energy (APE) is the domain-integrated PE minus the domain-integrated PE of the resting state. The resting state and interface positions used at initialization are given by ${\
mathit{\eta }}_{k-\mathrm{1}/\mathrm{2}}\left(t=\mathrm{0}\right)=max\left({z}_{k-\mathrm{1}/\mathrm{2}}^{\mathrm{0}},-D\right)$, where ${z}_{k-\mathrm{1}/\mathrm{2}}^{\mathrm{0}}$ is a constant
nominal position for each interface. In this adiabatic model, changes in the domain-integrated PE are exactly the changes in APE, with no approximations or ambiguity.
The NW2 configuration is set up as follows. The domain is a sector of a sphere with angular width 60^∘ and with a single basin and a re-entrant channel in the Southern Hemisphere. The basin is
bounded by solid coasts at 70^∘N and 70^∘S. Not extending to the poles avoids infinitesimal spherical coordinate cells as the meridians converge. We use a regular spherical grid rather than a
Mercator grid so that the placement of boundaries (extents of the grid) is exactly the same for all resolutions. The regular spherical grid means the cells are distorted with cell-wise average aspect
ratio $\mathrm{\Delta }y/\mathrm{\Delta }x\sim \mathrm{1.4}$, exceeding 2 only poleward of 60^∘N and 60^∘S. The full bathymetry is shown in Fig. 1a, with a cross-section along the Equator in Fig. 1
b. The depth of the continental shelf is 200m, and it has a nominal width of 2.5^∘. A cubic profile, of width $\mathrm{1}/\mathrm{8}$ of the shelf, connects the shelf to the beach (which has zero
depth out to $\mathrm{1}/\mathrm{8}$ of the shelf width). Another cubic profile of width 2.5^∘ connects the shelf to the abyss with nominal depth of 4000m. A large abyssal ridge of height 2000m
runs north–south down the middle of the basin with a cubic profile and radius of 20^∘. The re-entrant channel spans 60–40^∘S, and a semi-circular ridge of height 2000m, radius of 10^∘, and
thickness of 2^∘ is centered on the channel opening in the west to block the deep flow through the channel. These deep ridges are idealizations of the Mid-Atlantic Ridge and the Scotia Arc that acts
as a sill across the Drake Passage to remove momentum via form drag.
The wind stress is strictly zonal and fixed in time (Fig. 1b) and is an idealization of the mean zonal wind profile (see for example Fig. A1 of Chaudhuri et al., 2013). We construct the wind stress
from piecewise cubic functions that interpolate between the values 0, 0.2, −0.1, −0.02, −0.1, 0.1, and 0Pa at latitudes −70, −45, −15, 0, 15, 45, and 70^∘, respectively. Each interpolation node has
zero derivative so that both the wind stress and the curl of wind stress are zero at the north and south boundaries.
The initial stratification and vertical resolution are intimately linked (Fig. 1c). We use 15 layers and choose nominal thicknesses of 25, 50, 100, 125, 150, 175, 200, 225, 250, 300, 350, 400, 500,
550, and 600m. The adiabatic conditions relieve the model of resolving surface boundary layer processes, but finer resolution near the surface is preserved to accurately capture surface
intensification of mesoscale energy (Smith and Vallis, 2002). Actual initial thicknesses are the shallower of this nominal profile and whatever is clipped by topography to yield flat interfaces in
the interior. The reduced gravity at each interface has values of 10, 0.0021, 0.0039, 0.0054, 0.0058, 0.0058, 0.0057, 0.0053, 0.0048, 0.0042, 0.0037, 0.0031, 0.0024, 0.0017, and 0.0011ms^−2. The
first value corresponds to the gravitational acceleration at the surface. The implied density profile is approximately exponential with a depth scale of 1000m (approximate due to rounding input
parameter values to two significant digits).
The ratio between the first baroclinic deformation radius (L[D]) to the meridional and zonal grid spacing is shown in Fig. 2a and b, respectively. Assuming that to resolve eddies, ${L}_{\mathrm{D}}/\
mathrm{\Delta }x\ge \mathrm{2}$ (Hallberg, 2013), Fig. 2 highlights that the $\mathrm{1}/\mathrm{32}{}^{\circ }$ horizontal resolution simulation explicitly resolves mesoscale eddies, except on the
shelves and at very high latitudes of the Southern Ocean.
The model is initialized from rest at $\mathrm{1}/\mathrm{4}{}^{\circ }$ horizontal grid spacing, with initial conditions described in Sect. 2.2 and depicted in Fig. 1c. The $\mathrm{1}/\mathrm{4}{}^
{\circ }$ simulation is run to a quasi-steady state reached by about 3×10^4d, in which the total kinetic energy is no longer drifting. Next, the layer thicknesses are interpolated to the $\mathrm{1}
/\mathrm{8}{}^{\circ }$ horizontal grid, and the simulation is run again to a quasi-steady state for a few thousand days. Note that for expediency, velocities and transports are not interpolated but
rather reset to zero. This step introduces a mild shock, but the model quickly spins up mechanically, as seen in the recovery of kinetic energy levels in Fig. 3. This procedure is repeated for $\
mathrm{1}/\mathrm{16}$ and $\mathrm{1}/\mathrm{32}{}^{\circ }$ horizontal grids until the simulation has reached convergence (see Sect. 5 for further description).
As the model horizontal grid is refined, the total kinetic energy (KE) increases at each transition to finer spacing (Fig. 3); this behavior is expected since the finer dynamical modes contain more
kinetic energy. In addition, the available potential energy (APE) decreases at each transition to finer grids possibly because eddies are numerically better resolved and become more efficient at
extracting APE. The total energy of the system drops at each transition in grid spacing, since APE dominates the total energy reservoir, but the drop is less than for APE due to the increase in KE.
The circumpolar transport spins up in ∼10^4d (Fig. 4) and fluctuates with no discernible drift for the remaining 2×10^4d at $\mathrm{1}/\mathrm{4}{}^{\circ }$. There is a small reduction in
circumpolar transport at each refinement in grid spacing, just as for APE. Since the baroclinic component of circumpolar transport is related to APE, arguments about the improved efficiency of eddies
with refined grid spacing are relevant (Marshall and Radko, 2003).
The difference in barotropic transport stream function between sub-tropical and subpolar regions spins up in ∼3000–4000d in the $\mathrm{1}/\mathrm{4}{}^{\circ }$ simulation and fluctuates without
discernible drift for the remaining 2.5×10^4d (Fig. 5). This adjustment time is consistent with basin travel times for mid-latitude baroclinic Rossby waves (∼2cms^−1). The transition to finer grid
spacing does lead to an increase in the variability. However, the time mean of this transport index does not change significantly with grid spacing, as expected via Sverdrup theory.
The equatorial thermocline depth at 15^∘E adjusts on multiple timescales with large oscillations at the start of the $\mathrm{1}/\mathrm{4}{}^{\circ }$ simulation that damp out over ∼4000d (Fig. 6
). A statistical equilibrium depth is reached on the order of ∼10^4d. There is an adjustment at each transition in grid spacing, but it is smaller than the dynamical noise.
During the spinup, the model exhibits multiple timescales in the diagnostics described. The cost of the spinup to day 4.2×10^4 (covering $\mathrm{1}/\mathrm{4}$, $\mathrm{1}/\mathrm{8}$, and $\mathrm
{1}/\mathrm{16}{}^{\circ }$) is ∼17% of the cost of the 2800d segment at $\mathrm{1}/\mathrm{32}{}^{\circ }$ resolution. The initialization procedure of consecutively allowing adjustment and then
interpolating to finer grids is approximately 75× cheaper than spinning up the model solution entirely at $\mathrm{1}/\mathrm{32}{}^{\circ }$.
4Mean circulation and mesoscale turbulence
We illustrate the behavior of the model in terms of key properties of the flow, with a focus on transport and energy reservoirs. We compare the high-resolution NW2 with a $\mathrm{1}/\mathrm{32}{}^{\
circ }$ horizontal grid, in which mesoscale eddies are explicitly resolved in the majority of the domain, to lower-resolution NW2 with horizontal grids of $\mathrm{1}/\mathrm{4}$, $\mathrm{1}/\mathrm
{8}$, and $\mathrm{1}/\mathrm{16}{}^{\circ }$. No mesoscale eddy closures are used in any of the simulations beyond the Smagorinsky friction.
Most of the large-scale features of the finest-resolution configuration are recognizable in the coarsest-resolution solution. The time-mean circulation is represented by subtropical gyres in both
hemispheres, a subpolar gyre in the Northern Hemisphere (Fig. 7), and a series of circumpolar jets in the Southern Hemisphere reminiscent of the Antarctic Circumpolar Current (ACC) (Figs. 8 and 9).
As the grid spacing is refined, the strength of the western boundary currents and their extensions seems to increase (Fig. 7), while the circumpolar jet (Fig. 4) decreases slightly (proportionally).
The thermocline depth along the Equator also barely changes with resolution (Fig. 6). Overall, the character of the large-scale circulation is relatively invariant with resolution even though the
metrics of various features converge with finer resolution.
Some details of the circulation that differ across resolution include, for example, major differences in upper-ocean stratification in the Southern Ocean between the coarsest and finest resolutions.
The interfaces in the fine-resolution simulation are less steep, and the interface outcrops move southward as a result of re-stratification by eddies (Fig. 8a). As the grid spacing is refined, the
vertical extent of jets changes (Fig. 8). The time-mean zonal velocity shows either a change in the number of distinct jets or a migration of mean jet position. Most notable is the appearance of a
deep westward circulation in the channel, below the depth of the blocking topography (the Scotia Arc downstream of the passage).
Snapshots of depth-integrated KE reveal the eddying behavior of the simulations (Fig. 9). For the coarsest grid ($\mathrm{1}/\mathrm{4}{}^{\circ }$), the flow permits mesoscale eddies, in particular
at low latitudes, where the deformation scale is largest. As the grid spacing is refined, the first deformation radius is better resolved, and mesoscale eddies become more ubiquitous and widespread.
Note that a casual glance does not readily distinguish the $\mathrm{1}/\mathrm{16}$ and $\mathrm{1}/\mathrm{32}{}^{\circ }$ models.
The time mean of depth-integrated total KE (Fig. 10) becomes spatially smoother at finer grid spacing as eddies become increasingly ubiquitous. The same is true for sea surface height variance (not
shown), a frequently used indicator of eddy activity.
Defining convergence for a turbulent cascade is challenging when using a dynamic viscosity because finer grid spacing will permit ever more variability on finer scales. For the purposes of mesoscale
eddies, we are concerned with convergence as manifested by invariance of the large-scale properties as resolution changes, thus implying the upscale transfer of kinetic energy is not changing so that
details of the small scales are not affecting the large-scale solution.
The total mechanical energy of the system (APE + KE reservoirs) is dominated by the APE. APE is an integral metric of the system and the reservoir of energy that generates mesoscale eddies. In
equilibrium, there is a balance of APE generation by winds (pumping/heaving the isopycnals) and the conversion of APE to mesoscale eddy energy and the subsequent damping of mean and mesoscale energy
by bottom drag, Smagorinsky friction, and vertical viscosity. The equilibrium levels of energy reservoirs shown in Fig. 11 indicate a diminishing APE transition for each change in grid spacing, with
an empirical fit suggesting convergence towards a value that is about 3% lower than the value in the $\mathrm{1}/\mathrm{32}{}^{\circ }$ simulation. We suggest that this behavior implies we are
approaching convergence for resolving the mean APE-to-eddy energy pathway.
The kinetic energy increases less than linearly as grid spacing is refined (middle panel of Figs. 3 and 11); there is a 5-fold increase in the kinetic energy reservoir between the $\mathrm{1}/\mathrm
{4}$ and the $\mathrm{1}/\mathrm{32}{}^{\circ }$ simulations. A power law fit indicates convergence but not until kinetic energy is about another factor of 2 higher than in the $\mathrm{1}/\mathrm
{32}{}^{\circ }$ simulation.
To test convergence of the mesoscale, Fig. 12b shows the 500d averaged, zonally and depth-integrated mesoscale kinetic energy. The mesoscale kinetic energy is computed with a band-pass filter as $\
frac{\mathrm{1}}{\mathrm{2}}{\sum }_{n=\mathrm{1}}^{N}\left({\stackrel{\mathrm{̃}}{h}}_{n}\left({\stackrel{\mathrm{̃}}{u}}_{n}^{\mathrm{2}}+{\stackrel{\mathrm{̃}}{v}}_{n}^{\mathrm{2}}\right)-{\overline
{h}}_{n}\left({\overline{u}}_{n}^{\mathrm{2}}+{\overline{v}}_{n}^{\mathrm{2}}\right)\right)$. Here, $\stackrel{\mathrm{̃}}{\cdot }$ and $\overline{\cdot }$ denote low-pass filters, with filter scales
defined by the red and blue solid lines in Fig. 12a, respectively. For the respective filter operations, we use the Python package gcm-filters (Loose et al., 2022a) with a Gaussian filter shape.
As the horizontal grid spacing is refined from $\mathrm{1}/\mathrm{4}$ to $\mathrm{1}/\mathrm{32}{}^{\circ }$, we observe steadily growing mesoscale eddy activity in all regions of the domain (Fig.
12b), but the increase from $\mathrm{1}/\mathrm{16}$ to $\mathrm{1}/\mathrm{32}{}^{\circ }$ is smaller compared to the increase from $\mathrm{1}/\mathrm{4}$ to $\mathrm{1}/\mathrm{8}{}^{\circ }$ and
from $\mathrm{1}/\mathrm{8}$ to $\mathrm{1}/\mathrm{16}{}^{\circ }$. An exception is the energetic recirculation region north of Drake Passage near 38^∘S, where the mesoscale kinetic energy increases
considerably as we change the grid spacing from $\mathrm{1}/\mathrm{16}$ to $\mathrm{1}/\mathrm{32}{}^{\circ }$. We speculate that this increase is due to large-scale standing meanders that develop
north of Drake Passage due to more finely resolved topography (Kong and Jansen, 2020; Barthel et al., 2017).
The zonal wavenumber spectra of EKE at various latitudes (Fig. 13) show that at all latitudes, there is a clear gain in kinetic energy between the $\mathrm{1}/\mathrm{4}$ and the $\mathrm{1}/\mathrm
{32}{}^{\circ }$ simulations, at all wavenumbers. At higher resolutions, a pronounced peak in the EKE at scales at or slightly above the deformation scale develops. The peak is the result of an
increasingly resolved mesoscale eddy-driven inverse KE cascade. The notable exception is the ACC region, where there is less sensitivity to resolution. As hinted by the snapshots of KE, convergence
at low latitude is achieved faster than at high latitudes. The spectra generally suggest the large scales are more similar between $\mathrm{1}/\mathrm{16}$ and $\mathrm{1}/\mathrm{32}{}^{\circ }$,
although full convergence has not yet been obtained.
In this paper, we introduce an idealized ocean model configuration, NeverWorld2 (NW2), that resolves mesoscale eddy dynamics in a basin-scale context. NW2 is a stacked shallow-water model using the
MOM6 dynamical core (Adcroft et al., 2019), configured with a single cross-equatorial basin and a re-entering channel in the Southern Hemisphere with idealized geometry. Because NW2 is strictly
adiabatic, with no parameterizations in the vertical direction, the timescales of adjustment are controlled entirely by dynamics rather than far slower thermodynamic processes. The paper serves as an
introduction to the model and grid resolution hierarchy and to the datasets for use by the community.
For the purposes of analyzing the role of mesoscale eddies and deriving and testing parameterizations, we provide evidence that the finest grid spacing shown, $\mathrm{1}/\mathrm{32}{}^{\circ }$, is
practically converged. The large-scale metrics of APE and gyre transport, the latitudinal analysis of KE, and the spectral analysis of EKE all suggest convergence of the largest scales when comparing
solutions with the $\mathrm{1}/\mathrm{16}$ and $\mathrm{1}/\mathrm{32}{}^{\circ }$ grid spacings.
We have used the Smagorinsky dynamic biharmonic viscosity of Griffies and Hallberg (2000) for the momentum closure in these simulations, including for the finest grid spacing that defines our
converged “truth”. Previous work has shown a sensitivity of the details of the forward and inverse turbulent cascade to the form of dissipation (e.g., Smith and Vallis, 2002; Arbic and Flierl, 2004;
Thompson and Young, 2007; Arbic et al., 2012; Pietarila Graham and Ringler, 2013; Soufflet et al., 2016; Pearson et al., 2017; Bachman et al., 2017; Treguier et al., 1997). In refining grid spacing
here, we seek to converge on resolving the mechanism of interaction between the mesoscale eddies and the large-scale circulation, by showing it to be diminishingly dependent on the details of
dissipation near the bottom of the forward enstrophy cascade; this dissipation scale is more and more separated from the eddy production scale. This assumption could be tested by trying alternative
closures and evaluating what tuning is necessary to give the same upscale energy flux. We could have used one of several closures proposed as scale-aware schemes to use in the eddy-permitting regime
(e.g., Anstey and Zanna, 2017; Bachman et al., 2017). However, using them in the baseline configuration described here would hinder a fair evaluation of those schemes, which we plan to do in the near
future. Nevertheless, we acknowledge that the absolute magnitude of eddy energy, the span of the inverse cascade, and other metrics depend somewhat on the choice of viscous closure.
The model spatial-resolution hierarchy, from coarse to eddy-rich, allows for a clean and extensive analysis of the dynamics and energetics of the flow as a function of horizontal grid spacing, which
will be upcoming. The coarser grid configurations shown here can serve as a test bed for the evaluation of scale-aware mesoscale eddy parameterizations in the “gray” zone of eddy-permitting
resolution. Even coarser, non-eddying configurations, not shown here because they need an eddy parameterization to look sensible, will be used to evaluate parameterizations of subgrid mesoscale
One virtue of the NW2 model is that the adiabatic limit isolates mesoscale eddies from other processes. However, there are strong interactions between mesoscale eddies and the surface mixed layer
which have been recognized since early evaluations of mesoscale parameterizations (Danabasoglu et al., 1994). The dynamical core and algorithm used are the same as for a the full primitive-equation
general circulation model (Adcroft et al., 2019; Griffies et al., 2020), so adding diabatic processes is relatively straightforward. The NW2 model can be modified and developed further to explore the
large-scale overturning circulation (e.g., Wolfe and Cessi, 2009) or the connection with other parameterizations, for example, mixed-layer and sub-mesoscale parameterizations.
