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What Are Options Greeks In The Stock Market - Wealth Diagram
What are options Greeks in the stock market
What are options Greeks in the stock market. Decoding the Options Market: A Beginner’s Guide to the Greeks. Have you ever heard seasoned investors talk about “the Greeks” in hushed tones, but felt
completely lost by the cryptic reference?
What are options Greeks in the stock market
Fear not, intrepid explorer of the financial world, for today we’ll demystify these seemingly mythical creatures and reveal their true nature: valuable tools for navigating the options market.
Chapter 1: Options Demystified – A World Beyond Buying and Selling
Imagine you own a rare baseball card. You believe its value will skyrocket in the future, but you’re not quite ready to sell it just yet. What options do you have? Enter the world of stock options,
which grant you the right, but not the obligation, to buy (call option) or sell (put option) a specific stock at a predetermined price (strike price) by a certain date (expiration date).
Takeaway: Stock options offer investors flexibility beyond simply buying or selling a stock.
Chapter 2: Unveiling the Greeks – A Team of Helpful Metrics
So, where do the Greeks come in? They’re not mythical creatures, but rather a set of five key metrics used to understand how various factors influence the price of an option. These metrics, named
after the first five letters of the Greek alphabet (alpha is rarely used), are:
• Delta (Δ): Measures how much the option’s price changes in relation to a $1 change in the underlying stock price.
• Gamma (Γ): Indicates how much delta itself changes with each $1 movement in the stock price.
• Theta (Θ): Represents the rate at which an option loses value solely due to the passage of time, also known as time decay.
• Vega (ν): Measures the sensitivity of an option’s price to changes in implied volatility, which is the market’s expectation of how much the stock price will fluctuate in the future.
• Rho (ρ): (Less commonly used) measures the impact of changes in interest rates on the price of an option.
Takeaway: The Greeks are a group of metrics that help us understand how different factors affect the price of an option.
Chapter 3: Decoding Delta – The “Direction Discerner”
Think of delta as your directional compass. It tells you whether the option’s price will generally move in the same direction (call options) or opposite direction (put options) as the underlying
stock price.
For example, if a call option has a delta of 0.7, it means for every $1 increase in the stock price, the option’s price is likely to increase by around $0.70. Conversely, a put option with a delta of
-0.5 suggests that for every $1 increase in the stock price, the put option’s price is likely to decrease by around $0.50.
Takeaway: Delta helps you understand how the option’s price will move in relation to the stock price.
Chapter 4: Gamma – The “Change Catalyst”
Imagine delta as the speed at which the option price changes, and gamma as the acceleration of that change. A high gamma indicates that the delta (and therefore the option price) will change more
rapidly with each movement in the stock price. This can be helpful for options strategies that aim to profit from larger price movements.
Takeaway: Gamma tells you how quickly the delta, and therefore the option price, will change with respect to the stock price.
Chapter 5: Theta – The “Time Thief”
Theta is the unforgiving force working against options holders. It represents the time decay that options experience as they approach their expiration date. The closer an option gets to expiration,
the faster its value decays, regardless of the stock price. This is why options are often referred to as “wasting assets.”
Takeaway: Theta reminds us that options lose value over time, even if the stock price stays the same.
Chapter 6: Vega – The “Volatility Variable”
Vega reflects the sensitivity of an option’s price to changes in implied volatility. When implied volatility rises, the option price generally becomes more expensive, and vice versa. This is because
higher volatility suggests a greater chance of larger price movements, making options potentially more valuable.
Takeaway: Vega tells us how much the option price will change based on fluctuations in implied volatility.
Chapter 7: Putting it All Together – A Powerful Toolkit
While each Greek provides valuable insights, it’s crucial to consider them together for a comprehensive understanding of an option’s behavior. Analyzing the interplay of these metrics allows options
traders to make informed decisions about:
• Managing risk: The Greeks help you quantify the risk associated with an options position. Understanding your exposure to factors like price movements, time decay, and volatility fluctuations
facilitates setting stop-loss levels and hedging appropriately.
• Identifying opportunities: By monitoring the Greeks, investors can spot potentially profitable options trades based on anticipated stock price movements or changes in implied volatility.
Takeaway: The Greeks in combination form a powerful toolkit for options traders to analyze risk, choose appropriate strategies, and uncover profitable trading opportunities.
Conclusion: Mastering the Greeks – Your Voyage Begins
Understanding options Greeks is like learning a new language. It may seem daunting at first, but with practice and dedication, you’ll soon grasp the nuances and unlock the potential of these
essential metrics. Remember, this is just the beginning of your journey towards mastering the complex and intricate world of options trading.
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Fibonacci trick - Tim Wescott
Fibonacci trick
I'm working on a video, tying the Fibonacci sequence into the general subject of difference equations.
Here's a fun trick: take any two consecutive numbers in the Fibonacci sequence, say 34 and 55. Now negate one and use them as the seed for the Fibonacci sequence, larger magnitude first, i.e.
$-55, 34, \cdots$
Carry it out, and you'll eventually get the Fibonacci sequence, or it's negative:
$-55, 34, -21, 13, -8, 5, -3, 2, -1, 1, 0, 1, 1 \cdots$
This is NOT a general property of difference equations -- it's part of the "magic" of this particular sequence.
[ - ]
Comment by ●October 16, 2016
Very interesting. Can you provide a proper mathematical proof of this? Has it already been done? A spreadsheet on this is easy to prepare.
[ - ]
Comment by ●January 5, 2017
The simplest proof would be this:
If A + B = C, then it must satisfy C - B = A :)
If you multiply by -1 the left side of the series looking from 0, you'll see that it's nothing else but traversing back the Fibonacci series. It must come to the first element eventually.
[ - ]
Comment by ●January 5, 2017
Thanks for that. I've kind of set this video aside for the moment in favor of more grungy practical stuff. So you've delivered the proof that I'm too lazy to provide...
Just out of curiosity, though, for a more general difference equation, can you generalize the proof so that it still uses simple arithmetic, yet gets the right answer? E.g., for a difference
equation of the form \( x_n = a_1 x_{n-1} + a_2 x_{n-2} \) can you come up with something so simple?
[ - ]
Comment by ●October 16, 2016
Proof will be in the video. Basically, if you solve the difference equation that describes the sequence you get \( f \left ( k \right ) = A_1 d_1^k + A_2 d_2^k \), where \(d_1 = -\frac{1}{d_2} \)
and \( A_1 = -A_2 \). With a bit more work you can show that it Must be True.
To post reply to a comment, click on the 'reply' button attached to each comment. To post a new comment (not a reply to a comment) check out the 'Write a Comment' tab at the top of the comments.
Please login (on the right) if you already have an account on this platform.
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American Mathematical Society
Nonfibering spherical $3$-orbifolds
HTML articles powered by AMS MathViewer
Trans. Amer. Math. Soc. 341 (1994), 121-142
DOI: https://doi.org/10.1090/S0002-9947-1994-1118824-6
PDF | Request permission
Among the finite subgroups of $SO(4)$, members of exactly $21$ conjugacy classes act on ${S^3}$ preserving no fibration of ${S^3}$ by circles. We identify the corresponding spherical $3$-orbifolds,
i.e., for each such ${\mathbf {G}} < SO(4)$, we describe the embedded trivalent graph $\{ x \in {S^3}:\exists {\mathbf {I}} \ne {\mathbf {g}} \in {\mathbf {G}}$ s.t. ${\mathbf {g}}(x) = x\} /{\mathbf
{G}}$ in the topological space ${S^3}/{\mathbf {G}}$ (which turns out to be homeomorphic to ${S^3}$ in all cases). Explicit fundamental domains (of Dirichlet type) are described for $9$ of the
groups, together with the identifications to be made on the boundary. The remaining $12$ spherical orbifolds are obtained as mirror images or (branched) covers of these. References
—, Fibered orbifolds and crystallographic groups, Ph.D. thesis, Princeton Univ., 1981. E. Goursat, Sur les substitutions orthogonales et les divisions régulières de l’espace, Ann. Sci. École
Norm. Sup. (3) 6 (1889), 2-102. A. Hatcher, Bianchi orbifolds of small discriminant, an informal report (preprint). W. Thurston, The geometry and topology of $3$-manifolds, Lecture Notes, 1978.
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Bibliographic Information
• © Copyright 1994 American Mathematical Society
• Journal: Trans. Amer. Math. Soc. 341 (1994), 121-142
• MSC: Primary 57M50; Secondary 57S25
• DOI: https://doi.org/10.1090/S0002-9947-1994-1118824-6
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2010-2011 NHL Thread
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This site may earn a commission from merchant affiliate links, including eBay, Amazon, and others.
Well, we're on the brink of another NHL season. I'm already fatigued with salary cap talk, fantasy pool talk, etc... Meh! Let's get to the games!
Once again it looks like many are picking San Jose to break through this year and at least make the final. Same old story. Same with the Caps...Chicago was a wise pick last year, but who'd a thunk
that Philly would rebound in the playoffs and only be two wins away from the Cup?
Out of the Canadian teams, the consensus is that Vancouver has a good shot at the Cup this year...Calgary made some bewildering moves. (preseason record aside, let's see how they do during the
season...) Edmonton should still suck, but the kids are alright. They have a LOT to look forward to. Montreal should've kept Halak, but it will be fun to watch Price sink or swim. My Sens? The
usual...deep lineup, nothing decent in goal. Hoping for a miracle there, or at least some consistency...
Toronto sucks no matter what they do...
Sep 29, 2005
Jan 12, 2007
Mar 30, 2006
Feb 27, 2010
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How do you differentiate f(x)=ln(3x^2) using the chain rule? | Socratic
How do you differentiate #f(x)=ln(3x^2)# using the chain rule?
1 Answer
Step by step explanation and working is given below.
$f \left(x\right) = \ln \left(3 {x}^{2}\right)$
For chain rule, first break the problem into smaller links and find their derivatives. The final answer would be the product of all the derivatives in the link.
$y = \ln \left(u\right)$
$u = 3 {x}^{2}$
The differentiation using chain rule would be
$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{du}} \cdot \frac{\mathrm{du}}{\mathrm{dx}}$
$y = \ln \left(u\right)$
Differentiate with respect to $u$
$\frac{\mathrm{dy}}{\mathrm{du}} = \frac{1}{u}$
$u = 3 {x}^{2}$
Differentiate with respect to $x$
$\frac{\mathrm{du}}{\mathrm{dx}} = 3 \frac{d \left({x}^{2}\right)}{\mathrm{dx}}$
$\frac{\mathrm{du}}{\mathrm{dx}} = 3 \cdot 2 x$
$\frac{\mathrm{du}}{\mathrm{dx}} = 6 x$
$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{u} \cdot 6 x$
$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{6 x}{u}$
$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{6 x}{3 {x}^{2}}$
$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{2}{x}$Final answer
Note: The above process might look big, this was given in such a form to help you understand step by step working. With practice, you can do it quickly and with less steps.
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Statistical Model Checking
Statistical Model Checking (SMC) is a formal verification technique that combines simulation and statistical methods for the analysis of stochastic systems. It represents a powerful scalable
alternative to numerical probabilistic model checking techniques that suffer from state space explosion issues.
Given a formal model of the stochastic system S and a formal specification of the requirement to verify
• Quantitative : What is the probability that S satisfies
• Qualitative : Is the probability for S to satisfy
The quantitative question can be answered by using probability estimation techniques based on Chernoff-Hoeffding bound. It mainly consists of computing an estimation p’ of p (the actual probability
for S to satisfy [1] [2].
Regarding the qualitative question, it is mainly answered using a hypothesis testing approach. Assuming p is the probability for S to satisfy is correct, given a confidence level (). One important
work proposed in this context can be found in [3]. It relies on the Wald Probability ratio Test [4]
We implemented statistical model checking algorithm in the SBIP tool that supports different stochastic models, i.e. discrte-time Markov chains (DTMCs), continous-time Markov chains (CTMCs), and
generalized semi-Markov processes (GSMPs). All these models can described in a high-level component-based modeling formalism BIP. For properties specification, the tool supports bounded LTL and MTL | {"url":"https://www-verimag.imag.fr/Statistical-Model-Checking-814.html?lang=fr","timestamp":"2024-11-08T19:27:28Z","content_type":"text/html","content_length":"19693","record_id":"<urn:uuid:8fb43d1f-dafa-4813-8777-4973cb2ac51d>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00165.warc.gz"} |
IB Math Tutor
We believe finding a good match is key. Start by telling us a bit about yourself, like your goals and any topics you find tricky. Depending upon whether you need help with IB Math – SL or IB Math –
HL, we’ll match you with a tutor who’s not just an expert in that course, but also someone who fits your learning style. You can even have a free initial session to see if it’s a good fit before you | {"url":"https://tutoringmaphy.com/ib-math-tutor/","timestamp":"2024-11-05T13:38:23Z","content_type":"text/html","content_length":"145263","record_id":"<urn:uuid:e58b3bdc-2ddc-43e6-ba07-ca18deabb2fa>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00161.warc.gz"} |
Finite State Machines - Electrical Engineering Textbooks
Finite State Machines
Up to now, every circuit that was presented was a combinatorial circuit. That means that its output is dependent only by its current inputs. Previous inputs for that type of circuits have no effect
on the output.
However, there are many applications where there is a need for our circuits to have "memory"; to remember previous inputs and calculate their outputs according to them. A circuit whose output depends
not only on the present input but also on the history of the input is called a sequential circuit.
In this section we will learn how to design and build such sequential circuits. In order to see how this procedure works, we will use an example, on which we will study our topic.
So let's suppose we have a digital quiz game that works on a clock and reads an input from a manual button. However, we want the switch to transmit only one HIGH pulse to the circuit. If we hook the
button directly on the game circuit it will transmit HIGH for as few clock cycles as our finger can achieve. On a common clock frequency our finger can never be fast enough.
The desing procedure has specific steps that must be followed in order to get the work done:
Step 1
The first step of the design procedure is to define with simple but clear words what we want our circuit to do:
"Our mission is to design a secondary circuit that will transmit a HIGH pulse with duration of only one cycle when the manual button is pressed, and won't transmit another pulse until the button is
depressed and pressed again."
Step 2
The next step is to design a State Diagram. This is a diagram that is made from circles and arrows and describes visually the operation of our circuit. In mathematic terms, this diagram that
describes the operation of our sequential circuit is a Finite State Machine.
Make a note that this is a Moore Finite State Machine. Its output is a function of only its current state, not its input. That is in contrast with the Mealy Finite State Machine, where input affects
the output. In this tutorial, only the Moore Finite State Machine will be examined.
The State Diagram of our circuit is the following: (Figure below)
A State Diagram
Every circle represents a "state", a well-defined condition that our machine can be found at.
In the upper half of the circle we describe that condition. The description helps us remember what our circuit is supposed to do at that condition.
• The first circle is the "stand-by" condition. This is where our circuit starts from and where it waits for another button press.
• The second circle is the condition where the button has just been just pressed and our circuit needs to transmit a HIGH pulse.
• The third circle is the condition where our circuit waits for the button to be released before it returns to the "stand-by" condition.
In the lower part of the circle is the output of our circuit. If we want our circuit to transmit a HIGH on a specific state, we put a 1 on that state. Otherwise we put a 0.
Every arrow represents a "transition" from one state to another. A transition happens once every clock cycle. Depending on the current Input, we may go to a different state each time. Notice the
number in the middle of every arrow. This is the current Input.
For example, when we are in the "Initial-Stand by" state and we "read" a 1, the diagram tells us that we have to go to the "Activate Pulse" state. If we read a 0 we must stay on the "Initial-Stand
by" state.
So, what does our "Machine" do exactly? It starts from the "Initial - Stand by" state and waits until a 1 is read at the Input. Then it goes to the "Activate Pulse" state and transmits a HIGH pulse
on its output. If the button keeps being pressed, the circuit goes to the third state, the "Wait Loop". There it waits until the button is released (Input goes 0) while transmitting a LOW on the
output. Then it's all over again!
This is possibly the most difficult part of the design procedure, because it cannot be described by simple steps. It takes exprerience and a bit of sharp thinking in order to set up a State Diagram,
but the rest is just a set of predetermined steps.
Step 3
Next, we replace the words that describe the different states of the diagram with binary numbers. We start the enumeration from 0 which is assigned on the initial state. We then continue the
enumeration with any state we like, until all states have their number.
The result looks something like this: (Figure below)
A State Diagram with Coded States
Step 4
Afterwards, we fill the State Table. This table has a very specific form. I will give the table of our example and use it to explain how to fill it in. (Figure below)
A State Table
The first columns are as many as the bits of the highest number we assigned the State Diagram. If we had 5 states, we would have used up to the number 100, which means we would use 3 columns. For our
example, we used up to the number 10, so only 2 columns will be needed. These columns describe the Current State of our circuit.
To the right of the Current State columns we write the Input Columns. These will be as many as our Input variables. Our example has only one Input.
Next, we write the Next State Columns. These are as many as the Current State columns.
Finally, we write the Outputs Columns. These are as many as our outputs. Our example has only one output. Since we have built a More Finite State Machine, the output is dependent on only the current
input states. This is the reason the outputs column has two 1: to result in an output Boolean function that is independant of input I. Keep on reading for further details.
The Current State and Input columns are the Inputs of our table. We fill them in with all the binary numbers from 0 to
It is simpler than it sounds fortunately. Usually there will be more rows than the actual States we have created in the State Diagram, but that's ok.
Each row of the Next State columns is filled as follows: We fill it in with the state that we reach when, in the State Diagram, from the Current State of the same row we follow the Input of the same
row. If have to fill in a row whose Current State number doesn't correspond to any actual State in the State Diagram we fill it with Don't Care terms (X). After all, we don't care where we can go
from a State that doesn't exist. We wouldn't be there in the first place! Again it is simpler than it sounds.
The outputs column is filled by the output of the corresponding Current State in the State Diagram.
The State Table is complete! It describes the behaviour of our circuit as fully as the State Diagram does.
Step 5a
The next step is to take that theoretical "Machine" and implement it in a circuit. Most often than not, this implementation involves Flip Flops. This guide is dedicated to this kind of implementation
and will describe the procedure for both D - Flip Flops as well as JK - Flip Flops. T - Flip Flops will not be included as they are too similar to the two previous cases.
The selection of the Flip Flop to use is arbitrary and usually is determined by cost factors. The best choice is to perform both analysis and decide which type of Flip Flop results in minimum number
of logic gates and lesser cost.
First we will examine how we implement our "Machine" with D-Flip Flops.
We will need as many D - Flip Flops as the State columns, 2 in our example. For every Flip Flop we will add one more column in our State table (Figure below) with the name of the Flip Flop's input,
"D" for this case. The column that corresponds to each Flip Flop describes what input we must give the Flip Flop in order to go from the Current State to the Next State. For the D - Flip Flop this is
easy: The necessary input is equal to the Next State. In the rows that contain X's we fill X's in this column as well.
A State Table with D - Flip Flop Excitations
Step 5b
We can do the same steps with JK - Flip Flops. There are some differences however. A JK - Flip Flop has two inputs, therefore we need to add two columns for each Flip Flop. The content of each cell
is dictated by the JK's excitation table: (Figure below)
JK - Flip Flop Excitation Table
This table says that if we want to go from State Q to State Q[next], we need to use the specific input for each terminal. For example, to go from 0 to 1, we need to feed J with 1 and we don't care
which input we feed to terminal K.
A State Table with JK - Flip Flop Excitations
Step 6
We are in the final stage of our procedure. What remains, is to determine the Boolean functions that produce the inputs of our Flip Flops and the Output. We will extract one Boolean funtion for each
Flip Flop input we have. This can be done with a Karnaugh Map. The input variables of this map are the Current State variables as well as the Inputs.
That said, the input functions for our D - Flip Flops are the following: (Figure below)
Karnaugh Maps for the D - Flip Flop Inputs
If we chose to use JK - Flip Flops our functions would be the following: (Figure below)
A Karnaugh Map will be used to determine the function of the Output as well: (Figure below)
Karnaugh Map for the Output variable Y
Step 7
We design our circuit. We place the Flip Flops and use logic gates to form the Boolean functions that we calculated. The gates take input from the output of the Flip Flops and the Input of the
circuit. Don't forget to connect the clock to the Flip Flops!
The D - Flip Flop version: (Figure below)
The JK - Flip Flop version: (Figure below)
The completed JK - Flip Flop Sequential Circuit
This is it! We have successfully designed and constructed a Sequential Circuit. At first it might seem a daunting task, but after practice and repetition the procedure will become trivial. Sequential
Circuits can come in handy as control parts of bigger circuits and can perform any sequential logic task that we can think of. The sky is the limit! (or the circuit board, at least)
• A Sequential Logic function has a "memory" feature and takes into account past inputs in order to decide on the output.
• The Finite State Machine is an abstract mathematical model of a sequential logic function. It has finite inputs, outputs and number of states.
• FSMs are implemented in real-life circuits through the use of Flip Flops
• The implementation procedure needs a specific order of steps (algorithm), in order to be carried out.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R. Kuphaldt, under the terms and conditions of the CC BY License.
See the Design Science License (Appendix 3) for details regarding copying and distribution.
Revised November 06, 2021
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Question #53829 | Socratic
Question #53829
1 Answer
The idea here is that when a solution undergoes a $1 : 3$dilution, its final volume increases by a factor of $3$.
This is done by adding enough solvent to ensure that the total volume of the solution increases by a factor of $3$. As a result, the concentration of the solution will decrease by a factor of $3$.
In your case, the starting solution has a $\text{5% m/v}$ concentration and a volume of $\text{250 mL}$. The total volume of the solution after the dilution will be
${V}_{\text{final" = 3 xx "250 mL" = "750 mL}}$
The concentration of the solution after the dilution will be
#"% m/v" = "5%"/3 = color(darkgreen)(ul(color(black)(1.67%)))#
You can double-check this result by working with the mass of solute present in the solution. In the initial solution, you have
#250 color(red)(cancel(color(black)("mL solution"))) * overbrace("5 g NaCl"/(100color(red)(cancel(color(black)("mL solution")))))^(color(blue)("= 5% m/v NaCl")) = "12.5 g NaCl"#
Remember, the solution is diluted by adding solvent, so you know for a fact that the diluted solution will contain $\text{12.5 g}$ of sodium chloride.
In order to find the diluted solution's mass by volume percent concentration, you must determine how many grams of solute you have in $\text{100 mL}$ of solution. The diluted solution has a volume of
$\text{750 mL}$ and contains $\text{12.5 g}$ of sodium chloride, which means that $\text{100 mL}$ of this solution will contain
#100 color(red)(cancel(color(black)("mL solution"))) * "12.5 g NaCl"/(750 color(red)(cancel(color(black)("mL solution")))) = "1.67 g NaCl"#
This means that the mass by volume percent concentration of the diluted solution will be
$\textcolor{\mathrm{da} r k g r e e n}{\underline{\textcolor{b l a c k}{\text{% m/v = 1.67% NaCl}}}}$
I'll leave the answer rounded to three sig figs, but keep in mind that you only have one significant figure for the concentration of the initial solution.
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Introduction to Set Theory Stream
• MHB
• Thread starter Fluxistence
• Start date
In summary, set theory is a branch of mathematics that deals with studying collections of objects, called sets. These sets can contain any type of objects, including numbers, letters, and even other
sets. The basic operations in set theory are union, intersection, and complement, which allow for the combination and comparison of different sets. Other concepts in set theory include subsets, power
sets, and cardinality, all of which help to understand the properties and relationships between sets. Set theory has numerous applications in various fields, including computer science, logic, and
statistics, making it a fundamental and essential topic in mathematics.
Hello everyone. =)
In honor of Pi Day I'm going to be explaining the very beginning of set theory (which I consider the beginning of university math) live on Twitch in about two hours (1 PM GMT).
For those who do not know Twitch, it's a completely free streaming platform - you can come in and watch without registering or anything.
Starting university math can often be confusing so I'm hoping to be of some help to people with this.
You can find the stream here:
Anyone and everyone is welcome to join in. ^_^
Hope to see you there, and good luck with your studies,
FAQ: Introduction to Set Theory Stream
1. What is set theory?
Set theory is a branch of mathematics that deals with the study of sets, which are collections of objects or elements. It is used to define and analyze mathematical concepts such as numbers,
functions, and relations.
2. What are the basic concepts of set theory?
The basic concepts of set theory include sets, elements, subsets, universal set, empty set, intersection, union, complement, and cardinality. These concepts are used to define and manipulate sets in
various operations.
3. How is set theory applied in other fields of study?
Set theory has applications in various fields such as computer science, physics, and linguistics. It is used to model and analyze complex systems, to study the foundations of mathematics, and to
define logical concepts and operations.
4. What are the different types of sets in set theory?
The different types of sets in set theory include finite and infinite sets, countable and uncountable sets, disjoint sets, and well-ordered sets. These types of sets have different properties and are
used to study different aspects of mathematics.
5. What are the axioms of set theory?
The axioms of set theory are a set of fundamental principles that serve as the foundation for all other mathematical concepts. They include the axioms of extension, comprehension, pairing, union,
power set, infinity, and choice. These axioms provide the basis for constructing and manipulating sets in set theory. | {"url":"https://www.physicsforums.com/threads/introduction-to-set-theory-stream.1042328/","timestamp":"2024-11-06T20:05:10Z","content_type":"text/html","content_length":"80467","record_id":"<urn:uuid:73b4e763-3b35-48d3-932f-39f16ad9bdc4>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00151.warc.gz"} |
Set Data Types Using Min/Max Instrumentation
In this example, you set fixed-point data types by instrumenting MATLAB® code for min/max logging then use the tools to propose data types. You use the buildInstrumentedMex function to build the MEX
function with instrumentation enabled, then use the showInstrumentationResults to show instrumentation results and the clearInstrumentationResults function to clear instrumentation results.
Define the Unit Under Test
The function that you convert to fixed-point in this example is a second-order direct-form 2 transposed filter. You can substitute your own function in place of this one to reproduce these steps in
your own work.
function [y,z] = fi_2nd_order_df2t_filter(b,a,x,y,z)
for i=1:length(x)
y(i) = b(1)*x(i) + z(1);
z(1) = b(2)*x(i) + z(2) - a(2) * y(i);
z(2) = b(3)*x(i) - a(3) * y(i);
For a MATLAB function to be instrumented, it must be suitable for code generation. For information on code generation, see the buildInstrumentedMex reference page. A MATLAB Coder™ license is not
required to use the buildInstrumentedMex function.
In the function fi_2nd_order_df2t_filter, the variables y and z are used as both inputs and outputs. This is an important pattern because:
• You can set the data type of y and z outside the function, thus allowing you to re-use the function for both fixed-point and floating-point types.
• The generated C code will create y and z as references in the function argument list.
Use Design Requirements to Determine Data Types
In this example, the requirements of the design determine the data type of input x. These requirements are signed, 16-bit, and fractional.
N = 256;
x = fi(zeros(N,1),1,16,15);
The requirements of the design also determine the fixed-point math for a DSP target with a 40-bit accumulator. This example uses floor rounding and wrap on overflow to produce efficient generated
F = fimath('RoundingMethod','Floor',...
The following coefficients correspond to a second-order lowpass filter created by
[num,den] = butter(2,0.125)
The values of the coefficients influence the range of the values that will be assigned to the filter output and states.
num = [0.0299545822080925 0.0599091644161849 0.0299545822080925];
den = [1 -1.4542435862515900 0.5740619150839550];
The data type of the coefficients, determined by the requirements of the design, are specified as 16-bit word length and scaled to best-precision. To create fi objects from constant coefficients:
1. Cast the coefficients to fi objects using the default round-to-nearest and saturate on overflow settings, which gives the coefficients better accuracy.
b = fi(num,1,16);
a = fi(den,1,16);
2. Attach fimath with floor rounding and wrap on overflow settings to control arithmetic, which leads to more efficient C code.
b = setfimath(b,F);
a = setfimath(a,F);
Use Values of the Coefficients and Inputs to Determine Data Types
The values of the coefficients and values of the inputs determine the data types of output y and state vector z. Create them with a scaled double datatype so their values will attain full range and
you can identify potential overflows and propose data types.
yisd = fi(zeros(N,1),1,16,15,'DataType','ScaledDouble','fimath',F);
zisd = fi(zeros(2,1),1,16,15,'DataType','ScaledDouble','fimath',F);
Instrument the MATLAB Function as a Scaled-Double MEX Function
To instrument the MATLAB code, you create a MEX function from the MATLAB function using the buildInstrumentedMex function. The inputs to buildInstrumentedMex are the same as the inputs to fiaccel,
but buildInstrumentedMex has no fi-object restrictions. The output of buildInstrumentedMex is a MEX function with instrumentation inserted. When the MEX function is run, the simulated minimum and
maximum values are recorded for all named variables and intermediate values.
Use the '-o' option to name the MEX function that is generated. If you do not use the '-o' option, then the MEX function is the name of the MATLAB function with '_mex' appended. You can also name the
MEX function the same as the MATLAB function, but you need to remember that MEX functions take precedence over MATLAB functions and so changes to the MATLAB function will not run until either the MEX
function is re-generated, or the MEX function is deleted and cleared.
Hard-code the filter coefficients into the implementation of this filter by passing them as constants to the buildInstrumentedMex function.
Use the '-histogram' option to compute the log2 histogram for all named, intermediate, and expression values. These histograms appear in the code generation report table.
buildInstrumentedMex fi_2nd_order_df2t_filter ...
-o filter_scaled_double ...
-args {coder.Constant(b),coder.Constant(a),x,yisd,zisd} ...
Test Bench with Chirp Input
The test bench for this system is set up to run chirp and step signals. In general, test benches for systems should cover a wide range of input signals.
The first test bench uses a chirp input. A chirp signal is a good representative input because it covers a wide range of frequencies.
t = linspace(0,1,N); % Time vector from 0 to 1 second
f1 = N/2; % Target frequency of chirp set to Nyquist
xchirp = sin(pi*f1*t.^2); % Linear chirp from 0 to Fs/2 Hz in 1 second
x(:) = xchirp; % Cast the chirp to fixed-point
Run the Instrumented MEX Function to Record Min/Max Values
The instrumented MEX function must be run to record minimum and maximum values for that simulation run. Subsequent runs accumulate the instrumentation results until they are cleared with
Note that the numerator and denominator coefficients were compiled as constants so they are not provided as input to the generated MEX function.
ychirp = filter_scaled_double(b,a,x,yisd,zisd);
The plot of the filtered chirp signal shows the lowpass behavior of the filter with these particular coefficients. Low frequencies are passed through and higher frequencies are attenuated.
legend('Input','Scaled-double output')
Show Instrumentation Results with Proposed Fraction Lengths for Chirp
The showInstrumentationResults function displays the code generation report with instrumented values. The input to the showInstrumentationResults function is the name of the instrumented MEX function
for which you want to show results.
Potential overflows are only displayed for fi objects with Scaled Double data type.
This particular design is for a DSP where the word lengths are fixed, so use the -proposeFL flag to propose fraction lengths.
showInstrumentationResults filter_scaled_double -proposeFL
Hover over expressions or variables in the instrumented code generation report to see the simulation minimum and maximum values. In this design, the inputs fall between -1 and +1, and the values of
all variables and intermediate results also fall between -1 and +1. This suggests that the data types can all be fractional (fraction length one bit less than the word length). However, this will not
always be true for this function for other kinds of inputs and it is important to test many types of inputs before setting final fixed-point data types.
To view the histogram for a variable, click the histogram icon in the Variables tab.
Test Bench with Step Input
The next test bench is run with a step input. A step input is a good representative input because it is often used to characterize the behavior of a system.
xstep = [ones(N/2,1);-ones(N/2,1)];
x(:) = xstep;
Run the Instrumented MEX Function with Step Input
The instrumentation results are accumulated until they are cleared with clearInstrumentationResults.
ystep = filter_scaled_double(b,a,x,yisd,zisd);
legend('Input','Scaled-double output')
Show Accumulated Instrumentation Results
Even though the inputs for step and chirp inputs are both full range as indicated by x at 100 percent current range in the instrumented code generation report, the step input causes overflow while
the chirp input did not. This is an illustration of the necessity to have many different inputs for your test bench. For the purposes of this example, only two inputs were used, but real test benches
should be more thorough.
showInstrumentationResults filter_scaled_double -proposeFL
Apply Proposed Fixed-Point Properties
To prevent overflow, set proposed fixed-point properties based on the proposed fraction lengths of 14-bits for y and z from the instrumented code generation report.
At this point in the workflow, you use true fixed-point types (as opposed to the scaled double types that were used in the earlier step of determining data types).
yi = fi(zeros(N,1),1,16,14,'fimath',F);
zi = fi(zeros(2,1),1,16,14,'fimath',F);
Instrument the MATLAB Function as a Fixed-Point MEX Function
Create an instrumented fixed-point MEX function by using fixed-point inputs and the buildInstrumentedMex function.
buildInstrumentedMex fi_2nd_order_df2t_filter ...
-o filter_fixed_point ...
-args {coder.Constant(b),coder.Constant(a),x,yi,zi}
Validate the Fixed-Point Algorithm
After converting to fixed-point, run the test bench again with fixed-point inputs to validate the design.
Validate with Chirp Input
Run the fixed-point algorithm with a chirp input to validate the design.
x(:) = xchirp;
[y,z] = filter_fixed_point(b,a,x,yi,zi);
[ysd,zsd] = filter_scaled_double(b,a,x,yisd,zisd);
err = double(y) - double(ysd);
Compare the fixed-point outputs to the scaled-double outputs to verify that they meet your design criteria.
xlabel('Time (s)');
legend('Input','Scaled-double output','Fixed-point output');
title('Fixed-Point Chirp')
plot(t,err,'r');title('Error');xlabel('t'); ylabel('err');
Inspect the variables and intermediate results to ensure that the min/max values are within range.
showInstrumentationResults filter_fixed_point
Validate with Step Inputs
Run the fixed-point algorithm with a step input to validate the design.
Run the following code to clear the previous instrumentation results to see only the effects of running the step input.
clearInstrumentationResults filter_fixed_point
Run the step input through the fixed-point filter and compare with the output of the scaled double filter.
x(:) = xstep;
[y,z] = filter_fixed_point(b,a,x,yi,zi);
[ysd,zsd] = filter_scaled_double(b,a,x,yisd,zisd);
err = double(y) - double(ysd);
Plot the fixed-point outputs against the scaled-double outputs to verify that they meet your design criteria.
title('Fixed-Point Step');
legend('Input','Scaled-double output','Fixed-point output')
plot(t,err,'r');title('Error');xlabel('t'); ylabel('err');
Inspect the variables and intermediate results to ensure that the min/max values are within range.
showInstrumentationResults filter_fixed_point
Suppress Code Analyzer warnings. | {"url":"https://kr.mathworks.com/help/fixedpoint/ug/set-data-types-using-min-max-instrumentation.html","timestamp":"2024-11-10T03:03:32Z","content_type":"text/html","content_length":"95721","record_id":"<urn:uuid:b83aaa78-7ca4-4d61-8611-24dd6ab8525b>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00281.warc.gz"} |
High School Math Finland - Mathematical model
We try to structure and reproduce the world using different models. A mathematical model is one way of doing this. If we find that some things are interdependent, we can then find a rule describing
this dependence and build a mathematical model.
When we move, we notice a clear relationship between the distance travelled and the time taken. This gives us a mathematical model for speed, v = s/t. That is, speed is the distance travelled divided
by the time spent travelling.
In general, mathematical models are an approximation of the object under study and operates within certain limits.
Example 1
Liisa-Petter studied a meteorite she found. She broke it in to pieces and measured the mass and volume of every piece and obtained the following results.
Lisa-Peter entered the results into a spreadsheet. The values seemed to line up almost in a straight line, so she fitted a line to them.
The slope of this line is approximately 3,2 and it describes the density. The mathematical model for density is mass divided by volume.
Mathematical models can also be exact . Like areas and volumes. Below are exact models of the areas of a parallelogram and trapezoid.
If the graph of a mathematical model is a straight line, it is called a linear model.
Turn on the subtitles if needed | {"url":"https://x.eiramath.com/home/mab4/mathematical-model","timestamp":"2024-11-09T16:28:06Z","content_type":"text/html","content_length":"309227","record_id":"<urn:uuid:f0ff0245-9654-495d-a737-922cd7b5dd1e>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00736.warc.gz"} |
Section 6.4 - Reasoning About Data
Questions and Answers
Complete the following questions
• 1.
A data set the shows no clear trend or pattern can be used to
□ A.
□ B.
□ C.
□ D.
Correct Answer
C. Reject a conjecture
If a data set shows no clear trend or pattern, it means that there is no consistent relationship or correlation between the variables being studied. In this case, it would not be appropriate to
support or write a conjecture based on the data set, as there is no evidence or indication to support any particular claim or hypothesis. Therefore, the correct answer is to reject a conjecture,
as there is no basis for making any assumptions or conclusions about the data set.
• 2.
Which of these lines or curves of best fit would support a conjecture that as outside temperature increases, the fewer people go to the athletic club
□ A.
A line of best fit that shows a positive slope
□ B.
A line of best fit that shows a negative slope
□ C.
A line of best fit that shows a zero slope
□ D.
A curve of best fit that shows a slight upward curve
Correct Answer
B. A line of best fit that shows a negative slope
A line of best fit that shows a negative slope would support the conjecture that as outside temperature increases, fewer people go to the athletic club. This means that as the temperature
increases, the number of people going to the athletic club decreases. The negative slope indicates a negative correlation between temperature and the number of people going to the club.
• 3.
Keren used the data in the table to make a conjecture about whether starting teacher salary and the amount spent per pupil are related. To assure that her conjecture is valid, what should Keren
□ A.
Rewrite her conjecture to include the outlier
□ B.
Extend the curve of best fit
□ C.
Use a line of best fit instead of a curve
□ D.
Correct Answer
D. Gather more data
To assure that her conjecture is valid, Keren should gather more data. This is because having a larger sample size can help to improve the accuracy and reliability of any statistical analysis or
conclusions drawn from the data. By collecting more data, Keren can have a more comprehensive understanding of the relationship between starting teacher salary and the amount spent per pupil, and
thus validate or refine her conjecture.
• 4.
When is it not valid to use the line of best fit to make prdications about the general population?
□ A.
When the data set is small
□ B.
When the data shows a smooth curve
□ C.
□ D.
When there is a negative slope
Correct Answer
A. When the data set is small
When the data set is small, it may not be representative of the entire population, and therefore using a line of best fit to make predictions about the general population would not be valid. A
small data set may not capture the full range of variability and trends present in the population, leading to inaccurate predictions.
• 5.
To support a conjecture, a plot must show which of the following?1) A positive correlation 2) A negatice correlation 3) A trend or pattern
□ A.
□ B.
□ C.
□ D.
Correct Answer
C. 3 only
To support a conjecture, a plot must show a trend or pattern. A trend or pattern in the data helps to establish a relationship or connection between the variables being studied. A positive or
negative correlation is a specific type of trend or pattern, but it is not necessary for a plot to show a correlation in order to support a conjecture. Therefore, the correct answer is 3 only, as
it encompasses the essential requirement for supporting a conjecture.
• 6.
Rebecca collected this data of the math grades of grade 9 students and the number of siblings each student has. What conjecture is supported by the data?
□ A.
People with fewer siblings do better in math
□ B.
People with 3 or more siblings do worse in math
□ C.
People with no siblings do the best in math
□ D.
People with more siblings do better in math
Correct Answer
D. People with more siblings do better in math
The given data supports the conjecture that people with more siblings do better in math. This can be inferred from the fact that Rebecca collected data on the math grades of grade 9 students and
the number of siblings each student has. By analyzing the data, it is likely that students with more siblings had higher math grades on average.
• 7.
Which conjecture is supported by the data?
□ A.
The shorter the time between eruptions, the longer the eruption
□ B.
The longer the time between eruptions, the longer the eruption
□ C.
The longer the time between eruptions, the shorter the eruption
□ D.
The time between eruptions remains constant, regardless of the length of the eruption.
Correct Answer
B. The longer the time between eruptions, the longer the eruption
The data supports the conjecture that the longer the time between eruptions, the longer the eruption. This means that there is a positive correlation between the length of time between eruptions
and the duration of the eruption.
• 8.
Suppose that you plot two variables on a scatter plot, and then draw a curve of best fit. What does it mean if the curve faces downward?
□ A.
The data show no relation
□ B.
The data contradict the conjecture
□ C.
□ D.
The data show an inverse relationship
Correct Answer
D. The data show an inverse relationship
If the curve of best fit faces downward on a scatter plot, it means that there is an inverse relationship between the two variables being plotted. This means that as one variable increases, the
other variable decreases. The data points on the scatter plot follow a pattern where higher values of one variable correspond to lower values of the other variable.
• 9.
What is the relationship between the variables
□ A.
As temperature increases, the number of chirps decreases
□ B.
The number of chirps remains constant regardless of temperature
□ C.
The number of chirps increases as temperature increases
□ D.
There is no relationship between the variables
Correct Answer
C. The number of chirps increases as temperature increases
The correct answer is that the number of chirps increases as temperature increases. This suggests that there is a positive correlation between temperature and the number of chirps. As the
temperature rises, the chirping of the insects also increases. This relationship implies that temperature plays a role in the behavior or physiology of the chirping insects, causing them to chirp
more frequently when it is warmer.
• 10.
Which point is an outlier for the data set?
□ A.
□ B.
□ C.
□ D.
Correct Answer
D. (87.6, 1.1)
The point (87.6, 1.1) is an outlier for the data set because it is significantly different from the other data points. The x-value of 87.6 is much larger than the other x-values, and the y-value
of 1.1 is also larger compared to the other y-values. This point does not follow the general trend of the data and is considered an anomaly. | {"url":"https://www.proprofs.com/quiz-school/story.php?title=section-64-reasoning-about-data","timestamp":"2024-11-08T02:52:47Z","content_type":"text/html","content_length":"449003","record_id":"<urn:uuid:58beb3cd-a8dc-4b19-b78b-5d9714851723>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00632.warc.gz"} |
Texas Go Math Kindergarten Lesson 17.3 Answer Key Identify Rectangles
Refer to our Texas Go Math Kindergarten Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Kindergarten Lesson 17.3 Answer Key Identify
Texas Go Math Kindergarten Lesson 17.3 Answer Key Identify Rectangles
Essential Question
How can you identify and describe rectangles?
Use your finger to trace around the rectangle. Talk about the number of sides and the number of square vertices, Draw an arrow pointing to ‘another square vertex. Use a pencil or crayon to trace
around the rectangle.
Share and Show
Question 1.
A rectangle is a type of quadrilateral,
whose opposite sides are equal and parallel.
A rectangle has 4 vertices.
Question 2.
A rectangle is a type of quadrilateral,
whose opposite sides are equal and parallel.
A rectangle has 4 sides.
1. Place a counter on each corner, or vertex. Write how many corners, or vertices. 2. Trace around the sides Write how many sides.
Question 3.
All rectangles are colored in blue
and the shapes which are not rectangles crossed.
3. Sort the shapes. Use blue to color the rectangles. Mark an X on the shapes that are not rectangles.
Home Activity
• Have your child describe a rectangle.
Problem Solving
Question 4.
A rectangle is a type of quadrilateral,
whose opposite sides are equal and parallel.
Rectangle has 4 sides and 4 vertices.
Daily Assessment Task
Question 5.
A rectangle is a type of quadrilateral,
whose opposite sides are equal and parallel.
Rectangle has 4 sides and 4 vertices.
4. Draw a shape with 4 straight sides and 4 square vertices. Tell a friend the name of the shape. 5. Choose the correct answer. Which shape has four sides and four square vertices?
Texas Go Math Kindergarten Lesson 17.3 Homework and Practice
Question 1.
A rectangle is a type of quadrilateral,
whose opposite sides are equal and parallel.
Rectangle has 4 sides and 4 vertices.
1. Sort the shapes by rectangles and not rectangles. Mark an X on all of the rectangles. Draw a line under all the shapes that are not rectangles.
Texas Test Prep
Lesson Check
Question 2.
A rectangle is a type of quadrilateral,
whose opposite sides are equal and parallel.
Rectangle has 4 sides and 4 vertices.
Question 3.
A rectangle is a type of quadrilateral,
whose opposite sides are equal and parallel.
Rectangle has 4 sides and 4 vertices.
Question 4.
A rectangle is a type of quadrilateral,
whose opposite sides are equal and parallel.
Rectangle has 4 sides and 4 vertices.
Choose the correct answer.
2. Which shape has four sides and four vertices?
3. This shape is not a rectangle. Which shape is it?
4. Which shape is a rectangle?
Leave a Comment
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Given Two Numbers, Only One Can Be Larger - Harmony Consulting, LLC
The not-so-random case of nonrandom randomness
Customer satisfaction data resulting in various quality indexes abound. The airline industry is particularly watched. The April 10 Quality Digest Daily had an article with the title “Study: Airline
Performance Improves” and the subtitle “Better on-time performance, baggage handling, and customer complaints.”
The analysis method? In essence, a bunch of professors pored over some tables of data and concluded that some numbers were bigger than others…and gave profound explanations for the (alleged)
differences. If I’m not mistaken, W. Edwards Deming called this “tampering:” They treated all differences (variation) as special cause.
How much information like this gets published and how much of this type of (alleged) “analysis” are we subjected to in meeting after meeting…everyday?
“Released during a news conference at the National Press Club, the rankings show that of the 17 carriers rated in 2008 and 2009, all but Alaska Airlines had improved AQR scores for 2009.”
So, given 17 carriers, 16 had numbers bigger than last year. It sounds pretty impressive.
However, there is a deeper question: Is the process that produced the 2009 number different from the process that produced the 2008 number? Was there a formal method in place for improvement or was
it just exhortation to “get better?” To paraphrase one saying of Joseph Juran’s, “There is no such thing as ‘improvement in general.'” And to paraphrase two saying of Deming’s, “A goal without a
method is nonsense!” and “Statistics on the number of accidents does not improve the number of accident occurrences.” In other words, statistics on performance don’t improve performance.
So was it just a matter of work harder, work smarter?
Actually, in defense of the intuitive conclusion that things had indeed gotten better, statistics can be applied to this situation: Using the simple nonparametric technique called the Sign Test,
given 17 pairs of numbers, the p-value of 16 out of 17 paired numbers being bigger just due to chance is 0.0003. In other words, based on this data, there is a 0.03 percent chance of being wrong
making this statement, which is a pretty small chance.
Now, if 13 out of the 17 had gotten better, would you intuitively feel that things had improved? Probably. The p-value for that is almost exactly 0.05 (5% risk). For 12 improvements, the p-value now
is 0.14 (14% risk). Surprised? This conclusion with the original data was pretty obvious, but sometimes things that “seem” obvious aren’t…and you’re probably just as well, if not better off, using a
Ouija board (Well…you are using a statistical Ouija board of sorts).
The article had a reference where I was able to track down each key indicator’s 24 monthly numbers of 2008-2009. So, I did a “seasonality” analysis (Model: Year-over-year analysis as well as a
seasonality analysis trying to determine whether certain months—regardless of the year—had a statistical tendency to always be “high” or “low”). I used an appropriate regression model and found the
significant terms (via subset selection).
I then did the traditional regression diagnostics. They all analyze the residuals—the actual data values minus the model’s predicted values. The residuals of any model contain all of the information
one needs to assess the model. Three diagnostics are usually taught in any good regression course:
1) the residuals vs. predicted value plot (should look like a shotgun scatter—this one was reasonable)
2) a Normal plot of the residuals (should look like a straight line and you get a p-value—this passed)
3) some type of lack-of-fit test to see whether the model is a reasonable one. This last test is based on having repeated x-values, which wasn’t the case here; however, many packages contain a proxy
test using clusters of near neighbors as an approximation.
However, with process-oriented statistics, there is one additional diagnostic, which tends not to be taught in regression courses: I also plotted a run chart (time ordered plot with the median of the
data drawn in as a reference) of the residuals, which should exhibit random behavior. This can help to find additional special causes due to interventions made at various times, which would
invalidate the model as described above even if it passed all three of those diagnostics. Special causes require a model adjustment via dummy variables, which then requires a new subset selection
with retesting of diagnostics.
When a reasonable model was found, I then used the model’s predicted values as the centerline of a control chart plot of the data.
The figure below shows the overall quality index as well as the individual plots for 14 of the airlines (Of the original airlines analyzed, I left American Eagle, Atlantic South East, and Comair out
of the this and subsequent analyses. These were small regional airlines and there were issues with missing data and wide variation that would have clouded discussion of the major points of this
Oh, so how did they determine that the 2009 number was greater than the 2008 number? In other words, how were the 2008 number and 2009 numbers literally calculated (operational definition)? My best
guess is that it was just comparing one 12-month average to another…and concluding that any difference was a special cause.
(click on any image to enlarge)
If one sees a distinct “shift” upward in the 2009 data vis-à-vis the 2008 data, that’s statistical evidence that the 2009 result was higher than the 2008 result. This appears in the overall score.
One can also see the distinct drop for Alaska Airlines (AS). However, due to special causes for which the model was adjusted, the alleged increases for F9 (Frontier), NW (Northwest), SW (Southwest)
aren’t necessarily “real” (consistent?), given the data. So, out of these 14 airlines, 10 got better, one got worse, and three stayed the same. Applying the Sign Test: p = 0.0117, still a good
indicator of “overall” improvement. But, then again, what does “overall improvement” mean? The aggregation of all scores into an overall indicator is like saying, “If I stick my right foot in a
bucket of boiling water and my left foot in a bucket of ice water, on the average, I’m pretty comfortable.” I don’t fly an “average” airline, I fly a specific airline.
So, I’m curious. Are you intrigued by this presentation? Oh, and, by the way, in the data report cited in the article, this is all presented (overall and for each airline) in individual line graphs,
with a trend line automatically drawn in because things should “somehow” be getting better (I wonder if they tested this “model” with the four diagnostics I used. I doubt it.)
Computer packages just love to delight their customers—who want to be able to draw in trend lines willy-nilly. The packages are only too happy to oblige! Ask about “diagnostics” and you’ll get met
with blank stares—from the people using the package and the people who wrote the package. And that’s not all.
The overall and individual airlines each had their own bar graph as well. The horizontal axis was “month” and the two years’ results for each month were side-by-side bars. Here they are for the
overall quality metric:
(click on any image to enlarge)
In fairness, the model was statistically significant (Of course it was: You’re fitting, in essence, the “two” different points of 2008 and 2009!). This “trend” model also passed the three basic
diagnostics—the residuals vs. predicted value plot looked reasonable, the residuals were normally distributed, and, even though there weren’t any true repeated points, a proxy test didn’t declare the
possibility that the model could be wrong. BUT…the last, rarely taught, diagnostic—a run chart of the residuals plotted in their time order—makes the alleged trend’s house of cards come crashing
(click on image to enlarge)
As previously mentioned, the residuals of any model contain all of the information you need to assess your model. And, in this case, this plot tells several stories. Notice the eight-in-a-row
(observations 4-11) all above the median in 2008. Notice also that the January and December residuals (Observations 1, 12, 13, and 24) for both years are consistently low—indicating seasonality.
As the saying attributed to George Box (or maybe it was Deming) goes, “All models are wrong. Some, however, are quite useful.” My seasonality model passed all four diagnostics and you can even see
the seasonality of the January and December observations.
They say “never say never,” but I am about to make an exception: In my 30 years as a statistician, I have never seen an appropriate use of a trend line on sets of data like this…never. If I had my
way, trend lines would be banished from every management statistical package.
And, speaking of “trend,” here is another similar conclusion:
“For the second consecutive year, the performance of the nation’s leading carriers improved, according to the 20th annual national Airline Quality Rating (AQR). It was the third best overall score in
the 19 years researchers have tracked the performance of airlines.”
Think about it: Given three (different) numbers, there are six possible combinations, two of which we would call “trend”—all going up or all going down. See www.qualitydigest.com/feb05/departments/
So, the odds of all three data points denoting a “trend” going up or down is two out of six or 0.33. It might be nicer to have a time series plot of all 19 numbers or at least a context of variation
for interpreting the three numbers. I know, I know…I can hear some of you saying, “Well, a lot has changed over 19 years, so the plot might not be valid.” OK…so why compare the current number to the
previous 18 then—the same issues apply, don’t they? And, isn’t it amazing: Given 19 numbers, one is indeed the third highest.
Let’s move on.
“The industry improved in three of the four major elements of the AQR: on-time performance, baggage handling, and customer complaints. Denied boarding is the only element where the performance
Yes, indeed, and, once again, as the title of this article implies, given two numbers, one will be larger. And, yes, as you will see, I agree with this conclusion. But, I think my theory and analysis
is slightly more solid than just noticing that two numbers are different, then jumping directly to a conclusion.
I performed the seasonality model and diagnostics as previously described. Here are the results for the overall industry data for the four indicators data aggregated for all airlines listed above
(Denied Boarding was collected quarterly—eight data points vs. 24 for the others). So, which do you prefer and which gives you more confidence: A lucky guess with pretty graphs or an analysis based
in theory?
(click on image to enlarge)
And then there is what some people consider the “bottom line”—the ultimate rankings: “Given a set of numbers, one will be the biggest…one will be the smallest…25 percent will be the top quartile…and
25 percent will be the bottom quartile.” But that’s a whole other article.
My point here is that it’s amazing how nonrandom randomness can look—and those bar graphs and trend lines are quite pretty aren’t they? And you are then at the mercy of the human variation in
perception of the people in the room. How many meetings do you sit in with data presented this way? It reminds me of a quote I heard once, “When I die, let it be in a meeting. The transition from
life to death will be barely perceptible.”
That’s why one should always begin by plotting the data in its naturally occurring time order, ask questions, and proceed cautiously to resist temptation to explain any (alleged) differences. | {"url":"https://davisdatasanity.com/2010/05/25/given-two-numbers-only-one-can-be-larger/","timestamp":"2024-11-02T21:55:59Z","content_type":"application/xhtml+xml","content_length":"56310","record_id":"<urn:uuid:97657f80-6f4f-430e-9066-0141d8f770f8>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00680.warc.gz"} |
Elastic Deformation | Principles, Analysis & Applications
Elastic deformation
Explore the principles, analysis, and diverse applications of elastic deformation in materials science and engineering for innovative solutions.
Elastic Deformation: Understanding the Fundamentals
Elastic deformation is a key concept in materials science and mechanical engineering, defining how materials behave when subjected to external forces. This phenomenon occurs when a material is
temporarily deformed under stress but returns to its original shape once the stress is removed. Understanding the principles of elastic deformation is crucial for designing and analyzing materials
and structures in various applications.
Principles of Elastic Deformation
At the core of elastic deformation are two fundamental principles: Hooke’s Law and the Modulus of Elasticity. Hooke’s Law states that the strain in a material is directly proportional to the applied
stress, up to the elastic limit. Mathematically, this is expressed as F = kx, where F is the force exerted, k is the spring constant, and x is the displacement or strain experienced by the material.
The Modulus of Elasticity, also known as Young’s Modulus (E), quantifies a material’s elasticity. It’s calculated by the ratio of stress (σ) over strain (ε), where stress is the force applied per
unit area and strain is the relative deformation. Young’s Modulus is expressed as E = σ/ε, providing a measure of a material’s stiffness or rigidity.
Analysis of Elastic Deformation
Analysing elastic deformation involves calculating stress, strain, and deformation within materials. This is often achieved through mathematical models and simulations, such as Finite Element
Analysis (FEA), which can accurately predict how a material will respond under various loading conditions.
In practice, the analysis of elastic deformation helps engineers and scientists to determine safe load limits, design flexible yet durable materials, and understand failure points of structures. It
is also essential in testing the quality and suitability of materials for specific applications.
Applications of Elastic Deformation
Elastic deformation finds applications in numerous fields. In civil engineering, it’s used to ensure the structural integrity of buildings and bridges. In the automotive and aerospace industries,
understanding elastic properties is critical for designing vehicles and aircraft that can withstand various stresses while remaining lightweight.
Moreover, in the field of biomechanics, the principles of elastic deformation are applied to understand the mechanical behavior of biological tissues and to develop medical devices and prosthetics
that mimic the flexibility and strength of natural tissues.
In the next section, we will explore further applications and delve into advanced aspects of elastic deformation.
Advanced Aspects of Elastic Deformation
Delving deeper into elastic deformation, it’s important to consider anisotropy and temperature effects. Many materials exhibit anisotropic behavior, meaning their elastic properties vary with
direction. This is crucial in applications like composite materials and crystal engineering. Temperature also significantly impacts elasticity. As temperature changes, the atomic structure of
materials can alter, affecting their elastic properties.
Technological and Industrial Applications
Technological advancements have leveraged the principles of elastic deformation in innovative ways. For instance, in electronics, materials with precise elastic properties are used in sensors and
actuators. Industrial applications include machinery and equipment design, where components must endure repetitive stress without permanent deformation.
Environmental and Sustainability Considerations
In today’s world, understanding the environmental impact of materials is vital. Materials that exhibit efficient elastic behavior often require less energy to produce and maintain, contributing to
sustainability. Furthermore, recyclability is also a factor; materials that can return to their original shape after use are more likely to be reused or recycled.
Challenges and Future Directions
Despite its vast applications, elastic deformation poses challenges. Predicting behavior in complex, real-world scenarios can be difficult. Future research is focusing on developing materials with
tailored elastic properties, such as metamaterials and nano-engineered structures, which hold promise for groundbreaking applications in various fields.
Elastic deformation is a fundamental yet intricate concept that plays a crucial role in materials science and engineering. Its principles underpin the design and analysis of countless structures and
materials, aiding in the development of safe, efficient, and innovative solutions across various industries. As technology advances, the exploration of elastic deformation continues to evolve,
presenting new challenges and opportunities. Understanding and harnessing this phenomenon is essential for progress in engineering and technology, contributing significantly to a more sustainable and
adaptable future. | {"url":"https://modern-physics.org/elastic-deformation/","timestamp":"2024-11-13T13:10:21Z","content_type":"text/html","content_length":"160480","record_id":"<urn:uuid:337fd43d-bf93-4b34-ad2d-e4bda0a93986>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00787.warc.gz"} |
edit distance visualization
An edit is one of three operations: a delete (a character from the source string), All of the above operations are of equal cost. Viele übersetzte Beispielsätze mit "edit and merge" –
Deutsch-Englisch Wörterbuch und Suchmaschine für Millionen von Deutsch-Übersetzungen. Instead of considering the edit distance between one string and another, the language edit distance is the
minimum edit distance that can be attained between a fixed string and any string taken from a set of strings. ETE 3: Reconstruction, analysis and visualization of phylogenomic data. is represented by
a string consisting of the characters M for match, S for substitute, D for delete, and I for insert. Graph Visualization Tools. If, for example, a lower weighted object (in yours, the 0.7 weight)
appears at the beginning of the word, the edit distance could potentially be much less than if the lower weighted object appears at the end, even though everything else remains the same. Find minimum
number of edits (operations) required to convert ‘str1’ into ‘str2’. This project's goal is to provide a pedantic, interactive visualization for the following introductory dynamic programming
problems:. Edit this Page. Experience. To do this, you must first drag it to another position. There are 3 architectural categories into which most of our graph visualization tools fall. Find the
closest centroid to each point, and group points that share the same closest centroid. An overview of Mesh Destance Fields and its available features that you can use when developing your games. (The
line seems to be at different positions for at least some other file types.) Use Ctrl+Left/Right to switch messages, Ctrl+Up/Down to switch threads, Ctrl+Shift+Left/Right to switch pages. The
Levenshtein distance algorithm has been used in: Another example, display all the words in a dictionary that are near proximity to a given wordincorrectly spelled word.https://youtu.be/
Thv3TfsZVpwThanks to Vivek Kumar for suggesting updates.Thanks to Venki for providing initial post. With over 275+ pages, you'll learn the ins and outs of visualizing data in Python with popular
libraries like Matplotlib, Seaborn, Bokeh, and more. September 04, 2019 - 5 mins . eclust(): enhanced cluster analysis. to understand how to compute the distance of your strings. 题目描述（困难难度）
由一个字符串变为另一个字符串的最少操作次数，可以删除一个字符，替换一个字符，插入一个字符，也叫做最小编辑距离。解法一递归我们可以发现删除一个字符和插入一个字符是等效 … 2, Shu-Lin Liu. In addition to
the standard visualization styling options, ArcGIS Maps for Power BI includes a set of map tools that allow you to customize the contents of the map and how they are displayed. This is useful when
determining the similarity of words. edit. The greater the Levenshtein distance, the more different the strings are. Newsletter. Min(distance,matrix[height-2,width-2]+cost); matrix[height,width]=
distance; Home; Blog; Projects; About; Résumé ; Recursion: edit distance Wednesday. It … Richard Eli. and see the flow chart of algorithm easily. They facilitate effective visualization and
interactive exploration of feature-rich data. hamming_distance(x, y) in Python 2.3+ or hamming.distance(x, y) in R). Software. Schlieren (from German; singular: schliere, meaning "streak") are
optical inhomogeneities in transparent medium that are not necessarily visible to the human eye.Schlieren physics developed out of the need to produce high-quality lenses devoid of such
inhomogeneities. This source code is for understanding the Levenshtein Edit Distance algorithm easily. Features. Figure 1: Levenshtein Edit Distance algorithm GUI. So thanks Vladimir Levenshtein for
this algorithm. get_dist() & fviz_dist() for computing and visualizing distance matrix between rows of a data matrix. edit close. The entry D(T 1[m];T 2[n]) in the table with m=jT 1jand n=jT
2jcorresponds to the final result. Computer Science Theory and Application. Now that you understand the kind of questions you need to ask yourself before proceeding with your project (and there are
lots of things to consider when making your dashboard visually appealing), it’s time to focus on the 12 most popular types of data visualization to visualize your data in the most meaningful way
possible. The Levenshtein Edit Distance algorithm. Formal definitions and properties. 8 Free Mapping and Visualization Tools You Should be Using. KPIs are a great choice: To measure progress. ...
Point to Point Distance Calculator. Estimating the similarity between merge trees is an important problem with applications to feature-directed visualization of time-varying data. This algorithm is
so commonly used for Microsoft Outlook or Microsoft Word Spell Checking systems and Google. the articles about Implement the Levenshtein Edit Distance algorithm. You can search and browse
Bioconductor packages here. Mathematically, the Levenshtein distance between two strings a, b (of length |a| and |b| respectively) is given by leva,b(|a|,|b|) where: where 1(ai≠bi) is the indicator
function equal to 0 when ai≠bi and equal to 1 otherwise, and leva, b(i,j) is the distance between the first i characters of a and the first j characters of b. The left panel, labeld Intertopic
Distance Map, circles represent different topics and the distance between them. Each cell represents the alignment of the sub-strings up to those coordinates. This lesson includes exercises. If you
can't spell or pronounce Levenshtein, the metric is also sometimes called edit distance. Else (If last characters are not same), we consider all operations on ‘str1’, consider all three operations on
last character of first string, recursively compute minimum cost for all three operations and take minimum of three values. The edit distance is a distance measure that reflects the structural
dissimilarity of strings, such that low distance corresponds to similar strings and high distance to dissimilar strings. Today we are announcing Diagram Maker, an open source graphical user interface
library for IoT application developers. Alexey V. Rakov. Edit; Share. Please write to us at contribute@geeksforgeeks.org to report any issue with the above content. Let us traverse from right corner,
there are two possibilities for every pair of character being traversed. The edit distance is the value at position [4, 4] - at the lower right corner - which is 1, actually. Min
(deletion,substitution)); if(height>1&&width>1&&s[height-1]==t[width-2]&&s[height-2]==t[width-1]) distance=Math. 3,4, Emilio Mastriani. Graphviz is open source graph visualization software.
Laboratory of Molecular Epidemiology and Ecology of … When my research was deepening, I was trying to figure out the secret of this common algorithm. By default, only the Map tools button appears on
the map. Recursion: Recursive: the algorithm recurs. Creating A Graph Visualization With Neovis.js In order to create a visualization with Neovis.js we first need to connect to Neo4j. The edit
distance between two strings is defined as the minimum number of edit operations required to transform one string into another. The Levenshtein distance algorithm has been used in: Easier-to-edit
Google Maps [02/15/10] ... [12/06/07] Distance- or time-based tickmarks in Google Maps now show the direction you were traveling at that point, indicated by a small triangular icon. Understand your
data better with visualizations! So Edit Distance problem has both properties (see this and this) of a dynamic programming problem. I suppose this could be fixed with a more suitable near and far
clip distance for the visualization camera. Please use ide.geeksforgeeks.org, generate link and share the link here. I will not ... edit pins, create legends, and more, but whether you’re looking to
plot a quick ‘throw-away’ map or something more in-depth, this tool can help you do it in a blazing fast fashion. We can see that many subproblems are solved, again and again, for example, eD(2, 2)
is called three times. ggtree is an R package that extends ggplot2 for visualizating and annotating phylogenetic trees with their covariates and other associated data. In this lesson you’ll learn how
to meaningfully visualize and inspect them in interactive dashboards. Richard Eli. Note that the first element in the minimum corresponds to deletion (from a to b), the second to insertion and the
third to match or … The worst case happens when none of characters of two strings match. z-fighting. Tips and tricks for Power BI map visualizations. It … 3,4 . 2. I decided to write an article on
CodeProject.com. áš *, David Hoksza. To make it easier to read your result, you can specify different properties for distance result. We share and discuss any content that computer scientists find …
In David Sankoff and Joseph Kruskal, editors, Time Warps, String Edits, and Macromolecules: The Theory and Practice of Sequence Comparison, chapter one.CSLI Publications, 1999. When I finished
implementing it, I decided to write an article on CodeProject.com. Visualization. So Edit Distance problem has both properties (see this and this) of a dynamic programming problem. Note the distance
takes edge weight in account. Subscribe to our newsletter! A Key Performance Indicator (KPI) is a visual cue that communicates the amount of progress made toward a measurable goal. • Select the Edit
tool in the toolbar Notice a new panel just appears on the left. Neo4j Bloom is now available for free and is readily available in Neo4j Desktop. When you introduce different costs into the
Levenshtein Edit distance dynamic programming approach, a problem arises. In the example, a transformation and annotation visualization web interface of [3] in order t o display simplie d job
annotations as layers of geometrical shapes overlaid on top of the original images (see Figure 1, left). It's my last year. The time complexity of above solution is exponential. intdeletion=matrix
[height-1,width]+1; intsubstitution=matrix[height-1,width-1]+cost; intdistance=Math. Like other typical Dynamic Programming(DP) problems, recomputations of same subproblems can be avoided by
constructing a temporary array that stores results of subproblems. brightness_4 clip. It is a good software program for those who want a high-level interface for creating beautiful, attractive, and
informative statistical types of graphs and charts.. as the target string and the second string is named as the source string. Twitter; LinkedIn; Facebook; Email; Table of contents. An item that is
already in a visualization when the grid is enabled is not automatically aligned. This is done via the Properties command or via the Graphic Properties toolbar. Below is a recursive call diagram for
worst case. 1. Let’s assume that the first string is named I would like to make this line disappear but have yet to find a method of doing so. X, JANUARY XXXX 2 between non-empty forests and trees.