GMM, NL, NB, JS, EY, CYC, AA, MFJ, RWH, and LZ contributed to the development and analysis of NW2 configurations. GMM, NL, JS, EY, AA, and LZ led the manuscript, and SMG, BFK, MFJ, and HK contributed
to the manuscript.
The contact author has declared that none of the authors has any competing interests.
The statements, findings, conclusions, and recommendations are those of the author(s) and do not necessarily reflect the views of the National Oceanic and Atmospheric Administration or the US
Department of Commerce.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
We thank all the members of the Ocean Transport and Eddy Energy Climate Process Team for helpful discussions during the design, implementation, and evaluation of the NW2 simulations. We also thank
the editor, Qiang Wang, as well as Takaya Uchida and an anonymous reviewer for their constructive comments that helped to improve the quality and presentation of this paper. Gustavo M. Marques was
supported by National Science Foundation (NSF) grant OCE 1912420. Chiung-Yin Chang and Neeraja Bhamidipati were supported by award NA19OAR4310365 and Alistair Adcroft by award NA18OAR4320123, from
the National Oceanic and Atmospheric Administration (NOAA), US Department of Commerce. Laure Zanna and Elizabeth Yankovsky were supported by NSF grant OCE 1912357 and NOAA CVP NA19OAR4310364. Baylor
Fox-Kemper is supported by NOAA NA19OAR4310366. Laure Zanna is supported by NSF grant OCE 1912332. Malte F. Jansen is supported by NSF grant OCE 1912163. Jacob M. Steinberg is supported by NSF grant
OCE 1912302. Hemant Khatri acknowledges the support from Natural Environment Research Council (NERC) grant NE/T013494/1 and NOAA grant NA18OAR4320123. Stephen M. Griffies and Robert W. Hallberg
acknowledge support from the National Oceanic and Atmospheric Administration Geophysical Fluid Dynamics Laboratory. This material is also based upon work supported by the National Center for
Atmospheric Research (NCAR), which is a major facility sponsored by the NSF under cooperative agreement no. 1852977. Computing and data storage resources, including the Cheyenne supercomputer, were
provided by the Computational and Information Systems Laboratory at NCAR, under NCAR/CISL project number UNYU0004.
This research has been supported by the US Department of Commerce (grant no. NA18OAR4320123), the Division of Ocean Sciences (grant nos. 1912420, 1912332, 1912357, 1912163, and 1912302), the Division
of Atmospheric and Geospace Sciences (grant no. 1852977), and the Climate Program Office (grant nos. NA19OAR4310364, NA19OAR4310365, and NA19OAR4310366).
This paper was edited by Qiang Wang and reviewed by Takaya Uchida and one anonymous referee.
Adcroft, A., Anderson, W., Balaji, V., Blanton, C., Bushuk, M., Dufour, C. O., Dunne, J. P., Griffies, S. M., Hallberg, R. W., Harrison, M. J., Held, I., Jansen, M. F., John, J., Krasting, J. P.,
Langenhorst, A., Legg, S., Liang, Z., McHugh, C., Radhakrishnan, A., Reichl, B. G., Rosati, T., Samuels, B. L., Shao, A., Stouffer, R., Winton, M., Wittenberg, A. T., Xiang, B., Zadeh, N., and Zhang,
R.: The GFDL Global Ocean and Sea Ice Model OM4.0: Model Description and Simulation Features, J. Adv. Model. Earth Sy., 11, 3167–3211, https://doi.org/10.1029/2019MS001726, 2019.a, b, c, d
Anstey, J. A. and Zanna, L.: A deformation-based parametrization of ocean mesoscale eddy reynolds stresses, Ocean Model., 112, 99–111, https://doi.org/10.1016/j.ocemod.2017.02.004, 2017.a
Arakawa, A. and Hsu, Y.-J. G.: Energy Conserving and Potential-Enstrophy Dissipating Schemes for the Shallow Water Equations, Mon. Weather Rev., 118, 1960–1969, https://doi.org/10.1175/1520-0493
(1990)118<1960:ECAPED>2.0.CO;2, 1990.a
Arbic, B. K. and Flierl, G. R.: Baroclinically Unstable Geostrophic Turbulence in the Limits of Strong and Weak Bottom Ekman Friction: Application to Midocean Eddies, J. Phys. Oceanogr., 34,
2257–2273, https://doi.org/10.1175/1520-0485(2004)034<2257:BUGTIT>2.0.CO;2, 2004.a, b
Arbic, B. K., Scott, R. B., Flierl, G. R., Morten, A. J., Richman, J. G., and Shriver, J. F.: Nonlinear Cascades of Surface Oceanic Geostrophic Kinetic Energy in the Frequency Domain, J. Phys.
Oceanogr., 42, 1577–1600, https://doi.org/10.1175/JPO-D-11-0151.1, 2012.a
Bachman, S. D.: The GM+E closure: A framework for coupling backscatter with the Gent and McWilliams parameterization, Ocean Model.g, 136, 85–106, https://doi.org/10.1016/j.ocemod.2019.02.006, 2019.a
Bachman, S. D., Fox-Kemper, B., and Pearson, B.: A scale-aware subgrid model for quasi-geostrophic turbulence, J. Geophys. Res.-Oceans, 122, 1529–1554, https://doi.org/10.1002/2016JC012265, 2017.a,
b, c
Barthel, A., Hogg, A. M., Waterman, S., and Keating, S.: Jet–Topography Interactions Affect Energy Pathways to the Deep Southern Ocean, J. Phys. Oceanogr., 47, 1799–1816, https://doi.org/10.1175/
JPO-D-16-0220.1, 2017.a
Bhamidipati, N., Adcroft, A., Marques, G., and Abernathey, R.: ocean-eddy-cpt/NeverWorld2: NeverWorld2-description-paper (v1.1), Zenodo [code], https://doi.org/10.5281/zenodo.6993951, 2022.a
Cessi, P.: An Energy-Constrained Parameterization of Eddy Buoyancy Flux, J. Phys. Oceanogr., 38, 1807–1819, https://doi.org/10.1175/2007JPO3812.1, 2008.a
Chaudhuri, A. H., Ponte, R. M., Forget, G., and Heimbach, P.: A Comparison of Atmospheric Reanalysis Surface Products over the Ocean and Implications for Uncertainties in Air–Sea Boundary Forcing, J.
Climate, 26, 153–170, https://doi.org/10.1175/JCLI-D-12-00090.1, 2013.a
Chelton, D. B., deSzoeke, R. A., Schlax, M. G., El Naggar, K., and Siwertz, N.: Geographical Variability of the First Baroclinic Rossby Radius of Deformation, J. Phys. Oceanogr., 28, 433–460, https:/
/doi.org/10.1175/1520-0485(1998)028<0433:GVOTFB>2.0.CO;2, 1998.a
Danabasoglu, G., McWilliams, J. C., and Gent, P. R.: The Role of Mesoscale Tracer Transports in the Global Ocean Circulation, Science, 264, 1123–1126, https://doi.org/10.1126/science.264.5162.1123,
Eden, C. and Greatbatch, R. J.: Towards a mesoscale eddy closure, Ocean Model., 20, 223–239, https://doi.org/10.1016/j.ocemod.2007.09.002, 2008.a
Gent, P. R. and McWilliams, J. C.: Isopycnal Mixing in Ocean Circulation Models, J. Phys. Oceanogr., 20, 150–155, https://doi.org/10.1175/1520-0485(1990)020<0150:IMIOCM>2.0.CO;2, 1990.a
Gent, P. R., Willebrand, J., McDougall, T. J., and McWilliams, J. C.: Parameterizing Eddy-Induced Tracer Transports in Ocean Circulation Models, J. Phys. Oceanogr., 25, 463–474, https://doi.org/
10.1175/1520-0485(1995)025<0463:PEITTI>2.0.CO;2, 1995.a
Griffies, S. M. and Hallberg, R. W.: Biharmonic Friction with a Smagorinsky-Like Viscosity for Use in Large-Scale Eddy-Permitting Ocean Models, Mon. Weather Rev., 128, 2935–2946, https://doi.org/
10.1175/1520-0493(2000)128<2935:BFWASL>2.0.CO;2, 2000.a, b, c
Griffies, S. M., Adcroft, A., and Hallberg, R. W.: A Primer on the Vertical Lagrangian-Remap Method in Ocean Models Based on Finite Volume Generalized Vertical Coordinates, J. Adv. Model. Earth Sy.,
12, e2019MS001954, https://doi.org/10.1029/2019MS001954, 2020.a
Hallberg, R.: Using a resolution function to regulate parameterizations of oceanic mesoscale eddy effects, Ocean Model., 72, 92–103, https://doi.org/10.1016/j.ocemod.2013.08.007, 2013.a, b, c
Hewitt, H. T., Roberts, M., Mathiot, P., Biastoch, A., Blockley, E., Chassignet, E. P., Fox-Kemper, B., Hyder, P., Marshall, D. P., Popova, E., Treguier, A.-M., Zanna, L., Yool, A., Yu, Y., Beadling,
R., Bell, M., Kuhlbrodt, T., Arsouze, T., Bellucci, A., Castruccio, F., Gan, B., Putrasahan, D., Roberts, C. D., Van Roekel, L., and Zhang, Q.: Resolving and Parameterising the Ocean Mesoscale in
Earth System Models, Current Climate Change Reports, 6, 137–152, https://doi.org/10.1007/s40641-020-00164-w, 2020.a
Jansen, M. F. and Held, I. M.: Parameterizing subgrid-scale eddy effects using energetically consistent backscatter, Ocean Model., 80, 36–48, https://doi.org/10.1016/j.ocemod.2014.06.002, 2014.a
Jansen, M. F., Adcroft, A., Khani, S., and Kong, H.: Toward an Energetically Consistent, Resolution Aware Parameterization of Ocean Mesoscale Eddies, J. Adv. Model. Earth Sy., 11, 2844–2860, https://
doi.org/10.1029/2019MS001750, 2019.a, b
Khani, S., Jansen, M. F., and Adcroft, A.: Diagnosing subgrid mesoscale eddy fluxes with and without topography, J. Adv. Model. Earth Sy., 11, 3995–4015, https://doi.org/10.1029/2019MS001721, 2019.a
Kjellsson, J. and Zanna, L.: The Impact of Horizontal Resolution on Energy Transfers in Global Ocean Models, Fluids, 2, 45, https://doi.org/10.3390/fluids2030045, 2017.a
Kong, H. and Jansen, M. F.: The impact of topography and eddy parameterization on the simulated Southern Ocean circulation response to changes in surface wind stress, J. Phys. Oceanogr., 1, 825–843,
https://doi.org/10.1175/JPO-D-20-0142.1, 2020.a
Loose, N., Abernathey, R., Grooms, I., Busecke, J., Guillaumin, A., Yankovsky, E., Marques, G., Steinberg, J., Ross, A. S., Khatri, H., Bachman, S., Zanna, L., and Martin, P.: GCM-Filters: A Python
Package for Diffusion-based Spatial Filtering of Gridded Data, J. Open Source Softw., 7, 3947, https://doi.org/10.21105/joss.03947, 2022a.a
Loose, N., Bachman, S., GROOMS, I., and Jansen, M.: Diagnosing scale-dependent energy cycles in a high-resolution isopycnal ocean model, Earth and Space Science Open Archive [data set], https://
doi.org/10.1002/essoar.10511055.1, 2022b.a
Marques, G., Loose, N., Yankovsky, E., Steinberg, J., Chang, C.-Y., Bhamidipati, N., Adcroft, A., Fox-Kemper, B., Griffies, S., Hallberg, R., Jansen, M., Khatri, H., and Zanna, L.: NeverWorld2
Dataset, NeverWorld2 [data set], https://doi.org/10.26024/f130-ev71, 2022.a
Marshall, D. P. and Adcroft, A. J.: Parameterization of ocean eddies: Potential vorticity mixing, energetics and Arnold's first stability theorem, Ocean Model., 32, 188–204, https://doi.org/10.1016/
j.ocemod.2010.02.001, 2010.a
Marshall, D. P., Maddison, J. R., and Berloff, P. S.: A Framework for Parameterizing Eddy Potential Vorticity Fluxes, J. Phys. Oceanogr., 42, 539–557, https://doi.org/10.1175/JPO-D-11-048.1, 2012. a
Marshall, J. and Radko, T.: Residual-Mean Solutions for the Antarctic Circumpolar Current and Its Associated Overturning Circulation, J. Phys. Oceanogr., 33, 2341–2354, https://doi.org/10.1175/
1520-0485(2003)033<2341:RSFTAC>2.0.CO;2, 2003.a
McClean, J. L., Bader, D. C., Bryan, F. O., Maltrud, M. E., Dennis, J. M., Mirin, A. A., Jones, P. W., Kim, Y. Y., Ivanova, D. P., Vertenstein, M., Boyle, J. S., Jacob, R. L., Norton, N., Craig, A.,
and Worley, P. H.: A prototype two-decade fully-coupled fine-resolution CCSM simulation, Ocean Model., 39, 10–30, https://doi.org/10.1016/j.ocemod.2011.02.011, 2011.a
Pearson, B., Fox-Kemper, B., Bachman, S., and Bryan, F.: Evaluation of scale-aware subgrid mesoscale eddy models in a global eddy-rich model, Ocean Model., 115, 42–58, https://doi.org/10.1016/
j.ocemod.2017.05.007, 2017.a
Pietarila Graham, J. and Ringler, T.: A framework for the evaluation of turbulence closures used in mesoscale ocean large-eddy simulations, Ocean Model., 65, 25–39, https://doi.org/10.1016/
j.ocemod.2013.01.004, 2013.a
Smith, K. S. and Vallis, G. K.: The Scales and Equilibration of Midocean Eddies: Forced–Dissipative Flow, J. Phys. Oceanogr., 32, 1699–1720, https://doi.org/10.1175/1520-0485(2002)032<1699:TSAEOM>
2.0.CO;2, 2002.a, b, c
Soufflet, Y., Marchesiello, P., Lemarié, F., Jouanno, J., Capet, X., Debreu, L., and Benshila, R.: On effective resolution in ocean models, Ocean Model., 98, 36–50, https://doi.org/10.1016/
j.ocemod.2015.12.004, 2016.a
Thompson, A. F. and Young, W. R.: Two-Layer Baroclinic Eddy Heat Fluxes: Zonal Flows and Energy Balance, J. Atmos. Sci., 64, 3214–3231, https://doi.org/10.1175/JAS4000.1, 2007.a, b
Treguier, A. M., Held, I. M., and Larichev, V. D.: Parameterization of Quasigeostrophic Eddies in Primitive Equation Ocean Models, J. Phys. Oceanogr., 27, 567–580, https://doi.org/10.1175/1520-0485
(1997)027<0567:POQEIP>2.0.CO;2, 1997.a
Uchida, T., Rokem, A., Nicholas, T., Abernathey, R., Nouguier, F., Vanderplas, J., Martin, P., Mayer, A., Halchenko, Y., Wilson, G., Constantinou, N. C., Ponte, A., Squire, D., Busecke, J., Spring,
A., Pak, K., and Hoyer, S.: xgcm/xrft: v0.3.1-rc0, https://doi.org/10.5281/zenodo.5503856, 2021.a
Wolfe, C. L. and Cessi, P.: Overturning Circulation in an Eddy-Resolving Model: The Effect of the Pole-to-Pole Temperature Gradient, J. Phys. Oceanogr., 39, 125–142, https://doi.org/10.1175/
2008JPO3991.1, 2009.a, b
Yankovsky, E., Zanna, L., and Smith, K. S.: Influences of Mesoscale Ocean Eddies on Flow Vertical Structure in a Resolution-Based Model Hierarchy, Earth and Space Science Open Archive, https://
doi.org/10.1002/essoar.10511501.1, 2022.a
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seminars - The Minicourse of lattice algorithm
Traditional cryptography is ill-suited to modern security needs, arising from the outsourced storage and computation possibilities that the "cloud" offers. The minicourse of lattice algorithm is
centered around encryption and its advanced variants that are more suited to the cloud. We will introduce hard problems related to lattice and how to design protocols whose security provably relies
on the difficulty of hard problems. We will start from basic lattice algorithm, reduction techniques and move up to more and more advanced techniques to construct encryption suited to the cloud.
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What is a kilowatt hour [kWh], a unit of energy measurement
What is a kilowatt hour (unit)
The kilowatt hour is a unit of measurement of energy
Kilowatt hour (kWh) is a derived unit of energy equal to 3.6 megajoules, and expressed as power [kilowatt (kW)] multiplied by time [hour (h)]. The unit commonly appears on our electrical bills, and
as such, have become known as an electrical unit. However, the unit can be used to measure other types of energy.
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Archaeologists Unearth Ancient Babylonian Tablet Containing the Pythagorean Theorem, Predating Pythagoras by 1,000 Years
For centuries, the name Pythagoras has been synonymous with the Pythagorean theorem. This fundamental principle of geometry, taught to high school students the world over, states that in a
right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). It’s a cornerstone of mathematics, but did Pythagoras
actually discover it, or were there others before him who understood this fundamental relationship?
Recent research sheds light on the ancient Babylonians and their potential role in the origin of the Pythagorean theorem. While it is widely believed that Pythagoras, the Greek mathematician,
formulated this theorem around 2,500 years ago, evidence suggests that the concept might predate him by several centuries.
A study conducted in 2009 highlighted the possibility that the ancient Babylonians were well aware of the principles that would later be known as the Pythagorean theorem, although it was not referred
to by that name at the time. This theorem is also known by various names, including Pythagoras’ Theorem and Euclid I 47.