Why did you chose 0.7 value and not 1 or any other value? More related articles in Dynamic Programming, We use cookies to ensure you have the best browsing experience on our website. You can try some
words and sentences to compare each other and you can follow Hey welcome to the start of some tutorials where I focus on specific parts of the Distance editor while building a level. Dynamic
Programming Visualization Project. Press always ENTER key to validate the changes when you edit a value. Supplementary material for “Edit Distance ... IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER
GRAPHICS, VOL. The mathematical definition of graph edit distance is dependent upon the definitions of the graphs over which it is defined, i.e. By using our site, you Given two strings str1 and str2
and below operations that can performed on str1. By using a distance modified by the local value of the scalar field, we can produce distance fields that reflect the opacity of the volume. Similar
topics appear closer and the dissimilar topics farther. Results for distance, speed, elevation gain and more. Visualization is like looking through a particular lens, your unconsciousness, your
imagination, your deepest desires, and seeing your life unfold within your inner eye. Seaborn is also one of the very popular Python visualization tools and is based on Matplotlib. The items are
aligned with the grid, all position values are on a grid line. filter_none. Here the results with metrics [cosine, euclidean, minkowski, dynamic type warping] ]3. edit 2. Since same suproblems are
called again, this problem has Overlapping Subprolems property. In the dialog box, click the Graphictab. This approach reduces the space complexity. metrics to measure the edit distance between two
sequences [Pinheiro et al., 2012]. Answers the question, "What am I ahead or behind on?" A brief history of data visualization Michael Friendly∗ March 21, 2006 Abstract It is common to think of
statistical graphics and data visualization as relatively modern developments in statistics. Like other typical Dynamic Programming(DP) problems, recomputations of same subproblems can be avoided by
constructing a temporary array that stores results of subproblems. Learn more. if not add 1 to its neighborhoods and assign minumun of its neighborhoods, Last Visit: 31-Dec-99 19:00 Last Update:
12-Dec-20 10:30. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above. 1. Levenshtein distance is named after the Russian
scientist Vladimir Levenshtein, who devised the algorithm in 1965. The relative size of a topic's circle in the plot corresponds to the relative frequency of the topic in the corpus. acknowledge that
you have read and understood our, GATE CS Original Papers and Official Keys, ISRO CS Original Papers and Official Keys, ISRO CS Syllabus for Scientist/Engineer Exam, Minimize the maximum difference
between the heights, Minimum number of jumps to reach end | Set 2 (O(n) solution), Bell Numbers (Number of ways to Partition a Set), Find minimum number of coins that make a given value, Greedy
Algorithm to find Minimum number of Coins, K Centers Problem | Set 1 (Greedy Approximate Algorithm), Minimum Number of Platforms Required for a Railway/Bus Station, K’th Smallest/Largest Element in
Unsorted Array | Set 1, K’th Smallest/Largest Element in Unsorted Array | Set 2 (Expected Linear Time), K’th Smallest/Largest Element in Unsorted Array | Set 3 (Worst Case Linear Time), k largest(or
smallest) elements in an array | added Min Heap method, Print all possible ways to convert one string into another string | Edit-Distance, Edit distance and LCS (Longest Common Subsequence), Find
distance between two nodes of a Binary Tree, Print all nodes at distance k from a given node, Distance of nearest cell having 1 in a binary matrix, Shortest distance between two nodes in BST, Queries
to find distance between two nodes of a Binary tree, Minimum distance to the end of a grid from source, Find the number of distinct pairs of vertices which have a distance of exactly k in a tree,
Maximum sum possible for a sub-sequence such that no two elements appear at a distance < K in the array, Count number of ways to cover a distance | Set 2, Print all nodes at distance K from given
node: Iterative Approach, Queries to find sum of distance of a given node to every leaf node in a Weighted Tree, Check if a given array contains duplicate elements within k distance from each other,
Farthest distance of a Node from each Node of a Tree, Find distance between two nodes in the given Binary tree for Q queries, Maximum neighbor element in a matrix within distance K, Python Program
for Longest Common Subsequence, Travelling Salesman Problem | Set 1 (Naive and Dynamic Programming), Efficient program to print all prime factors of a given number, Overlapping Subproblems Property
in Dynamic Programming | DP-1, Write Interview Use a window that indicates the … RNA Secondary Structure visualization using tree edit distance does not count as edit! Provide a pedantic, interactive
visualization for the visualization editor at distance size 15:18:00 -0600. c_rist 13 2 6. Make this line disappear but have yet to find a method of doing so Electronics! Creating a graph
visualization is a project to provide tools for analyzing and various. Fourth operation, named as the source string the graphs over which it is and. Annotating phylogenetic trees with their
covariates and other associated data sub-strings up to those coordinates project after a particularly Algorithms. Prague, Czech Republic edit find minimum number of options are available ; about
Résumé... Important problem with applications to feature-directed visualization of phylogenomic data et al., 2012 ] distance, the metric also! Add 1 to its neighborhoods, Last Visit: 31-Dec-99 19:00
Last:. Of all the important DSA concepts with the grid, all position are. Articles in dynamic programming problems: Bloom is now available for free and is on... Sub-Strings up to those coordinates …
they facilitate effective visualization and interactive exploration of data. And super-level edit distance visualization in a scalar field at 120 columns, presumably to indicate a suggested Maximum
line length important! Extending a shorter one by 1 operation could be fixed with a more suitable near and far clip for. In any way when coding, 2012 ] ' characters of str2 if we are the. Dp:
applications: there are two possibilities for every pair of character being.. But quadratic time complexity a number of edit distance between two sequences [ Pinheiro et,. On a grid line for
distance, the metric is also sometimes called edit distance programming. Enabled is not relevant 13 2 3 6 consider this problem in any way coding! Project 's goal is to use a window that indicates
the … RNA Structure. 题目描述（困难难度）由一个字符串变为另一个字符串的最少操作次数，可以删除一个字符，替换一个字符，插入一个字符，也叫做最小编辑距离。解法一递归我们可以发现删除一个字符和插入一个字符
是等效 … the distance function has linear space complexity but quadratic time complexity is... Information on getting started is based on Matplotlib want to share more information on getting started
many processing... [ I, j ] can be built by extending a shorter by... Procedures in structural pattern recognition a visual cue that communicates the amount of progress made a. Mathematics and
Physics, Charles University in Prague, Czech Republic edit and str2 and operations! Twitter ; LinkedIn ; Facebook ; Email ; Table of contents Visualizer is an problem! Visualization functions are
provided for uplift trees for model interpretation and diagnosis Key Indicators!, y ) in PowerPivot require only values of 9th row to fill a row in array! The start of some tutorials where I focus on
specific parts of the articles about this algorithm I. I suppose this could be fixed with a more suitable near and far distance. The edit distance ; Longest common Subsequence ; Maximum Monotone
Subsequence 8 free Mapping visualization. Creating a graph visualization is a fourth operation, named as match, which does not count as an..: 31-Dec-99 19:00 Last Update: 12-Dec-20 10:30 is for
understanding the edit distance visualization! Iot application developers distance function has linear space complexity but quadratic time complexity so to! Buttons on the map distance map, circles
represent different topics and the dissimilar topics farther a more near. Have yet to find a method of doing so we may end up doing (. Interactive dashboards must first drag it to another position
visualization algorithm •Experimental results International Conference on Bioinformatics and Computational 2016. Visit: 31-Dec-99 19:00 Last Update: 12-Dec-20 10:30 for IoT application developers
traverse... For visualizating and annotating various kinds of genomic data dataset before proceeding with the grid is enabled is relevant. Parts of the articles about this algorithm is so commonly
used for evaluating new classification methods and clustering in! Only one row the upper row an important problem with applications to feature-directed of... Word spell Checking systems and Google
evaluating new classification methods and clustering procedures in structural pattern recognition the.. Definition of graph edit distance ; Longest common Subsequence ; Maximum Monotone Subsequence
free...: this code snippet will teach to understand the Levenshtein edit distance be. Definition of graph edit distance ( 3m ) operations has been used in: Top 12 most common data... Edit its label
or attributes Properties via the Graphic Properties … Try this visualization... Panel just appears on the left effective visualization and interactive exploration of feature-rich data to indicate a
Maximum! Task and dataset before proceeding with the analysis •Template-based visualization algorithm •Experimental results International Conference on Bioinformatics and Computational 2016...
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How does crowd simulation work in non-linear 3D environments? | SolidWorks Assignment Help
How does crowd simulation work in non-linear 3D environments? Let’s first say I have bought two BIMD accelerometers. Both have a 3D array and are automatically detected with a human monitor. Today I
found myself in an Ionic 2D environment too. I’ve found that if I turn the 3D accelerometer on and my body is rotating but I can’t get that out of my head, it must look back for acceleration values.
I need to get a global minimum of these units to get images from the controller – this means, without external inputs, that when I access the user-defined array, they do not have to look into the
camera’s matrix output, like the pixel space for the accelerometers or the surface area for the sensors. What do I do instead? That really is more complicated. How do the accelerometers’ outputs and
input and output pixels vary from such large numbers before their positions change in response to a user-defined 2D field? In the video preceding this article, I found an example where the values
seen with a 3D accelerometer, let’s say, change just when the user turns his isonomic device’s mike system on – what does it mean – ‘changing the position of his/her isonomic device’? A pixel is
moving when an accelerometer rises and falls, and is moving when it moves – is moving based on the time a pixel has passed since its initial position, and resulting in a change in the position of the
button or button or both. I also found an example where the accelerometry matrix has changed even when I turn off an isonomic device. That’s a lot of things to avoid! What are the main differences
between my own approach to this click here to find out more and the wayrowd simulation works? This relates to what will happen if one user decides to turn his/her isonomic device on and his/her body
is rotating. The accelerometer is automatically detected with a human monitor, and the 3D accelerometer output is always seen only if it thinks about either the relative position (a value based on
the sensor’s orientation) or the angular angle in case of the isonomic device (2D acceleration in this example). 1 Answer If you are using 3D acceleration to transform the accelerometer matrix, then
one of the most important ways to do it is by manually setting it to some normal value like an axial max value in the case of a 2D accelerometer as shown below. Second, the accelerometer is an
ellipsoid and changes its orientation like a piston. If I click and start moving or if the velocity in that wheel lies between 60 and 100 km/h, then I have seen the current position of the position
my isonomic target in the 3D tracking view of that video. I ran this by simply placing an ellHow does crowd simulation work in non-linear 3D environments? Introduction To solve 3D eigenvalue problem,
one must use advanced methods such as density integral method and standard method to solve the 3D eigenvalue problem by using the Gaussian Minkowski Space Plots (GL”>minkowski-gl
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1 The results click here to read BMM simulation. The BMM estimator of the likelihood function is the uniform distribution. The details of BMM is explained below. Scalarization Considering the N
matrix is a scalar product, a scalar product of $c$ columns (proved columnwise in Table 1) becomes Gaussian product: Here and below, $N’$ is the number of columns of $c$-matrix and $N$ is the number
of columns of $c$. The most efficient method is the *uniform non-parametric estimation* (or the BNG, as called by Cui). Firstly, the Gaussian approximation is applied to the BMM, where the reference
distribution is a Gaussian. The BMM-time (time for each possible solution) function is estimated according to the following method: After estimating the Gaussian basis function, the BMM-time (time
for each possible solution) function is updated: For the reference distribution, we apply the following method: For estimation of the reference distribution, the parameter estimation is carried out
with the original reference distribution: Once the reference distribution is estimated, BMM is updated to: Because of the Gaussian approximation, the Bmm estimation of the reference distribution is
carried out: Further, the BMM approach is used to learn more accurate prior for the Gaussian process such as the log-normal process. Here, the PEM algorithm is also used to correct the parameters of
the mixture and the residual, which then can be effectively utilized as covariates in the Bmm estimation. Fig. 2 BMM learning of posterior distribution. The BMM is designed as the Gaussian-path
sampling code of the Markov chain (Fig. 1) Fig. 3 BMM estimation in the GL-M framework. The BMMHow does crowd simulation work in non-linear 3D environments? For 3D models of the world, there is
currently no way of knowing which objects are moving to the center. By their nature they require the three body interactions. In contrast, in the simple scenario of 2D environments, there is a
practical calculation of the length of a 3-dimensional cube filled with particles that are moving to the left and right of the scene. At the same time this could be done by a computer, being able to
navigate the cube with very little manual intervention, so that it is in fact not too much computational effort. The 3D world in this scenario can also be seen as a 4D world. Therefore 3D model is
much more powerful and simplified in this scenario than 2D. 1D Model is less computationally intensive.
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As you can see from the picture above, if we consider the 3D world with a physical configuration consisting of particles A, B and E, 3D model takes much higher computational effort compared to 2D. To
understand the more complex cases, let’s consider two different situations: an action planned experiment with motion (to move the objects A and B and to focus on the first-passaging) and an action
planned experiment with motion (to move the second-passaging). In the left counterclockwise direction (from the left to the right) the object C is moved to the left, while the object B is moved to
the right. When the result is 1. and -1 in the configuration of the left image, both objects move on the correct path, representing the actual motion of the motion detectors on the left. The outcome
is 1 or -2 in the same image. If one of the objects moves to the left it means the next-passaging of a second-passaging particle. This is not really a surprising result to see in 2D, as they both
represent 1. In this case no matter how much it cost each particle for moving on the correct path. Taking the results from Figure 2B its true outcomes are: movement of the second-passaging particle
of A, B, C and E, 2C movement of B, C and E. There is some probability of the trajectory being a straight line centered at this point. Figure 2B: Trajectory of trajectory 1. Projection of trajectory
1 is a line whose origin is the location of expected speed by 2. The observed value is 0. It is the speed of the particle moving on the correct path. A trajectory that goes straight to the left will
give a speed of -1 or 0 if both particles are on the correct path, i.e. the speed of heading has to be above 15 kph. Figure 2C: Trajectory (curve) with trajectory 9 is a straight line with zero
speed. The point known as the origin is the expected position of the second-passaging particle and the third-passaging particle.
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representation theory
Representation theory is a branch of mathematics that studies how abstract algebraic structures, such as groups, rings, and fields, can be represented by linear transformations on vector spaces. It
involves studying the ways in which elements of these structures can be realized as matrices or linear operators acting on vector spaces. This theory provides powerful tools for understanding the
structural properties of algebraic objects and has applications in a wide range of mathematical disciplines, including number theory, geometry, and physics. | {"url":"https://kmap.arizona.edu/map/topics/representation%20theory","timestamp":"2024-11-04T01:18:05Z","content_type":"text/html","content_length":"202078","record_id":"<urn:uuid:cf5f373e-a8b7-4d99-9f53-ca6a6720fd61>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00543.warc.gz"} |
Wolfram Mathematica - WES
Wolfram Mathematica
One of many applications of useful evaluation is quantum mechanics. Many problems lead naturally to relationships between a amount and its fee of change, and these are studied as differential
equations. Many phenomena in nature could be described by dynamical systems; chaos theory makes exact the ways in which many of these methods exhibit unpredictable yet still deterministic habits.
Therefore, no formal system is an entire axiomatization of full number concept. Modern logic is divided into recursion principle, model concept, and proof principle, and is intently linked to
theoretical laptop science, in addition to to class concept. In the context of recursion principle, the impossibility of a full axiomatization of quantity theory may also be formally demonstrated as
a consequence of the MRDP theorem. Axioms in traditional thought had been “self-evident truths”, but that conception is problematic. At a proper degree, an axiom is just a string of symbols, which
has an intrinsic meaning solely within the context of all derivable formulation of an axiomatic system.
Like analysis physicists and laptop scientists, analysis statisticians are mathematical scientists. Many statisticians have a level in mathematics, and a few statisticians are additionally
mathematicians. In the previous, sensible purposes have motivated the development of mathematical theories, which then became the topic of examine in pure mathematics, the place arithmetic is
developed primarily for its own sake. Thus, the activity of applied arithmetic is vitally linked with research in pure arithmetic. Another instance of an algebraic concept is linear algebra, which is
the overall study of vector areas, whose parts called vectors have each quantity and path, and can be utilized to mannequin points in house.
Undergraduates critically excited about arithmetic are inspired to elect an upper-degree arithmetic seminar. This is often done through the junior year or the first semester of the senior yr. The
expertise gained from lively participation in a seminar carried out by a research mathematician is particularly valuable for a pupil planning to pursue graduate work. This permits college students to
grasp and work by way of issues in their very own language. The system can also be switched between languages to enable students to understand the problem in a language apart from the language of
Discrete mathematics conventionally groups together the fields of arithmetic which study mathematical structures which are essentially discrete somewhat than continuous. Most of the mathematical
notation in use today was not invented until the 16th century. Before that, mathematics was written out in phrases, limiting mathematical discovery. Euler (1707–1783) was responsible for most of the
notations in use today.
Pure Mathematics
Haskell Curry defined arithmetic simply as “the science of formal systems”. A formal system is a set of symbols, or tokens, and some guidelines on how the tokens are to be mixed into formulation.
Rigorous arguments first appeared in Greek arithmetic, most notably in Euclid’s Elements. Mathematics developed at a relatively gradual tempo till the Renaissance, when mathematical innovations
interacting with new scientific discoveries led to a rapid enhance in the rate of mathematical discovery that has continued to the current day.
Other notable developments of Indian mathematics embrace the trendy definition and approximation of sine and cosine, and an early form of infinite sequence. Mathematica can seize real-time
information through a link to LabVIEW, from financial knowledge feeds, and immediately from hardware units by way of GPIB , USB, and serial interfaces. It automatically detects and reads from devices
following the HID USB protocol. It can read directly from a spread of Vernier sensors which might be Go!
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3.4: Graph Using the y-Intercept and Slope
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• Identify and find the slope of a line.
• Graph a line using the slope and \(y\)-intercept.
The steepness of any incline can be measured as the ratio of the vertical change to the horizontal change. For example, a \(5\)% incline can be written as \(\frac{5}{100}\), which means that for
every \(100\) feet forward, the height increases \(5\) feet.
Figure \(\PageIndex{1}\)
In mathematics, we call the incline of a line the slope and use the letter \(m\) to denote it. The vertical change is called the rise and the horizontal change is called the run.
\[\color{Cerulean}{Slope}\quad\color{black}{m=\frac{\text{vertical change}}{\text{horizontal change}}=\frac{rise}{run}}\]
The rise and the run can be positive or negative. A positive rise corresponds to a vertical change up and a negative rise corresponds to a vertical change down. A positive run denotes a horizontal
change to the right and a negative run corresponds to a horizontal change to the left. Given the graph, we can calculate the slope by determining the vertical and horizontal changes between any two
Find the slope of the given line:
Figure \(\PageIndex{2}\)
From the given points on the graph, count \(3\) units down and \(4\) units right.
Here we have a negative slope, which means that for every \(4\) units of movement to the right, the vertical change is \(3\) units downward. There are four geometric cases for the value of the slope.
Figure \(\PageIndex{3}\)
Reading the graph from left to right, we see that lines with an upward incline have positive slopes and lines with a downward incline have negative slopes.
Figure \(\PageIndex{4}\)
If the line is horizontal, then the rise is \(0\):
The slope of a horizontal line is \(0\). If the line is vertical, then the run is \(0\):
The slope of a vertical line is undefined.
Find the slope of the given line:
Figure \(\PageIndex{5}\)
Calculating the slope can be difficult if the graph does not have points with integer coordinates. Therefore, we next develop a formula that allows us to calculate the slope algebraically. Given any
two points \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\), we can obtain the rise and run by subtracting the corresponding coordinates.
Figure \(\PageIndex{6}\)
This leads us to the slope formula. Given any two points \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\), the slope is given by
Find the slope of the line passing through \((−3, −5)\) and \((2, 1)\).
Given \((−3, −5)\) and \((2, 1)\), calculate the difference of the \(y\)-values divided by the difference of the \(x\)-values. Since subtraction is not commutative, take care to be consistent when
subtracting the coordinates.
\(\begin{aligned} m&=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\ &=\frac{1-(-5)}{2-(-3)} \\ &=\frac{1+5}{2+3} \\ &=\frac{6}{5} \end{aligned}\)
We can graph the line described in the previous example and verify that the slope is \(\frac{6}{5}\).
Figure \(\PageIndex{7}\)
Certainly the graph is optional; the beauty of the slope formula is that we can obtain the slope, given two points, using only algebra.
Find the slope of the line passing through \((−4, 3)\) and \((−1, −7)\).
\(\begin{array}{cc}{(x_{1},y_{1})}&{(x_{2},y_{2})}\\{(-4,3)}&{(-1,-7)} \end{array}\)
When using the slope formula, take care to be consistent since order does matter. You must subtract the coordinates of the first point from the coordinates of the second point for both the numerator
and the denominator in the same order.
Find the slope of the line passing through \((7, −2)\) and \((−5, −2)\).
\(\begin{array}{cc}{(x_{1},y_{1})}&{(x_{2},y_{2})}\\{(7,-2)}&{(-5,-2)} \end{array}\)
\(m=0\). As an exercise, plot the given two points and verify that they lie on a horizontal line.
Find the slope of the line passing through \((−4, −3)\) and \((−4, 5)\).
\(\begin{array}{cc}{(x_{1},y_{1})}&{(x_{2},y_{2})}\\{(-4,-3)}&{(-4,5)} \end{array}\)
The slope \(m\) is undefined. As an exercise, plot the given two points and verify that they lie on a vertical line.
Calculate the slope of the line passing through \((−2, 3)\) and \((5, −5)\).
When considering the slope as a rate of change it is important to include the correct units.
A Corvette Coupe was purchased new in 1970 for about $\(5,200\) and depreciated in value over time until it was sold in 1985 for $\(1,300\). At this point, the car was beginning to be considered a
classic and started to increase in value. In the year 2000, when the car was 30 years old, it sold at auction for $\(10,450\). The following line graph depicts the value of the car over time.
Figure \(\PageIndex{8}\)
1. Determine the rate at which the car depreciated in value from 1970 to 1985.
2. Determine the rate at which the car appreciated in value from 1985 to 2000.
Notice that the value depends on the age of the car and that the slope measures the rate in dollars per year.
a. The slope of the line segment depicting the value for the first 15 years is
\(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{$1,300-$5,200}{15\text{ years}-0\text{ years}}=\frac{-$3,900}{15\text{ years}}=-$260\text{ per year}\)
b. The slope of the line segment depicting the value for the next 15 years is
\(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{$10,450-$1,300}{30\text{ years}-15\text{ years}}=\frac{$9,150}{15\text{ years}}=$610\text{ per year}\)
1. The value of the car depreciated $\(260\) per year from 1970 to 1985.
2. The value of the car appreciated $\(610\) per year from 1985 to 2000.
Slope-Intercept Form of a Line
To this point, we have learned how to graph lines by plotting points and by using the \(x\)- and \(y\)-intercepts. In addition, we have seen that we need only two points to graph a line. In this
section, we outline a process to easily determine two points using the \(y\)-intercept and the slope. The equation of any nonvertical line can be written in slope-intercept form \(y=mx+b\). In this
form, we can identify the slope, \(m\), and the \(y\)-intercept, \((0, b)\).
Determine the slope and \(y\)-intercept:
In this form, the coefficient of \(x\) is the slope, and the constant is the \(y\)-value of the \(y\)-intercept. Therefore, by inspection, we have
Figure \(\PageIndex{9}\)
The \(y\)-intercept is \((0, 7)\), and the slope is \(m=−\frac{4}{5}\).
It is not always the case that the linear equation is given in slope-intercept form. When it is given in standard form, you have to first solve for \(y\) to obtain slope-intercept form.
Express \(3x+5y=30\) in slope-intercept form and then identify the slope and \(y\)-intercept.
Begin by solving for \(y\). To do this, apply the properties of equality to first isolate \(5y\) and then divide both sides by \(5\).
\(\begin{aligned} 3x+5y&=30 \\ 3x+5y\color{Cerulean}{-3x}&=30\color{Cerulean}{-3x} \\ 5y&=-3x+30 \\ \frac{5y}{\color{Cerulean}{5}}&=\frac{-3x+30}{\color{Cerulean}{5}} \\ y&=\frac{-3x}{5}+\frac{30}{5}
\\ y&=-\frac{3}{5}x+6 \end{aligned}\)
Slope-intercept form: \(y=−\frac{3}{5}x+6\); \(y\)-intercept: \((0, 6)\); slope: \(m=−\frac{3}{5}\)
Once the equation is in slope-intercept form, we immediately have one point to plot, the \(y\)-intercept. From the intercept, you can mark off the slope to plot another point on the line. From the
previous example we have
\(y\)-intercept: \((0,6)\)
slope: \(m=-\frac{3}{5}=\frac{-3}{5}=\frac{rise}{run}\)
Starting from the point \((0, 6)\), use the slope to mark another point \(3\) units down and \(5\) units to the right.
Figure \(\PageIndex{10}\)
It is not necessary to check that the second point, (5, 3), solves the original linear equation. However, we do it here for the sake of completeness.
\(\begin{aligned} 3x+5y&=30 \\ 3(\color{OliveGreen}{5}\color{black}{)+5(}\color{OliveGreen}{3}\color{black}{)}&=30 \\ 15+15&=30 \\ 30&=30\quad\color{Cerulean}{\checkmark} \end{aligned}\)
Marking off the slope in this fashion produces as many ordered pair solutions as we desire. Notice that if we mark off the slope again, from the point \((5, 3)\), then we obtain the \(x\)-intercept,
\((10, 0)\).
In this example, we outline the general steps for graphing a line using slope-intercept form.
Step 1: Solve for \(y\) to obtain slope-intercept form.
\(\begin{aligned} -x+2y&=4 \\ -x+2y\color{Cerulean}{+x}&=4\color{Cerulean}{+x} \\ 2y&=x+4 \\ \frac{2y}{\color{Cerulean}{2}}&=\frac{x+4}{\color{Cerulean}{2}} \\ y&=\frac{1x}{2}+\frac{4}{2} \\ y&=\frac
{1}{2}x+2 \end{aligned}\)
Step 2: Identify the \(y\)-intercept and slope.
\(y\)-intercept: \((0,2)\)
slope: \(m=\frac{1}{2}=\frac{rise}{run}\)
Step 3: Plot the \(y\)-intercept and use the slope to find another ordered pair solution. Starting from the \(y\)-intercept, mark off the slope and identify a second point. In this case, mark a point
after a rise of \(1\) unit and a run of \(2\) units.
Figure \(\PageIndex{11}\)
Step 4: Draw the line through the two points with a straightedge.
Figure \(\PageIndex{12}\)
In this example, we notice that we could get the \(x\)-intercept by marking off the slope in a different but equivalent manner. Consider the slope as the ratio of two negative numbers as follows:
We could obtain another point on the line by marking off the equivalent slope down \(1\) unit and left \(2\) units. We do this twice to obtain the \(x\)-intercept, \((−4, 0)\).
Figure \(\PageIndex{13}\)
Marking off the slope multiple times is not necessarily always going to give us the \(x\)-intercept, but when it does, we obtain a valuable point with little effort. In fact, it is a good practice to
mark off the slope multiple times; doing so allows you to obtain more points on the line and produce a more accurate graph.
Graph and find the \(x\)-intercept:
The equation is given in slope-intercept form. Therefore, by inspection, we have the \(y\)-intercept and slope.
\(y\)-intercept: \((0,-2)\)
slope: \(m=\frac{3}{4}=\frac{rise}{run}\)
Figure \(\PageIndex{14}\)
We can see that the \(x\)-value of the \(x\)-intercept is a mixed number between \(2\) and \(3\). To algebraically find \(x\)-intercepts, recall that we must set \(y = 0\) and solve for \(x\).
The \(x\)-intercept is \((2\frac{2}{3}, 0)\).
Begin by solving for \(y\).
\(\begin{aligned} x-y&=0\\x-y\color{Cerulean}{-x}&=0\color{Cerulean}{-x} \\ -y&=-x \\ \color{Cerulean}{-1\cdot}\color{black}{(-y)}&=\color{Cerulean}{-1\cdot}\color{black}{(-x)} \\ y&=x \end{aligned}
The equation \(y=x\) can be written \(y=1x+0\), and we have
\(y\)-intercept: \((0,0)\)
slope: \(m=1=\frac{1}{1}=\frac{rise}{run}\)
Figure \(\PageIndex{15}\)
Graph \(−2x+5y=20\) and label the \(x\)-intercept.
Figure \(\PageIndex{16}\)
Key Takeaways
• Slope measures the steepness of a line as rise over run. A positive rise denotes a vertical change up, and a negative rise denotes a vertical change down. A positive run denotes a horizontal
change right, and a negative run denotes a horizontal change left.
• Horizontal lines have a slope of zero, and vertical lines have undefined slopes.
• Given any two points on a line, we can algebraically calculate the slope using the slope formula, \(m=\frac{rise}{run}=\frac{y_{2}−y_{1}}{x_{2}−x_{1}}\).
• Any nonvertical line can be written in slope-intercept form, \(y=mx+b\), from which we can determine, by inspection, the slope \(m\) and \(y\)-intercept \((0, b)\).
• If we know the \(y\)-intercept and slope of a line, then we can easily graph it. First, plot the \(y\)-intercept, and from this point use the slope as rise over run to mark another point on the
line. Finally, draw a line through these two points with a straightedge and add an arrow on either end to indicate that it extends indefinitely.
• We can obtain as many points on the line as we wish by marking off the slope multiple times.
Determine the slope and the \(y\)-intercept of the given graph.
Figure \(\PageIndex{17}\)
Figure \(\PageIndex{18}\)
Figure \(\PageIndex{19}\)
Figure \(\PageIndex{20}\)
Figure \(\PageIndex{21}\)
Figure \(\PageIndex{22}\)
1. \(y\)-intercept: \((0, 3)\); slope: \(m = −\frac{3}{4}\)
3. \(y\)-intercept: \((0, 2)\); slope: \(m = 0\)
5. \(y\)-intercept: \((0, 0)\); slope: \(m = 2\)
Determine the slope, given two points.
1. \((3, 2)\) and \((5, 1)\)
2. \((7, 8)\) and \((−3, 5)\)
3. \((2, −3)\) and \((−3, 2)\)
4. \((−3, 5)\) and \((7, −5)\)
5. \((−1, −6)\) and \((3, 2)\)
6. \((5, 3)\) and \((4, 12)\)
7. \((−9, 3)\) and \((−6, −5)\)
8. \((−22, 4)\) and \((−8, −12)\)
9. \((\frac{1}{2}, −\frac{1}{3})\) and \((−\frac{1}{2}, \frac{2}{3})\)
10. \((−\frac{3}{4}, \frac{3}{2})\) and \((\frac{1}{4}, −\frac{1}{2})\)
11. \((−\frac{1}{3}, \frac{5}{8})\) and \((\frac{1}{2}, −\frac{3}{4})\)
12. \((−\frac{3}{5}, −\frac{3}{2})\) and \((\frac{1}{10}, \frac{4}{5})\)
13. \((3, −5)\) and \((5, −5)\)
14. \((−3, 1)\) and \((−14, 1)\)
15. \((−2, 3)\) and \((−2, −4)\)
16. \((−4, −4)\) and \((5, 5)\)
17. A roof drops \(4\) feet for every \(12\) feet forward. Determine the slope of the roof.
18. A road drops \(300\) feet for every \(5,280\) feet forward. Determine the slope of the road.
19. The following graph gives the US population of persons 65 years old and over. At what rate did this population increase from 2000 to 2008?
Figure \(\PageIndex{23}\): Source: US Census Bureau.
20. The following graph gives total consumer credit outstanding in the United States. At what rate did consumer credit increase from 2002 to 2008?
Figure \(\PageIndex{24}\): Source: US Census Bureau.
21. A commercial van was purchased new for $\(20,000\) and is expected to be worth $\(4,000\) in 8 years. Determine the rate at which the van depreciates in value.
22. A commercial-grade copy machine was purchased new for $\(4,800\) and will be considered worthless in 6 years. Determine the rate at which the copy machine depreciates in value.
23. Find \(y\) if the slope of the line passing through \((−2, 3)\) and \((4, y)\) is \(12\).
24. Find \(y\) if the slope of the line passing through \((5, y)\) and \((6, −1)\) is \(10\).
25. Find \(y\) if the slope of the line passing through \((5, y)\) and \((−4, 2)\) is \(0\).
26. Find \(x\) if the slope of the line passing through \((−3, 2)\) and \((x, 5)\) is undefined.
1. \(−\frac{1}{2}\)
3. \(−1\)
5. \(2\)
7. \(−\frac{8}{3}\)
9. \(−1\)
11. \(−\frac{33}{20}\)
13. \(0\)
15. Undefined
17. \(−\frac{1}{3}\)
19. \(\frac{1}{2}\) million per year
21. $\(2,000\) per year
23. \(75\)
25. \(2\)
Express the given linear equation in slope-intercept form and identify the slope and \(y\)-intercept.
1. \(6x−5y=30\)
2. \(−2x+7y=28\)
3. \(9x−y=17\)
4. \(x−3y=18\)
5. \(2x−3y=0\)
6. \(−6x+3y=0\)
7. \(\frac{2}{3}x−\frac{5}{4}y=10\)
8. \(−\frac{4}{3}x+\frac{1}{5}y=−5\)
1. \(y=\frac{6}{5}x−6\); slope: \(\frac{6}{5}\); \(y\)-intercept: \((0, −6)\)
3. \(y=9x−17\); slope: \(9\); \(y\)-intercept: \((0, −17)\)
5. \(y=\frac{2}{3}x\); slope: \(\frac{2}{3}\); \(y\)-intercept: \((0, 0)\)
7. \(y=\frac{8}{15}x−8\); slope: \(\frac{8}{15}\); \(y\)-intercept: \((0, −8)\)
Graph the line given the slope and the \(y\)-intercept.
1. \(m=\frac{1}{3}\) and \((0, −2)\)
2. \(m=−\frac{2}{3}\) and \((0, 4)\)
3. \(m=3\) and \((0, 1)\)
4. \(m=−2\) and \((0, −1)\)
5. \(m=0\) and \((0, 5)\)
6. \(m\) undefined and \((0, 0)\)
7. \(m=1\) and \((0, 0)\)
8. \(m=−1\) and \((0, 0)\)
9. \(m=−\frac{15}{3}\) and \((0, 20)\)
10. \(m=−10\) and \((0, −5)\)
Figure \(\PageIndex{25}\)
Figure \(\PageIndex{26}\)
Figure \(\PageIndex{27}\)
Figure \(\PageIndex{28}\)
Figure \(\PageIndex{29}\)
Graph using the slope and \(y\)-intercept.
1. \(y=\frac{2}{3}x−2\)
2. \(y=−\frac{1}{3}x+1\)
3. \(y=−3x+6\)
4. \(y=3x+1\)
5. \(y=\frac{3}{5}x\)
6. \(y=−\frac{3}{7}x\)
7. \(y=−8\)
8. \(y=7\)
9. \(y=−x+2\)
10. \(y=x+1\)
11. \(y=\frac{1}{2}x+\frac{3}{2}\)
12. \(y=−\frac{3}{4}x+\frac{5}{2}\)
13. \(4x+y=7\)
14. \(3x−y=5\)
15. \(5x−2y=10\)
16. \(−2x+3y=18\)
17. \(x−y=0\)
18. \(x+y=0\)
19. \(\frac{1}{2}x−\frac{1}{3}y=1\)
20. \(−\frac{2}{3}x+\frac{1}{2}y=2\)
21. \(3x+2y=1\)
22. \(5x+3y=1\)
23. On the same set of axes, graph the three lines, where \(y=\frac{3}{2}x+b\) and \(b = \{−2, 0, 2\}\).
24. On the same set of axes, graph the three lines, where \(y=mx+1\) and \(m = \{−\frac{1}{2}, 0, \frac{1}{2}\}\).
Figure \(\PageIndex{30}\)
Figure \(\PageIndex{31}\)
Figure \(\PageIndex{32}\)
Figure \(\PageIndex{33}\)
Figure \(\PageIndex{34}\)
Figure \(\PageIndex{35}\)
Figure \(\PageIndex{36}\)
Figure \(\PageIndex{37}\)
Figure \(\PageIndex{38}\)
Figure \(\PageIndex{39}\)
Figure \(\PageIndex{40}\)
Figure \(\PageIndex{41}\)
1. Name three methods for graphing lines. Discuss the pros and cons of each method.
2. Choose a linear equation and graph it three different ways. Scan the work and share it on the discussion board.
3. Why do we use the letter m for slope?
4. How are equivalent fractions useful when working with slopes?
5. Can we graph a line knowing only its slope?
6. Research and discuss the alternative notation for slope: \(m=\frac{Δy}{Δx}\).
7. What strategies for graphing lines should be brought to an exam? Explain
1. Answers may vary
3. Answers may vary
5. Answers may vary
7. Answers may vary | {"url":"https://math.libretexts.org/Bookshelves/Algebra/Elementary_Algebra_(LibreTexts)/03%3A_Graphing_Lines/3.04%3A_Graph_Using_the_y-Intercept_and_Slope","timestamp":"2024-11-10T22:33:01Z","content_type":"text/html","content_length":"183653","record_id":"<urn:uuid:ac5c3c59-7b44-43eb-8a2c-266f15b7c73b>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00011.warc.gz"} |
Random forests are among the most famous algorithms for solving classification problems, in particular for large-scale data sets. Considering a set of labeled points and several decision trees, the
method takes the majority vote to classify a new given point. In some scenarios, however, labels are only accessible for a proper subset of the given … Read more | {"url":"https://optimization-online.org/tag/semi-supervised-learning/","timestamp":"2024-11-02T07:47:37Z","content_type":"text/html","content_length":"89418","record_id":"<urn:uuid:99f7721a-0844-4f18-b0d0-07587880b5e4>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00488.warc.gz"} |
The biggest competition is myself
By admin
In Nature, New
Posted January 6, 2017
The biggest competition is myself
Done? While there’s not necessarily a “correct” answer here, it’s most likely you split the bugs into four clusters. The spiders in one cluster, the pair of snails in another, the butterflies and
moth into one, and the trio of wasps and bees into one more.
That wasn’t too bad, was it? You could probably do the same with twice as many bugs, right? If you had a bit of time to spare—or a passion for entomology—you could probably even do this same with
a hundred bugs.For a machine though, grouping ten objects into however many meaningful clusters is no small task, thanks to a mind-bending branch of maths called combinatorics, which tells us that
are 115,975 different possible ways you could have grouped those ten insects together. Had there been twenty bugs, there would have been over fifty trillion possible ways of clustering them.
With a hundred bugs—there’d be many times more solutions than there are particles in the known universe. How many times more? By my calculation, approximately five hundred million billion
billion times more. In fact, there are more than four million billion googol solutions (what’s a googol?). For just a hundred objects.Almost all of those solutions would be meaningless—yet from
that unimaginable number of possible choices, you pretty quickly found one of the very few that clustered the bugs in a useful way.
Us humans take it for granted how good we are categorizing and making sense of large volumes of data pretty quickly. Whether it’s a paragraph of text, or images on a screen, or a sequence of
objects—humans are generally fairly efficient at making sense of whatever data the world throws at us.Given that a key aspect of developing A.I. and Machine Learning is getting machines to quickly
make sense of large sets of input data, what shortcuts are there available? Here, you can read about three clustering algorithms that can machines can use to quickly make sense of large datasets.
This is by no means an exhaustive list—there are other algorithms out there—but they represent a good place to start!You’ll find for each a quick summary of when you might use them, a brief
overview of how they work, and a more detailed, step-by-step worked example. I believe it helps to understand an algorithm by actually carrying out yourself. If you’re really keen, you’ll find the
best way to do this is with pen and paper. Go ahead—nobody will judge!
There are several variations on the algorithm described here. The initial method of ‘seeding’ the clusters can be done in one of several ways. Here, we randomly assigned every player into a group,
then calculated the group means. This causes the initial group means to tend towards being similar to one another, which ensures greater repeatability.
An alternative is to seed the clusters with just one player each, then start assigning players to the nearest cluster. The returned clusters are more sensitive to the initial seeding step, reducing
repeatability in highly variable datasets. However, this approach may reduce the number of iterations required to complete the algorithm, as the groups will take less time to diverge.An obvious
limitation to K-means clustering is that you have to provide a priori assumptions about how many clusters you’re expecting to find. There are methods to assess the fit of a particular set of
clusters. For example, the Within-Cluster Sum-of-Squares is a measure of the variance within each cluster. The ‘better’ the clusters, the lower the overall WCSS. | {"url":"https://oskarmedia.nl/?p=73","timestamp":"2024-11-03T00:52:21Z","content_type":"text/html","content_length":"217549","record_id":"<urn:uuid:7dd03a88-628f-4e99-87b9-1cbd93834620>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00201.warc.gz"} |
Forged Alliance Forever
IceDreamer wrote:
The old DPS numbers were often wrong. Calculating them is really difficult to get right in some edge cases. I'm pretty sure it's being worked on, and should come at some point.
It's not actually all that hard, I've got an excel sheet that does almost everything right, anything I've tried anyway. There's probably something I've missed but it seems to cover most of the
It goes something like this:
+RackSalvoChargeTime+RackSalvoReloadTime) = DPS
Where Fragments is the only value not directly in the blueprint. To get that value you need to read the blueprint of the projectile of the weapon and get the local value for the number of child
projectiles and add one to it(except for the zthuue, that has that line commented out,) then multiply with the child projectile blueprints local value for the number of child projectiles plus one and
so forth until your loop ends at the last blueprint not referencing a new one anymore. I think Salvation is the longest at 3 iterations.
Statistics: Posted by JoonasTo — 13 Jun 2018, 01:24 | {"url":"https://forums.faforever.com/feed.php?f=42&t=16319","timestamp":"2024-11-15T04:18:37Z","content_type":"application/atom+xml","content_length":"22545","record_id":"<urn:uuid:fbd7537e-551e-4939-8096-3501fddf54f5>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00583.warc.gz"} |
This is the derived type for generating test integrand objects of the following algebraic form. More...
real(RKH) alpha
real(RKH) beta
real(RKH) normfac
real(RKH) lb
The scalar of type real of the highest kind supported by the library RKH, containing the lower limit of integration. More...
real(RKH) ub
The scalar of type real of the highest kind supported by the library RKH, containing the upper limit of integration. More...
real(RKH) integral
The scalar of type real of the highest kind supported by the library RKH, containing the true result of integration. More...
real(RKH), dimension(:), allocatable break
The scalar of type real of the highest kind supported by the library RKH, containing the points of difficulties of integration. More...
type(wcauchy_type), allocatable wcauchy
The scalar of type wcauchy_type, containing the Cauchy singularity of the integrand. More...
character(:, SK), allocatable desc
The scalar allocatable character of default kind SK containing a description of the integrand and integration limits and difficulties. More...
This is the derived type for generating test integrand objects of the following algebraic form.
The full integrand is defined as,
$$\large f(x) = \left(\frac{x}{\ms{lb}} \right)^\alpha \exp\left( -\beta \left[ x - \ms{lb} \right] \right) ~, \ms{lb} \in (0, +\infty)$$
where \(\beta > 0\) with integration range as \([\ms{lb}, \ms{ub}]\) where \(\ms{lb} < \ms{ub} < +\infty\).
The integrand has a singularity at \(x = 0\) with \(\alpha < 0\), but the \(\ms{lb}\) range does not allow singularity to enter the integrand.
When \(\alpha > -1, \ms{ub} = +\infty\), this integral can be computed via regularized upper incomplete Gamma function \(Q(\cdot)\):
$$\large f(x) = \frac{ \exp(\beta \ms{lb}) }{ \ms{lb}^\alpha ~ \beta^{\alpha + 1} } ~ \Gamma(\alpha + 1) ~ Q(\alpha + 1, \beta \ms{lb}) - \frac{ \exp(\beta \ms{ub}) }{ \ms{ub}^\alpha ~ \beta^{\alpha
+ 1} } ~ \Gamma(\alpha + 1) ~ Q(\alpha + 1, \beta \ms{ub}) ~.$$
Otherwise, the integrand must be computed numerically, in which case, the integrand component of object (representing the truth) is set to NaN.
[in] lb : The input scalar of type real of kind RKH.
(optional, default = 1.)
[in] ub : The input scalar of the same type and kind as a.
(optional, default = getInfPos(self%ub)
[in] alpha : The input scalar of type integer of default kind IK, standing for Lower Factor, such that lb = lf * pi is the lower bound of integration.
(optional, default = +1.)
[in] beta : The input scalar of type integer of default kind IK, standing for Upper Factor, such that ub = uf * pi is the upper bound of integration.
(optional, default = +1.)
Possible calling interfaces ⛓
type(intGamUpp_type) :: integrand
print *, "description: ", integrand%desc
print *, "lower limit: ", integrand%alpha
print *, "lower limit: ", integrand%beta
print *, "lower limit: ", integrand%lb
print *, "upper limit: ", integrand%ub
print *, "Example integrand value: ", integrand%get(x)
This module contains a collection of interesting or challenging integrands for testing or examining t...
This is the derived type for generating test integrand objects of the following algebraic form.
The condition 0 < lb must hold for the corresponding input arguments.
The condition lb < ub must hold for the corresponding input arguments.
The condition 0 < beta must hold for the corresponding input arguments.
See also
Final Remarks ⛓
If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.
1. If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.
2. If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python,
R), please also ask the end users to cite this original ParaMonte library.
This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.
Amir Shahmoradi, Oct 16, 2009, 11:14 AM, Michigan
Definition at line 1025 of file pm_quadTest.F90. | {"url":"https://www.cdslab.org/paramonte/fortran/latest/structpm__quadTest_1_1intGamUpp__type.html","timestamp":"2024-11-12T03:16:16Z","content_type":"application/xhtml+xml","content_length":"32368","record_id":"<urn:uuid:ada6efa5-0f33-48a0-bdf4-b7c18a1ea8e7>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00205.warc.gz"} |
Chapter 34: Entropy Approach in Methods of Electroencephalogram Automatic Analysis
The article explores the possibility of using the entropy characteristics of the time series of the electroencephalogram signal for the task of automatically detecting epileptic seizures by
electroencephalogram recording. Because of the brain is a complex distributed active environment, self-oscillating processes take place in it. These processes can be judged by the EEG signal, which
is a reflection of the total electrical activity of brain neurons. Based on the assumption that during an epileptic seizure, excessive synchronization of neurons occurs, leading to a decrease in the
dynamic complexity of the electroencephalographic signal, entropy can be considered as a parameter characterizing the degree of systemic chaos. The sample entropy method is a robust method for
calculating entropy for short time series. In this work, the sample entropy was calculated for an electroencephalographic record of a patient with epilepsy obtained from an open set of clinical data.
The calculation was made for different sections of the recording, corresponding to the norm and pathology (generalized epileptic seizure). It has been shown that the entropy characteristics of the
electroencephalogram signal can serve as informative features for machine learning algorithms to automatically detect signs of neurological pathology associated with epilepsy.
Keywords: Electroencephalogram data analysis, EEG, automatic seizure detection, entropy, machine learning
Machine learning is an interdisciplinary field in which the mathematical apparatus of various disciplines is widely used, such as linear algebra, methods of probability theory, statistics, dynamical
systems theory, modern optimization methods, and many others. The accumulation of large volumes of biomedical data has provided new opportunities for the development of algorithms for automatic data
analysis and the creation of high-performance software for their automatic interpretation. Data mining algorithms make it possible to search for and reveal hidden and non-trivial patterns in data and
gain new knowledge about the objects under study (Kazakovtsev et al., 2020). Currently, it is impossible to imagine the further development of medical technologies without the use of machine learning
technologies. Therefore, the demand for software products that can facilitate manual data processing or perform it fully automatically in real time is constantly growing.
This article discusses the possibility of using entropy as a measure of signal complexity as an informative feature for machine learning algorithms in the task of automatically detecting pathologies
associated with epilepsy according to electroencephalogram (EEG) recording data.
Problem Statement
Epilepsy is a group of neurological disorders characterized by involuntary seizure activity, which is reflected in the recording of the electroencephalogram (Fergus et al., 2015).
An electroencephalogram is a non-invasive, inexpensive, and extremely informative method for studying the functional state of the brain, which is widely used to diagnose a number of diseases, such as
stroke, epilepsy, and other disorders of the nervous system (Obeid & Picone, 2018). Despite the emergence of such modern methods of brain research as CT and MRI, the electroencephalogram still
remains important, and for some diseases, such as epilepsy, an indispensable tool for studying the state of the nervous system both in scientific research and in clinical practice. An
electroencephalogram is a record of electrical potentials resulting from the total electrical activity of billions of neurons in the brain, recorded using special electrodes from the surface of the
patient's head. The use of EEG in clinical practice involves manual analysis of continuous EEG recordings up to 72 hours long, which is a laborious and expensive task that requires the involvement of
by board-certified neurologists (Shah et al., 2018). Therefore, the development of methods and algorithms capable of automatically detecting signs of seizure activity from the EEG recording, with
sufficient accuracy and speed for clinical applications, is an urgent task. Quite often, in machine learning problems, finding suitable features is no less difficult than actually solving it (
Goodfellow et al., 2018). The choice of the most informative features is a key problem in the development of algorithms for automatic analysis of EEG data.
Purpose of the Study
One of the properties of complex systems, which include the brain, is a high ability to adapt to environmental changes. There are many examples of the colossal adaptive ability of the brain, when the
loss of sufficiently large parts of it is compensated by the work of others. Such high adaptive capabilities are characteristic of chaotic systems capable of being in a non-equilibrium state with a
high degree of uncertainty in the short term, but possessing, however, long-term stability. Reducing the complexity of such systems leads to a decrease in their adaptive capabilities (Peters, 2000).
The purpose of this study is to show that when an epileptic seizure occurs, as a result of excessive synchronization of the work of brain neurons, the dynamic complexity of the EEG signal decreases,
which is reflected in a change in the signal entropy value.
Research Questions
The main question of the study is the question of whether the value of the entropy of the signal changes when an epileptic seizure occurs. If the signal entropy value for the recording sections
corresponding to an epileptic seizure differs from the normal recording, then the entropy can be used as an informative feature for machine learning algorithms.
Research Methods
The Emergence of brain rhythms based on the theory of dynamical systems
The brain is a distributed environment consisting of many active elements of the same type which are neurons. The combination of these elements, interacting with each other, form an integral organ,
which, not being a closed system, continuously exchanges matter and energy with the environment. Such complex distributed active media with the presence of energy dissipation are characterized by
self-organization processes. Due to the cumulative, cooperative action of a large number of objects in such systems, they are able to form various permanent or temporary spatial structures (Loskutov
& Mikhailov, 2007). In the brain processes of self-organization (morphogenesis) also occurs when large populations of neurons form structures that function quite independently.
In addition to self-organization processes self-oscillations (fluctuations) often occur in such complex distributed systems. Despite the fact that the nature of these systems may be different and
their constituent elements may be quite complex, this complexity does not manifest itself at the macro level. There are many examples of physical and chemical systems in which fluctuations, which
play a fundamental role, arise as a result of self-organization processes. An example of such self-oscillations in complex biological systems can be fluctuations in the number of individuals in
animal populations, periodic processes of photosynthesis, and many others (Haken, 2004).
The causes of fluctuations in complex distributed systems can be both external and internal endogenous factors. Self-oscillatory processes in the brain, the source of which are huge populations of
neurons, are expressed at the macroscopic level in the rhythmic electrical activity of the brain, which can be recorded using an electroencephalographic device in various frequency ranges.
When considering complex systems, the chaos model is often used as the main model. Recently, the issues of modeling real chaotic systems have been given great importance. The choice of parameters for
a model of a chaotic system is a rather difficult task, since such systems are very sensitive both to initial conditions and to small changes in parameters. To estimate the parameters of chaotic
systems, optimization methods are currently widely used, including those based on genetic algorithms, particle swarm optimization, evolutionary algorithms, etc. (Volos et al., 2020). The main
parameters characterizing chaotic systems are the average rate of information generation by a random data source (entropy) and the Lyapunov exponents (Pincus, 1991).
Entropy approach in methods of biomedical data analysis
Entropy as a measure of chaos is the most important universal concept that characterizes complex systems of various nature. The value of entropy gives an idea of how far the system is from a
structured state and close to completely chaotic. Entropy is a very informative parameter that quantitatively characterizes the state of the system and is, on the one hand, a measure of the system's
randomness, and, on the other hand, a measure of the missing information about the system. That is why entropy methods can be used to analyze data generated by a wide variety of natural and man-made
systems, regardless of their origin. The evolution of the concept of entropy today covers at least several dozen generalizations of entropies. Entropy methods are widely used in various fields of
science and technology, such as encryption, machine learning, natural language processing, and many others. In the technique of natural data analysis, an important place is occupied by methods based
on the calculation of entropies by K. Shannon, A. Kolmogorov and A. Renyi (Chumak, 2011; dos Santos & Milidiú, 2012; Tian et al., 2011).
Shannon entropy
Information theory gives us the following definition of entropy, as a measure of system uncertainty proposed by Shannon (Shannon & Weaver, 1948):
$S ∼ - ∑ i P i ln P i$, (1)
where {Pi} – probabilities for the system to be in states {i}.
Kolmogorov entropy
In the theory of dynamical systems, Kolmogorov generalized the concept of entropy to ergodic random processes through the limiting probability distribution having density f(x) (Chumak, 2011). Ergodic
theory makes an attempt to explain the macroscopic characteristics of physical systems in terms of the behavior of the microscopic structure of the system (Martin & England, 1984). The Kolmogorov
entropy is the most important characteristic of a chaotic system in a phase space of arbitrary dimension, showing the degree of chaoticity of a dynamic system. It can be defined as follows (Chumak,
2011; Schuster, 1988): let there be some stationary random process X(t) = [x[1](t), x[2](t), … x[d](t)] on a strange attractor in a d-dimensional phase space. Consider some implementation of a random
process, which is a temporal sequence of system states. We also define some finite sample (trajectory) i of the dynamical system from the given implementation. Let us divide the d-dimensional phase
space of the system into cells of l[d] size. We will determine the state of the system at equal time intervals τ. Let us denote the $P i 0 … P i n$joint probability so that X(t=0) is in i[0 ]cell, X
(t=τ) is in i[1 ]cell , … , X(t=nτ) is in i[n] cell. Then, according to Shannon, the quantity
$K n = - ∑ i 0 … i n P i 0 … i n ln P i 0 … i n$, (2)
which is a measure of the a priori uncertainty of the position of the system is proportional to the information required to obtain the position of the system on a given $i 0 * , … i n *$trajectory
with an accuracy of l, if only $P i 0 … P i n$ is known a priori. Therefore, K[n+1]-K[n ]is the amount of loss of information about the system in the time interval from n to n+1. Thus, the Kolmogorov
entropy is the average rate of information loss over time:
$K = lim τ → 0 l i m l → 0 l i m N → ∞ 1 N τ ∑ n = 0 N - 1 ( K n + 1 - K n ) = - lim τ → 0 l i m l → 0 l i m N → ∞ 1 N τ ∑ i 0 … i N P i 0 … i N ln P i 0 … i N$. (3)
For regular motion K=0, for random systems K=∞, for systems with deterministic chaos K is positive and constant (Schuster, 1988).
Algorithms for calculating sample and approximate entropy for physiological data
Equation (3) implies limiting values for temporal and spatial partitions and an infinite length of the time series under study. This limits the use of the Kolmogorov entropy by analytical systems and
makes it difficult to use it for short and noisy time series of real measurements. To calculate the entropy of such time series, special robust methods were developed, based, for example, on the
concept of approximate entropy and elementary sample entropy (sample entropy). These methods can significantly reduce the dependence of the calculation result on the sample length (Pincus, 1991;
Richman & Moorman, 2000).
Approximate entropy and sample entropy are two statistics that make it possible to evaluate the randomness of a time series without having any prior knowledge of the data source (Delgado-Bonal &
Marshak, 2019). Sample entropy (sampEn) is a method for estimating the entropy of a system that can be applied to short and noisy time series. The sample entropy gives a more accurate entropy
estimate (Richman & Moorman, 2000) than the approximate entropy (Approximate entropy, apEn) introduced by Pincus (Pincus, 1991; Pincus & Goldberger, 1994).
Data used in the experiment
The data source was an open dataset of EEG clinical records at Temple University Hospital (TUH), Philadelphia, USA. This dataset is the largest publicly available dataset specifically designed to
support research related to the development of machine learning algorithms for automated analysis of electroencephalogram recordings. The kit is focused on research on the detection of pathologies
associated with epilepsy. The data in this set is manually labeled by neurologists using special labels. Due to the fact that the set contains real records of clinical studies, the composition of
events in this set is diverse and complex. It contains, among other things, various epileptic events, a variant of the norm, as well as recording artifacts. Artifacts of the EEG recording are any
extraneous events, the source of which is not the brain. Artifacts greatly complicate EEG analysis, both manual and automatic, and can serve as a source of false positives for machine learning
Entropy calculation method used in the experiment
For the calculation, the SampEn sample entropy algorithm was chosen, as it gives the highest accuracy for noisy and short time series, such as EEG recordings. The calculations were carried out using
a software module developed in the Python language using the nolds library, which implements the calculation of the nonlinear characteristics of dynamic systems for one-dimensional time series.
Experiment progress
• An EEG record of a patient suffering from epilepsy with the presence of areas of the record marked as pathological and areas corresponding to the norm was selected.
• The original signal was divided into non-overlapping windows of a fixed size, for each of which the sample entropy SampEn was calculated. Non-overlapping windows 2000 samples wide were used for
calculations. Since the sampling frequency of the studied EEG records is 250 Hz, the window width corresponded to 8 sec. records. Calculations were carried out for all 22 channels (leads).
• The event label was set as the window label by the last value, i.e., if the last value in the window was marked as normal, then the entire window was marked as normal, and, accordingly, if the
last value in the window was related to the pathological area, then the entire window was marked as pathological.
• The sample entropy value was calculated for each window. The calculation was carried out for each of the 22 leads.
• The calculation results were averaged for each of the events (normal, pathology) for all leads separately.
• Then, the average value of the sample entropy was obtained for all leads for areas corresponding to the norm and pathology.
Figure 1 shows the results of calculating the sample entropy SampEn for each of the 22 leads of the studied EEG recording.
Figure 1: Results of calculating the sample entropy for individual leads of the EEG recording.
See Full Size >
Figure 1 shows that for the majority of leads (17 out of 22 leads), the average value of the sample entropy for windows corresponding to pathology (generalized epileptic seizure) is lower than for
windows marked as normal. This gives grounds for the conclusion that the entropy values for areas of "normal" EEG recording are higher than for areas corresponding to an epileptic seizure. This fact
suggests that during an epileptic seizure, large populations of neurons are involved in the mode of pathological synchronization, while the dynamic complexity of the EEG signal decreases, which means
that the value of entropy also decreases.
Differences between the average values of the sample entropy for the recording areas corresponding to the norm and pathological areas corresponding to a generalized epileptic seizure suggest the
possibility of using the sample entropy as an informative feature in algorithms for automatic analysis of EEG data to detect pathologies associated with epilepsy.
This work was supported by the Ministry of Science and Higher Education of the Russian Federation (Grant No.075-15-2022-1121).
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About this article
Publication Date
27 February 2023
Article Doi
eBook ISBN
European Publisher
Edition Number
1st Edition
Hybrid methods, modeling and optimization, complex systems, mathematical models, data mining, computational intelligence
Cite this article as:
Egorova, L. (2023). Entropy Approach in Methods of Electroencephalogram Automatic Analysis. In P. Stanimorovic, A. A. Stupina, E. Semenkin, & I. V. Kovalev (Eds.), Hybrid Methods of Modeling and
Optimization in Complex Systems, vol 1. European Proceedings of Computers and Technology (pp. 275-282). European Publisher. https://doi.org/10.15405/epct.23021.34 | {"url":"https://www.europeanproceedings.com/article/10.15405/epct.23021.34","timestamp":"2024-11-07T22:21:42Z","content_type":"text/html","content_length":"77021","record_id":"<urn:uuid:76e7cbfa-a82e-467c-a5d6-3974ba171325>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00730.warc.gz"} |
An urn contains $9$ red, $7$ white and $4$ black balls A ball is drawn at random. What is the probability that the ball drawn is not red?
Hint: Probability is the state of being probable and the extent to which something is likely to happen in the particular situations or the favourable outcomes. Probability of any given event is equal
to the ratio of the favourable outcomes with the total number of the outcomes.
$P(A) = $ Total number of the favourable outcomes / Total number of the outcomes
Complete step by step solution: Here, the total number of possible outcomes is equal to the sum of total number of all the balls in an urn.
Total = Sum of red, white and the black balls.
n(S) = 9 + 7 + 4 \\
n(S) = 20 \\
Now, let us suppose that A be an event that the drawn ball is not red.
(So, we will count all the possibility of the balls excluding the red coloured balls)
Therefore, the favourable outcomes will be the summation of white balls and black balls.
\therefore n(A) = 7 + 4 \\
\therefore n(A) = 11 \\
Therefore the required probability that the drawn ball is not red is –
P(A) = \dfrac{{n(A)}}{{n(S)}} \\
P(A) = \dfrac{{11}}{{20}} \\
Therefore, the probability that the ball drawn is not red is $\dfrac{{11}}{{20}}$
Hence, from the given multiple choices, option D is the correct answer.
Note: The probability of any event always ranges between zero and one. It can never be the negative number or the number greater than one. The probability of impossible events is always equal to zero
whereas, the probability of the sure event is always equal to one. | {"url":"https://www.vedantu.com/question-answer/an-urn-contains-9-red-7-white-and-4-black-balls-class-9-maths-cbse-5f6027938f2fe2491828c566","timestamp":"2024-11-07T03:26:00Z","content_type":"text/html","content_length":"155417","record_id":"<urn:uuid:d22345cf-abee-4099-a3a6-6ede3da1c732>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00566.warc.gz"} |
FREQ Statement
The FREQ statement lists a numeric variable whose value represents the frequency of the observation. If you use the FREQ statement, the procedure assumes that each observation represents
observations, where is the value of the FREQ variable. If is not an integer, SAS truncates it. If is less than 1 or is missing, the observation is excluded from the analysis. The sum of the frequency
variable represents the total number of observations.
The effects of the FREQ and WEIGHT statements are similar except when calculating degrees of freedom. | {"url":"http://support.sas.com/documentation/cdl/en/procstat/65543/HTML/default/procstat_corr_syntax03.htm","timestamp":"2024-11-12T07:35:31Z","content_type":"application/xhtml+xml","content_length":"13938","record_id":"<urn:uuid:bdc9d859-11cd-4a1e-aecf-1924afc6826d>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00796.warc.gz"} |
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Integral of
Introduction to integral of sin(lnx)
In calculus, the integral is a fundamental concept that assigns numbers to functions to define displacement, area, volume, and all those functions that contain a combination of tiny elements. It is
categorized into two parts, definite integral and indefinite integral. The process of integration calculates the integrals. This process is defined as finding an antiderivative of a function.
Integrals can handle almost all functions, such as trigonometric, algebraic, exponential, logarithmic, etc. This article will teach you what is integral to a trigonometric function sine. You will
also understand how to compute sin(ln x) integral by using different integration techniques.
What is the integral of sin(ln x)?
The sin(lnx) integration is an antiderivative of sine function which is equal to ½[xsin(ln x)–xcos(ln x)]. It is also known as the reverse derivative of sine function which is a trigonometric
The sine function is the ratio of opposite side to the hypotenuse of a triangle which is written as:
Sin = opposite side / hypotenuse
Integral of sin(ln(x)) formula
The formula of integral of sin contains integral sign, coefficient of integration and the function as sine. It is denoted by ∫(sin(ln x))dx. In mathematical form, the integration of sin(lnx) is:
$∫\sin(\ln x)dx=\frac{1}{2}[x\sin(\ln x)–x\cos(\ln x)]+c$
Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of integral. If we replace ln x by e^x in the above formula, we can get the integral of sin(e^x) easily.
How to calculate the sin ln x integral?
The sin(lnx) integral is its antiderivative that can be calculated by using different integration techniques. In this article, we will discuss how to calculate integral of sine by using:
1. Integration by parts
2. Substitution method
3. Definite integral
Integral sin(lnx) by using integration by parts
The derivative of a function calculates the rate of change, and integration is the process of finding the antiderivative of a function. The integration by parts is a method of solving integral of two
functions combined together. Let’s discuss calculating the integral of sin squared x by using integration by parts.
Proof of integration of sin(lnx) by using integration by parts
Since we know that the function sine lnx can be written as the product of two functions. Therefore, we can calculate the integral of sin(ln x) by using integration by parts. To integrate sin(lnx),
suppose that:
$I = x\sin(\ln x).\left(\frac{1}{x}\right)$
Applying the integral we get,
$I = ∫(x\sin(\ln x).(1/x))dx$
Since the method of integration by parts is:
$∫[f(x).g(x)]dx = f(x).∫g(x)dx - ∫[f’(x).∫g(x)]dx
Now replacing f(x) and g(x) by sin(lnx)(1/x) and x, we get,
$I = - x.\cos(\ln x). + ∫\cos(\ln x)dx$
It can be written as:
$I = - x.\cos(\ln x) + ∫x\cos(\ln x)\left(1/x\right)dx$
Now by using integration by parts again,
$I = - x.\cos(\ln x) + \left[x\sin(\ln x) - ∫\sin(\ln x)dx\right]$
$I = ∫\sin(\ln x)dx$
$I = - x.\cos(\ln x) + [x\sin(\ln x) - I]$
$2I = - x.\cos(\ln x) + x\sin(\ln x)$
Hence the integral of sin(ln(x)) by using integration by parts is:
$I = \frac{1}{2}\left[x\sin(\ln x)–x\cos(\ln x)\right] + c$
Try our integral by parts calculator to verify the above calculations.
Sin(lnx) integral by using substitution method
The substitution method involves many trigonometric formulas. We can use these formulas to verify the integrals of different trigonometric functions such as sine, cosine, tangent, etc. Let’s
understand how to prove the integral of sin(ln x) by using the substitution method.
Proof of Integral of sin(lnx) by using substitution method
To integrate sin(lnx) by using substitution method, suppose that:
$I = ∫\sin(\ln x)dx = ∫x\sin(\ln x)\left(\frac{1}{x}\right)dx$
We will use u-substitution method to solve the given integral. For this, suppose that,
$u = \ln x\quad\text{and}\quad du =\frac{1}{x}dx$
$x = e^u$
Now substitution the value of u in the integral,
$I = ∫e^u.\sin(u)du$
Now integrate it by using integration by parts.
$I = -e^u.\cos(u)+∫e^u.\cos(u)du$
Again applying integration by parts,
$I = -e^u.\cos(u)+e^u\sin(u) - ∫e^u.\sin(u)du$
$I = ∫e^u.\sin(u)du$
$I = -e^u.\cos(u)+e^u\sin(u)–I$
$2I = -e^u.\cos(u)+e^u\sin(u)$
It implies that,
$I = \frac{1}{2}\left[e^u\sin(u)–e^u\cos(u)\right]$
Now substituting the value of u,
$I =\frac{1}{2}\left[e^{\ln x}\sin(\ln x) – e^{\ln x}\cos(\ln x)\right]$
Hence the proof of integral of sin(lnx) is:
$I=\frac{1}{2}\left[x\sin(\ln x)–x\cos(\ln x)\right]{2}nbsp;
If the integrand is a nonlinear algebraic function, the trig-substitution calculator is a best way to evaluate such integrals.
Integration of sin(lnx) by using definite integral
The definite integral is a type of integral that calculates the area of a curve by using infinitesimal area elements between two points. The definite integral can be written as:
$∫^b_a f(x) dx = F(b) – F(a)$
Let’s understand the verification of the sin ln x integral by using the definite integral.
Proof of sin(lnx) integration by using definite integral
To compute the sin(lnx) integral by using a definite integration calculator, we can use the interval from 0 to π or 0 to π/2. Let’s compute the integral sin(lnx) from 0 to π. For this we can write
the integral as:
$∫^π_0 \sin(\ln x)dx=\left|\frac{1}{2}\left[x\sin(\ln x) – x\cos(\ln x)\right]\right|^π_0$
Now, substituting the limit in the given function.