The real surprise comes from deciphering ancient Babylonian tablets, which suggest that they were already using the principles of the theorem. One notable tablet, IM 67118, is estimated to date back
to 1770 BC, a time long before Pythagoras was born.
This tablet, along with another one dating back to 1800-1600 BC, contained diagrams and mathematical notations that, when interpreted using the Babylonians’ base 60 counting system, unveiled their
understanding of the Pythagorean theorem.
The groundbreaking discovery on the Babylonian tablet IM 67118 revealed a crucial piece of information. It displayed the formula d = √2, where “d” represents the length of the diagonal of a square,
and “2” signifies the ratio of the diagonal to the side length. This discovery implies that the Babylonians were already aware of the Pythagorean theorem or, at the very least, a special case of it
related to the diagonal of a square.
Bruce Ratner, a mathematician, emphasized the significance of this finding. He stated, “The conclusion is inescapable. The Babylonians knew the relation between the length of the diagonal of a square
and its side: d = √2.” This particular number, the square root of 2, is known to be irrational and holds a central role in the Pythagorean theorem.
What’s even more intriguing is that this knowledge would have predated Pythagoras by over a millennium. The scarcity of written records from that time makes it challenging to definitively establish
who first conceived these mathematical concepts. It’s possible that Pythagoras’ contributions were passed down orally through generations, and the Greeks eventually attributed the theorem to him out
of respect for their leader.
This revelation raises an important question about the history and evolution of mathematical knowledge. Did Pythagoras truly discover the Pythagorean theorem, or was he merely a custodian of ancient
wisdom that was already in circulation long before his time? The answer to this question could reshape our understanding of the development of mathematics and the transmission of knowledge in ancient
In conclusion, while Pythagoras is undeniably celebrated for his contributions to mathematics, it appears that he might not have been the original author of the Pythagorean theorem. The ancient
Babylonians, with their advanced mathematical understanding and the discovery of tablets predating Pythagoras, challenged the conventional wisdom regarding the origins of this fundamental geometric
As we continue to explore the annals of history, we must remain open to the possibility that knowledge often emerges from collective wisdom, rather than the genius of a single individual. The
question of whether Pythagoras truly discovered the Pythagorean theorem invites us to reevaluate the foundations of mathematical history.
The research findings were made available in the Journal of Targeting, Measurement, and Analysis for Marketing on September 15, 2009.
Study Abstract:
While most individuals who have delved into geometry can recall some aspect of the Pythagorean Theorem, the intriguing tale of Pythagoras and his renowned theorem remains relatively obscure. This
article aims to shed light on key plot points within this narrative. The famous theorem is known by various appellations, some rooted in the context of its time, such as the Pythagorean Theorem,
Pythagoras’ Theorem, and notably, Euclid I 47.
Widely regarded as the most iconic proposition in mathematics, it also ranks as the fourth most aesthetically pleasing equation. Surprisingly, there exist more than 371 distinct proofs of the
Pythagorean Theorem, initially compiled in a book in 1927.
These proofs include contributions from notable figures like a 12-year-old Einstein, who would later employ the theorem in his groundbreaking work on relativity, as well as Leonardo da Vinci and the
U.S. President, James A. Garfield. Pythagoras is forever associated with the discovery and proof of this eponymous theorem, despite the absence of concrete evidence supporting his direct involvement.
In fact, historical records indicate that Babylonian mathematicians had uncovered and proven the Pythagorean Theorem a millennium before Pythagoras’ birth.
In the ancient city of Trichy, veiled by the mist of centuries, stands a testament to
Throughout ancient history, tales of giants have woven themselves into the fabric of cultures worldwide. These
In the heart of India, nestled in the rugged hills of Maharashtra, lie the ancient Bhaja
Our history books often paint a picture of ancient civilizations as primitive and technologically underdeveloped. This
In a world filled with hidden secrets and forbidden places, few can match the enigmatic allure
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Empirical study on relationship between sports analytics and success in regular season and postseason in Major League Baseball
In this paper, we study the relationship between sports analytics and success in regular season and postseason in Major League Baseball via the empirical data of 2014-2017. The categories of
analytics belief, the number of analytics staff, and the total number of research staff employed by MLB teams are examined. Conditional probabilities, correlations, and various regression models are
used to analyze the data. It is shown that the use of sports analytics might have some positive impact on the success of teams in the regular season, but not in the postseason. After taking into
account the team payroll, we apply partial correlations and partial F tests to analyze the data again. It is found that the use of sports analytics, with team payroll already in the regression model,
might still be a good indicator of success in the regular season, but not in the postseason. Moreover, it is shown that both the team payroll and the use of sports analytics are not good indicators
of success in the postseason. The predictive modeling of decision trees is also developed, under different kinds of input and target variables, to classify MLB teams into no playoffs or playoffs. It
is interesting to note that 87 wins (or 0.537 winning percentage) in a regular season may well be the threshold of advancing into the postseason.
In recent years, sports analytics has been very popular in professional baseball teams among other professional sports in North America. The movie “Money Ball” vividly portrayed a true story of the
magical use of sabermetrics. A team under financial constraint in Major League Baseball (MLB) was formed. Nevertheless, it thrived and successfully competed with high payroll teams in the league.
Many baseball management teams nowadays have allocated plenty of financial resources to sports analytics in order to improve teams’ performance. In particular, they wish to analyze the statistics of
their players and those of other teams’ players to devise more strategic game plans to give their team advantages of winning games. One could argue that the sports analytics belief in a team is an
indicator of management style or management quality. One would then hypothesize that greater management quality, all else being equal, translates into more wins and hence greater success in both
regular season and postseason. One might even desire that this eventually leads to winning a championship of the professional sport. But some management teams do not believe in the notion of sports
analytics at all. They still utilize the traditional methods to set up game plans, using their professional knowledge and experience to guide their decisions rather than taking full advantage of the
statistics of all players. Other management teams, however, are skeptical or gradually buy in the idea of sports analytics.
In this paper, we study the relationship between sports analytics and the performance of MLB teams. In particular, we look at the final standings of both regular season and postseason of 2014-2017.
To assess the use of sports analytics in MLB teams, we will examine three aspects. The first aspect comes from a source outside of the teams, which is the ESPN sports analytics categorization of
teams listed in The Great Analytics Rankings (2015). The sports analytics belief in MLB teams was categorized in five levels: All-in, Believers, One-foot-in, Skeptics, and Non-believers. The second
and third aspects come from the source inside of the teams. We look at the number of analytics staff hired and devoted to the analytics department, if any, of each team. In addition, we investigate
the total number of research staff worked in the baseball operations of each team. The research work may include analytics, statistical data analysis, mathematical modeling, data science, data
architecture, decision sciences, informatics, performance science, research and development, etc. The categories of analytics belief and these numbers of analytics staff and research staff might
reflect teams’ commitment and their potential shift into more reliance on analytics and other innovative research. We will evaluate how the use of sports analytics influences the number of games won
in the regular season, thereby affecting the chances of advancing to the postseason. Once a team is in postseason, we will study further how the use of sports analytics influences its chances of
getting into the last 4 teams and last 2 teams of playoffs, and eventually its chances of winning the championship of World Series.
Through the historical data from 1977 to 2008, Schwartz and Zarrow (2009) showed that the team payroll had great influence in regular season, but not in postseason. They also tested several other
potential indicators of postseason success and found that none of them was a significant predictor. They concluded that the success in October (i.e., playoffs) was a truly random event.
It makes sense to see that teams of high payroll in general would perform better than teams of low payroll. Teams of high payroll could afford to recruit more talented and experienced players that
lead to better chances of winning games. In other words, team payroll could be an indicator of the quality of players’ talent, which is an input into producing wins and success in both regular season
and postseason. To confirm this belief, we will calculate the Pearson sample correlation coefficients between the success of teams and their team payroll for both regular season and postseason of
2014-2017. The Pearson sample correlation coefficients between the success of teams and their use of sports analytics are calculated as well. These results are shown in Section 3.
Binary logistic regression models and multiple linear regression models are applied to test the significance of the levels of sports analytics on the success in regular season, while ordinal logistic
regression models are applied to test the significance of the levels of sports analytics on the success in postseason. The findings are also given in Section 3.
In addition to the factor of team payroll, does the use of sports analytics play an important role in explaining the success of a team (in terms of winning more games in regular season and the
advancement level in postseason towards a championship of World Series)? We will tackle this issue from two perspectives. First, as the team payroll is an essential component of a team’s success, it
is necessary to control this factor while calculating the correlation coefficient. Therefore, we have to calculate the partial correlation coefficient between the success of a team and its use of
sports analytics, after the team payroll is accounted for. Second, we will use the partial F tests to evaluate the significance of adding the use of sports analytics in explaining the success of a
team, after the factor of team payroll is already in the regression model. Moreover, we apply the model utility F test to show that both the team payroll and the use of sports analytics are not good
indicators of success in the postseason. These results are shown in Section 4.
In Section 5, the predictive modeling of decision trees will be developed to classify MLB teams into no playoffs or playoffs. We will consider the situations where the number of games won in a
regular season is available or not as an input variable. It is interesting to note that 87 wins (or 0.537 winning percentage) in a regular season may well be the threshold of advancing into the
postseason. A decision tree shows that teams with pitchers’ salary at least $29.7 million and analytics belief either All_in or Believers have significantly higher chances of advancing to playoffs.
Summary and concluding comments will be presented in Section 6.
The MLB teams’ categories of analytics belief are listed in The Great Analytics Rankings. The standings, playoffs and payrolls of the MLB teams in 2014-2017 can be found from the corresponding
websites shown on the references. The numbers of analytics staff employed by teams can be tracked down from the Baseball America Directories (2014-2017). The total numbers of research staff can also
be tracked down and added up from the Baseball America Directories.
3Effects of sports analytics on regular season and postseason
The MLB teams’ analytics belief can be classified in five levels (or categories) as mentioned previously. To simplify the study of conditional probabilities below, we place the levels of sports
analytics belief into two groups: BELIEVERS (All-in, Believers) and NON-BELIEVERS (One-foot-in, Skeptics, Non-believers). There are 16 teams identified as BELIEVERS and the other 14 teams as
3.1Conditional probabilities
The last four teams in the postseason of 2014-2017, respectively, were Baltimore Orioles (BELIEVER), St. Louis Cardinals(BELIEVER), Kansas City Royals(BELIEVER, advanced to World Series), and San
Francisco(NON-BELIEVER, the champion of World Series); Chicago Cubs(BELIEVER), Toronto Blue Jays(BELIEVER), New York Mets(BELIEVER, advanced to World Series), and Kansas City Royals(BELIEVER, the
champion of World Series); Los Angeles Dodgers(BELIEVER), Toronto Blue Jays(BELIEVER), Cleveland Indians(BELIEVER, advanced to World Series), and Chicago Cubs (BELIEVER, the champion of World
Series); Chicago Cubs(BELIEVER), New York Yankees(BELIEVER), Los Angeles Dodgers(BELIEVER, advanced to World Series), and Houston Astros(BELIEVER, the champion of World Series). The distribution of
BELIEVERS and NON-BELIEVERS teams in no playoffs and playoffs for 2014-2017 is displayed in Fig. 1.
Fig. 1.
Based on Fig. 1, the conditional probabilities below show the chances of getting into playoffs if the team is a BELIEVER or a NON-BELIEVER. Once the team is in playoffs, the conditional probabilities
show its chances of advancing to different stages of playoffs. The entries in the parentheses are the corresponding conditional probabilities occurred in 2014-2017.
P(Playoffs | BELIEVERS) ≈ (44%, 56%, 50%, 44%)
P(Playoffs | NON-BELIEVERS) ≈ (21%, 7%, 14%, 21%)
P(Last 4 teams of playoffs | BELIEVERS and playoffs) ≈ (43%, 44%, 50%, 57%)
P(Last 4 teams of playoffs | NON-BELIEVERS and playoffs) ≈ (33%, 0%, 0%, 0%)
P(Last 2 teams of playoffs | BELIEVERS and playoffs) ≈ (14%, 22%, 25%, 29%)
P(Last 2 teams of playoffs | NON-BELIEVERS and playoffs) ≈ (33%, 0%, 0%, 0%)
P(Champion of World Series | BELIEVERS and playoffs) ≈ (0%, 11%, 13%, 14%)
P(Champion of World Series | NON-BELIEVERS and playoffs) ≈ (33%, 0%, 0%, 0%)
Through the data of 2014-2017, we can see that the chances (44%, 56%, 50%, 44%) of advancing to playoffs for a BELIEVER team were higher than those (21%, 7%, 14%, 21%) for a NON-BELIEVER team. Once
the team was in playoffs, the chances (43%, 44%, 50%, 57%) of advancing to the last 4 teams of playoffs for a BELIEVER team were also higher than those (33%, 0%, 0%, 0%) for a NON-BELIEVER team.
Similar patterns remain for the data of 2015-2017, when comparing a BELIEVER team and a NON-BELIEVER team for advancing to the last 2 teams of playoffs and for becoming the champion of World Series.
The data of 2014, however, shows the opposite pattern that a NON-BELIEVER team had a higher chance of advancing to the last 2 teams of playoffs than a BELIEVER team (33% vs 14%) and had a higher
chance of becoming the champion of World Series (33% vs 0%). This happened because of the fact that the champion of World Series in 2014 was San Francisco which was categorized as a NON-BELIEVER
To test the relationship between a team’s category of analytics belief (BELIEVER or NON-BELIEVER) and whether it advances to the postseason, we apply the chi-square test for testing the independence
of these two characteristics. We obtain the observed chi-square value=1.674, 8.105, 4.286, 1.674 and p-value=0.196, 0.004, 0.038, 0.196 for 2014-2017, respectively. Notice that both p-values for
2014 and 2017 are 0.196. Therefore, with 5% level of significance, there is insufficient evidence to reject the null hypothesis that these two characteristics were independent for these two years,
i.e., a team advancing to the postseason of 2014 and 2017 was unrelated to the category of its analytics belief. However, opposite conclusion occurred for 2015 and 2016 as the corresponding p-values
(0.004 and 0.038) are less than 0.05. The opposite conclusion obtained is related to the fact that 30% (higher percentage) of the playoffs teams were NON-BELIEVERS in 2014 and 2017, while there were
only 10% in 2015 and 20% in 2016.
3.1.2Analytics staff
According to the information listed in the Baseball America Directories, the distribution of teams having analytics staff=0, 1, 2 or more, in no playoffs and playoffs for 2014-2017 is presented in
Fig. 2. It appears that teams having no analytics staff were in much higher proportion than teams having 1 or at least 2 analytics staff in both no playoffs and playoffs for these four years. The
following conditional probabilities show the chances of getting into playoffs and the chances of advancing to different stages of playoffs for teams having different numbers of analytics staff.
P(Playoffs | Analytics Staff (A)=0) ≈ (30%, 40%, 40%, 39%)
P(Playoffs | A=1) ≈ (33%, 11%, 33%, 0%)
P(Playoffs | A ≥ 2) ≈ (100%, 100%, 0%, 20%)
P(Last 4 teams of playoffs | Playoffs and A=0) ≈ (29%, 38%, 38%, 44%)
P(Last 4 teams of playoffs | Playoffs and A=1) ≈ (50%, 0%, 50%, 0%)
P(Last 4 teams of playoffs | Playoffs and A ≥ 2) ≈ (100%, 100%, NA, 0%)
P(Last 2 teams of playoffs | Playoffs and A=0) ≈ (14%, 13%, 25%, 22%)
P(Last 2 teams of playoffs | Playoffs and A=1) ≈ (0%, 0%, 0%, NA)
P(Last 2 teams of playoffs | Playoffs and A ≥ 2) ≈ (100%, 100%, NA, 0%)
P(Champion of World Series | Playoffs and A=0) ≈ (14%, 0%, 13%, 11%)
P(Champion of World Series | Playoffs and A=1) ≈ (0%, 0%, 0%, NA)
P(Champion of World Series | Playoffs and A ≥ 2) ≈ (0%, 100%, NA, 0%)
Fig. 2.
If there was no team under the condition of the conditional probability, then NA (not applicable) would be given. There were many MLB teams having no analytics department and hence no analytics staff
for 2014-2017. Kansas City Royals, with four analytics staff, was a very successful team in 2014 and 2015. It advanced to the World Series but lost in 2014. Nevertheless, it became the champion of
World Series in 2015. Other than Kansas City Royals, all other teams advanced to the World Series from 2014-2017 didn’t have any analytics staff. Among those last four teams of playoffs from 2014 to
2017, only Baltimore Orioles and Toronto Blue Jays each had one analytics staff. Besides the extraordinary performance of Kansas City Royals in 2014 and 2015, teams having analytics staff indicated
no higher percentage of success in advancing to playoffs or during postseason in almost all conditional probabilities shown above.
3.1.3Research staff
Again from the Baseball America Directories, we compile the list of research staff (including analytics staff) employed by teams. The distribution of teams having research staff=High(7-5), Medium
(4-3), Low(2-1), None(0), in no playoffs and playoffs for 2014-2017 is given in Fig. 3. The conditional probabilities below show the chances of getting into playoffs and the chances of advancing to
different stages of playoffs for teams having different numbers of research staff.
P(Playoffs | Research Staff (R)=High) ≈ (0%, 50%, 20%, 50%)
P(Playoffs | R=Medium) ≈ (67%, 75%, 20%, 63%)
P(Playoffs | R=Low) ≈ (40%, 20%, 43%, 20%)
P(Playoffs | R=None) ≈ (20%, 33%, 33%, 0%)
P(Last 4 teams of playoffs | Playoffs and R=High) ≈ (NA, 0%, 0%, 67%)
P(Last 4 teams of playoffs | Playoffs and R= Medium)≈(100%, 67%, 100%, 20%)
P(Last 4 teams of playoffs | Playoffs and R=Low) ≈ (33%, 67%, 33%, 50%)
P(Last 4 teams of playoffs | Playoffs and R=None) ≈ (0%, 0%, 50%, NA)
P(Last 2 teams of playoffs | Playoffs and R=High) ≈ (NA, 0%, 0%, 0%)
P(Last 2 teams of playoffs | Playoffs and R=Medium)≈(50%, 33%, 100%, 20%)
P(Last 2 teams of playoffs | Playoffs and R=Low) ≈ (17%, 33%, 17%, 50%)
P(Last 2 teams of playoffs | Playoffs and R=None) ≈ (0%, 0%, 0%, NA)
P(Champion of World Series | Playoffs and R=High) ≈ (NA, 0%, 0%, 0%)
P(Champion of World Series | Playoffs and R=Medium) ≈ (0%, 33%, 0%, 20%)
P(Champion of World Series | Playoffs and R=Low) ≈ (17%, 0%, 17%, 0%)
P(Champion of World Series | Playoffs and R=None) ≈ (0%, 0%, 0%, NA)
Fig. 3.
Three years out of four from 2014 to 2017, teams with research staff=Medium had higher percentages of advancing to playoffs than teams with research staff=High, Low, or None. Once teams were in
playoffs, similar pattern occurred (3 years out of four) that teams with research staff=Medium had the same or higher percentages of advancing to the last 4 teams as well as the last 2 teams of
playoffs than teams with research staff in other groups. Two years out of four from 2014 to 2017, teams with research staff=Medium had higher percentages of becoming the champion of World Series
than teams with research staff in other groups. It appears that teams with 3 or 4 research staff (Medium group) performed most consistently than teams in other groups.
3.2Correlation coefficients
To study the correlations between different variables, we define the following notation:
• W— Wins, i.e., number of games won in a regular season;
• P— Team payroll;
• B— Categories of analytics belief: 4(All-in), 3(Believers), 2(One-foot-in), 1(Skeptics), and 0(Non-believers);
• A— Number of analytics staff;
• R— Number of research staff (including analytics staff);
• C— Levels of playoffs towards the championship of World Series: 5(Champion), 4(Third round game but lost), 3(Second round game but lost), 2(First round game but lost), 1(Wild card game but lost),
and 0(No playoffs);
• C*— C but removing 0(No playoffs).
The Pearson sample correlation coefficients for various pairs of variables for 2014-2017 are computed and displayed in Table 1. For example, r(W and P)=0.38 (0.04) means that the Pearson sample
correlation coefficient for assessing the linear relationship between W (number of games won in a regular season) and P (team payroll) is 0.38 with p-value=0.04. Hence, the correlation between W
and P is positive and significant at α = 5%. We are mostly interested in those pairs of variables whose correlations are significant (i.e., p-values < 0.05).
Table 1
Pearson sample correlation coefficient r with p-value in parentheses
Relationship 2014 2015 2016 2017
W and P 0.38 (0.04) 0.28 (0.13) 0.61 (0.00) 0.35 (0.06)
W and B 0.29 (0.12) 0.61 (0.00) 0.44 (0.01) 0.43 (0.02)
W and A 0.20 (0.29) –0.01 (0.94) –0.23 (0.22) –0.13 (0.50)
W and R –0.01 (0.98) 0.31 (0.09) 0.06 (0.77) 0.39 (0.03)
W and C 0.67 (0.00) 0.71 (0.00) 0.75 (0.00) 0.83 (0.00)
W and C* –0.15 (0.69) 0.13 (0.73) 0.71 (0.02) 0.68 (0.03)
P and B –0.01 (0.96) 0.14 (0.45) 0.24 (0.20) 0.20 (0.28)
P and A –0.15 (0.43) –0.24 (0.19) –0.31 (0.10) –0.02 (0.91)
P and R –0.27 (0.14) –0.27 (0.14) –0.20 (0.29) 0.06 (0.75)
P and C 0.34 (0.07) 0.15 (0.43) 0.43 (0.02) 0.45 (0.01)
P and C* 0.16 (0.67) –0.26 (0.48) 0.06 (0.86) 0.40 (0.25)
B and C 0.12 (0.52) 0.37 (0.04) 0.37 (0.04) 0.37 (0.04)
B and C* –0.31 (0.39) –0.38 (0.29) 0.53 (0.11) 0.61 (0.06)
B and A 0.13 (0.49) –0.05 (0.80) –0.14 (0.45) –0.08 (0.69)
B and R 0.48 (0.01) 0.38 (0.04) 0.39 (0.03) 0.35 (0.06)
A and C 0.39 (0.03) 0.34 (0.07) –0.25 (0.18) –0.20 (0.30)
A and C* 0.42 (0.23) 0.66 (0.04) –0.21 (0.57) –0.14 (0.70)
A and R 0.46 (0.01) 0.44 (0.02) 0.33 (0.07) 0.24 (0.21)
R and C 0.16 (0.40) 0.27 (0.16) –0.11 (0.57) 0.27 (0.14)
R and C* 0.42 (0.23) 0.33 (0.36) 0.17 (0.63) –0.10 (0.79)
The number of games won in a regular season (W) and levels of playoffs towards the championship of World Series (C) were moderately to strongly positively correlated (r=0.67, 0.71, 0.75, 0.83) for
2014-2017. It was because teams of fewer wins would likely not advance to playoffs and hence their value of C would be zero. Because of the zero value of C mostly coming from teams of fewer wins, the
correlation between W and C tends to be positive. To see the actual effect of the number of games won in a regular season on the success in the postseason, we need to consider only those teams in the
postseason, i.e., removing those teams with zero value of C. After ignoring all zeros of C (i.e., considering C*), W and levels of playoffs towards the championship of World Series in the postseason
(C*) show no significant correlation at α = 5% for 2014 and 2015, but a significantly positive correlation (r=0.71, 0.68) for 2016 and 2017. It seems that once a team is in playoffs, its standing in
the regular season has a random effect on its success in the postseason. Teams of higher standings in the regular season might not generate any advantages to move forward in the postseason.
W and categories of analytics belief (B) were significantly positively correlated (r=0.61, 0.44, 0.43) for 2015-2017.
B and C show positive correlations (r=0.37, 0.37, 0.37) for 2015-2017. This might indicate that the categorization of analytics belief on teams could reflect the commitment made by teams that, in
turn, would contribute some positive impact on the success of winning a championship of World Series. To see the actual effect of the categories of analytics belief on the success in the postseason,
we consider only those teams in the postseason. B and C* show no significant correlations for 2014-2017 as all p-values are greater than 0.05. It seems that once a team is in playoffs, the analytics
belief and commitment made by a team has no effect on its success in the postseason.
B and number of research staff (R) indicate a positive correlation as demonstrated by r=0.48, 0.38, 0.39 for 2014-2016 and r=0.35 (p-value=0.06) in 2017.
The number of analytics staff (A) and R display some positive correlation (r=0.46, 0.44) for 2014-2015 and r=0.33 (p-value=0.07) for 2016, but not for 2017.
A and C had a positive correlation (r=0.39) in 2014, but not in 2015-2017. A and C* had a positive correlation (r=0.66) in 2015, but not in the other three years.
R, however, does not show any significant positive or negative correlation with C or C*. R and W had a positive correlation (r=0.39) in 2017 and r=0.31 (p-value=0.09) in 2015.
Table 2
β[0] β[1] β[2] β[3] e^β[1] e^β[2] e^β[3] p-value
2014 –1.90 0.51 0.82 –0.32 1.66 2.27 0.73 0.31
(1.04) (0.39) (0.65) (0.37) (0.77, 3.58) (0.63, 8.09) (0.35, 1.51)
2015 –4.06 1.22 0.33 –0.13 3.39 1.38 0.88 0.03
(1.59) (0.53) (0.57) (0.29) (1.19, 9.66) (0.46, 4.19) (0.50, 1.56)
2016 –2.54 1.15 –0.54 –0.55 3.14 0.58 0.58 0.03
(1.42) (0.58) (0.80) (0.34) (1.01, 9.79) (0.12, 2.77) (0.29, 1.13)
2017 –1.87 0.11 –0.99 0.45 1.12 0.37 1.56 0.09
(1.06) (0.35) (0.74) (0.24) (0.56, 2.23) (0.09, 1.58) (0.98, 2.51)
It is interesting to note that the team payroll (P) and W were positively correlated (r=0.38, 0.61) in 2014 and 2016, and had r=0.35 (p-value=0.06) in 2017. In addition, P and C had a positive
correlation (r=0.43, 0.45) in 2016-2017 and had r=0.34 (p-value=0.07) in 2014. This might reflect that financial resources did have some positive impact on the journey of winning a championship
of World Series. To see the actual effect of the team payroll on the success in the postseason, we need to consider C*. P and C* do not show any significant positive correlation for 2014-2017. It
seems that once a team is in playoffs, its team payroll has no linear effect on its success in the postseason.
3.3Binary logistic regression models
We employ binary logistic regression models to assess the relationship between the success of advancing to playoffs and the use of sports analytics (categories of analytics belief, number of
analytics staff, and number of research staff) for the data of 2014-2017. The response variable Y is the advancement to playoffs, which has the value of 1 if the team advances to playoffs and 0 if
not. The continuous explanatory variable X[1] is the categories of analytics belief, which has the value of 4 if the team is All-in, 3 if Believers, 2 if One-foot-in, 1 if Skeptics, and 0 if
Non-believers. The variables X[2] and X[3] represent the number of analytics staff and number of research staff, respectively. The equation of the binary logistic regression model is
where log is the natural logarithm,
is the probability that the team advances to playoffs (i.e.,
= 1)),
/(1 -
) is the odds,
= 0, 1, 2, 3, are regression parameters to be estimated. The statistical software Minitab version 18 was used to formulate the binary logistic regression models. It produced the estimated values of
= 0, 1, 2, 3, with their standard error, odds ratios (
) and their 95% confidence interval (CI), and
-value of the model for the data of 2014-2017. The results are displayed in
Table 2
We can see from Table 2 that the p-values are both 0.03 for 2015 and 2016. Therefore there is sufficient evidence, with 5% level of significance, that the categories of analytics belief, numbers of
analytics staff and research staff employed were associated with the success of a team advancing to playoffs for 2015 and 2016. However, there is insufficient evidence to indicate this association
for 2014. The evidence is less convincing for 2017 as p-value=0.09 is greater than 5% but slightly less than 10%.
Table 3
Goodness-of-fit test
Pearson Deviance Hosmer-Lemeshow
2014 0.27 0.12 0.43
2015 0.53 0.30 0.33
2016 0.31 0.29 0.09
2017 0.31 0.20 0.90
The associated goodness-of-fit tests give the corresponding p-values in Table 3.
With the p-values of the goodness-of-fit tests shown in Table 3, there is insufficient evidence (with 5% level of significance) to claim that the binary logistic regression models do not fit the data
adequately for 2014-2017. However, the p-value of Hosmer-Lemeshow test for 2016 is slightly less than 10%.
Table 4
β[0] β[1] β[2] β[3] β[4] β[5] β[6] F p-value
2014 75.32 7.43 12.20 7.67 –0.67 1.30 –0.97 1.38 0.26
(6.71) (7.90) (7.54) (7.64) (7.64) (3.07) (1.82)
2015 66.31 16.63 19.37 12.74 5.50 –1.33 1.36 3.06 0.02
(6.31) (7.21) (7.05) (7.23) (7.16) (2.41) (1.24)
2016 76.83 8.05 10.46 5.64 –4.45 –1.89 0.02 1.78 0.15
(7.71) (8.09) (7.98) (8.43) (8.23) (2.16) (1.43)
2017 67.34 13.16 13.28 4.96 9.73 –3.68 2.08 1.88 0.13
(7.86) (8.58) (8.72) (8.67) (8.93) (2.65) (1.18)
3.4Multiple linear regression models
We also apply multiple linear regression models to assess the relationship between the number of games won in a regular season and the use of sports analytics (categories of analytics belief, number
of analytics staff, and number of research staff) for the data of 2014-2017. The response variable Y is the number of games won in a regular season. The categories of analytics belief are treated as
a categorical explanatory variable, using four indicator variables X[1], X[2], X[3] and X[4] for the first four categories of analytics belief (All-in, Believers, One-foot-in, and Skeptics with
Non-believers as the baseline). X[5] is the number of analytics staff, and X[6] is the number of research staff. The equation of the multiple linear regression model is
) is the expected value of Y.
is 1 if the category of analytics belief is All-in and 0 otherwise;
is 1 if Believers and 0 otherwise;
is 1 if One-foot-in and 0 otherwise;
is 1 if Skeptics and 0 otherwise. The regression parameters
= 0, 1, 2, . . . , 6, are to be estimated.
Minitab yielded the estimated values of parameters β[i], i = 0, 1, 2, . . . , 6, with their standard error, observed F value, and p-value of the model for the data of 2014-2017. These results are
presented in Table 4.
More games won in a regular season should increase the chances of moving forward to playoffs. Consequently, the number of games won in a regular season should very likely relate to the success of
advancing to playoffs. By comparing 0.05 with the p-values in Table 4, we obtain the same conclusions as those in the previous section for 2014, 2015 and 2017. For 2016, the binary logistic
regression model shows that, with α = 5%, the set of explanatory variables was useful for predicting the chances of advancement to playoffs. But the same set of explanatory variables was not useful
for predicting the number of games won in that regular season, as shown in the multiple linear regression model. As advancing to playoffs is an important goal for teams, it seems that the binary
logistic regression model is more appropriate to formulate the relationship between the success in a regular season and those explanatory variables.
Table 5
α[1] α[2] α[3] α[4] β e^β 95% CI p-value
X=Categories of –0.36 1.49 2.50 3.31 –0.35 0.70 (0.39, 1.28) 0.25
analytics belief (0.98) (1.01) (1.06) (1.12) (0.31)
X= Number of –1.23 0.62 1.68 2.58 –0.56 0.57 (0.31, 1.06) 0.07
analytics staff (0.40) (0.35) (0.45) (0.60) (0.32)
X=Number of –0.99 0.86 1.85 2.66 –0.20 0.82 (0.58, 1.16) 0.22
research staff (0.52) (0.51) (0.57) (0.68) (0.17)
3.5Ordinal logistic regression models
To assess the relationship between the success of a team in the postseason and the use of sports analytics, we apply ordinal logistic regression models. Since there are only ten teams competing in
playoffs each year, we have combined four years’ data together for data analysis. The response variable Y is the levels of playoffs in the postseason towards the championship of World Series, which
is defined previously as C* (i.e., C without considering the value of 0). It is because those teams in playoffs have nonzero values of C. Consequently, there are five different values (5, 4, 3, 2, 1)
for five different levels of playoffs towards the championship of World Series. This response variable has a natural order (i.e., champion(5) > third round(4) > second round(3) > first round(2) >
wild card(1)) and can be classified as an ordinal variable. The continuous explanatory variable (or predictor) X is either (1) categories of analytics belief, (2) number of analytics staff, or (3)
number of research staff. When X is the categories of analytics belief, it is defined as 4 if the team is All-in, 3 if Believers, 2 if One-foot-in, 1 if Skeptics, and 0 if Non-believers. As the
response variable Y has five levels, Minitab used level 5 as the reference and formulated only four logit equations. Each equation has a unique constant, but the parameter of the predictor X is the
same for all equations. The ordinal logistic regression model is
where log is the natural logarithm,
is the cumulative probability that the team advances up to and including level
of the playoffs. For example,
= 1) ,
= 1) +
= 2) ,
= 1) +
= 2) +
= 3), and so on. The intercepts
= 1, 2, 3, 4, and the common parameter
are to be estimated. Hence the ordinal logistic regression model assumes that the effect of the predictor X is common across all levels of the response variable. Minitab produced the estimated values
= 1, 2, 3, 4, and the common parameter
with their standard error, odds ratio (
) and its 95% CI, and
-value of the ordinal logistic regression model for the combined data of 2014-2017. These results are shown in
Table 5
To assess the relationship between the response variable and the predictor in the ordinal logistic regression model, we test the null hypothesis that β is zero. All p-values in Table 5 are greater
than 0.05. Therefore, with 5% level of significance, we conclude that no significant relationship exists between the success of a team in the postseason and any of the three analytics indicators
(categories of analytics belief, number of analytics staff, and number of research staff) for the combined data of 2014-2017. Note that the evidence is less convincing when X=number of analytics
staff as the p-value=0.07.
Table 6
Goodness-of-fit test
Pearson Deviance
X=Categories of analytics belief 0.43 0.27
X=Number of analytics staff 0.87 0.73
X=Number of research staff 0.73 0.52
The associated goodness-of-fit tests give the corresponding p-values in Table 6.
Based on the p-values shown in Table 6, there is insufficient evidence, with 5% level of significance, to claim that the ordinal logistic regression models do not fit these combined four years’ data
4Effect of sports analytics after team payroll is controlled
4.1Partial correlation coefficients
4.1.1Categories of analytics belief
While the team payroll is controlled, the correlation between the success of a team and its category of analytics belief can be revealed by the partial correlation coefficient. To calculate the
partial correlation coefficient (r[X,Y|Z]) between variables X and Y while variable Z is controlled, one may refer to the following formula given by Kutner et al. (2005).
is the Pearson sample correlation coefficient between X and Y, and so on.
We use formula (1) and the values of correlation coefficients listed in Table 1 to calculate the partial correlation coefficients. The partial correlation coefficients between the number of games won
in a regular season (W) and the categories of analytics belief (B), while team payroll (P) is controlled, are 0.32, 0.60, 0.38, and 0.39 for 2014-2017, respectively. The positive values of the
partial correlation coefficients indicate that moderately positive partial correlation existed between W and B for these four years. It suggests that the categories of analytics belief in MLB teams
have some positive effect on the success of teams in the regular season, after the team payroll is taken into account.
However, the partial correlation coefficients between the levels of playoffs in the postseason towards the championship of World Series (C*) and B, while P is controlled, are -0.31, - 0.36, 0.53, and
0.59 for 2014-2017, respectively. The negative and positive values of the partial correlation coefficients indicate that random effect existed between C* and B for these four years. It suggests that
the categories of analytics belief in playoffs teams have random effect on their success in the postseason, after the team payroll is taken into account.