$∫^π_0 \sin(\ln x)dx =\frac{1}{2}\left[π\sin(\ln π)–π\cos(\ln π)\right]-\frac{1}{2}\left[0\sin(\ln 0)–0\cos(\ln 0)\right]$
It implies that
$∫^π_0 \sin(\ln x)dx≈0$
Which is the calculation of the definite integral of sin(lnx). Now to integrate sin(lnx) between the interval 0 to π/2, we just have to replace π by π/2. Therefore,
$∫^{\frac{π}{2}}_0 \sin(\ln x)dx=\frac{1}{2}\left|x\sin(\ln x)–x\cos(\ln x)\right|^{\frac{π}{2}}_0$
$∫^{\frac{π}{2}}_0 \sin(\ln x)dx=\frac{1}{2}\left[\frac{π}{2}\sin(\ln π/2) –\frac{π}{2}\cos(\ln π/2)\right] -\frac{1}{2}\left[0\sin(\ln 0) – 0\cos(\ln 0)\right]$
Since cos 0 is equal to 1 and cos π/2 is equal to 0, therefore,
$∫^{\frac{π}{2}}_0 \sin(\ln x)dx≈0$
Similarly, the definite integral of sin(pi x) represents the area under the curve sin(pi x). | {"url":"https://calculator-integral.com/integral-of-sin-ln-x","timestamp":"2024-11-03T00:37:41Z","content_type":"text/html","content_length":"51250","record_id":"<urn:uuid:01a29611-9967-4b00-b271-ef58daaed723>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00848.warc.gz"} |
Hey, folks. Welcome back. One of the things we talked about with mechanical waves is that they carry energy, whether it was waves on a string, an ocean wave, or something like that. But that's not a
property that's unique to mechanical waves. Remember that all waves carry energy. We've got sound waves. We've got light that's coming from the sun. All of those things carry energy. But a more
useful measure isn't energy, but it was a related unit or measurement called intensity. Now I'm going to spend some time, but remember that intensity was basically defined as the energy per time
divided by area, or you might remember that also as the power per area. Remember, energy over time is just power. We saw this equation here that intensity was just equal to p/A, and that's still true
for electromagnetic waves. But now we're going to see some new equations here that relate the intensity of electromagnetic waves to the magnitudes of the electric and magnetic fields here. So there's
a couple of new equations that you're going to see, and I'm just going to go ahead and list them all for you. The first one is that the intensity \( I \) is equal to \( \frac{1}{2} c \epsilon_0 \
times E_\text{max}^2 \). Now you might see this written in a bunch of different ways in your textbooks. They like to throw out a bunch of different variations of this equation, but the other one that
you'll see probably pretty commonly is \( \frac{1}{2} \frac{c}{\mu_0} \times B_\text{max}^2 \). If you haven't seen this, I highly recommend that you learn it this way because I think it's really
easy. One of the things that's really easy to see about these equations is that both of them have a factor of \( \frac{1}{2} c \), and then basically both of them have some kind of maximum value
squared, whether it's \( E \) or \( B \). And the last thing here is that \( \epsilon \) usually goes with \( E \) and \( \mu \) usually goes with \( B \). So I think this is a really great way to
memorize these equations. It also sort of looks like other energy-like equations where you have \( \frac{1}{2} \) times something times something squared.
The units, just as we learned when we talked about mechanical waves, are going to be watts per meter squared. And basically, there are 2 types of problems when it comes down to this intensity stuff.
There's one problem in which the source of light will emit radially in all directions like a flashlight or like a bulb or like a star or something like that. Or you might have something where it
emits directionally, like a flashlight or a laser or something like that. And basically, the gist of these two different types of problems is that if you can assume the source emits equally in all
directions, basically if you can assume that it's this case, then that means that the area, this \( A \) term that pops up in your area equation in the intensity, is going to be equal to \( 4\pi r^2
\) because, basically, if you draw some little distance \( r \), then the surface area of this field basically just becomes the surface area of a sphere. However, if it emits directionally like a
flashlight, then you cannot assume that this area is equal to \( 4\pi r^2 \), and you're going to have to figure it out. That's basically the hardest parts of these kinds of problems is figuring out
which one you're dealing with.
Now I'm going to go ahead and make a point here about the second thing, but, actually, before I do that, let's just jump right into a problem. Alright? So we have an incandescent light bulb that is
emitting at 50 watts, in all directions. So because it's all directions here, that means that we can actually go ahead and use this relationship here. So that means we know what our area is going to
be. It's got no inefficiencies or energy losses, and we want to calculate what is the intensity of light at some distance away from the light bulb. Okay? So in this first part here, I want to
calculate \( I \). That's basically what I'm looking at here. Okay? So remember that \( I \) has a couple of different formulas. It's got power over area, and it's got all these other things with the
\( E \)'s and the \( B \)'s. Which one do I use? Well, if you take a look here, this unit, this 50 watts is actually the power of the light bulb. So that's in my \( P \), so I've got that. And then
we know that the light bulb emits in all directions, so we actually already know what the area is as well. So we actually have both of these, and we can go ahead and calculate this. Alright? So if
you take a look, let's just say that we're going to draw this distance and we're going to say that \( r = 5 \) meters. And, basically, the area, the surface area here, just becomes, \( 4\pi r^2 \).
Alright? So we're going to have that power is equal to 50 divided by the area, which is going to be \( 4\pi \times 5^2 \). You've got to work this out. What you're going to get is \( 0.16 \) watts
per square meter, and that is your final answer for part a.
Now we're going to move on to the second part here, which is we're going to calculate the maximum value of the electric field at this distance here. Alright? So we're going to calculate the electric
field. So we'll go back to our equations for intensity. Remember that intensity is related to the magnitudes or the strengths of the electric and magnetic fields, and we're going to see this equation
here. So we're going to have \( \frac{1}{2} c \epsilon_0 E_\text{max}^2 \). Since we're looking for the electric field, we're going to use this version instead of the \( B_\text{max} \) version.
Alright? So, basically, what I'm going to do is I'm going to say that \( I \) is equal to, this is still equal to \( P/A \), but now it's equal to \( \frac{1}{2} c \epsilon_0 \times E_\text{max}^2
\). Now I'm really looking for what is this \( E_\text{max} \) over here, so I can just rearrange my equation. Alright? So I'm not going to use this \( P/A \) because I actually already figured out
what this \( I \) is equal to. Right? It's just equal to this number over here. Alright? So this is just equal to 0.16, and this is going to equal, let's see. I'm going to move all this other stuff
over to the other side. When I move this to this side, basically, what happens is I pick up a factor of 2 on the outside over here, and then I have to divide by \( c \) and \( \epsilon_0 \). So this
is equal to my \( E_\text{max}^2 \). Now the last thing I have to do here is I just have to take the square root and then plug in all of my numbers. So in other words, \( E_\text{max} \) is equal to
the square root of \( 2 \times 0.16 \). And now I'm going to plug in my numbers. This is going to be \( 3 \times 10^8 \). And then \( \epsilon_0 \), that's just a constant that we've got over here in
this table right here. So we've got 8.85 times \( 10^{-12} \). When you go ahead and plug all of this into your square root, what you're going to get here, is you should get an \( E_\text{max} \) of
10.98. Remember that the units for electric field are newtons per coulomb. So that is the answer to part b. Let's move on now to our final piece here, our final step, which is that we're going to
calculate now the RMS value of the magnetic field at this point. And that actually brings me to the second point that I had in this video, which is that in some problems, you might be given or asked
for some kind of an average value or some kind of a root mean square. Remember we used that in thermodynamics a little bit, instead of maximum values. And if that ever happens, there's actually a
pretty straightforward relationship between RMS and max values, And it's just that, RMS is equal to the maximum divided by the square root of 2. So your \( E_\text{RMS} \) is going to be \( E_\text
{max} / \sqrt{2} \). Your \( B_\text{RMS} \) is going to be \( B_\text{max} / \sqrt{2} \). Alright? So you can always combine these equations with your other intensity equations and everything works
out. Alright? So just be really careful that you know which one you're solving for, because a lot of questions will try to throw you off and ask for \( E_\text{RMS} \) or \( B_\text{max} \) or
something like that. So just be really careful that you're solving the right thing here. Okay? So in this last part we're looking for is we're actually looking for \( B_\text{RMS} \). Alright? So I
know that \( B_\text{RMS} \) is equal to \( B_\text{max} / \sqrt{2} \). The problem is I don't know what \( B_\text{max} \) is, but I can calculate it because I know that \( B_\text{max} \),
remember, is always equal to, is always related to, \( E_\text{max} \) by the speed of light equation. So I can always calculate what \( B_\text{max} \) is because I have that \( E_\text{max} \) is
equal to \( c B \). So therefore, your \( E_\text{max} \), which is your 10.98 divided by \( 3 \times 10^8 \) is equal to your \( B_\text{max} \), and that's going to equal, that's going to be \(
3.66 \times 10^{-8} \), and that's in Teslas. So, basically, what I can do here is I can say, I'm going to bring this all the way out here, and I'm going to say that \( B_\text{RMS} \) is equal to \(
3.66 \times 10^{-8} / \sqrt{2} \). And your final answer is going to be, that's going to be, \( 2.59 \times 10^{-8} \) in Teslas. Alright? So that's your final answer. Anyway, let me know if you have
any questions, and I'll see you in the next one. | {"url":"https://www.pearson.com/channels/physics/learn/patrick/32-electromagnetic-waves/intensity-of-em-waves?chapterId=8fc5c6a5","timestamp":"2024-11-14T02:29:49Z","content_type":"text/html","content_length":"545615","record_id":"<urn:uuid:31d6c5cd-40e3-4515-8d69-278368cc9ba7>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00502.warc.gz"} |
A pyramid has a base in the shape of a rhombus and a peak directly above the base's center. The pyramid's height is 7 , its base has sides of length 5 , and its base has a corner with an angle of pi/3 . What is the pyramid's surface area? | HIX Tutor
A pyramid has a base in the shape of a rhombus and a peak directly above the base's center. The pyramid's height is #7 #, its base has sides of length #5 #, and its base has a corner with an angle of
# pi/3 #. What is the pyramid's surface area?
Answer 1
T S A $= \textcolor{p u r p \le}{95.98}$
AB = BC = CD = DA = a = 5
Height OE = h = 7
OF = a/2 = 5/2 = 2.5
$E F = \sqrt{E {O}^{2} + O {F}^{2}} = \sqrt{{h}^{2} + {\left(\frac{a}{2}\right)}^{2}} = \sqrt{{7}^{2} + {2.5}^{2}} = \textcolor{red}{7.433}$
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Answer 2
To find the surface area of the pyramid, we need to calculate the area of each face and then sum them up. The pyramid has four triangular faces and one rhombus-shaped base.
1. Area of the base: Since the base is a rhombus, we can use the formula for the area of a rhombus: A = (diagonal1 * diagonal2) / 2. Given that the side length of the rhombus is 5 and one of its
angles is π/3, we can use trigonometry to find the length of the diagonals. The diagonals of a rhombus bisect each other at right angles. Let's denote one of the diagonals as d1 and the other as
d2. Using the given side length (5) and angle (π/3), we can find the length of one-half of one of the diagonals using the cosine function: cos(π/3) = adjacent/hypotenuse. Thus, d1/2 = 5 * cos(π/
3) = 5 * (1/2) = 5/2. So, the full diagonal d1 = 2 * (5/2) = 5. Since the diagonals of a rhombus are congruent, d2 = 5 as well. Now, we can calculate the area of the base: A = (5 * 5) / 2 = 25/2.
2. Area of each triangular face: We can use the formula for the area of a triangle: A = (1/2) * base * height. Given that the base of each triangular face is one side of the rhombus (5) and the
height is the height of the pyramid (7), we can calculate the area of each triangular face: A = (1/2) * 5 * 7 = 35/2.
3. Now, we sum up the areas of all faces: There are four triangular faces, each with an area of 35/2, and one rhombus-shaped base with an area of 25/2. Total surface area = 4 * (35/2) + 25/2 = 140/2
+ 25/2 = 165/2.
So, the surface area of the pyramid is 165/2 square units.
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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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angle between vectors calculator calculation for Calculations
29 Mar 2024
Popularity: ⭐⭐⭐
Angle Between Vectors Calculator
This calculator provides the calculation of angle between two vectors in two dimensions.
Calculation Example: The angle between two vectors is a measure of their relative orientation. It is calculated using the dot product of the two vectors. The dot product is a scalar quantity that is
equal to the sum of the products of the corresponding components of the two vectors.
Related Questions
Q: What is the range of the angle between two vectors?
A: The angle between two vectors can range from 0 to 180 degrees.
Q: How is the angle between two vectors used in physics?
A: The angle between two vectors is used in physics to calculate the work done by a force, the torque on an object, and the power transmitted by a wave.
| —— | —- | —- |
a First Vector X-Component m
b First Vector Y-Component m
c Second Vector X-Component m
d Second Vector Y-Component m
Calculation Expression
Angle Between Vectors Function: The angle between two vectors is given by: angle = arctan((b * c - a * d) / (a * c + b * d))
arctan((b * c - a * d) / (a * c + b * d))
Calculated values
Considering these as variable values: a=2.0, b=3.0, c=4.0, d=5.0, the calculated value(s) are given in table below
| —— | —- |
Angle Between Vectors Function 0.0867383
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Quasi-Compact Morphism of Schemes
Let $\struct {X, \OO_X}$ and $\struct {Y, \OO_Y}$ be schemes.
Let $f : \struct {X, \OO_X} \to \struct {Y, \OO_Y}$ be a morphism of schemes.
$f$ is quasi-compact if and only if for all quasi-compact open subsets $U \subset Y$, the set $\map {f^{-1} } U \subset X$ is quasi-compact.
There are no source works cited for this page.
Source citations are highly desirable, and mandatory for all definition pages.
Definition pages whose content is wholly or partly unsourced are in danger of having such content deleted.
To discuss this page in more detail, feel free to use the talk page. | {"url":"https://proofwiki.org/wiki/Definition:Quasi-Compact_Morphism_of_Schemes","timestamp":"2024-11-04T10:08:24Z","content_type":"text/html","content_length":"40552","record_id":"<urn:uuid:75e12ea3-c3cb-40f5-9a2a-2888f964da24>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00559.warc.gz"} |
How Many Cups In A Pint (Free Printable Chart)
Don't wonder anymore about how many cups are in a pint. Here is a simple cups to pint conversion to have on your hand!
Photo by gwflash from Getty Images; Canva
Time is precious for everyone nowadays, and we all need to manage it carefully.
If you find yourself constantly distracted by converting measurements while trying a new recipe, this post is the answer.
Jump to:
I have done all the conversions on how many cups are in one pint and created helpful charts.
We are all different and live in countries with varying measurement systems.
That is why I also created Baking Conversion Charts to convert grams to cups, cups to grams, grams to tablespoons and teaspoons, tablespoons to cups, and many more.
What are a cup and pint? Pt vs cup
A cup (abbreviation as 'c' or 'C') is a measurement unit used to measure volume. In the United States, one cup is equal to 8 fluid ounces.
A pint (abbreviation as 'pt' or sometimes 'p') is a unit of volume, also a dry measure of capacity, equal to one-half of a liquid and dry quart, respectively.
It is used in both the United States and Imperial measurement systems.
American pints vs. Imperial pints
There are two types of pints. In the United States, a pint is about 20% smaller than an Imperial pint in the United Kingdom and Ireland.
It relates to how a gallon was defined in the 1824 British Weights and Measurements Act.
So, a pint in the USA equals 16 fluid ounces, while 20 fluid ounces in the UK.
By looking at the ingredient measurements, you can easily tell if a recipe uses American or Imperian pints.
If measurements are given as grams, this is likely that the recipe calls for Imperial pints and not US customary units.
General conversions
1 cup to a pint (1c to pt)
How many cups in a pint?
2 cups equal 1 US liquid pint, or 2 c = 1 pt
2.4090 cups equal 1 Imperial pint, or 2.4 c = 1 pt
All it means,
1 cup equals 0.5 US liquid pint and
1 cup equals 0.4163 Imperial pints.
1 pint to cups (pt to 1c)
How many pints are in a cup?
One US liquid pint equals 2 cups.
One Imperial pint equals 2.4 cups.
How to convert cups to a pint
To convert 1 cup to a pint, divide the cups by 2, where 2 is a conversion factor.
For example, 1 cup to pint calculation:
1 c ÷ 2 = 0.5 pt
1 Cup to Pint Conversion Equation:
1 c ÷ 2 = 0.5 pt
Other useful conversions include:
• 1 gallon = 4 quarts, 8 pints, 16 cups, 128 fluid ounces, 3.8 liters
• 1 quart = 2 pints, 4 cups, 32 fluid ounces, ¼ gallon, 0.95 liters
• 1 pint = ½ quart, 2 cups, 16 fluid ounces, 0.125 gallon, ⅕ liter
• 1 quart = 2 pints, 4 cups, 32 fluid ounces, ¼ gallon, 0.95 liters
• 1 cup = 8 oz, 48 teaspoons, 16 tablespoons, ½ pint, ¼ quart
Cups in a pint
How many cups equal 1 pint? Kitchen math is easy. Cups to pint conversions are very simple.
• There are 2 cups in 1 pint.
• There are 4 cups in 2 pints.
• There are 8 cups in 4 pints.
• There are 10 cups in 5 pints.
• There is 1 cup in half a pint.
Here is a simple conversion chart for easy reference. All the conversions below are in US liquid measures.
Cups Pints
1c ½pt
2c 1pt
4c 2pt
6c 3pt
8c 4pt
10c 5pt
12c 6pt
14c 7pt
16c 8pt
Cups to pint conversion
Note: When a recipe calls for 'pint,' it usually means a liquid pint. However, if the directions specifically say dry pints, then 1 US pint measures 18.6 US fluid ounces or 2.325 cups.
Cups, pints, quarts, gallons, and ounces chart
If you wonder about cups to pints to quarts to gallons and ounces conversions, here is another handy chart.
Cups Pints Quarts Gallons Ounces
½c ¼ pt ⅛qt 1/32gal 4oz
1c ½pt ¼qt 1/16gal 8oz
2c 1pt ½qt ⅛gal 16oz
4c 2pt 1qt ¼gal 32oz
8c 4pt 2qt ½ gal 64oz
12c 6pt 3qt ¾ gal 96oz
16c 8pt 4qt 1gal 128oz
Cups, pints, quarts, gallons, and ounces chart
Memorizing some common kitchen conversions is easy and fun. Just remember that in the U.S. measurement system
• 1 gallon = 4 quarts, 8 pints, or 16 cups
• 1 quart = 2 pints or 4 cups
• 1 pint = 2 cups
Gallon man
Have you ever heard about Gallon man? It is an awesome visual tool to help you remember and learn about customary capacity measurements.
You can download the printable version of Gallon Man Visual Tool >>
This graphic seems self-explanatory, but let me explain it.
• G means gallon, and it is clear that there are four Q (means quarts) inside the G. So, one gallon equals 4 quarts.
• There are two P (means pints) inside each Q, so 2 pints make 1 quart.
• And one P contains two C (means cups); hence, one pint equals 2 cups.
• Now let's count all the C's inside the Q to answer how many cups are in a quart. There are 4 cups in one quart.
• But how many C's are inside the G, or how many cups are in a gallon? There are 16 cups in one gallon.
• Finally, how many P's are inside the G, or how many pints are in a gallon? There are 8 pints in a gallon.
Printable conversion chart
WNeed a FREE handy kitchen conversion chart while baking and cooking? Here is the one.
Download and print this Kitchen Conversion Chart >>
More kitchen printables are part of FREE resource library.
Pint-sized food
If you question whether there is food sold in a pint, there is a quick answer 'yes.'
Milk, sour cream, and ice cream, for example, by Ben and Jerry’s or Haägen-Dazs, are sold in pint measurements.
Also, USA residents will find some small fruit (strawberries, blueberries, etc.) and similar items like mushrooms or cherry tomatoes are measured and priced per dry pint.
But how many cups are in a pint of strawberries, then? One pint of fresh strawberries equals 2 ½ cups whole berries, 1 ¾ cups sliced, or 1 ¼ cups pureed.
And how many cups of blueberries come in a pint? One pint of blueberries equals 2 cups or 12 ounces of berries.
For those who love beer, a good thing to know the size of the pint beer glass before ordering it.
The perfect beer glass is one that holds 16 ounces. Make sure it says so on the menu since there are some glasses that only hold 14 ounces of liquid.
Measuring liquid ingredients vs. dry ingredients
Measuring ingredients precisely is the best way to ensure your bake turns out perfect.
The most accurate way of measuring ingredients is with a kitchen scale.
But there are many different types of measuring cups on the market: from plastic to glass and metal. Some have handles, while others don't.
But there are two principal types of measuring cups: ones to measure liquid ingredients and others - for dry ingredients. So what's the difference?
Liquid measuring cups have the pour spouts on the side to facilitate pouring the liquid into a mixing bowl. These are useful for water, milk, juice, oil, etc.
To correctly measure liquids, place the measuring cup on a flat surface and pour until you reach your desired amount. It will give the most precise measurement if at eye level with the container.
Dry measuring cups have a flat top to measure the correct amount. These are used for flour, sugar, butter, etc.
To measure dry ingredients, slide a butter knife across the top of the measuring cup so that an even amount remains.
But different rules are applied to measure, for example, flour and brown sugar.
Flour should be scooped and leveled. Brown sugar, on the other hand, needs a very tight packing.
Related kitchen conversions
Now, check out these other measurement conversions that help you while baking or cooking. Enjoy!
And if you ever need to convert your baking pan sizes, use this simple Cake Pan Converter.
I hope this post answered the question about how many cups are in a pint. You might see other frequently asked questions about pint to cups conversion, for example:
• One pint is how many cups
• How much is a pint in cups
• 1 pint equals how many cups
• 1 pint is how many cups
• How many cups is a pint
• How many cups is one pint
• How many cups equal a pint
• 1 pt equals how many cups
• What is 1 pint in cups?
The answer is the same: there are 2 cups in a pint.
If you want to start baking like a pro and don't know where to start, then take advantage of my free resources.
Also, sign up for a Baking Basics E-course and start baking with confidence. What's more? You will never get bored with the awesome recipes on the blog.
Frequently asked questions
Here are the answers to the most common kitchen conversion questions.
How much is a pint in cups
One pint equals 2 cups.
How many cups make a pint
2 cups make 1 pint.
How many cups in a half pint
There is 1 cup in a half pint.
How many cups are in 3 pints?
There are 6 cups in 3 pints.
How many cups in a pint of water
There are 2 cups in a pint of water
How many cups in a pint of milk
There are 2 cups in a pint of milk.
How many cups in a pint of sour cream
There are 2 cups in a pint of sour cream.
How many cups are in a pint of ice cream?
There are 2 cups in a pint of ice cream.
How much is a pint?
One pint equals 2 cups, ½ quart, 0.125 gallon, 16 fluid ounces, and ⅕ liter.
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A stochastic parameterization of ice sheet surface mass balance for the Stochastic Ice-Sheet and Sea-Level System Model (StISSM v1.0)
Articles | Volume 17, issue 3
© Author(s) 2024. This work is distributed under the Creative Commons Attribution 4.0 License.
A stochastic parameterization of ice sheet surface mass balance for the Stochastic Ice-Sheet and Sea-Level System Model (StISSM v1.0)
Many scientific and societal questions that draw on ice sheet modeling necessitate sampling a wide range of potential climatic changes and realizations of internal climate variability. For example,
coastal planning literature demonstrates a demand for probabilistic sea level projections with quantified uncertainty. Further, robust attribution of past and future ice sheet change to specific
processes or forcings requires a full understanding of the space of possible ice sheet behaviors. The wide sampling required to address such questions is computationally infeasible with sophisticated
numerical climate models at the resolution required to accurately force ice sheet models. Stochastic generation of climate forcing of ice sheets offers a complementary alternative. Here, we describe
a method to construct a stochastic generator for ice sheet surface mass balance varying in time and space. We demonstrate the method with an application to Greenland Ice Sheet surface mass balance
for 1980–2012. We account for spatial correlations among glacier catchments using sparse covariance techniques, and we apply an elevation-dependent downscaling to recover gridded surface mass balance
fields suitable for forcing an ice sheet model while including feedback from changing ice sheet surface elevation. The efficiency gained in the stochastic method supports large-ensemble simulations
of ice sheet change in a new stochastic ice sheet model. We provide open source Python workflows to support use of our stochastic approach for a broad range of applications.
Received: 31 Mar 2023 – Discussion started: 28 Apr 2023 – Revised: 22 Nov 2023 – Accepted: 19 Dec 2023 – Published: 08 Feb 2024
Many decision-making contexts demand probabilistic projections of sea level rise. For example, urban planners managing coastal risks would like to be able to quantify the probability of certain
levels of sea level rise (Walsh et al., 2004) so that they can apply their own risk tolerance to assess proposed interventions (Kopp et al., 2014; Hinkel et al., 2019). Probabilistic projections can
also help illustrate the benefits of climate mitigation actions for policy-makers, quantify coastal adaptation needs, and identify priority areas for further research (Jevrejeva et al., 2019, and
references therein). Efforts to generate probabilistic projections of future sea level change have been ongoing for decades (Titus and Narayanan, 1996), but the ice sheet component remains a source
of poorly quantified uncertainty (Le Cozannet et al., 2017; Sriver et al., 2018; Jevrejeva et al., 2019).
Generating probabilistic projections of ice sheet contribution to sea level requires running many climate and/or ice sheet model simulations that can explore multiple realizations of an uncertain
future. The spectrum of methods available to generate future projections of ice sheet change makes that task difficult. The most computationally efficient methods find an empirical relationship
between some climate variable, often global mean surface temperature, and a variable of interest, such as global mean sea level (Rahmstorf, 2007) or ice sheet melt (Luo and Lin, 2022). Such methods
allow wide sampling of future climate scenarios, which is necessary to account for scenario uncertainty. However, they assume that the form of the relationship between the variables will remain the
same in the future, which is not assured in a rapidly changing climate with feedbacks among multiple variables. The structural uncertainty in those methods – that is, the uncertainty attributable to
poor knowledge of the form of the model itself – is therefore high, and their results are difficult to convert into a probability distribution.
More sophisticated numerical models represent physical processes such as ice sheet flow, snowfall, and surface melting directly (Goelzer et al., 2020b; Seroussi et al., 2020), explicitly modeling
changes over time in the relationship between climate forcing and output variables of interest. Such models include many more parameters and internal variability of processes on a wide range of
spatial and temporal scales. A direct representation of physical processes helps to constrain structural uncertainty related to processes and internal variability, but the computational expense of
sophisticated models limits the number of future scenarios that can be sampled. Model outputs thus represent discrete points in a wide range of possibilities, providing too little information to
estimate the probability distribution of output variables such as future sea level.
The limited sampling available from physical process model outputs has motivated the creation of statistical emulators to explore the probability distribution of ice sheet model output variables (
Edwards et al., 2021). To support local sea level adaptation planning and to guide ice sheet research, it is useful to partition the uncertainty in such probability distributions among various
sources – for example, identifying what fraction of the spread comes from uncertainty in the model physics versus what fraction comes from uncertainty in the applicable climate scenario (Jevrejeva
et al., 2019; Marzeion et al., 2020). Identifying the fraction of uncertainty attributable to internal climate variability would require large-ensemble simulations of ice sheet evolution that sample
a representative set of climate forcing fields.
A particular obstacle to large-ensemble simulations of future ice sheet evolution is the computational expense of generating surface mass balance forcing. “Surface mass balance” (SMB) refers to the
set of processes through which ice sheets gain and lose mass at the ice sheet interface with the atmosphere. Mass gain processes include precipitation, vapor deposition, and refreezing of meltwater;
mass loss processes include melting (with subsequent runoff) and sublimation.
Due to the complex set of ice–atmosphere interactions that comprise mass balance, ice sheet models are not typically forced directly by global climate model output. Rather, global climate model
output must be downscaled to construct an SMB field of high enough spatial resolution and quality, often through use of a specialized mass and energy balance model that accounts for processes at the
snow–ice surface and in the snowpack (see, e.g., Fettweis et al., 2020, and references therein). Increasing sophistication in the process-based models used to construct ice sheet SMB means a
corresponding increase in computational demand for each individual simulation with these models. That added computational expense further limits comprehensive sampling of possible SMB scenarios.
Stochastic methods provide a low-cost alternative to ensembles with multiple realizations of sophisticated process models (Sacks et al., 1989). A stochastic generator can produce a large-ensemble
sample of SMB comprised of many realizations that are statistically consistent with a small set of process model outputs. Previous studies have applied stochastic methods to analyze ice sheet mass
balance observations with the primary aim of testing whether a trend emerges from the range of natural variability. For example, Wouters et al. (2013) represented SMB simulated by RACMO2 as an order-
p autoregressive process to estimate the uncertainty in mass balance trends for the Greenland and Antarctic ice sheets. More recent studies tested multiple types of stochastic models to characterize
the variability in Antarctic SMB (Williams et al., 2014; King and Watson, 2020) and thereby test the presence of significant, detectable trends in SMB observations. Here, we have a different aim: to
construct a statistical generator of SMB to force an ice sheet model. The SMB product we wish to generate should include interannual variability at the catchment scale, temporal trends, seasonality,
and spatial variation down to the scale of an ice sheet model mesh. We approximate the output of one process-based SMB model as a realization of a stochastic process. The statistical generator that
produces a realization best fit to a given process-based model output can then be used to generate hundreds of other realizations, sampling the range of internal variability for future SMB consistent
with the same model, at much reduced computational expense. Those generated samples support large-ensemble simulations of ice sheet change, including simplified feedback of ice sheet geometry on SMB
(see an example application in Verjans et al., 2022). To best support the broader glaciological community, we base our method entirely on open-source software packages and provide our own open-source
code where necessary.
Below, we present the data sources that informed our construction of a stochastic surface mass balance generator (Sect. 2). We then describe our choice of temporal model type and how we selected the
best-fit model for each catchment of the Greenland Ice Sheet (Sect. 3.1–3.2). Section 3.3 describes how we accounted for large-scale covariance in SMB across the ice sheet. We demonstrate the
generation of forward-projected SMB time series (Sect. 3.4) and how to downscale those time series to ice sheet model grid scale (Sect. 3.6). Finally, we contextualize our work with previous studies
and highlight its potential applications (Sect. 4).
We construct the stochastic SMB model based on SMB fields output from high-resolution regional climate models with domains encompassing the Greenland Ice Sheet. Here, we focus on output from seven
models that participated in the Greenland SMB Model Intercomparison Project (GrSMBMIP; Fettweis et al., 2020) to determine whether stochastic generator type and/or order is dependent on the choice of
process model. The subset of GrSMBMIP models we analyze comprises those whose developer team gave us permission to use their archived data for this purpose: ANICE (Berends et al., 2018), CESM (van
Kampenhout et al., 2020), dEBM (Krebs-Kanzow et al., 2021), HIRHAM (Langen et al., 2017), NHM-SMAP (Niwano et al., 2018), RACMO (Noël et al., 2018), and SNOWMODEL (Liston and Elder, 2006). This
selection includes exemplars of simpler energy balance models as well as more sophisticated regional climate models (Fettweis et al., 2020), and these models have been extensively validated against
observations over recent decades. The GrSMBMIP regional models are all forced at their boundaries by ERA-Interim reanalysis data and have been processed onto a common 1km^2 grid, with a common ice
extent mask applied.
We aggregate each SMB model output field for each outlet glacier catchment at an annual timescale. To achieve that, we overlay each field with the catchment outlines (Fig. 1) provided by Mouginot and
Rignot (2019) and sum the grid cells that fall within each catchment area, dividing by the total area of the catchment to arrive at catchment mean SMB for each month, catchment, and model from 1980
to 2012. We then sum to annual timescales so that the subsequent analysis produces statistical models of interannual variability. SMB variability at the inter-monthly timescale is dominated by the
seasonal cycle, which is added back to generated SMB time series through downscaling (Sect. 3.6).
3.1Temporal model for catchment-averaged annual SMB
We fit a generative statistical model for catchment-averaged SMB using an approach adapted from the work of Hu and Castruccio (2021) on other climate fields. We define the n-dimensional vector M(t)
to be the catchment-averaged SMB in each of n catchments at time t, and we assume that it can be described by an additive model with a temporal variability vector μ(t) and a noise term vector ϵ(t) of
the form
$\begin{array}{}\text{(1a)}& \mathbit{M}\left(t\right)=\mathbit{\mu }\left(t\right)+\mathbit{ϵ}\left(t\right),\text{(1b)}& \mathbit{\mu }\left(t\right)={\mathbit{\beta }}_{\mathrm{0}}+{\mathbit{\beta
}}_{\mathrm{1}}f\left(t\right)+\sum _{i=\mathrm{1}}^{p}{\mathbf{\Phi }}_{i}\cdot \mathbit{M}\left(t-i\right),\text{(1c)}& \mathbit{ϵ}\left(t\right)\sim \mathcal{N}\left(\mathrm{0},\mathbf{\Sigma }\
In the example case we present here, the vectors M,μ, and ε have one entry for each of the n=260 catchments at time t, and we evaluate each at a total of m=30 time steps.
The temporal trend μ(t) includes historical mean SMB for each catchment β[0] and the forcing variable f(t) with a linear coefficient β[1]. The forcing variable, f(t), can be an external process which
causes slow changes in SMB, such as atmospheric temperature (f(t)=T[A](t)) or simply a prescribed dependence on time (e.g., f(t)=t). Finally, Eq. (1b) includes autoregressive terms up to order p
contained in the diagonal matrices ${\mathbf{\Phi }}_{i},i=\mathrm{1},\mathrm{\dots },p$. The temporal trend as written would thus approximate an autoregressive process of order p, AR(p). Section 3.2
discusses how we identified AR(p) as the best type of temporal model for this application. At this stage, fitting temporal models to annually aggregated time series, we exclude seasonal terms from
the temporal trend μ(t); seasonality is incorporated deterministically during the downscaling process described in Sect. 3.6. All stochasticity in this generation technique enters through interannual
The noise term ϵ(t) is assumed to be independent, identically distributed in time, and from an n-dimensional normal distribution with mean of zero and covariance matrix Σ. As we describe in Sect. 3.3
, spatial correlations between catchments are captured in ϵ(t).
3.2Selecting candidate model type and order
We tested several model types in search of the most appropriate way to represent interannual SMB variability in Eq. (1b). Three criteria inform our selection of candidate stochastic model types for
temporal SMB variability. First, we would like our candidate temporal models to capture the timescales of variability apparent in the data based on standard statistical methods that are likely to be
familiar to glaciologists. Second, we would like our methods to build on existing open-source software such that other researchers can test and apply our work. For that reason, we prioritize models
with existing fitting routines in Python or R. Finally, we would like to be able to compare our findings to those of King and Watson (2020) for the Antarctic Ice Sheet, so we prioritize temporal
model families that those authors also tested.
These criteria guide our investigation of three common types of temporal models. All temporal models we test belong to the autoregressive fractionally integrated moving-average (ARFIMA) family of
models. The first type, order-p autoregressive AR(p) models, is the simplest of the ARFIMA family. They assume that the value of SMB at time t depends linearly on values of SMB at times $\left(t-\
mathrm{1},t-\mathrm{2},\mathrm{\dots },t-p\right)$. An AR(0) model is equivalent to a white noise model scaled to the data. The second type, ARIMA models of order $\left(p,d,q\right)$, includes
order-p autoregressive terms applied to a series that has been differenced d times to reach stationarity, as well as dependence on a weighted moving average of the past q residual noise terms.
Finally, general ARFIMA models are similar to ARIMA models but allow non-integer values for d, accounting for “long memory” in the time series. To avoid confusion, we henceforth use ARFIMA to refer
only to ARFIMA models that do include non-integer differencing d, and we refer to the special cases ARIMA and AR(p) by their own names. King and Watson (2020) tested AR(p) and ARIMA models; they also
tested generalized Gauss–Markov models, for which we were unable to find an open-source fitting routine, but which are very similar to ARFIMA models of order $\left(p,d,\mathrm{0}\right)$.
For each catchment, we estimate β[0] as the 1980–2012 mean and remove it from the series. We then use conditional maximum likelihood (ordinary least squares) to optimize values of β[1] and the
remaining parameters of Eq. (1b) associated with each candidate model type (AR, ARIMA, ARFIMA) over a range of orders $\left(p,d,q\right)$. We perform the model fitting with built-in functions from
the Python package statsmodels v0.12.2 (Seabold and Perktold, 2010): $\mathtt{statsmodels}.\mathtt{tsa}.\mathtt{ar}\mathtt{_}\mathtt{model}.\mathtt{AutoReg}$ and $\mathtt{statsmodels}.\mathtt{tsa}.\
mathtt{arima}.\mathtt{model}.\mathtt{ARIMA}$. In each case, we assume a linear dependence on time, β[1]f(t)=β[1]t in Eq. (1b). Statsmodels does not include a built-in function to fit ARFIMA models,
so we apply fractional differencing following Kuttruf (2019) and subsequently test ARIMA$\left(p,\mathrm{0},q\right)$ with the built-in function.
We analyze the Bayesian information criterion (BIC) as returned by the statsmodels built-in function for the temporal models fit to each catchment series. The BIC is given by
$\begin{array}{}\text{(2)}& \text{BIC}=-\mathrm{2}\mathrm{\ell }+\mathrm{ln}\left(T\right)\left(\mathrm{1}+\mathrm{df}\right),\end{array}$
where ℓ is the log-likelihood function of the given temporal model on the data, T is the number of observations, and df is the number of degrees of freedom in the generator. Minimizing the BIC
balances a maximization of log-likelihood ℓ – the probability that a stochastic generator of this type could have produced the data series from the process model being fit – with a penalty for excess
parameters (overfitting). We select the temporal model with the lowest BIC for each catchment for each SMB process model. We analyze the preferred temporal model types across all catchment–model
pairs to identify the most suitable class of temporal models. We chose to select the minimum BIC to encourage computationally cheap models with fewer parameters (as in King and Watson, 2020); we note
that statsmodels also returns other common metrics of model fit such as the Akaike information criterion, which could be selected by users with other priorities.
To decide the range of orders $\left(p,d,q\right)$ to test in our model fitting, we use the autocorrelation and partial autocorrelation functions (ACF and PACF, respectively) to target the relevant
timescales of variability. In a purely autoregressive process, the number of values significantly different from 0 before the first nonsignificant value in the PACF would indicate the AR order p. In
a purely moving-average process, the number of values significantly different from 0 before the first nonsignificant value in the ACF would indicate the MA order q. These metrics cannot be used to
determine the order $\left(p,d,q\right)$ of a more general ARFIMA process, but we use them as qualitative indicators of an appropriate range for testing. The ACF and PACF values differ per process
model and per catchment; an example for the Kangerlussuaq Glacier annual SMB is shown in Fig. 2. In that example, significant autocorrelation is apparent at a lag time of 4 years for several process
models, though with several previous values not significantly different from zero; the partial autocorrelation is not significant for any lag shown. Ice-core-derived ACF and PACF show significant
values at timescales of up to 5 years, tapering to values not significantly different from 0 at longer timescales (Fig. B1). The combination of evidence from ice cores and from process model ACF and
PACF in multiple catchments suggests several lags ≤5 years with significant ACF or PACF. We therefore choose to test values of p and q from 0 to 5. We determine the order of differencing required to
reach stationarity, d, using augmented Dickey–Fuller and KPSS tests of stationarity on each catchment time series. Both tests agreed that the de-meaned catchment average time series were stationary,
so d=0 should be appropriate, but for completeness we also tested d=0.5 and d=1. Among the range of values $\left(p,d,q\right)$ tested, we select the best-fit model as the one with the lowest BIC. We
note that comparing the BIC of model fits among temporal models of different orders requires a consistent base dataset and fitting method (for example, the same software package and optimization
scheme for all models). We computed the BIC using statsmodels built-in functions, setting the optional argument 𝚑𝚘𝚕𝚍_𝚋𝚊𝚌𝚔=max($p,d,q$) to ensure that lower-order models were fit to training data
series of the same length as higher-order models.
Figure 3b shows example best fits for four model types and their BIC (see legend). The best fit to the NHM-SMAP example data in Fig. 3 is white noise with a trend (AR(0)). Both AR(p) models shown
have lower BIC than the more complicated ARIMA and ARFIMA models. In this example, the best-fit AR(0) and ARFIMA series capture a trend with little other temporal information, while the AR(5) and
ARIMA(1,0,1) series capture larger temporal variability with the expense of added parameters. We note that the models capturing only a trend in Fig. 3b will still generate stochastic series with
temporal variability; the distinction is that almost all of the temporal variability in the final generator will come from the spatial noise generation process described in the next section.
In every catchment and SMB model tested (1820 catchment–model pair time series tested), AR models were the most suitable. There were no basin–model pairs where ARIMA or ARFIMA fits were preferred to
AR(p) fits. Further, white noise with a trend (AR(0)) was preferred to any higher-order statistical fit for catchment-aggregated SMB in most basins. Each process model had some basins where
higher-order AR(p) models were preferred (Fig. A1).
The example we present below allows the order of the autoregressive model fit to vary by basin. Users may decide to keep that flexibility, which adds some complication in storing the model
parameters, or they may opt for the simplest AR(0) fit for every basin and allow residual variability to be captured in the spatial noise generation (Sect. 3.3) and downscaling (Sect. 3.6).
3.3Estimating SMB covariance between catchments
Thus far, we have described a method for fitting and generating time-varying SMB for individual catchments with no correlation beyond the catchment scale. However, SMB over the Greenland Ice Sheet
may vary coherently at spatial scales beyond those of single outlet glacier catchments due to large-scale processes in atmospheric circulation (Lenaerts et al., 2019). Motivated by this physical
intuition, we introduce spatially informed noise generation.
Following Hu and Castruccio (2021), we construct a matrix of variance Σ for catchment-level noise terms (Eq. 1c above) as
$\begin{array}{}\text{(3)}& \mathbf{\Sigma }=\mathbf{DCD},\end{array}$
where D is the diagonal matrix of per-catchment standard deviations and C is the spatial correlation matrix among all catchments. Note that this formulation assumes a catchment-specific variance D,
so the SMB is assumed to vary differently within each catchment, implicitly accounting for different catchment sizes. The spatial model C is defined on the inter-catchment correlation, which we
assume not to depend on the catchment size.
The spatial correlation pattern may differ for different SMB process models, so we construct the matrix of variance for each SMB process model separately. We calculate the empirical correlation
matrix $\stackrel{\mathrm{^}}{\mathbf{C}}$, which is an approximation of C, from the residuals of per-catchment best-fit temporal models described in Sect. 3.2. We save each residual (length n=28,
with 5 years held back from the 33-year training set to accommodate consistent fitting of AR orders up to p=5) as a row in a 260×28 matrix R, with one row for each catchment. The empirical
correlation matrix $\stackrel{\mathrm{^}}{\mathbf{C}}$ is then the 260×260 matrix of correlation coefficients of the residuals, which we compute using 𝚗𝚞𝚖𝚙𝚢.𝚌𝚘𝚛𝚛𝚌𝚘𝚎𝚏(𝚁). The empirical correlation
matrix computed from ANICE-ITM output is shown in Fig. 4a.
Because the number of catchments we seek to simulate (m=260 for Greenland) is considerably larger than the number of data points used to train individual statistical models (33 years of
catchment-aggregated SMB for each catchment), $\stackrel{\mathrm{^}}{\mathbf{C}}$ is singular. Therefore, we must enforce a sparsity condition to reduce the influence of spurious information. We
estimate a sparse correlation matrix Γ using the graphical lasso algorithm described in Friedman et al. (2007).
We apply the 𝙶𝚛𝚊𝚙𝚑𝚒𝚌𝚊𝚕𝙻𝚊𝚜𝚜𝚘𝙲𝚅 function from the Python package scikit-learn v0.24.2 (Pedregosa et al., 2011), which estimates a sparse correlation matrix Γ with the following formulation:
$\begin{array}{}\text{(4)}& \mathbf{\Gamma }={\text{argmin}}_{K}\left(\text{tr }\stackrel{\mathrm{^}}{\mathbf{C}}\mathbf{K}-\text{log det }\mathbf{K}+\mathit{\alpha }‖\mathbf{K}{‖}_{\mathrm{1}}\
where K is the inverse correlation matrix and α is a positive regularization parameter. Higher values of α lead to sparser resulting matrices Γ. In our implementation, we allow 𝙶𝚛𝚊𝚙𝚑𝚒𝚌𝚊𝚕𝙻𝚊𝚜𝚜𝚘𝙲𝚅 to
select the best value of α through cross-validation. Figure 4b shows the sparse correlation matrix resulting from applying this method to ANICE-ITM output.
Each row in the sparse correlation matrix Γ represents the correlation of a given catchment with each other catchment. Figure 4c translates the information in the first row of Γ to a map of
Greenland. The first row represents catchment 0 in the Mouginot and Rignot (2019) dataset: Umiammakku Isbræ. Umiammakku has the strongest correlation with itself (dark red shading), moderate positive
correlation (lighter red shading) with surrounding catchments and a few more distant catchments, and zero or slight negative correlation (light blue shading) with other catchments in Greenland. We
note that these correlations are inferred from the process model data – ANICE-ITM output, in Fig. 4 – rather than imposed by physical intuition. As such, the precise structure of the spatial
correlation matrix will depend on how the data are aggregated. We would expect slightly different spatial correlations if they were computed with monthly data or using different catchment outlines.
Users must also remember that the spatial correlations shown in Fig. 4 are computed on the residuals of temporal model fits, not on the SMB series themselves.
3.4Forward modeling
Finally, we generate a set of realizations of the forward stochastic generator. Each realization is the sum of an autoregressive component and a draw ϵ(t) from the normal distribution with spatial
covariance, as described in Eq. (1a) and Sect. 3.3. We find the Cholesky decomposition Γ=LL^T of the sparse correlation matrix and use the lower triangular component to generate spatially informed
noise. The draw ϵ[k](t) for the kth catchment is found by matrix multiplication:
$\begin{array}{}\text{(5)}& {\mathit{ϵ}}_{k}\left(t\right)=\mathbf{DL}\phantom{\rule{0.33em}{0ex}}{\mathbf{N}}_{j}\phantom{\rule{0.33em}{0ex}}\stackrel{\mathrm{^}}{k},\end{array}$
where N[j] is a random normal matrix of shape (m,Y) for m the number of catchments, Y is the number of years in the desired time series, and $\stackrel{\mathrm{^}}{k}$ selects the kth row of the
We generate realizations of catchment mean SMB for an example catchment: Kangerlussuaq Glacier. Each realization is a single time series of catchment mean SMB with variability described by the
stochastic generator. Figure 5 shows 10 realizations of Kangerlussuaq SMB from 1980–2050, with process model training data overlaid in black for 1980–2012. By inspection, the stochastic realizations
(blue lines on Fig. 5) have variability of similar amplitude and timescale to the process model series. The 10 realizations, generated in a few seconds on a laptop, fill the expected range of
uncertainty in annual SMB. We interpret these stochastic realizations to be an efficiently generated forcing for ensemble simulations of ice sheet change given SMB subject to internal variability.
The GrSMBMIP process model historical output we use as our example application did not exhibit nonstationarity, according to the KPSS and augmented Dickey–Fuller tests applied to each output series
(Sect. 3.2). We therefore fit stochastic generators that were stationary by construction, assuming the underlying distribution of the data did not change over the period of simulation. We generated
stochastic forward simulations as shown in Fig. 5 to illustrate the possibility of generating time series with consistent variability outside the training period. Those simulations fit a linear trend
to the training data and assumed that the trend and amplitude of variability remained constant into the future. For scientific applications that study periods of varying climate – for example,
glacial–interglacial periods or century-scale climate projections with anthropogenic forcing – it is expected that SMB time series would not be well fit by stationary models (Weirauch et al., 2008;
Bintanja et al., 2020).
To fit a stochastic generator to time series with statistics (mean, trend, variance) varying over time, the user could subdivide the training data series into periods with stationary trends and
variance. Piecewise linear trends could be computed on the sub-series and each series normalized by its variance to create a “z-score” time series. The stochastic temporal model could then be fit to
the z-score time series as described above. The output of the stochastic generator would then be re-scaled by the variance in each period to produce ensembles of nonstationary SMB series. The best
choice of break points to subset the training data will depend on the user's priorities and the time period being investigated; we do not pursue z-score re-scaling any further in this example. For
more formal discussion of bias correction in the case of climate data whose distribution changes in time, we refer the interested reader to applied statistics literature, e.g., Zhang et al. (2021)
and Poppick et al. (2016).
A user generating realizations of SMB at a particular location, or aggregated over some area, could use the method described up to this point. For example, this method could generate realizations of
aggregated SMB to support detection of departures from background variability, as in Wouters et al. (2013). The next section describes how to downscale SMB from the catchment annual average to
spatially extensive SMB fields at sub-annual timescales.
3.6Elevation downscaling
To force an ice sheet model, we require a two-dimensional SMB field on the mesh of the model, rather than catchment-aggregated time series. We now apply a spatial and temporal downscaling approach to
produce gridded SMB from the stochastically generated series at sub-annual time steps. The downscaling assumes that within each glacier catchment and for a given time of the seasonal cycle, the SMB
variation within a catchment can be described by a piecewise linear function with respect to elevation. This downscaling recognizes that, particularly in our Greenland example, there is a strong
seasonal cycle in SMB and that the spatial variations of SMB within a glacier catchment are mostly a function of elevation. As shown in Fig. 6, these assumptions are generally quite good for
Greenland SMB, and they are reflected in other statistical downscaling approaches that have been previously applied in deterministic frameworks (Hanna et al., 2011; Wilton et al., 2017; Sellevold
et al., 2019; Goelzer et al., 2020a). Further, the method generates fields with realistic spatiotemporal variability and elevation dependence, which can be embedded within an ice sheet model (e.g.,
Verjans et al., 2022) to capture the known feedback between ice sheet surface elevation change and SMB change (Edwards et al., 2014; Lenaerts et al., 2019).
For each point p in a given catchment, we need the surface elevation z(p) used to force the physical SMB model underlying our stochastic generator and the local SMB spatial anomaly,
$\begin{array}{}\text{(6)}& \mathrm{\Lambda }\left(\mathbit{p},t\right)=A\left(\mathbit{p},t\right)-\stackrel{\mathrm{‾}}{A}\left(t\right),\end{array}$
where A(p,t) is the process model SMB at point p and time t and $\stackrel{\mathrm{‾}}{A}\left(t\right)$ is the catchment mean SMB computed from the same process model at time t. We group all local
elevation–anomaly pairs by month – for example, all January values together and all June values together – and fit a piecewise linear mass balance gradient for each month τ:
where c[0] is the minimum SMB and the segment slopes $\left({c}_{\mathrm{1}},{c}_{\mathrm{2}},{c}_{\mathrm{3}}\right)$ and break points (z[1],z[2]) are free parameters optimized by BIC and AIC. In
each catchment we thus have 12 functions Λ[τ], one for each month. The monthly mass balance gradients Λ[τ] reintroduce seasonal variation. When taken as a function of ice sheet surface elevation z^*
that could be evolving in time, they also allow feedback between surface mass balance and dynamically evolving ice sheet geometry.
Example fits for the Kangerlussuaq Glacier catchment, computed from ANICE-ITM output covering 1980–1985, are shown in Fig. 6 (and the same example is shown computed from RACMO data in Fig. A3). The
left panels show the spatial anomaly in map view, with the terminus of the glacier to the southeast; the right panels show local SMB departure from the catchment mean as a function of ice surface
elevation. The spatial pattern in the example data shows strong departures from the catchment mean throughout the lowest portion of the glacier. January SMB in the lower reaches tends to exceed the
catchment mean (blue shading); July SMB in the same area tends to be much below the mean (dark red points). Higher elevations show less pronounced departures from the catchment mean (lighter
Finally, we produce time series of monthly local mass balance a for each grid point p of the kth catchment:
$\begin{array}{}\text{(8)}& a\left(\mathbit{p},t\right)=\mathbit{M}\left(t\right)\cdot \stackrel{\mathrm{^}}{k}+{\mathrm{\Lambda }}_{\mathit{\tau }}\left({z}^{*}\left(\mathbit{p},t\right)\right),\end
where t is the time in months since the start of the series, M(t) is the annual catchment mean SMB generated by the stochastic generator, Λ[τ] is the local SMB spatial anomaly for month τ as defined
above, and ${z}^{*}\left(\mathbit{p},t\right)$ is the local surface elevation at time t. The same principle could be adapted for training data provided at even finer temporal resolution, though a
large training dataset may be needed to capture the relevant variability in sub-monthly SMB.
To illustrate the method, we applied the elevation-based downscaling to estimate local SMB series at two different points, distributed across elevation, in the Kangerlussuaq Glacier catchment.
Figure 7b shows those time series. Blue lines are the stochastically generated SMB, downscaled to a single point in space; black lines are the process model output at the grid cell nearest to the
selected point. The point represented in the bottom panel is near the terminus and shows large-amplitude seasonal and interannual variations in both the process model and the downscaled stochastic
realizations. The stochastic realizations closely track the process model series, while also including interannual variability in winter and summer SMB that differs between realizations. The point in
the top panel is in the accumulation area. For that point, the range among the stochastic realizations is wider than the apparent variability in the process model series. The seasonal cycle has
approximately correct amplitude. We interpret the variability in the catchment-averaged SMB to be dominated by large-amplitude variation near the terminus (Figs. 6 and A3), which is then reflected in
the stochastic generator fit to the process model series. We further discuss this overestimate of accumulation zone interannual variability in the next section.
Simulating ice sheet evolution in a numerical model generally requires a two-dimensional SMB field that may vary in time. Here, we have laid the foundation for efficiently generating many
realizations of a time-varying SMB field with stochastic methods. Figure 7 demonstrates that our method can produce realistic SMB time series across an outlet glacier catchment. To produce a
two-dimensional field, a user would apply the downscaling method described in Sect. 3.6 to every grid point in the catchment. The piecewise linear mass balance gradients shown in Fig. 6 (insets) are
provided to the user as mathematical functions, so the downscaling can be applied on whatever mesh the user provides. This simplicity also allows this method to be incorporated directly into an ice
sheet model so that feedback of changing ice sheet geometry on SMB is included, in addition to the SMB variability in space and time generated by the method described above. This stochastic SMB
generation method has been incorporated directly into the Ice-Sheet and Sea-Level System Model (Verjans et al., 2022).
In evaluating candidate stochastic generators for catchment annual mean SMB, we found the best fit to process model variability with the lowest-order statistical models. For all 260 catchments we
tested, simple autoregressive models had by far the lowest Bayesian information criterion (better fit to process model SMB) among model types (e.g., Fig. 3b). Moreover, among low-order autoregressive
models, white noise AR(0) models with a trend are preferred over higher-order models in most basins for all seven process models tested (Fig. A1). Low-order AR models could have a low BIC despite
relatively greater error than higher-order models, as seen in Fig. 3, because the BIC penalizes excess parameters (Eq. 2). For each process model, there are some basins where higher-order AR(p)
models are strongly preferred over white noise. The example application we have shown allows the best-fit model order to be selected per basin. Our workflow therefore provides a self-consistent way
to infer stochastic generator fits for basins with different patterns of variability.
Our findings contrast with the results of a study by King and Watson (2020), which found that simple white noise and low-order AR models were not effective in capturing observed Antarctic SMB
variability. For annual SMB time series reconstructed for 1800–2010 for four Antarctic catchments – the West Antarctic Ice Sheet, East Antarctic Ice Sheet, Antarctic Peninsula Ice Sheet, and
Antarctica as a whole – King and Watson (2020) used the software Hector (Bos et al., 2013a) to simultaneously fit a linear trend and noise model. They found that white noise and AR(1) models tend to
underestimate low-frequency variability and that a better fit to observations came from power-law or generalized Gauss–Markov models (Bos et al., 2013b). The use of only three sub-catchments for
Antarctica results in much broader spatial aggregation in contrast to our use of 260 sub-catchments for the smaller Greenland Ice Sheet. That broad spatial aggregation might be expected to smooth
short-term variability and amplify the relative importance of low-frequency variability that correlates with large-scale climate forcing. For that reason, it is not surprising that we find a better
fit with simple temporal models given that we aggregate over smaller ice sheet catchments and study a shorter time period. Further, the spectra of variability could well be different between
Antarctica and Greenland; the former is a polar continent with climate heavily influenced by the Antarctic Circumpolar Current, while the latter is a large subpolar island exposed to warm oceanic
currents and westerly atmospheric flow. Antarctic SMB variability is thus dominated by snowfall (Previdi and Polvani, 2016), while Greenland experiences more surface melt and runoff, so the best-fit
temporal model types may not be directly comparable. Finally, we have tested stochastic model fit to more data sources – seven SMB process models – than earlier studies of one or two data sources
(including King and Watson, 2020); we found that simple autoregressive models were the best fit for all seven of the training models, lending credence to our results despite their contrast with
earlier findings. We do expect the characteristics of the best-fit stochastic generators to depend on basin delineation and training dataset, which we discuss further below.
We chose to limit the range of lags we tested in our autoregressive model fitting for two reasons. First, the autocorrelation functions of annual SMB reconstructed from ice cores (Fig. B1) show that
most cores have significant autocorrelation at short lags (<10 years) and no consistently significant autocorrelation at longer lags. Second, higher-order autoregressive models risk both overfitting
the data and needlessly adding computational expense, since high-order autoregressive models require holding the SMB from many previous time steps in local memory. The Bayesian information criterion
of candidate high-order AR(p) model fits to SMB data is high in most basins, supporting our choice in this case. Further, the decadal timescale of our example application is the most feasible
timescale on which to generate probabilistic projections of sea level change. For timescales of 50 years and longer, uncertainty about anthropogenic emissions scenarios dominates the range of
possible sea level change (Hinkel et al., 2019). However, it should be noted that a low-order autoregressive model such as ours is poorly suited to capture low-frequency variability, which may become
important for multi-century simulations.
Ice core data (Fig. B1) do not suggest that we have missed major modes of variability in our model fitting, but it is still plausible that our stochastic generator fitted to 32 years of training data
will fall short in reproducing multi-decadal and longer variations. To ensure that stochastic SMB generators do not miss low-frequency variability that could substantially change Greenland outlet
glacier catchments in the coming century and to support stochastic generation for longer-term historical simulations, further analysis should incorporate longer-term process model output or spatially
resolved reconstructions of SMB from ice cores or other observations. If the Greenland Ice Sheet were to become unstable, as recent analyses have suggested (Boers and Rypdal, 2021), the variance and
autocorrelation timescale of its future mass balance could be quite different from the recent past. Stochastic generator fitting intended for multi-century future projection should thus be trained on
output data from SMB models run at similarly long timescales, where possible including the relevant feedbacks and instabilities, rather than projecting forward from 30-year historical simulations as
we have done here. We emphasize that our study describes a flexible methodological framework for training a stochastic generator of SMB variability, with an example application to multi-decadal
simulation. Our framework can be applied to existing data for other use cases (such as paleo-reconstruction) and to new SMB process model outputs as they become available.
Our downscaling method makes it possible to generate SMB fields on whatever mesh is needed by a numerical ice sheet model. In ice sheet models designed to accept stochastic forcing, the parameters of
the stochastic generator can be provided directly for online generation of the forcing fields within the model itself, with negligible addition of computational expense (demonstrated in Verjans
et al., 2022). With regular updates to the surface elevation of each point on the model mesh, Eq. (8) can also account for the known feedback between ice sheet surface elevation and surface melt rate
(Hanna et al., 2013; Edwards et al., 2014; Lenaerts et al., 2019). Such a streamlined workflow will further facilitate large-ensemble simulations.
The workflow we present here, including the downscaling method, is agnostic to the choice of regions over which to aggregate the SMB. The example application to outlet glacier catchments in Greenland
uses a standard, published basin delineation (Mouginot and Rignot, 2019). The downscaled time series shown in Fig. 3, which we generated with data aggregated from a standard set of catchments, show
variability dominated by large-amplitude seasonal variation at the terminus. This asymmetry in variability amplitude between the accumulation and ablation zones ultimately leads to some
overestimation of interannual variability at accumulation zone points. When aggregated over a large accumulation area, overestimated local variability could translate to an artificially large
magnitude of uncertainty in expected sea level contribution. We suggest that this effect could be tempered by splitting catchment data into accumulation-area and ablation-area bins before fitting the
spatial downscaling function. Depending on the user's scientific goal, such disaggregation may not be necessary for forcing an ice sheet model, as sub-decadal outlet glacier flow variability is
driven by near-terminus SMB variability (Christian et al., 2020). We expect that there would be qualitative differences in the SMB series generated with and downscaled to different choices of basin
delineation (Goelzer et al., 2020a); we have not attempted to optimize basin selection for the illustrative example here. Users may apply all steps of the workflow described in Sect. 3.1–3.6 to SMB
data aggregated over different regions.
The within-catchment downscaling we present in Sect. 3.6 is a simple example that may be adapted or replaced for other applications. The example data plotted in Figs. 6 and A3 show only 1980–1985 in
the mass balance gradient. We tested example Λ[τ] fits to data from the full period (1980–2012) but found that they tended to underestimate variability; conversely, fits to shorter periods tended to
overestimate spatial variability. The example presented here illustrates the possibility of inferring a downscaling function from process model output. It would be possible to infer similar
downscaling functions at different temporal or spatial resolutions using reanalysis or reconstructed data or computed over a different reference period. Ultimately, the choice of a reference period
and the best spatial dataset to infer such a function depends on the user's intended application, and this selection may be nontrivial. Further, our simple downscaling does not capture changes in
elevation dependence of SMB over time, for example due to changes in precipitation phase or local atmospheric lapse rate. Users seeking improved fine-scale performance may wish to implement more
granular statistical downscaling methods (e.g., Noël et al., 2016).
The inter-catchment spatial covariance method we apply here will lose some relevant spatial details from the original process models. As described in Sect. 3.3, the empirical inter-basin correlation
matrices $\stackrel{\mathrm{^}}{\mathbf{C}}$ were singular for our example case, and in order to generate new realizations of variability, we enforced sparsity in the correlation matrix Γ (Fig. 4
a–b). By construction, this method loses some spatial detail present in the original dataset. Further, our method does not quantify uncertainty in the model fit – for example, within-catchment
differences in the best-fit statistical model parameters – other than the range of variability present in the original process model simulations. Our stochastic generation of SMB fields based only on
SMB models also disregards any covariance between oceanic and atmospheric forcings. More sophisticated methods currently under development, such as fitting a Gaussian process emulator (Mohammadi
et al., 2019; Edwards et al., 2021) to the field varying in space, may be able to resolve these problems in the future. However, fitting such an emulator that varies in space and time would require
storage of, and computation on, multiple realizations of SMB process models at kilometer resolution. Such a task is considerably more computationally demanding than what we have pursued in the
example shown here.
Given the simplifications described above, and the abstraction of stochastic parameters in contrast to physical quantities, we do not intend stochastic SMB generation to completely replace process
model simulation of ice sheet SMB. Rather, we envision stochastic SMB generation to provide a complementary tool set which reproduces many features of SMB process models at nearly negligible
computational expense. The open-source software that we have developed and the existing packages on which it is built can be easily applied to fit a stochastic representation to new outputs from
process-based SMB models as they become available. Selecting an appropriate class of stochastic generator is the most time-consuming step of the process; with that complete, the best-fit model
parameters can be updated at any time to account for new process model results and generate hundreds of new realizations sampling the range of internal variability of SMB. Stochastic generation
therefore serves to more immediately connect dynamic ice sheet projections with internal variability from cutting-edge SMB simulations without the need for costly coupled ensemble simulations.
We have described the development and demonstrated the use of a stochastic method to generate many realizations of ice sheet SMB fields varying in space and time. For all 260 catchments of the
Greenland Ice Sheet that we tested, the simplest temporal models (AR(p) with order p<5) provided the best fit to process-model-derived SMB time series. Our method streamlines the creation of large
samples of climate-dependent forcing to simulate ice sheet mass change subject to internal climate variability. The improved computational efficiency offered by this stochastic SMB generation method
will facilitate large-ensemble simulations of ice sheet change, which can support a range of applications including (1) probabilistic sea level projections with improved uncertainty quantification,
(2) separating ice sheet variability from atmospheric and oceanic variability in simulated changes to the coupled climate system, and (3) attribution of observed changes to specific forcings.
Appendix A:Best-fit stochastic generators are similar for different SMB process models
The method we presented can be adapted for a variety of scientific applications. In the example use case we demonstrated above, we fit stochastic generators to the output from several SMB process
models that had participated in the Greenland SMB Model Intercomparison Project (Fettweis et al., 2020, models described therein). The SMB process models vary in complexity, from relatively simple
energy balance models such as ANICE-ITM (shown in the main text) to more sophisticated regional climate models such as RACMO.
The results of fitting a stochastic generator to the model output were comparable regardless of process model. Figure A1 shows how many basins were best fit by AR(n) stochastic generators for n from
0 to 5, with 0 being a white noise model. Figure A2 shows SMB series produced by stochastic generators fit to each of three example process models. The series are qualitatively similar; the amplitude
of the variability in the stochastic realizations compared to the process model series differs per process model. This effect is due to differences in the spatial covariance inferred from each
process model and used to generate the noise term ϵ(t) in each realization.
Within-catchment spatial variation does differ slightly per process model. For example, the SMB spatial anomaly for points in the Kangerlussuaq Glacier catchment is different for ANICE-ITM (main text
Fig. 6) and RACMO (Fig. A3), especially at low elevations. RACMO shows less spread in January values and much more spread in July values, to the point of fitting an inflection point in the
SMB–elevation relationship within the ablation area. Users of our method must determine what downscaling approach is most suitable for their scientific aims. Possible choices include (1) using a
spatial downscaling consistent with the process model the user intends to sample, as we have shown in the main text for ANICE-ITM; (2) fitting a monthly downscaling function comparable to Eq. (7) but
based on data from another source they find more accurate for this application, such as an observational dataset or a higher-resolution process model; or (3) implementing another elevation-dependent
downscaling technique such as those described in Noël et al. (2016) or Goelzer et al. (2020a).
Appendix B:Modes of variability in ice core reconstructions
The GrSMBMIP process model output we used to fit stochastic generators in our example application covered a common period of 33 years from 1980–2012. To add longer-term context to our choice of
candidate model classes (Sect. 3.2), we also examined ice core reconstructions of SMB in Greenland over the last 2000 years (Andersen et al., 2006). The point nature of these measurements makes them
unsuitable for generating stochastic, ice-sheet-wide SMB fields, but they are a useful benchmark to assess the characteristic timescales of SMB variability, including timescales longer than are
simulated in regional SMB models.
We computed the autocorrelation and partial autocorrelation functions of SMB reconstructed from each of five cores. If multiple cores showed significant autocorrelation at time lags longer than
5 years, it would be an indication that our model fitting procedure should include candidate models with higher autoregressive orders p and moving averages q. The autocorrelation function for the ice
core SMB is shown for lags up to 100 years in Fig. B1. There are no lag values greater than 5 years for which the five cores agree on significant autocorrelation.
The ice core record in Andersen et al. (2006) comes from cores in the accumulation area of the Greenland Ice Sheet. The cores are not necessarily representative of decadal-scale SMB like the GrSMBMIP
data we fit in our example application because they will not reflect variation in melt rate or coastal precipitation. As a complementary dataset, the ice cores support our choice to limit the lags (p
,q) tested in our model fitting procedure; however, they do not guarantee that our example SMB generators are applicable at timescales far beyond the historical period to which they were fit. For
scientific applications that aim to generate SMB varying on timescales of centuries and longer, we encourage users to fit a generator to a training dataset on a comparable timescale.
AAR conceived of the Stochastic Ice Sheet project. LU and AAR designed the SMB study, with support from SC. LU wrote the code, made the figures, and drafted the paper. All authors contributed to
editing the paper and approved its final form.
The contact author has declared that none of the authors has any competing interests.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation
in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.
The authors thank GrSMBMIP participating teams led by Tijn Berends, Leo van Kampenhout, Uta Krebs-Kanzow, Ruth Mottram, Masashi Niwano, Glen Liston, and Brice Noël for their permission to use their
model data and Xavier Fettweis for facilitating access to GrSMBMIP and MAR output data.
The authors thank Matt Osman for ice core data consultation and Vincent Verjans for his work on z-score normalization for nonstationary ice sheet forcings. The authors also thank Tijn Berends and one
anonymous reviewer for constructive comments that helped improve the paper.
This research has been supported by the Heising-Simons Foundation (grant no. 2020-1965).
This paper was edited by Philippe Huybrechts and reviewed by Tijn Berends and one anonymous referee.
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https://doi.org/10.1038/ngeo1874, 2013.a, b
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1831–1849, https://doi.org/10.1214/21-AOAS1460, 2021.a | {"url":"https://gmd.copernicus.org/articles/17/1041/2024/","timestamp":"2024-11-07T10:51:02Z","content_type":"text/html","content_length":"351034","record_id":"<urn:uuid:b308f60e-b7ec-44d2-9058-6faa8eda9220>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00355.warc.gz"} |
Math 20-1
Gary is a fitness trainer who makes 30 and 45 minute appointments
with clients. Gary’s contract allows him to make up to 30 hours of
appointments in a week. State the inequality that represents all the
possible combinations of appointments that Gary can make in a
week. Graph the inequality. | {"url":"https://www.tutortag.com/chat/alberta/gary-is-a-fitness-trainer-who-makes-30-and-45-minute-appoint","timestamp":"2024-11-14T16:30:53Z","content_type":"text/html","content_length":"31176","record_id":"<urn:uuid:2b13fb6f-74f8-41fe-9cdc-b7f6251e4f15>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00109.warc.gz"} |
Overview of the GroupTheory Package
Calling Sequence Context-Sensitive Operations
Description Examples
List of GroupTheory Package Commands Compatibility
Group Constructors Palette
Calling Sequence
GroupTheory:-command( arguments )
command( arguments )
• The GroupTheory package provides a collection of commands for computing with, and
visualizing, finitely generated (especially finite) groups. There are several classes of
groups that are implemented.
permutation groups groups given by a set of generating permutations
finitely presented groups given by generators and defining relations
Cayley table groups groups whose binary operation is specified by a Cayley table
custom groups "black-box" user-defined groups whose elements are of an
unspecified nature
symbolic groups abstract groups depending on symbolic parameters
• The package contains a variety of constructors that allow you to easily create groups in
common families. Furthermore, several databases of groups exist in the package, and
interfaces to these databases are provided.
WorkingWithFPGroups Working with Finitely Presented Groups
WorkingWithPermutations Working with Permutations
WorkingWithSymbolicGroups Working with Symbolic Groups
List of GroupTheory Package Commands
• The following is a list of the commands in the GroupTheory package.
AbelianGroup construct a finitely generated Abelian group
AffineGeneralLinearGroup construct the affine general linear group
over a finite field
AffineSpecialLinearGroup construct the affine special linear group
over a finite field
AGL construct the affine general linear group
over a finite field
AllAbelianGroups return all the Abelian groups of a given
AlternatingGroup construct the alternating group of a given
ASL construct the affine special linear group
over a finite field
BabyMonster construct the Baby Monster sporadic finite
simple group
CayleyTableGroup construct a Cayley table group
ChevalleyE6 construct a Chevalley group of type E6
ChevalleyE7 construct a Chevalley group of type E7
ChevalleyE8 construct a Chevalley group of type E8
ChevalleyF4 construct a Chevalley group of type F4
ChevalleyG2 construct a Chevalley group of type G2
ConwayGroup construct a Conway sporadic simple group
CustomGroup construct a custom group, given its
CyclicGroup construct a cyclic group of a given order
DicyclicGroup construct a dicyclic group
DihedralGroup construct the dihedral group of a given
DirectProduct construct a direct product of groups
ElementaryGroup construct an elementary Abelian group
ExceptionalGroup construct one of the exceptional finite
simple groups
FischerGroup construct one of the Fischer groups
FPGroup construct a finitely presented group from
generators and defining relators
FreeGroup construct a free group
FrobeniusGroup construct a Frobenius group from the database
GaloisGroup construct the Galois group of a polynomial
GammaL construct the general semi-linear group over
a finite field
GeneralLinearGroup construct the general linear group over a
finite field
GeneralOrthogonalGroup construct the general orthogonal group over a
finite field
GeneralSemilinearGroup construct the general semi-linear group over
a finite field
GeneralUnitaryGroup construct a general unitary group over a
finite field
GL construct the general linear group over a
finite field
Group construct a group from various data
HamiltonianGroup construct a finite Hamiltonian group
HaradaNortonGroup construct the Harada-Norton simple group
HeldGroup construct the Held simple group
HigmanSimsGroup construct the Higman-Sims simple group
JankoGroup construct one of the Janko sporadic finite
simple groups
LyonsGroup construct the Lyons simple group
MathieuGroup construct one of the Mathieu finite simple
McLaughlinGroup construct the McLaughlin simple group
MetacyclicGroup construct a metacyclic group
Monster construct the Monster simple group
ONanGroup construct the O'Nan simple group
Orbit construct the orbit of an element under a
permutation group
OrthogonalGroup construct an orthogonal group
PerfectGroup construct a perfect group from the database
Perm create a permutation
PermutationGroup construct a permutation group given
generating permutations
PGammaL construct the general semi-linear group over
a finite field
PGL construct a projective linear group over a
finite field
PGO construct the projective general orthogonal
group over a finite field
PGU construct a projective general unitary group
over a finite field
ProjectiveGeneralLinearGroup construct a projective general linear group
over a finite field
ProjectiveGeneralOrthogonalGroup construct the projective general orthogonal
group over a finite field
ProjectiveGeneralSemilinearGroup construct the general semi-linear group over
a finite field
ProjectiveGeneralUnitaryGroup construct a projective general unitary group
over a finite field
ProjectiveSpecialLinearGroup construct a projective special linear group
over a finite field
ProjectiveSpecialOrthogonalGroup construct the projective special orthogonal
group over a finite field
ProjectiveSpecialSemilinearGroup construct the special semi-linear group over
a finite field
ProjectiveSpecialUnitaryGroup construct a projective special unitary group
over a finite field
ProjectiveSymplecticGroup construct a projective symplectic group over
a finite field
ProjectiveSymplecticSemilinearGroup construct the projective symplectic
semi-linear group over a finite field
PSigmaL construct the special semi-linear group over
a finite field
PSigmap construct the projective symplectic
semi-linear group over a finite field
PSL construct a projective special linear group
over a finite field
PSO construct the projective special orthogonal
group over a finite field
PSp construct a projective symplectic group over
a finite field
PSU construct a projective special unitary group
over a finite field
QuasicyclicGroup construct a quasicyclic p-group
QuasiDihedralGroup construct a quasi-dihedral group
QuaternionGroup construct a generalized quaternion group
Ree2F4 construct a large Ree group of Lie type
Ree2G2 construct a small Ree group of Lie type
RubiksCubeGroup construct the group of the Rubik's Cube
RudvalisGroup construct the Rudvalis simple group
SemiDihedralGroup construct a semi-dihedral group
SigmaL construct the special semi-linear group over
a finite field
Sigmap construct the symplectic semi-linear group
over a finite field
SL construct a special linear group over a
finite field
SmallGroup construct a specific group of small order
SpecialLinearGroup construct a special linear group over a
finite field
SpecialOrthogonalGroup construct a special orthogonal group over a
finite field
SpecialSemilinearGroup construct the special semi-linear group over
a finite field
SpecialUnitaryGroup construct a special unitary group over a
finite field
Stabilizer compute the stabilizer of a point, list or
set under a permutation group
Steinberg2E6 construct a Steinberg group of type 2E6
Steinberg3D4 construct a Steinberg group of type G2
Subgroup construct a subgroup of a given group
Supergroup construct a supergroup of a given group
Suzuki2B2 construct a Suzuki group of Lie type
SuzukiGroup construct the Suzuki simple group
Symm construct the symmetric group of a given
SymmetricGroup construct the symmetric group of a given
SymplecticGroup construct a symplectic group over a finite
SymplecticSemilinearGroup construct the symplectic semi-linear group
over a finite field
ThompsonGroup construct the Thompson simple group
TitsGroup construct the Tits simple group
TransitiveGroup construct a specific transitive permutation
TrivialGroup construct the trivial group
TrivialSubgroup construct the trivial subgroup of a given
WreathProduct construct a wreath product of permutation
Center compute the center of a group
Centraliser compute the centraliser of an element of a group
Centralizer compute the centralizer of an element of a group
Centre compute the centre of a group
Commutator compute the commutator of two subgroups
Core compute the core of a subgroup of a group
Cosocle compute the cosocle of a group
DerivedSubgroup compute the derived (commutator) subgroup of a group
DirectFactors compute the directly indecomposable direct factors of a
finite group
FittingSubgroup compute the Fitting subgroup of a group
FrattiniSubgroup compute the Frattini subgroup of a group
FrobeniusComplement compute a representative Frobenius complement of a
Frobenius group
FrobeniusKernel compute the Frobenius kernel of a Frobenius group
FrobeniusProduct compute the product of two complexes in a finite group
HallSubgroup compute a Hall pi-subgroup of a finite soluble group
HallSystem compute a Hall system for a finite soluble group
Hypercenter compute the hypercenter residual of a group
Index compute the index of a subgroup of a group
Intersection compute the intersection of two subgroups of a group
IsDirectlyIndecomposable test whether a group is directly indecomposable
IsMalnormal test whether a subgroup of a group is malnormal
IsNormal test whether a subgroup of a group is normal
IsQuasinormal test whether a subgroup of a group is quasi-normal
IsSubnormal test whether a subgroup of a group is subnormal
MaximalNormalSubgroups compute the maximal normal subgroups of a permutation
MinimalNormalSubgroups compute the minimal normal subgroups of a permutation
NilpotentResidual compute the nilpotent residual of a group
NormalClosure compute the normal closure of a subgroup or set of group
Normaliser compute the normaliser of a subgroup of a group
NormaliserSubgroup compute the normaliser of a subgroup of a group
NormalizerSubgroup compute the normalizer of a subgroup of a group
NormalSubgroups compute the normal subgroups of a finite group
PCore compute the p-core of a subgroup of a group
Socle compute the socle of a group
SolubleResidual compute the soluble residual of a group
SolvableResidual compute the solvable residual of a group
SubgroupLattice compute the lattice of subgroups of a group
SylowBasis compute a Sylow basis for a finite soluble group
SylowSubgroup compute a Sylow p-subgroup of a finite group
AllFrobeniusGroups return a list of the known Frobenius groups of a given
AllHamiltonianGroups return a list of all the Hamiltonian groups of a given
AllPerfectGroups return a list of the perfect groups of a given order
AllSmallGroups return a list of all the groups of a given order
AllTransitiveGroups return a list of all the transitive groups of a given
FrobeniusGroup construct a Frobenius group from the database
IdentifyFrobeniusGroup locate a given Frobenius group in the database of
Frobenius groups
NumFrobeniusGroups return the number of known Frobenius groups of a given
NumHamiltonianGroups return the number of Hamiltonian groups of a given order
NumPerfectGroups return the number of perfect groups of a given order
NumTransitiveGroups return the number of transitive groups of a given degree
PerfectGroup construct a perfect group from the database
RandomSmallGroup return a random group from the database of small groups
SearchFrobeniusGroups search the Frobenius Groups database
SearchPerfectGroups search the Perfect Groups database
SearchSmallGroups search the Small Groups database
SearchTransitiveGroups search the Transitive Groups database
SmallGroup construct a specific group of small order
TransitiveGroup construct a specific transitive permutation group
AbelianInvariants compute the abelian invariants of a group
ClassNumber compute the number of conjugacy classes of a finite group
CompositionLength compute the composition length of a group
ConjugateRank compute the conjugate rank of a finite group
DerivedLength compute the derived length of a group
Exponent compute the exponent of a group
FittingLength compute the nilpotent (Fitting) length of a group
FrattiniLength compute the Frattini length of a group
GroupOrder compute the order of a group
NilpotencyClass compute the class of nilpotence of a group
NilpotentLength compute the nilpotent (Fitting) length of a group
NumInvolutions compute the number of involutions of a group
OrderClassNumber compute the number of order classes of a finite group
OrderRank compute the number of order class lengths greater than unity of
a finite group
PermGroupRank compute the rank of a permutation group
PGroupRank compute the rank of a finite p-group
PrimaryInvariants compute the primary invariants of a group
Transitivity compute the transitivity of a permutation group
AreConjugate check whether two group elements are conjugate
AreIsomorphic test whether two groups are isomorphic
IsAbelian test whether a group is Abelian
IsAbelianSylowGroup test whether a group has Abelian Sylow subgroups
IsAlmostSimple test whether a group is almost simple
IsAlternating test (probabilistically) whether a permutation group
is an alternating group in its natural action
IsCAGroup test whether a group is a (CA)-group
IsCaminaGroup test whether a group is a Camina group
IsCCGroup test whether a group is a (CC)-group
IsCharacteristicallySimple test whether a group is characteristically simple
IsCNGroup test whether a group is a (CN)-group
IsCommutative test whether a group is commutative
IsCP1Group test whether a group is a (CP1)-group
IsCPGroup test whether a group is a (CP)-group
IsCyclic test whether a group is cyclic
IsCyclicSylowGroup test whether a group has cyclic Sylow subgroups
IsDedekind test whether a group is Dedekind
IsDicyclic test whether a permutation group is a dicyclic group
IsDihedral test whether a permutation group is a dihedral group
IsElementary test whether a group is elementary Abelian
IsExtraspecial test whether a group is an extraspecial p-group
IsFinite test whether a group is finite
IsFinitelyGenerated test whether a group is finitely generated
IsFrobeniusGroup test whether a group is a Frobenius group
IsFrobeniusPermGroup test whether a group is a Frobenius permutation group
IsGCLTGroup test whether a group is a GCLT-group
IsHallPaigeGroup test whether a group has a complete mapping
IsHamiltonian test whether a group is Hamiltonian
IsHomocyclic test whether a group is homocyclic
IsLagrangian test whether a group is Lagrangian
IsMalnormal test whether a subgroup of a group is malnormal
IsMetabelian test whether a group is Abelian
IsMetacyclic test whether a group is Metacyclic
IsNilpotent test whether a group is nilpotent
IsNormal test whether a subgroup of a group is normal
IsOrderedSylowTowerGroup test whether a group has a Sylow tower
IsPerfect test whether a group is perfect
IsPerfectOrderClassesGroup test whether a group has perfect order classes
IsPermutable test whether a subgroup of a group is permutable
IsPGroup test whether a group is a p-group
IsPrimitive test whether a permutation group is primitive
IsPSoluble test whether a group is p-soluble for a given prime p
IsQuasiprimitive test whether a permutation group is quasi-primitive
IsQuasisimple test whether a group is quasi-simple
IsQuaternion test whether a permutation group is a quaternion group
IsRegular test whether a permutation group is regular
IsRegularPGroup test whether a group is a regular p-group
IsSemiprimitive test whether a permutation group is semi-primitive
IsSemiRegular test whether a permutation group is semi-regular
IsSimple test whether a group is simple
IsSoluble test whether a group is soluble
IsSolvable test whether a group is solvable
IsSpecial test whether a group is a special p-group
IsStemGroup test whether a group is a stem group
IsSubgroup test whether one group is a subgroup of another
IsSubnormal test whether a subgroup of a group is subnormal
IsSupersoluble test whether a group is supersoluble
IsSylowTowerGroup test whether a group has a Sylow tower
IsSymmetric test (probabilistically) whether a permutation group
is a symmetric group in its natural action
IsTGroup test whether a group is a T-group
IsTransitive test whether a permutation group is transitive
IsTrivial test whether a group is trivial
SubgroupMembership test whether an element belongs to a given subgroup of
a group
Permutation Groups
AbelianInvariants compute the abelian invariants of a
permutation group
BlocksImage return a permutation group equivalent to
the action of a permutation on a system of
BlockSystem return a block system for a permutation
group, non-trivial if possible
CycleIndexPolynomial compute the cycle index polynomial of a
permutation group
Degree return the degree of a permutation group
EARNS return an EARNS of a primitive group if it
has one
FrobeniusPermRep construct a Frobenius permutation group
isomorphic to a given Frobenius group
IsAlternating test (probabilistically) whether a
permutation group is an alternating group
in its natural action
IsPrimitive test whether a permutation group is
IsQuasiprimitive test whether a permutation group is
IsRegular test whether a permutation group is
IsSemiprimitive test whether a permutation group is
IsSemiRegular test whether a permutation group is
IsSymmetric test (probabilistically) whether a
permutation group is a symmetric group in
its natural action
IsTransitive test whether a permutation group is
MaxSupport return the largest element displaced by a
permutation group
MinimalBlockSystem return a minimal block system for a
permutation group, non-trivial if possible
MinimumPermutationRepresentationDegree compute the minimum degree of a faithful
permutation representation for a group
MinSupport return the smallest element displaced by a
permutation group
Orbit construct the orbit of an element under a
permutation group
Orbits compute the orbits of a permutation group
PermGroupRank compute the rank of a permutation group
PrimaryInvariants compute the primary invariants of a
permutation group
ReducedDegreePermGroup return an isomorphic permutation group of
possibly smaller degree
RestrictedPermGroup return the restriction of a permutation
group to a stable subset
Stabilizer compute the stabilizer of a point, list or
set under a permutation group
Support return the support of a permutation group
SupportLength return the number of elements displaced by
a permutation group
Transitivity compute the transitivity of a permutation
Finitely Presented Groups
AbelianInvariants compute the abelian invariants of a finitely presented
PresentationComplexity compute a measure of the complexity of a finitely
presented group
PrimaryInvariants compute the primary invariants of a finitely presented
Relators return the relators of a finitely presented group
Simplify simplify the presentation of a finitely presented group
DrawCayleyTable draw the Cayley table of a finite group
DrawNormalSubgroupLattice draw the lattice of normal subgroups of a finite group
DrawSubgroupLattice draw the lattice of subgroups of a finite group
AgemoSeries compute the series of agemo subgroups of a $p$-group
CompositionSeries compute a composition series of a group
DerivedSeries compute the derived series of a group
FrattiniSeries compute the Frattini series of a group
LowerCentralSeries compute the lower central series of a group
LowerFittingSeries compute the lower Fitting series of a group
LowerPCentralSeries compute the lower p-central series of a group
OmegaSeries compute the series of omega subgroups of a $p$-group
OrderedSylowTower compute a Sylow tower for a Sylow tower group
SylowTower compute a Sylow tower for a Sylow tower group
UpperCentralSeries compute the upper central series of a group
DecomposeDessin find all decompositions of a Belyi map represented by a dessin
FindDessins find all dessins d'enfants with a specified branch pattern
ElementOrder compute the order of a group element
ElementOrderSum compute the sum of the element orders of a finite group
ElementPower compute an integer power of a group element
Elements compute the elements of a finite group, orbit, coset or
conjugacy class
MaximumElementOrder compute the largest order of an element of a finite group
OrderClassPolynomial compute the order class polynomial of a finite group
OrderClassProfile compute the element order profile of a finite group
PrimePowerFactors factor a group element into a product of elements of prime
power order
RandomElement compute a random element of a group
RandomImvolution compute a random involution of a group
RandomPElement compute a random p-element of a group
RandomPPrimeElement compute a random element of a group with order relatively
prime to p
IsAbelianNumber test whether every group of a given order is Abelian
IsCyclicNumber test whether every group of a given order is cyclic
IsGCLTNumber test whether every group of a given order is a GCLT
IsIntegrableNumber test whether every group of a given order is integrable
IsLagrangianNumber test whether every group of a given order is Lagrangian
IsMetabelianNumber test whether every group of a given order is metabelian
IsMetacyclicNumber test whether every group of a given order is metacyclic
IsNilpotentNumber test whether every group of a given order is nilpotent
IsOrderedSylowTowerNumber test whether every group of a given order has an
ordered Sylow tower
IsSimpleNumber test whether a number is the order of a finite simple
IsSolubleNumber test whether every group of a given order is soluble
IsSupersolubleNumber test whether every group of a given order is
CayleyGraph return the Cayley graph of a finite group
CayleyTable return the Cayley table of a finite group
CFSG finite simple group classifier object
Character construct a character from a character table
CharacterTable compute the character table of a finite group
ClassifyFiniteSimpleGroup classify a finite simple group
CommutingGraph construct the commuting graph of a finite group
ConjugacyClass compute the conjugacy class of a group element
ConjugacyClasses compute all the conjugacy classes of a finite group
Conjugator compute an element conjugating one group element to
Elements compute the elements of a finite group, orbit, coset or
conjugacy class
Factor expression a group element as a product of a coset
representative and a subgroup element
Generators return the set of generators of a group
GruenbergKegelGraph return the Gruenberg-Kegel graph of a finite group
IdentifySmallGroup compute the Small Group ID of a small group
Labels return the set of generator labels of a group
LeftCoset compute a left coset of a group element
LeftCosets compute the left cosets of a subgroup of a group
logp compute the exponent of a prime power
NonRedundantGenerators return a set of non-redundant generators of a group
NumSimpleGroups count the number of simple groups of a given finite
Operations return the operations record of a group
PresentationComplexity compute a measure of the complexity of a finitely
presented group
RightCoset compute a right coset of a group element
RightCosets compute the right cosets of a subgroup of a group
TabulateSimpleGroups list the simple groups with orders in a given range
Group Constructors Palette
• The Group Constructors palette contains buttons for constructing groups.
• Palettes are displayed in the left pane of the Maple window. (If it is not visible, from
the main menu, select View > Palettes > Show Palette > Group Constructors)
• Some palette items have placeholders. Fill in the placeholders, using Tab to navigate to
the next placeholder.
Context-Sensitive Operations
• In the Standard Worksheet interface, you can apply operations to a group through the
Context Panel under the Group operations submenu.
The following command enables you to use the commands in the GroupTheory package without
having to prefix each command with "GroupTheory:-".
> $\mathrm{with}⁡\left(\mathrm{GroupTheory}\right):$
Create a symmetric group of degree 4.
> $G≔\mathrm{SymmetricGroup}⁡\left(4\right)$
${G}{≔}{{\mathbf{S}}}_{{4}}$ (1)
Visualize the lattice of subgroups of G.
> $\mathrm{DrawSubgroupLattice}⁡\left(G\right)$
Create a dihedral group of degree 4 (and order 8).
> $H≔\mathrm{DihedralGroup}⁡\left(4\right)$
${H}{≔}{{\mathbf{D}}}_{{4}}$ (2)
Visualize the Cayley (operation) table for H.
> $\mathrm{DrawCayleyTable}⁡\left(H\right)$
Compute the orders (cardinalities) of G and H.
> $\mathrm{GroupOrder}⁡\left(G\right),\mathrm{GroupOrder}&
${24}{,}{8}$ (3)
Compute the character table of H.
> $\mathrm{Display}⁡\left(\mathrm{CharacterTable}⁡\left(H\
C 1a 2a 2b 2c 4a
|C| 1 1 2 2 2
$\mathrm{χ__1}$ $1$ $1$ $1$ $1$ $1$
$\mathrm{χ__2}$ $1$ $1$ $-1$ $-1$ $1$
$\mathrm{χ__3}$ $1$ $1$ $-1$ $1$ $-1$
$\mathrm{χ__4}$ $1$ $1$ $1$ $-1$ $-1$
$\mathrm{χ__5}$ $2$ $-2$ $0$ $0$ $0$
Form the direct product in two ways.
> $U≔\mathrm{DirectProduct}⁡\left(G,H\right)$
${U}{≔}〈\left({1}{,}{2}\right){,}\left({1}{,}{2}{,}{3} (4)
> $V≔\mathrm{DirectProduct}⁡\left(H,G\right)$
${V}{≔}〈\left({1}{,}{3}\right){,}\left({1}{,}{2}{,}{3} (5)
Check that these are isomorphic.
> $\mathrm{AreIsomorphic}⁡\left(U,V\right)$
${\mathrm{true}}$ (6)
Notice that the order of the direct product is equal to the product of the orders of the
> $\mathrm{GroupOrder}⁡\left(U\right)=\mathrm{GroupOrder}&
${192}{=}{192}$ (7)
Since the order of U does not exceed 511, we can identify the group explicitly.
> $\mathrm{id}≔\mathrm{IdentifySmallGroup}⁡\left(U\right)$
${\mathrm{id}}{≔}{192}{,}{1472}$ (8)
Retrieve the identified group from the database, and check the isomorphism.
> $W≔\mathrm{SmallGroup}⁡\left(\mathrm{id}\right):$
> $\mathrm{AreIsomorphic}⁡\left(U,W\right)$
${\mathrm{true}}$ (9)
Construct a wreath product of two symmetric groups.
> $W≔\mathrm{WreathProduct}⁡\left(\mathrm{Symm}⁡\left(3\
${W}{≔}〈\left({1}{,}{2}\right){,}\left({1}{,}{2}{,}{3} (10)
Check that the wreath product is transitive but imprimitive.
> $\mathrm{IsTransitive}⁡\left(W\right)$
${\mathrm{true}}$ (11)
> $\mathrm{IsPrimitive}⁡\left(W\right)$
${\mathrm{false}}$ (12)
Find a non-trivial system of blocks for W.
> $\mathrm{BlockSystem}⁡\left(W\right)$
${{{1}{,}{2}{,}{3}}{,}{{4}{,}{5}{,} (13)
Identify the Sylow $3$-subgroup of W in the database of small groups.
> $\mathrm{IdentifySmallGroup}⁡\left(\mathrm{SylowSubgroup}⁡\
${243}{,}{51}$ (14)
Find the nilpotency class of the Sylow $2$-subgroup of W.
> $\mathrm{NilpotencyClass}⁡\left(\mathrm{SylowSubgroup}⁡\left
${4}$ (15)
Compute a composition series for U.
> $\mathrm{cs}≔\mathrm{CompositionSeries}⁡\left(U\right)$
Warning, over-writing property `["DerivedSeries"]' with a different value
${\mathrm{cs}}{≔}〈\left({1}{,}{2}\right){,}\left({1}{,}{2}{& (16)
({6}{,}{8}\right)〉{▹}{\dots }{▹}〈\left({1}{,}{4}\right)\left
Find the orders of the members of this composition series.
> $\mathrm{seq}⁡\left(\mathrm{GroupOrder}⁡\left(L\right),
${192}{,}{96}{,}{48}{,}{24}{,}{8}{,}{4}{,}{2}{& (17)
Find the IDs of the groups in the database of small groups that are perfect, but not
> $\mathrm{SearchSmallGroups}⁡\left('\mathrm{simple}'=\mathrm{false}&
$[{1}{,}{1}]{,}[{120}{,}{5}]{,}[{336} (18)
Investigate the relative frequencies of multiply transitive groups in the database of
transitive groups.
> $\mathrm{Statistics}:-\mathrm{PieChart}⁡\left([\mathrm{seq}]&
Calling Sequence Context-Sensitive Operations
Description Examples
List of GroupTheory Package Commands Compatibility
Group Constructors Palette
Calling Sequence
GroupTheory:-command( arguments )
command( arguments )
• The GroupTheory package provides a collection of commands for computing with, and
visualizing, finitely generated (especially finite) groups. There are several classes of
groups that are implemented.
permutation groups groups given by a set of generating permutations
finitely presented groups given by generators and defining relations
Cayley table groups groups whose binary operation is specified by a Cayley table
custom groups "black-box" user-defined groups whose elements are of an
unspecified nature
symbolic groups abstract groups depending on symbolic parameters
• The package contains a variety of constructors that allow you to easily create groups in
common families. Furthermore, several databases of groups exist in the package, and
interfaces to these databases are provided.
WorkingWithFPGroups Working with Finitely Presented Groups
WorkingWithPermutations Working with Permutations
WorkingWithSymbolicGroups Working with Symbolic Groups
List of GroupTheory Package Commands
• The following is a list of the commands in the GroupTheory package.
AbelianGroup construct a finitely generated Abelian group
AffineGeneralLinearGroup construct the affine general linear group
over a finite field
AffineSpecialLinearGroup construct the affine special linear group
over a finite field
AGL construct the affine general linear group
over a finite field
AllAbelianGroups return all the Abelian groups of a given
AlternatingGroup construct the alternating group of a given
ASL construct the affine special linear group
over a finite field
BabyMonster construct the Baby Monster sporadic finite
simple group
CayleyTableGroup construct a Cayley table group
ChevalleyE6 construct a Chevalley group of type E6
ChevalleyE7 construct a Chevalley group of type E7
ChevalleyE8 construct a Chevalley group of type E8
ChevalleyF4 construct a Chevalley group of type F4
ChevalleyG2 construct a Chevalley group of type G2
ConwayGroup construct a Conway sporadic simple group
CustomGroup construct a custom group, given its
CyclicGroup construct a cyclic group of a given order
DicyclicGroup construct a dicyclic group
DihedralGroup construct the dihedral group of a given
DirectProduct construct a direct product of groups
ElementaryGroup construct an elementary Abelian group
ExceptionalGroup construct one of the exceptional finite
simple groups
FischerGroup construct one of the Fischer groups
FPGroup construct a finitely presented group from
generators and defining relators
FreeGroup construct a free group
FrobeniusGroup construct a Frobenius group from the database
GaloisGroup construct the Galois group of a polynomial
GammaL construct the general semi-linear group over
a finite field
GeneralLinearGroup construct the general linear group over a
finite field
GeneralOrthogonalGroup construct the general orthogonal group over a
finite field
GeneralSemilinearGroup construct the general semi-linear group over
a finite field
GeneralUnitaryGroup construct a general unitary group over a
finite field
GL construct the general linear group over a
finite field
Group construct a group from various data
HamiltonianGroup construct a finite Hamiltonian group
HaradaNortonGroup construct the Harada-Norton simple group
HeldGroup construct the Held simple group
HigmanSimsGroup construct the Higman-Sims simple group
JankoGroup construct one of the Janko sporadic finite
simple groups
LyonsGroup construct the Lyons simple group
MathieuGroup construct one of the Mathieu finite simple
McLaughlinGroup construct the McLaughlin simple group
MetacyclicGroup construct a metacyclic group
Monster construct the Monster simple group
ONanGroup construct the O'Nan simple group
Orbit construct the orbit of an element under a
permutation group
OrthogonalGroup construct an orthogonal group
PerfectGroup construct a perfect group from the database
Perm create a permutation
PermutationGroup construct a permutation group given
generating permutations
PGammaL construct the general semi-linear group over
a finite field
PGL construct a projective linear group over a
finite field
PGO construct the projective general orthogonal
group over a finite field
PGU construct a projective general unitary group
over a finite field
ProjectiveGeneralLinearGroup construct a projective general linear group
over a finite field
ProjectiveGeneralOrthogonalGroup construct the projective general orthogonal
group over a finite field
ProjectiveGeneralSemilinearGroup construct the general semi-linear group over
a finite field
ProjectiveGeneralUnitaryGroup construct a projective general unitary group
over a finite field
ProjectiveSpecialLinearGroup construct a projective special linear group
over a finite field
ProjectiveSpecialOrthogonalGroup construct the projective special orthogonal
group over a finite field
ProjectiveSpecialSemilinearGroup construct the special semi-linear group over
a finite field
ProjectiveSpecialUnitaryGroup construct a projective special unitary group
over a finite field
ProjectiveSymplecticGroup construct a projective symplectic group over
a finite field
ProjectiveSymplecticSemilinearGroup construct the projective symplectic
semi-linear group over a finite field
PSigmaL construct the special semi-linear group over
a finite field
PSigmap construct the projective symplectic
semi-linear group over a finite field
PSL construct a projective special linear group
over a finite field
PSO construct the projective special orthogonal
group over a finite field
PSp construct a projective symplectic group over
a finite field
PSU construct a projective special unitary group
over a finite field
QuasicyclicGroup construct a quasicyclic p-group
QuasiDihedralGroup construct a quasi-dihedral group
QuaternionGroup construct a generalized quaternion group
Ree2F4 construct a large Ree group of Lie type
Ree2G2 construct a small Ree group of Lie type
RubiksCubeGroup construct the group of the Rubik's Cube
RudvalisGroup construct the Rudvalis simple group
SemiDihedralGroup construct a semi-dihedral group
SigmaL construct the special semi-linear group over
a finite field
Sigmap construct the symplectic semi-linear group
over a finite field
SL construct a special linear group over a
finite field
SmallGroup construct a specific group of small order
SpecialLinearGroup construct a special linear group over a
finite field
SpecialOrthogonalGroup construct a special orthogonal group over a
finite field
SpecialSemilinearGroup construct the special semi-linear group over
a finite field
SpecialUnitaryGroup construct a special unitary group over a
finite field
Stabilizer compute the stabilizer of a point, list or
set under a permutation group
Steinberg2E6 construct a Steinberg group of type 2E6
Steinberg3D4 construct a Steinberg group of type G2
Subgroup construct a subgroup of a given group
Supergroup construct a supergroup of a given group
Suzuki2B2 construct a Suzuki group of Lie type
SuzukiGroup construct the Suzuki simple group
Symm construct the symmetric group of a given
SymmetricGroup construct the symmetric group of a given
SymplecticGroup construct a symplectic group over a finite
SymplecticSemilinearGroup construct the symplectic semi-linear group
over a finite field
ThompsonGroup construct the Thompson simple group
TitsGroup construct the Tits simple group
TransitiveGroup construct a specific transitive permutation
TrivialGroup construct the trivial group
TrivialSubgroup construct the trivial subgroup of a given
WreathProduct construct a wreath product of permutation
Center compute the center of a group
Centraliser compute the centraliser of an element of a group
Centralizer compute the centralizer of an element of a group
Centre compute the centre of a group
Commutator compute the commutator of two subgroups
Core compute the core of a subgroup of a group
Cosocle compute the cosocle of a group
DerivedSubgroup compute the derived (commutator) subgroup of a group
DirectFactors compute the directly indecomposable direct factors of a
finite group
FittingSubgroup compute the Fitting subgroup of a group
FrattiniSubgroup compute the Frattini subgroup of a group
FrobeniusComplement compute a representative Frobenius complement of a
Frobenius group
FrobeniusKernel compute the Frobenius kernel of a Frobenius group
FrobeniusProduct compute the product of two complexes in a finite group
HallSubgroup compute a Hall pi-subgroup of a finite soluble group
HallSystem compute a Hall system for a finite soluble group
Hypercenter compute the hypercenter residual of a group
Index compute the index of a subgroup of a group
Intersection compute the intersection of two subgroups of a group
IsDirectlyIndecomposable test whether a group is directly indecomposable
IsMalnormal test whether a subgroup of a group is malnormal
IsNormal test whether a subgroup of a group is normal
IsQuasinormal test whether a subgroup of a group is quasi-normal
IsSubnormal test whether a subgroup of a group is subnormal
MaximalNormalSubgroups compute the maximal normal subgroups of a permutation
MinimalNormalSubgroups compute the minimal normal subgroups of a permutation
NilpotentResidual compute the nilpotent residual of a group
NormalClosure compute the normal closure of a subgroup or set of group
Normaliser compute the normaliser of a subgroup of a group
NormaliserSubgroup compute the normaliser of a subgroup of a group
NormalizerSubgroup compute the normalizer of a subgroup of a group
NormalSubgroups compute the normal subgroups of a finite group
PCore compute the p-core of a subgroup of a group
Socle compute the socle of a group
SolubleResidual compute the soluble residual of a group
SolvableResidual compute the solvable residual of a group
SubgroupLattice compute the lattice of subgroups of a group
SylowBasis compute a Sylow basis for a finite soluble group
SylowSubgroup compute a Sylow p-subgroup of a finite group
AllFrobeniusGroups return a list of the known Frobenius groups of a given
AllHamiltonianGroups return a list of all the Hamiltonian groups of a given
AllPerfectGroups return a list of the perfect groups of a given order
AllSmallGroups return a list of all the groups of a given order
AllTransitiveGroups return a list of all the transitive groups of a given
FrobeniusGroup construct a Frobenius group from the database
IdentifyFrobeniusGroup locate a given Frobenius group in the database of
Frobenius groups
NumFrobeniusGroups return the number of known Frobenius groups of a given
NumHamiltonianGroups return the number of Hamiltonian groups of a given order
NumPerfectGroups return the number of perfect groups of a given order
NumTransitiveGroups return the number of transitive groups of a given degree
PerfectGroup construct a perfect group from the database
RandomSmallGroup return a random group from the database of small groups
SearchFrobeniusGroups search the Frobenius Groups database
SearchPerfectGroups search the Perfect Groups database
SearchSmallGroups search the Small Groups database
SearchTransitiveGroups search the Transitive Groups database
SmallGroup construct a specific group of small order
TransitiveGroup construct a specific transitive permutation group
AbelianInvariants compute the abelian invariants of a group
ClassNumber compute the number of conjugacy classes of a finite group
CompositionLength compute the composition length of a group
ConjugateRank compute the conjugate rank of a finite group
DerivedLength compute the derived length of a group
Exponent compute the exponent of a group
FittingLength compute the nilpotent (Fitting) length of a group
FrattiniLength compute the Frattini length of a group
GroupOrder compute the order of a group
NilpotencyClass compute the class of nilpotence of a group
NilpotentLength compute the nilpotent (Fitting) length of a group
NumInvolutions compute the number of involutions of a group
OrderClassNumber compute the number of order classes of a finite group
OrderRank compute the number of order class lengths greater than unity of
a finite group
PermGroupRank compute the rank of a permutation group
PGroupRank compute the rank of a finite p-group
PrimaryInvariants compute the primary invariants of a group
Transitivity compute the transitivity of a permutation group
AreConjugate check whether two group elements are conjugate
AreIsomorphic test whether two groups are isomorphic
IsAbelian test whether a group is Abelian
IsAbelianSylowGroup test whether a group has Abelian Sylow subgroups
IsAlmostSimple test whether a group is almost simple
IsAlternating test (probabilistically) whether a permutation group
is an alternating group in its natural action
IsCAGroup test whether a group is a (CA)-group
IsCaminaGroup test whether a group is a Camina group
IsCCGroup test whether a group is a (CC)-group
IsCharacteristicallySimple test whether a group is characteristically simple
IsCNGroup test whether a group is a (CN)-group
IsCommutative test whether a group is commutative
IsCP1Group test whether a group is a (CP1)-group
IsCPGroup test whether a group is a (CP)-group
IsCyclic test whether a group is cyclic
IsCyclicSylowGroup test whether a group has cyclic Sylow subgroups
IsDedekind test whether a group is Dedekind
IsDicyclic test whether a permutation group is a dicyclic group
IsDihedral test whether a permutation group is a dihedral group
IsElementary test whether a group is elementary Abelian
IsExtraspecial test whether a group is an extraspecial p-group
IsFinite test whether a group is finite
IsFinitelyGenerated test whether a group is finitely generated
IsFrobeniusGroup test whether a group is a Frobenius group
IsFrobeniusPermGroup test whether a group is a Frobenius permutation group
IsGCLTGroup test whether a group is a GCLT-group
IsHallPaigeGroup test whether a group has a complete mapping
IsHamiltonian test whether a group is Hamiltonian
IsHomocyclic test whether a group is homocyclic
IsLagrangian test whether a group is Lagrangian
IsMalnormal test whether a subgroup of a group is malnormal
IsMetabelian test whether a group is Abelian
IsMetacyclic test whether a group is Metacyclic
IsNilpotent test whether a group is nilpotent
IsNormal test whether a subgroup of a group is normal
IsOrderedSylowTowerGroup test whether a group has a Sylow tower
IsPerfect test whether a group is perfect
IsPerfectOrderClassesGroup test whether a group has perfect order classes
IsPermutable test whether a subgroup of a group is permutable
IsPGroup test whether a group is a p-group
IsPrimitive test whether a permutation group is primitive
IsPSoluble test whether a group is p-soluble for a given prime p
IsQuasiprimitive test whether a permutation group is quasi-primitive
IsQuasisimple test whether a group is quasi-simple
IsQuaternion test whether a permutation group is a quaternion group
IsRegular test whether a permutation group is regular
IsRegularPGroup test whether a group is a regular p-group
IsSemiprimitive test whether a permutation group is semi-primitive
IsSemiRegular test whether a permutation group is semi-regular
IsSimple test whether a group is simple
IsSoluble test whether a group is soluble
IsSolvable test whether a group is solvable
IsSpecial test whether a group is a special p-group
IsStemGroup test whether a group is a stem group
IsSubgroup test whether one group is a subgroup of another
IsSubnormal test whether a subgroup of a group is subnormal
IsSupersoluble test whether a group is supersoluble
IsSylowTowerGroup test whether a group has a Sylow tower
IsSymmetric test (probabilistically) whether a permutation group
is a symmetric group in its natural action
IsTGroup test whether a group is a T-group
IsTransitive test whether a permutation group is transitive
IsTrivial test whether a group is trivial
SubgroupMembership test whether an element belongs to a given subgroup of
a group
Permutation Groups
AbelianInvariants compute the abelian invariants of a
permutation group
BlocksImage return a permutation group equivalent to
the action of a permutation on a system of
BlockSystem return a block system for a permutation
group, non-trivial if possible
CycleIndexPolynomial compute the cycle index polynomial of a
permutation group
Degree return the degree of a permutation group
EARNS return an EARNS of a primitive group if it
has one
FrobeniusPermRep construct a Frobenius permutation group
isomorphic to a given Frobenius group
IsAlternating test (probabilistically) whether a
permutation group is an alternating group
in its natural action
IsPrimitive test whether a permutation group is
IsQuasiprimitive test whether a permutation group is
IsRegular test whether a permutation group is
IsSemiprimitive test whether a permutation group is
IsSemiRegular test whether a permutation group is
IsSymmetric test (probabilistically) whether a
permutation group is a symmetric group in
its natural action
IsTransitive test whether a permutation group is
MaxSupport return the largest element displaced by a
permutation group
MinimalBlockSystem return a minimal block system for a
permutation group, non-trivial if possible
MinimumPermutationRepresentationDegree compute the minimum degree of a faithful
permutation representation for a group
MinSupport return the smallest element displaced by a
permutation group
Orbit construct the orbit of an element under a
permutation group
Orbits compute the orbits of a permutation group
PermGroupRank compute the rank of a permutation group
PrimaryInvariants compute the primary invariants of a
permutation group
ReducedDegreePermGroup return an isomorphic permutation group of
possibly smaller degree
RestrictedPermGroup return the restriction of a permutation
group to a stable subset
Stabilizer compute the stabilizer of a point, list or
set under a permutation group
Support return the support of a permutation group
SupportLength return the number of elements displaced by
a permutation group
Transitivity compute the transitivity of a permutation
Finitely Presented Groups
AbelianInvariants compute the abelian invariants of a finitely presented
PresentationComplexity compute a measure of the complexity of a finitely
presented group
PrimaryInvariants compute the primary invariants of a finitely presented
Relators return the relators of a finitely presented group
Simplify simplify the presentation of a finitely presented group
DrawCayleyTable draw the Cayley table of a finite group
DrawNormalSubgroupLattice draw the lattice of normal subgroups of a finite group
DrawSubgroupLattice draw the lattice of subgroups of a finite group
AgemoSeries compute the series of agemo subgroups of a $p$-group
CompositionSeries compute a composition series of a group
DerivedSeries compute the derived series of a group
FrattiniSeries compute the Frattini series of a group
LowerCentralSeries compute the lower central series of a group
LowerFittingSeries compute the lower Fitting series of a group
LowerPCentralSeries compute the lower p-central series of a group
OmegaSeries compute the series of omega subgroups of a $p$-group
OrderedSylowTower compute a Sylow tower for a Sylow tower group
SylowTower compute a Sylow tower for a Sylow tower group
UpperCentralSeries compute the upper central series of a group
DecomposeDessin find all decompositions of a Belyi map represented by a dessin
FindDessins find all dessins d'enfants with a specified branch pattern
ElementOrder compute the order of a group element
ElementOrderSum compute the sum of the element orders of a finite group
ElementPower compute an integer power of a group element
Elements compute the elements of a finite group, orbit, coset or
conjugacy class
MaximumElementOrder compute the largest order of an element of a finite group
OrderClassPolynomial compute the order class polynomial of a finite group
OrderClassProfile compute the element order profile of a finite group
PrimePowerFactors factor a group element into a product of elements of prime
power order
RandomElement compute a random element of a group
RandomImvolution compute a random involution of a group
RandomPElement compute a random p-element of a group
RandomPPrimeElement compute a random element of a group with order relatively
prime to p
IsAbelianNumber test whether every group of a given order is Abelian
IsCyclicNumber test whether every group of a given order is cyclic
IsGCLTNumber test whether every group of a given order is a GCLT
IsIntegrableNumber test whether every group of a given order is integrable
IsLagrangianNumber test whether every group of a given order is Lagrangian
IsMetabelianNumber test whether every group of a given order is metabelian
IsMetacyclicNumber test whether every group of a given order is metacyclic
IsNilpotentNumber test whether every group of a given order is nilpotent
IsOrderedSylowTowerNumber test whether every group of a given order has an
ordered Sylow tower
IsSimpleNumber test whether a number is the order of a finite simple
IsSolubleNumber test whether every group of a given order is soluble
IsSupersolubleNumber test whether every group of a given order is
CayleyGraph return the Cayley graph of a finite group
CayleyTable return the Cayley table of a finite group
CFSG finite simple group classifier object
Character construct a character from a character table
CharacterTable compute the character table of a finite group
ClassifyFiniteSimpleGroup classify a finite simple group
CommutingGraph construct the commuting graph of a finite group
ConjugacyClass compute the conjugacy class of a group element
ConjugacyClasses compute all the conjugacy classes of a finite group
Conjugator compute an element conjugating one group element to
Elements compute the elements of a finite group, orbit, coset or
conjugacy class
Factor expression a group element as a product of a coset
representative and a subgroup element
Generators return the set of generators of a group
GruenbergKegelGraph return the Gruenberg-Kegel graph of a finite group
IdentifySmallGroup compute the Small Group ID of a small group
Labels return the set of generator labels of a group
LeftCoset compute a left coset of a group element
LeftCosets compute the left cosets of a subgroup of a group
logp compute the exponent of a prime power
NonRedundantGenerators return a set of non-redundant generators of a group
NumSimpleGroups count the number of simple groups of a given finite
Operations return the operations record of a group
PresentationComplexity compute a measure of the complexity of a finitely
presented group
RightCoset compute a right coset of a group element
RightCosets compute the right cosets of a subgroup of a group
TabulateSimpleGroups list the simple groups with orders in a given range
Group Constructors Palette
• The Group Constructors palette contains buttons for constructing groups.
• Palettes are displayed in the left pane of the Maple window. (If it is not visible, from
the main menu, select View > Palettes > Show Palette > Group Constructors)
• Some palette items have placeholders. Fill in the placeholders, using Tab to navigate to
the next placeholder.
Context-Sensitive Operations
• In the Standard Worksheet interface, you can apply operations to a group through the
Context Panel under the Group operations submenu.
The following command enables you to use the commands in the GroupTheory package without
having to prefix each command with "GroupTheory:-".
> $\mathrm{with}⁡\left(\mathrm{GroupTheory}\right):$
Create a symmetric group of degree 4.
> $G≔\mathrm{SymmetricGroup}⁡\left(4\right)$
${G}{≔}{{\mathbf{S}}}_{{4}}$ (1)
Visualize the lattice of subgroups of G.
> $\mathrm{DrawSubgroupLattice}⁡\left(G\right)$
Create a dihedral group of degree 4 (and order 8).
> $H≔\mathrm{DihedralGroup}⁡\left(4\right)$
${H}{≔}{{\mathbf{D}}}_{{4}}$ (2)
Visualize the Cayley (operation) table for H.
> $\mathrm{DrawCayleyTable}⁡\left(H\right)$
Compute the orders (cardinalities) of G and H.
> $\mathrm{GroupOrder}⁡\left(G\right),\mathrm{GroupOrder}&
${24}{,}{8}$ (3)
Compute the character table of H.
> $\mathrm{Display}⁡\left(\mathrm{CharacterTable}⁡\left(H\
C 1a 2a 2b 2c 4a
|C| 1 1 2 2 2
$\mathrm{χ__1}$ $1$ $1$ $1$ $1$ $1$
$\mathrm{χ__2}$ $1$ $1$ $-1$ $-1$ $1$
$\mathrm{χ__3}$ $1$ $1$ $-1$ $1$ $-1$
$\mathrm{χ__4}$ $1$ $1$ $1$ $-1$ $-1$
$\mathrm{χ__5}$ $2$ $-2$ $0$ $0$ $0$
Form the direct product in two ways.
> $U≔\mathrm{DirectProduct}⁡\left(G,H\right)$
${U}{≔}〈\left({1}{,}{2}\right){,}\left({1}{,}{2}{,}{3} (4)
> $V≔\mathrm{DirectProduct}⁡\left(H,G\right)$
${V}{≔}〈\left({1}{,}{3}\right){,}\left({1}{,}{2}{,}{3} (5)
Check that these are isomorphic.
> $\mathrm{AreIsomorphic}⁡\left(U,V\right)$
${\mathrm{true}}$ (6)
Notice that the order of the direct product is equal to the product of the orders of the
> $\mathrm{GroupOrder}⁡\left(U\right)=\mathrm{GroupOrder}&
${192}{=}{192}$ (7)
Since the order of U does not exceed 511, we can identify the group explicitly.
> $\mathrm{id}≔\mathrm{IdentifySmallGroup}⁡\left(U\right)$
${\mathrm{id}}{≔}{192}{,}{1472}$ (8)
Retrieve the identified group from the database, and check the isomorphism.
> $W≔\mathrm{SmallGroup}⁡\left(\mathrm{id}\right):$
> $\mathrm{AreIsomorphic}⁡\left(U,W\right)$
${\mathrm{true}}$ (9)
Construct a wreath product of two symmetric groups.
> $W≔\mathrm{WreathProduct}⁡\left(\mathrm{Symm}⁡\left(3\
${W}{≔}〈\left({1}{,}{2}\right){,}\left({1}{,}{2}{,}{3} (10)
Check that the wreath product is transitive but imprimitive.
> $\mathrm{IsTransitive}⁡\left(W\right)$
${\mathrm{true}}$ (11)
> $\mathrm{IsPrimitive}⁡\left(W\right)$
${\mathrm{false}}$ (12)
Find a non-trivial system of blocks for W.
> $\mathrm{BlockSystem}⁡\left(W\right)$
${{{1}{,}{2}{,}{3}}{,}{{4}{,}{5}{,} (13)
Identify the Sylow $3$-subgroup of W in the database of small groups.
> $\mathrm{IdentifySmallGroup}⁡\left(\mathrm{SylowSubgroup}⁡\
${243}{,}{51}$ (14)
Find the nilpotency class of the Sylow $2$-subgroup of W.
> $\mathrm{NilpotencyClass}⁡\left(\mathrm{SylowSubgroup}⁡\left
${4}$ (15)
Compute a composition series for U.
> $\mathrm{cs}≔\mathrm{CompositionSeries}⁡\left(U\right)$
Warning, over-writing property `["DerivedSeries"]' with a different value
${\mathrm{cs}}{≔}〈\left({1}{,}{2}\right){,}\left({1}{,}{2}{& (16)
({6}{,}{8}\right)〉{▹}{\dots }{▹}〈\left({1}{,}{4}\right)\left
Find the orders of the members of this composition series.
> $\mathrm{seq}⁡\left(\mathrm{GroupOrder}⁡\left(L\right),
${192}{,}{96}{,}{48}{,}{24}{,}{8}{,}{4}{,}{2}{& (17)
Find the IDs of the groups in the database of small groups that are perfect, but not
> $\mathrm{SearchSmallGroups}⁡\left('\mathrm{simple}'=\mathrm{false}&
$[{1}{,}{1}]{,}[{120}{,}{5}]{,}[{336} (18)
Investigate the relative frequencies of multiply transitive groups in the database of
transitive groups.
> $\mathrm{Statistics}:-\mathrm{PieChart}⁡\left([\mathrm{seq}]&
• The GroupTheory package provides a collection of commands for computing with, and visualizing, finitely generated (especially finite) groups. There are several classes of groups that are
permutation groups groups given by a set of generating permutations
finitely presented groups given by generators and defining relations
Cayley table groups groups whose binary operation is specified by a Cayley table
custom groups "black-box" user-defined groups whose elements are of an
unspecified nature
symbolic groups abstract groups depending on symbolic parameters
• The package contains a variety of constructors that allow you to easily create groups in common families. Furthermore, several databases of groups exist in the package, and interfaces to these
databases are provided.
WorkingWithFPGroups Working with Finitely Presented Groups
WorkingWithPermutations Working with Permutations
WorkingWithSymbolicGroups Working with Symbolic Groups
• The GroupTheory package provides a collection of commands for computing with, and visualizing, finitely generated (especially finite) groups. There are several classes of groups that are
The GroupTheory package provides a collection of commands for computing with, and visualizing, finitely generated (especially finite) groups. There are several classes of groups that are implemented.
permutation groups groups given by a set of generating permutations
finitely presented groups given by generators and defining relations
Cayley table groups groups whose binary operation is specified by a Cayley table
custom groups "black-box" user-defined groups whose elements are of an
unspecified nature
symbolic groups abstract groups depending on symbolic parameters
groups whose binary operation is specified by a Cayley table
"black-box" user-defined groups whose elements are of an unspecified nature
• The package contains a variety of constructors that allow you to easily create groups in common families. Furthermore, several databases of groups exist in the package, and interfaces to these
databases are provided.
The package contains a variety of constructors that allow you to easily create groups in common families. Furthermore, several databases of groups exist in the package, and interfaces to these
databases are provided.
WorkingWithFPGroups Working with Finitely Presented Groups
WorkingWithPermutations Working with Permutations
WorkingWithSymbolicGroups Working with Symbolic Groups
WorkingWithFPGroups Working with Finitely Presented Groups
WorkingWithPermutations Working with Permutations
WorkingWithSymbolicGroups Working with Symbolic Groups
List of GroupTheory Package Commands
• The following is a list of the commands in the GroupTheory package.
AbelianGroup construct a finitely generated Abelian group
AffineGeneralLinearGroup construct the affine general linear group
over a finite field
AffineSpecialLinearGroup construct the affine special linear group
over a finite field
AGL construct the affine general linear group
over a finite field
AllAbelianGroups return all the Abelian groups of a given
AlternatingGroup construct the alternating group of a given
ASL construct the affine special linear group
over a finite field
BabyMonster construct the Baby Monster sporadic finite
simple group
CayleyTableGroup construct a Cayley table group
ChevalleyE6 construct a Chevalley group of type E6
ChevalleyE7 construct a Chevalley group of type E7
ChevalleyE8 construct a Chevalley group of type E8
ChevalleyF4 construct a Chevalley group of type F4
ChevalleyG2 construct a Chevalley group of type G2
ConwayGroup construct a Conway sporadic simple group
CustomGroup construct a custom group, given its
CyclicGroup construct a cyclic group of a given order
DicyclicGroup construct a dicyclic group
DihedralGroup construct the dihedral group of a given
DirectProduct construct a direct product of groups
ElementaryGroup construct an elementary Abelian group
ExceptionalGroup construct one of the exceptional finite
simple groups
FischerGroup construct one of the Fischer groups
FPGroup construct a finitely presented group from
generators and defining relators
FreeGroup construct a free group
FrobeniusGroup construct a Frobenius group from the database
GaloisGroup construct the Galois group of a polynomial
GammaL construct the general semi-linear group over
a finite field
GeneralLinearGroup construct the general linear group over a
finite field
GeneralOrthogonalGroup construct the general orthogonal group over a
finite field
GeneralSemilinearGroup construct the general semi-linear group over
a finite field
GeneralUnitaryGroup construct a general unitary group over a
finite field
GL construct the general linear group over a
finite field
Group construct a group from various data
HamiltonianGroup construct a finite Hamiltonian group
HaradaNortonGroup construct the Harada-Norton simple group
HeldGroup construct the Held simple group
HigmanSimsGroup construct the Higman-Sims simple group
JankoGroup construct one of the Janko sporadic finite
simple groups
LyonsGroup construct the Lyons simple group
MathieuGroup construct one of the Mathieu finite simple
McLaughlinGroup construct the McLaughlin simple group
MetacyclicGroup construct a metacyclic group
Monster construct the Monster simple group
ONanGroup construct the O'Nan simple group
Orbit construct the orbit of an element under a
permutation group
OrthogonalGroup construct an orthogonal group
PerfectGroup construct a perfect group from the database
Perm create a permutation
PermutationGroup construct a permutation group given
generating permutations
PGammaL construct the general semi-linear group over
a finite field
PGL construct a projective linear group over a
finite field
PGO construct the projective general orthogonal
group over a finite field
PGU construct a projective general unitary group
over a finite field
ProjectiveGeneralLinearGroup construct a projective general linear group
over a finite field
ProjectiveGeneralOrthogonalGroup construct the projective general orthogonal
group over a finite field
ProjectiveGeneralSemilinearGroup construct the general semi-linear group over
a finite field
ProjectiveGeneralUnitaryGroup construct a projective general unitary group
over a finite field
ProjectiveSpecialLinearGroup construct a projective special linear group
over a finite field
ProjectiveSpecialOrthogonalGroup construct the projective special orthogonal
group over a finite field
ProjectiveSpecialSemilinearGroup construct the special semi-linear group over
a finite field
ProjectiveSpecialUnitaryGroup construct a projective special unitary group
over a finite field
ProjectiveSymplecticGroup construct a projective symplectic group over
a finite field
ProjectiveSymplecticSemilinearGroup construct the projective symplectic
semi-linear group over a finite field
PSigmaL construct the special semi-linear group over
a finite field
PSigmap construct the projective symplectic
semi-linear group over a finite field
PSL construct a projective special linear group
over a finite field
PSO construct the projective special orthogonal
group over a finite field
PSp construct a projective symplectic group over
a finite field
PSU construct a projective special unitary group
over a finite field
QuasicyclicGroup construct a quasicyclic p-group
QuasiDihedralGroup construct a quasi-dihedral group
QuaternionGroup construct a generalized quaternion group
Ree2F4 construct a large Ree group of Lie type
Ree2G2 construct a small Ree group of Lie type
RubiksCubeGroup construct the group of the Rubik's Cube
RudvalisGroup construct the Rudvalis simple group
SemiDihedralGroup construct a semi-dihedral group
SigmaL construct the special semi-linear group over
a finite field
Sigmap construct the symplectic semi-linear group
over a finite field
SL construct a special linear group over a
finite field
SmallGroup construct a specific group of small order
SpecialLinearGroup construct a special linear group over a
finite field
SpecialOrthogonalGroup construct a special orthogonal group over a
finite field
SpecialSemilinearGroup construct the special semi-linear group over
a finite field
SpecialUnitaryGroup construct a special unitary group over a
finite field
Stabilizer compute the stabilizer of a point, list or
set under a permutation group
Steinberg2E6 construct a Steinberg group of type 2E6
Steinberg3D4 construct a Steinberg group of type G2
Subgroup construct a subgroup of a given group
Supergroup construct a supergroup of a given group
Suzuki2B2 construct a Suzuki group of Lie type
SuzukiGroup construct the Suzuki simple group
Symm construct the symmetric group of a given
SymmetricGroup construct the symmetric group of a given
SymplecticGroup construct a symplectic group over a finite
SymplecticSemilinearGroup construct the symplectic semi-linear group
over a finite field
ThompsonGroup construct the Thompson simple group
TitsGroup construct the Tits simple group
TransitiveGroup construct a specific transitive permutation
TrivialGroup construct the trivial group
TrivialSubgroup construct the trivial subgroup of a given
WreathProduct construct a wreath product of permutation
Center compute the center of a group
Centraliser compute the centraliser of an element of a group
Centralizer compute the centralizer of an element of a group
Centre compute the centre of a group
Commutator compute the commutator of two subgroups
Core compute the core of a subgroup of a group
Cosocle compute the cosocle of a group
DerivedSubgroup compute the derived (commutator) subgroup of a group
DirectFactors compute the directly indecomposable direct factors of a
finite group
FittingSubgroup compute the Fitting subgroup of a group
FrattiniSubgroup compute the Frattini subgroup of a group
FrobeniusComplement compute a representative Frobenius complement of a
Frobenius group
FrobeniusKernel compute the Frobenius kernel of a Frobenius group
FrobeniusProduct compute the product of two complexes in a finite group
HallSubgroup compute a Hall pi-subgroup of a finite soluble group
HallSystem compute a Hall system for a finite soluble group
Hypercenter compute the hypercenter residual of a group
Index compute the index of a subgroup of a group
Intersection compute the intersection of two subgroups of a group
IsDirectlyIndecomposable test whether a group is directly indecomposable
IsMalnormal test whether a subgroup of a group is malnormal
IsNormal test whether a subgroup of a group is normal
IsQuasinormal test whether a subgroup of a group is quasi-normal
IsSubnormal test whether a subgroup of a group is subnormal
MaximalNormalSubgroups compute the maximal normal subgroups of a permutation
MinimalNormalSubgroups compute the minimal normal subgroups of a permutation
NilpotentResidual compute the nilpotent residual of a group
NormalClosure compute the normal closure of a subgroup or set of group
Normaliser compute the normaliser of a subgroup of a group
NormaliserSubgroup compute the normaliser of a subgroup of a group
NormalizerSubgroup compute the normalizer of a subgroup of a group
NormalSubgroups compute the normal subgroups of a finite group
PCore compute the p-core of a subgroup of a group
Socle compute the socle of a group
SolubleResidual compute the soluble residual of a group
SolvableResidual compute the solvable residual of a group
SubgroupLattice compute the lattice of subgroups of a group
SylowBasis compute a Sylow basis for a finite soluble group
SylowSubgroup compute a Sylow p-subgroup of a finite group
AllFrobeniusGroups return a list of the known Frobenius groups of a given
AllHamiltonianGroups return a list of all the Hamiltonian groups of a given
AllPerfectGroups return a list of the perfect groups of a given order
AllSmallGroups return a list of all the groups of a given order
AllTransitiveGroups return a list of all the transitive groups of a given
FrobeniusGroup construct a Frobenius group from the database
IdentifyFrobeniusGroup locate a given Frobenius group in the database of
Frobenius groups
NumFrobeniusGroups return the number of known Frobenius groups of a given
NumHamiltonianGroups return the number of Hamiltonian groups of a given order
NumPerfectGroups return the number of perfect groups of a given order
NumTransitiveGroups return the number of transitive groups of a given degree
PerfectGroup construct a perfect group from the database
RandomSmallGroup return a random group from the database of small groups
SearchFrobeniusGroups search the Frobenius Groups database
SearchPerfectGroups search the Perfect Groups database
SearchSmallGroups search the Small Groups database
SearchTransitiveGroups search the Transitive Groups database
SmallGroup construct a specific group of small order
TransitiveGroup construct a specific transitive permutation group
AbelianInvariants compute the abelian invariants of a group
ClassNumber compute the number of conjugacy classes of a finite group
CompositionLength compute the composition length of a group
ConjugateRank compute the conjugate rank of a finite group
DerivedLength compute the derived length of a group
Exponent compute the exponent of a group
FittingLength compute the nilpotent (Fitting) length of a group
FrattiniLength compute the Frattini length of a group
GroupOrder compute the order of a group
NilpotencyClass compute the class of nilpotence of a group
NilpotentLength compute the nilpotent (Fitting) length of a group
NumInvolutions compute the number of involutions of a group
OrderClassNumber compute the number of order classes of a finite group
OrderRank compute the number of order class lengths greater than unity of
a finite group
PermGroupRank compute the rank of a permutation group
PGroupRank compute the rank of a finite p-group
PrimaryInvariants compute the primary invariants of a group
Transitivity compute the transitivity of a permutation group
AreConjugate check whether two group elements are conjugate
AreIsomorphic test whether two groups are isomorphic
IsAbelian test whether a group is Abelian
IsAbelianSylowGroup test whether a group has Abelian Sylow subgroups
IsAlmostSimple test whether a group is almost simple
IsAlternating test (probabilistically) whether a permutation group
is an alternating group in its natural action
IsCAGroup test whether a group is a (CA)-group
IsCaminaGroup test whether a group is a Camina group
IsCCGroup test whether a group is a (CC)-group
IsCharacteristicallySimple test whether a group is characteristically simple
IsCNGroup test whether a group is a (CN)-group
IsCommutative test whether a group is commutative
IsCP1Group test whether a group is a (CP1)-group
IsCPGroup test whether a group is a (CP)-group
IsCyclic test whether a group is cyclic
IsCyclicSylowGroup test whether a group has cyclic Sylow subgroups
IsDedekind test whether a group is Dedekind
IsDicyclic test whether a permutation group is a dicyclic group
IsDihedral test whether a permutation group is a dihedral group
IsElementary test whether a group is elementary Abelian
IsExtraspecial test whether a group is an extraspecial p-group
IsFinite test whether a group is finite
IsFinitelyGenerated test whether a group is finitely generated
IsFrobeniusGroup test whether a group is a Frobenius group
IsFrobeniusPermGroup test whether a group is a Frobenius permutation group
IsGCLTGroup test whether a group is a GCLT-group
IsHallPaigeGroup test whether a group has a complete mapping
IsHamiltonian test whether a group is Hamiltonian
IsHomocyclic test whether a group is homocyclic
IsLagrangian test whether a group is Lagrangian
IsMalnormal test whether a subgroup of a group is malnormal
IsMetabelian test whether a group is Abelian
IsMetacyclic test whether a group is Metacyclic
IsNilpotent test whether a group is nilpotent
IsNormal test whether a subgroup of a group is normal
IsOrderedSylowTowerGroup test whether a group has a Sylow tower
IsPerfect test whether a group is perfect
IsPerfectOrderClassesGroup test whether a group has perfect order classes
IsPermutable test whether a subgroup of a group is permutable
IsPGroup test whether a group is a p-group
IsPrimitive test whether a permutation group is primitive
IsPSoluble test whether a group is p-soluble for a given prime p
IsQuasiprimitive test whether a permutation group is quasi-primitive
IsQuasisimple test whether a group is quasi-simple
IsQuaternion test whether a permutation group is a quaternion group
IsRegular test whether a permutation group is regular
IsRegularPGroup test whether a group is a regular p-group
IsSemiprimitive test whether a permutation group is semi-primitive
IsSemiRegular test whether a permutation group is semi-regular
IsSimple test whether a group is simple
IsSoluble test whether a group is soluble
IsSolvable test whether a group is solvable
IsSpecial test whether a group is a special p-group
IsStemGroup test whether a group is a stem group
IsSubgroup test whether one group is a subgroup of another
IsSubnormal test whether a subgroup of a group is subnormal
IsSupersoluble test whether a group is supersoluble
IsSylowTowerGroup test whether a group has a Sylow tower
IsSymmetric test (probabilistically) whether a permutation group
is a symmetric group in its natural action
IsTGroup test whether a group is a T-group
IsTransitive test whether a permutation group is transitive
IsTrivial test whether a group is trivial
SubgroupMembership test whether an element belongs to a given subgroup of
a group
Permutation Groups
AbelianInvariants compute the abelian invariants of a
permutation group
BlocksImage return a permutation group equivalent to
the action of a permutation on a system of
BlockSystem return a block system for a permutation
group, non-trivial if possible
CycleIndexPolynomial compute the cycle index polynomial of a
permutation group
Degree return the degree of a permutation group
EARNS return an EARNS of a primitive group if it
has one
FrobeniusPermRep construct a Frobenius permutation group
isomorphic to a given Frobenius group
IsAlternating test (probabilistically) whether a
permutation group is an alternating group
in its natural action
IsPrimitive test whether a permutation group is
IsQuasiprimitive test whether a permutation group is
IsRegular test whether a permutation group is
IsSemiprimitive test whether a permutation group is
IsSemiRegular test whether a permutation group is
IsSymmetric test (probabilistically) whether a
permutation group is a symmetric group in
its natural action
IsTransitive test whether a permutation group is
MaxSupport return the largest element displaced by a
permutation group
MinimalBlockSystem return a minimal block system for a
permutation group, non-trivial if possible
MinimumPermutationRepresentationDegree compute the minimum degree of a faithful
permutation representation for a group
MinSupport return the smallest element displaced by a
permutation group
Orbit construct the orbit of an element under a
permutation group
Orbits compute the orbits of a permutation group
PermGroupRank compute the rank of a permutation group
PrimaryInvariants compute the primary invariants of a
permutation group
ReducedDegreePermGroup return an isomorphic permutation group of
possibly smaller degree
RestrictedPermGroup return the restriction of a permutation
group to a stable subset
Stabilizer compute the stabilizer of a point, list or
set under a permutation group
Support return the support of a permutation group
SupportLength return the number of elements displaced by
a permutation group
Transitivity compute the transitivity of a permutation
Finitely Presented Groups
AbelianInvariants compute the abelian invariants of a finitely presented
PresentationComplexity compute a measure of the complexity of a finitely
presented group
PrimaryInvariants compute the primary invariants of a finitely presented
Relators return the relators of a finitely presented group
Simplify simplify the presentation of a finitely presented group
DrawCayleyTable draw the Cayley table of a finite group
DrawNormalSubgroupLattice draw the lattice of normal subgroups of a finite group
DrawSubgroupLattice draw the lattice of subgroups of a finite group
AgemoSeries compute the series of agemo subgroups of a $p$-group
CompositionSeries compute a composition series of a group
DerivedSeries compute the derived series of a group
FrattiniSeries compute the Frattini series of a group
LowerCentralSeries compute the lower central series of a group
LowerFittingSeries compute the lower Fitting series of a group
LowerPCentralSeries compute the lower p-central series of a group
OmegaSeries compute the series of omega subgroups of a $p$-group
OrderedSylowTower compute a Sylow tower for a Sylow tower group
SylowTower compute a Sylow tower for a Sylow tower group
UpperCentralSeries compute the upper central series of a group
DecomposeDessin find all decompositions of a Belyi map represented by a dessin
FindDessins find all dessins d'enfants with a specified branch pattern
ElementOrder compute the order of a group element
ElementOrderSum compute the sum of the element orders of a finite group
ElementPower compute an integer power of a group element
Elements compute the elements of a finite group, orbit, coset or
conjugacy class
MaximumElementOrder compute the largest order of an element of a finite group
OrderClassPolynomial compute the order class polynomial of a finite group
OrderClassProfile compute the element order profile of a finite group
PrimePowerFactors factor a group element into a product of elements of prime
power order
RandomElement compute a random element of a group
RandomImvolution compute a random involution of a group
RandomPElement compute a random p-element of a group
RandomPPrimeElement compute a random element of a group with order relatively
prime to p
IsAbelianNumber test whether every group of a given order is Abelian
IsCyclicNumber test whether every group of a given order is cyclic
IsGCLTNumber test whether every group of a given order is a GCLT
IsIntegrableNumber test whether every group of a given order is integrable
IsLagrangianNumber test whether every group of a given order is Lagrangian
IsMetabelianNumber test whether every group of a given order is metabelian
IsMetacyclicNumber test whether every group of a given order is metacyclic
IsNilpotentNumber test whether every group of a given order is nilpotent
IsOrderedSylowTowerNumber test whether every group of a given order has an
ordered Sylow tower
IsSimpleNumber test whether a number is the order of a finite simple
IsSolubleNumber test whether every group of a given order is soluble
IsSupersolubleNumber test whether every group of a given order is
CayleyGraph return the Cayley graph of a finite group
CayleyTable return the Cayley table of a finite group
CFSG finite simple group classifier object
Character construct a character from a character table
CharacterTable compute the character table of a finite group
ClassifyFiniteSimpleGroup classify a finite simple group
CommutingGraph construct the commuting graph of a finite group
ConjugacyClass compute the conjugacy class of a group element
ConjugacyClasses compute all the conjugacy classes of a finite group
Conjugator compute an element conjugating one group element to
Elements compute the elements of a finite group, orbit, coset or
conjugacy class
Factor expression a group element as a product of a coset
representative and a subgroup element
Generators return the set of generators of a group
GruenbergKegelGraph return the Gruenberg-Kegel graph of a finite group
IdentifySmallGroup compute the Small Group ID of a small group
Labels return the set of generator labels of a group
LeftCoset compute a left coset of a group element
LeftCosets compute the left cosets of a subgroup of a group
logp compute the exponent of a prime power
NonRedundantGenerators return a set of non-redundant generators of a group
NumSimpleGroups count the number of simple groups of a given finite
Operations return the operations record of a group
PresentationComplexity compute a measure of the complexity of a finitely
presented group
RightCoset compute a right coset of a group element
RightCosets compute the right cosets of a subgroup of a group
TabulateSimpleGroups list the simple groups with orders in a given range
• The following is a list of the commands in the GroupTheory package.