4.1.2Number of analytics staff
Likewise, the partial correlation coefficients between W and the number of analytics staff (A), while P is controlled, are obtained as 0.28, 0.06, -0.05, and -0.13 for 2014-2017, respectively. The
above positive and negative values suggest that the numbers of analytics staff employed by MLB teams have random effect on the success of teams in the regular season, after the team payroll is taken
into account.
In addition, the partial correlation coefficients between C* and A, while P is controlled, are 0.45, 0.64, - 0.20, and -0.14 for 2014-2017, respectively. Again, the positive and negative values
suggest that the numbers of analytics staff employed by playoffs teams have random effect on their success in the postseason, after the team payroll is taken into account.
4.1.3Number of research staff
The partial correlation coefficients between W and the number of research staff (R), while P is controlled, are 0.10, 0.42, 0.23, and 0.39 for 2014-2017, respectively. The above positive values
suggest that the numbers of research staff hired by MLB teams have some positive effect on the success of teams in the regular season, while the team payroll is controlled.
However, the partial correlation coefficients between C* and R, while P is controlled, are 0.49, 0.28, 0.19, and -0.14 for 2014-2017, respectively. The positive and negative values suggest that the
numbers of research staff hired by playoffs teams have random effect (perhaps slightly positive effect) on their success in the postseason, while the team payroll is controlled.
4.2Partial F tests
When the team payroll is used in the regression model as an explanatory variable to predict the number of games won in a regular season, does the addition of the information of (1) categories of
analytics belief, (2) number of analytics staff, or (3) number of research staff significantly improve the predictability of the regression model?
4.2.1Categories of analytics belief
To investigate the above question, we first consider the reduced model consisting of the team payroll (logarithmic value) X[1] as the only continuous explanatory variable. Then we consider the full
model consisting of team payroll (logarithmic value) X[1] and the categories of analytics belief as a categorical explanatory variable using four indicator variables X[2], X[3], X[4], and X[5]. The
response variable Y is the number of games won in a regular season. The model equations are:
is 1 if the category of analytics belief is All-in and 0 otherwise;
is 1 if Believers and 0 otherwise;
is 1 if One-foot-in and 0 otherwise;
is 1 if Skeptics and 0 otherwise. The regression parameters
are to be estimated.
To test the null hypothesis that β2*=β3*=β4*=β5*=0 , we utilize the partial F test to determine whether the difference between the sums of squared residuals for the reduced and full models is so
large that it is unlikely to occur by chance. Minitab computed all the parameter estimates as well as the sums of squared residuals for the reduced and full models. The results are given in Table 7.
The test statistic is
are the sum of squared residuals for the reduced model and the full model, respectively. F follows an F distribution with 4 and 24 degrees of freedom. Examining the
-values in the first part of
Table 7
, we conclude (with 5% level of significance) that once the team payroll was already in the model, the addition of the information of the categories of analytics belief was not statistically
significantly useful in the regression model for predicting the number of games won in a regular season for 2014, 2016 and 2017. However, we obtain the conclusion of statistical significance for
Table 7
(1) Categories of analytics belief
β[0] β[1] β0* β1* β2* β3* β4* β5* SSE[r] SSE[f] F p
2014 –94 9.46 –73 8.07 4.97 10.4 4.67 –2.31 2302 1746 1.91 0.14
(83) (4.5) (81) (4.4) (6.7) (7.0) (7.1) (7.0)
2015 –80 8.62 –25 4.99 18.2 17.7 10.7 4.76 2901 1779 3.79 0.02
(100) (5.4) (91) (4.9) (6.8) (7.2) (7.2) (7.1)
2016 –303 20.5 –267 18.6 3.25 1.01 –0.17 –8.78 2002 1458 2.24 0.10
(90) (4.8) (90) (4.9) (6.3) (6.8) (6.7) (6.5)
2017 –145 12.0 –139 11.3 14.5 6.3 2.0 5.2 3450 26.76 1.74 0.17
(125) (6.6) (124) (6.7) (8.3) (8.8) (8.7) (8.7)
(2) Number of analytics staff
β0* β1* β2* SSE[r] SSE[f] F p
2014 –112 (81.8) 10.35 (4.41) 3.08 (2.07) 2302 2129 2.21 0.15
2015 –87 (105) 9.01 (5.60) 0.72 (2.42) 2901 2891 0.09 0.77
2016 –296 (96.9) 20.13 (5.15) –0.38 (1.67) 2002 1998 0.05 0.82
2017 –144 (0.26) 12.03 (6.69) –1.73 (2.42) 3450 3386 0.51 0.48
(3) Number of research staff
β0* β1* β2* SSE[r] SSE[f] F p
2014 –113 (88.3) 10.41 (4.74) 0.81 (1.24) 2302 2267 0.42 0.52
2015 –149 (94.7) 12.1 (5.06) 2.84 (1.09) 2901 2321 6.75 0.02
2016 –330 (91.4) 21.8 (4.85) 1.14 (0.89) 2002 1885 1.67 0.21
2017 –131 (117) 10.99 (6.24) 2.07 (0.95) 3450 2676 1.74 0.17
4.2.2Number of analytics staff
The reduced model remains unchanged. The full model becomes
is the number of analytics staff hired. The null hypothesis becomes
. The numerator and denominator degrees of freedom of the test statistic F are changed to 1 and 27, respectively. The
-values in the second part of
Table 7
are much greater than 0.05. Thus we conclude that the addition of the information of the number of analytics staff is not useful in the regression model for predicting the number of games won in a
regular season, once the team payroll is already in the model.
4.2.3Number of research staff
The reduced model remains unchanged. The variable X[2] in the full model shown in Section 4.2.2 is changed to the number of research staff. Nonetheless, the corresponding null hypothesis, the
numerator and denominator degrees of freedom of F remain intact. Inspecting the p-values in the third part of Table 7, we conclude (with α = 5%) that once the team payroll was already in the model,
the addition of the information of the number of research staff was not statistically significantly useful in the regression model for predicting the number of games won in a regular season for 2014,
2016 and 2017. However, we obtain the conclusion of statistical significance for 2015.
4.3Model utility F test
To test if both the team payroll and the use of sports analytics are good predictors of the success in playoffs towards the championship of World Series, we apply the model utility F test to the
combined postseasonal data of 2014-2017. Thus there are altogether forty data points for analysis, instead of ten data points in each year. The response variable Y is the levels of playoffs in the
postseason towards the championship of World Series: 5(Champion), 4(Third round game but lost), 3(Second round game but lost), 2(First round game but lost), 1(Wild card game but lost). For
simplicity, we assume that Y is a continuous variable. The four continuous explanatory variables are the team payroll (logarithmic value) X[1], categories of analytics belief X[2], number of
analytics staff X[3], and number of research staff X[4]. The variable X[2] has the values: 4(All-in), 3(Believers), 2(One-foot-in), 1(Skeptics), and 0(Non-believers); it is assumed to be a continuous
variable. The equation of the multiple linear regression model is
Minitab calculated the estimated values of the regression parameters β[i]’s with their standard error, observed F value, and p-value of the model for the combined data of 2014-2017. The results are
given in Table 8.
Fig. 4.
The model utility F test is to test the null hypothesis that β[1] = β[2] = β[3] = β[4] = 0 . As the p-value is 0.12, we conclude (with 5% level of significance) that the team payroll, categories of
analytics belief, number of analytics staff, and number of research staff are not good predictors in the multiple linear regression model for the success in playoffs. This conclusion agrees with
Schwartz and Zarrow (2009) that the success in October (i.e., playoffs) can be viewed as a truly random event.
Table 8
β[0] β[1] β[2] β[3] β[4] F p-value
–16.9 0.97 0.26 0.49 0.07 1.99 0.12
(12) (0.63) (0.20) (0.22) (0.12)
5Decision trees
The predictive modeling of decision trees will be used to classify MLB teams for their success in the regular season and postseason. There are 30 teams in MLB, and only 10 teams move on to playoffs
each year. The data collected in one year, however, are not adequate to come up with meaningful predictive models. Rather, we will combine four years’ data (2014-2017) to have 120 teams (or
instances) to build the predictive models.
5.1Wins as an input variable available
5.1.1Binary target variable
The target (or response) variable is the advancement to playoffs, which has the value of 1 if the team advances to playoffs and 0 if not. The input variables are catchers’ salary, infielders’ salary,
outfielders’ salary, pitchers’ salary, team payroll, and Wins (number of games won in a regular season). In addition, the input variables include the number of analytics staff, the number of research
staff as well as four indicator variables for the categories of analytics belief: All-in, Believers, One-foot-in, and Skeptics with Non-believers as the baseline. SAS Enterprise Miner workstation
14.2 with interactive mode was used to implement the algorithms to build the predictive model. The decision tree is given in Fig. 4.
Node 1 shows that the data are randomly selected into the training model (60 teams) and the validation model (60 teams). Among the 60 teams in both the training and validation models, 66.67% of them
were not in playoffs whereas 33.33% of them were.
The decision tree shows that Wins is the variable first chosen among all input variables mentioned above to split the tree. Since there are no missing observations in our data set, the corresponding
test condition is < 86.5 or ≥ 86.5 games. This means that the decision tree classifies teams in each of the training and validation models into 2 groups, one group in Node 3 with teams winning less
than 86.5 games (i.e., less than or equal to 86 games) in a regular season and the other group in Node 4 with teams winning at least 86.5 games (i.e., greater than or equal to 87 games) in a regular
season. The 60 teams in the training model are separated into 41 teams in Node 3 and 19 teams in Node 4. Likewise, the 60 teams in the validation model are separated into 40 teams in Node 3 and 20
teams in Node 4. In Node 3, 2.44% of the 41 teams in the training model and 2.50% of the 40 teams in the validation model advanced to playoffs. In Node 4, however, all 19 teams (100%) in the training
model and 95% of the 20 teams in the validation model advanced to playoffs. SAS then chose the input variable Wins again to split Node 3 with the new test condition: < 83.5 or ≥ 83.5 games. For those
teams winning less than 83.5 games in a regular season in Node 5, none of the 36 teams in the training model and none of the 33 teams in the validation model advanced to playoffs. For those teams
winning at least 83.5 games (but less than 86.5 games), there was only 1 team out of 5 teams (20%) in the training model and also only 1 team out of 7 teams (14.29%) in the validation model advanced
to playoffs.
Therefore, from the decision tree in Fig. 4, we may conclude that teams winning 87 games or more is very highly likely to advance to playoffs, teams winning between 84 and 86 games (inclusive) still
have a small chance of moving on to playoffs, and teams winning 83 games or fewer in a regular season have no chance of advancing to playoffs. As 162 games are played by each team in a regular
season, 87 wins translate to 0.537 winning percentage and 83 wins to 0.512 winning percentage.
The misclassification rates for the training model and validation model are 1.67% and 3.33%, respectively. In addition, the average squared errors are 0.0133 and 0.0313 for the training model and
validation model, respectively. As a result, we conclude that the test conditions (< 83.5 games and ≥ 86.5 games) of the input variable Wins are excellent criteria to classify MLB teams into no
playoffs or playoffs.
5.1.2Ordinal target variable
With all the input variables given in Section 5.1.1, the target variable is considered to be the levels of playoffs towards the championship of World Series. The target variable has 6 levels: 0(No
playoffs), 1(Wild card game but lost), 2(First round game but lost), 3(Second round game but lost), 4(Third round game but lost), and 5(Champion). It will be treated as an ordinal variable. The
outcome of a decision tree is displayed in Fig. 5.
In Node 1, all 120 teams are randomly divided approximately 50-50 into the training and validation models according to their stages towards the championship of World Series. In fact, 59 teams and 61
teams go to the training and validation models, respectively. The input variable Wins with the same test condition (< 86.5 or ≥ 86.5 games) is first chosen among all the given input variables to
split the decision tree. Among the 59 teams in the training model, 40 teams go to Node 3 as their wins are less than 86.5 games and 19 teams go to Node 4 as their wins are 86.5 games or more.
Likewise, among the 61 teams in the validation model, 41 teams go to Node 3 and 20 teams go to Node 4. Wins less than 86.5 games in a regular season is an excellent predictor to classify 100% of the
teams in training model and 95.12% of the teams in validation model as target 0 (i.e., no playoffs). There is only one team (2.44%) in the validation model having target 1 (i.e., wildcard game but
lost), and also only one team having target 2 (i.e., first round game but lost).
Node 4 shows that none of the 19 teams in the training model and only 1 team out of 20 teams (5%) in the validation model has target 0(no playoffs). It means that 38 of 39 teams (97.4%) with 86.5
wins or more in a regular season go to playoffs. As four teams out of the ten playoffs teams go to the first round game but lost, target 2 in Node 4 has the highest percentage for both models, 42.11%
of teams in the training model and 35% of teams in the validation model. Apart from this observation, however, both training and validation models do not display any distinct patterns on different
stages towards the championship of World Series.
Fig. 5.
Fig. 6.
The input variable All_in is then chosen to further split Node 4. In Node 5 (when teams are not All_in), it looks like more teams have lower target values (1 and 2) in the training model and teams
spread evenly with slightly higher target values in the validation model. In Node 6 (when teams are All_in), however, it seems that teams spread more widely across the targets with three spikes at
targets 1, 3, 5 in the training model, and teams have lower targets (1 and 2) in the validation model.
Afterwards, the input variable pitchers’ salary with the test condition (< 55,764,782 or ≥ 55,764,782) is used to split Node 5 further. For not All_in teams winning at least 87 games in a regular
season and pitchers’ salary at least $55.7 million (approximately), Node 8 shows that they have higher targets (2-5) in the validation model and concentrate on target 2 in the training model. Node 7,
however, shows that these not All_in teams with pitchers’ salary less than $55.7 million (approximately) spread more evenly across the targets in both the validation and training models.
The misclassification rates for the training and validation models are 11.86% and 31.15%, respectively. The average squared errors for the training and validation models are 0.0232 and 0.0752,
respectively. This decision tree yields moderately accurate predictions for classifying MLB teams into different stages of playoffs towards the championship of World Series.
5.2Wins as an input variable not available
At the beginning of a MLB season in late March, the information of the number of games won in a regular season is certainly not available. In this situation, one may still wish to build a decision
tree to classify teams into no playoffs (target 0) or playoffs (target 1). As the target variable is binary, we can use the same procedure as shown in Section 5.1.1 but without Wins as an input
variable. The decision tree is presented in Fig. 6.
Node 1 shows that the data are randomly selected into the training model (60 teams) and validation model (60 teams). The first input variable chosen is pitchers’ salary with the test condition (<
29,766,607 or ≥ 29,766,607). Under this test condition, the 60 teams in each of the training and validation models are separated into 23 teams in Node 3 and 37 teams in Node 4 in their respective
model. In Node 3 (pitchers’ salary < ≈ $29.7 million), none of the 23 teams in the training model and 17.39% of the 23 teams in the validation model advanced to playoffs. In Node 4 (pitchers’ salary
≥ ≈ $29.7 million), however, 54.05% of the 37 teams in the training model and 43.24% of the 37 teams in the validation model advanced to playoffs. Consequently, pitchers’ salary with a threshold of
approximately $29.7 million played an important role to identify whether or not the team had a higher chance of moving on to playoffs.
Under Node 4, further classification takes place. The second input variable chosen is All_in with the test condition: 1(yes) or 0(no). Under this test condition, the 37 teams in the training model
are separated into 11 teams in Node 5 and 26 teams in Node 6. Likewise, the 37 teams in the validation model are separated into 7 teams in Node 5 and 30 teams in Node 6. For teams that were All_in,
Node 5 shows that 72.73% of the 11 teams in the training model and 71.43% of the 7 teams in the validation model advanced to playoffs. However, Node 6 shows that 46.15% of the 26 teams in the
training model and 36.67% of the 30 teams in the validation model advanced to playoffs. In addition to spending at least $29.7 million on pitchers’ salary, the All_in teams would significantly
increase their chances of advancing to playoffs. For teams that were not All_in but were Believers instead, Node 7 indicates that their chances of advancing to playoffs were 66.67% and 61.54% in the
training and validation models, respectively. These percentages were much higher than those (35.29% and 17.65%) in Node 8 for teams that were not Believers either.
The misclassification rates for the training model and the validation model are 20% and 23.33%, respectively. The average squared errors produce 0.1344 and 0.1923 for the training and validation
models, respectively. This decision tree yields moderately accurate predictions for classifying MLB teams into no playoffs or playoffs.
6Summary and concluding comments
The relationship between the use of sports analytics and the success in regular season and postseason in MLB has been studied through the empirical data of 2014-2017. The use of sports analytics can
be examined by (1) the categories of analytics belief given by ESPN in The Great Analytics Rankings, (2) the number of analytics staff worked in the analytics department, and (3) the total number of
research staff (including analytics staff) employed by a team. Several good indicators are identified to be useful for predicting the success in the regular season. They are Wins (number of games won
in a regular season), categories of analytics belief, and team payroll (in particular, the pitchers’ salary).
Fig. 4 illustrates that for a MLB team winning 87 games or more in a regular season, the chance of the team advancing to playoffs is at least 95%. On the contrary, for a MLB team winning 83 games or
fewer in a regular season, it has no chance of getting into playoffs. Teams that win between 84 and 86 games (inclusive) in a regular season have about 14-20% chances of moving on to playoffs.
Winning 87 games out of 162 games in a regular season translates to the winning percentage of 0.537. This winning percentage can be regarded as an important goal that teams would like to achieve in a
regular season.
From Fig. 1, we can see that on average 77.5% of the playoffs teams came from the category of BELIEVERS (All-in, Believers) and only 22.5% came from the category of NON-BELIEVERS (One-foot-in,
Skeptics, Non-believers). Moreover, we obtain the results that about 48% of the BELIEVERS’ teams and about 16% of the NON-BELIEVERS’ teams moved on to playoffs during 2014-2017. These outcomes may
encourage MLB teams which are currently in the level of One-foot-in, Skeptics or Non-believers to re-consider their engagement with the sports analytics.