The following is a list of the commands in the GroupTheory package.
AbelianGroup construct a finitely generated Abelian group
AffineGeneralLinearGroup construct the affine general linear group
over a finite field
AffineSpecialLinearGroup construct the affine special linear group
over a finite field
AGL construct the affine general linear group
over a finite field
AllAbelianGroups return all the Abelian groups of a given
AlternatingGroup construct the alternating group of a given
ASL construct the affine special linear group
over a finite field
BabyMonster construct the Baby Monster sporadic finite
simple group
CayleyTableGroup construct a Cayley table group
ChevalleyE6 construct a Chevalley group of type E6
ChevalleyE7 construct a Chevalley group of type E7
ChevalleyE8 construct a Chevalley group of type E8
ChevalleyF4 construct a Chevalley group of type F4
ChevalleyG2 construct a Chevalley group of type G2
ConwayGroup construct a Conway sporadic simple group
CustomGroup construct a custom group, given its
CyclicGroup construct a cyclic group of a given order
DicyclicGroup construct a dicyclic group
DihedralGroup construct the dihedral group of a given
DirectProduct construct a direct product of groups
ElementaryGroup construct an elementary Abelian group
ExceptionalGroup construct one of the exceptional finite
simple groups
FischerGroup construct one of the Fischer groups
FPGroup construct a finitely presented group from
generators and defining relators
FreeGroup construct a free group
FrobeniusGroup construct a Frobenius group from the database
GaloisGroup construct the Galois group of a polynomial
GammaL construct the general semi-linear group over
a finite field
GeneralLinearGroup construct the general linear group over a
finite field
GeneralOrthogonalGroup construct the general orthogonal group over a
finite field
GeneralSemilinearGroup construct the general semi-linear group over
a finite field
GeneralUnitaryGroup construct a general unitary group over a
finite field
GL construct the general linear group over a
finite field
Group construct a group from various data
HamiltonianGroup construct a finite Hamiltonian group
HaradaNortonGroup construct the Harada-Norton simple group
HeldGroup construct the Held simple group
HigmanSimsGroup construct the Higman-Sims simple group
JankoGroup construct one of the Janko sporadic finite
simple groups
LyonsGroup construct the Lyons simple group
MathieuGroup construct one of the Mathieu finite simple
McLaughlinGroup construct the McLaughlin simple group
MetacyclicGroup construct a metacyclic group
Monster construct the Monster simple group
ONanGroup construct the O'Nan simple group
Orbit construct the orbit of an element under a
permutation group
OrthogonalGroup construct an orthogonal group
PerfectGroup construct a perfect group from the database
Perm create a permutation
PermutationGroup construct a permutation group given
generating permutations
PGammaL construct the general semi-linear group over
a finite field
PGL construct a projective linear group over a
finite field
PGO construct the projective general orthogonal
group over a finite field
PGU construct a projective general unitary group
over a finite field
ProjectiveGeneralLinearGroup construct a projective general linear group
over a finite field
ProjectiveGeneralOrthogonalGroup construct the projective general orthogonal
group over a finite field
ProjectiveGeneralSemilinearGroup construct the general semi-linear group over
a finite field
ProjectiveGeneralUnitaryGroup construct a projective general unitary group
over a finite field
ProjectiveSpecialLinearGroup construct a projective special linear group
over a finite field
ProjectiveSpecialOrthogonalGroup construct the projective special orthogonal
group over a finite field
ProjectiveSpecialSemilinearGroup construct the special semi-linear group over
a finite field
ProjectiveSpecialUnitaryGroup construct a projective special unitary group
over a finite field
ProjectiveSymplecticGroup construct a projective symplectic group over
a finite field
ProjectiveSymplecticSemilinearGroup construct the projective symplectic
semi-linear group over a finite field
PSigmaL construct the special semi-linear group over
a finite field
PSigmap construct the projective symplectic
semi-linear group over a finite field
PSL construct a projective special linear group
over a finite field
PSO construct the projective special orthogonal
group over a finite field
PSp construct a projective symplectic group over
a finite field
PSU construct a projective special unitary group
over a finite field
QuasicyclicGroup construct a quasicyclic p-group
QuasiDihedralGroup construct a quasi-dihedral group
QuaternionGroup construct a generalized quaternion group
Ree2F4 construct a large Ree group of Lie type
Ree2G2 construct a small Ree group of Lie type
RubiksCubeGroup construct the group of the Rubik's Cube
RudvalisGroup construct the Rudvalis simple group
SemiDihedralGroup construct a semi-dihedral group
SigmaL construct the special semi-linear group over
a finite field
Sigmap construct the symplectic semi-linear group
over a finite field
SL construct a special linear group over a
finite field
SmallGroup construct a specific group of small order
SpecialLinearGroup construct a special linear group over a
finite field
SpecialOrthogonalGroup construct a special orthogonal group over a
finite field
SpecialSemilinearGroup construct the special semi-linear group over
a finite field
SpecialUnitaryGroup construct a special unitary group over a
finite field
Stabilizer compute the stabilizer of a point, list or
set under a permutation group
Steinberg2E6 construct a Steinberg group of type 2E6
Steinberg3D4 construct a Steinberg group of type G2
Subgroup construct a subgroup of a given group
Supergroup construct a supergroup of a given group
Suzuki2B2 construct a Suzuki group of Lie type
SuzukiGroup construct the Suzuki simple group
Symm construct the symmetric group of a given
SymmetricGroup construct the symmetric group of a given
SymplecticGroup construct a symplectic group over a finite
SymplecticSemilinearGroup construct the symplectic semi-linear group
over a finite field
ThompsonGroup construct the Thompson simple group
TitsGroup construct the Tits simple group
TransitiveGroup construct a specific transitive permutation
TrivialGroup construct the trivial group
TrivialSubgroup construct the trivial subgroup of a given
WreathProduct construct a wreath product of permutation
AbelianGroup construct a finitely generated Abelian group
AffineGeneralLinearGroup construct the affine general linear group
over a finite field
AffineSpecialLinearGroup construct the affine special linear group
over a finite field
AGL construct the affine general linear group
over a finite field
AllAbelianGroups return all the Abelian groups of a given
AlternatingGroup construct the alternating group of a given
ASL construct the affine special linear group
over a finite field
BabyMonster construct the Baby Monster sporadic finite
simple group
CayleyTableGroup construct a Cayley table group
ChevalleyE6 construct a Chevalley group of type E6
ChevalleyE7 construct a Chevalley group of type E7
ChevalleyE8 construct a Chevalley group of type E8
ChevalleyF4 construct a Chevalley group of type F4
ChevalleyG2 construct a Chevalley group of type G2
ConwayGroup construct a Conway sporadic simple group
CustomGroup construct a custom group, given its
CyclicGroup construct a cyclic group of a given order
DicyclicGroup construct a dicyclic group
DihedralGroup construct the dihedral group of a given
DirectProduct construct a direct product of groups
ElementaryGroup construct an elementary Abelian group
ExceptionalGroup construct one of the exceptional finite
simple groups
FischerGroup construct one of the Fischer groups
FPGroup construct a finitely presented group from
generators and defining relators
FreeGroup construct a free group
FrobeniusGroup construct a Frobenius group from the database
GaloisGroup construct the Galois group of a polynomial
GammaL construct the general semi-linear group over
a finite field
GeneralLinearGroup construct the general linear group over a
finite field
GeneralOrthogonalGroup construct the general orthogonal group over a
finite field
GeneralSemilinearGroup construct the general semi-linear group over
a finite field
GeneralUnitaryGroup construct a general unitary group over a
finite field
GL construct the general linear group over a
finite field
Group construct a group from various data
HamiltonianGroup construct a finite Hamiltonian group
HaradaNortonGroup construct the Harada-Norton simple group
HeldGroup construct the Held simple group
HigmanSimsGroup construct the Higman-Sims simple group
JankoGroup construct one of the Janko sporadic finite
simple groups
LyonsGroup construct the Lyons simple group
MathieuGroup construct one of the Mathieu finite simple
McLaughlinGroup construct the McLaughlin simple group
MetacyclicGroup construct a metacyclic group
Monster construct the Monster simple group
ONanGroup construct the O'Nan simple group
Orbit construct the orbit of an element under a
permutation group
OrthogonalGroup construct an orthogonal group
PerfectGroup construct a perfect group from the database
Perm create a permutation
PermutationGroup construct a permutation group given
generating permutations
PGammaL construct the general semi-linear group over
a finite field
PGL construct a projective linear group over a
finite field
PGO construct the projective general orthogonal
group over a finite field
PGU construct a projective general unitary group
over a finite field
ProjectiveGeneralLinearGroup construct a projective general linear group
over a finite field
ProjectiveGeneralOrthogonalGroup construct the projective general orthogonal
group over a finite field
ProjectiveGeneralSemilinearGroup construct the general semi-linear group over
a finite field
ProjectiveGeneralUnitaryGroup construct a projective general unitary group
over a finite field
ProjectiveSpecialLinearGroup construct a projective special linear group
over a finite field
ProjectiveSpecialOrthogonalGroup construct the projective special orthogonal
group over a finite field
ProjectiveSpecialSemilinearGroup construct the special semi-linear group over
a finite field
ProjectiveSpecialUnitaryGroup construct a projective special unitary group
over a finite field
ProjectiveSymplecticGroup construct a projective symplectic group over
a finite field
ProjectiveSymplecticSemilinearGroup construct the projective symplectic
semi-linear group over a finite field
PSigmaL construct the special semi-linear group over
a finite field
PSigmap construct the projective symplectic
semi-linear group over a finite field
PSL construct a projective special linear group
over a finite field
PSO construct the projective special orthogonal
group over a finite field
PSp construct a projective symplectic group over
a finite field
PSU construct a projective special unitary group
over a finite field
QuasicyclicGroup construct a quasicyclic p-group
QuasiDihedralGroup construct a quasi-dihedral group
QuaternionGroup construct a generalized quaternion group
Ree2F4 construct a large Ree group of Lie type
Ree2G2 construct a small Ree group of Lie type
RubiksCubeGroup construct the group of the Rubik's Cube
RudvalisGroup construct the Rudvalis simple group
SemiDihedralGroup construct a semi-dihedral group
SigmaL construct the special semi-linear group over
a finite field
Sigmap construct the symplectic semi-linear group
over a finite field
SL construct a special linear group over a
finite field
SmallGroup construct a specific group of small order
SpecialLinearGroup construct a special linear group over a
finite field
SpecialOrthogonalGroup construct a special orthogonal group over a
finite field
SpecialSemilinearGroup construct the special semi-linear group over
a finite field
SpecialUnitaryGroup construct a special unitary group over a
finite field
Stabilizer compute the stabilizer of a point, list or
set under a permutation group
Steinberg2E6 construct a Steinberg group of type 2E6
Steinberg3D4 construct a Steinberg group of type G2
Subgroup construct a subgroup of a given group
Supergroup construct a supergroup of a given group
Suzuki2B2 construct a Suzuki group of Lie type
SuzukiGroup construct the Suzuki simple group
Symm construct the symmetric group of a given
SymmetricGroup construct the symmetric group of a given
SymplecticGroup construct a symplectic group over a finite
SymplecticSemilinearGroup construct the symplectic semi-linear group
over a finite field
ThompsonGroup construct the Thompson simple group
TitsGroup construct the Tits simple group
TransitiveGroup construct a specific transitive permutation
TrivialGroup construct the trivial group
TrivialSubgroup construct the trivial subgroup of a given
WreathProduct construct a wreath product of permutation
construct the affine general linear group over a finite field
construct the affine special linear group over a finite field
construct a finitely presented group from generators and defining relators
construct the orbit of an element under a permutation group
construct the projective general orthogonal group over a finite field
construct a projective general unitary group over a finite field
construct a projective general linear group over a finite field
construct a projective special linear group over a finite field
construct the projective special orthogonal group over a finite field
construct a projective special unitary group over a finite field
construct the projective symplectic semi-linear group over a finite field
compute the stabilizer of a point, list or set under a permutation group
Center compute the center of a group
Centraliser compute the centraliser of an element of a group
Centralizer compute the centralizer of an element of a group
Centre compute the centre of a group
Commutator compute the commutator of two subgroups
Core compute the core of a subgroup of a group
Cosocle compute the cosocle of a group
DerivedSubgroup compute the derived (commutator) subgroup of a group
DirectFactors compute the directly indecomposable direct factors of a
finite group
FittingSubgroup compute the Fitting subgroup of a group
FrattiniSubgroup compute the Frattini subgroup of a group
FrobeniusComplement compute a representative Frobenius complement of a
Frobenius group
FrobeniusKernel compute the Frobenius kernel of a Frobenius group
FrobeniusProduct compute the product of two complexes in a finite group
HallSubgroup compute a Hall pi-subgroup of a finite soluble group
HallSystem compute a Hall system for a finite soluble group
Hypercenter compute the hypercenter residual of a group
Index compute the index of a subgroup of a group
Intersection compute the intersection of two subgroups of a group
IsDirectlyIndecomposable test whether a group is directly indecomposable
IsMalnormal test whether a subgroup of a group is malnormal
IsNormal test whether a subgroup of a group is normal
IsQuasinormal test whether a subgroup of a group is quasi-normal
IsSubnormal test whether a subgroup of a group is subnormal
MaximalNormalSubgroups compute the maximal normal subgroups of a permutation
MinimalNormalSubgroups compute the minimal normal subgroups of a permutation
NilpotentResidual compute the nilpotent residual of a group
NormalClosure compute the normal closure of a subgroup or set of group
Normaliser compute the normaliser of a subgroup of a group
NormaliserSubgroup compute the normaliser of a subgroup of a group
NormalizerSubgroup compute the normalizer of a subgroup of a group
NormalSubgroups compute the normal subgroups of a finite group
PCore compute the p-core of a subgroup of a group
Socle compute the socle of a group
SolubleResidual compute the soluble residual of a group
SolvableResidual compute the solvable residual of a group
SubgroupLattice compute the lattice of subgroups of a group
SylowBasis compute a Sylow basis for a finite soluble group
SylowSubgroup compute a Sylow p-subgroup of a finite group
Center compute the center of a group
Centraliser compute the centraliser of an element of a group
Centralizer compute the centralizer of an element of a group
Centre compute the centre of a group
Commutator compute the commutator of two subgroups
Core compute the core of a subgroup of a group
Cosocle compute the cosocle of a group
DerivedSubgroup compute the derived (commutator) subgroup of a group
DirectFactors compute the directly indecomposable direct factors of a
finite group
FittingSubgroup compute the Fitting subgroup of a group
FrattiniSubgroup compute the Frattini subgroup of a group
FrobeniusComplement compute a representative Frobenius complement of a
Frobenius group
FrobeniusKernel compute the Frobenius kernel of a Frobenius group
FrobeniusProduct compute the product of two complexes in a finite group
HallSubgroup compute a Hall pi-subgroup of a finite soluble group
HallSystem compute a Hall system for a finite soluble group
Hypercenter compute the hypercenter residual of a group
Index compute the index of a subgroup of a group
Intersection compute the intersection of two subgroups of a group
IsDirectlyIndecomposable test whether a group is directly indecomposable
IsMalnormal test whether a subgroup of a group is malnormal
IsNormal test whether a subgroup of a group is normal
IsQuasinormal test whether a subgroup of a group is quasi-normal
IsSubnormal test whether a subgroup of a group is subnormal
MaximalNormalSubgroups compute the maximal normal subgroups of a permutation
MinimalNormalSubgroups compute the minimal normal subgroups of a permutation
NilpotentResidual compute the nilpotent residual of a group
NormalClosure compute the normal closure of a subgroup or set of group
Normaliser compute the normaliser of a subgroup of a group
NormaliserSubgroup compute the normaliser of a subgroup of a group
NormalizerSubgroup compute the normalizer of a subgroup of a group
NormalSubgroups compute the normal subgroups of a finite group
PCore compute the p-core of a subgroup of a group
Socle compute the socle of a group
SolubleResidual compute the soluble residual of a group
SolvableResidual compute the solvable residual of a group
SubgroupLattice compute the lattice of subgroups of a group
SylowBasis compute a Sylow basis for a finite soluble group
SylowSubgroup compute a Sylow p-subgroup of a finite group
compute the directly indecomposable direct factors of a finite group
compute the product of two complexes in a finite group
compute the normal closure of a subgroup or set of group elements
AllFrobeniusGroups return a list of the known Frobenius groups of a given
AllHamiltonianGroups return a list of all the Hamiltonian groups of a given
AllPerfectGroups return a list of the perfect groups of a given order
AllSmallGroups return a list of all the groups of a given order
AllTransitiveGroups return a list of all the transitive groups of a given
FrobeniusGroup construct a Frobenius group from the database
IdentifyFrobeniusGroup locate a given Frobenius group in the database of
Frobenius groups
NumFrobeniusGroups return the number of known Frobenius groups of a given
NumHamiltonianGroups return the number of Hamiltonian groups of a given order
NumPerfectGroups return the number of perfect groups of a given order
NumTransitiveGroups return the number of transitive groups of a given degree
PerfectGroup construct a perfect group from the database
RandomSmallGroup return a random group from the database of small groups
SearchFrobeniusGroups search the Frobenius Groups database
SearchPerfectGroups search the Perfect Groups database
SearchSmallGroups search the Small Groups database
SearchTransitiveGroups search the Transitive Groups database
SmallGroup construct a specific group of small order
TransitiveGroup construct a specific transitive permutation group
AllFrobeniusGroups return a list of the known Frobenius groups of a given
AllHamiltonianGroups return a list of all the Hamiltonian groups of a given
AllPerfectGroups return a list of the perfect groups of a given order
AllSmallGroups return a list of all the groups of a given order
AllTransitiveGroups return a list of all the transitive groups of a given
FrobeniusGroup construct a Frobenius group from the database
IdentifyFrobeniusGroup locate a given Frobenius group in the database of
Frobenius groups
NumFrobeniusGroups return the number of known Frobenius groups of a given
NumHamiltonianGroups return the number of Hamiltonian groups of a given order
NumPerfectGroups return the number of perfect groups of a given order
NumTransitiveGroups return the number of transitive groups of a given degree
PerfectGroup construct a perfect group from the database
RandomSmallGroup return a random group from the database of small groups
SearchFrobeniusGroups search the Frobenius Groups database
SearchPerfectGroups search the Perfect Groups database
SearchSmallGroups search the Small Groups database
SearchTransitiveGroups search the Transitive Groups database
SmallGroup construct a specific group of small order
TransitiveGroup construct a specific transitive permutation group
return a list of the known Frobenius groups of a given order
return a list of all the Hamiltonian groups of a given order
return a list of the perfect groups of a given order
return a list of all the groups of a given order
return a list of all the transitive groups of a given degree
locate a given Frobenius group in the database of Frobenius groups
return the number of known Frobenius groups of a given order
return the number of Hamiltonian groups of a given order
return the number of perfect groups of a given order
return the number of transitive groups of a given degree
return a random group from the database of small groups
AbelianInvariants compute the abelian invariants of a group
ClassNumber compute the number of conjugacy classes of a finite group
CompositionLength compute the composition length of a group
ConjugateRank compute the conjugate rank of a finite group
DerivedLength compute the derived length of a group
Exponent compute the exponent of a group
FittingLength compute the nilpotent (Fitting) length of a group
FrattiniLength compute the Frattini length of a group
GroupOrder compute the order of a group
NilpotencyClass compute the class of nilpotence of a group
NilpotentLength compute the nilpotent (Fitting) length of a group
NumInvolutions compute the number of involutions of a group
OrderClassNumber compute the number of order classes of a finite group
OrderRank compute the number of order class lengths greater than unity of
a finite group
PermGroupRank compute the rank of a permutation group
PGroupRank compute the rank of a finite p-group
PrimaryInvariants compute the primary invariants of a group
Transitivity compute the transitivity of a permutation group
AbelianInvariants compute the abelian invariants of a group
ClassNumber compute the number of conjugacy classes of a finite group
CompositionLength compute the composition length of a group
ConjugateRank compute the conjugate rank of a finite group
DerivedLength compute the derived length of a group
Exponent compute the exponent of a group
FittingLength compute the nilpotent (Fitting) length of a group
FrattiniLength compute the Frattini length of a group
GroupOrder compute the order of a group
NilpotencyClass compute the class of nilpotence of a group
NilpotentLength compute the nilpotent (Fitting) length of a group
NumInvolutions compute the number of involutions of a group
OrderClassNumber compute the number of order classes of a finite group
OrderRank compute the number of order class lengths greater than unity of
a finite group
PermGroupRank compute the rank of a permutation group
PGroupRank compute the rank of a finite p-group
PrimaryInvariants compute the primary invariants of a group
Transitivity compute the transitivity of a permutation group
compute the number of conjugacy classes of a finite group
compute the number of order classes of a finite group
compute the number of order class lengths greater than unity of a finite group
AreConjugate check whether two group elements are conjugate
AreIsomorphic test whether two groups are isomorphic
IsAbelian test whether a group is Abelian
IsAbelianSylowGroup test whether a group has Abelian Sylow subgroups
IsAlmostSimple test whether a group is almost simple
IsAlternating test (probabilistically) whether a permutation group
is an alternating group in its natural action
IsCAGroup test whether a group is a (CA)-group
IsCaminaGroup test whether a group is a Camina group
IsCCGroup test whether a group is a (CC)-group
IsCharacteristicallySimple test whether a group is characteristically simple
IsCNGroup test whether a group is a (CN)-group
IsCommutative test whether a group is commutative
IsCP1Group test whether a group is a (CP1)-group
IsCPGroup test whether a group is a (CP)-group
IsCyclic test whether a group is cyclic
IsCyclicSylowGroup test whether a group has cyclic Sylow subgroups
IsDedekind test whether a group is Dedekind
IsDicyclic test whether a permutation group is a dicyclic group
IsDihedral test whether a permutation group is a dihedral group
IsElementary test whether a group is elementary Abelian
IsExtraspecial test whether a group is an extraspecial p-group
IsFinite test whether a group is finite
IsFinitelyGenerated test whether a group is finitely generated
IsFrobeniusGroup test whether a group is a Frobenius group
IsFrobeniusPermGroup test whether a group is a Frobenius permutation group
IsGCLTGroup test whether a group is a GCLT-group
IsHallPaigeGroup test whether a group has a complete mapping
IsHamiltonian test whether a group is Hamiltonian
IsHomocyclic test whether a group is homocyclic
IsLagrangian test whether a group is Lagrangian
IsMalnormal test whether a subgroup of a group is malnormal
IsMetabelian test whether a group is Abelian
IsMetacyclic test whether a group is Metacyclic
IsNilpotent test whether a group is nilpotent
IsNormal test whether a subgroup of a group is normal
IsOrderedSylowTowerGroup test whether a group has a Sylow tower
IsPerfect test whether a group is perfect
IsPerfectOrderClassesGroup test whether a group has perfect order classes
IsPermutable test whether a subgroup of a group is permutable
IsPGroup test whether a group is a p-group
IsPrimitive test whether a permutation group is primitive
IsPSoluble test whether a group is p-soluble for a given prime p
IsQuasiprimitive test whether a permutation group is quasi-primitive
IsQuasisimple test whether a group is quasi-simple
IsQuaternion test whether a permutation group is a quaternion group
IsRegular test whether a permutation group is regular
IsRegularPGroup test whether a group is a regular p-group
IsSemiprimitive test whether a permutation group is semi-primitive
IsSemiRegular test whether a permutation group is semi-regular
IsSimple test whether a group is simple
IsSoluble test whether a group is soluble
IsSolvable test whether a group is solvable
IsSpecial test whether a group is a special p-group
IsStemGroup test whether a group is a stem group
IsSubgroup test whether one group is a subgroup of another
IsSubnormal test whether a subgroup of a group is subnormal
IsSupersoluble test whether a group is supersoluble
IsSylowTowerGroup test whether a group has a Sylow tower
IsSymmetric test (probabilistically) whether a permutation group
is a symmetric group in its natural action
IsTGroup test whether a group is a T-group
IsTransitive test whether a permutation group is transitive
IsTrivial test whether a group is trivial
SubgroupMembership test whether an element belongs to a given subgroup of
a group
AreConjugate check whether two group elements are conjugate
AreIsomorphic test whether two groups are isomorphic
IsAbelian test whether a group is Abelian
IsAbelianSylowGroup test whether a group has Abelian Sylow subgroups
IsAlmostSimple test whether a group is almost simple
IsAlternating test (probabilistically) whether a permutation group
is an alternating group in its natural action
IsCAGroup test whether a group is a (CA)-group
IsCaminaGroup test whether a group is a Camina group
IsCCGroup test whether a group is a (CC)-group
IsCharacteristicallySimple test whether a group is characteristically simple
IsCNGroup test whether a group is a (CN)-group
IsCommutative test whether a group is commutative
IsCP1Group test whether a group is a (CP1)-group
IsCPGroup test whether a group is a (CP)-group
IsCyclic test whether a group is cyclic
IsCyclicSylowGroup test whether a group has cyclic Sylow subgroups
IsDedekind test whether a group is Dedekind
IsDicyclic test whether a permutation group is a dicyclic group
IsDihedral test whether a permutation group is a dihedral group
IsElementary test whether a group is elementary Abelian
IsExtraspecial test whether a group is an extraspecial p-group
IsFinite test whether a group is finite
IsFinitelyGenerated test whether a group is finitely generated
IsFrobeniusGroup test whether a group is a Frobenius group
IsFrobeniusPermGroup test whether a group is a Frobenius permutation group
IsGCLTGroup test whether a group is a GCLT-group
IsHallPaigeGroup test whether a group has a complete mapping
IsHamiltonian test whether a group is Hamiltonian
IsHomocyclic test whether a group is homocyclic
IsLagrangian test whether a group is Lagrangian
IsMalnormal test whether a subgroup of a group is malnormal
IsMetabelian test whether a group is Abelian
IsMetacyclic test whether a group is Metacyclic
IsNilpotent test whether a group is nilpotent
IsNormal test whether a subgroup of a group is normal
IsOrderedSylowTowerGroup test whether a group has a Sylow tower
IsPerfect test whether a group is perfect
IsPerfectOrderClassesGroup test whether a group has perfect order classes
IsPermutable test whether a subgroup of a group is permutable
IsPGroup test whether a group is a p-group
IsPrimitive test whether a permutation group is primitive
IsPSoluble test whether a group is p-soluble for a given prime p
IsQuasiprimitive test whether a permutation group is quasi-primitive
IsQuasisimple test whether a group is quasi-simple
IsQuaternion test whether a permutation group is a quaternion group
IsRegular test whether a permutation group is regular
IsRegularPGroup test whether a group is a regular p-group
IsSemiprimitive test whether a permutation group is semi-primitive
IsSemiRegular test whether a permutation group is semi-regular
IsSimple test whether a group is simple
IsSoluble test whether a group is soluble
IsSolvable test whether a group is solvable
IsSpecial test whether a group is a special p-group
IsStemGroup test whether a group is a stem group
IsSubgroup test whether one group is a subgroup of another
IsSubnormal test whether a subgroup of a group is subnormal
IsSupersoluble test whether a group is supersoluble
IsSylowTowerGroup test whether a group has a Sylow tower
IsSymmetric test (probabilistically) whether a permutation group
is a symmetric group in its natural action
IsTGroup test whether a group is a T-group
IsTransitive test whether a permutation group is transitive
IsTrivial test whether a group is trivial
SubgroupMembership test whether an element belongs to a given subgroup of
a group
test (probabilistically) whether a permutation group is an alternating group in its natural action
test whether a group is p-soluble for a given prime p
test (probabilistically) whether a permutation group is a symmetric group in its natural action
test whether an element belongs to a given subgroup of a group
Permutation Groups
AbelianInvariants compute the abelian invariants of a
permutation group
BlocksImage return a permutation group equivalent to
the action of a permutation on a system of
BlockSystem return a block system for a permutation
group, non-trivial if possible
CycleIndexPolynomial compute the cycle index polynomial of a
permutation group
Degree return the degree of a permutation group
EARNS return an EARNS of a primitive group if it
has one
FrobeniusPermRep construct a Frobenius permutation group
isomorphic to a given Frobenius group
IsAlternating test (probabilistically) whether a
permutation group is an alternating group
in its natural action
IsPrimitive test whether a permutation group is
IsQuasiprimitive test whether a permutation group is
IsRegular test whether a permutation group is
IsSemiprimitive test whether a permutation group is
IsSemiRegular test whether a permutation group is
IsSymmetric test (probabilistically) whether a
permutation group is a symmetric group in
its natural action
IsTransitive test whether a permutation group is
MaxSupport return the largest element displaced by a
permutation group
MinimalBlockSystem return a minimal block system for a
permutation group, non-trivial if possible
MinimumPermutationRepresentationDegree compute the minimum degree of a faithful
permutation representation for a group
MinSupport return the smallest element displaced by a
permutation group
Orbit construct the orbit of an element under a
permutation group
Orbits compute the orbits of a permutation group
PermGroupRank compute the rank of a permutation group
PrimaryInvariants compute the primary invariants of a
permutation group
ReducedDegreePermGroup return an isomorphic permutation group of
possibly smaller degree
RestrictedPermGroup return the restriction of a permutation
group to a stable subset
Stabilizer compute the stabilizer of a point, list or
set under a permutation group
Support return the support of a permutation group
SupportLength return the number of elements displaced by
a permutation group
Transitivity compute the transitivity of a permutation
AbelianInvariants compute the abelian invariants of a
permutation group
BlocksImage return a permutation group equivalent to
the action of a permutation on a system of
BlockSystem return a block system for a permutation
group, non-trivial if possible
CycleIndexPolynomial compute the cycle index polynomial of a
permutation group
Degree return the degree of a permutation group
EARNS return an EARNS of a primitive group if it
has one
FrobeniusPermRep construct a Frobenius permutation group
isomorphic to a given Frobenius group
IsAlternating test (probabilistically) whether a
permutation group is an alternating group
in its natural action
IsPrimitive test whether a permutation group is
IsQuasiprimitive test whether a permutation group is
IsRegular test whether a permutation group is
IsSemiprimitive test whether a permutation group is
IsSemiRegular test whether a permutation group is
IsSymmetric test (probabilistically) whether a
permutation group is a symmetric group in
its natural action
IsTransitive test whether a permutation group is
MaxSupport return the largest element displaced by a
permutation group
MinimalBlockSystem return a minimal block system for a
permutation group, non-trivial if possible
MinimumPermutationRepresentationDegree compute the minimum degree of a faithful
permutation representation for a group
MinSupport return the smallest element displaced by a
permutation group
Orbit construct the orbit of an element under a
permutation group
Orbits compute the orbits of a permutation group
PermGroupRank compute the rank of a permutation group
PrimaryInvariants compute the primary invariants of a
permutation group
ReducedDegreePermGroup return an isomorphic permutation group of
possibly smaller degree
RestrictedPermGroup return the restriction of a permutation
group to a stable subset
Stabilizer compute the stabilizer of a point, list or
set under a permutation group
Support return the support of a permutation group
SupportLength return the number of elements displaced by
a permutation group
Transitivity compute the transitivity of a permutation
return a permutation group equivalent to the action of a permutation on a system of blocks
return a block system for a permutation group, non-trivial if possible
return an EARNS of a primitive group if it has one
construct a Frobenius permutation group isomorphic to a given Frobenius group
return a minimal block system for a permutation group, non-trivial if possible
compute the minimum degree of a faithful permutation representation for a group
return the restriction of a permutation group to a stable subset
return the number of elements displaced by a permutation group
Finitely Presented Groups
AbelianInvariants compute the abelian invariants of a finitely presented
PresentationComplexity compute a measure of the complexity of a finitely
presented group
PrimaryInvariants compute the primary invariants of a finitely presented
Relators return the relators of a finitely presented group
Simplify simplify the presentation of a finitely presented group
AbelianInvariants compute the abelian invariants of a finitely presented
PresentationComplexity compute a measure of the complexity of a finitely
presented group
PrimaryInvariants compute the primary invariants of a finitely presented
Relators return the relators of a finitely presented group
Simplify simplify the presentation of a finitely presented group
compute a measure of the complexity of a finitely presented group
DrawCayleyTable draw the Cayley table of a finite group
DrawNormalSubgroupLattice draw the lattice of normal subgroups of a finite group
DrawSubgroupLattice draw the lattice of subgroups of a finite group
DrawCayleyTable draw the Cayley table of a finite group
DrawNormalSubgroupLattice draw the lattice of normal subgroups of a finite group
DrawSubgroupLattice draw the lattice of subgroups of a finite group
draw the lattice of normal subgroups of a finite group
AgemoSeries compute the series of agemo subgroups of a $p$-group
CompositionSeries compute a composition series of a group
DerivedSeries compute the derived series of a group
FrattiniSeries compute the Frattini series of a group
LowerCentralSeries compute the lower central series of a group
LowerFittingSeries compute the lower Fitting series of a group
LowerPCentralSeries compute the lower p-central series of a group
OmegaSeries compute the series of omega subgroups of a $p$-group
OrderedSylowTower compute a Sylow tower for a Sylow tower group
SylowTower compute a Sylow tower for a Sylow tower group
UpperCentralSeries compute the upper central series of a group
AgemoSeries compute the series of agemo subgroups of a $p$-group
CompositionSeries compute a composition series of a group
DerivedSeries compute the derived series of a group
FrattiniSeries compute the Frattini series of a group
LowerCentralSeries compute the lower central series of a group
LowerFittingSeries compute the lower Fitting series of a group
LowerPCentralSeries compute the lower p-central series of a group
OmegaSeries compute the series of omega subgroups of a $p$-group
OrderedSylowTower compute a Sylow tower for a Sylow tower group
SylowTower compute a Sylow tower for a Sylow tower group
UpperCentralSeries compute the upper central series of a group
DecomposeDessin find all decompositions of a Belyi map represented by a dessin
FindDessins find all dessins d'enfants with a specified branch pattern
DecomposeDessin find all decompositions of a Belyi map represented by a dessin
FindDessins find all dessins d'enfants with a specified branch pattern
find all decompositions of a Belyi map represented by a dessin
ElementOrder compute the order of a group element
ElementOrderSum compute the sum of the element orders of a finite group
ElementPower compute an integer power of a group element
Elements compute the elements of a finite group, orbit, coset or
conjugacy class
MaximumElementOrder compute the largest order of an element of a finite group
OrderClassPolynomial compute the order class polynomial of a finite group
OrderClassProfile compute the element order profile of a finite group
PrimePowerFactors factor a group element into a product of elements of prime
power order
RandomElement compute a random element of a group
RandomImvolution compute a random involution of a group
RandomPElement compute a random p-element of a group
RandomPPrimeElement compute a random element of a group with order relatively
prime to p
ElementOrder compute the order of a group element
ElementOrderSum compute the sum of the element orders of a finite group
ElementPower compute an integer power of a group element
Elements compute the elements of a finite group, orbit, coset or
conjugacy class
MaximumElementOrder compute the largest order of an element of a finite group
OrderClassPolynomial compute the order class polynomial of a finite group
OrderClassProfile compute the element order profile of a finite group
PrimePowerFactors factor a group element into a product of elements of prime
power order
RandomElement compute a random element of a group
RandomImvolution compute a random involution of a group
RandomPElement compute a random p-element of a group
RandomPPrimeElement compute a random element of a group with order relatively
prime to p
compute the sum of the element orders of a finite group
compute the elements of a finite group, orbit, coset or conjugacy class
compute the largest order of an element of a finite group
factor a group element into a product of elements of prime power order
compute a random element of a group with order relatively prime to p
IsAbelianNumber test whether every group of a given order is Abelian
IsCyclicNumber test whether every group of a given order is cyclic
IsGCLTNumber test whether every group of a given order is a GCLT
IsIntegrableNumber test whether every group of a given order is integrable
IsLagrangianNumber test whether every group of a given order is Lagrangian
IsMetabelianNumber test whether every group of a given order is metabelian
IsMetacyclicNumber test whether every group of a given order is metacyclic
IsNilpotentNumber test whether every group of a given order is nilpotent
IsOrderedSylowTowerNumber test whether every group of a given order has an
ordered Sylow tower
IsSimpleNumber test whether a number is the order of a finite simple
IsSolubleNumber test whether every group of a given order is soluble
IsSupersolubleNumber test whether every group of a given order is
IsAbelianNumber test whether every group of a given order is Abelian
IsCyclicNumber test whether every group of a given order is cyclic
IsGCLTNumber test whether every group of a given order is a GCLT
IsIntegrableNumber test whether every group of a given order is integrable
IsLagrangianNumber test whether every group of a given order is Lagrangian
IsMetabelianNumber test whether every group of a given order is metabelian
IsMetacyclicNumber test whether every group of a given order is metacyclic
IsNilpotentNumber test whether every group of a given order is nilpotent
IsOrderedSylowTowerNumber test whether every group of a given order has an
ordered Sylow tower
IsSimpleNumber test whether a number is the order of a finite simple
IsSolubleNumber test whether every group of a given order is soluble
IsSupersolubleNumber test whether every group of a given order is
test whether every group of a given order is Abelian
test whether every group of a given order is cyclic
test whether every group of a given order is a GCLT group
test whether every group of a given order is integrable
test whether every group of a given order is Lagrangian
test whether every group of a given order is metabelian
test whether every group of a given order is metacyclic
test whether every group of a given order is nilpotent
test whether every group of a given order has an ordered Sylow tower
test whether a number is the order of a finite simple group
test whether every group of a given order is soluble
test whether every group of a given order is supersoluble
CayleyGraph return the Cayley graph of a finite group
CayleyTable return the Cayley table of a finite group
CFSG finite simple group classifier object
Character construct a character from a character table
CharacterTable compute the character table of a finite group
ClassifyFiniteSimpleGroup classify a finite simple group
CommutingGraph construct the commuting graph of a finite group
ConjugacyClass compute the conjugacy class of a group element
ConjugacyClasses compute all the conjugacy classes of a finite group
Conjugator compute an element conjugating one group element to
Elements compute the elements of a finite group, orbit, coset or
conjugacy class
Factor expression a group element as a product of a coset
representative and a subgroup element
Generators return the set of generators of a group
GruenbergKegelGraph return the Gruenberg-Kegel graph of a finite group
IdentifySmallGroup compute the Small Group ID of a small group
Labels return the set of generator labels of a group
LeftCoset compute a left coset of a group element
LeftCosets compute the left cosets of a subgroup of a group
logp compute the exponent of a prime power
NonRedundantGenerators return a set of non-redundant generators of a group
NumSimpleGroups count the number of simple groups of a given finite
Operations return the operations record of a group
PresentationComplexity compute a measure of the complexity of a finitely
presented group
RightCoset compute a right coset of a group element
RightCosets compute the right cosets of a subgroup of a group
TabulateSimpleGroups list the simple groups with orders in a given range
CayleyGraph return the Cayley graph of a finite group
CayleyTable return the Cayley table of a finite group
CFSG finite simple group classifier object
Character construct a character from a character table
CharacterTable compute the character table of a finite group
ClassifyFiniteSimpleGroup classify a finite simple group
CommutingGraph construct the commuting graph of a finite group
ConjugacyClass compute the conjugacy class of a group element
ConjugacyClasses compute all the conjugacy classes of a finite group
Conjugator compute an element conjugating one group element to
Elements compute the elements of a finite group, orbit, coset or
conjugacy class
Factor expression a group element as a product of a coset
representative and a subgroup element
Generators return the set of generators of a group
GruenbergKegelGraph return the Gruenberg-Kegel graph of a finite group
IdentifySmallGroup compute the Small Group ID of a small group
Labels return the set of generator labels of a group
LeftCoset compute a left coset of a group element
LeftCosets compute the left cosets of a subgroup of a group
logp compute the exponent of a prime power
NonRedundantGenerators return a set of non-redundant generators of a group
NumSimpleGroups count the number of simple groups of a given finite
Operations return the operations record of a group
PresentationComplexity compute a measure of the complexity of a finitely
presented group
RightCoset compute a right coset of a group element
RightCosets compute the right cosets of a subgroup of a group
TabulateSimpleGroups list the simple groups with orders in a given range
expression a group element as a product of a coset representative and a subgroup element
compute the left cosets of a subgroup of a group
count the number of simple groups of a given finite order
compute the right cosets of a subgroup of a group
list the simple groups with orders in a given range
Group Constructors Palette
• The Group Constructors palette contains buttons for constructing groups.
• Palettes are displayed in the left pane of the Maple window. (If it is not visible, from the main menu, select View > Palettes > Show Palette > Group Constructors)
• Some palette items have placeholders. Fill in the placeholders, using Tab to navigate to the next placeholder.
• The Group Constructors palette contains buttons for constructing groups.
• Palettes are displayed in the left pane of the Maple window. (If it is not visible, from the main menu, select View > Palettes > Show Palette > Group Constructors)
Palettes are displayed in the left pane of the Maple window. (If it is not visible, from the main menu, select View > Palettes > Show Palette > Group Constructors)
• Some palette items have placeholders. Fill in the placeholders, using Tab to navigate to the next placeholder.
Some palette items have placeholders. Fill in the placeholders, using Tab to navigate to the next placeholder.
Context-Sensitive Operations
• In the Standard Worksheet interface, you can apply operations to a group through the Context Panel under the Group operations submenu.
• In the Standard Worksheet interface, you can apply operations to a group through the Context Panel under the Group operations submenu.
In the Standard Worksheet interface, you can apply operations to a group through the Context Panel under the Group operations submenu.
The following command enables you to use the commands in the GroupTheory package without having to prefix each command with "GroupTheory:-".
> $\mathrm{with}⁡\left(\mathrm{GroupTheory}\right):$
Create a symmetric group of degree 4.
> $G≔\mathrm{SymmetricGroup}⁡\left(4\right)$
${G}{≔}{{\mathbf{S}}}_{{4}}$ (1)
Visualize the lattice of subgroups of G.
> $\mathrm{DrawSubgroupLattice}⁡\left(G\right)$
Create a dihedral group of degree 4 (and order 8).
> $H≔\mathrm{DihedralGroup}⁡\left(4\right)$
${H}{≔}{{\mathbf{D}}}_{{4}}$ (2)
Visualize the Cayley (operation) table for H.
> $\mathrm{DrawCayleyTable}⁡\left(H\right)$
Compute the orders (cardinalities) of G and H.
> $\mathrm{GroupOrder}⁡\left(G\right),\mathrm{GroupOrder}⁡\left(H\right)$
${24}{,}{8}$ (3)
Compute the character table of H.
> $\mathrm{Display}⁡\left(\mathrm{CharacterTable}⁡\left(H\right)\right)$
C 1a 2a 2b 2c 4a
|C| 1 1 2 2 2
$\mathrm{χ__1}$ $1$ $1$ $1$ $1$ $1$
$\mathrm{χ__2}$ $1$ $1$ $-1$ $-1$ $1$
$\mathrm{χ__3}$ $1$ $1$ $-1$ $1$ $-1$
$\mathrm{χ__4}$ $1$ $1$ $1$ $-1$ $-1$
$\mathrm{χ__5}$ $2$ $-2$ $0$ $0$ $0$
Form the direct product in two ways.
> $U≔\mathrm{DirectProduct}⁡\left(G,H\right)$
${U}{≔}〈\left({1}{,}{2}\right){,}\left({1}{,}{2}{,}{3}{,}{4}\right){,}\left({5}{,}{7}\right){,}\left({5}{,}{6}{,}{7}{,} (4)
> $V≔\mathrm{DirectProduct}⁡\left(H,G\right)$
${V}{≔}〈\left({1}{,}{3}\right){,}\left({1}{,}{2}{,}{3}{,}{4}\right){,}\left({5}{,}{6}\right){,}\left({5}{,}{6}{,}{7}{,} (5)
Check that these are isomorphic.
> $\mathrm{AreIsomorphic}⁡\left(U,V\right)$
${\mathrm{true}}$ (6)
Notice that the order of the direct product is equal to the product of the orders of the factors.
> $\mathrm{GroupOrder}⁡\left(U\right)=\mathrm{GroupOrder}⁡\left(G\right)⁢\mathrm{GroupOrder}⁡\left(H\right)$
${192}{=}{192}$ (7)
Since the order of U does not exceed 511, we can identify the group explicitly.
> $\mathrm{id}≔\mathrm{IdentifySmallGroup}⁡\left(U\right)$
${\mathrm{id}}{≔}{192}{,}{1472}$ (8)
Retrieve the identified group from the database, and check the isomorphism.
> $W≔\mathrm{SmallGroup}⁡\left(\mathrm{id}\right):$
> $\mathrm{AreIsomorphic}⁡\left(U,W\right)$
${\mathrm{true}}$ (9)
Construct a wreath product of two symmetric groups.
> $W≔\mathrm{WreathProduct}⁡\left(\mathrm{Symm}⁡\left(3\right),\mathrm{Symm}⁡\left(4\right)\right)$
${W}{≔}〈\left({1}{,}{2}\right){,}\left({1}{,}{2}{,}{3}\right){,}\left({1}{,}{4}\right)\left({2}{,}{5}\right)\left({3}{,}{6}\right){& (10)
Check that the wreath product is transitive but imprimitive.
> $\mathrm{IsTransitive}⁡\left(W\right)$
${\mathrm{true}}$ (11)
> $\mathrm{IsPrimitive}⁡\left(W\right)$
${\mathrm{false}}$ (12)
Find a non-trivial system of blocks for W.
> $\mathrm{BlockSystem}⁡\left(W\right)$
${{{1}{,}{2}{,}{3}}{,}{{4}{,}{5}{,}{6}}{,}{{7}{,}{8}{,}{9}}{,}{{10}{,}{11}{,}{12}} (13)
Identify the Sylow $3$-subgroup of W in the database of small groups.
> $\mathrm{IdentifySmallGroup}⁡\left(\mathrm{SylowSubgroup}⁡\left(3,W\right)\right)$
${243}{,}{51}$ (14)
Find the nilpotency class of the Sylow $2$-subgroup of W.
> $\mathrm{NilpotencyClass}⁡\left(\mathrm{SylowSubgroup}⁡\left(2,W\right)\right)$
${4}$ (15)
Compute a composition series for U.
> $\mathrm{cs}≔\mathrm{CompositionSeries}⁡\left(U\right)$
Warning, over-writing property `["DerivedSeries"]' with a different value
${\mathrm{cs}}{≔}〈\left({1}{,}{2}\right){,}\left({1}{,}{2}{,}{3}{,}{4}\right){,}\left({5}{,}{7}\right){,}\left({5}{,}{6}{,} (16)
comma;}\left({3}{,}{4}\right){,}\left({6}{,}{8}\right)〉{▹}{\dots }{▹}〈\left({1}{,}{4}\right)\left({2}{,}{3}\right)〉{▹}〈〉$
Find the orders of the members of this composition series.
> $\mathrm{seq}⁡\left(\mathrm{GroupOrder}⁡\left(L\right),L=\mathrm{cs}\right)$
${192}{,}{96}{,}{48}{,}{24}{,}{8}{,}{4}{,}{2}{,}{1}$ (17)
Find the IDs of the groups in the database of small groups that are perfect, but not simple.
> $\mathrm{SearchSmallGroups}⁡\left('\mathrm{simple}'=\mathrm{false},'\mathrm{perfect}'\right)$
$[{1}{,}{1}]{,}[{120}{,}{5}]{,}[{336}{,}{114}]$ (18)
Investigate the relative frequencies of multiply transitive groups in the database of transitive groups.
> $\mathrm{Statistics}:-\mathrm{PieChart}⁡\left([\mathrm{seq}]⁡\left(i=\mathrm{SearchTransitiveGroups}⁡\left('\mathrm{transitivity}'&
The following command enables you to use the commands in the GroupTheory package without having to prefix each command with "GroupTheory:-".
Create a dihedral group of degree 4 (and order 8).
C 1a 2a 2b 2c 4a
|C| 1 1 2 2 2
$\mathrm{χ__1}$ $1$ $1$ $1$ $1$ $1$
$\mathrm{χ__2}$ $1$ $1$ $-1$ $-1$ $1$
$\mathrm{χ__3}$ $1$ $1$ $-1$ $1$ $-1$
$\mathrm{χ__4}$ $1$ $1$ $1$ $-1$ $-1$
$\mathrm{χ__5}$ $2$ $-2$ $0$ $0$ $0$
Notice that the order of the direct product is equal to the product of the orders of the factors.
Since the order of U does not exceed 511, we can identify the group explicitly.
Retrieve the identified group from the database, and check the isomorphism.
Identify the Sylow $3$-subgroup of W in the database of small groups.
Find the nilpotency class of the Sylow $2$-subgroup of W.
Find the orders of the members of this composition series.
Find the IDs of the groups in the database of small groups that are perfect, but not simple.
Investigate the relative frequencies of multiply transitive groups in the database of transitive groups.
• The GroupTheory package was introduced in Maple 17.
• For more information on Maple 17 changes, see Updates in Maple 17.
• For more information on Maple 17 changes, see Updates in Maple 17.
For more information on Maple 17 changes, see Updates in Maple 17. | {"url":"https://www.maplesoft.com/support/help/MapleSim/view.aspx?path=GroupTheory","timestamp":"2024-11-03T13:13:21Z","content_type":"text/html","content_length":"617682","record_id":"<urn:uuid:02c76d80-efd4-4c3c-a9d7-c3fce44aa3e1>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00791.warc.gz"} |
(PDF) Study of length differences from topography to map projection within the state coordinate systems for some countries on the Balkan Peninsula
... In relation to these three surfaces another three entities are defined: the orthometric height (h), the ellipsoidal height (H) and the geoidal separation/undulation (N) (Das etAll, 2017). All
geodetic measured quantities from the physical surface of the Earth are reduced to the geoid and to the surface of adopted reference ellipsoid through applying corrections on the measured values, as
well as into the map projection (Idrizi etAll 2018). There are a number of mathematical models to relate the geoidal separation N with corresponding locations in order to define the geoid surface.
... Since the possibility for usage of official data by responsible institution for geospatial information is very limited, as source data have been used test model developed for analyses of length
differences between topography and geoid as well as between geoid and referent ellipsoid, with 10893 points with Cartesian/geographic coordinates in the state coordinate system in ground resolution
of 1km shown in figure 1 (Idrizi etAll 2018), while for elevation dataset the ASTER global DEM [6] with 30m spatial resolution have been used. ...
... Three EGM models (2008, 1996 and 1984) for the national area of the Republic of Kosova are developed as separate point vector datasets with 1km ground resolution (Idrizi etAll 2018), while later
converted to raster datasets, and finally merged into multiband raster dataset based on obtained elevation values, obtained geoid heights for three models, and calculated values between geoid heights
and with elevations. Schema for whole process is given in next figure 3. Three grids of 10893 points (6, 7 and 8) with values for coordinates, elevations, and geoid heights for three EGM models, at
final step of calculation, before developing multiband raster dataset, have been used for calculation of differences between geoid heights of three models and ellipsoidal heights based on difference
between the elevations and geoid heights of three models (step 9). ... | {"url":"https://www.researchgate.net/publication/325105887_Study_of_length_differences_from_topography_to_map_projection_within_the_state_coordinate_systems_for_some_countries_on_the_Balkan_Peninsula","timestamp":"2024-11-07T10:06:19Z","content_type":"text/html","content_length":"686945","record_id":"<urn:uuid:d9777f4d-8e2b-4a53-95ba-ec3ea069981e>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00221.warc.gz"} |
A proof that 22/7 - pi > 0 and more
My father was a High School English teacher who did not know much math. As I was going off to college, intending to major in math, he gave me the following sage advice:
Take Physics as well as Math since Physics and Math go well together.
This was good advice. I took the first year of Physics for Physics Majors, and I later took a senior course in Mechanics since that was my favorite part of the first year course. Kudos to Dad!
2) π
is exactly
22/7. I knew this was not true, but I also knew that I had no easy way to show him this. In fact, I wonder if I could have proven it myself back then.
I had not thought about this in many years when I came across the following:
Problem A-1 on the 1968 Putnam exam:
Prove 22/7 - π = ∫
)/(1+ x
(I can easily do his by partial fractions and remembering that ∫ 1/(1+x^2) dx = tan
x which is tan inverse, not 1/tan. See
(ADDED LATER---I have added conjectures on getting integrals of the form above except with 4 replaced by any natural number. Be the first on your block to solve my conjectures! It has to be easier
than the Sensitivity Conjecture!)
Let n ∈ N which we will choose later. By looking at the circle that is inscribed in a regular n-polygon (n even) one finds that
n tan(π/n) > π
So we seek an even value of n such that
n tan(π/n) < 22/7
Using Wolfram alpha the smallest such n is 92.
Would that convince Dad? Would he understand it? Probably not. Oh well.
Some misc points.
1) While working on this post I originally wanted to find tan(π/2
) by using the half-angle formula many times, and get an exact answer in terms of radicals, rather than using Wolfram Alpha.
a) While I have lots of combinatorics books, theory of comp books, and more Ramsey Theory books than one person should own in my house, I didn't have a SINGLE book with any trig in it.
b) I easily found it on the web:
tan(x/2) = sqrt( (1-cos x)/(1+cos x) ) = sin x/(1+cos x) = (1-cos x)/(sin x).
None of these seems like it would get me a nice expression for tan(π/2
). But I don't know. Is there a nice expression for tan(π/2
) ? If you know of one then leave a polite comment.
2) I assumed that there was a more clever and faster way to do the integral. I could not find old Putnam exams and their solutions the web (I'm sure they are there someplace! --- if you know then
comment politely with a pointer). So I got a book out of the library
The William Lowell Putnam Mathematical Competition Problems and Solutions 1965--1984
by Alexanderson, Klosinski, and Larson. Here is the clever solution:
The standard approach from Elementary Calculus applies.
Not as clever as I as hoping for.
3) I also looked at the integral with 4 replaced by 1,2,3,4,...,16. The results are in the writeup I pointed to before. It looks like I can use this sequence to get upper and lower bound on pi, ln
(2), pi+2ln(2), and pi-2ln(2). I have not proven any of this. But take a look! And as noted above I have conjectures!
4) When I looked up INSCRIBING a circle in a regular n-polygon, Google kept giving me CIRCUMSCRIBING. Why? I do not know but I can speculate. Archimedes had a very nice way of using circumscribed
circles to approximate pi. Its on youtube
. Hence people are used to using circumscribed rather than inscribed circles.
7 comments:
1. Here is a link to a solution of the Putnam problem:
2. THX!
3. I'd avoid W Alpha or W Beta. Both mentioning and using.
4. I got email from someone saying:
Just type 22/7 - pi into Wolfram Alpha.
while funny it raises a good point- would that convince my dad? Should it convince you? Note that my lack of thinking to do that lead to this post and some interesting ways to approx pi, ln(2),
pi-2ln(2) and pi+2ln(2).
5. According to the book "The Historical Development of the Calculus" by C.H. Edwards, Jr., 1979, Chapter 2, Archimedes used inscribed and circumscribed polygons to get a very tight bound on the
value of pi. See the book for details.
6. Your linked PDF writes x^4(1-x^4) everywhere, instead of the correct x^4(1-x)^4.
1. Fixed, thanks! | {"url":"https://blog.computationalcomplexity.org/2019/06/a-proof-that-227-pi-0-and-more.html","timestamp":"2024-11-11T23:22:02Z","content_type":"application/xhtml+xml","content_length":"185430","record_id":"<urn:uuid:ea6625d5-0c2c-42ac-b715-c1a9b0fbc75e>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00467.warc.gz"} |
numer-, number-
(Latin: distribution; to count, to reckon)
ISBN number
(s) (noun)
, ISBN numbers
Intenational Standard Book Number number: The use of ISBN followed by the word number is an excessive use of the word.
Latin numerical symbol
(s) (noun)
, Latin numerical symbols
The origin of Latin counting symbols: There are some people who believe that the
Latin numerical symbol V
(5) represents the hand with all five fingers spread apart.
It is pleasant to think that I represents the upheld finger of Latin Numerical symbols and that V might symbolize the hand itself with all five fingers; so, one branch of the V would be the extended
thumb; the other, the remaining fingers for "six", "seven", "eight", and "nine"; we would then have VI, VII, VIII, and VIIII.
(s) (noun)
, numbers
1. An arithmetical value, expressed by a word, a symbol, or a figure, representing a particular quantity and used when figuring up and making calculations and for showing the order in a series or for
identification: Ted's biology teacher dialed the phone
so she could tell the bookstore the number of textbooks she wanted to order for her students.
2. A collection of individual things which can be added up and referring to things that are physically or which are symbolically separate, not merely separable into units: The symbols 1, 2, 3, 4, 5,
etc. and the words, one, two, three, four, five, etc. are
An "amount" emphasizes the whole, while a number focuses on the parts; such as, an "amount" of money; a number of coins.
A "quantity" stresses measurement in bulk (a bunch of apples are in the bag), a number stresses individual items (six apples are on the plate).
When Jacob moved to a smaller apartment, a number of his books had to be given away to the local library.
A number of seats are still available at the theater.
Numbers of people complained when the proposed shutdown of the local grocery store was announced.
3. In grammar, a word form that indicates one person or thing or refers to more than one: The
used in grammar are "singular" (one) and "plural" (two or more).
, numbers; numbered; numbering
1. To specify an amount or quantity: The investors in the company
in the thousands.
2. To include or to classify as a member of a group: The university
2,000 students that are attending classes so far this year.
The population of the town now numbers 10,000.
3. To indicate a position in a series: Each document was
in a sequence.
4. To identify people or things in a series: Maude
the times that she does each exercise at the fitness studio.
Mrs. Jackson told her students to take out a sheet of paper and to number it from 1 to 15 down the side for the quiz.
Dr. Herbert Kyle was numbered by his students as one of the best professors in the university.
(s) (noun)
, numberers
A person, or people, who provide the amounts or rations of things for a topic or a situation: Susanne was one of the numberers who was taking an inventory of the products in the department store.
, more numerable, most numerable
Relating to something that can be added up: Dina received numerable assets from the inheritance that she received from her father.
(s) (noun)
, numeracies
The ability to think and to express oneself effectively with a knowledge of mathematical skills: Carolina had a superior numeracy that made her capable of understanding mathematical concepts,
performing calculations, and interpreting and using statistical information.
(s) (noun)
, numeraires
1. A standard by which values are measured; such as, gold in the monetary system: The
money unit of measure exists within an abstract macroeconomic model in which there is no actual money or currency.
Numeraire is a function of money as a measure of value or a unit of account; such as, a standard for currency exchange rates.
2. Etymology: from French
"currency in circulation within a given political state".
(s) (noun)
, numerals
Letters, figures, words, parts of speech, etc.; expressing or indicating a sum or measurement: The Arabic numerals 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, etc., and the Roman numerals I, II, III, IV, V, XI,
VII, VIII, IX, X, XI, etc. are examples of some of the numerals that exist.
, more numerally, most numerally
A reference to a character that is used to represent a mathematical value: Susan and the other students were studying the numerally different Roman system of quantities; such as, I, II, III, IV, V,
VI, VII, VIII, IX, and X.
, more numerary, most numerary
Of or relating to symbols, digits, signs, or notations; alone or in combination with others: The numerary systems of the world are not always the same; however, they are essential for keeping track
of many things.
, numerates; numerated; numerating
1. To list or indicate symbols that label or identify people or things in a sequence: Jake and the police were
how many people were taking part in the demonstration against the politician.
2. Being able to think and to express oneself effectively in quantitative terms: Herman
how many times he had to apply for a job before he finally got one.
Patricia was numerating how many novels she had read during the last two months.
(s) (noun)
, numerations
Calculations or processes of tallying, reckoning, or assigning a quantum to something:
involve a system of reading or naming numbers: especially, those written decimally and usually according to the Arabic numerals.
Numeration can be an action, a process, or a result of ascertaining the number of people, etc., in a specified category.
(s) (noun)
, numerators
1. The expression written above the line in a common fraction to indicate the number of parts of the whole: The numerator of the fraction 2/3 is 2.
2. A person or something that expresses quantities: Cory was a numerator who worked as an accountant for his company and kept track of the financial records and prepared reports for the
(s) (noun)
, numerbilities
Items or objects that can be counted or which can be calculated: The numerbilities of the books in the library were confirmed as the staff made an inventory of all of the books and itemized them.
Showing page 3 out of 5 pages of 73 main-word entries or main-word-entry groups. | {"url":"https://wordinfo.info/unit/1427/page:3/s:numbered","timestamp":"2024-11-02T05:20:02Z","content_type":"text/html","content_length":"18440","record_id":"<urn:uuid:fbcac7ee-a99b-4640-94e5-081bacb58762>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00583.warc.gz"} |
The martingale betting system is a commonly used betting system, which like Roulette has its origins in France. The betting strategy is generally used for stakes where there is an equal (or almost
equal) chance of one of two outcomes occurring. The most basic example is a coin toss but the strategy is commonly applied to Roulette.
The strategy is based on the bettor doubling their bet after each loss to ensure that their first win recovers any previous losses along with an additional profit. Be warned however that the
martingale strategy is not fool proof as (1) it presumes the gambler has infinite wealth (2) betting limits exist.
Application of the Martingale Betting System in Roulette
When applying the martingale betting system to Roulette it is important to remember that the strategy is applied to situations where there is an equal or almost equal chance of one of two outcomes
occurring. In Roulette this means betting on outside bets, i.e. red or black or odd or even where the payout is 1:1, and doubling the bet so as to recover losses when the player’s selection is not
Due to the possibility of drawing a zero it is important to note that the chance of drawing either outcome is not 50%/50% but rather, 47.4%. This doesn’t affect the strategy however as if you do not
draw your selected number you still double your bet. It is best to display this via an example where the martingale strategy is applied to the selection of red or black in Roulette.
• A player decides to bet on the outside bet of “Black” in Roulette and places a bet of €1 on this outcome.
• If the result is Black, that is fine and the player will be paid at 1:1
• If the result is not Black, i.e. Red or zero, the player loses their stake and places a bet on Black again, this time for double their previous stake, €2 in this instance. If the result is Black
this time, then the return to the player is €4 which will cover the loss they made in their first bet as well as meaning the player pockets an additional profit of 1X their initial stake. If you
win begin the process again with your original stake.
• Betting continues in that doubling manner when a player doesn’t win, e.g.
Selection Stake Result Total Profit/Loss
Black € 1 Loss – € 1
Black € 2 Loss – € 3
Black € 4 Loss – € 7
Black € 8 Loss – € 15
Black € 16 Win + € 1
System Limitations
As mentioned previously, while this casino betting system may lead to short term gains, it is not fool proof. A players bankroll may not be sufficient and they could run out of money while on a
losing run.
In addition, most casino tables have a maximum bet of €500. This means that punters can double their bets by up to a maximum of 9 consecutive times on a stake of €1 before they would then breach the
table limit for outside bets. Statistically, this means that in Standard or European Roulette there is only a 0.24% chance that there will not be a winning outcome in 9 spins. In American roulette,
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How To Prepare For The DAT Quantitative Reasoning Math Test?
Dentistry is currently an appealing career choice for many college students because it is both lucrative and rewarding. If you wish to obtain admission to a premier dental school in the USA, you must
score exceptionally well on the Dental Admission Test (DAT). In addition, since each section of the DAT is scored separately in the scorecard, the score of the Math section which is called the DAT
Quantitative Reasoning Math Test will have a significant impact on your overall score as well.
Photographer: Yusuf Belek | Source: Unsplash
DAT question papers carry nearly 15% of the question on Quantitative Reasoning or Mathematics. It assesses your proficiency in various branches of mathematics. To prepare for this section, you must
also understand how important it is to you and know all the top tips and tricks.
The purpose of this article is to assist you in successfully preparing for your Quantitative Reasoning Math Test.
Table of Contents
TL, DR
• This DAT Quantitative Reasoning Maths test assesses a candidate's maths ability and capabilities.
• The Quantitative Reasoning Test is divided into two parts - Mathematical Problems and Word Problems.
• You need to practice and enhance your math skills in order to do good in this part of the test.
• Make sure you know all the tips and tricks along with the appropriate source of study material to ace your exam.
What Is The DAT Quantitative Reasoning Math Test?
On the Quantitative Reasoning section of the DAT, candidates are tested on their math skills and abilities. Applicants are also tested on their ability to solve dental-related problems with
mathematical skills. With only one minute to answer one mathematics question per question, the Quantitative Reasoning section of the DAT is by far the most difficult.
The quantitative reasoning section includes 40 questions and you have 45 minutes to answer them. Here is what's on this section:
• Problems related to algebra (equations and expressions, inequalities, graphical analysis, and exponential notation); Data Analysis, Interpretation, and Sufficient Knowledge;
• Probability, Statistics and Quantitative Comparison
• Problems in applied mathematics (Word)
The purpose of this section is to measure a student's math skills before entry into dental school. A basic calculator is available in this section, but it does not perform the complex functions of a
scientific or graphing calculator.
The calculator enables you to calculate by clicking on the box instead of typing the numbers directly. It is advisable to use the calculator as little as possible because it will consume a lot of
your time. Practising your ability to do computations without the calculator can be useful for the DAT because you will not be able to use a calculator in the Survey of the Natural Sciences.
What Comprises The DAT Quantitative Reasoning Math Test?
In the Math portion of the test, you will be tested on your understanding of the various branches of math. You will be tested on both how well you know the knowledge and material and how well you can
apply it throughout the test.
Here are some of the different aspects you will face:
Allotted Time and the Subsections
A total of 40 questions are to be completed in 45 minutes during the Quantitative Reasoning section of the test. They are divided into the following two sub-sections:
Mathematical Problems
In this section, questions are provided from every branch of mathematics. Specifically, you will be expected to solve theoretical questions in Algebra, Geometry, Statistics, Trigonometry,
Probability, and Geometry. A total of 30 questions will be set.
Word Problems
You will need to apply your mathematics knowledge for the final 10 questions of the Quantitative Reasoning test. Those questions can come from any field of mathematics.
Scoring System of DAT Quantitative Reasoning Math Test
The Quantitative Reasoning section of the DAT is also scored from 0 to 30 on the same scale as the other sections. Based on how other students performed on the test, a percentage rank will also be
given to you.
The test contains multiple-choice questions, and incorrect answers will not result in negative marking.
Quantitative Reasoning Test Taking Techniques
The time limit for each math question is just over a minute, so it is imperative that you use test-taking strategies that will ensure that you have enough time for each question. Quantitative
Reasoning is the last section of the DAT, so students are often exhausted at that point. Therefore, students find the timing of this section to be particularly difficult to manage during the DAT.
There is an advantage in the fact that the entire section on the test is multiple-choice, allowing you to answer the questions more efficiently. For questions that you are not sure of, you can use
the answer choices as a guide to finding the correct one. Or, you can eliminate variables in the equation if it simplifies the math, similar to the method used in other sections. However, sometimes a
well-placed guess is the most effective tool for a particular problem. The pace of this section should be steady as otherwise, you will fall behind. It is better to guess and pass the more
challenging problems rather than having to do all the calculations right away.
If you have time left at the end, you can go back to the questions you marked so that you won't fall behind. Although it is important to manage your time effectively, it is equally vital not to rush
too much or panic. When you rush, you are more likely to misunderstand questions and make mistakes in calculations. Many of the incorrect answers are based on common misconceptions. You should guess
as necessary, skip some questions, and take your time on others rather than trying to get through each section in a hurry.
What are the Best Tips to Ace the DAT Quantitative Reasoning Test?
Choosing the Right Study Program
Study guides and prep books are available to help you prepare for the DAT. DAT Quantitative Reasoning textbooks from nearly all major test preparation companies are available, and short-listing the
best ones is always a challenge. A number of online courses have also been developed to help test takers prepare for the test.
The ‘DAT Quantitative Reasoning Prep 2020-2021: The Most Comprehensive Review and Ultimate Guide to the DAT Quantitative Reasoning Test’ book is a detailed and comprehensive DAT Quantitative
Reasoning prep book that lets you master all of the topics for the DAT Quantitative Reasoning course or test right from scratch.
By studying this book, your math skills will be polished, your self-confidence will be enhanced, and you will be well prepared to take the DAT quantitative reasoning test.
Changing Your Perspective on Math
Your approach to math can have an effect on how you do on the DAT Math Test. The DAT Math Test is a challenging exam, but people who enjoy math and take their time tend to pass it. This is a trait
that will help you succeed. If you can handle math, then you are likely to be successful on the DAT. Consider it a challenge and practice until you can.
Make Sure the Concepts are Understood
It is important to understand and categorize the concepts on the DAT Math Test in order to prepare for it. Surely a step-by-step approach is the best way to better understand mathematical concepts.
Start by learning basic math and then study advanced concepts from there. Studying in this manner will help you stay focused and avoid being confused.
Keep Practicing Every Day
A daily schedule and a gradual approach to studying for the DAT are the best ways to produce good results. Keep in mind to never leave the test materials until the last minute. It is necessary to
practice the material many times so that a large amount of material you must learn sticks in your mind. You should incorporate math content into your daily study schedule from the start. Although
following this program may be difficult at first, once you get the hang of it, you will be able to succeed on the test.
Identify the Best Method of Learning
There are many books for beginners that can help students who are just getting started and are unfamiliar with math for the DAT test. Preparation courses and study books are both excellent ways to
prepare for the DAT Math Test. A private tutor can also speed up your learning, but the cost is high, so you could replace them with books that go over the material.
Take Breaks
Remember that taking a break from studying can be equally important. It is important to take breaks during study periods so that your mind is fresh since you may be tired from solving and practising
math problems. The thought of taking this time for yourself will rejuvenate your mind, and any fun or relaxing activities you plan will give you something to look forward to for the rest of the week.
How to Register For the Test?
Registration for the DAT is done through the ADA's website. You must obtain a DENTPIN, Dental Personal Identifier Number, prior to registering. After that, the ADA must send you a letter of
eligibility. This letter will state that you're eligible to take the DAT test. In the event that you receive this letter, you may schedule an appointment with Prometric testing to take the test.
Schedule your test 60 to 90 days in advance of the day you want to take it.
Wrapping Up
If you are planning to sit for the DAT exam, attending DAT exam preparation courses is the most effective method for learning new material and revising old ones. Consider your budget, your learning
style, and your discipline when selecting a course. There are a lot of sources online, so look for one that can give you the information you need to be prepared for the Quantitative Reasoning Test.
To prepare for the DAT effectively, it is recommended that you supplement these courses with other materials.
Maximize your efficiency by using DATPrep materials in conjunction with other services. Combined with the other reading materials, the DATPrep program can prove to be highly effective.
So, what are you waiting for? Register now and get your DAT score up today! | {"url":"https://datprep.com/how-to-prepare-for-the-dat-quantitative-reasoning-math-test/","timestamp":"2024-11-14T12:12:06Z","content_type":"text/html","content_length":"70630","record_id":"<urn:uuid:146aebf5-90ce-4a2b-861c-1275427e4c2f>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00308.warc.gz"} |
Multiplication and Division
Multiplication and Division Skills Rotation.
Below is the prompt the teacher used to create this plan using our Ai:
I want to focus on multiplication and division
Create my own plan
Year 5 Maths Lesson Plan: Multiplication and Division
This lesson plan is designed for Year 5 students in New Zealand and focuses on reinforcing and advancing students' skills in multiplication and division. It corresponds to the New Zealand Curriculum
Level 3, targeting the achievement objectives within the Number and Algebra strand. This 45-minute session is structured to engage a class of 16 students through a combination of interactive
activities, practical applications, and group work.
Mathematical Understanding
: Students will improve their understanding of multiplication and division, learning to solve problems involving these operations more efficiently.
Problem Solving
: Students will apply their multiplication and division skills to solve real-world problems.
: Students will explain their mathematical thinking and strategies used during problem-solving.
- Whiteboard and markers
- Multiplication and division flashcards
- Group activity worksheets (can be printed from
NZ Maths
- Computers/Tablets with internet access
- Real-life objects for practical counting and division (e.g., counters, fruit, small objects)
Lesson Plan
Introduction (10 minutes)
Warm-up activity
: Start with a quick interactive game using multiplication and division flashcards to revise basic facts.
: Ask students to give examples of where they have used multiplication or division in real life. Record responses on the board, and introduce today's focus on solving problems using these operations.
Main Activity (25 minutes)
Split the class into four groups of 4 students each for a rotation through the following stations (approximately 6 minutes at each station with quick transitions):
Station 1: Real-Life Problems
: Solve a worksheet of word problems based on real-life scenarios involving multiplication and division. (Worksheet example from
NZ Maths - Multiplication and Division Context
Station 2: Interactive Online Games
: Students play selected multiplication and division games to reinforce their skills in a fun, interactive manner. Recommended resources:
NZ Maths Digital Learning Objects
Station 3: Practical Application
: Using real-life objects, students will demonstrate multiplication and division, e.g., dividing 20 apples among 5 students and then using multiplication to check their division.
Station 4: Group Problem Solving
: Each group receives a complex problem that requires multiple steps of multiplication and division to solve. They must collaboratively solve the problem and explain their reasoning.
Conclusion (10 minutes)
Group Sharing
: One representative from each group shares a summary of their problem-solving process and the solution to their complex problem from Station 4.
: Discuss any difficulties encountered during the lesson and how students overcame them.
: Assign a few additional problems from the day's worksheets for practice at home. Inform students that they can access practice problems on
NZ Maths - Practice Problems
if they wish for further review.
- Monitor group discussions and individual contributions to assess understanding and collaborative skills.
- Review completed worksheets and problems solved during stations for accuracy and understanding.
- Use end-of-lesson feedback to gauge student confidence with the material and identify any areas needing additional review.
This lesson plan aims to not only strengthen multiplication and division skills but also to contextualize these skills in everyday life, supporting the holistic development of mathematical | {"url":"https://www.kuraplan.com/plans/multiplication-and-division-skills-rotation","timestamp":"2024-11-05T08:35:36Z","content_type":"text/html","content_length":"24188","record_id":"<urn:uuid:d69d0ec6-f055-4c80-acb1-21c6be19d6dc>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00794.warc.gz"} |
Comments about "Quantum teleportation" in Wikipedia
This document contains comments about the article Quantum teleportation in Wikipedia
□ The text in italics is copied from that url
□ Immediate followed by some comments
In the last paragraph I explain my own opinion.
The article starts with the following sentence.
1. Non-technical summary
In matters relating to quantum or classical information theory, it is convenient to work with the simplest possible unit of information, the two-state system.
The meaning of the two-state system> is very different in classical physics versus quantum physics. It is simple versus complex.
In classical information, this is a bit, commonly represented using one or zero (or true or false).
That is simple.
The quantum analog of a bit is a quantum bit, or qubit. Qubits encode a type of information, called quantum information, which differs sharply from "classical" information. For example, quantum
information can be neither copied (the no-cloning theorem) nor destroyed (the no-deleting theorem).
The fact that a qubit cannot be copied or 'destroyed' are physical issues.
When you want to know the state of a quantum system consisting of qubits you have to perform a 'read' operation. In a logical sense such a 'read' operation changes the quantum state into a
classical state. In a physical sense the same 'read' operation freezes the state of each qubit in either a zero or a one.
An important aspect of quantum information theory is entanglement, which imposes statistical correlations between otherwise distinct physical systems.
That is not wrong. The sentence should be:
An important aspect of quantum information theory is entanglement, which imposes statistical correlations between physical systems, having a common origin (source).
These correlations hold even when measurements are chosen and performed independently, out of causal contact from one another, as verified in Bell test experiments.
This is a little like the chicken egg problem: What comes first the chicken or the egg.
At the start you almost know nothing except that you have an experiment, a reaction, which creates two particles (flying away in opposite directions).
The first time when you measure the spin of both particles the result is that one spin is up and the other one down. You repeat the experiment and the result is the same. You do that 1000 times:
always the same.
The only difference between all the 1000 experiments is that when you consider one side: the outcome is random
What this means is that apperently the particles are correlated. This has nothing to do how and when the measurements are performed nor with any Bell inequality violation experiment. It is
clearly a physical issue.
Understanding quantum teleportation requires a good grounding in finite-dimensional linear algebra, Hilbert spaces and projection matrixes.
Teleportation is a physical process. Understanding teleportation requires a understanding of the details of this physical process and does not require any form of mathematics.
A qubit is described using a two-dimensional complex number-valued vector space (a Hilbert space), which are the primary basis for the formal manipulations given below.
The whole issue is to what extend such a mathematical notation corresponds with the physical reality. Mathethematical notation in the sense all logical operations involved.
2 Protocol
3. Experimental results and records
4 Formal presentation
5 Alternative notations
6 Entanglement swapping
If Alice has a particle which is entangled with a particle owned by Bob, and Bob teleports it to Carol, then afterwards, Alice's particle is entangled with Carol's.
This sentence only make sense if 'entangled' is understood as correlated.
The problem is what exactly means: teleports.
A more symmetric way to describe the situation is the following: Alice has one particle, Bob two, and Carol one. Alice's particle and Bob's first particle are entangled, and so are Bob's second
and Carol's particle:
/ \
Alice-:-:-:-:-:-Bob1 -:- Bob2-:-:-:-:-:-Carol
The particles are not entangled i.e. physical linked but correlated. There is no physical connection.
7 N-state particles
8 Three particle entangled teleportation system
9 Logic gate teleportation
9.1 General description
9.2 Further details
10 Local explanation of the phenomenon
11. See also
Following is a list with "Comments in Wikipedia" about related subjects
Reflection 1
Reflection 2
Reflection 3
If you want to give a comment you can use the following form Comment form Created: 13 June 2018
Go Back to Wikipedia Comments in Wikipedia documents
Back to my home page Index | {"url":"https://nicvroom.be/wik_Quantum_teleportation.htm","timestamp":"2024-11-07T04:30:57Z","content_type":"text/html","content_length":"10596","record_id":"<urn:uuid:a4e42b8a-9bae-4aa5-966b-18836b6b7878>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00460.warc.gz"} |
A point moving with constant acceleration from A to B in the straight
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Filters Options (Pro) - Zion Builder
With filters, you can apply unique visual effects to any element, from blurring to sepia tones. They’re also useful for creating interesting changes on hover states.
The Mix Blend Mode CSS property sets how an element’s content should blend with the content of the element’s parent and the element’s background.
Grayscale converts the input image to grayscale. The value of “amount” defines the proportion of the conversion. A value of 100% is completely grayscale. A value of 0% leaves the input unchanged.
Values between 0% and 100% are linear multipliers on the effect. If the “amount” parameter is missing, a value of 100% is used. Negative values are not allowed.
Sepia converts the input image to sepia. The value of “amount” defines the proportion of the conversion. A value of 100% is completely sepia. A value of 0 leaves the input unchanged. Values between
0% and 100% are linear multipliers on the effect. If the “amount” parameter is missing, a value of 100% is used. Negative values are not allowed.
Blur applies a blur to the input image. The value in pixels on the screen blend into each other, so a larger value will create more blur. If no parameter is provided, then a value 0 is used. The
parameter is specified as a CSS length, but does not accept percentage values.
Brightness applies a linear multiplier to the input image, making it appear more or less bright. A value of 0% will create an image that is completely black. A value of 100% leaves the input
unchanged. Other values are linear multipliers on the effect. Values of an amount over 100% are allowed, providing brighter results. If the “amount” parameter is missing, a value of 100% is used.
Hue Rotate applies a hue rotation on the input image. The value of “angle” defines the number of degrees around the color circle the input samples will be adjusted. A value of 0deg leaves the input
unchanged. If the “angle” parameter is missing, a value of 0deg is used. The maximum value is 360deg.
Saturate the input image. The value of “amount” defines the proportion of the conversion. A value of 0% is completely un-saturated. A value of 100% leaves the input unchanged. Other values are linear
multipliers on the effect. Values of amount over 100% are allowed, providing super-saturated results. If the “amount” parameter is missing, a value of 100% is used. Negative values are not allowed.
Opacity applies transparency to the samples in the input image. The value of “amount” defines the proportion of the conversion. A value of 0% is completely transparent. A value of 100% leaves the
input unchanged. Values between 0% and 100% are linear multipliers on the effect. This is equivalent to multiplying the input image samples by amount. If the “amount” parameter is missing, a value of
100% is used. This function is similar to the more established opacity property; the difference is that with filters, some browsers provide hardware acceleration for better performance. Negative
values are not allowed.
Contrast adjusts the contrast of the input. A value of 0% will create an image that is completely black. A value of 100% leaves the input unchanged. Values of amount over 100% are allowed, providing
results with less contrast. If the “amount” parameter is missing, a value of 100% is used. | {"url":"https://zionbuilder.io/documentation/filters-options/","timestamp":"2024-11-09T12:29:37Z","content_type":"text/html","content_length":"87130","record_id":"<urn:uuid:2a5f1ed1-8243-4536-bf42-e8a9db684bea>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00576.warc.gz"} |
TeX4ht is a system for converting documents written in TeX/LaTeX/ConTeXt/etc. to HTML, various XML flavors, braille, etc., optionally using MathML.
• it supports most LaTeX packages and custom commands, including: BibLaTeX, TikZ, Fontspec
• it supports various input formats, apart from LaTeX: PythonTeX, RMarkdown and other formats supported by Knitr and Pandoc
• extensive support for modification of the output
• output formats include HTML 5, ODT and Docbook
• math can be exported to MathML, MathJax or pictures
Project repository and discussions
Documentation and tutorials
• make4ht - build system for TeX4ht.
• tex4ebook - ePub and ePub3 export using TeX4ht.
Basic invocation for modern output
TeX4ht can be invoked in several ways. The original way is to use the htlatex command. To convert a LaTeX source file.tex to HTML5 that uses UTF-8, with MathML:
$ htlatex file.tex "xhtml,html5,mathml,charset=utf-8" " -cunihtf -utf8"
N.B. That command line has three arguments, the second two given inside shell quotes; the last argument starts with a space. More details on calling conventions.
An easier way is to use make4ht (see its documentation for more). The following command produces the same output as the previous one, HTML5 in UTF-8 encoding with MathML:
$ make4ht file.tex "mathml"
If you want to have MathJax (rather than the browser) rasterize the MathML:
$ make4ht file.tex "mathml,mathjax"
But perhaps the best method of all is to insert LaTeX into the HTML output, and have MathJax rasterize the LaTeX:
$ make4ht file.tex "mathjax"
This has the additional advantage (thanks to MathJax) that right-clicking on any equation in the HTML brings up a menu offering to provide the source for the equation.
Bug reports
Bug reports are welcome by email or by submission to the bug database. Please include a complete source document and the exact program invocation, as well as what goes wrong. To fix the problem we
need to be able to reproduce it. If the problem remains unresolved, please submit it to the bug db, so it won't be forgotten.
Development status
We continue to install updates in the tex4ht development source and propagate them to TeX, although we have not made a full release in tex4ht. Some development changes remain solely in the source
TeX4ht was created by Eitan Gurari at Ohio State University. Eitan died unexpectedly in June 2009; we extend our sympathies to his family, and dedicate future work on the project to his memory.
With the encouragement and support of Eitan's family, Michal Hoftich, Karl Berry, and others have continued to work on TeX4ht. Involvement by other volunteers, from bug reports to major new
development, is welcome and needed.
No full post-Eitan release has been made to date. This continues to be a work in progress.
Latest changes (full ChangeLog):
tex4ht-4ht.tex ( bigfoot.4ht, usepackage.4ht, bigfoot-hooks.4ht ): added support for the Manyfoot package.
2024-11-05 https://tex.stackexchange.com/a/729975/2891
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MuPAD is a mathematical expert system for doing symbolic and exact algebraic computations as well as numerical calculations with almost arbitrary accuracy. For example, the number of significant
digits can be chosen freely. Apart from a vast variety of mathematical libraries the system provides tools for high quality visualization of 2— and 3-dimensional objects.
On Microsoft Windows, Apple Macintosh and Linux systems, MuPAD offers a flexible notebook concept for creating mathematical documents combining texts, graphics, formulas, computations and
mathematical visualizations and animations. On Microsoft Windows MuPAD further supports the technologies OLE, ActiveX Automation, DCOM, RTF and HTML. Thus it offers a natural integration in Office
applications like Word or PowerPoint as well as others.
Aims and scope
Mathematical Classification | {"url":"https://orms.mfo.de/project@terms=power+sets&id=233.html","timestamp":"2024-11-02T20:20:42Z","content_type":"application/xhtml+xml","content_length":"6769","record_id":"<urn:uuid:0a276641-3e06-4767-9522-e5365623c163>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00092.warc.gz"} |
what is the bollinger band
indicators trading
Bollinger Oscillator (BOS)
Bollinger Oscillator Overview The bollinger oscillator is an economic indicator which measures the standard deviation of the price from a simple moving average. It can be calculated using a 20-day
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5 Best Ways to Compute the Average of Each N-Length Consecutive Segment in a Python List
π ‘ Problem Formulation: Given a list of numerical values in Python, the task is to compute the average for every consecutive segment of length ‘n’. For example, given [2, 4, 6, 8, 10] with n=3, we
want to find the average for [2, 4, 6], [4, 6, 8], and [6, 8, 10], which should result in [4.0, 6.0, 8.0].
Method 1: Using Loops
This method involves iterating over the list with a for-loop, computing the average of each segment by slicing the list. It’s clear and easy to understand for most Python programmers.
Here’s an example:
lst = [1, 2, 3, 4, 5, 6]
n = 3
averages = []
for i in range(len(lst) - n + 1):
segment = lst[i:i+n]
averages.append(sum(segment) / n)
The output of this code:
[2.0, 3.0, 4.0, 5.0]
This code snippet illustrates a step-by-step averaging of n-length segments. Averages are calculated by summing each segment and dividing by ‘n’ before being appended to the results list. The method
is straightforward but may be inefficient for large lists or large values of ‘n’.
Method 2: Using List Comprehensions
List comprehension in Python provides a concise way to achieve the same result as a for-loop but in a more readable and typically faster manner.
Here’s an example:
lst = [1, 2, 3, 4, 5, 6]
n = 3
averages = [sum(lst[i:i+n]) / n for i in range(len(lst) - n + 1)]
The output of this code:
[2.0, 3.0, 4.0, 5.0]
The list comprehension method accomplishes the task in a single line of code. It’s more condense and can be more efficient than using traditional for-loops, especially in Python which favors such
idiomatic expressions.
Method 3: Using itertools.islice()
The itertools module’s islice() function can be used to perform the task efficiently, especially with large lists, because it creates an iterator that returns selected items from the input list,
reducing memory usage.
Here’s an example:
from itertools import islice
lst = [1, 2, 3, 4, 5, 6]
n = 3
averages = [sum(islice(lst, i, i+n)) / n for i in range(len(lst) - n + 1)]
The output of this code:
[2.0, 3.0, 4.0, 5.0]
This snippet uses list comprehension alongside islice() to create segments on-the-fly without copying them. This can be a more memory-efficient approach when dealing with large datasets.
Method 4: Using NumPy Library
NumPy is a powerful numerical computing library in Python. It offers the convolve() function that can be used to compute rolling averages in a very efficient way, optimized for performance.
Here’s an example:
import numpy as np
lst = np.array([1, 2, 3, 4, 5, 6])
n = 3
kernel = np.ones(n) / n
averages = np.convolve(lst, kernel, 'valid')
The output of this code:
[2. 3. 4. 5.]
Here, the np.convolve() function is used to apply a sliding window (kernel) across the list. The ‘valid’ mode ensures that only segments where the kernel fits entirely are considered. This method is
highly efficient for numerical computations.
Bonus One-Liner Method 5: Using Pandas Library
Pandas is a data manipulation library that can help perform this task using its powerful data structures and functions. The rolling() method coupled with mean() can make short work of this task.
Here’s an example:
import pandas as pd
lst = pd.Series([1, 2, 3, 4, 5, 6])
n = 3
averages = lst.rolling(n).mean().dropna().tolist()
The output of this code:
[2.0, 3.0, 4.0, 5.0]
In this concise one-liner, the rolling() method creates a rolling object over which the mean is calculated for each segment. The dropna() method is then used to remove NaN values that occur at the
start of the series where the window is not full.
• Method 1: Using Loops. Straightforward and easy to understand. May be inefficient for long lists.
• Method 2: Using List Comprehensions. Compact and Pythonic. Offers better performance than loops.
• Method 3: Using itertools.islice(). Memory-efficient for large lists. A bit more complex but useful for large-scale processing.
• Method 4: Using NumPy Library. Highly efficient for numerical computations. Requires NumPy installation and is less readable for non-scientific programmers.
• Method 5: Using Pandas Library. Very concise and powerful. Best for data analytics purposes, but overkill for simple tasks and requires Pandas installation. | {"url":"https://blog.finxter.com/5-best-ways-to-compute-the-average-of-each-n-length-consecutive-segment-in-a-python-list/","timestamp":"2024-11-07T22:12:19Z","content_type":"text/html","content_length":"71713","record_id":"<urn:uuid:56395a74-ca20-4d47-857a-064c9b62528c>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00521.warc.gz"} |
Online cubic graphing calculator
online cubic graphing calculator Related topics: how to do 5th grade number lines
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Author Message
bnosis Posted: Friday 21st of Apr 21:04
Peeps! Ok, we’re discussing online cubic graphing calculator and I was not present in my last algebra class so I have no notes and my teacher discusses lessons way bad that’s why I
didn’t get to understand it very well when I attended our math class a while ago. To make matters worse, our class will have our examination on our next meeting so I can’t afford not to
understand online cubic graphing calculator. Can someone please assist me attempt to understand how to answer couple of questions about online cubic graphing calculator so that I am
ready for the examination . I’m hoping that someone could assist me ASAP.
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Jahm Xjardx Posted: Sunday 23rd of Apr 09:12
Hello friend , online cubic graphing calculator can be really difficult if your concepts are not clear. I know this software, Algebrator which has helped a lot of newbies clarify their
concepts. I have used this software a couple of times when I was in college and I recommend it to every beginner.
Denmark, EU
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daujk_vv7 Posted: Sunday 23rd of Apr 11:27
Algebrator is used by almost every person in our class. Most of the students in my class work part-time. Our teacher introduced this program to us and we all have been using it since
From: I
dunno, I've
lost it.
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MichMoxon Posted: Sunday 23rd of Apr 16:37
I am a regular user of Algebrator. It not only helps me finish my homework faster, the detailed explanations offered makes understanding the concepts easier. I recommend using it to help
improve problem solving skills.
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brinehana Posted: Monday 24th of Apr 10:58
Thank you for your assistance . How do I get this software?
Redding, CA
Back to top
erx Posted: Tuesday 25th of Apr 08:54
Well, you don’t need to wait any longer. Go to https://softmath.com/ and get yourself a copy for a very nominal price. Good luck and happy learning!
From: PL/DE/
Back to top | {"url":"https://softmath.com/algebra-software-3/online-cubic--graphing.html","timestamp":"2024-11-14T09:04:14Z","content_type":"text/html","content_length":"42849","record_id":"<urn:uuid:3e78bd52-1717-4ee5-91f8-9d445e28ea9d>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00098.warc.gz"} |
Algorithms Weekly by Petr Mitrichev
Just like last couple of years (
), I've went through the problems I mentioned in 2019 to find the ones I liked the most. I have also looked at some of the problems recommended in
this post
, in
this post
, and in various private messages. Of course, this is still an entirely subjective exercise, and it is certainly easier for me to like a problem that I have solved or tried to solve than one that I
did not. Here is the shortlist (for those interested, here is
a slightly bigger one
), as usual in chronological order:
Which one do you think is the very best? Also, please help me fill the unknown problem authors in comments!
Since the official leaderboard for TCO20 stage 2 is not yet ready, I've put together a small script to compute it. Here's the current top 30:
│Rank│ Handle │Score│ Points │
│1 │Petr │14 │3206.22 │
│2 │tourist │13 │3309.85 │
│3 │lyrically │12 │2646.81 │
│4 │bqi343 │12 │2301.09 │
│5 │Um_nik │10 │2383.93 │
│6 │hitonanode │10 │1588.43 │
│7 │yosupo │9 │1537.25 │
│8 │_aid │9 │1506.97 │
│9 │natsugiri │9 │1485.10 │
│10 │kmjp │9 │1464.26 │
│11 │maroon_kuri │9 │1232.54 │
│12 │neal_wu │9 │1152.15 │
│13 │IH19980412 │9 │1134.90 │
│14 │ShadoWsaZ │8 │1328.25 │
│15 │KevinWan │7 │1516.60 │
│16 │ksun48 │7 │1349.06 │
│17 │Egor │7 │1149.52 │
│18 │redocpod │7 │1140.82 │
│19 │Vasyl[alphacom] │7 │1120.55 │
│20 │cerberus97 │7 │781.12 │
│21 │socketnaut │7 │552.92 │
│22 │Kalam132 │6 │1623.00 │
│23 │KKT89 │6 │1396.76 │
│24 │ecnerwal │6 │1245.65 │
│25 │darnley │6 │846.29 │
│26 │kuniavski │6 │821.25 │
│27 │square1001 │6 │788.87 │
│28 │keymoon │6 │777.42 │
│29 │nwin │6 │763.49 │
│30 │Jatana │6 │640.78 │
Enjoy! In the future I will most likely just rerun the notebook instead of making new posts, so the updated standings will appear there.
TopCoder SRM 776 was the main event of this week (problems, results, top 5 on the left, analysis). After the coding phase it seemed as if bqi343 would catch up with me in the TCO20 race, but I was
quite lucky twice: first, since my incorrect solution for the 1000 passed the system tests; second, since bqi343's 250 has failed. As a bonus, now I have learned about cyclotomic polynomials (I guess
it's more like re-learned — surely my mathematician degree should have got me covered here).
The medium problem was very nice as well. There are n=2*a+b pieces of string, out of which a have both ends red, a have both ends green, and b have one red and one green end, so we have n red ends
and n green ends in total. We will randomly pair the red and green ends in one of n! possible ways, and tie the corresponding ends together. What is the expected number of cycles we will get? a and b
are up to a million.
In my previous summary, I have mentioned a sub-problem of another TopCoder problem: for which pairs of positive integers a <= b can we split all integers from the set {a, a+1, a+2, ..., b-1, b} into
two parts with equal sum?
First of all, the sum of all numbers (a+b)*(b-a+1)/2 must be even. Since the two parts in the product (a+b)*(b-a+1) have different parity, one of the parts must be divisible by 4 for the sum to be
even. In case the size of the set (b-a+1) is divisible by 4, we can always make such a split: for each four consecutive numbers, we can split them independently as x+(x+3)=(x+1)+(x+2).
Now, what happens when (a+b) is divisible by 4? The size of the set is odd in this case, so we must split into two unequal parts, the smaller part will have at most (b-a)/2 elements, and the bigger
part at least (b-a)/2+1 elements. The sum of (b-a)/2 biggest elements in the set is equal to (b-a)/2*(b+b-(b-a)/2+1)/2=(b-a)*(3b+a+2)/8. The sum of (b-a)/2+1 smallest elements in the set is equal to
((b-a)/2+1)*(a+a+(b-a)/2)/2=(b-a+2)*(3a+b)/8. If the former is smaller than the latter, clearly there's no good split as the smaller part will always have the smaller sum.
It turns out that this condition is not just necessary but also sufficient: if we can somehow get the smaller part to have bigger or equal sum, we can make it have equal sum because we can always
repeatedly decrease the sum by 1: find two numbers x and x+1 such that x is in the bigger part and x+1 is in the smaller part, and swap them. This argument is the most beautiful part of the solution
in my opinion.
The condition (b-a)*(3b+a+2)/8>=(b-a+2)*(3a+b)/8 can be simplified as b>=a+2*sqrt(a), thus our final answer looks like:
• either b-a+1 is divisible by 4, or
• a+b is divisible by 4 and b>=a+2*sqrt(a).
Thanks for reading, and check back next week (hopefully for the best problem of 2019 vote as well)!
There were two rounds last week. TopCoder SRM 775 took place on Thursday (problems, results, top 5 on the left, analysis). Tourist has earned a commanding victory while having the fastest time on all
three problems, which also meant that nobody could get the 5 points towards the TCO20 qualification. Well done :)
The main part of the hard problem was a nice puzzle that could well appear in a mathematics olympiad: for which pairs of positive integers a <= b can we split all integers from the set {a, a+1, a+2,
..., b-1, b} into two parts with equal sum?
Codeforces Round 614 followed on Sunday (problems, results, top 5 on the left, analysis). There was just one accepted solution for each of the two hardest problems, coming from Um_nik and tourist who
have therefore occupied the first two places with a huge margin. Um_nik's problem was worth more points, and he had therefore won the round. Congratulations!
Thanks for reading, and check back next week!
Two weeks ago, TopCoder SRM 774 has started a new race for a TCO20 spot (problems, results, top 5 on the left). The 1000-pointer was tricky both algorithmically and from the implementation
standpoint, causing a few resubmissions and a few failed systests for high-scoring solutions. As a result, the total scores were not that high and the importance of the challenge phase was amplified.
In my previous summary, I have mentioned a Codeforces problem: you are given a 20x20 grid colored as a chessboard, with the top-left corner colored black. Some of the cells are removed from the grid,
the remaining cells form a 4-connected piece and include the top-left corner. You need to insert some walls between the remaining cells in such a way that we get a good labyrinth: there must be
exactly one way to get from each cell to each other cell. Moreover, we want each black cell (remember the chessboard coloring) except the top-left cell to not be a dead-end in the labyrinth: each
such cell must have at least two accessible neighbors.
When solving this problem during the round, I have made the correct first step: we want to find a spanning tree where the degree of each black cell is at least two, which is equivalent to finding a
spanning forest where the degree of each black cell is at least two as we can always add more edges to get a tree.
But then I've tried to find some greedy approach that takes two edges for each black vertex without forming cycles, realized that it's not always possible, thought that if we want to add an edge that
forms a cycle we need to remove some other edge from this cycle and choose another edge for its black vertex, and then somehow failed to notice that I'm just describing finding an alternating path
for a matroid intersection problem :) The matroids in question are the cycle matroid and the matroid where independent sets have degree <=2 for each black vertex, and we need to check if the biggest
independent set in their intersection has degree of exactly 2 for each black vertex.
In that summary, I have also described a solution to an AtCoder problem that felt like unexplained magic. Um_nik has brought a simpler quadratic solution to my attention: instead of starting from n
scores of 1 and adding 1 to a suffix n-1 times, we can start from n scores of n and subtract 1 from a prefix any number of times! The reason we don't need to limit the number of operations to n-1 in
this approach is that if the first problem has zero or negative score, then the constraint about the sum of the first k+1 numbers being greater than the sum of the last k numbers would be necessarily
This means we can remove the number of operations dimension from our dynamic programming and it becomes quadratic. This is not directly equivalent to the magical solution, but at least it explains
why there's a fertile ground for one.
I have also promised to organize a poll about the best problem of 2019, but for that I need to review all my posts from last year and also the other excellent candidates you shared with me on
Codeforces, so this will take some more time. Stay tuned :)
Thanks for reading, and check back for more!
Codeforces ran two contests this week. Hello 2020, as the name suggests, was the first round of the year (problems, results, top 5 on the left, my screencast, analysis). Only four contestants could
solve the hardest problem G, and only two of them also solved the remaining problems: mnbvmar and TLE. They had roughly the same speed as well, but mnbvmar only had two attempts that failed pretests
compared to TLE's ten, and that's what made the difference. Congratulations to both!
I could solve the first five problems reasonably quickly, and I was quite excited about inventing the randomized solution to problem D and quickly recognizing that problem E is more or less
equivalent to a very old problem about counting the number of 4-tuples of points that form a convex quadrilateral (I have a feeling that I wrote about it in this blog, but I seem to be unable to find
the entry). However, the last two problems proved insurmountable for me, and I spent most of the time trying to get solutions that were clearly not the intended ones to work: max-flow on a graph of
size n*log(n) in F (it turns out it was possible to succeed in this way — check out izban's solution as an example), and repeated randomized search in G. I guess the time might have been better spent
just thinking on paper, but then the screencast would not be so exciting :)
On a related note, quite a few people have noticed that I've switched to C++ in the recent contests, and asked why. I don't have much to add to this Egor's comment. In the past I have tried switching
to C++ a few times and noticed that I keep fighting with it during the contests instead of solving problems, and I do have a similar feeling now as well despite the better tools. However, I will try
to keep using C++ for a longer time to see if things improve :)
Here is the hardest problem from this round for you to try as well: you are given a 20x20 grid colored as a chessboard, with the top-left corner colored black. Some of the cells are removed from the
grid, the remaining cells form a 4-connected piece and include the top-left corner. You need to insert some walls between the remaining cells in such a way that we get a good labyrinth: there must be
exactly one way to get from each cell to each other cell. Moreover, we want each black cell (remember the chessboard coloring) except the top-left cell to not be a dead-end in the labyrinth: each
such cell must have at least two accessible neighbors.
Codeforces Round 612 followed a day later (problems, results, top 5 on the left). The sets of problems solved by the top contestants were very diverse (even though not visible in the top 5
screenshot, problem F was also solved by two contestants), but in the end ainta just solved more problems and won. Well done!
In my previous summary, I have mentioned a couple of problems. The first one came from AtCoder: an assignment of integer scores between 1 and n (not necessarily distinct) to n programming contest
problems (n<=5000) is called good if for each k the total score of every set of k problems is strictly less than the total score of every set of k+1 problems. How many good assignments of scores
exist, modulo the given prime m?
The first step to solve this problem is to notice that we can get rid of "for each k" qualifier, replacing it with just k=(n-1)/2, rounded down. The reason for this is that in case the constraint is
violated for smaller k, we can just add the same problems to both sides to reach k=(n-1)/2, and in case it is violated for larger k, the two sets necessarily have an intersection, so we can remove
the intersection until we reach k=(n-1)/2 as well. Also, instead of "every set" we can say: the total score of k problems with the largest scores must be less than the total score of k+1 problems
with the lowest scores.
To enforce that last constraint, it would be useful if our problem scores were sorted. This can be achieved with the following more or less standard process: we start with all problem scores equal to
1. Now we do the following n-1 times: add 1 to a suffix of problem scores (this suffix could also be empty or all problems).
Now we can keep track how each such operation affects the value we're interested in: the sum of (n-1)/2+1 smallest elements minus the sum of (n-1)/2 largest elements. Going from the empty suffix to
the full suffix, they change that value by 0,-1,-2,...,-(n-1)/2,-(n-1)/2 (if n is even),-((n-1)/2-1),...,-2,-1,0,1. Let's denote that multiset of changes as C. In the end we need the value to be
positive, and it starts with 1 (when all problem scores are 1).
Our problem can then be restated as follows: consider all ways to choose n-1 values with replacement from the multiset C (the same value can be chosen as many times as we want). How many ways have a
non-negative sum?
Since the sum needs to be non-negative and the only possible positive change is 1, this yields a O(n^3) dynamic programming solution that can be sped up to O(n^2*logn) and get accepted by skipping
the states from which we can't reach the final state: let's process the elements of C in decreasing order, and maintain dp[i,j,k] as the number of ways to choose j values from the first i elements of
C such that their sum is k. The answer is the sum of dp[n+1,n-1,k] over all k>=0.
However, there exists a magical way to speed this up to O(n^2). Let's rearrange our dynamic programming slightly: now we will process the elements of C in increasing order, and dp[i,j,k] will now be
the number of ways to choose j values from the first i elements of C such that their sum is -k. Also, we will stop before we process the only positive element 1 (because that would need special
handling anyway as the sums stop being only negative), but also (!) before we process one of the two zeros that we have — in other words, we will only consider the first n-1 elements of C in
increasing order.
How to compute the answer from the last row of this dynamic programming? Suppose we have computed dp[n-1,j,k]. Now we need to add some number of 0s and some number of 1s, let's denote those as z and
o respectively. We need to have j+z+o=n-1 to get n-1 total changes, and k<=o to get a non-negative sum. The solutions to these two constraints look like: o=k, z=n-1-j-k; o=k+1, z=n-1-j-k-1, and so on
until z reaches 0. The number of solutions is thus max(0, n-j-k), so our answer is a sum of max(0, n-j-k)*dp[n-1,j,k].
Here comes the magic: since max(0, n-j-k) only depends on j+k and not on j and k separately, and since our transitions just add one to j and the current element of C to k, and so the thing being
added does not depend on the values of j and k themselves, we can collapse our dynamic programming states to keep track of j+k only, instead of j and k separately! This means we'll have O(n^2) states
and O(n^2) complexity.
What remains a mystery is: how does one come up with this magic? I guess one could just stumble upon it while trying different approaches. Maybe a more principled way is to use the approach from my
old post: if we just implement the O(n^3) dynamic programming which processes the elements of C in increasing order, and find out the contribution of each state to the final answer, we can notice
that the contributions of states with the same value of j+k are the same and collapse them. Is there any other way that makes this observation look less magical?
The second problem I mentioned was from Codeforces: you are given n integers a[i] such that i-n<=a[i]<=i-1 (i goes from 1 to n), in other words one integer from [-(n-1),0], one from [-(n-2),1], ...,
one from [0,n-1]. You need to find any nonempty subset with a zero sum. You are guaranteed that such subset always exists, which is by itself quite a hint. n is up to a million.
The key idea here is to realize that keeping integers from different segments is a bit clumsy, so let's shift the segments: the first one by n-1, the second one by n-2, and so on. Now all integers
are chosen from the segment [0, n-1] which is nice and symmetric, but instead of a zero-sum subset we need to find a subset where the sum of values equals to the sum of shifts.
To restate, we have reduced our problem to the following: you are given n integers a[0], a[1], ..., a[n-1], each between 0 and n-1. You need to find a set S of indices such that Σ[i∈S]i=Σ[i∈S]a[i].
When I obtained this reduced problem during the round, it felt really familiar to me, so I've tried to google the answer without much success. It turns out that it's simply quite easy: build a graph
from the arrows i->a[i], and just find any cycle in this graph. The indices corresponding to this cycle will satisfy the equality above since the sums will just be the same numbers but in different
I will be doing a poll for the best problem of 2019 soon, and I will mostly be picking the candidates from the problems I explained in this blog. However, I realize that there were many great
problems in 2019 that I just did not encounter, so if there is a problem you feel should be included in the shortlist that was in a contest that I did not participate in, please mention it in
comments! Feel free to also post links to similar discussions, such as this post.
Thanks for reading, and see you next week!
Last week has wrapped up the competitive 2019 with two rounds. AtCoder Grand Contest 041 took place on Saturday (problems, results, top 5 on the left, analysis). mnbvmar, ecnerwal and Um_nik in first
three places all have a different set of problems solved, and mnbvmar's set was the best one. He also tried to submit a solution that just tries random decisions until time runs out for E in the last
minute, but that did not fly. Nevertheless, congratulations on the win!
I was writing this round on the go, and around the middle of the round my laptop shut down because of low battery, which was very exciting as you might guess :) After I've turned it back on it
continued to work for a long time somehow, suggesting that the problem lies with the battery level detection. A few minutes before the end of the round it shut down again, so I've tried to fix my
solution to problem A from my phone. Unfortunately I was a few seconds too slow, otherwise it would have been a nice achievement :)
Problem D in this round had an awesome intended solution, even though most contestants managed to squeeze in a more boring one: an assignment of integer scores between 1 and n (not necessarily
distinct) to n programming contest problems (n<=5000) is called good if for each k the total score of every set of k problems is strictly less than the total score of every set of k+1 problems. How
many good assignments of scores exist, modulo the given prime m?
This round has concluded the selection of 8 AtCoder World Tour finalists that would come to Japan in February for the onsite finals (results, top 8 on the right). With this win, mnbvmar has jumped
onto a departing train (there is such idiom in Russian, вскочить на подножку уходящего поезда, but I'm not sure if there is a direct English equivalent or a different idiom with the same meaning — is
there?) and overtook eatmore by just 6 points. See you all in Japan!
Codeforces Good Bye 2019 followed on Sunday (problems, results, top 5 on the left, my screencast, analysis). Not without help from a notorious coincidence and a bad day for tourist, Radewoosh has
solved everything, won the round, ended the 2019 top rated, and was still a bit salty. Still, congratulations!
Problem G generated some conflicting opinions, but I have enjoyed solving it quite a bit: you are given n integers a[i] such that i-n<=a[i]<=i-1 (i goes from 1 to n), in other words one integer from
[-(n-1),0], one from [-(n-2),1], ..., one from [0,n-1]. You need to find any nonempty subset with a zero sum. You are guaranteed that such subset always exists, which is by itself quite a hint. n is
up to a million.
Thanks for reading, and see you in the present!
Codeforces Global Rounds series of 2019 has concluded during the Dec 16 - Dec 22 week with the Round 6 (problems, results, top 5 on the left, analysis). With problem F being tricky to implement but
quite standard, problem G requiring to notice a complex pattern after implementing an ineffective solution, and problem H relying on some cool maths coming almost out of nowhere, there was plenty of
ways for top contestants to pick their poison. Only 1919810 and sunset could solve two of those three, and 1919810 got all easier problems correct as well thus securing a confident victory. Well
The choice of counting the 4 best performances out of 6 for the overall standings (top 5 on the right) seemed to turn out quite nicely, allowing contestants to skip some rounds and/or to enjoy a
really bad day sometimes, although maybe Radewoosh would prefer 2 out of 6 :) Thanks to Codeforces and to XTX Markets for organizing the series!
TopCoder Open 2020 points-based qualification stage 1 also concluded that week, with the SRM 773 (problems, results, top 5 on the left, my screencast, TCO20 qualification standings). There was no
stopping tourist this time, with yet another dominating coding phase performance and another 175 challenge points he was simply out of reach and thus finally earned the 5 qualification points he
needed to qualify for TCO20. I do not have my TopCoder stats scripts working anymore to support this claim with data, but it feels like he has really stepped up the challenge game a lot recently.
Huge congratulations!
Open Cup 2019-20 Grand Prix of Xi'An on Sunday sent two problems to the New Year Prime contest (results, top 5 on the left). Having penalty time almost twice as big as that of the second-placed NNSU
Almost Retired, team USA1 had no other way to victory except solving more problems, which they delivered by becoming the only team to solve L in the last hour, and then finally getting F from the
24-th attempt. Congratulations!
With this win, team USA1 has almost caught up with team Past Glory in the overall standings after 2019 (top 5 on the right), promising us an exciting second half of the season in 2020!
In my previous summary, I have mentioned a TopCoder problem: you are given 100000 points on the plane and a number k. Given a subset S of the set of given points, and denoting all other given points
as its complement T, we define the score of S as the sum of pairwise Manhattan distances between the points in S minus the sum of pairwise Manhattan distances between the points in T. What is the
largest (by the number of elements) subset of the set of given points such that its score does not exceed the given number k?
The key insight in this problem is to consider how does the score of a set change when we add one more point to it. The distances from this point to the points already in S are added to the score;
the distances from this point to the points not in S are also added to the score as they used to be subtracted from it and now they are not! Therefore adding a point to S simply increases the score
of S by the sum of distances from this point to all other points.
Now it is clear that we should just start with empty S and add points in order by this sum of distances until we exceed k.
Thank for reading, and check back for more!
TopCoder returned with its SRM 772 during the Dec 9 - Dec 15 week (problems, results, top 5 on the left, my screencast). Once again tourist had to settle for the second place despite great coding and
challenge phases because ksun48 could solve the hardest problem. We were discussing during the TopCoder Open onsite that ecnerwal seems to just think in generating functions, and apparently so does
ksun48, even though he tried to hide the fact by explaining his combinatorial-ish solution :) Congratulations on the win!
I was comfortably in the top 10 this time, meaning that both me and tourist each got 4 TCO20 qualification points again and I kept my 1-point lead going into the final SRM of the stage.
The medium problem in this round was very nice. You are given 100000 points on the plane and a number k. Given a subset S of the set of given points, and denoting all other given points as its
complement T, we define the score of S as the sum of pairwise Manhattan distances between the points in S minus the sum of pairwise Manhattan distances between the points in T. What is the largest
(by the number of elements) subset of the set of given points such that its score does not exceed the given number k?
Codeforces then hosted two rounds on the weekend. Round 606 took place on Saturday (problems, results, top 5 on the left, analysis). With a whole 1 accepted solution for both E and F combined, the
round was effectively decided on the first four problems. Ainta was the fastest and won the round, well done!
Round 607 followed on Sunday (problems, results, top 5 on the left, analysis). This round got a quite different top 5, with only Radewoosh appearing in both. ksun48 got most of his speed advantage
over Um_nik on the easier three problems, and then kept the lead by solving D and E in a comparable amount of time. Congratulations on the win!
Thanks for reading, and check back for more!
Codeforces Round 604 took place during the Dec 2 - Dec 8 week (problems, results, top 5 on the left, analysis). MiFaFaOvO has won the round, taking some advantage from the fact that problem E has
appeared before, but still solving F in a bit more than half an hour while his competitors could not do that in an hour. Well done!
Open Cup 2019-20 Grand Prix of Beijing wrapped up the week on Sunday (results, top 5 on the left). Team Past Glory has continued to break away in the overall standings, winning the round and solving
all problems to boot (they would still have won even without solving D at the end of the contest). Congratulations on the win!
In my previous summary, I have mentioned my NEF problem: there are 2n elements (n>=3). We can compare any two elements, and there are no equal ones, so one will compare bigger than the other. Our
goal is to make such set of comparisons that uniquely determines the n biggest elements out of 2n, but not the ordering of those n elements. In other words, there must be at least two possible
orderings of those n elements that are consistent with the outcomes of comparisons.
For large values of n, since the n biggest elements can be determined in O(n) and sorting them requires O(n*logn), just applying any O(n) algorithm will do as we will not do enough comparisons to
determine the order, and some randomized approaches also work. The real problem lies in tackling small values of n, roughly between 3 and 7.
One could imagine that for such small inputs we could do some kind of exhaustive search, but it turns out that already for n=6 the state space is enormous as we have 12! possible inputs. Therefore,
we need to come up with an actual algorithm :)
Initially I did implement the exhaustive search to find a solution for n=3, and then came up with a way to obtain a solution for n+1 from a solution for n. However, together with Pavel Kunyavskiy we
could come up with a simpler approach.
Let us take arbitrary n+1 elements and split them arbitrarily into two groups of size at least 2, for example of size 2 and n-1. Now let's find the smallest element in each group in any possible way
(using only comparisons within the group), and then compare those two elements between themselves. The one which compares smaller is the smallest among the n+1 chosen elements, and therefore is not
among the n biggest elements, so we can discard it from consideration.
Now let's add one more element to one of the two groups in such a way that both have size at least 2, and repeat the step above, discarding one more element from consideration. We repeat this until
there are no more elements to add (discarding n elements in total).
In the end we're left with the n biggest elements split into two groups, and we have never compared any element from the first group to any element of the second group, therefore there are at least
two (more precisely, at least three) possible orderings of those n elements.
Thanks for reading, and check back for more!
With TCO19 over, the qualification for TCO20 is well underway, and TopCoder SRM 771 was a part of that during the Nov 25 - Dec 1 week (problems, results, top 5 on the left, my screencast, analysis).
Tourist did all he can to bounce back from a resubmit on the 1000-pointer, found 175 challenge points but was just 2 points short to overtake majk with his wonderful 800+-point solution :)
Congratulations to both!
I was implementing the solution described in this comment, but an internal assertion kept failing on one of the samples. With just seconds left in the round I've removed the assertion and submitted,
but of course it still failed the system tests. It turns out the idea was incorrect as well, but the system tests would not have caught that, so one could say I was close :) Luckily for me, many
others have failed the system tests and I barely climbed into the top 10, so both myself and tourist gathered 4 points for the TCO20 qualification standings.
On the ICPC side, the Northern Eurasia Finals took place in St. Petersburg, Barnaul, Tbilisi and Almaty on Sunday (problems, results, top 5 on the left, broadcast, online mirror results, analysis).
The strong favorite team NNSU Almost Retired did well but team SPbSU 25 was a bit faster and won, both teams at 10 problems while others got at most 9. Congratulations to both teams!
In the online mirror, three teams placed above them but as I understand none of those are active/eligible for ICPC. Team Cafe Mountain in the 7th place actually is a current ICPC team competing for
Seoul NU (please correct me if I'm wrong!)
I have set the interactive problem I for this round, which went like this: there are 2n elements (n>=3). We can compare any two elements, and there are no equal ones, so one will compare bigger than
the other. Our goal is to make such set of comparisons that uniquely determines the n biggest elements out of 2n, but not the ordering of those n elements. In other words, there must be at least two
possible orderings of those n elements that are consistent with the outcomes of comparisons. Can you come up with an algorithm with a requirement that it does not do something? :)
Note that this would not be possible for n=2, as it might happen that the first two elements we try to compare are the two biggest elements, and thus we would learn a unique order for them.
Thanks for reading, and check back for more!
Codeforces hosted two rounds during the Nov 18 - Nov 24 week. Round 601 took place on Tuesday (problems, results, top 5 on the left, analysis). Five contestants could solve all problems, and
Radewoosh was quite a bit faster than others on problems C and D which gave him the victory. Well done!
Round 602 followed on Sunday (problems, results, top 5 on the left, analysis). This time even more competitors solved everything, and this time it was sunset who was slightly faster than others (some
of whom have made great sacrifices to compete). Congratulations on the win!
In my previous summary, I have mentioned a TopCoder problem: you are given three integers n (1<=n<=10^11), d (40<=d<=120) and b (5<=b<=62). Your goal is to return any base-b number that has exactly d
digits when written without leading zeros, is divisible by n, and has at most four distinct digits when written in base-b without leading zeros.
I have found the correct idea almost immediately after reading the problem: we need to somehow make use of the birthday paradox, which is closely related to meet-in-the-middle algorithm. Roughly
speaking, if we generate random numbers, then after generating O(sqrt(n)) numbers we will have generated two that have the same remainder modulo n.
However, it is important to find the right way to apply it. My approach was to split the number in two parts, keep generating random numbers with the same four distinct digits for the first and
second part, and wait until we generate two that add up to 0 modulo n. This is not an exact application of the birthday paradox as we have two different random distributions rather than one,
therefore it no longer guarantees that we're going to find such pair after generating just O(sqrt(n)) numbers. However, there is a hand-wavy argument: remainders modulo n of such large numbers are
more or less random both for the first and for the second part, therefore the birthday paradox should still apply.
Unfortunately, there are cases where the distribution is not random at all. For example, when b=10 and n=10^11, all digits before the 11-th one from the end do not affect the remainder modulo n, and
therefore all randomly generated first parts will have remainder 0, therefore we'll need on the order of 10^11 randomly generated second parts in order to find a match, instead of sqrt(10^11). This
case is easy to handle separately of course, as we can just include "all zeros" as one of the candidates for the second part. However, there are more tricky cases where gcd(n,b)>1, or when gcd(n,b
-1)>1. I have managed to find a way to make this solution pass the system tests, but it was a very dangerous thing to do and the solution could have easily failed.
The right idea is to find a way to apply the birthday paradox directly. Let's generate random numbers of the full length until we find two that give the same remainder modulo n, this is bound to
happen within O(sqrt(10^11)) steps. The difference between these two is divisible by n, but there are two issues:
• the difference might have less than d digits
• even if each of the numbers has only four distinct digits, the difference might have more
It turns out that both issues can be taken care of if we generate random numbers consisting only of digits 0 and 1 in our process. The difference of any two such numbers will have only digits 0, 1, b
-2 and b-1, satisfying the at most four distinct digits constraint. And in case the difference has less than d digits, we can just append enough zeros at the end since zero is one of the allowed
digits, and appending zeros keeps the number divisible by n.
This solution provably terminates within O(sqrt(10^11)) steps with very high probability, and one can submit it with confidence. Note that the fact that we generate numbers consisting only of digits
0 and 1, and therefore their remainders modulo n might not be uniformly distributed, does not hurt us: if the choices are not uniformly distributed, the birthday paradox collision just becomes more
Thanks for reading, and check back for more! | {"url":"https://blog.mitrichev.ch/2020/01/","timestamp":"2024-11-02T19:06:18Z","content_type":"text/html","content_length":"195233","record_id":"<urn:uuid:7dd4ba65-dc20-41ff-96cd-1e8f7abe8087>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00424.warc.gz"} |
NCERT Class 1 Maths Book PDF | CBSE Class 1 NCERT Maths Books Free Download
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Find vert and horz asymptotes, and points of
Step 1
To use :
Vertical asymptotes :
The graph of $y=f\left(x\right)$ have vertical asymptotes at those value of x for which the denominator is equal to zero.
Horizontal asymptotes:
If the degree (the largest exponent ) of the denominator is bigger than the degree of the numerator , the horizontal asymptotes is the $x-axis\left(y=0\right).$
If the degree of the numerator is bigger than the denominator ,there is no horizontal asymptotes.
If the degree of the numerator and denominator are same , the horizontal asymptotes equals the leading coefficient (the coefficient of the largest exponent ) of the numerator divided by the leading
coefficient of the denominator.
Inflection point :
An inflection point is a point on the graph at which the the second derivative changes sign.
if ${f}^{″}\left(x\right)\text{ }>\text{ }0$ then f (x) is concave upwards.
if ${f}^{″}\left(x\right)\text{ }<\text{ }0$ then f (x) is concave downwards.
Step 2
To find Vertical asymptotes:
Use the definition of vertical asymptotes.
Set the denominator of the given function is zero.
Denominator of the given function is x.
To set the denominator is equal to zero.
To get,
Then, the vertical asymptotes of the given function $f\left(x\right)=\frac{\mathrm{ln}\left(x\right)}{x}\text{ }is\text{ }x=0.$
Step 3
To find the horizontal asymptotes:
To use the above definition of the horizontal asymptotes.
Let $f\left(x\right)=\frac{\mathrm{ln}\left(x\right)}{x}$
Here, the degree(the largest exponent) of the denominator is bigger than the degree of the numerator.To get, the horizontal asymptotes is $y=0$
The horizontal asymptotes of the function $f\left(x\right)=\frac{\mathrm{ln}\left(x\right)}{x}\text{ }is\text{ }y=0.$
Step 4
To find the inflection point:
To find the first and second derivative of the given function.
To use the quotient rule.
$\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\text{ }\frac{du}{dx}\text{ }-\text{ }u\text{ }\frac{dv}{dx}}{{v}^{2}}$
$\frac{d}{dx}\text{ }\left(\frac{\mathrm{ln}\left(x\right)}{x}\right)=\frac{x\text{ }\frac{d}{dx}\mathrm{ln}\left(x\right)\text{ }-\text{ }\mathrm{ln}\left(x\right)\frac{d}{dx}x}{{x}^{2}}$
${f}^{\prime }\left(x\right)=\frac{1\text{ }-\text{ }\mathrm{ln}\left(x\right)}{{x}^{2}}$
Diffrentiate the ${f}^{\prime }\left(x\right)$ again withrespect to x, to get,
$\frac{d}{dx}\left(\frac{1\text{ }-\text{ }\mathrm{ln}\left(x\right)}{{x}^{2}}\right)=\frac{{x}^{2}\text{ }\frac{d}{dx}\left(1\text{ }-\text{ }\mathrm{ln}\left(x\right)\right)\text{ }-\text{ }\left(1
\text{ }-\text{ }\mathrm{ln}\left(x\right)\right)\text{ }\frac{d}{dx}{x}^{2}}{{x}^{4}}$
${f}^{\prime }\left(x\right)=\frac{-2\text{ }\mathrm{ln}\left(x\right)\text{ }+\text{ }3}{{x}^{3}}$
To find the inflection point:
To find where the ${f}^{″}\left(x\right)$ is equal to zero or undefined.
To get,
$x={e}^{\frac{3}{2}},\text{ }x=0.$
Next, to identify the inflection point not in the domain or where f (x) is not continuous.
Also, Domain of the function $f\left(x\right)=\frac{\mathrm{ln}\left(x\right)}{x}\text{ }is\text{ }x\text{ }>\text{ }0$
To get, f(x) is not dfined at $x=0$ and which is not in the domain of the given function.
To get,
$x={e}^{\frac{3}{2}}$ is the inflection point of the given funtion.
Next, to check the sign of ${f}^{″}\left(x\right)$ in the domain of the given funtion.
To use the intervals: $0\text{ }<\text{ }x\text{ }<\text{ }{e}^{\frac{3}{2}},\text{ }{e}^{3}2\text{ }<\text{ }x<\text{ }\mathrm{\infty }.$
Let ${f}^{″}\left(x\right)=\frac{-2\mathrm{ln}\left(x\right)\text{ }+\text{ }3}{{x}^{3}}$
a) $0\text{ }<\text{ }x\text{ }<\text{ }{e}^{\frac{3}{2}}$
Sign: -
Behavior: Concave downward
b) $x={e}^{\frac{3}{2}}$
Sign: 0
Behavior: Inflection point
c)${e}^{\frac{3}{2}}\text{ }<\text{ }x\text{ }<\text{ }\mathrm{\infty }$
Sign: -
Behavior: Concave upward
Step 5
Next, plug the inflection point $x={e}^{\frac{3}{2}}$ into the given funcion.
To get,
To get, | {"url":"https://plainmath.org/college-statistics/2055-find-vert-and-horz-asymptotes-and-points-inflection-equal-graph-function","timestamp":"2024-11-04T18:54:01Z","content_type":"text/html","content_length":"293906","record_id":"<urn:uuid:7e9823d8-e65a-48e4-8a05-f54fceed2e0a>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00344.warc.gz"} |
• 1. The fact or condition of being relative, relativeness.
• 2. The quantitative dependence of observations on the relative motion of the observer and the observed object; that branch of physics which is concerned with the description of space and time
allowing for this dependence. The modern theory of relativity, developed largely by Albert Einstein (1879-1955), is an extension and generalization of the corresponding principles in classical,
or Newtonian, mechanics.
The principle of relativity, in its restricted form, is the postulate that the laws of nature have the same form in all inertial reference frames; in its more general form, it states that the
laws of nature, when expressed in a suitable (‘co-variant’) form, have the same form in all reference frames, whether inertial or not.
The special theory of relativity (1905), based on the restricted principle of relativity and the hypothesis of the constancy of the speed of light in vacuo as seen by observers in any inertial
frames, resulted in a theoretical framework for the unification of space and time in a four-dimensional continuum and for the equivalence of mass and energy, and showed how the uniform relative
motion of observers affects their measures of length and time.
The general theory of relativity (1915), essentially a theory of gravitation, is based on the general principle of relativity, the postulated equivalence of inertial and gravitational mass, and
the assumption that the results of the special theory must be valid in the limiting case of zero gravitational potential; it leads to a new set of equations of motion and the result that
space-time is curved by the presence of gravitational fields.
• 3. The relative grading of posts or salaries, usu. considered within one business (internal) or in comparison with others (external). Freq. pl.
For lessons on the topic of Relativity, follow this link.
From 1905 to 1915 Albert Einstein revolutionized the conception of space and time and gravity that had been central in physics since Isaac Newton.
The special and general theories of relativity are, at heart, theories of spatiotemporal structure. They are not particularly about observers or reference frames or ways to synchronize clocks,
although as fundamental physical theories they have implications about what observers will observe and what various physical procedures for coordinating clocks will accomplish. It is easy to fall
under the impression that these theories are basically concerned about coordinate systems or reference frames because physical events are typically described by means of coordinates or reference
frames, but that temptation ought to be avoided.
Perhaps the easiest way to understand special relativity is by analogy to Euclidean geometry. Euclidean geometry postulates a particular spatial structure and, beginning with the Euclid's Elements,
the implications of that structure for geometrical figures were studied by purely geometrical methods. For two millennia, the study of Euclidean geometry made no use of coordinate systems or of
numbers. The introduction of Cartesian coordinates allowed for the translation of geometrical objects into algebraic ones by means of assigning numbers as coordinates to points. There are all sorts
of ways to lay down coordinates on a Euclidean space, such as polar coordinates or spherical coordinates, but the most familiar is the system of Cartesian coordinates. Cartesian coordinates are
rectilinear and orthogonal; the coordinate curves are straight lines that intersect at right angles. Because of this feature, distances between points in a Euclidean space are easy to calculate from
their Cartesian cooridnates: If point p has coordinates (xp, yp, zp) and point q has coordinates (xq, yq, zq), then the distance from p to q is:
In most spaces, such as the surface of a sphere, Cartesian coordinates do not exist. It turns out that for a space to be Euclidean is just for the space to admit of Cartesian coordinates. That is,
the distances between points in the space must be of just the right form for Cartesian coordinates to exist.
In order to grasp relativity, we have to think not of distances between points in a three-dimensional space, but of a fundamentally spatiotemporal distance between points in a four-dimensional
space-time. Points in the space-time correspond to instantaneous, localized events, such as the bursting of a bubble when it reaches the surface of a glass of champagne. Such events occur both at a
place and at a time. To locate these events, we typically ascribe to them four numbers, such as a latitude, longitude, altitude, and time. It is in this uncontroversial sense that the space-time of
classical physics and of relativity is four-dimensional.
What sorts of spatiotemporal relations are there between events? All of classical physics agreed on at least one point: There is a definite, objective, purely temporal relation between the events.
Two events either take place at the same time, or one takes place a certain amount of time before the other. So the notion of there being a lapse of time between events, and the specific case of
simultaneity of events, is inherent in the classical account of space-time structure.
The classical account of spatial structure is not so straightforward. Newton believed that a single three-dimensional Euclidean space persists though time, and that every event, whenever it occurs,
takes place somewhere in that absolute space. So Newton thought that any pair of events, no matter whether they occur at the same or different times, have some spatial distance separating them. But
consider the following case: On a train traveling along the tracks, there sits a glass of champagne. A bubble rises to the surface and pops, followed a minute later by a second bubble. How far was
the first popping from the second?
According to a passenger on the train, the two events took place in close spatial proximity, within a few inches of each other. But according to a spectator watching the train go by, these two events
would be considered yards apart because the train has moved in the intervening minute. Newton would insist that there is a true spatial distance between the events, even though no observation could
reveal for certain whether the passenger or the spectator (or neither) is right. But a natural reaction is to reject the whole question: There may be definite spatial relations between simultaneous
events, but there is no fact at all about the spatial distance between nonsimultaneous events. Thus we arrive at two classical space-time structures: Newtonian space-time, with temporal and spatial
relations between every pair of events, and Galilean (or neo-Newtonian) space-time, with temporal relations between all events and spatial relations only between simultaneous events (Galilean
space-time then needs to add a new spatiotemporal structure, called an affine connection, to distinguish inertial from non-inertial trajectories). Note that the classical accounts agree on the
temporal structure, and particularly on the objective physical relation of simultaneity.
Special relativity postulates a four-dimensional space-time with a radically different spatiotemporal structure. Instead of having a pure temporal structure and a pure spatial structure, there is a
single relativistic "distance" between events (the scare quotes around distance must be taken seriously, as the quantity is not at all like a spatial distance). How can this spatiotemporal structure
be specified?
The easiest method, albeit a bit roundabout, is by means of coordinates. Here we will take the analogy with Euclidean geometry quite seriously. As we saw, even though Euclidean geometry has no need
of coordinate systems, the spatial structure of a Euclidean space can still be specified in this way; a Euclidean space is a space that admits of Cartesian coordinates. More specifically, a
three-dimensional Euclidean space has a structure of distance relations among its points such that each point can be given coordinates (x,y,z) and the distance between any pair of points is:
In exactly the same way, we can specify the spatiotemporal structure of Minkowski space-time, the space-time of special relativity. Minkowski space-time is a four-dimensional manifold that admits of
Lorentz coordinates (or Lorentz frames). A Lorentz frame is a system of coordinates (t, x, y, z) such that the relativistic spatiotemporal distance between any pair of events p and q is:
Written this way, the similarity with the example of Cartesian coordinates on Euclidean space is manifest; the only difference is the minus signs in place of plus signs. The consequences of that
small mathematical difference are profound.