From Fig. 2, we obtain the results that about 37%, 22%, and 27% of teams with the number of analytics staff= 0, 1, and 2 or more, respectively, advanced to playoffs during 2014-2017. In addition,
we obtain from Fig. 3 that about 33%, 55%, 31%, and 23% of teams with the number of research staff=High(7-5), Medium(4-3), Low(2-1), and None(0), respectively, advanced to playoffs during these
four years. These results might suggest that analytics staff alone may not be able to show the positive effect on the teams. Therefore, other research staff such as the ones working in the areas of
mathematical modeling, data science, informatics, research and development, etc. should also be employed to complement the work of analytics staff.
Total team payroll has long been identified as an essential component for the success of a team in the regular season. From the decision tree in Fig. 6, we can see that teams with pitchers’ salary at
least $29.7 million would have about 72% and 64% chances of advancing to playoffs if their categories of analytics belief are All_in and Believers, respectively. This area with the payroll amount
might provide management teams with some guidelines on where and how much they need to allocate their financial resources to players for a successful regular season.
So far there haven’t been any good predictors found for the success in the postseason. It seems that once teams have advanced to the postseason, the playoffs teams start with a clean slate. Perhaps
the factors of the excitement of playoffs, more media coverage, and high expectations from baseball fans might transform the playoffs teams to different levels of intensity and eagerness to compete.
However, it is very difficult to quantify these factors.
The limitation of this study is that it involves only four years’ empirical data. Teams that are rebuilding might not maximize wins in a season or two. The improvement of teams’ performance and
optimizing the wins might take place in the medium to long term. Consequently, more data are necessary for further study of the long-term effect of the sports analytics on the success of teams in
regular season and postseason. In order to control for teams that are rebuilding in a season or are competing for the playoffs, one might consider the preseason projected wins from some projection
systems such as PECOTA, Steamer, and ZiPS. If a team’s belief in analytics has a positive impact on the team, then it should outperform its projected wins. This notion is worth pursuing in the
future. The changes of MLB teams’ categories of analytics belief, such as changing from One-foot-in to Believers, should be made known and updated in The Great Analytics Rankings. This updated
information would be crucial for future data analysis.
The authors are grateful to a referee for many valuable suggestions.
1 Baseball-reference.com, 2014-2017. ‘MLB Standings ’. URL: https://www.baseball-reference.com/leagues/MLB/2014-standings.shtml (Substitute 2015, 2016, and 2017 for 2014 to get the corresponding
2 Espn.com, 2015. ‘The Great Analytics Rankings’. URL: https://www.espn.com/espn/feature/story/_/id/12331388/the-great-analytics-rankings
3 Kutner, M ., Nachtsheim, C ., Neter, J ., Li, W. , (2005) . Applied Linear Statistical Models. 5th ed. McGrawHill, pp. 271.
Sbnation.com, 2014, 2015, 2016, 2017. ‘MLB Playoffs’. URLs: https://www.sbnation.com/mlb/2014/9/29/6859873/2014-mlb-Playoffs-schedule-bracket-format https://www.sbnation.com/mlb/2015/10/6/9460931/
4 2015-mlb-Playoffs-schedule-postseason-bracket-results-teams https://www.sbnation.com/mlb/2016/10/4/13100224/2016-mlb-Playoffs-schedule-postseason-bracket-results-teams https://www.sbnation.com/mlb
5 Schwartz, N ., Zarrow, J. , (2009) . An analysis of the impact of team payroll on regular season and postseason success in Major League Baseball, Undergraduate Economic Review 5: (1), Article 3.
URL: http://digitalcommons.iwu.edu/uer/vol5/iss1/3
6 Spotrac.com, 2014-2017. ‘MLB Team Payroll’. URL: https://www.spotrac.com/mlb/payroll/2014/ (Substitute 2015, 2016, and 2017 for 2014 to get the corresponding URL.)
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How can I create a functions that returns functions in Haskell?
In Haskell, you can create functions that return other functions as their result. To do this, you simply specify the type of the returned function in the type signature of the outer function.
Here's an example:
1 addFunction :: Int -> (Int -> Int)
2 addFunction x y = x + y
In this example, addFunction takes an Int value x as an argument and returns a function (Int -> Int) that takes an Int value y and returns their sum x + y.
To use this function, you can call addFunction with an Int argument and then immediately apply the resulting function to another Int argument:
1 addFive = addFunction 5
2 result = addFive 3
In this example, addFive is assigned the result of calling addFunction 5, which is the function that takes an Int value and returns its sum with 5. When we call addFive 3, it returns 8, which is the
sum of 5 and 3.
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Eviatar B. Procaccia, Sarai Hernandez-Torres: The chemical distance in random interlacements in the low-intensity regime
Abstract: In this talk, I will present a new proof of the sharpness of the phase transition for Random interlacements is a Poissonian soup of doubly-infinite random walk trajectories on
We consider the time constant u ↓ 0. In this high-dimensional case, we prove a sharp upper bound (of order
Based on
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Mathematics contributes to the school curriculum by developing students’ abilities to solve problems, to calculate, to reason logically, algebraically, and geometrically and to make sense of data.
Mathematics is important for students in many other areas of study, particularly Science and Technology. It is also important in everyday life, in many forms of employment and in decision-making.
As a subject in its own right, Mathematics presents frequent opportunities for creativity. It can stimulate moments of wonder; especially when problems are solved or when more elegant solutions to
problems are discovered. Mathematics is one means of making knowledge useful. Mathematics enables students to build a secure framework of mathematical reasoning, which they can use and apply with
confidence. The power of mathematical reasoning lies in its use of precise and concise forms of language, symbolism and representation to reveal and explore general relationships.
Our curriculum map is sequenced with fewer topics each week, term or year, putting depth before breadth. We find that spending longer on each topic enables pupils to really think and talk about the
mathematics they are learning. We sequence concepts and methods so that previously learnt ideas can be connected to new learning, supporting students in understanding the coherent and connected
nature of the subject, and ensuring they consolidate learning by continually using and applying it in a variety of contexts. We believe that all of mathematics can be appreciated more fully once a
student has a deep appreciation of the number system, therefore we put number sense and place value first to ensure that all understanding builds. We use the Mastery system in Maths to further
develop the number system.
Maths is taught six lessons per fortnight in Years 7 and 8. Both years are taught in mixed ability groups.
Year 7
│Term 1 │Term 2 │Term 3 │
│ │Angles │Primes, factors and multiples│
│Number systems and the axioms │ │ │
│ │Classifying 2-D shapes │Fractions │
│Order of operations │ │ │
│ │Coordinates │Ratio │
│Positive and negative numbers │ │ │
│ │Area of 2-D shapes │Percentages │
│Expressions Equations Inequalities│ │ │
│ │Transforming 2-D figures and Construction │Project work │
Year 8
│Term 1 │Term 2 │Term 3 │
│LCM, HCF and Prime Factorisation │ │ │
│ │ │Sequences │
│Standard form │ │ │
│ │Fractions, Decimals and Percentages│Probability │
│Powers, roots │ │ │
│ │Ratio and Proportion │Transformations: Reflection, Translation, Rotation and Enlargement │
│Area of 2D Shapes │ │ │
│ │Expressions │Probability │
│Circumference of circles │ │ │
│ │Equations │Averages │
│Properties of 3D shapes │ │ │
│ │Graphs y = mx+c │Angles │
│Volume Cubes and Cuboids │ │ │
│ │ │Project work │
│Surface area of Cubes and Cuboids│ │ │
How can parents support?
• Access Google Classroom for up to date homework information.
• For revision materials access videos and resources on Corbett Maths, Hegarty Maths and Just Maths
• Be positive! Children pick up on signals from adults who display a negative attitude towards the subject. This can have a significant impact on the way they view the subject and can impact their
success in it.
• Try to provide a quiet, well-lit place for your child to study, away from TV, video games, mobile phone
• Have your child explain what he or she is doing in maths and try to develop a conversation with your child around maths
• If we have contacted home to offer extra support or revision sessions for your child, please support us and encourage them to attend the activities
• Ensure your son/ daughter has the correct equipment including a scientific calculator, pen, pencil, ruler, a pair of compasses and protractor. All of these can be purchased from school.
• Some students prefer using a textbook or revision guide to work through. Revision guides will be available through school at certain points in the year. If you wish to purchase a textbook to use
at home, please ensure you buy one for the new curriculum.
Course Maths
Examining Board EDEXCEL
Specification Link
Mathematics is a creative and highly inter-connected discipline that has been developed over centuries, providing the solution to some of history’s most intriguing problems. It is
essential to everyday life, critical to science, technology and engineering, and necessary for financial literacy and most forms of employment.
Why study this
subject? A high-quality mathematics education therefore provides a foundation for understanding the world, the ability to reason mathematically, an appreciation of the beauty and power of
mathematics, and a sense of enjoyment and curiosity about the subject.
There are six topic strands in Maths GCSE:
Strands Ratio, Proportion and Rates of Change
Geometry and Measures
The qualification consists of three equally-weighted written examination papers at either Foundation tier or Higher tier. The main topic areas are:
• Number
• Algebra
• Ratio, proportion and rates of change
• Geometry and measures
• Probability
• Statistics
The school will decide on the tier of entry for each student at the end of Year 9 based on assessments and teacher knowledge.
We regularly assess KS4 pupils at the end of each reporting window. Pupils will sit at least two exam papers that are both calculator and non-calculator. Throughout the academic
year in Years 9, and 10, Year 11 pupils will sit ‘Pre Public Examinations’, known as PPEs. This will include pupils sitting examinations in the main hall, replicating real exam
Pupils are given weekly homework that is either exam related, extending classroom learning or consolidating classwork via Hegarty Maths.
Students can progress from this qualification to Level 3 qualifications (A Level) in numerate disciplines, such as:
• Core Mathematics
• A level Mathematics and A Level Further Mathematics
• Sciences
Next steps - • Geography
Careers/HE • Psychology
courses • Economics
• other qualifications that require mathematical skills, knowledge and understanding.
This course provides a strong foundation for further academic and vocational study and for employment. The lin beow shows the numerous careers with maths
PIXL Maths App:
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Course A level Mathematics
Examining Board Edexcel
Specification Link
• Develop key employability skills
• problem-solving, communication, logical reasoning and resilience
• Leads to versatile qualifications, well-respected by employers and higher education
Why study this • Increase knowledge and understanding of mathematical techniques and their applications​
subject? • Excellent preparation for a wide range of university courses
• Support the study of other A levels
• Stimulating and challenging courses
Pure Mathematics
Key Content Statistics
Unit 1 & Unit 2 Pure Maths
Unit 3 Mechanics and Statistics
The AS is assessed using 2 exam papers.
All papers carry equal weighting.
Assessment AS-
Paper 1: Pure Maths,
Paper 2: Statistics and Mechanics.
Paper 1: Pure Mathematics 1
Assessment A-Level Paper 2- Pure Mathematics 2
Paper 3- Mechanics and Statistics
Next steps - Medical, Games Design, Internet Security, Financial, Cryptography, Programming, Communications, Aircraft Modelling, Fluid Flows, Acoustic Software Development, Electronics, Civil
Careers/HE courses Engineering, Quantum Physics, Astronomy, Forensics, DNA sequencing, Data Science, Psychology, Law, Economics, Climate Change Environmental Modelling
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Course A level Further Mathematics
Examining Board Edexcel
Specification Link
• Understand mathematics and mathematical systems in ways that promote confidence, foster enjoyment, and provide a deep foundation for progress to further study.
• Extend your range of mathematical skills and techniques.
• Understand coherence and advancement in mathematics and how different concepts of mathematics are related.
Why study this • Apply mathematics in other fields of study and be informed of the relevance of mathematics to the world of work and to situations in society in general.
subject? • Use mathematical knowledge to construct mathematical models to make coherent and reasoned decisions in solving problems both in a variety of contexts and communicate the
mathematical rationale for these decisions clearly and conclude on solution strategy.
• Leads to versatile qualifications, well-respected by employers and higher education.
• Gain transferable skills: cognitive, creativity, decision making, reasoning, critical thinking, ICT literacy, collaborative, relationship-building skill, problem solving,
interpersonal and intrapersonal skill, self management and development.
Core Pure Mathematics 1 & 2: Proof, Complexnumbers, Matrices, Further algebra and functions, Further calculus, Further vectors, Polar coordinates, Hyperbolic functions, Differential
Key Content Decision Mathematics 1: Algorithms and graph theory, Algorithms on graphs, Critical path analysis, Linear programming.
Further Statistics 1: Discrete probability distributions, Poisson & binomial distributions, Geometric and negative binomial distributions, Hypothesis Testing, Central Limit Theorem,
Chi Squared Tests, Probability generating functions, Quality of tests
Core Pure Mathematics 1
Unit 1 & Unit 2
Core Pure Mathematics 2
Further Statistics 1
Unit 3
Decision Mathematics 1
The AS is assessed using 2 exam papers.
Paper 1: Core Pure Mathematics (*Paper code: 8FM0/01)
Assessment Paper 2: Further Statistics 1 and Decision Mathematics 1 (*Paper codes: 8FM0/2F)
Each paper is: 1 hour and 40 minutes written examination
50% of the qualification - 80 marks.
Paper 1: Core Pure Mathematics 1 (*Paper code: 9FM0/01) Paper 2: Core Pure Mathematics 2 (*Paper code: 9FM0/02)
Paper 3B: Further Statistics 1 (*Paper codes: 9FM0/3B)
Assessment Paper 3D :Decision Mathematics 1 (*Paper codes: 9FM0/3D)
Each paper is: 1 hour and 30 minutes written examination
25% of the qualification 75 marks
Next steps - Actuarial Science, Aeronautical Engineering, Astronomy, Biochemistry, Biomedical Sciences (including Medical Science), Chemical Engineering, Chemistry, Civil Engineering, Computer
Careers/HE Science, Dentistry, Electrical/Electronic Engineering, Engineering (General), Law – facilitating subjects at A-level are useful when applying for Law, Materials Science (including
courses Biomedical Materials Science), Mathematics, Mechanical Engineering, Medicine, Optometry (Ophthalmic Optics), Physics, Quantum Physics, Veterinary Science
Dr Frost Maths
Suggested links Physics and Maths Tutor
to resources
Advanced Maths Support Program (AMSP)
Crash Maths
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Augustin Bariant
Symmetric cryptography expert at ANSSI
WHO AM I
Former student at École Polytechnique, I completed a double degree with KTH Royal Institute of Technology in Stockholm in 2021. In march 2021, I started a Ph.D in symmetric cryptography in the team
of Inria de Paris, under the supervision of
Gaëtan Leurent
. There, I mainly worked on boomerang attacks against AES-based ciphers and algebraic attacks against arithmetization-oriented primitives. I also codesigned LeMac, which is the fastest MAC in the
literature on modern processors to this day. I defended my PhD on the 27/06/2024.
I am currently working as an expert in the cryptography laboratory of ANSSI, and am still performing research. I am mainly interested in other types of attacks on SPN ciphers, e.g. impossible
differential or square attacks, and I am still actively involved in the algebraic cryptanalysis of arithmetization-oriented primitives. I also enjoy designing and breaking crypto challenges in CTF
competitions. You can find my resumé in french
• Title: Analysis of AES-based and arithmetization-oriented symmetric cryptography primitives.
• Defense: On the 27/06/2024 at École Normale Supérieure Paris.
• Tutor in mathematics for first-year students (MPSI) in the preparatory school Lycée Louis Le Grand.
• Teaching assistant (TP) for the first year class LU1IN011 at Sorbonne Université (SU): Introduction to Programming in Python.
• Teaching assistant (TP) for the second year class LU2IN019 at SU: Functional Programming.
• Teaching assistant (TD+TP) for the third year class LU3IN024 at SU: Introduction to Cryptology.
• Teaching assistant (TD+TP) for the third year class LU2IN017 at SU: Web Technologies.
• Teaching assistant (TP) for the first year class LU1IN011 at SU: Introduction to Programming in Python.
• Teaching assistant (TP) for the second year class LU2IN019 at SU: Functional Programming.
1. A. Bariant, N. David, G. Leurent, Cryptanalysis of Forkciphers, IACR Transactions on Symmetric Cryptology (ToSC) 2020, volume 1.
2. A. Bariant, C. Bouvier, G. Leurent, L. Perrin, Algebraic Attacks against Some Arithmetization-Oriented Primitives, IACR Transactions on Symmetric Cryptology (ToSC) 2022, volume 3.
3. A. Bariant, G. Leurent, Truncated Boomerang Attacks and Application to AES-based Ciphers, EUROCRYPT 2023.
4. A. Bariant, A. Boeuf, A. Lemoine, I. Manterola Ayala, M. Øygarden, L. Perrin, H. Raddum, The Algebraic Freelunch: Efficient Gröbner Basis Attacks Against Arithmetization-Oriented Primitives,
CRYPTO 2024.
5. A. Bariant, J. Baudrin, G. Leurent, C. Pernot, L. Perrin, T. Peyrin, Fast AES-Based Universal Hash Functions and MACs: Featuring LeMac and PetitMac, IACR Transactions on Symmetric Cryptology
(ToSC) 2024, volume 2.
6. A. Bariant, A Univariate Attack on a Full Ciminion Instance, Selected Areas in Cryptography (SAC) 2024.
7. A. Bariant, O. Dunkelman, N. Keller, G. Leurent, V. Mollimard, Improved Boomerang Attacks on 6-Round AES, EPRINT 2024.
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Part 1: Implementing Linear Regression in C. | by Zephania Reuben | Jun, 2024 - Artificial Intelligence Article
When exploring machine learning, followers usually encounter fairly a number of sources centered on high-level languages resembling Python. Nonetheless, for these intrigued by the effectivity and
vitality of C, the journey takes a specific path. C, known as a middle-level language, strikes a steadiness between the simplicity of high-level programming and the robust memory manipulation
capabilities of low-level languages. This distinctive combination makes C an exquisite different for implementing machine learning algorithms, providing every precision and administration over the
computational processes.
However, smart implementations of machine learning and deep learning algorithms in C may be elusive. On this text, we embark on an thrilling exploration of a simple linear regression model to predict
scholar effectivity based mostly totally on study hours using C, strolling through the strategy step-by-step. This hands-on methodology will not solely deepen your understanding however moreover
showcase the flexibleness and robustness of C throughout the realm of machine learning.