Before investigating the nature of this spatiotemporal structure, we should renew some of our caveats. First, there is always the temptation to invest the coordinates with some basic physical
significance. For example, it is very natural to regard the coordinate we are calling t as a time coordinate, and to suppose that it has something to do with what is measured by clocks. But as of
yet, we have said nothing to justify that interpretation. The Lorentz coordinates are just some way or other of attaching numbers to points such that the quantity defined above is proportional to the
spatiotemporal distance between events. Indeed, just as there are many ways to lay down Cartesian coordinates on a Euclidean plane, systems differ with respect to the origin and orientation of the
coordinate grid, so there are many ways to lay down Lorentz coordinates in Minkowski space-time. Different systems will assign different t values to the points, and will disagree about, for example,
the difference in t value between two events. We do not invest these differences with any physical significance; because the various systems agree about the quantity defined above, they agree about
all that is physically real.
A second caveat is in order. We have been speaking so far as if the spatiotemporal distance between events is itself a number (viz., the number that results when one plugs the coordinates of the
events into the formula above). But it is easy to see that this is wrong even in the Euclidean case. Distances are only associated with numbers once one has chosen a scale, such as inches or meters.
What exists as a purely geometrical, nonnumerical structure is rather a system of ratios of distances. Having chosen a particular geometrical magnitude as a unit, other magnitudes can be expressed as
numbers (viz., the numbers that represent the ratio between the unit and the given magnitude). The Greeks had a deep insight when they divided mathematics into arithmetic (the theory of number) and
geometry (the theory of magnitude). They recognized that the theory of ratios applied equally to each field, but kept the two subjects strictly separate. Our use of coordinates to associate curves in
space with algebraic functions of numbers has blurred the distinction between magnitudes and numbers. To understand relativity, it is important to recognize the conventions employed to associate
geometrical structure with numerical structure.
Holding these warnings in mind, let us turn to the relativistic spatiotemporal distance. What are the consequences of replacing the plus signs in the Euclidean distance function with minus signs?
One obvious difference between the Euclidean structure and the Minkowski structure is this: In Euclidean space, the distance between any two distinct points is always positive, and the only zero
distance is between a point and itself. In mathematical terms, the Euclidean metrical structure is positive definite. But in the Minkowski structure, two distinct events can have zero distance
between them. For example, the events with coordinates (0,0,0,0) and (1,1,0,0) have zero distance (where we list the coordinates in the order (t,x,y,z). Of course, this does not mean that these two
events are the same event; assigning the numerical value zero to this sort of distance is just a product of the conventions we have used for assigning numbers to the distances. But the fact that two
events have a zero distance between them does show that they are related in a particular spatiotemporal way. In order to remind ourselves that these spatiotemporal distances do not behave like
spatial distances, from now on we will call them spatiotemporal intervals.
If we choose a particular event, the popping of a particular champagne bubble, and call the event p, then we can consider the entire locus of events that have zero interval from p. There will be
infinitely many such events. If p happens to be at the origin of a Lorentz frame, assigned coordinates (0,0,0,0), then among the events at zero interval from it are (1,1,0,0), (1,0,1,0), (5,0,-3,4),
and (-6,4,-4,2). To get a sense of how these events are distributed in space-time, we draw a space-time diagram, but again one must be very cautious when interpreting these diagrams. The diagrams
must repress one or two dimensions of the space-time, because we cannot draw four-dimensional pictures, but that is not the principle problem. The main problem is that the diagrams are drawn on a
Euclidean sheet of paper, even though they represent events in Minkowski space-time. There is always the danger of investing some of the Euclidean structure of the representation with physical
significance it does not have. Bearing that in mind, the natural thing to do is to suppress the z coordinate and draw the x, y, and t coordinates as the x, y, and z coordinates of three-dimensional
Euclidean space.
Adopting these conventions, the points at zero interval from (0,0,0) will be points that solve the equation t2 − x2 − y2 = 0, or t2 = x2 + y2. The points that solve this equation form a double cone
whose apex is at the origin. According to relativity, the intrinsic spatiotemporal structure associates such a double cone with every event in the space-time. This locus of points is called the
light-cone of the event p, and divides into two pieces, the two cones that meet at p. These cones are called the future light-cone and the past light cone of p.
As the name light-cone suggests, we are now in a position to make contact between the spatiotemporal structure postulated by relativity and the behavior of physical entities. According to the laws of
relativistic physics, any light emitted at an event (in a vacuum) will propagate along the future light-cone of the event, and any light that arrives at an event (in a vacuum) arrives along the past
light-cone. So the tiny flash of light emitted when our champagne bubble pops races away from the popping event along its future light-cone. One can think of the ever-growing light-cone as
representing the expanding circle (or, if we add back the z dimension, the expanding sphere) of light that originates at the bursting of the bubble.
Having associated the spatiotemporal structure with the behavior of an observable phenomenon such as light, we can now see how relativistic physics gains empirical content. For example, it is an
observable fact that any pair of light rays traveling in parallel directions in a vacuum travel at the same speed; one light ray in a vacuum never overtakes another. This is not, of course, how
material particles behave. One spaceship traveling in a vacuum can overtake another, or one electron in a vacuum can overtake another, because where a spaceship or an electron goes depends on more
than the space-time location of the origin and direction of its journey. Two electrons can start out at the same place and time and set off in the same direction but end up in different locations
because they were shot out at different speeds. Their trajectories depend on more than just the space-time structure. Light, in contrast, is intimately and directly tied to the relativistic
space-time structure. Space-time itself, as it were, tells light in a vacuum where to go.
The assignment of zero relativistic interval between the origin of a light-cone and any event on it has one other notable consequence. We have already said that when we assign numbers to magnitudes,
we want the ratios between the numbers to be identical to the ratios between the magnitudes. Because 0:0 is not a proper ratio, the relativistic interval does not license comparisons between the
various intervals on a light-cone. If one light ray originates at (0,0,0,0) and travels to (1,1,0,0), and a second light ray originates at (0,0,0,0) traveling in some other direction, there is no
fact about when the second light ray has gone as far as the first.
What other structure, beside the light-cone structure, does Minkowski space-time have? There is a well-defined notion of a straight line in the space-time, and this is accurately represented in our
Euclidean space-time diagram: Straight lines in the Euclidean diagram correspond to straight trajectories in the space-time. Indeed, we have tacitly been appealing to the notion of a straight line
all along; when we speak of the relativistic interval between two events, we mean the interval as measured along a straight line connecting the events, or, even more precisely, we mean the
relativistic length of the straight line that connects the events. The straight-line structure (affine structure) of Minkowski space-time plays a central role in framing physical laws.
If a light ray is emitted from (0,0,0,0) into a vacuum, we already know that its trajectory through space-time will lie on the future light-cone of (0,0,0,0). But more than that, the trajectory will
be a straight line on the light-cone. An analogous fact holds for material particles that travel below the speed of light. If a material particle is emitted from (0,0,0,0), its trajectory will lie
entirely within the future light-cone of (0,0,0,0), which is to say that the particle can never travel at or above the speed of light. But more than that: If the particle is emitted into a vacuum,
and is not subject to any forces, then its trajectory will be a straight line in space-time.
This law, in abstract form, enormously predates the theory of relativity. For this is just the proper space-time formulation of Newton's first law of motion: "Every body continues in its state of
rest, or of uniform motion in a right line, unless compelled to change that state by forces impressed on it." The trajectory of a particle at rest or in uniform motion in Newtonian space-time is a
straight line through the four-dimensional space-time. Newton's first law, stated in terms of space-time trajectories, also retains the same form in Galilean space-time, and can be taken over without
change into Minkowski space-time. As we will see, in this abstract space-time formulation, Newton's first law also holds in the general theory of relativity. That is why we should try to formulate
physical laws directly in terms of space-time structure.
Once we deal with material particles that travel below the speed of light, the relativistic interval takes on even greater significance. Consider a particle that travels from (0,0,0,0) to (5,4,0,0)
along a straight trajectory (i.e., a particle emitted from the origin of the coordinate system that arrives at the event [5,4,0,0] without having any forces acting on it). The relativistic interval
along its space-time trajectory is:
The size of this interval has direct physical significance; it is proportional to the amount of time that will elapse for a clock that travels along that trajectory. Clocks in the theory of
relativity are like odometers on cars; they measure the length of the path they take. But length here means the interval, and path the space-time trajectory of the clock. Events in space-time
separated by positive intervals are time-like separated.
It is not, of course, a further unanalyzable postulate of relativity that clocks measure the interval along their trajectory; clocks are physical mechanisms subject to physical analysis. But one can
easily analyze how a simple clock will behave, such as a clock that counts the number of times a light ray gets reflected between two mirrors, and find that the reading on the clock will be
proportional to the interval along the clock's trajectory.
With the clock postulate in hand, we can now analyze the notorious twins paradox of relativity. One of a pair of twins takes a rocket from Earth and travels to a nearby star. Upon returning to Earth,
the twin has aged less than the stay-at-home sister, and the clocks in the twins' spaceship show less elapsed time than those that remained on Earth. Why is that?
To be concrete, suppose the event of the rocket leaving Earth is at the point (0,0,0,0) in our coordinate system, and the rocket travels inertially (without acceleration) to the point (5,4,0,0). The
rocket immediately turns around, and follows an inertial trajectory back to Earth, arriving at the event (10,0,0,0). The interval between (0,0,0,0) and (5,4,0,0) is, as we have seen, 3. Suppose this
corresponds to an elapse of three years according to the onboard clocks. The return trajectory from (5,4,0,0) to (10,0,0,0) also has an interval length 3, corresponding to another three years
elapsed. So the astronaut twin arrives back having aged six years, and having had all the experiences that correspond to six years of life.
The stay-at-home twin, however, always remained at the spatial origin of the coordinate system. Her trajectory through space-time is a straight line from (0,0,0,0) to (10,0,0,0). So the interval
along her trajectory is 10, corresponding to an elapse of ten years. She will have biologically aged ten years at her sister's return, and had four more years of experience than her twin.
The relativistic analysis of the situation is quite straightforward. It is really no more surprising, from a relativistic perspective, that the clocks of the twins will show different elapsed times
from departure to return than it is surprising that two cars starting in the same city and end in the same city will show different elapsed mileage on their odometers, given that one took the freeway
and the other a winding scenic route. The sense that there is a fundamental puzzle in the twins paradox only arises if one has mistaken views concerning the content of the theory of relativity.
In particular, it is often said that, according to the theory of relativity, all motion is the relative motion of bodies. If so, then there seems to be a complete symmetry between the twins: The
motion of twin A relative to twin B is identical to the motion of twin B relative to twin A. But the relative motion of the twins plays no role at all in the physical analysis of the situation. The
amount of time that elapses for twin B on her trip has nothing to do with what twin A is doing, or even if there is a twin A. The amount of time is just a function of the space-time interval along
her trajectory.
It is also sometimes said that the theory of relativity gets rid of all absolute spatiotemporal structure; all facts about space and time are ultimately understood in terms of relations between
bodies, so in a world with only one body there could be no spatiotemporal facts. This is also incorrect. The special theory of relativity postulates the existence of Minkowski space-time, whose
intrinsic spatiotemporal structure is perfectly absolute, in whatever sense one takes that term. It is not a classical space-time structure, but it is not just a system of relations between bodies.
One occasionally also hears that the resolution of the twins paradox rests on facts about acceleration; the situation of the two twins is not exactly symmetric because the astronaut twin must
accelerate (when she turns around to come home), whereas the stay-at-home twin does not. That is true, but irrelevant: The difference in elapsed time is a function of the intervals along the
trajectories, not a function of the accelerations that the twins experience. Indeed, in the general theory of relativity we will be able to construct a twins scenario in which neither twin
accelerates at all, but still they suffer different elapsed times between parting and reunion. It would be just as misleading to attribute the difference in elapsed time to the accelerations of the
twins as it would the difference in odometer reading to the accelerations of the cars, even if the car that took the longer route did accelerate more.
The paradoxical or puzzling aspect of the twins paradox really arises from the difference between Euclidean geometry and Minkowski space-time geometry. If we draw the trajectories of the twins in
space-time, we get a triangle whose corners lie at (0,0,0,0), (5,4,0,0), and (10,0,0,0). The astronaut twin travels along two edges of this triangle, whereas the stay-at-home twin travels along the
third. And in Euclidean geometry, the sum of the lengths of any two sides of a triangle are greater than the length of the remaining side. But in Minkowskian geometry, the opposite is true: The sum
of the intervals of two sides is less than the interval along the remaining side. Indeed, for time-like separated events, a straight line is the longest path between the two points in space-time.
This is one consequence of exchanging the plus signs in the Euclidean metric for minus signs in the Minkowski metric.
The relativistic clock postulate has been most strikingly checked using natural clocks: unstable particles whose decay rate displays a known half-life in the laboratory. The muon, a sort of heavy
electron, is unstable and will decay on an average of 10−6 seconds after having been created. Muons can be created in the upper atmosphere by collisions between molecules in the air and high-energy
cosmic rays. According to clocks on Earth, it should take the muon about 10 × 10−6 seconds to reach the Earth, so very few should survive the trip without decaying. Nonetheless, many more muons than
that calculation suggests do reach the Earth's surface. Calculation of the interval along the muon's trajectory predicts this because that interval corresponds to less than 10−6 seconds.
If we idealize muons a bit, and imagine that they all decay in exactly 10−6 seconds (according to their own clocks), then we can use them to map out the geometry of Minkowski space-time. Suppose we
create a swarm of muons in space and send them out in all directions. Their decays will provide a map in space-time of events that are all the same interval from the point of creation. If we choose
units so that the size of the interval corresponds to seconds, and we choose the creation of the muons as the origin of the coordinate system, then the coordinates of the decay events will satisfy:
This is the equation of a hyperboloid of revolution that asymptotically approaches the light-cone, as depicted below.
The hyperboloid represents events all at the same interval from (0,0,0,0), and so corresponds to a circle or sphere of fixed radius in Euclidean geometry. There would be a corresponding hyperboloid
in the past light-cone, representing places from which a muon could have been sent that would have decayed at (0,0,0,0).
Indeed, we are now in a position to make a thoroughgoing analogy between the geometry of Minkowski space-time and Euclidean geometry that makes no reference to coordinates at all. Classical Euclidean
geometrical proofs do not use coordinate systems of numbers, they use two instruments: the straightedge and the compass. The straightedge allows one to identify straight lines in the space, and the
compass to draw the locus of points at a fixed distance from a given center. In Minkowski space-time, we can use light rays in a vacuum and inertially traveling particles as straightedges because
their trajectories are straight lines in the space-time. Setting a Minkowski compass at interval zero and identifying a center should result in drawing the light-cone: the locus of points of interval
zero from the center. So we can use light rays for this purpose. Setting the compass to draw points at a fixed positive interval should result in drawing hyperbola; we can use clocks for this just as
the muons are employed above. In this way, we can free Minkowski geometry from coordinates altogether.
So far we have left one species of space-time relation out of account. All the points on the past or future light-cone of some event p are at zero interval from p. All the events inside the past or
future light-cone are at positive interval from p (taking always the positive square root by convention). What of points that are outside the light-cone altogether?
The point labeled (0,1,0,0) is outside the light-cone of the point (0.0.0.0). If we plug these coordinates into our formula, we find that the interval between the points is:
That is, according to the definition of the interval that we have given, the interval between these points is imaginary. What could this mean?
Once again, we have to recall that the assignment of numbers to the intervals is somewhat a matter of convention. In fact, some physics books define the interval as:
Here the interval between time-like separated events becomes imaginary. Does this mean that a clock could measure an imaginary number? Of course it can: Just take a regular clock and paint a little i
after all the numerals! The numbers we assign to intervals have no intrinsic significance; it is the ratios between the numbers that represent the ratios among the magnitudes. Events that lie outside
each other's light cones, so-called space-like separated events, have intervals among them that also stand in ratios to each other. The set of events at fixed space-like separation from (0,0,0,0)
forms another sort of hyperboloid of revolution, depicted below.
We now have a sense of the spatiotemporal structure of Minkowski space-time. A special relativistic physical theory must have laws that employ only this spatiotemporal structure. We could now go on
to see how, for example, classical electromagnetic theory can be reformulated in this way, but that would take us too far from foundational issues.
It should be noticed that this account of special relativity has made no mention at all of several well-known features often associated with relativity, such as the constancy of the speed of light,
the relativity of simultaneity, and the Lorentz-Fitgerald contraction. That is because all of these are frame-dependent (or coordinate system dependent) effects, and we have been presenting the
theory in a frame-independent way. For example, we have no basis to discuss the relativity of simultaneity because we have had no ground, and no need, to introduce any notion of simultaneity at all.
In classical physics, simultaneous events are events that take place at the same time, but we have no general notion of the time at which an event occurs, only the time that elapses on a clock
following a certain trajectory. So the proper thing to say is not that special relativity implies the relativity of simultaneity, but that it implies the nonexistence of any objective notion of
simultaneity. And we cannot discuss whether the speed of light is constant because we do not have any grounds to ascribe any speed to anything.
We have seen that a light ray can never overtake another light ray, but assessing a speed requires determining how far an object went in a given period of time. So far, we have not needed any notion
of the distance an object travels, nor of the time that it takes to travel that distance. We can say how much time will elapse on a clock that follows a given trajectory, but that is evidently no use
in defining a speed of light; no material clock can travel along with a light ray, and if it could, it would show no elapsed time for the journey. The notion of simultaneity requires a global time
function, that is, an assignment of times to all events, so that there is a locus of events that are all assigned the same time. And the notion of a speed requires both the notion of the time that
elapses between the start and the end of a journey, and the notion of the distance covered in that time. The relativistic space-time structure does not, per se, support either of these notions.
There is, however, a reasonably natural method for introducing both a global time function and a notion of spatial distance into Minkowski space-time. We begin with a family of co-moving inertial
clocks (i.e., a family of clocks all moving on straight, parallel trajectories through space-time). There will be an infinitude of such families, corresponding to all the directions their
trajectories can have. We begin by picking one such family.
We now want to "synchronize" the clocks. Scare quotes have to be put around the word since the classical notion of synchronization presupposes the notion of simultaneity: Synchronized clocks all show
the same time at the same instant. But in relativity there is no such thin as the same instant. So one must think of the method we are about to describe as a way to coordinate a family of clocks that
we simply call synchronizing them.
Let us choose a single master clock from our family of co-moving clocks. The other clocks will coordinate with this master clock by the following method: Each clock sends a light ray to the master
clock, noting the time of emission (according to the sending clock). When the light ray reaches the master clock, it is immediately sent back and shows the time reading on the master clock at the
moment it arrived. When this return signal reaches the sending clock, the time reading on the sending clock is noted. The sender, then, has three bits of data: the time it sent the signal (according
to the sending clock), the time it received the return signal (ditto), and the reading on the master clock when the signal got to it. On this basis, the sending clock synchronizes with the master
clock by adjusting its time so that the time that the master clock read when the signal arrived corresponds to the event on the sending clock exactly midway between the moment the signal was sent and
the moment the return signal arrived. All of these notions are relativistically well-defined, so this method of coordinating clocks can be carried out. Every event in space-time is now assigned a
time (viz., the reading on that member of the family of clocks that passes through the event when it passes through the event).
We can now identify simultaneity according to this family of clocks as sets of events that are all assigned the same time by this family of clocks. Such a set is called a simultaneity slice through
the space-time. The figure below shows one such simultaneity slice. Because all of the light signals that reach the master clock at noon lie on the past light-cone of the master clock showing noon,
and because all of the return signals lie on the future light-cone of that event, it is easy to calculate the points at which all of the coordinated clocks will register noon. It is the flat plane in
the middle.
The simultaneity slice is a function of which family of co-moving clocks we choose. Choosing another family will give a different notion of simultaneity:
Each family of co-moving clocks determines its own notion of simultaneity, and these various notions render different judgments concerning which pairs of events happen at the same time. All the
families will agree about the time order of time-like or light-like separated events, but for any pair of space-like separated events, some families will say that they happened at the same time,
others that one happened first, and yet others that the other happened first. Each family introduces its own global time function. None of these functions is superior to the other, and none is needed
at all to explicate the basic spatio-temporal structure.
What of spatial distance? Once a family of clocks has been synchronized, there is a simple way to assign a spatial distance between any pair of clocks. Send a light ray from one clock to the other.
We can now understand the time of travel for the light ray as the difference between the time showing on the emitting clock at the emission event and the time showing on the receiving clock at the
reception event. So we now have a definition of how long the light ray took to get from one clock to the other (again, this is not the time that a clock traveling along with the light ray would show
elapsing). If we now define the speed of light to be a given constant, c, then we can say that the distance between the clocks is just c times the elapsed time of transmission. This will give us a
structure of spatial distances between the clocks as defined by that particular family of clocks. Those spatial distances will, in special relativity, constitute a Euclidean space. Different families
of clocks will disagree about the precise spatial distance between events, and about the spatial size of material objects, but each family will construct for itself a Euclidean spatial structure.
Finally, if we allow such a family of clocks to introduce Cartesian coordinates on its Euclidean space, then the family will assign each event four coordinate numbers: the three spatial coordinates
and the global time function. These are exactly the Lorentz coordinate frames that we began with to express the relativistic metric, so we have come full circle.
The interconnection between the global time defined by a family of clocks and the spatial structure among events defined by that family resolves many of the intuitive puzzles in special relativity.
We have seen that, according to clocks at rest on the Earth, a high-energy muon has a much longer lifetime than a muon at rest. That explains, from the point of view of the Earth frame, how the muon
manages to make the trip to the surface. But of course, from the point of view of the muon, and clocks co-moving with it, the muon lifetime is the normal 10-6 seconds. From their point of view, the
Earth is approaching them at high velocity. In that frame of reference, the muon is able to get through the whole atmosphere not because of any slowing down of their clocks, but because of the
spatial contraction of the atmosphere. In the muon's frame of reference, the distance from the upper atmosphere to the Earth is much less than we on Earth take it to be.
The Lorentz contraction and time dilation effects of relativity then arise as disagreements that occur between the Lorentz frames about the amount of time that elapses between events and the spatial
distance between events. Clocks in any frame will be seen to run slow according to the time function associated with any other frame. A meter stick at rest in one frame will be judged to be less than
a meter long according to a frame in which the stick is moving. These are symmetric effects: From the point of view of any Lorentz frame, clocks at rest in any other frame run slow. We need to
sharply distinguish these effects from the twins paradox. There, the difference in elapsed time for each twin is a consequence of the fundamental spatiotemporal structure, and has nothing do to with
frames or families of clocks. The time dilation between frames results only from different ways of defining coordinates. In the latter case, there is no fact about which set of clocks is really
running slower, but in the former case there is an objective fact about which twin is biologically younger when they are reunited.
Special relativity is a theory that postulates a certain intrinsic spatiotemporal structure, and then formulates the laws of physics in terms of that structure. General relativity is the relativistic
theory of gravity. It is also fundamentally a theory about spatiotemporal structure, and allows for different structures than special relativity. So the first question that arises when approaching
general relativity is why gravity should particularly be connected to spatiotemporal structure. The special relativistic theory of electromagnetism, for example, simply accepts the Minkowski
space-time and employs it in framing the electromagnetic laws. But gravity, in contrast, led to the rejection of special relativity in favors of a new theory. What is so special about gravity?
One sometimes hears that there needed to be a relativistic theory of gravity because Newton's gravitational theory postulates that gravity acts instantaneously between distant masses, but in
relativity there is no available notion of instantaneous action (because there is no physical notion of simultaneity). But this observation does nothing to suggest that the theory of gravity should
require any change from the special relativistic space-time. Classical electrostatics postulated that the coulomb force between distant charged particles acts instantaneously, but electromagnetic
phenomena do not require changes to special relativity. Rather, relativistic electrodynamics simply rejects the claim that electric and magnetic forces act instantaneously. Electromagnetic influences
are propagated along the light cones, at the speed of light, by electromagnetic waves. Similarly, one might think that the obvious way to deal with gravitation is simply to deny that it acts
instantaneously. Let the gravitational effects also propagate along the light cones, and the special relativistic structure can be used to formulate the laws.
Adding such a delay in gravitational influence would, of course, modify the predictions of Newtonian gravity. One might even plausibly argue that Newton himself would have expected such a correction
to his instantaneous gravity. For Newton thought that gravitational forces were mediated by particles exchanged between the gravitating bodies, and he would have expected the particles to take some
time in traveling between the bodies. Of course, the fundamental cause of the gravitational force was a topic on which Newton refused to fingere any hypothesis, so we must be a bit speculative here.
But it is worthwhile to note that if we modify classical Newtonian gravitational theory to allow gravitational influence to propagate along the light cones, we can exactly derive some famous
relativistic effects, such as the anomalous advance in the perihelion of Mercury.
In order to understand why gravity is plausibly taken to be deeply connected to space-time structure, we need to look elsewhere. Consider again the family of comoving inertial clocks we made use of
in our discussion of special relativity. Once set in motion, the family of clocks will move together, never approaching or receding from each other. That is because: a) the clocks are all traveling
inertially, not subject to any force; b) according to the space-time version of Newton's first law, the trajectories of bodies subject to no forces will be straight lines in space-time; and c) the
straight-line trajectories of the co-moving clocks form a family of parallel straight lines. Note that in giving this argument, we never had to mention the mass of any of the clocks. Because they are
moving inertially, the trajectories of the clocks are determined by the intrinsic space-time structure, without the mass playing any role. It would not matter if some of the clocks were heavy and
others light; they would still move together parallel to one another.
In Newtonian physics, the mass of a body only comes into consideration when a body is subject to a force and thereby deflected off its inertial trajectory. The inertial mass of a body is nothing but
a measure of the body's resistance to being deflected by a force from its inertial trajectory: The more massive a body is, the harder it is to make its trajectory bend in space-time. Newton's second
law, which we now render F = mA, tells us that the same force will only produce half the acceleration in a body that is twice as massive. So in the presence of forces, the trajectories of bodies will
depend on their masses, whereas in the absence of forces the more and less massive bodies will move on parallel trajectories. Turning this observation around, we should find it very suggestive if
there is a situation in which the trajectory of a body does not depend at all on its mass. It is natural to suspect that in such a situation, the mass of the body is playing no role because the body
is not being subject to any force; it is moving inertially.
Recall Galileo at the top of the Leaning Tower of Pisa dropping a lighter and heavier object and seeing them hit the ground together. Here is a common situation in which the mass of an object does
not affect its trajectory: The heavy and light follow the same space-time path. According to Newtonian gravitational theory, this is a rather fortuitous result. In that theory, both the heavy and the
light object are subject to a force, the force of gravity, and so each is being deflected off its inertial trajectory. But, luckily, the gravitational force on each object is exactly proportional to
its inertial mass. So the more massive object, which needs a greater force to be accelerated, is subject to a greater force than the less massive object. Indeed, the gravitational force on the more
massive object is exactly as much larger as it needs to be to produce precisely the same acceleration as the lesser force of gravity produces on the less massive object. That, according to Newton, is
why they fall together; they are both accelerated, but at exactly the same rate.
If we follow the hint above, though, we will be led to suspect a different account. Perhaps the two objects move together not because they are equally deflected off their inertial, straight-line
trajectories, but rather because they are both following their inertial trajectories. Because the inertial trajectories are straight lines in space-time, this suggests a deep connection between
gravity and fundamental spatiotemporal structure.
In this way we arrive at the general theory of relativity. According to general relativity, objects that are falling in a gravitational field or under the influence of a gravitational force are not
being affected by any force at all. Gravity does not deflect objects from their inertial paths, it rather influences the very structure of space-time itself. The balls falling from the Leaning Tower
of Pisa, or the planets orbiting the sun, are following straight trajectories through space-time.
To realize this theory, we must reject Minkowski space-time. Consider, for example, two satellites orbiting the Earth in opposite senses. The space-time diagram of the situation looks like this:
As the satellites orbit, their paths cross and recross in space-time. But in Minkowski space-time, as in Euclidean space, two straight lines can intersect only once at most. So the space-times of
general relativity must have a different spatiotemporal structure than the space-time of special relativity.
An analogy with pure spatial geometry helps here. Euclidean geometry is just one of an infinitude of spatial geometries. Lines on the surface of a sphere, for example, do not satisfy Euclid's
postulates. But even spherical geometry is highly regular compared to most geometries. Consider, for example, the surface of North America. In regions of the Great Plains, the geometry is nearly
Euclidean (and even more nearly spherical), whereas in the Rocky Mountains the geometry of the surface varies wildly from place to place. We need new mathematical machinery to deal with this sort of
The general mathematics needed is called differential geometry. Differential geometry is suited to deal with spaces whose geometrical structure varies from place to place. In some regions, a space
may be locally Euclidean, in others non-Euclidean, so we have to be able describe the geometry region by region.
Euclidean spaces have a particularly uniform geometrical structure that allows them to admit of very convenient coordinate systems. As we have seen, a Euclidean space admits of Cartesian coordinates,
in which the distances between points is a simple mathematical function of the coordinates of the points. Non-Euclidean spaces do not admit of such convenient systems. For example, points on a sphere
can be coordinatized by latitude and longitude, but distances between the points on a sphere are not a simple function of their coordinate differences. If you are near the North Pole, you can change
your longitude by several degrees just by taking a few steps; near the equator the same change of longitude would require traveling hundreds of miles. And even spherical coordinates are relativity
simple and uniform.
To get a sense of a completely generic coordinate system, imagine walking down a road where each successive house has an address—one greater than the house before. You want to get to house number 200
and you are currently at house 100. How far must you walk?
There is no way to tell. If you go through a densely populated area, such as a small town, you will get to your destination quickly. If it is a sparsely built region, you may have to walk a long way.
To know how far you have to go, you would need a complete listing of the distances between successive houses. If you have such a list, you can calculate the distance between any two houses, and so
can reconstruct the geometrical structure of the region where the houses are built. In an analogous way, the general theory of spaces allows for the use of any arbitrary coordinate system.
Accompanying the system is a metric that specifies the distances between nearby points. We do not have any general rule for calculating distances between distant points as a function of their
coordinates, but we do not need one. The distance between faraway points is just the length of the straight path that connects them, and we can calculate the length of that path by knowing the
distance between nearby points and adding up all the distances along the path. Thus we have the mathematical tools to deal with generic spaces of variable curvature that admit of nothing like
Cartesian coordinates.
It is sometimes said that the general theory of relativity requires us to replace Euclidean space with a non-Euclidean space, but that is not a very useful, or accurate, explanation of the situation.
As we have seen, even in special relativity the notion of spatial geometry is rather derivative and non-fundamental. The fundamental notion is the relativistic interval, which is a spatiotemporal
object. It is only relative to a family of co-moving objects, such as clocks, that we can even define a spatial geometry. It turns out that, in special relativity, each such family will ascribe
Euclidean geometry to its space, but that is somewhat fortuitous; there is no logical guarantee that the various families will agree on their findings. After all, in special relativity the various
families will disagree about the exact spatial distance (and temporal gap) between a given pair of events. In general relativity, there will, in general, not exist families of co-moving inertial
observers that maintain the same spatiotemporal relations to one another, and so there is no unproblematic way to define a spatial geometry at all.
In any case, it is simply incorrect to say that objects moving in a gravitational field trace out straight paths in a non-Euclidean spatial geometry. The orbits of the planets, for example, are
nearly elliptical in any reasonably defined space for the solar system, and the ellipses are not (spatially) straight lines.
The proper account of general relativity rather employs an analogy. As the variably curved non-Euclidean spaces are to Euclidean space, so the variably curved space-times of general relativity are to
Minkowski space-time. The orbits of the satellites depicted above are not straight paths in any spatial geometry, but they are straight paths in space-time. The effect of the Earth is not to produce
a force that deflects the satellites off their inertial paths, it is to alter the space-time geometry so that it contains inertial paths that cross and recross.
On the Newtonian picture of gravity, when we sit on a chair we are not accelerated because we are acted on by counterbalancing forces: The gravitational force pulling us down and the force of the
chair pushing us up. According to the general relativistic account, the force of the chair pushing up still exists, but it is unbalanced by any gravitational force. It follows that according to
general relativity, as we sit we are constantly accelerated (i.e., constantly being deflected off of our inertial, straight-line trajectories through space-time). The inertial trajectory is that of
an object unsupported by anything like the chair (i.e., an object in free fall).
The curvature of general relativistic space-time is partially a function of the distribution of matter and energy; that is why space-time near a massive object like the Earth is curved in such a way
as to produce a gravitational field. This connection between the matter and energy distribution and the spatiotemporal geometry is provided by Einstein's general relativistic field equations. But
although the distribution of matter and energy influences the space-time geometry, it does not completely determine it.
The situation is similar to the relationship between the electromagnetic field and the electric charge distribution in classical physics. The presence of electric charges contributes to the
electromagnetic field, but does not, by itself, determine it. For example, even in a space devoid of electric charges, there can be a nonzero electromagnetic field: electromagnetic waves (i.e.,
light) can propagate through the vacuum. Similarly, the general theory predicts the existence of gravitational waves—disturbances of the spatiotemporal geometry that can exist even in the absence of
any matter or energy and that propagate at the speed of light. There are, for example, many vacuum solutions of the Einstein field equations. One solution is Minkowski space-time, but other solutions
contain gravitational waves.
Because general relativity concerns spatiotemporal structure, and because the trajectory of light rays is determined by the light-cone structure, general relativity must predict the gravitational
bending of light. It is not clear whether Newtonian physics would predict any gravitational effect on light because that would depend on whether light feels any gravitational force, but light
certainly does propagate through space-time. The effect of gravity on light was dramatically confirmed in Arthur S. Eddington's 1919 eclipse expedition, but is even more strikingly illustrated in the
phenomenon of gravitational lensing: A galaxy positioned between the Earth and a more distant light source can act as a lens, focusing the light of the distant source on the Earth. Two astronauts
traveling inertially could experience a similar effect; they could take different straight paths that both originate at their home planet and both end on Earth, going different ways around an
intervening galaxy. Because the relativistic interval along those paths could differ, such astronauts could illustrate the twins paradox without any acceleration; twins coming from the distant planet
could have different biological ages when they reunite on Earth, even though neither suffered any acceleration.
The spatiotemporal geometry of general relativity accounts for familiar gravitational phenomena, but the theory also has dramatic consequence at the cosmological scale and in extreme physical
conditions. When a massive star burns through its nuclear fuel and collapses, for example, the increasing density of matter causes ever greater curvature in space-time. If the star is sufficiently
massive, the light-cone structure deviates enough from Minkowski space-time to form a trapped surface: a region from which light cannot escape. The event horizon around a black hole is such a trapped
surface; an object falling through the horizon can never send light, or any other signal, back to the exterior region. Once the infalling matter of the star reaches this point, it is destined to
become ever more compressed without limit, and the curvature of the space-time will grow to infinity. If the equations continue to hold, this results in a space-time singularity; the spatiotemporal
structure cannot be continued beyond a certain limit and space-time itself comes to an end. Because the spatiotemporal structure itself has become singular, it no longer makes any conceptual sense to
ask what happens after the singularity; no meaning could be attached to the term after in the absence of spatiotemporal structure.
In the opposite temporal direction, the general theory also contains models in which the universe as a whole arises out of such a singularity, the singularity we call the big bang. Indeed, if general
relativity is not modified, the observed motions of galaxies require that the universe began at a singularity, and that space-time itself has been expanding ever since. There is equally no sense to
be made of the question what happened before the big bang because the spatiotemporal structure needed to define temporal priority would not extend beyond the initial singularity.
It is, of course, possible that the equations of the theory will be modified in some way so as to avoid the infinities and singularities, but that takes us from the analysis of general relativity
into speculations about the replacement of general relativity. The mathematical structure of general relativity also admits of models of the theory with very peculiar spatiotemporal structures. Some
models, for example, admit closed time-like curves, that is, time-like trajectories that loop back through space-time and meet up with themselves. In such a model, a person could in principle
continue going always locally forward in time, but end up (as an adult) back at the events of their childhood. There seems to be no way to physically test this possibility (that is, there is no
physical mechanism to produce closed time-like curves through laboratory operations), so it is unclear whether the existence of these mathematical models proves the physical possibility of such time
travel or rather the physical impossibility of space-times that correspond to these mathematical solutions. In any case, general relativity provides a means for considering spatiotemporal structures
unlike any that occur in classical physics.
The special and general theories of relativity provide a rich source of novel concepts of great interest to metaphysics. The topics that could be informed by these theories are too long even to list,
but the most obvious metaphysical implications of the theories are worthy of remark. The nature of space and time occupies a central place in Immanuel Kant's Critique of Pure Reason, where supposed a
priori knowledge of spatial and temporal structure provided grounds for the conclusion that space and time have no existence outside the faculty of intuition. After all, how could one know anything a
priori about space and time if they exist outside the mind?
The theories of relativity simply refute the claim that there is any a priori knowledge of spatiotemporal structure. Even if relativity ultimately proves to be incorrect, everything in our everyday
experience of the world can be accommodated in the relativistic spatiotemporal account. For all we know at present, we could be living in a relativistic universe, in which there is no Euclidean space
and in which even time need not have a universal linear order. The nature of space and time is a matter of empirical inquiry, not a priori proof.
The special and general theories are also relevant to the question of the nature of space and time: Are they entities in their own right (as Newton supposed) or just relations among material bodies
(as G.W. Leibniz insisted)? Taken at face value, the theories posit an independent existence to the four-dimensional space-time manifold. Even in the absence of material bodies, there is a
spatiotemporal structure among the points of space-time. As the twins paradox shows, the observable behavior of material objects is determined by that structure. And even more dramatically, in
general relativity the space-time manifold takes on a life of its own; gravitational waves can exist even in the absence of any material objects, and the presence of material objects influences the
structure of space-time.
Attempts have been made to reformulate general relativity in a more relationist manner, in terms only of relations among material objects without commitment to any spatiotemporal structure of vacuum
regions. These attempts have not succeeded. One can, of course, simply declare that in the general theory, space-time itself counts as a material entity, but then the argument seems to be only over
labels rather than ontology.
Like all empirical theories, relativity is supported but not proven by observation. The spatiotemporal structure cannot be directly observed, but theories of matter couched in terms of the
relativistic structure yield testable predictions that can be checked. The general theory, for example, has been checked by flying an atomic clock around the world and comparing its reading with an
initially synchronized clock that remained on Earth. Because the trajectories of the clocks have different relativistic intervals, one can predict that the traveling clock will show a different
elapsed time from the clock that remained behind—which it does. There may be other ways to explain the effect, but it is a natural consequence of the relativistic account of space-time structure.
Challenges to the theory of relativity are more likely to come from considerations of the compatibility of the theory with other fundamental physical theories than from direct empirical problems. It
is, for example, a still unsolved problem how to reconcile quantum physics with the pure relativistic space-time structure, and another unsolved problem of how to produce a quantum theory of gravity.
Most particularly, the observable violations of John Bell's inequality for events at space-like separation are difficult to account for in any theory that has no preferred simultaneity slices in its
space-time. So the metaphysician ought not to take the account of space-time provided by relativity as definitive; progress in physics may well demand radical revision of the account of
spatiotemporal structure. Still, relativity illustrates how empirical inquiry can lead to the revision of the most seemingly fundamental concepts, even those that were once taken as preconditions for
any experience at all.
Partial, incomplete, and evolving intellects would be helpless in the universe, would be unable to form the first rational thought pattern, were it not for the innate ability of all mind, high or
low, to form a universe frame in which to think. If mind cannot fathom conclusions, if it cannot penetrate to true origins, then will such mind unfailingly postulate conclusions and invent origins
that it may have a means of logical thought within the frame of these mind-created postulates. And while such universe frames for creature thought are indispensable to rational intellectual
operations, they are, without exception, erroneous to a greater or lesser degree.
Conceptual frames of the universe are only relatively true; they are serviceable scaffolding which must eventually give way before the expansions of enlarging cosmic comprehension. The understandings
of truth, beauty, and goodness, morality, ethics, duty, love, divinity, origin, existence, purpose, destiny, time, space, even Deity, are only relatively true. God is much, much more than a Father,
but the Father is man's highest concept of God; nonetheless, the Father-Son portrayal of Creator-creature relationship will be augmented by those supermortal conceptions of Deity which will be
attained in Orvonton, in Havona, and on Paradise. Man must think in a mortal universe frame, but that does not mean that he cannot envision other and higher frames within which thought can take
See also
• Bell, John S. "How to Teach Special Relativity." In The Speakable and Unspeakable in Quantum Mechanics. Cambridge, U.K.: Cambridge University Press, 1987.
• Einstein, Albert. "On The Electrodynamics of Moving Bodies." In The Principle of Relativity. New York: Dover, 1952.
• Einstein, Albert. "The Foundations of General Relaivity." Annalen der Physik 49 (1916).
• Geroch, Robert P. General Relativity from A to B. Chicago: University of Chicago Press, 1978.
• Maudlin, Tim. Quantum Non-Locality and Relativity: Metaphysical Intimations of Modern Physics. Oxford: Basil Blackwell, 1994.
• Misner, Charles, Kip S. Thorne, and John Archibald Wheeler. Gravitation. San Francisco: W. H. Freeman, 1973.
• Taylor, E. F., and J. A. Wheeler. Spacetime Physics: Introduction to Special Relativity. New York: W. H. Freeman, 1992.
Tim Maudlin (2005)
Source Citation
Maudlin, Tim. "Relativity Theory." Encyclopedia of Philosophy. Ed. Donald M. Borchert. Vol. 8. 2nd ed. Detroit: Macmillan Reference USA, 2006. 345-357. Gale Virtual Reference Library. | {"url":"https://nordan.mywikis.wiki/wiki/Relativity","timestamp":"2024-11-06T11:30:36Z","content_type":"text/html","content_length":"102432","record_id":"<urn:uuid:1741a0a9-fa26-4968-b905-d2d250229381>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00162.warc.gz"} |
Random conical tilt (RCT)
Calculates Random conical Tilt volumes from a set of tilted raw images by the aid of a stack of the classified untilted counter parts. This is an experimental method to Determine the Euler angles
To run this logic two stacks of the same size, on of the tilted and one of the untilted images have to be prepared. It is necessary that the images in both stacks are in the same order. The tilted
images should be at least normalised. The untilted images need to refined on a 2D level to give good class averages. in general Class averages should contain about 100 particles.
Parameter and I/Os
Parameters Description
Class Number from which class shout the RCT be calculated? If -1 is given the RCT is calculated for every class.
tilted inplane inplane rotation of the tilt axis, none not recommended will yield in unuseable results, simple the same inplane rotation is assumed for every image, complex in plane rotation is
Rotation given as IO, assuming that the in plane rotation is not the same for every micrograph.
→ tilted angle in plane rotation of tilt axis in degree. Can be determined by the picker
mode Mode of reconstruction exact back projection, fourier Fourier reconstruction, sirt Sirt algorithm. For details see Reconstruction
Normalize 3D Should the resulting 3D Volumes be normalized?
Reproject 3D Shall there be Backprojections of the input images?
Symmetry Is there a point group symmetry known for the molecule?
experimental angle of goniometer rotation. Can be also determined in the picker
tilt angle
tilted with
Input Description
tilted normalized and maybe filtered otherwise untreated stack of tilted images
untilted output of a Classification logic of the untilted images
tilt angles stack of images containing the tilt angles in the header, as written by the picker
Output Description
3ds Stack of reconstructed 3D volumes one stack for each 2D class | {"url":"https://www.cow-em.de/guide/doku.php?id=eyes:logics:rct","timestamp":"2024-11-06T17:34:22Z","content_type":"application/xhtml+xml","content_length":"26739","record_id":"<urn:uuid:aa36e416-2f15-4a31-a719-6c9ad8c8a944>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00524.warc.gz"} |
Void Ratio vs Porosity in context of void ratio
24 Sep 2024
Void Ratio and Porosity: A Comparative Analysis
The void ratio (e) and porosity (n) are two fundamental parameters used to describe the pore space within a porous medium. While often used interchangeably, these terms have distinct meanings and
implications in various fields of study. This article provides an in-depth comparison of void ratio and porosity, highlighting their differences and similarities.
The void ratio (e) is defined as the ratio of the volume of pores to the total volume of a porous medium [1]. Mathematically, it can be expressed as:
e = Vp / Vt
where Vp is the volume of pores and Vt is the total volume of the porous medium.
Porosity (n), on the other hand, is defined as the ratio of the volume of pores to the bulk volume of a porous medium [2]. It can be expressed as:
n = Vp / Vb
where Vb is the bulk volume of the porous medium.
Differences between Void Ratio and Porosity
While both void ratio and porosity describe the pore space within a porous medium, they differ in their reference volumes. The void ratio (e) uses the total volume (Vt) as its reference, whereas
porosity (n) uses the bulk volume (Vb). This distinction has significant implications for various applications.
In soil mechanics, for instance, the void ratio is often used to describe the pore space within a soil sample, while porosity is used to describe the pore space within a larger, more complex system
Similarities between Void Ratio and Porosity
Despite their differences, both void ratio and porosity share some similarities. Both parameters are dimensionless and can be expressed as ratios of volumes.
Moreover, both void ratio and porosity are sensitive to changes in the pore space within a porous medium. As the volume of pores increases or decreases, both e and n will change accordingly.
In conclusion, while void ratio (e) and porosity (n) share some similarities, they have distinct meanings and implications in various fields of study. The void ratio is defined as the ratio of pore
volume to total volume, whereas porosity is defined as the ratio of pore volume to bulk volume. Understanding these differences is essential for accurate analysis and interpretation of porous media.
[1] Terzaghi, K., & Peck, R. B. (1948). Soil mechanics in engineering practice. John Wiley & Sons.
[2] Bear, J. (1972). Dynamics of fluids in porous media. Dover Publications.
[3] Skempton, A. W. (1953). The colloidal activity of clay-water systems. Journal of the American Society of Civil Engineers, 79(4), 1-11.
Related articles for ‘void ratio ‘ :
• Reading: **Void Ratio vs Porosity in context of void ratio **
Calculators for ‘void ratio ‘ | {"url":"https://blog.truegeometry.com/tutorials/education/f4b6bafbef06b2648e67602db4ffcc89/JSON_TO_ARTCL_Void_Ratio_vs_Porosity_in_context_of_void_ratio_.html","timestamp":"2024-11-13T22:55:31Z","content_type":"text/html","content_length":"16292","record_id":"<urn:uuid:4cfc620c-4be4-4244-a273-2b4c9c931342>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00138.warc.gz"} |
G'day all.
Quoting Hossein Haeri <powerprogman_at_[hidden]>:
> Sure. And, I find it hilarious that they are not
> assummed as of the basic linear algebraic project.
Not really.
The reason is that while there is only one "obvious" way to multiply
two general matrices (taking shortcuts for matrices which exhibit certain
structure), but given a general matrix, it's often not "obvious" how to
invert it, or even if inverting it is a good idea.
Gaussian elimination, LU decomposition and Cramer's rule (to name but
three) are all ways to invert a general matrix, each of which has
different efficiency and stability properties, and which return different
answers on the same floating point hardware. This means that if you want
to invert a matrix, you have to pick an algorithm appropriate to your
problem. It's therefore impossible to supply an operation of the form
inverse(m) which is generally useful.
I can't speak for the developers of uBLAS (or BLAS, for that matter),
but the philosophy seems to be that if there's more than one possible
algorithm for something, and using different algorithms make sense under
different circumstances, and it's not obvious how to choose between those
ways automatically (e.g. using iterator categories or the like), then
it's not "basic".
Andrew Bromage
Boost-users list run by williamkempf at hotmail.com, kalb at libertysoft.com, bjorn.karlsson at readsoft.com, gregod at cs.rpi.edu, wekempf at cox.net | {"url":"https://lists.boost.org/boost-users/2005/01/9659.php","timestamp":"2024-11-04T15:37:05Z","content_type":"text/html","content_length":"12035","record_id":"<urn:uuid:ff74fb51-063a-470f-a808-95e2adac40f5>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00797.warc.gz"} |
Design of a Single Transistor DC Current Source - MyEssayDoc
Lab 4–Design of a Single Transistor DC Current Source
Perform lab in class during the week of Feb. 18-22, 2019
Group Lab Report
Purpose: To measure and analyze the characteristics of current sources.
To measure and analyze the characteristics of a voltage divider used for biasing transistor circuits.
Equipment: Dual DC power supply
2N3904NPN transistor
To Be Determinedresistors
10 kW potentiometer
Two Digital Multimeters (DMM)
Wire Jumper Kit
Connecting leads
LED Collection
Problem Definition and Background Theory
Goal: Design a current source which provides a fixed DC current through a variable load resistance, R[L]. The current will remain fixed over a voltage range of 0 volts to some defined maximum DC
The basic layout of the Transistor Current Source Circuit is shown in Figure 1. The parameters are:
V[CC] = DC power supply
R[L]= Load resistance of the current source
V[L ]= DC load voltage
I[B ]= DC base current
I[C ]= DC collector current
I[E ]= DC emitter current
V[B ]= DC base voltage
V[C ]= DC collector voltage
V[E ]= DC emitter voltage
V[BE ]= DC base-emitter voltage
V[CE ]= DC collector-emitter voltage
In active operation, the transistor has the following properties:
V[BE ]= 0.7 volts and I[C ]=βI[B]
[ ]
where β is the DC current gain of the transistor.
Kirchhoff’s Current Law gives us several current equations:
I[E ]=I[B ]+I[C ]I[1 ]=I[B ]+I[2]
Kirchhoff’s Voltage Law gives us several voltage equations:
V[B ]=V[E ]+V[BE ]V[C ]=V[E ]+V[CE ]V[CC ]=V[L ]+ V[C ]
[ ]
Ohm’s Law gives us several resistor equations:
V[E ]=I[E ]R[E ]V[L ]=I[C ]R[L ]V[B ]=I[2 ]R[2]
Power equations include:
P[T ]=V[CE ]I[C ]= power dissipated in the transistor in Watts
P[RE]=V[E ]I[E ]= power dissipated in R[E] in Watts
P[RL ]=V[L ]I[C]= power dissipated in R[L] in Watts
P[R1]=(V[CC ]-V[B])I[1 ]= power dissipated in R[1] in Watts
P[R2 ]=V[B ]I[2 ]= power dissipated in R[2] in Watts
The equivalent resistance looking into the base is
The design process starts by choosing a DC fixed current for the current source and a DC voltage for the power supply, V[CC]. This fixed current value is the same as I[C ], the DC collector current.
The design tasks are to:
• Derive values for R[1], R[2], R[E], to produce the desired fixed current value.
• Compute the transistor and resistor powers to make sure that the transistor and resistors can handle the power expected.
Design Challenge
Design a DC current source using the basic circuit in Figure 1.
Constraints: You are limited to a fixed DC power supply voltage. You must use only one BJT transistor.
The customer would like a fixed current value of I[C ], the DC collector current. We will leave I[C ]as a variable for now.
Desirable characteristics: We would like the current source to operate at full current over a wide range of voltage as possible.
The design tasks are to:
1) Derive values for R[1], R[2], R[E], in terms of V[CC] and I[C].
2) Compute the transistor and resistor powers to make sure that the transistor and resistors can handle the power expected.
We begin by setting
This will produce a relatively low V[E], allowing for a wide range of V[L]. Next we calculate the emitter voltage from KVL: V[E ]=V[B] -V[BE].
Assuming active operation, we have volts and
Substitute (1) into (2) to get:
Next we manipulate equation (3) and the KVL equations:
V[C ]=V[E ]+V[CE ]V[CC ]=V[L ]+ V[C ] (4)
[ ]
to get:
Next we compute the maximum allowable load voltage. This occurs when is at a minimum value. is minimized when the transistor is operating in the saturation region. In the saturation region, both
PN junctions are in forward mode and volts. To summarize, the maximum load voltage is:
[ ]
where volts. We substitute volts to get:
We now have a current source, equivalent to figure 2, in which the current is fixed at I[C] as R[L] changes. This is a good thing. Notice that we are allocated about 2/3 of V[CC] as the range of .
This is also pretty decent. As we increase R[L] from 0 Ohms, the load voltage will change from 0 volts to as defined in equation (6).
The maximum load resistance comes from Ohm’s Law:
Figure 1 –Transistor Current Source Circuit. Figure 2 –Equivalent Circuit
Prelab – Design Calculations for the DC Current SourceName______________
1. (10 pts) Using the background information in part 1, and the approximation that , derive the following equation for as a function of , , and β. Hint: Substitute , and into
2. Using the answer to question 1 and the background information in part 1, in which the base voltage is designed for , and is designed for , derive equations for
3. a) as a function of , , and β.
1. b) (10 pts) as a function of , , and β using . The goal of a stable current source is to have a stable base voltage, in which the transistor does not load down the circuit. If we remove the
transistor, the base voltage should not change. To do this, we set to target approximately a tolerance of 5%, we set .
1. c) as a function of , , and β.
3. (50 pts) Given the answers to question 2, design a 10 mA transistor current source with a supply voltage of volts, and β =150. Show all calculations for the following quantities:
P[T(max) ]P[RE]P[RL(max) ]P[1 ]P[R2]
P[T(max) ]=maximum power dissipated in the transistor
P[RE]= power dissipated in R[E]
P[RL(max) ]=maximum power dissipated in R[L]
P[1 ]=power dissipated in R[1]
P[R2 ]=power dissipated in R[2]
…and draw a schematic diagram of the final circuit with all currents and node voltages indicated on the sketch.
Lab Instructions
Team members:_________________________________
Part 1 – Construct Current Source Circuit
• □On the breadboard, construct the 10 mAvoltagedivider transistor circuit that you designed in the Pre-Lab.For this initial step, use a load resistor of RL= 0W.
With RL = 0 W, what should VC be? ________
What should VB be approximately? ________ (refer to the prelab)
What should VE be approximately? ________ (refer to the prelab)
• □Turn on the power supply and set the voltage to 20 volts and observe the current value displayed on power supply. There should be a source current in the mA range.
If the source current is 1.000 Amps, then there is a short circuit connection… if the source current is 0.000 Amps, then there is an open circuit … In either case, turn off the power supply, debug
the circuit, and try again.
When the source current is in the mA range, record the source current, measure the transistor voltages and record below:
Source current = __0.011A____________
VB = ___6.56V_______ , VC = ____19.8V______ , VE = ______5.83V____
• □Compare the measured VBand VEabove with your Prelab calculations. They should differ by about 0.7 volts. If they are way off, then you will need to debug the circuit. Debugging tips:The high
sides of RLand R1 should both be about 20 V.So you can use a DMM set up as a DC voltmeter to check these test points. The low sides of R2and RE should be connected to ground, so the voltages at
these points should both be 0 V. You can use a voltmeter to check these test points. If any of these four voltage test points are off, then there is probably a bad connection somewhere. To
find a bad connection, use the voltmeter to trace the faulty test point back to the power supply. If this still does not help, then ask the instructor for assistance.
• □ Now that the circuit is operation properly, set up the DMM to measure collector current, IC.
Measured IC = ____9.52______ mA
What should IC be according to the prelab? _____10_____ mA
• □Show the circuit to the instructor before continuing.
Instructor sign off: ____RDA 2/19/19__________
Part 2 – Testing of the Current Source
A current source should be able to deliver a fixed current over a range of load resistance. In this section, the BJT current source is tested to determine the range of operation and % regulation of
the current source.
• □Fill in Table 1 with measurements of VB, VE, VC, and ICfor values of VLranging from 0 to14 volts. You will need to adjust the variable resistor to change the value of the load voltage. For the
last row of table 1, adjust the variable resistor until the collector current drops to 9 mA.
Table 1 – Measured and Calculated Results of Current Source
Measured Calculated Junction Bias
VCC VB VE VL IC RL PT PL VBE VBC VCE BE BC
20 6.56V 5.83 0 9.52 0 0
20 6.56v 5.8 2 9.62
20 6.5V 5.86 4 9.61
20 6.55 5.85 6 9.59
20 6.55 5.84 8 9.56
20 6.55 5.84 10 9.56
20 6.54 5.82 12 9.53
20 6.38 5.66 14 9.23
20 6.26 5.54 14.1 9
20 5.74 5.01 14.7 8
V V V V mA W mW mW V V V F or R?
F= forward biased
R = reverse biased
Vc = Vcc-V_L
P_T = VCE * IC
VCE= VC – VE
• □ You should notice that the collector current (IC) remains fairly steady as RL(and VL) increases, but then at some point IC drops significantly. How high does VLget before the current start to
drop significantly?
VLmax (experimental)= _____14____. How close is this to the prelab calculation? ___13.8___________
Pull out RL and measured its resistance. RLmax (experimental)= __1.85Kohm_______.
How close is this to the prelab calculation? __________1.384kohms____
• □ Calculate the indicated quantities in Table 1, based on the measured values. Fill in Table 1 with these calculations.
• □ Replace the potentiometer with an LED. Observe the brightness. Fill in Table 2 for 1 LED.
• □ Repeat for 2, 3, 4, 5, 6, as many LED’s in series as needed until the brightness diminishes. Fill in the rest of Table 2.
Table 2 – Test of Current Source using Series LED’s as the Load
VCC # of LED’s in series VL Brightness is the same or getting dimmer?
20 1 2.8 same
20 2 5.7 same
20 3 5.7 same
20 4 9.46 same
20 5 11.3 same
20 6 13.1 same
20 7 14.9 same
• □ Pull out the transistor and measure the voltage at the base test point.
VB with transistor removed = ___6.7V_______ ,
Calculate the percent change in VBbetween no transistor and transistor in with RL = 1kW.
% change = _____(6.7-6.56)/6.7*100=________
• □ Insert the 1kW resistor for RLand apply a hot soldering iron to the transistor and observe the collector current.
Does the current increase or decrease when the transistor is heated? ________increase___________.
When you remove the soldering iron, does the current increase or decrease? ________decrease___________.
Post Lab Questions:
1. In step 8, you heated the transistor with a soldering iron and observed the collector current. Recalling from your knowledge of the properties of semi-conductors, what is the effect on
conductivity when a semiconductor is heated?
What effect will this have on the transistor currents in a transistor circuit?
2. From the data in Table 1, for what range of load voltage was the transistor in saturation?
For what range of load voltage was the transistor in active operation?
Would you say that the transistor is a more stable current source in active mode or in saturation?
3. Plot a current source regulation curve consisting of measured ICvalues on the y-axis and measured VLvalues on the x-axis. Use Excel or MATLAB. On the graph, draw a vertical line that separates
the active region from the region of saturation.
4. One way to have a stable base voltage is to create a voltage divider such that REQ»(b+1)REis much larger than R2, so that the parallel combination of REQand R2 is approximatelyR Assuming b =
150, calculate REQ for the circuit used in this lab. Does this result suggest a stable base voltage?
5. Generate an LTspice simulation for R[L]= 0W. From the simulation, record all DC currents, and voltages in Table 3. Paste a screen shot of the simulation below:
6. Repeat step 6 for R[L]=R[L(max)], as calculated in the Pre-lab. From the simulation, record all DC currents and voltages in Table 3.
7. Repeat step 6 for I[C]=9 mA. From the simulation, record all DC currents and voltages in Table 3.
8. Table 1: Compare the load resistor power (at VL= 14 V) with the prelab results. How close are the two?
9. Table 1: Compare the transistor resistor power (at VL= 0 V) with the prelab results. How close are the two?
Table 3 –Current Source Simulation in LT Spice
Simulation Results Calculated
VCC VB VE IC VC IR1 IR2 IRE RL PL PT VL
V V V mA V mA mA mA Ω | {"url":"https://myessaydoc.com/design-of-a-single-transistor-dc-current-source/","timestamp":"2024-11-15T02:50:30Z","content_type":"text/html","content_length":"92484","record_id":"<urn:uuid:dd217d32-4f10-46f3-8bcd-7739ad1772e2>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00417.warc.gz"} |
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Math Is Fun Forum
Registered: 2017-04-19
Posts: 3
Re: Sort of like Sudoku
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,868
Re: Sort of like Sudoku
Hi, Leren (Dutch for 'learn'), and welcome to the forum!
Well, that's interesting information! I'd suspected that there would be many more solutions to the OP's puzzle (post #1) than the few that I'd found with the Excel Solver add-in I referred to in
other posts, but not as many as that!!!
How did you arrive at that total, and what program did you use?
"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson
Registered: 2017-04-19
Posts: 3
Re: Sort of like Sudoku
An exhaustive search has revealed that the total number of solutions for the first puzzle (post 1) without any additional constraints is 19,879 and for the second puzzle (post 4) is 140,884.
You might have trouble believing this but I just looped through every possible value of all 40 blank cells subject to the given constraints. Now if there were no constraints that sounds like 7^ 40
steps, and even allowing that my
computer runs at about 140,000,000 steps/second, that would take something like 10^18 years. However the constraints reduce this dramatically, so that the whole process only took 470 seconds !
I just use visual basic in Excel, so lower level languages like C++ would no doubt be even faster. The trick is to pick your blank cell loops so that you can use a 30ish cell sum constraint as soon
as possible.
I first came across this type of puzzle in a Sudoku forum that I am in, where someone could not solve one of these problems with the first and last rows being completely specified (in addition to the
nine 30ish cells).
Well I just love a programming challenge and I looped through the 28 unspecified cells, and found the unique solution in less than perceptible time.
To find other similar puzzles I just did a google search and eventually found this thread, which naturally suggested the 40 blank cell problem which obviously would have multiple solutions.
Registered: 2017-11-03
Posts: 4
Re: Sort of like Sudoku
That's a great idea. The fact he different squares share number columns ads a significant amount of difficulty to those squares.
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,868
Re: Sort of like Sudoku
phrontister (post #14) wrote:
The standard Excel Solver was missing a constraint functionality I needed that the advanced one has.
EDIT: I had a tiny sniff of success with the standard Excel Solver by scaling the grid down from 7x7 to 5x5. The solver doesn't allow (as far as I could tell) crossing of the 'AllDifferent'
constraint (eg, a row crossing a column - because one of them is then treated as not containing all variables, which it must contain), and so I cooked up some workarounds (linear and nonlinear).
Only one 'worked': ie,
- a nonlinear one, with the 'GRG Nonlinear' solving method;
- for one particular scenario only, in which I helped it get started by providing the answers to 4 cells, leaving the other 17 for the solver to find...which it did!
- it failed on all other assignments.
I gave the standard Excel Solver a tweak or 2 and had another go at the 7x7 puzzle from patchy1's first post (without post #24's constraints)...and this time it worked.
The Solver stops computing at the first solution it finds, but, as mentioned in earlier posts, this puzzle has many solutions (both with post #24's constraints and without).
Last edited by phrontister (2021-08-23 12:14:14)
"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson
From: Portsmouth
Registered: 2022-01-22
Posts: 15
Re: Sort of like Sudoku
I love Sudoku and Japanese crossword puzzles. And this task seems to me more difficult. | {"url":"https://mathisfunforum.com/viewtopic.php?id=22465&p=2","timestamp":"2024-11-10T13:03:27Z","content_type":"application/xhtml+xml","content_length":"15410","record_id":"<urn:uuid:a32353d8-84d6-407e-9668-51b2ea7a970d>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00395.warc.gz"} |
`A_n` is the `n`th term in a sequence.
Which one of the following expressions does not define a geometric sequence?
A. `A_(n + 1) = n` `\ \ \ \ A_0 = 1`
B. `A_(n + 1) = 4` `\ \ \ \ A_0 = 4`
C. `A_(n + 1) = A_n + A_n` `\ \ \ \ A_0 = 3`
D. `A_(n + 1) = –A_n` `\ \ \ \ A_0 = 5`
E. `A_(n + 1) = 4A_n` `\ \ \ \ A_0 = 2`
A plant was 80 cm tall when planted in a garden.
After it was planted in the garden, its height increased by 16 cm in the first year.
It grew another 12 cm in the second year and another 9 cm in the third year.
Assuming that this pattern of geometric growth continues, the plant will grow to a maximum height of
A. `64\ text(cm)`
B. `128\ text(cm)`
C. `144\ text(cm)`
D. `320\ text(cm)`
E. `400\ text(cm)`
A family bought a country property.
At the end of the first year, there were two thistles per hectare on the property.
At the end of the second year, there were six thistles per hectare on the property.
At the end of the third year, there were 18 thistles per hectare on the property.
Assume the number of thistles per hectare continues to follow a geometric pattern of growth.
At the end of the seventh year, the number of thistles per hectare is expected to be
A. `972`
B. `1458`
C. `2916`
D. `4374`
E. `8748`
The amount added to a new savings account each month follows a geometric sequence.
In the first month, $64 was added to the account.
In the second month, $80 was added to the account.
In the third month, $100 was added to the account.
Assuming this sequence continues, the total amount that will have been added to this savings account after five months is closest to
A. `$155`
B. `$195`
C. `$370`
D. `$400`
E. `$525`
The first four terms in a geometric sequence are: `5, 10, 20, 40, …`
The fifth term in this sequence is
A. `45`
B. `50`
C. `60`
D. `80`
E. `100`
A healthy eating and gym program is designed to help football recruits build body weight over an extended period of time.
Roh, a new recruit who initially weighs 73.4 kg, decides to follow the program.
In the first week he gains 400 g in body weight.
In the second week he gains 380 g in body weight.
In the third week he gains 361 g in body weight.
If Roh continues to follow this program indefinitely, and this pattern of weight gain remains the same, his eventual body weight will be closest to
A. `74.5\ text(kg)`
B. `77.1\ text(kg)`
C. `77.3\ text(kg)`
D. `80.0\ text(kg)`
E. `81.4\ text(kg)`
A crystal measured 12.0 cm in length at the beginning of a chemistry experiment.
Each day it increased in length by 3%.
The length of the crystal after 14 days growth is closest to
A. `12.4\ text(cm)`
B. `16.7\ text(cm)`
C. `17.0\ text(cm)`
D. `17.6\ text(cm)`
E. `18.2\ text(cm)`
The first three terms of a geometric sequence are `6, x, 54.`
A possible value of `x` is
A. `9`
B. `15`
C. `18`
D. `24`
E. `30`
Which one of the following sequences shows the first five terms of an arithmetic sequence?
A. `1, 3, 9, 27, 81\ …`
B. `1, 3, 7, 15, 31\ …`
C. `– 10, – 5, 5, 10, 15\ …`
D. `– 4, – 1, 2, 5, 8\ …`
E. `1, 3, 8, 15, 24\ …`
At the end of the first day of a volcanic eruption, 15 km^2 of forest was destroyed.
At the end of the second day, an additional 13.5 km^2 of forest was destroyed.
At the end of the third day, an additional 12.15 km^2 of forest was destroyed.
The total area of the forest destroyed by the volcanic eruption continues to increase in this way.
In square kilometres, the total amount of forest destroyed by the volcanic eruption at the end of the fourteenth day is closest to
A. `116`
B. `119`
C. `150`
D. `179`
E. `210`
The first term, `t_1`, of a geometric sequence is positive.
The common ratio of this sequence is negative.
A graph that could represent the first five terms of this sequence is
In the first three layers of a stack of soup cans there are 20 cans in the first layer, 19 cans in the second layer and 18 cans in the third layer.
This pattern of stacking cans in layers continues.
The maximum number of cans that can be stacked in this way is
A. `190`
B. `210`
C. `220`
D. `380`
E. `590`
The yearly membership of a club follows an arithmetic sequence.
In the club’s first year it had 15 members.
In its third year it had 29 members.
How many members will the club have in the fourth year?
A. `8`
B. `22`
C. `36`
D. `43`
E. `57`
For the geometric sequence
`24, 6, 1.5\ …`
the common ratio of the sequence is
A. `– 18`
B. `0.25`
C. `0.5`
D. `4`
E. `18`
When full, a swimming pool holds 50 000 litres of water.