1. Basic understanding of Programming in C language.
2. Basic understanding of Algebra.
3. Basic understanding of Calculus (i.e derivatives)
Understanding Linear Regression
Linear regression fashions the connection between two variables, the place one (dependent variable) is predicted from the other (neutral variable) through a linear equation. In our case, the equation
is represented as y = wx + b, the place:
• y is the anticipated effectivity,
• x is the number of study hours,
• w is the burden (slope of the highway),
• b is the bias (y-intercept).
Step-by-Step Implementation
1. Initialization and Info Definition:
static double X[] = {2, 4, 6, 8}; // Unbiased variable (study hours)
static double y[] = {20, 40, 60, 80}; // Dependent variable (effectivity marks)
static double weight = 0;
static double bias = 0;
Proper right here, X and y characterize our teaching information. weight and bias are parameters initialized to zero, which the model might be taught all through teaching.
2.Prediction Carry out:
static double* predict(double inputs[], int measurement, double weight, double bias) {
double* y_predicted = (double*)malloc(measurement * sizeof(double));
// Calculate predictions y = wx + b
for (int i = 0; i < measurement; i++) {
y_predicted[i] = inputs[i] * weight + bias;
return y_predicted;
This carry out computes predicted values (y_predicted) based mostly totally on current weight and bias.
3. Worth Carry out:
The payment carry out measures the error between the anticipated values and the exact values. On this case, we use the Suggest Squared Error (MSE).
Mathematical expression for value Carry out
The place:
• J(w,b) is the related payment.
• m is the number of information components.
• y^i is the anticipated price for the i-th information degree.
• yi is the exact price for the i-th information degree.
static double value(double inputs[], double labels[], int measurement, double weight, double bias) {
double loss_value = 0;
double sum_loss = 0;
double* y_predicted = predict(inputs, measurement, weight, bias);
for (int i = 0; i < measurement; i++) {
loss_value = (labels[i] - y_predicted[i]) * (labels[i] - y_predicted[i]);
sum_loss += loss_value;
return sum_loss / (2 * measurement);
Proper right here, value carry out computes the indicate squared error, which quantifies the excellence between exact labels and predicted y_predicted.
4. Gradient Capabilities for Optimization:
Gradients of Weights
The gradient of the burden is the partial spinoff of the related payment carry out with respect to the burden. It signifies how rather a lot the burden must be adjusted to chop again the related
Mathematical expression for gradients of weights
C Code Implementation
static double weight_grad(double inputs[], double labels[], int measurement) {
double grad = 0;
double* y_predicted = predict(inputs, measurement, weight, bias);
for (int i = 0; i < measurement; i++) {
grad += (y_predicted[i] - labels[i]) * inputs[i];
return grad / measurement;
Gradients of Bias
The gradient of the bias is the partial spinoff of the related payment carry out with respect to the bias. It signifies how rather a lot the bias must be adjusted to chop again the related payment.
Mathematical expression for gradients of bias
C Code Implementation
static double bias_grad(double inputs[], double labels[], int measurement) {
double grad = 0;
double* y_predicted = predict(inputs, measurement, weight, bias);
for (int i = 0; i < measurement; i++) {
grad += (y_predicted[i] - labels[i]);
return grad / measurement;
These options compute gradients (derivatives) of the related payment carry out with respect to weight and bias, important for updating these parameters all through teaching.
5. Teaching and Testing:
All through teaching, we change the burden and bias iteratively using the gradients. The coaching value controls the size of the steps we absorb course of minimizing the related payment.
Weight change:
Mathematical expression for weight change
Bias change:
Mathematical expression for bias change
The place:
• α is the coaching value.
• ∂J(w,b)/∂w is the gradient of the related payment carry out J(w,b) with respect to the burden w.
• ∂J(w,b)/∂b is the gradient of the related payment carry out J(w,b) with respect to the bias b.
C Code Implementation
void test() {
int measurement;
printf("Let's test our linear modeln");
printf("Enter the size of your information (Number of information components):n");
scanf("%d", &measurement);
double inputs[size];
for (int i = 0; i < measurement; i++) {
printf("Enter number of hour(s) for information : %d n", i + 1);
scanf("%lf", &inputs[i]);
double* predictions = predict(inputs, measurement, weight, bias);
printf("Prediction for inputsnn");
for (int i = 0; i < measurement; i++) {
printf("%lf hrs : %lf marks(performances)n", inputs[i], predictions[i]);
int most essential(void) {
int epoch = 100000;
double learning_rate = 0.0001;
int measurement = sizeof(X) / sizeof(X[0]);
double loss = 0;
double grad_w = 0;
double grad_b = 0;
for (int i = 1; i <= epoch; i++) {
loss = value(X, y, measurement, weight, bias);
grad_w = weight_grad(X, y, measurement);
grad_b = bias_grad(X, y, measurement);
weight = weight - learning_rate * grad_w;
bias = bias - learning_rate * grad_b;
printf("Epoch %d ---- Loss: %lf n", i, loss);
printf("Weight: %lf, Bias: %lf, Grad_W: %lf, Grad_B: %lfn", weight, bias, grad_w, grad_b);
printf("Model Loss: %lf n", loss);
printf("Optimum Weight: %lf n", weight);
printf("Optimum Bias: %lf n", bias);
This most essential carry out initializes parameters, trains the model using gradient descent, and evaluates its effectivity using value carry out, throughout the subsequent half we’re going to
uncover completely different evaluation metrics. The test carry out is used to test how the model is able to appropriately predict the effectivity from the hours equipped by particular person.
Implementing machine learning algorithms in C provides a deeper understanding of their inside workings and computational effectivity. This occasion demonstrates the equipment of linear regression,
from information illustration to parameter optimization, in C. By leveraging low-level languages, followers can enhance their programming experience and grasp the foundations of machine learning
algorithms additional comprehensively.
By mastering such implementations, builders can leverage the robustness of low-level languages to type out superior machine learning challenges efficiently.
github: https://github.com/nsomazr/ml-in-c/blob/main/linear-regression/example_linear_model.c
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Interpretation: How we interpret our assessment results
Because the simulation is a well-specified situation, preservice teacher performance can be scored using checklists keyed to the decomposition of the practice, as well as to if/how the preservice
teacher takes up specific things that the student does or says. Each item in the checklist includes criteria for proficient performance and when possible the elements of the checklist are organized
to correspond with the likely sequence of the work. Using checklists allows for appraisal as the simulation unfolds which supports focused observation of the performance and enables efficient
What is assessed: Simulation
• Initial elicitation (e.g., asks the student what he or she did or thought about when solving the problem)
• Follow-up questions
□ Elicits specific steps of the student’s process (e.g., where the 8 comes from)
□ Probes the student’s understanding of the process and understanding mathematical ideas (e.g., meaning of the “.3”)
□ Attends to the student’s ideas (e.g., attends to and takes up specific ideas that the student talks about)
• Tone and manner (e.g., prompts the student fluently)
• Use of mathematical language in ways that are accessible, accurate, and precise
What is assessed: Interview
• Explanation the elicited student thinking (e.g., process and understanding)
Generation of a mathematically sound follow-up problem to confirm the student’s process
• Anticipation of the student’s response to a given follow-up problem based on evidence from the interaction with the student
• Anticipation of the student’s understanding of particular parts of the process on the follow-up problem using evidence from the interaction
• Explanation of the generalizability of the methods
• Use of mathematical language in ways that are accessible, accurate, and precise
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Microwave Integrated Circuits
Microwave Integrated Circuits. Instructor: Prof. Jayanta Mukherjee, Department of Electrical Engineering, IIT Bombay. This course is designed to introduce the field of Microwave Engineering to
students, engineers and academics. Since at microwave frequencies, the distributed circuit effects become very prominent, new circuit theories based on Maxwell's laws have to be introduced. Further
new circuit design techniques as well as new circuit elements are also introduced. The first part of the course deals with the basics of theory. In the later part, the design of various microwave
devices like couplers, circulators, filters and amplifiers is introduced. (from nptel.ac.in)
Microwave Integrated Circuits
Instructor: Prof. Jayanta Mukherjee, Department of Electrical Engineering, IIT Bombay. This course is designed to introduce the field of Microwave Engineering.
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Understand advanced mathematics? | Since 1989
What is it like to understand advanced mathematics?
• You can answer many seemingly difficult questions quickly. But you are not very impressed by what can look like magic, because you know the trick. The trick is that your brain can quickly decide
if a question is answerable by one of a few powerful general purpose “machines” (e.g., continuity arguments, the correspondences between geometric and algebraic objects, linear algebra, ways to
reduce the infinite to the finite through various forms of compactness) combined with specific facts you have learned about your area. The number of fundamental ideas and techniques that people
use to solve problems is, perhaps surprisingly, pretty small — see http://www.tricki.org/tricki/map for a partial list, maintained by Timothy Gowers.
• You are often confident that something is true long before you have an airtight proof for it (this happens especially often in geometry).The main reason is that you have a large catalogue of
connections between concepts, and you can quickly intuit that if X were to be false, that would create tensions with other things you know to be true, so you are inclined to believe X is probably
true to maintain the harmony of the conceptual space. It’s not so much that you can imagine the situation perfectly, but you can quickly imagine many other things that are logically connected to
• You are comfortable with feeling like you have no deep understanding of the problem you are studying. Indeed, when you do have a deep understanding, you have solved the problem and it is time to
do something else. This makes the total time you spend in life reveling in your mastery of something quite brief. One of the main skills of research scientists of any type is knowing how to work
comfortably and productively in a state of confusion. More on this in the next few bullets.
• Your intuitive thinking about a problem is productive and usefully structured, wasting little time on being aimlessly puzzled. For example, when answering a question about a high-dimensional
space (e.g., whether a certain kind of rotation of a five-dimensional object has a “fixed point” which does not move during the rotation), you do not spend much time straining to visualize those
things that do not have obvious analogues in two and three dimensions. (Violating this principle is a huge source of frustration for beginning maths students who don’t know that they shouldn’t be
straining to visualize things for which they don’t seem to have the visualizing machinery.) Instead…
• When trying to understand a new thing, you automatically focus on very simple examples that are easy to think about, and then you leverage intuition about the examples into more impressive
insights. For example, you might imagine two- and three-dimensional rotations that are analogous to the one you really care about, and think about whether they clearly do or don’t have the
desired property. Then you think about what was important to the examples and try to distill those ideas into symbols. Often, you see that the key idea in the symbolic manipulations doesn’t
depend on anything about two or three dimensions, and you know how to answer your hard question.As you get more mathematically advanced, the examples you consider easy are actually complex
insights built up from many easier examples; the “simple case” you think about now took you two years to become comfortable with. But at any given stage, you do not strain to obtain a magical
illumination about something intractable; you work to reduce it to the things that feel friendly.
• To me, the biggest misconception that non-mathematicians have about how mathematicians work is that there is some mysterious mental faculty that is used to crack a research problem all at once.
It’s true that sometimes you can solve a problem by pattern-matching, where you see the standard tool that will work; the first bullet above is about that phenomenon. This is nice, but not
fundamentally more impressive than other confluences of memory and intuition that occur in normal life, as when you remember a trick to use for hanging a picture frame or notice that you once saw
a painting of the street you’re now looking at.In any case, by the time a problem gets to be a research problem, it’s almost guaranteed that simple pattern matching won’t finish it. So in one’s
professional work, the process is piecemeal: you think a few moves ahead, trying out possible attacks from your arsenal on simple examples relating to the problem, trying to establish partial
results, or looking to make analogies with other ideas you understand. This is the same way that you solve difficult problems in your first real maths courses in university and in competitions.
What happens as you get more advanced is simply that the arsenal grows larger, the thinking gets somewhat faster due to practice, and you have more examples to try. Sometimes, during this
process, a sudden insight comes, but it would not be possible without the painstaking groundwork [http://terrytao.wordpress.com/ca… ].Indeed, most of the bullet points here summarize feelings
familiar to many serious students of mathematics who are in the middle of their undergraduate careers; as you learn more mathematics, these experiences apply to “bigger” things but have the same
fundamental flavor.
• You go up in abstraction, “higher and higher”. The main object of study yesterday becomes just an example or a tiny part of what you are considering today. For example, in calculus classes you
think about functions or curves. In functional analysis or algebraic geometry, you think of spaces whose points are functions or curves — that is, you “zoom out” so that every function is just a
point in a space, surrounded by many other “nearby” functions. Using this kind of zooming out technique, you can say very complex things in short sentences — things that, if unpacked and said at
the zoomed-in level, would take up pages. Abstracting and compressing in this way makes it possible to consider extremely complicated issues with one’s limited memory and processing power.
• The particularly “abstract” or “technical” parts of many other subjects seem quite accessible because they boil down to maths you already know. You generally feel confident about your ability to
learn most quantitative ideas and techniques. A theoretical physicist friend likes to say, only partly in jest, that there should be books titled “______ for Mathematicians”, where _____ is
something generally believed to be difficult (quantum chemistry, general relativity, securities pricing, formal epistemology). Those books would be short and pithy, because many key concepts in
those subjects are ones that mathematicians are well equipped to understand. Often, those parts can be explained more briefly and elegantly than they usually are if the explanation can assume a
knowledge of maths and a facility with abstraction.Learning the domain-specific elements of a different field can still be hard — for instance, physical intuition and economic intuition seem to
rely on tricks of the brain that are not learned through mathematical training alone. But the quantitative and logical techniques you sharpen as a mathematician allow you to take many shortcuts
that make learning other fields easier, as long as you are willing to be humble and modify those mathematical habits that are not useful in the new field.
• You move easily among multiple seemingly very different ways of representing a problem. For example, most problems and concepts have more algebraic representations (closer in spirit to an
algorithm) and more geometric ones (closer in spirit to a picture). You go back and forth between them naturally, using whichever one is more helpful at the moment.Indeed, some of the most
powerful ideas in mathematics (e.g., duality,Galois theory, algebraic geometry) provide “dictionaries” for moving between “worlds” in ways that, ex ante, are very surprising. For example, Galois
theory allows us to use our understanding of symmetries of shapes (e.g., rigid motions of an octagon) to understand why you can solve any fourth-degree polynomial equation in closed form, but not
any fifth-degree polynomial equation. Once you know these threads between different parts of the universe, you can use them like wormholes to extricate yourself from a place where you would
otherwise be stuck. The next two bullets expand on this.
• Spoiled by the power of your best tools, you tend to shy away from messy calculations or long, case-by-case arguments unless they are absolutely unavoidable. Mathematicians develop a powerful
attachment to elegance and depth, which are in tension with, if not directly opposed to, mechanical calculation. Mathematicians will often spend days figuring out why a result follows easily from
some very deep and general pattern that is already well-understood, rather than from a string of calculations. Indeed, you tend to choose problems motivated by how likely it is that there will be
some “clean” insight in them, as opposed to a detailed but ultimately unenlightening proof by exhaustively enumerating a bunch of possibilities. (Nevertheless, detailed calculation of an example
is often a crucial part of beginning to see what is really going on in a problem; and, depending on the field,some calculation often plays an essential role even in the best proof of a result.)In
A Mathematician’s Apology [http://www.math.ualberta.ca/~mss…, the most poetic book I know on what it is “like” to be a mathematician], G.H. Hardy wrote:”In both [these example] theorems (and in
the theorems, of course, I include the proofs) there is a very high degree of unexpectedness, combined with inevitability and economy. The arguments take so odd and surprising a form; the
weapons used seem so childishly simple when compared with the far-reaching results; but there is no escape from the conclusions. There are no complications of detail—one line of attack is enough
in each case; and this is true too of the proofs of many much more difficult theorems, the full appreciation of which demands quite a high degree of technical proficiency. We do not want many
‘variations’ in the proof of a mathematical theorem: ‘enumeration of cases’, indeed, is one of the duller forms of mathematical argument. A mathematical proof should resemble a simple and
clear-cut constellation, not a scattered cluster in the Milky Way.”
“[A solution to a difficult chess problem] is quite genuine mathematics, and has its merits; but it is just that ‘proof by enumeration of cases’ (and of cases which do not, at bottom, differ at
all profoundly) which a real mathematician tends to despise.”
• You develop a strong aesthetic preference for powerful and general ideas that connect hundreds of difficult questions, as opposed to resolutions of particular puzzles. Mathematicians don’t really
care about “the answer” to any particular question; even the most sought-after theorems, like Fermat’s Last Theorem, are only tantalizing because their difficulty tells us that we have to develop
very good tools and understand very new things to have a shot at proving them. It is what we get in the process, and not the answer per se, that is the valuable thing. The accomplishment a
mathematician seeks is finding a new dictionary or wormhole between different parts of the conceptual universe. As a result, many mathematicians do not focus on deriving the practical or
computational implications of their studies (which can be a drawback of the hyper-abstract approach!); instead, they simply want to find the most powerful and general connections. Timothy Gowers
has some interesting comments on this issue, and disagreements within the mathematical community about it [http://www.dpmms.cam.ac.uk/~wtg1… ].
• Understanding something abstract or proving that something is true becomes a task a lot like building something. You think: “First I will lay this foundation, then I will build this framework
using these familiar pieces, but leave the walls to fill in later, then I will test the beams…” All these steps have mathematical analogues, and structuring things in a modular way allows you to
spend several days thinking about something you do not understand without feeling lost or frustrated. (I should say, “without feeling unbearably lost and frustrated”; some amount of these
feelings is inevitable, but the key is to reduce them to a tolearable degree.)Andrew Wiles, who proved Fermat’s Last Theorem, used an “exploring” metaphor:
“Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark unexplored mansion. You enter the first room of the mansion and it’s completely dark. You
stumble around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it’s
all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they’re momentary,
sometimes over a period of a day or two, they are the culmination of—and couldn’t exist without—the many months of stumbling around in the dark that proceed them.” [http://www.pbs.org/wgbh/nova/
phy… ]
• In listening to a seminar or while reading a paper, you don’t get stuck as much as you used to in youth because you are good at modularizing a conceptual space, taking certain calculations or
arguments you don’t understand as “black boxes”, and considering their implications anyway. You can sometimes make statements you know are true and have good intuition for, without understanding
all the details. You can often detect where the delicate or interesting part of something is based on only a very high-level explanation. (I first saw these phenomena highlighted by Ravi Vakil,
who offers insightful advice on being a mathematics student: http://math.stanford.edu/~vakil/… .)
• You are good at generating your own definitions and your own questions in thinking about some new kind of abstraction.
One of the things one learns fairly late in a typical mathematical education (often only at the stage of starting to do research) is how to make good, useful definitions. Something I’ve reliably
heard from people who know parts of mathematics well but never went on to be professional mathematicians (i.e., write articles about new mathematics for a living) is that they were good at
proving difficult propositions that were stated in a textbook exercise, but would be lost if presented with a mathematical structure and asked to find and prove some interesting facts about it.
Concretely, the ability to do this amounts to being good at making definitions and, using the newly defined concepts, formulating precise results that other mathematicians find intriguing or
enlightening.This kind of challenge is like being given a world and asked to find events in it that come together to form a good detective story. You have to figure out who the characters should
be (the concepts and objects you define) and what the interesting mystery might be. To do these things, you use analogies with other detective stories (mathematical theories) that you know and a
taste for what is surprising or deep. How this process works is perhaps the most difficult aspect of mathematical work to describe precisely but also the thing that I would guess is the strongest
thing that mathematicians have in common.
• You are easily annoyed by imprecision in talking about the quantitative or logical. This is mostly because you are trained to quickly think about counterexamples that make an imprecise claim seem
obviously false.
• On the other hand, you are very comfortable with intentional imprecision or “hand-waving” in areas you know, because you know how to fill in the details. Terence Tao is very eloquent about this
here [ http://terrytao.wordpress.com/ca… ]:”[After learning to think rigorously, comes the] ‘post-rigorous’ stage, in which one has grown comfortable with all the rigorous foundations of one’s
chosen field, and is now ready to revisit and refine one’s pre-rigorous intuition on the subject, but this time with the intuition solidly buttressed by rigorous theory. (For instance, in this
stage one would be able to quickly and accurately perform computations in vector calculus by using analogies with scalar calculus, or informal and semi-rigorous use of infinitesimals, big-O
notation, and so forth, and be able to convert all such calculations into a rigorous argument whenever required.) The emphasis is now on applications, intuition, and the ‘big picture’. This stage
usually occupies the late graduate years and beyond.”In particular, an idea that took hours to understand correctly the first time (“for any arbitrarily small epsilon I can find a small delta so
that this statement is true”) becomes such a basic element of your later thinking that you don’t give it conscious thought.
• Before wrapping up, it is worth mentioning that mathematicians are not immune to the limitations faced by most others. They are not typically intellectual superheroes. For instance, they often
become resistant to new ideas and uncomfortable with ways of thinking (even about mathematics) that are not their own. They can be defensive about intellectual turf, dismissive of others, or
petty in their disputes. Above, I have tried to summarize how the mathematical way of thinking feels and works at its best, without focusing on personality flaws of mathematicians or on the
politics of various mathematical fields. These issues are worthy of their own long answers!
• You are humble about your knowledge because you are aware of how weak maths is, and you are comfortable with the fact that you can say nothing intelligent about most problems. There are only very
few mathematical questions to which we have reasonably insightful answers. There are even fewer questions, obviously, to which any givenmathematician can give a good answer. After two or three
years of a standard university curriculum, a good maths undergraduate can effortlessly write down hundreds of mathematical questions to which the very best mathematicians could not venture even a
tentative answer. (The theoretical computer scientist Richard Lipton lists some examples of potentially “deep” ignorance here:http://rjlipton.wordpress.com/20…) This makes it more comfortable to
be stumped by most problems; a sense that you know roughly what questions are tractable and which are currently far beyond our abilities is humbling, but also frees you from being very
intimidated, because you do know you are familiar with the most powerful apparatus we have for dealing with these kinds of problems.
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Giving to the Department of Mathematics
The generous support of our alumni and friends gives us a significant edge. When you give to the Department of Mathematics, you allow us to: enhance our efforts to attract top students and faculty,
provide exceptional opportunities for our graduate and undergraduate students, expand outstanding teaching and public outreach programs.
Graduate Education
• Support our graduate students’ research projects, their travel to research conferences, and expenses associated with the organization of math-related events.
Undergraduate Education
• Support research projects of math undergraduate students, their travel to research conferences, scholarships, and expenses associated with the organization of math-related events.
Actuarial Science Fund
• For the Advancement of the Actuarial Science Program, as Directed by the Head of the Actuarial Program in the Department of Mathematics.
David Goss Technology & Academic Innovation Stimulus Fund
• Upon David Goss's passing, his wife, Rita Eppler-Goss, established a memorial fund in his honor: the David Goss Technology and Academic Innovation Stimulus Fund. Its purpose is to support
academics and researchers in the Department of Mathematics with their entrepreneurial academic endeavors.
Math Department Activities: Chair’s Discretionary Fund
• Supports a vast array of departmental activities and initiatives, including math-related social events, such as receptions for the Radó and Zassenhaus lectures and the Young Mathematicians
Conference, graduate-student summer stipends, and recruiting talented math majors.
Outreach-Diversity: BAMM Fund
● Our outreach group, BAMM (Buckeye Aha! Math Moments), started in 2018 by Erika Roldan, then a PhD student at Ohio State. This fund supports outreach programming of BAMM to increase the public
awareness, appreciation, and enjoyment of mathematics, including in underrepresented populations
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NCERT Class 6 Maths Chapter 3 Exercise 3.11 Number Play PDF
NCERT Solutions for Class 6 Maths Number Play Chapter 3 Exercise 3.11 - FREE PDF Download
NCERT Solutions for Class 6 Maths Chapter 3, Exercise 3.11, titled "Simple Estimation," simplifies the process of understanding estimation and number patterns. This exercise introduces students to
methods of rounding numbers and estimating sums, differences, and products. The step-by-step solutions provided make it easier for students to grasp the concepts of estimation, helping them apply
these techniques effectively in everyday situations. These solutions enhance students’ problem-solving skills by offering clear and concise explanations for each problem. The NCERT Solutions for
Class 6 Maths ensures that students develop a solid foundation in maths through easy-to-understand methods.
1. NCERT Solutions for Class 6 Maths Number Play Chapter 3 Exercise 3.11 - FREE PDF Download
2. Glance on NCERT Solutions Maths Chapter 3 Exercise 3.11 Class 6 | Vedantu
3. Access NCERT Solutions for Maths Class 6 Chapter 3 - Number Play Exercise 3.11
4. Benefits of NCERT Solutions for Class 6 Maths Chapter 3 Exercise 3.11 Number Play
5. Class 6 Maths Chapter 3: Exercises Breakdown
6. Important Study Material Links for Maths Chapter 3 Class 6
8. Chapter-wise NCERT Solutions Class 6 Maths
9. Related Important Links for Class 6 Maths
Download the FREE PDF of NCERT Solutions for Class 6 Maths Chapter 3, Exercise 3.11, crafted by Vedantu’s expert teachers, and prepare thoroughly in line with the CBSE Class 6 Maths syllabus.
Glance on NCERT Solutions Maths Chapter 3 Exercise 3.11 Class 6 | Vedantu
• In NCERT Solutions for Class 6 Maths Chapter 3, Exercise 3.11 - Simple Estimation, students are introduced to the concept of estimation and rounding numbers for easier calculations.
• This exercise focuses on understanding how to approximate sums, differences, and products, making calculations faster and more efficient.
• The solutions offer a variety of examples and practice problems, helping students develop the skill of estimating accurately in different scenarios.
• By working through these solutions, students gain a strong understanding of estimation techniques, which are essential for solving real-world maths problems and preparing them for more advanced
mathematical concepts.
FAQs on NCERT Solutions for Class 6 Maths Chapter 3 - Number Play Exercise 3.11
1. What does Exercise 3.11 in Chapter 3 of Class 6 Maths cover?
Exercise 3.11 in Chapter 3 of Class 6 Maths focuses on Simple Estimation. It teaches students how to estimate numbers by rounding them off to the nearest ten, hundred, or a thousand. This exercise
helps in understanding how estimation can be used to simplify complex calculations in real-life scenarios, such as approximating sums, differences, and products. The goal is to introduce students to
the practical application of estimation techniques, making it easier to handle larger numbers quickly and efficiently.
2. How do NCERT solutions help in solving Exercise 3.11?
The solutions provide step-by-step guidance to help students understand and solve the problems accurately.
3. Are the NCERT solutions for Chapter 3 - Number Play suitable for self-study?
Yes, the solutions are designed in a simple and easy-to-understand format, making them ideal for self-study.
4. Do NCERT solutions for Exercise 3.11 follow the CBSE syllabus?
Yes, all the solutions are based on the NCERT textbook and strictly follow the CBSE curriculum.
5. How can NCERT solutions for Number Play Exercise 3.11 improve my exam preparation for Class 6 Maths?
By providing accurate and detailed solutions, these resources help students practice well and prepare effectively for exams.
6. Can NCERT Solutions for Class 6 Maths Chapter 3 - Number Play Exercise 3.11 help me solve similar number pattern problems?
Yes, the solutions explain the methods clearly, enabling students to apply the same techniques to similar problems.
7. Where can I find NCERT solutions for Class 6 Maths Chapter 3 Exercise 3.11?
Students can download the free PDF of NCERT Solutions for Chapter 3, Exercise 3.1 from the Vedantu Website. These solutions also offer helpful tips and methods to simplify complex concepts, ensuring
that students build a strong foundation in mathematics.
8. What are the main concepts taught in Chapter 3 - Number Play?
Chapter 3 teaches number patterns, sequences, and how to recognize and extend these patterns.
9. Are there any shortcuts provided in the NCERT solutions for solving Exercise 3.11?
The solutions explain standard methods step by step but focus on ensuring understanding rather than shortcuts.
10. How do the NCERT Solutions for Class 6 Maths Chapter 3 - Number Play Exercise 3.11 help in improving logical thinking?
By working through the simple estimation in Exercise 3.11, students enhance their logical reasoning and analytical skills.
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Indexed annuity example
The folks at Fidelity crunched some numbers to show how performance limiting indexed annuities can be. They point out, for example, that in 2013, the S&P 500 surged by 32% (including dividends A
Fixed Index Annuity is a tax-favored accumulation product issued by an insurance company. It shares features with fixed deferred interest rate annuities; however, with an index annuity, the annual
growth is bench-marked to a stock market index (e.g., Nasdaq, NYSE, S&P500) rather than an interest rate.
We illustrate the methodology with examples of equity annuity contracts with opposite sensitivities to vega risk. Discover the world's 18 May 2017 Yes. The problem with this type of crediting has to
do with the volatility of the S&P from month to month. Even in a year where the S&P is up a 600 at time 4 (index 150). So in total the claim amounts to 1380. Without a Stability Clause the accounts
of this example are very simple: - at time 2 the reinsurer An indexed annuity (sometimes referred to as a fixed indexed annuity or FIA) is a Some use monthly index movements while others calculate
an average Indexed annuities track a market index like the S&P 500 and calculate earnings based on one of three models: annual reset, high water mark, or point-to-point. For Male, female and joint
index annuity rates. Best indexed annuity rates. Calculate an index annuity rate using our calculator.
2 Nov 2016 “With indexed annuities and indexed universal life insurance, the Here is an example: You deposit $100,000 into an indexed annuity that will
4 May 2016 The rules that govern the performance credited to an indexed annuity account tend to be relatively simple and intuitive. A hypothetical example 15 Aug 2018 At his Scottsdale office,
Vazirani pulls out sample ads for fixed-indexed annuities. One features a chart showing how a $100,000 investment 7 Dec 2009 The equity-indexed annuity (EIA) was introduced in 1995 and are many ways
insurance companies calculate your index-linked returns. 18 Mar 2015 How Fixed Annuities Calculate Returns And Their Real (Interest Rate Spread) Cost. When a premium contribution is made to fixed
annuity, the 2 Nov 2016 “With indexed annuities and indexed universal life insurance, the Here is an example: You deposit $100,000 into an indexed annuity that will 1 Nov 2016 So for example, if an
underlying index returns 10% in a given period, but the annuity has a cap rate of 7%, the annuity holder will receive a
The folks at Fidelity crunched some numbers to show how performance limiting indexed annuities can be. They point out, for example, that in 2013, the S&P 500 surged by 32% (including dividends
1 Nov 2016 So for example, if an underlying index returns 10% in a given period, but the annuity has a cap rate of 7%, the annuity holder will receive a Indexed Annuity Historical Performance
Example. This historical indexed annuity example illustrates a monthly averaging strategy using the S&P 500 as the tracking index. The green line is the annuity performance and the red line shows the
returns of the S&P 500, excluding dividends. For example, a Company XYZ indexed annuity might pay the investor 85% of the annual increase in the S&P 500, guaranteeing a minimum of 3% per year and a
maximum of 9%. So if the index is up 10% in a year, the annuity will pay 85% of this, or 8.5%. But if the index is up 25%, the annuity will pay only the maximum 9% that year. An indexed annuity is a
type of variable annuity contract that delivers cash flows to the annuitant based on the return of a stock index, usually the S&P 500. Indexed annuities give people the opportunity to enhance their
annuity income, but fees and caps may limit the potential upside actually returned. Fixed-Indexed Annuities — A Hypothetical Example . The following example will illustrate how the various methods of
computing the amount credited to a contract might operate. Assumptions: Initial annuity Investment: $50,000 Date of annuity Investment: May 30th, Year 0. Market Index, May 30th, Year 0: 1,422; Market
Index, May 30th, Year 1: 1,600 An indexed annuity (a.k.a. fixed indexed annuity or FIA) is a tax-deferred retirement savings vehicle that provides the guarantee of a fixed return plus the potential
for a higher variable return based on market performance. The structure of a FIA is based on that of a simple fixed annuity,
The folks at Fidelity crunched some numbers to show how performance limiting indexed annuities can be. They point out, for example, that in 2013, the S&P 500 surged by 32% (including dividends
Here are the following example that illustrate how the various methods of computing the amount credited to a contract might operate. Indexed Rate Annuity Calculator. Many indexed annuities credit
interest annually based upon the performance of an index, limited to an annual cap rate. 17 Feb 2020 A Fixed Index Annuity is a tax-favored accumulation product issued by an For example, if the
participation rate is 25% and the stock market
For example, if the participation rate is 80 percent and the index gained 10 percent, the annuity would be credited with 80 percent of the 10-percent gain, or 8 percent. Spread/Margin/Asset Fee Some
index annuities use this in place of or in addition to a participation rate.
15 Aug 2018 At his Scottsdale office, Vazirani pulls out sample ads for fixed-indexed annuities. One features a chart showing how a $100,000 investment 7 Dec 2009 The equity-indexed annuity (EIA)
was introduced in 1995 and are many ways insurance companies calculate your index-linked returns. 18 Mar 2015 How Fixed Annuities Calculate Returns And Their Real (Interest Rate Spread) Cost. When a
premium contribution is made to fixed annuity, the 2 Nov 2016 “With indexed annuities and indexed universal life insurance, the Here is an example: You deposit $100,000 into an indexed annuity that
2 Nov 2016 “With indexed annuities and indexed universal life insurance, the Here is an example: You deposit $100,000 into an indexed annuity that will 1 Nov 2016 So for example, if an underlying
index returns 10% in a given period, but the annuity has a cap rate of 7%, the annuity holder will receive a Indexed Annuity Historical Performance Example. This historical indexed annuity example
illustrates a monthly averaging strategy using the S&P 500 as the tracking index. The green line is the annuity performance and the red line shows the returns of the S&P 500, excluding dividends. For
example, a Company XYZ indexed annuity might pay the investor 85% of the annual increase in the S&P 500, guaranteeing a minimum of 3% per year and a maximum of 9%. So if the index is up 10% in a
year, the annuity will pay 85% of this, or 8.5%. But if the index is up 25%, the annuity will pay only the maximum 9% that year. An indexed annuity is a type of variable annuity contract that
delivers cash flows to the annuitant based on the return of a stock index, usually the S&P 500. Indexed annuities give people the opportunity to enhance their annuity income, but fees and caps may
limit the potential upside actually returned. Fixed-Indexed Annuities — A Hypothetical Example . The following example will illustrate how the various methods of computing the amount credited to a
contract might operate. Assumptions: Initial annuity Investment: $50,000 Date of annuity Investment: May 30th, Year 0. Market Index, May 30th, Year 0: 1,422; Market Index, May 30th, Year 1: 1,600 An
indexed annuity (a.k.a. fixed indexed annuity or FIA) is a tax-deferred retirement savings vehicle that provides the guarantee of a fixed return plus the potential for a higher variable return based
on market performance. The structure of a FIA is based on that of a simple fixed annuity,
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Data in 2D and 3D
Edit and select Data from a grid and draw a 2D or 3D Graph. You may view 3D data in the 3D viewer and make Contour plots and Cross-sections through Contour plots. Some statistical properties of the
data may be calculated. You may also draw a histogram or the (cumulative, normalized) distribution function of selected data. You may apply a Chi-square test or a Kolmogorov-Smirnov test to compare
two distributions.
Chi-square test 1 and 2
The Chi-square test is used to test differences between binned distributions. The selected Y-values in grid 1 and grid 2 are compared. In the first test the Y-values in grid 1 are assumed to
represent a theoretical distribution. Chi-squared is defined as
where Ni is the number in the i-th bin and ni is the number expected according to some known distribution. Both distributions should contain the same total number of events. The probability (0<P<1)
that the numbers Ni are drawn from the expected distribution is calculated. It is an incomplete Gamma function of Chi-squared. In the second test two measured data sets (in grid 1 and grid 2) are
compared. Chi-squared is then defined as
where Ni and Mi are the numbers in the i-th bin of the distributions calculated in grid 1 and grid 2. The total numbers should be the same. The probability P that the two distributions are drawn from
the same underlying distribution is calculated. A small value of P indicates that the two distributions are probably different.
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