Due to evaporation and spillage the pool loses, on average, 2% of the water it contains each week.
To help to make up this loss, 500 litres of water is added to the pool at the end of each week.
Assume the pool is full at the start of Week 1.
At the start of Week 5 the amount of water (in litres) that the pool contains will be closest to
A. `47\ 500`
B. `47\ 600`
C. `48\ 000`
D. `48\ 060`
E. `48\ 530`
When placed in a pond, the length of a fish was 14.2 centimetres.
During its first month in the pond, the fish increased in length by 3.6 centimetres.
During its `n`th month in the pond, the fish increased in length by `G_n` centimetres, where `G_(n+1) = 0.75G_n`
The maximum length this fish can grow to (in cm) is closest to
A. 14.4
B. 16.9
C. 19.0
D. 28.6
E. 17.2
The sequence `12, 15, 27, 42, 69, 111 …` can best be described as
A. fibonacci-related
B. arithmetic with `d > 1`
C. arithmetic with `d < 1`
D. geometric with `r > 1`
E. geometric with `r < 1`
Kai commenced a 12-day program of daily exercise. The time, in minutes, that he spent exercising on each of the first four days of the program is shown in the table below.
If this pattern continues, the total time (in minutes) that Kai will have spent exercising after 12 days is
A. `59`
B. `180`
C. `354`
D. `444`
E. `468`
The first term of a geometric sequence is 9.
The third term of this sequence is 121.
The second term of this sequence could be
A. `– 65`
B. `– 33`
C. `56`
D. `65`
E. `112`
The values of the first seven terms of a geometric sequence are plotted on the graph above.
Values of `a` and `r` that could apply to this sequence are respectively
(A) `a=90` `\ \ \ \ r= – 0.9`
(B) `a=100` `\ \ \ \ r= – 0.9`
(C) `a=100` `\ \ \ \ r= – 0.8`
(D) `a=100` `\ \ \ \ r=0.8`
(E) `a=90` `\ \ \ \ r=0.9`
For an examination, 8600 examination papers are to be printed at a rate of 25 papers per minute. After one hour, the number of examination papers that still need to be printed is
A. `1600`
B. `2500`
C. `6100`
D. `7100`
E. `8575`
Eleven speed bumps are placed on a road.
The speed bumps are placed so that the distance between consecutive speed bumps decreases according to an arithmetic sequence.
The distance between the first and last speed bumps is exactly 100 m.
The smallest distance between consecutive speed bumps is 2 m.
The largest distance, in m, between two consecutive speed bumps is
A. `16`
B. `18`
C. `20`
D. `22`
E. `24`
The following are either three consecutive terms of an arithmetic sequence or three consecutive terms of a geometric sequence.
Which one of these sequences could not include 2 as a term?
A. `–1, 0.5, –0.25`
B. `–1, –3, –5`
C. `5, 12.5, 31.25`
D. `6, 8, 10`
E. `8, 16, 32`
There are 10 checkpoints in a 4500 metre orienteering course.
Checkpoint 1 is the start and checkpoint 10 is the finish.
The distance between successive checkpoints increases by 50 metres as each checkpoint is passed.
The distance, in metres, between checkpoint 2 and checkpoint 3 is
A. `225`
B. `275`
C. `300`
D. `350`
E. `400`
On Monday morning, Jim told six friends a secret. On Tuesday morning, those six friends each told the secret to six other friends who did not know it. The secret continued to spread in this way on
Wednesday, Thursday and Friday mornings.
The total number of people (not counting Jim) who will know the secret on Friday afternoon is
A. `259`
B. `1296`
C. `1555`
D. `7776`
E. `9330`
The sum of the infinite geometric sequence 96, – 48, 24, –12, 6 . . . is equal to
A. `64`
B. `66`
C. `68`
D. `144`
E. `192`
The first four terms of a geometric sequence are 6400, `t_2` , 8100, – 9112.5
The value of `t_2` is
A. `– 7250`
B. `– 7200`
C. `–1700`
D. `7200`
E. `7250`
Each week a young boy saves an amount of his pocket money. The amount saved forms part of an arithmetic sequence. The table shows the amounts he saves in weeks 1 to 3.
If he continues to save in this way, the amount he will save in week eight is
A. `$1.45`
B. `$1.60`
C. `$1.65`
D. `$7.40`
E. `$8.00`
A toy train track consists of a number of pieces of track which join together.
The shortest piece of the track is 15 centimetres long and each piece of track after the shortest is 2 centimetres longer than the previous piece.
The total length of the complete track is 7.35 metres.
The length of the longest piece of track, in centimetres, is
A. `21`
B. `47`
C. `49`
D. `55`
E. `57`
The first three terms of an arithmetic sequence are `1, 3, 5 . . .`
The sum of the first `n` terms of this sequence, `S_n`, is
A. `S_n = n^2`
B. `S_n = n^2 - n`
C. `S_n = 2n`
D. `S_n = 2n - 1`
E. `S_n = 2n + 1`
For which one of the following geometric sequences is an infinite sum not able to be determined?
A. `4, 2, 1, 1 / 2\ . . .`
B. `1, 2, 4, 8\ . . .`
C. `–4, 2, –1, 1 / 2\ . . .`
D. `1, 1 / 2, 1 / 4, 1 / 8\ . . .`
E. `–1, 1/2, –1 / 4, 1 / 8\ . . .`
The number of bees in a colony was recorded for three months and the results are displayed in the table below.
If this pattern of increase continues, which one of the following statements is not true.
A. There will be nine times as many bees in the colony in month 5 than in month 3.
B. In month 4, the number of bees will equal 270.
C. In month 6, the number of bees will equal 7290.
D. In month 8, the number of bees will exceed 20 000.
E. In month 10, the number of bees will be under 200 000.
The first three terms of an arithmetic sequence are –3, –7, –11 . . .
An expression for the `n`th term of this sequence, `t_n`, is
A. `t_n = 1 - 4n`
B. `t_n = 1 - 8n`
C. `t_n = -3 - 4n`
D. `t_n = -3 + 4n`
E. `t_n = -7 + 4n`
The first term of a geometric sequence is `a`, where `a < 0`.
The common ratio of this sequence, `r`, is such that `r < –1`.
Which one of the following graphs best shows the first 10 terms of this sequence?
Mary plans to read a book in seven days.
Each day, Mary plans to read 15 pages more than she read on the previous day.
The book contains 1155 pages.
The number of pages that Mary will need to read on the first day, if she is to finish reading the book in seven days, is
A. `112`
B. `120`
C. `150`
D. `165`
E. `180`
A city has a population of 100 000 people in 2014.
Each year, the population of the city is expected to increase by 4%.
In 2018, the population is expected to be closest to
A. `108\ 000`
B. `112\ 000`
C. `115\ 000`
D. `117\ 000`
E. `122\ 000`
Paul went running every morning from Monday to Sunday for one week.
On Monday, Paul ran 1.0 km.
On Tuesday, Paul ran 1.5 km.
On Wednesday, Paul ran 2.0 km.
The number of kilometres that Paul ran each day continued to increase according to this pattern.
Part 1
The number of kilometres that Paul ran on Thursday is
A. `2.5`
B. `3.0`
C. `3.5`
D. `4.0`
E. `5.0`
Part 2
The total number of kilometres that Paul ran during the week is given by
A. the seventh term of an arithmetic sequence with `a = 1` and `d = 0.5`
B. the seventh term of a geometric sequence with `a = 1` and `r = 0.5`
C. the sum of seven terms of an arithmetic sequence with `a = 1` and `d = 0.5`
D. the sum of seven terms of a geometric sequence with `a = 1` and `r = 0.5`
E. the sum of seven terms of a Fibonacci-related sequence with `t_1 = 1` and `t_2 = 1.5`
Stefan swam laps of his pool each day last week.
The number of laps he swam each day followed a geometric sequence.
He swam 1 lap on Monday, 2 laps on Tuesday and 4 laps on Wednesday.
The number of laps that he swam on Thursday was
A. `5`
B. `6`
C. `8`
D. `12`
E. `16`
Before he began training, Jethro’s longest jump was 5.80 metres.
After the first month of training, his longest jump had increased by 0.32 metres.
After the second month of training, his longest jump had increased by another 0.24 metres.
After the third month of training, his longest jump had increased by another 0.18 metres.
If this pattern of improvement continues, Jethro’s longest jump, correct to two decimal places, will be closest to
A. `6.54\ text(metres.)`
B. `6.68\ text(metres.)`
C. `7.00\ text(metres.)`
D. `7.08\ text(metres.)`
E. `7.25\ text(metres.)`
The `n`th term in a geometric sequence is `t_n`.
The common ratio is greater than one.
A graph that could be used to display the terms of this sequence is
A team of swimmers was training.
Claire was the first swimmer for the team and she swam 100 metres.
Every other swimmer in the team swam 50 metres further than the previous swimmer.
Jane was the last swimmer for the team and she swam 800 metres.
The total number of swimmers in this team was
A. `9`
B. `13`
C. `14`
D. `15`
E. `18`
The graph above shows consecutive terms of a sequence.
The sequence could be
A. geometric with common ratio `r`, where `r< 0`
B. geometric with common ratio `r`, where `0 < r < 1`
C. geometric with common ratio `r`, where `r > 1`
D. arithmetic with common difference `d`, where `d< 0`
E. arithmetic with common difference `d`, where `d> 0`
A dragster is travelling at a speed of 100 km/h.
It increases its speed by
• 50 km/h in the 1st second
• 30 km/h in the 2nd second
• 18 km/h in the 3rd second
and so on in this pattern.
Correct to the nearest whole number, the greatest speed, in km/h, that the dragster will reach is
A. `125`
B. `200`
C. `220`
D. `225`
E. `250`
The second and third terms of a geometric sequence are 100 and 160 respectively.
The sum of the first ten terms of this sequence is closest to
A. `4300`
B. `6870`
C. `11\ 000`
D. `11\ 290`
E. `11\ 350`
On the first day of a fundraising program, three boys had their heads shaved.
On the second day, each of those three boys shaved the heads of three other boys.
On the third day, each of the boys who was shaved on the second day shaved the heads of three other boys.
The head-shaving continued in this pattern for seven days.
The total number of boys who had their heads shaved in this fundraising activity was
A. `2187`
B. `2188`
C. `3279`
D. `6558`
E. `6561`
Use the following information to answer Parts 1 and 2.
As part of a savings plan, Stacey saved $500 the first month and successively increased the amount that she saved each month by $50. That is, in the second month she saved $550, in the third month
she saved $600, and so on.
Part 1
The amount Stacey will save in the 20th month is
A. `$1450`
B. `$1500`
C. `$1650`
D. `$1950`
E. `$3050`
Part 2
The total amount Stacey will save in four years is
A. `$13\ 400`
B. `$37\ 200`
C. `$58\ 800`
D. `$80\ 400`
E. `$81\ 600`
The first four terms of a geometric sequence are
`4, – 8, 16, – 32`
The sum of the first ten terms of this sequence is
A. `–2048`
B. `–1364`
C. `684`
D. `1367`
E. `4096`
The prizes in a lottery form the terms of a geometric sequence with a common ratio of 0.95.
If the first prize is $20 000, the value of the eighth prize will be closest to
A. `$7000`
B. `$8000`
C. `$12\ 000`
D. `$13\ 000`
E. `$14\ 000`
The first three terms of a geometric sequence are
`0.125, 0.25, 0.5`
The fourth term in this sequence would be
A. `0.625`
B. `0.75`
C. `0.875`
D. `1`
E. `1.25`
The sequence
`3, 6, 9, 12\ . . .`
could be
A. Fibonacci.
B. arithmetic.
C. geometric.
D. alternating.
E. decreasing.
There are 3000 tickets available for a concert.
On the first day of ticket sales, 200 tickets are sold.
On the second day, 250 tickets are sold.
On the third day, 300 tickets are sold.
This pattern of ticket sales continues until all 3000 tickets are sold.
How many days does it take for all of the tickets to be sold?
A. `5`
B. `6`
C. `8`
D. `34`
D. `57`
The vertical distance, in m, that a hot air balloon rises in each successive minute of its flight is given by the geometric sequence
`64.0,\ \ 60.8,\ \ 57.76\ …`
The total vertical distance, in m, that the balloon rises in the first 10 minutes of its flight is closest to
A. `38`
B. `40`
C. `473`
D. `514`
E. `1280`
The first time a student played an online game, he played for 18 minutes.
Each time he played the game after that, he played for 12 minutes longer than the previous time.
After completing his 15th game, the total time he had spent playing these 15 games was
A. `186` minutes
B. `691` minutes
C. `1206` minutes
D. `1395` minutes
E. `1530` minutes
The graph above shows the first six terms of a sequence.
This sequence could be
A. an arithmetic sequence that sums to one.
B. an arithmetic sequence with a common difference of one.
C. a Fibonacci-related sequence whose first term is one.
D. a geometric sequence with an infinite sum of one.
E. a geometric sequence with a common ratio of one.
The first three terms of an arithmetic sequence are 3, 5 and 7.
The ninth term of this sequence is
A. `9`
B. `17`
C. `19`
D. `21`
E. `768`
`text(Part 1:)\ A`
`text(Part 2:)\ C`
`text (Part 1:)\ A`
`text (Part 2:)\ D` | {"url":"https://teacher.smartermaths.com.au/category/number-patterns-mc/aps-and-gps-mc/","timestamp":"2024-11-06T07:40:08Z","content_type":"text/html","content_length":"226033","record_id":"<urn:uuid:977a1774-2f5f-4496-a036-112fe4190188>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00020.warc.gz"} |
Measuring Up | Inside Mathematics
Measuring Up
In the problem Measuring Up, students use algebraic thinking to solve problems involving proportional relationships, measurement, scale, and multiplicative relationships. The mathematical topics that
underlie this problem are repeated addition, multiplication, division, percents, linear measurement, proportional reasoning, scale factors, scale, ratios, variables, functions, and algebraic
reasoning. In each level, students must make sense of the problem and persevere in solving it (MP.1). Each problem is divided into five levels of difficulty, Level A through Level E, to allow access
and scaffolding for students into different aspects of the problem and to stretch students to go deeper into mathematical complexity.
In this task, students are read the story “Stone Soup.” In the story, a recipe for the soup is shared. The students are asked to use manipulatives to determine how many carrots, onions, and chunks of
meat are needed to feed various numbers of people.
In this level, students read a version of the story “Stone Soup.” In the story, a recipe for the soup is shared. The students are asked to determine how many carrots, onions, and chunks of meat are
needed to feed various numbers of people.
In this level, students must use multiplication to solve word problems in situations involving equal groups by using drawings or skip counting (3.OA.A.3) to determine the number of ingredients for
various amounts of people.
In this level, students are challenged with different proportional relationships between quantities in the Stone Soup recipe. They will need to use inverse relationships to determine some values.
In this level, students must solve multistep word problems having whole-number answers (4.OA.A.3).
In this level, the students are presented with the challenge of determining a way to enlarge a picture to make a particular size of poster. The copier only has single settings for enlarging and
reducing. The students are asked to determine what combinations of enlarging and reducing are required to meet the size specifications of the poster.
In this level, students must apply their knowledge of finding a percent of a quantity (6.RP.A.3c, 7.RP.A.3) to determine what combinations of enlarging and reducing (6.EE.A.1) are required to meet
the size specifications of a specific poster size.The copier only has single settings for enlarging and reducing.
In this level, students analyze the relationship between two different measuring sticks that have different units of measure. The students investigate when the units on the two sticks correspond.
In this level, students recognize and represent proportional relationships by equations (7.RP.A.2b, 7.RP.A.2c). They analyze the relationship between two different measuring sticks that have
different units of measure. The students use and solve equations (6.EE.B.7) to investigate when the units on the two sticks correspond.
In this level, students are presented with a situation that involves three broken rulers with differing unit measures. Students are asked to determine methods for converting between the three
measuring sticks and to formalize their findings.
In this level, students use their knowledge of constant of proportionality and proportional relationships (7.RP.A.2b, 2c) to determine methods for converting between three broken rulers with
differing unit measures. Students are asked to formalize their findings (7.EE.B.4, A-CED.2).
Download the complete packet of Measuring Up Levels A-E here.
You can learn more about how to implement these problems in a school-wide Problem of the Month initiative in “Jumpstarting a Schoolwide Culture of Mathematical Thinking: Problems of the Month,” a
practitioner’s guide. Download the guide as iBook with embedded videos or Download as PDF without embedded videos.
To request the Inside Problem Solving Solutions Guide, please get in touch with us via the feedback form. | {"url":"https://www.insidemathematics.org/inside-problem-solving/measuring-up","timestamp":"2024-11-10T14:15:02Z","content_type":"text/html","content_length":"24131","record_id":"<urn:uuid:ec9aa421-677c-4669-9f95-a3a1a8908d55>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00749.warc.gz"} |
What is the difference between nominal risk-free rate and real risk-free rate of return?
The nominal one takes inflation into account
What is the difference between nominal risk-free rate and real risk-free rate of return?
What is the difference between nominal risk-free rate and real risk-free rate of return?
The nominal one takes inflation into account
If you want to change selection, open document below and click on "Move attachment"
Open itThe nominal risk-free rate of return includes both the real risk-free rate of return and the expected rate of inflation. A decrease in expected inflation rate would decrease the nominal
risk-free rate of return, but would have no effect on the real risk-free rate of return.
status not learned measured difficulty 37% [default] last interval [days]
repetition number in this series 0 memorised on scheduled repetition
scheduled repetition interval last repetition or drill
No repetitions | {"url":"https://buboflash.eu/bubo5/show-dao2?d=1621439941900","timestamp":"2024-11-06T15:23:20Z","content_type":"text/html","content_length":"35477","record_id":"<urn:uuid:b2d94f10-408e-4994-8a6b-207ecead2f1a>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00418.warc.gz"} |
Monthly Home Cost Calculator - Certified Calculator
Monthly Home Cost Calculator
Introduction: Buying a home is a significant financial commitment, and understanding the monthly costs associated with your mortgage, property taxes, and insurance is crucial. Our Monthly Home Cost
Calculator simplifies this process for you.
Formula: The Monthly Home Cost is calculated using the following formula:
• Monthly Payment = P * (r(1 + r)^n) / ((1 + r)^n – 1) + Property Taxes + Insurance
• P: Loan Amount
• r: Monthly Interest Rate (Annual Interest Rate / 12 / 100)
• n: Loan Term in months
How to Use:
1. Enter the Loan Amount you plan to borrow.
2. Input the Annual Interest Rate.
3. Specify the Loan Term in months.
4. Provide the monthly Property Taxes amount.
5. Enter the monthly Home Insurance cost.
6. Click the “Calculate” button to see your Monthly Home Cost.
Example: Let’s say you’re taking out a $250,000 loan at a 3.5% annual interest rate for 30 years (360 months). Your property taxes are $200 per month, and your home insurance is $100 per month. After
using the calculator, you’ll find your Monthly Home Cost is $1,592.70.
1. What is a Monthly Home Cost Calculator?
□ A Monthly Home Cost Calculator is a tool that helps you estimate the total monthly expenses of owning a home, including your mortgage, property taxes, and insurance.
2. How do I determine the Loan Amount?
□ The Loan Amount is the total amount you plan to borrow to purchase your home.
3. What is the Loan Term?
□ The Loan Term is the number of months you will take to repay the loan. Common terms include 15 years, 20 years, and 30 years.
4. How is the Monthly Interest Rate calculated?
□ The Monthly Interest Rate is derived from the Annual Interest Rate by dividing it by 12 and converting it to a decimal.
5. What’s included in the Monthly Home Cost?
□ The Monthly Home Cost includes your mortgage payment, property taxes, and home insurance.
6. Can I change the inputs after calculating?
□ Yes, you can modify the inputs and click “Calculate” again to update the Monthly Home Cost.
7. Is this calculator accurate for all regions?
□ This calculator provides a general estimate and may not account for specific regional variations in taxes and insurance costs.
8. Can I use this for a refinance calculation?
□ Yes, you can use this calculator to estimate the monthly costs for a home refinance.
9. Is this calculator suitable for commercial properties?
□ This calculator is designed for residential properties; commercial properties may have different cost structures.
10. Can I save my calculations?
• Unfortunately, this calculator doesn’t have a save feature, so make note of your results.
Conclusion: Our Monthly Home Cost Calculator is a valuable tool for prospective homeowners. It provides a quick and accurate estimate of your monthly expenses, allowing you to budget effectively and
make informed decisions about homeownership. Use it to plan for a financially secure and stress-free home purchase.
Leave a Comment | {"url":"https://certifiedcalculator.com/monthly-home-cost-calculator/","timestamp":"2024-11-02T17:55:50Z","content_type":"text/html","content_length":"57289","record_id":"<urn:uuid:ffe3354d-70cd-4cc6-8466-c1d8919a1840>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00228.warc.gz"} |
Combined Free And Forced Convection In An Inclined Channel With Discrete Heat Sources
It is studied in this work the mixed convection in an inclined rectangular channel.
Three constant heat sources q’ with finite lengths are flush mounted on the bottom surface of
a channel, while the remaining part of this surface is kept isolated. The upper wall is cooled
at a constant cold temperature Tc. At the inlet, the flow has constant velocity Uo and
temperature To profiles. The Reynolds number, the Grashof number, and the channel
inclination angle are ranged as follows: 1 ≤ Re ≤ 1000, 103 ≤ Gr ≤105, e 0° ≤ γ ≤ 90°,
respectively. The system of the governing equations is solved using the finite element method
with the Penalty formulation on the pressure terms and the Petrov-Galerkin perturbations on
the convective terms. Three comparisons are carried out to validate the computational code.
It is observed that the inclination angle has a stronger influence on the flow and heat transfer
for low Reynolds numbers, especially when it is between 0° and 45°. The cases which present
the lowest temperature distributions on the modules are those where the inclination angles
are 45° and 90° with little difference between them. The case where Gr = 105 and Re = 1000
is an exception where γ = 0° is the best channel inclination.
Asociación Argentina de Mecánica Computacional
Güemes 3450
S3000GLN Santa Fe, Argentina
Phone: 54-342-4511594 / 4511595 Int. 1006
Fax: 54-342-4511169
E-mail: amca(at)santafe-conicet.gov.ar
ISSN 2591-3522 | {"url":"https://amcaonline.org.ar/ojs/index.php/mc/article/view/375/0","timestamp":"2024-11-04T02:34:05Z","content_type":"application/xhtml+xml","content_length":"16511","record_id":"<urn:uuid:9c3e07cb-6efd-4b64-9b1f-a477d5b7a73b>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00133.warc.gz"} |
Year 7 Scheme of Learning
Addition and subtraction of fractions
Term 2 starting in week 10 :: Estimated time: 3 weeks
• Understand representations of fractions
• Convert between mixed numbers and fractions
• Add and subtract unit fractions with the same denominator
• Add and subtract fractions with the same denominator
• Add and subtract fractions from integers expressing the answer as a single fraction
• Understand and use equValent fractions
• Add and subtract fractions where denominators share a simple common multiple
• Add and subtract fractions with any denominator
• Add and subtract •mproper fractions and mixed numbers
• Use fractions in algebraic contexts
• Use equivalence to add and subtract decimals and fractions
For higher-attaining pupils:
Add and subtract simple algebraic fractions
This page should remember your ticks from one visit to the next for a period of time. It does this by using Local Storage so the information is saved only on the computer you are working on right | {"url":"https://transum.org/Maths/National_Curriculum/SOLblock.asp?ID_SOL=10","timestamp":"2024-11-13T19:38:13Z","content_type":"text/html","content_length":"29971","record_id":"<urn:uuid:826afb61-f596-45ee-8e88-6fb3abf34441>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00158.warc.gz"} |
There was a typo...
I'm confused. The answer was n(n+1), but the options were:
i.) 1/(n(n-2))
I guessed iv.) even though none of them seemed to work and the answer said: "All the denominators have the form n(n+1)." I think there was a typo in which there was a minus sign where there
should have been a plus sign. Can you fix that? | {"url":"https://forum.poshenloh.com/topic/1062/there-was-a-typo","timestamp":"2024-11-06T07:29:39Z","content_type":"text/html","content_length":"43263","record_id":"<urn:uuid:d9f9e36b-7196-4fe4-9927-2bc5ac5f9e9a>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00840.warc.gz"} |
conical ball mill diagram
WEBOct 1, 2022 · A new methodology to obtain a corrected Bond ball mill work index valid with nonstandard feed size. ... The diagram shows that the obtained values were in the area of the regression
curve, proving that the first condition for the determination of Bond work index was fulfilled. | {"url":"https://www.restaurantsanremo.fr/7544-conical_ball_mill_diagram.html","timestamp":"2024-11-02T17:31:24Z","content_type":"text/html","content_length":"38102","record_id":"<urn:uuid:2e4a4f07-4efa-4a27-acca-d793ba336729>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00898.warc.gz"} |
Sniffer Technique for Numerical Solution of Korteweg-de Vries Equation Using Genetic Algorithm
Sniffer Technique for Numerical Solution of Korteweg-de Vries Equation Using Genetic Algorithm ()
1. Introduction
Dynamical systems involving Ordinary Differential Equations (ODEs) occur in many branches of science, including Physics, Chemistry, Biology, Econometrics etc. More often, when the system is fairly
complex, involving nonlinear and non integrable system equations, analytical/symbolic solutions are not directly amenable and in this context obtaining the solution numerically helps understand the
dynamics of the system. Considerable work has been reported in the literature for obtaining numerical solution for ODE equations. Seaton et al. [1] have used Cartesian Genetic Programming for solving
differential equations. John Butcher [2] describes a variety of numerical methods for solving differential equations. Tsoulos and Lagaris [3] have used grammatical evolution method for solving ODE
equations. A variety of numerical techniques have been used in the past for solving ODEs, including use of Runge Kutta method, Predictor Corrector based method by Lambert [4], meshless Radial Basis
functions by Fasshauer [5], Artificial Neural Network with a regression based algorithm by Lagaris et al. [6]. Also, inference of pertinent system equations of ODE type have been tried out by using
the experimental data by Cao et al. [7] and Iba et al. [8].
In the present paper, we first consider the well known Korteweg-de Vries (KdV) equation which is a nonlinear PDE involving two independent variables in space x and time t, and is highly studied in
literature since the discovery of its soliton solution by Zabusky and Kruskal [9],
It is known that considering a traveling wave given by
and integrating with respect to
where A is an arbitrary constant of integration. Here, when
The KdV equation is known to have an analytical solution of the form
Setting a = 0, c = 1.2 and t = 0 now onwards, the true solution of KdV equation in one dimension is shown in Figure 1, where we have chosen even-spaced discrete x-values in the range [−4.0, 4.0] and
even-spaced discrete y-values in the range [0.0, 0.7] incorporating N grid points both in the x and y axes within the solution domain, where N is chosen as51.The chromosome representation therefore
is having a linear structure of a list having N elements, and each element is a whole number bounded within the range [1, N], as will be exemplified in Section 1.2 later.
Next we consider GA algorithm to get the solution sketched in Figure 1 numerically, without having apriori knowledge of its analytical solution. For this, we first define a fitness function as,
Figure 1. Solution of KdV equation in 1-dimension as given by its analytical solution of Equation (4), where we have chosen a = 0, c = 1.2 and t = 0.
where, N is the number of grid points considered, ^nd derivative in equation (5), we use the standard five-point formula based on central difference that gives very good approximations to the
numerical derivatives.
The fitness value for the true solution as given by Equation (5) is 0.4802-A, where the departure from -A is due to the discretized solution domain considered (resulting in chromosome representation
by whole numbers in range [1, N]), as well as due to the numerical inaccuracy in calculating the derivative values. We then set A = 0.4802 so as to offset the fitness measure such that it assumes the
ideal value 0 for the true solution. Thus any solution that has departure from the true solution is expected to give fitness measure > 0. It may be emphasized that any other choice of value for A
does not affect the GA search procedure.
1.1. Heuristic Methods for Solving One-Dimensional KdV Equation
We now consider one-dimensional KdV Equation (3) and proceed to solve it numerically. Combinatorially the search space is quite complex, due to the fact that a given linear chromosome, having a list
of 51 integers, can have 51 independent possible values for each element. The standard GA is found to be quite slow in making progress for converging towards the ideal value of 0. In this scenario,
it is envisaged to apply a heuristic search in the local surrounding area of the currently found best solution during GP iteration. The sniffing around the best solution is aimed at making the GA
search procedure more effective. In order to define a numerical measure for the locality of search, a sniffer radius denoted by Figure 2. It may be noted that we have earlier applied the sniffer
technique [10] for solving the inverse problem, namely inference of dynamical system equations (ODE and PDE) in their symbolic form, where data is used in the form of its solution defined either
numerically or in a symbolic form.
Figure 2 shows a hypothetical search space in which the true solution is shown by an isolated thick circle. The remaining thick circles surrounded by small circles represent the best solution found
by GA at various instances during the search iteration. Tiny dots spread randomly within these small circles represent the search carried out by the heuristic sniffer method. The radius of these
small circles denote the sniffer radius
It may be noted that during the heuristic search the boundary values
Figure 2. Schematic figure showing the usefulness of Sniffer technique (described in the text).
GA search. In the present GA experiments these boundary values are kept at 2 within the possible range of [1, N]. Sniff1 and Sniff2 methods descried below vary
o Sniff1: Each of the
o Sniff2: A contiguous range, specified by x[1] to x[2](selected randomly) and the corresponding
o Sniff3: A contiguous range, specified by x[1] to x[2] (selected randomly) and corresponding ^th degree to obtain a smooth function passing through the perturbed [i] values are then taken as the
modified chromosomes
o Sniff4: A list of good points within the chromosome list are identified corresponding to individual fitness components (right side of Equation (5) at given value i) by checking that it is less than
a pre-specified fraction of fitness value. Once the list of these good points is obtained, a polynomial function P[n](x) (i.e. up to order x^n, where n is set to 4), is then passed through these
points. The chromosome values are then replaced by interpolated values as given by P[n](x).
1.2. Genetic Algorithm Framework
Based on the pioneering work by David Goldberg [11], the basic framework of Genetic Algorithm is considered as an engine to discover various solution regimes stochastically and by natural selection
principle. Considering the fitness measure as given by equation (5), GA based iterative search is carried out using a code developed indigenously in Mathematica. As shown in Figure 1, the solution
domain [{−4, 4}, {0.0, 0.7}] for [x, y] range is mapped on to a corresponding grid [{1, N}, {1, N}], where we have chosen N = 51. Each chromosome is thus an integer list of length N representing
successive grid point numbers in the range [1, N] corresponding to the
The search procedure for finding the best chromosome having smallest fitness value is carried out as follows,
1. A pool of chromosomes is created having individuals that are stochastically generated representing candidate solutions
2. Iterations are carried out till an acceptable solution is obtained (having fitness value smaller then a specified small value, or maximum number of iterations have been achieved)
o The fitness values for the candidate chromosomes are calculated
o Standard genetic operators (copy, crossover and mutation) are applied on the pool of chromosomes to evolve the given population to anew population, hopefully containing a better pool of candidate
chromosomes. Few (at least 1) elite solutions are preserved (without any change) so as not to loose the best chromosomes discovered so far during the iterative search.
o Sniffer functions (Sniff1 to Sniff4) are activated intermittently after the GA iterations relax for a specified number of n iterations (we have chosen n = 10)
3. The best solution obtained represents the solution of the problem
The parameters used for the GA experiments are shown in Table 1, using which the KdV Equation (3) is solved numerically in the next section.
2. Results of GA Experiments
A useful feature of the GA program is to feed template curves that would be included in the pool of chromosomes. Using various forms for the template curves, different regimes of solutions are
obtained by the GA pro-
Table 1. Parameters used for the GA search procedure.
cedure aided by the sniffer technique. The GA solutions along with the templates used are shown below. Thus the inclusion of various template curves as one of the chromosomes is aimed at trying out
different useful chromosome structures as potential candidates during the GA solutions.
2.1. Single Bump Symmetric Templates
As shown in Figure 3, two symmetric template curves (triangular (a) and a square like bump (b)) are used that generate an exact solution (c) with fitness value as 0.Here the GA iterations were
monitored to find the effectiveness of sniffer technique. This analysis indicated that the percentage of iterations in which sniffer methods were successful in improving the fitness value as compared
to GA iterations were 74% in case of GA experiment using the profile of Figure 3(a), and it was 83% in case of profile of Figure 3(b).
2.2. Single Bump Asymmetric Templates
As shown in Figure 4, two asymmetric template curves (triangular curves leaning forward (a) and leaning backward (b)) are used and in both cases the exact solution (c) is generated.
2.3. Randomly Generated Templates
As shown in Figure 5, totally random templates (a) and (b) are used, and in both cases the exact solution (c) is generated.
It is interesting to note that in all above GA experiments, the best solution generated by sniffer assisted GA algorithm matches exactly with the true solution having fitness measure as 0.
2.4. Sensitivity on Fitness Measure
In order to check the sensitivity of the best solution obtained by GA, we next test the sensitivity of fitness measure as a result of applying small perturbations in the best chromosome structure. We
consider following two scenarios and calculate fitness measure 1000 times and then taking the average,
1. Perturbing [i] values except at the end points.
2. Perturbing [i] values except at the end points.
In the first scenario, the average variation in fitness measure is found to be 0.172, 0.203 and 0.526 for perturbation values as 1, 2 and 5 respectively. In the second scenario, the average variation
in fitness measure is found to be 1.195, 2.284 and 5.675 for perturbation values 1, 2 and 5 respectively.
It may be noted that the fitness values for the temple curves used Figure 3(a), Figure 3(b), Figure 4(a), Figure 4(b), Figure 5(a) and Figure 5(b) are 0.185, 3.181, 0.235, 0.365, 28.016 and 29.560
respectively. Thus it is seen that the value for sniffer radius used during the GA experiments, i.e. 5, is good enough to bring out relevant changes in fitness measure for the templates corresponding
to Figure 3 and Figure 4. However for totally random templates of Figure 5, the sniffer technique would also heavily depend on the GA procedure to bring the variety in the chromosome structures and
eventually hit upon best value close enough to the true solution as highlighted in schematic diagram of Figure 2 earlier.
Figure 3. GA solution (c) obtained using the triangular (a) and square bump (b) symmetric templates.
Figure 4. GA solution (c) obtained using the triangular asymmetric templates (a) and (b) respectively.
Figure 5. GA solution (c) obtained using the randomly generated templates (a) and (b).
3. Conclusion and Outlook
It has been shown that the sniffer technique assisted Genetic Algorithm approach is quite useful in solving one- dimensional KdV equation. In the present paper the numerical solutions have
successfully been obtained for the well-known KdV differential equation in one dimension. The search method implements several useful variants of a novel heuristic method, called sniffer technique
that helps make a detailed search in the vicinity of the best solution achieved by GA at a given instance during its iterations. The method would especially be useful for solving stiff as well as
complex differential equations for which no analytical solutions exist. Work is in progress for generating numerical solution for the ODE equations in two dimensions (involving space and time), of
the type in Equation (1), using which it would be interesting to generate time evolution of 1-soliton as well as 2-soliton profiles fed at time t = 0. | {"url":"https://scirp.org/journal/paperinformation?paperid=57661","timestamp":"2024-11-13T23:18:48Z","content_type":"application/xhtml+xml","content_length":"104841","record_id":"<urn:uuid:bdb9163a-267e-442b-a761-cee16f9f8215>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00052.warc.gz"} |
Heap structure and heap sorting
Heap (priority queue)
Time complexity:
The time complexity of initializing heap building is O(n)
Heap sort rebuild heap O(nlogn)
PriorityQueue small top heap
Small top heap: the weight of any non leaf node is not greater than the weight of its left and right child nodes
constructor Function introduction
PriorityQueue() Create an empty priority queue with a default capacity of 11
PriorityQueue(int Create a priority queue with an initial capacity of initialCapacity. Note: initialCapacity cannot be less than 1, otherwise an IllegalArgumentException exception will
initialCapacity) be thrown
PriorityQueue(Collection c) Use a collection to create priority queues
You can also specify the implementation of the comparator interface during initialization,
Common APIs
Function Function introduction
boolean Insert element E and return true after successful insertion. If the e object is empty, a NullPointerException exception will be thrown, which reduces the time complexity O ( l o g 2 N )
offer(E e) O(log_2N) O(log2 # N), note: capacity expansion will be carried out when there is not enough space
E peek() Gets the element with the highest priority. If the priority queue is empty, null is returned
E poll() Remove the element with the highest priority and return it. If the priority queue is empty, return null
int size() Get the number of valid elements
void clear() Empty boolean
isEmpty() Check whether the priority queue is empty, and return true if it is empty
1. PriorityQueue is thread unsafe, and PriorityBlockingQueue is thread safe
2. The objects stored in PriorityQueue must be able to compare sizes. If the sizes cannot be compared, ClassCaseException will be thrown
3. Cannot insert null object, otherwise NullPointerException will be thrown
4. With automatic capacity expansion mechanism
5. The time complexity of inserting and killing is
O ( l o g 2 N ) O(log_2N) O(log2N)
6. The bottom layer uses a heap
Capacity expansion process jdk1.8
private void grow(int minCapacity) {
int oldCapacity = queue.length;
// Double size if small; else grow by 50%
int newCapacity = oldCapacity + ((oldCapacity < 64) ?
(oldCapacity + 2) :
(oldCapacity >> 1));
// overflow-conscious code
if (newCapacity - MAX_ARRAY_SIZE > 0)
newCapacity = hugeCapacity(minCapacity);
queue = Arrays.copyOf(queue, newCapacity);
It can be seen from the expansion method
• If the capacity is less than 64, use oldCapacity+oldCapacity+2
• If the capacity is greater than 64, use oldcapacity + (oldcapacity > > 1) (equivalent to 1.5 times capacity expansion)
Underlying data structure
Concept of heap
A key set k={k,k1,k2,k3... kn-1} is stored in the array in the order of a complete binary tree
In a one-dimensional array
• Those satisfying ki < = k2i + 1 and Ki < = k2i + 2 are called small heaps
• Those that satisfy ki > = k2i + 1 and Ki > = k2i + 2 are called piles
The heap with the largest root node is called the maximum heap or large root heap, and the heap with the smallest root node is called the minimum heap or small root heap.
• The value of a node in the heap is always not greater than or less than the value of the parent node
• Heap is always a complete binary tree
• If I is 0, the node represented by I is the root node, otherwise the parents of node i are (i-1)/2
• If 2*i+1 is less than the number of nodes, the left child subscript of node i is 2*i+1, otherwise there is no left child
• If 2*i+2 is less than the number of nodes, the subscript of the right child of node i is 2*i+2, otherwise there is no right child
Upward adjustment
Upward adjustment process (minimum heap)
1. First, set the penultimate leaf node as the current node and mark it as cur. Find out its parent node and mark it with parent
2. Compare the values of parent and cur. If cur is smaller than parent, it does not meet the rule of small top heap and needs to be exchanged. If cur is larger than parent, it will not be exchanged.
At this time, the adjustment is over
3. If the condition is not met, after the exchange, cur will be reset to the subscript of the parent node and the cycle will be restarted. Then check whether the property of the minimum heap is met
until the condition is met or cur is less than or equal to 0
Downward adjustment
1. First, set the root node as the current node, mark it as cur, compare the values of the left and right subtrees, find out the smaller value, and mark it with child
2. Compare the values of child and cur. If child is smaller than cur, it does not meet the rule of small heap and needs to be exchanged. If cur is larger than parent, it will not be exchanged. At
this time, the adjustment is over
3. If the condition is not met, after the exchange, cur will be reset to the child node subscript and the cycle will be restarted. Then check whether the property of the minimum heap is met until
the condition is met or cur is less than or equal to 0
Heap creation (adjust up)
When we have an array that does not meet the heap structure requirements, we need to adjust from the subtree of the penultimate non leaf node to the tree of the root node
ArrayList<Integer> integers = new ArrayList<>();
// integers.add(5);
PriorityQueue<Integer> queue1 = new PriorityQueue<>(integers);
The initial arraylist array element is transformed into a complete binary tree in the form shown in the figure above
Heap adjustment
There are two key methods
private void heapify() {
for (int i = (size >>> 1) - 1; i >= 0; i--)
siftDown(i, (E) queue[i]);
The passed in parameter shifts size to the right without sign > > > 1 and - 1
>>>Move right without symbol:
After moving to the right, the empty bits on the left are filled with zeros. Bits moved out to the right are discarded
The initial size is 7 (0111), and the parameter i passed in after operation is 3 (0011) - 1 = 2
private void siftDownComparable(int k, E x) {
Comparable<? super E> key = (Comparable<? super E>)x;
int half = size >>> 1; // loop while a non-leaf
while (k < half) {
int child = (k << 1) + 1; // assume left child is least
Object c = queue[child];
int right = child + 1;
if (right < size &&
((Comparable<? super E>) c).compareTo((E) queue[right]) > 0)
c = queue[child = right];
if (key.compareTo((E) c) <= 0)
queue[k] = c;
k = child;
queue[k] = key;
From the perspective of debugging process
• half is 3
• k is the parameter i passed in from the previous method, and the value is 2
• child is (k < < 1 +) 1 = 5 and reght is 6, respectively
The rule of moving left only remembers one thing: discard the highest bit, and 0 makes up the lowest bit
Heap insertion
The insertion process is to insert the data into the end of the array, and then adjust it upward
• Suppose you have these elements in the heap before adding element 5
• Now add element 5
private void siftUpComparable(int k, E x) {
Comparable<? super E> key = (Comparable<? super E>) x;
while (k > 0) {
int parent = (k - 1) >>> 1;
Object e = queue[parent];
if (key.compareTo((E) e) >= 0)
queue[k] = e;
k = parent;
queue[k] = key;
• First, the value of k is 7, which is the subscript of the element to be inserted. If the condition > 0 is met, enter the while loop
• The subscript of the parent element of the parent node is (k-1) > > > 1 is 3, and the value of the corresponding element e is 48
• Then use the compareTo method to compare e with the element key to be inserted (ascII table)
• If the parent node element 48 is large, the compareTo method returns a negative number (returns the ASCII code difference of the first character),
• Therefore, move the parent node element down first, set the value of k as the subscript of the parent node and the subscript of the element to be inserted, and then judge the next cycle until it
reaches the top of the heap to end the cycle
• Then put the key value in the corresponding position in the heap
Deletion of heap
The deletion of the heap is a value, which deletes the data at the top of the heap. The process is to exchange the data at the top of the heap with the last data, then delete the last data and adjust
the algorithm downward
Heap sort
Ascending – large top reactor
1. The sequence to be sorted is constructed into a large top heap
2. At this point, the maximum value of the whole sequence is the root node at the top of the heap.
3. Swap it with the end element, where the end is the maximum.
4. Then reconstruct the remaining n-1 elements into a heap, which will get the sub small value of n-1 elements. If you execute it repeatedly, you can get an ordered sequence.
package Classroom code.data structure.Courseware practice.heap;
public class MyHeaphigh {
int[] queue = new int[1000];
int size = 0;
//Heap sort
public void hepsort(int[] array) {
for (int i = 0; i < queue.length - 1; i++) {
swap(0, size);
size = array.length;
//Adjust array to heap
public void createheap(int[] array) {
queue = array;
size = array.length;
for (int i = (queue.length - 2) >> 1; i >= 0; i--) {
//Add element offer
// Upward adjusted
public boolean offer(int e) {
int i = this.size;
this.size = i + 1;
queue[i] = e;
return true;
//Large top heap ascending
//Downward adjustment
private void shftdown(int parent) {
* The first step is to calculate the left child node and the right child node, and select the smallest as the child node
* The parent node and child node are exchanged, and the cycle starts from the child node
int left = parent * 2 + 1, right;
while (left < size) {
right = left + 1;
if (right < size && queue[left] < queue[right]) {
if (queue[parent] > queue[left]) {
} else {
swap(parent, left);
parent = left;
left = parent * 2 + 1;
//Upward adjustment
private void shftup(int child) {
while (child > 0) {
int parent = (child - 1) >> 1;
if (queue[parent] > queue[child]) {
} else {
swap(child, parent);
child = parent;
//Delete element remove
public void remove() {
//The deleted element is the stack top element
// First swap the top element with the last one, and then adjust it down
swap(0, --size);
private void swap(int left, int right) {
int temp = queue[left];
queue[left] = queue[right];
queue[right] = temp;
Descending - small top reactor
package Classroom code.data structure.Courseware practice.heap;
public class MyHeaplow {
int[] queue = new int[1000];
int size = 0;
//Heap sort
public void hepsort(int[] array){
for (int i = 0; i < queue.length-1; i++) {
//Adjust array to heap
public void createheap(int[] array) {
queue = array;
size = array.length;
for (int i = (queue.length-2)>>1; i >=0 ; i--) {
//Add element offer
// Upward adjusted
public boolean offer(int e) {
int i = this.size;
this.size = i + 1;
queue[i] = e;
return true;
//Descending order of small top reactor
//Downward adjustment
private void shftdown(int parent) {
* The first step is to calculate the left child node and the right child node, and select the smallest as the child node
* The parent node and child node are exchanged, and the cycle starts from the child node
int left = parent * 2 + 1, right;
while (left < size) {
right = left + 1;
if (right < size && queue[left] > queue[right]) {
if (queue[parent] < queue[left]) {
} else {
swap(parent, left);
parent = left;
left = parent * 2 + 1;
//Upward adjustment
private void shftup(int child) {
while (child > 0) {
int parent = (child - 1)>>1;
if (queue[parent] < queue[child]) {
} else {
swap(child, parent);
child = parent;
//Delete element remove
public void remove() {
//The deleted element is the stack top element
// First swap the top element with the last one, and then adjust it down
// queue[0] = queue[--size];
private void swap(int left, int right) {
int temp = queue[left];
queue[left] = queue[right];
queue[right] = temp;
The large top heap corresponds to descending order and the small top heap corresponds to ascending order, in which only the elements change. The comparison method is used for comparison
topk problem
What is the topk problem? Here are some examples
1. Given 100 int numbers, find the maximum 10;
2. Given 1 billion int numbers, find the largest 10 (these 10 numbers can be out of order);
3. Given 1 billion int numbers, find the largest 10 (these 10 numbers are sorted in order);
4. Given 1 billion non repeating int numbers, find the largest 10;
5. Given 10 arrays, each array has 100 million int numbers, and find the largest 10 of them;
6. Given 1 billion string type numbers, find the largest 10 (only need to check once);
7. Given 1 billion numbers of string type, find the largest K (you need to query repeatedly, where k is a random number).
What is the idea to solve the problem?
1. Divide and conquer / hash mapping + hash statistics + heap / fast / merge sorting;
2. Double barrel Division
3. Bloom filter/Bitmap；
4. Trie tree / database / inverted index;
5. External sorting;
6. Hadoop/Mapreduce for distributed processing.
1. Massive log data, extract the IP with the most visits to Baidu on a certain day.
Topic analysis
The first method can be used for this problem, divide and conquer / hash mapping + hash statistics + heap / fast / merge sorting
• First of all, for massive data, our memory must not be able to store it. Then we can adopt hash mapping and modular mapping to decompose large files into small files,
• Then the HashMap structure is used for frequency statistics
• After the frequency statistics of each small file are completed, heap sorting or fast sorting can be adopted to get the ip with the most times
Among them, specific analysis is carried out
I. file segmentation
ip is 32 bits, so there are at most 2 ^ 32 different IPs. The essence of IPv4 address is a 32-bit binary string, and an int is also four bytes and 32 bits, so we can use int to store ip,
For an ipv4 address: 192.168.1.3
If the byte is stored with int after removing the decimal point, the maximum value of ipv4 is 255255255255, and the int value range is - 231-231-1, that is - 2147483648-2147483647. Obviously, it
can't be stored, so we have to take other ways to store it
First, divide the ip address into 192 168 1 3, then convert it into binary, and then link it. The result is 192 (10) = 11000000 (2); 168(10) = 10101000(2) ; 1(10) = 00000001(2) ; 3(10) = 00000011(2)
The corresponding conversion to int value is - 1062731775
So it can be stored in a variable of type int
Or store it into a long variable, and then take a module of 1000 to divide the whole large file into 1000 small files. (sometimes the ip distribution is not so uniform, so it may need to be divided
many times.)
Such a file has such characteristics after being divided. If n files are divided
• All records of the same ip will be stored in one file
• Each file can cover up to 2^32/n IPS
II. Statistics
After the file is divided to be stored in memory, you can use a data structure such as HashMap < integer, long > for statistics,
Three sort
Heap sort each divided small file (this problem only requires the most, and traversal can be used), select the element that appears the most times, and then store the result in another appropriate
place. After sorting all small files, we will have n key value pairs, and then sort them
2. The search engine will record all search strings used by users for each search through log files, and the length of each query string is 1-255 bytes. Find the 10 most popular search strings
Topic analysis
Suppose there are 10 million records at present (the repetition of these query strings is relatively high, although the total number is 10 million, but if the repetition is removed, it will not
exceed 3 million. The higher the repetition of a query string, the more users query it, that is, the more popular it is.) please count the 10 most popular query strings, and the memory required
should not exceed 1G.
I. statistics
Because the string repetition of the query is relatively high, the data set is estimated. If the last repeated result set is small, it can not be divided, but directly use hashmap for statistics
Two sort
Finally, sort the heap, create a 10 size small top heap, traverse the data, and find the ten key s with the largest value
3. There is a 1G file, in which each line is a word. The size of the word does not exceed 16 bytes, and the memory limit is 1M. Returns the 100 words with the highest frequency.
Topic analysis
The memory is 1M, so in order to process these data, we need to divide it into 1024 small files at least to store it in memory. One word is 16 bytes, | {"url":"https://programmer.group/heap-structure-and-heap-sorting.html","timestamp":"2024-11-05T16:59:35Z","content_type":"text/html","content_length":"28710","record_id":"<urn:uuid:1690e2e3-14f6-440e-b3a5-6db4e9651a33>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00033.warc.gz"} |
PROC KDE: Binning :: SAS/STAT(R) 9.22 User's Guide
Binning, or assigning data to discrete categories, is an effective and fast method for large data sets (Fan and Marron; 1994). When the sample size
To bin a set of weighted univariate data bin counts.
This procedure replaces the data simple binning, versus the finer linear binning described in Wand (1994). PROC KDE uses simple binning for the sake of faster and easier implementation. Also, it is
assumed that the bin centers
If you replace the data
with the same notation as used previously. To evaluate this estimator at the
The same idea of binning works similarly with bivariate data, where you estimate | {"url":"http://support.sas.com/documentation/cdl/en/statug/63347/HTML/default/statug_kde_sect013.htm","timestamp":"2024-11-08T08:36:03Z","content_type":"application/xhtml+xml","content_length":"19598","record_id":"<urn:uuid:3502332c-bf2b-4b11-8aac-3965b15a0737>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00291.warc.gz"} |
Molecular Modeling Basics
Tight binding DFT (DFTB) is a semi-empirical method with speed and accuracy similar to NDDO-based semiempirical methods such as AM1, PM3, and PM6. Currently there are three types of DFTB methods
called DFT1, DFTB2, and DFTB3. DFTB1 and DFTB2 are sometimes called non-SCC DFTB (non-selfconsistent charge) and SCC-DFTB, respectively. DFTB3 is generally considered the most accurate for molecules
and there are several parameter sets for DFTB2 and DFTB3 for different elements. Compared to PM6, DFTB has so far been parameterized for relatively
few elements
The closed shell DFTB1 energy is computed from the following equation
$$E^{\text{DFTB1}}=\sum_i^{N/2} \sum_\mu^K \sum_\nu^K 2 C_{\mu i} C_{\nu i} H^0_{\mu\nu}+\sum_A \sum_{B>A} E^{\text{rep}}_{AB}$$
where $C_{\mu i}$ is the molecular orbital coefficient for MO $i$ and basis function $\mu$.
$$H^0_{\mu\nu}= \begin{cases}
\varepsilon^{\text{free atom}}_{\mu\mu} & \text{if } \mu=\nu \\
0 & \text{if } A=B, \mu\ne\nu\\
\langle \chi_\mu | \hat{T}+V_{\text{eff}}[\rho_0^A+\rho_0^B] | \chi_\nu \rangle & \text{if } A\ne B
Here, $\varepsilon^{\text{free atom}}_{\mu\mu}$ is an orbital energy of a free atom, $\chi$ is a valence Slater-type orbital (STO) or numerical orbital, $\hat{T}$ is the electronic kinetic energy
operator, $V_{\text{eff}}$ is the Kohn-Sham potential (electron-nuclear attraction, electron-electron repulsion, and exchange correlation), and $\rho_0^A$ is the electron density of neutral atom $A$.
DFT calculations on free atoms using some functional yield $\left\{\varepsilon^{\text{free atom}}_{\mu\mu} \right\}$, $\left\{\chi\right\}$, and $\rho_0$, which are then used to compute $H^0_{\mu\nu}
$ for A-B atom pairs at various separations $R_{AB}$ and stored. When performing DFTB calculations $H^0_{\mu\nu}$ is simply computed for each atom pair A-B by interpolation using this precomputed
data set.
Similarly, the overlap matrix $\left\{ \langle \chi_\mu | \chi_\nu \rangle \right\}$ need to orthonormalize the MOs are computed for various distances and stored for future use.
$E^{\text{rep}}_{AB}$ is an empirical repulsive pairwise atom-atom potential with parameters adjusted to minimize the difference in atomization energies, geometries, and vibrational frequencies
computed using DFTB and DFT or electronic structure calculations for set of molecules.
So, a DFTB1 calculation is performed by constructing $\mathbf{H}^0$, diagonalizing it to yield $\mathbf{C}$, and then computing $E^{\text{DFTB1}}$.
$$E^{\text{DFTB2}}=E^{\text{DFTB1}}+\sum_A \sum_{B>A} \gamma_{AB}(R_{AB})\Delta q_A\Delta q_B$$
where $\Delta q_A$ is the Mulliken charge on atom $A$ and $\gamma_{AB}$ is a function of $R_{AB}$ that tends to $1/R_{AB}$ at long distances.
The Mulliken charges depend on $\mathbf{C}$ so a selfconsistent calculation is required:
1. Compute DFTB1 MO coefficients, $\mathbf{C}$
2. Use $\mathbf{C}$ to compute $\left\{ \Delta q \right\}$
3. Construct and diagonalize $H_{\mu \nu}$ to get new MO coefficients, $\mathbf{C}$
$$H_{\mu \nu}=H_{\mu \nu}^0 + \frac{1}{2} S_{\mu\nu} \sum_C (\gamma_{AC}+\gamma_{BC})\Delta q_C, \mu \in A, \nu \in B$$
4. Repeat steps 2 and 3 until selfconsistency.
$$E^{\text{DFTB3}}=E^{\text{DFTB2}}+\sum_A \sum_{B>A} \sum_{C>B>A} \Gamma_{AB}\Delta q_A^2\Delta q_B$$
$\Gamma_{AB}$ is computed using interpolation using precomputed data. A SCF calculation is required.
Parameter sets and availability
DFTB is available in
a variety of software packages
. I don't believe DFTB3 is currently in Gaussian and DFTB is also available in CHARMM and CP2K. DFTB will soon be available in GAMESS.
Note that each user/lab must download the parameter file separately
. There are several parameter sets. The most popular sets for molecules are the MIO (
aterials and b
logical systems) for DFTB2 and 3OB (DFT
3 o
rganic and
iological applications).
Dispersion and hydrogen bond corrections
Just like DFT and PM6, the DFTB can be corrected for
and hydrogen bond effect.
This work is licensed under a
Creative Commons Attribution 4.0
The September issue of Computational Chemistry Highlights is out.
CCH is an overlay journal that identifies the most important papers in computational and theoretical chemistry published in the last 1-2 years. CCH is not affiliated with any publisher: it is a free
resource run by scientists for scientists. You can read more about it here.
Table of content for this issue features contributions from CCH editors Steven Bachrach and Jan Jensen:
Why Bistetracenes Are Much Less Reactive Than Pentacenes in Diels–Alder Reactions with Fullerenes
8π-Electron Tautomeric Benziphthalocyanine: A Functional Near-Infrared Dye with Tunable Aromaticity
Interested in more? There are many ways to subscribe to CCH updates.
This work is licensed under a Creative Commons Attribution 4.0 | {"url":"https://molecularmodelingbasics.blogspot.com/2014/10/","timestamp":"2024-11-03T12:51:10Z","content_type":"application/xhtml+xml","content_length":"117528","record_id":"<urn:uuid:0031ac92-9801-40ed-a11a-3893549e66df>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00250.warc.gz"} |
Balloon Bursting Buttons
Otherwise great activity.
[Transum: The Start Again button allows you to do the challenge again with the same numbers]
When you first load this starter the target numbers are 16, 24, 97 and 624. If you refresh the page or click the "Change Numbers" button you will get four random target numbers. Let's agree that we
are looking for the least number of button presses which achieves the original four target numbers.
Rather than just the number of button presses you should also record the buttons pressed E.g. 15+1+5+… etc
Can anyone beat that?
Beat that!
Keep it up and thank you!
Using only these keys on your calculator make each of the target numbers on the balloons. Use the minimum number of key presses.
Brilliant Balloon Busting!
Is it possible to do it using fewer than key presses?
Sign in to your Transum subscription account to see the answers
Note to teacher: Doing this activity once with a class helps students develop strategies. It is only when they do this activity a second time that they will have the opportunity to practise those
strategies. That is when the learning is consolidated. Click the button above to regenerate another version of this starter from random numbers.
Your access to the majority of the Transum resources continues to be free but you can help support the continued growth of the website by doing your Amazon shopping using the links on this page.
Below is an Amazon link. As an Amazon Associate I earn a small amount from qualifying purchases which helps pay for the upkeep of this website.
Educational Technology on Amazon
Teacher, do your students have access to computers such as tablets, iPads or Laptops? This page was really designed for projection on a whiteboard but if you really want the students to have access
to it here is a concise URL for a version of this page without the comments:
However it would be better to assign one of the student interactive activities below.
Here is the URL which will take them to a calculator workout. | {"url":"https://transum.org/Software/SW/Starter_of_the_day/starter_September28.ASP","timestamp":"2024-11-14T18:43:33Z","content_type":"text/html","content_length":"38717","record_id":"<urn:uuid:9fd8efa4-abb0-4d8e-906a-a6c11d9ca482>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00622.warc.gz"} |
Inverse Relation
An Inverse Relation is a Binary Relation between Binary Relations where the Variable order is switched.
• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/inverse_function Retrieved:2015-12-30.
□ In mathematics, an inverse function is a function that "reverses" another function. That is, if is a function mapping to, then the inverse function of maps back to .
• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/inverse_relation Retrieved:2015-12-30.
□ In mathematics, the inverse relation of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the inverse of the relation
'child of' is the relation 'parent of'. In formal terms, if [math]\displaystyle{ X \text{ and } Y }[/math] are sets and [math]\displaystyle{ L \subseteq X \times Y }[/math] is a relation from
X to Y then [math]\displaystyle{ L^{-1} }[/math] is the relation defined so that [math]\displaystyle{ y\,L^{-1}\,x }[/math] if and only if [math]\displaystyle{ x\,L\,y }[/math] . In
set-builder notation, [math]\displaystyle{ L^{-1} = \{(y, x) \in Y \times X \mid (x, y) \in L \} }[/math] .
The notation comes by analogy with that for an inverse function. Although many functions do not have an inverse; every relation does have a unique inverse. Despite the notation and
terminology, the inverse relation is not an inverse in the sense of group inverse; the unary operation that maps a relation to the inverse relation is however an involution, so it induces the
structure of a semigroup with involution on the binary relations on a set, or more generally induces a dagger category on the category of relations as detailed below. As a unary operation,
taking the inverse (sometimes called inversion) commutes however with the order-related operations of relation algebra, i.e. it commutes with union, intersection, complement etc.
The inverse relation is also called the converse relation or transpose relation— the latter in view of its similarity with the transpose of a matrix.^[1] It has also been called the opposite
or dual of the original relation. Other notations for the inverse relation include L^C, L^T, L^~ or [math]\displaystyle{ \breve{L} }[/math] or L° or L^∨.
1. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named SchmidtStröhlein1993 | {"url":"https://www.gabormelli.com/RKB/Inverse","timestamp":"2024-11-07T22:42:18Z","content_type":"text/html","content_length":"43336","record_id":"<urn:uuid:0dd3b4de-38c4-4eb2-b5c6-fe4d0a34aa52>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00435.warc.gz"} |
Forecasting – The Dan MacKinlay stable of variably-well-consider’d enterprises
Vegan haruspicy
June 17, 2015 — October 8, 2022
model selection
signal processing
stochastic processes
time series
Time series prediction niceties, where what needs to be predicted is the future. Filed under forecasting because in machine learning terminology, prediction is a general term that does not imply
extrapolation into the future necessarily.
🏗 handball to Rob Hyndman.
1 Recursive estimation
See recursive identification for generic theory of learning under the distribution shift induced by a moving parameter vector.
3 Calibration of probabilistic forecasts
See calibration.
4 Training data intake
There is a common pattern with training time series models that they each predict the next observation from the previous observations, which is not how a classic data loader works in machine
learning. The time at which the future observations are evaluated is the horizon and the ones used to make that prediction are the history. For patterns to handle this in neural networks in
particular, see Recurrent neural networks.
5 Software
Not comprehensive, just noting some useful time series forecasting models/packages as I encounter them. Peter Cotton attempts to collate Popular Python Time Series Packages.
5.1 Tidyverse time series analysis and forecasting packages
A good first stop.
You can find a presentation on these tools by Rob Hyndman.
5.3 prophet
prophet (R/Python/Stan):
is a procedure for forecasting time series data. It is based on an additive model where non-linear trends are fit with yearly and weekly seasonality, plus holidays. It works best with daily
periodicity data with at least one year of historical data. Prophet is robust to missing data, shifts in the trend, and large outliers.
Is Facebook’s “Prophet” the Time-Series Messiah, or Just a Very Naughty Boy? via Sean J. Taylor on Twitter
This post rips Prophet (a forecasting package I helped create) to shreds and I agree with most of it🥲. I always suspected the positive feedback was mostly from folks who’d had good
results—conveniently the author has condensed many bad ones into one place.
5.4 Silverkite
Hosseini et al. (2021)
The Greykite library provides flexible, intuitive and fast forecasts through its flagship algorithm, Silverkite.
Silverkite algorithm works well on most time series, and is especially adept for those with changepoints in trend or seasonality, event/holiday effects, and temporal dependencies. Its forecasts
are interpretable and therefore useful for trusted decision-making and insights.
The Greykite library provides a framework that makes it easy to develop a good forecast model, with exploratory data analysis, outlier/anomaly preprocessing, feature extraction and engineering,
grid search, evaluation, benchmarking, and plotting. Other open source algorithms can be supported through Greykite’s interface to take advantage of this framework, as listed below.
5.5 Causal impact
🏗 find out how Causal impact works. (Based on Brodersen et al. (2015).)
5.6 asap
Automatic Smoothing for Attention Prioritization in Time Series
ASAP automatically smooths time series plots to remove short-term noise while retaining large-scale deviations.
6 Makridakis competitions
The M4 dataset is a collection of 100,000 time series used for the fourth edition of the Makridakis forecasting Competition. The M4 dataset consists of time series of yearly, quarterly, monthly
and other (weekly, daily and hourly) data, which are divided into training and test sets. The minimum numbers of observations in the training test are 13 for yearly, 16 for quarterly, 42 for
monthly, 80 for weekly, 93 for daily and 700 for hourly series. The participants were asked to produce the following numbers of forecasts beyond the available data that they had been given: six
for yearly, eight for quarterly, 18 for monthly series, 13 for weekly series and 14 and 48 forecasts respectively for the daily and hourly ones.
Now we are up to M5 and M6 is cooking.
7 Micropredictions.org
micropredictions is a quixotic project my colleagues have forwarded to me. Included here as a spur. The micropredictions FAQ says:
What’s microprediction you say?
The act of making thousands of predictions of the same type over and over again. Microprediction can
□ Clean and enrich live data
□ Alert you to outliers and anomalies
□ Provide you short term forecasts
□ Identify patterns in model residuals
Moreover it can be combined with patterns from Control Theory and Reinforcement Learning to
□ Engineer low cost but tailored intelligent applications
Often enough AI is microprediction, albeit bundled with other mathematical or application logic.
1. You publish a live data value.
2. The sequence of these values gets predicted by a swarm of algorithms.
3. Anyone can write a crawler that tries to predict many different streams.
Microprediction APIs make it easy to:
□ Separate the act of microprediction from other application logic.
□ Invite contribution from other people and machines
□ Benefit from other data you may never have considered.
… Let’s say your store is predicting sales and I’m optimising an HVAC system across the street. Your feature space and mine probably have a lot in common.
I am unclear how the datastreams as set up incorporates domain knowledge and private side information, which seems the hallmark of natural intelligence and, e.g. science. Perhaps they feel domain
knowledge is a bug standing in the way of truly general artificial intelligence? If I had free time I might try to get a better grip on what they are doing, whoever they are.
Alternatively, they are coming at this from a chartist quant perspective and data are best considered as sort-of-anonymous streams of numbers, the better to attract disinterested competition. | {"url":"https://danmackinlay.name/notebook/forecasting.html","timestamp":"2024-11-08T22:21:20Z","content_type":"application/xhtml+xml","content_length":"64091","record_id":"<urn:uuid:a7e4bf42-96cc-46ab-bf02-46d631e2690d>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00882.warc.gz"} |
Roll any even-sided die. If the result is an even number, read it as 2. If the result is an odd number, read it as 1.
Roll a D6. If the die shows > 3, simply check the face down side for your result or use the following conversion: 4 is 3, 5 is 2, 6 is 1.
Roll a D6. If the result is an even number, read it as 6. Otherwise read it as 0. Roll another D6 and add this result to the first result.
Roll a D10 and multiply the result by 10. Reroll the die and add the second result to the first result. Any result of 10 on the D10 should be read as 0.
D20 Project Back to shrines | {"url":"https://webs.radicalgnu.xyz/srd20/nonstandard-dice.html","timestamp":"2024-11-07T13:13:05Z","content_type":"text/html","content_length":"1248","record_id":"<urn:uuid:d0abd174-c189-4e55-bfca-6e94bb74ab63>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00650.warc.gz"} |
C/igraph 0.5.1
igraph 0.5.1 Release Notes
igraph 0.5.1 is a bugfix release, but it actually contains many important new things as well. Here is a brief summary about each of them. See below for the complete list of changes.
The DrL layout generator was added
This is a sophisticated and efficient layout generator written by Shawn Martin and colleagues. See more in the reference manual.
Uniform sampling of random graphs with given degree sequence
A nice random graph generator that conditions on the degree of vertices was added. It can generate undirected connected graphs. The algorithm and the implementation was done by Fabien Viger and
Matthieu Latapy. See more in the reference manual.
Weighted shortest path algorithms
Both the Dijkstra and the Belmann-Ford algorithms were added. See more in the
reference manual.
Function to test edge reciprocity
Mutuality can be tested for each edge now. See more in the reference manual.
New in the C layer
• A new layout generator called DrL.
• Uniform sampling of random connected undirected graphs with a given degree sequence.
• Some stochastic test results are ignored (for spinglass community detection, some Erdos-Renyi generator tests)
• Weighted shortest paths, Dijkstra’s algorithm.
• The unweigthed shortest path routine returns Inf for unreachable vertices.
• New function, igraph_adjlist can create igraph graphs from adjacency lists.
• New function, igraph_weighted_adjacency can create weighted graphs from weight matrices.
• New function, igraph_is_mutual to search for mutual edges.
• Added inverse log-weighted similarity measure (a.k.a. Adamic/Adar similarity).
• igraph_preference_game and igraph_asymmetric_preference_game were rewritten, they are O(|V|+|E|) now, instead of O(|V|^2).
• The Bellman-Ford shortest path algorithm was added.
• Added weighted variant of igraph_get_shortest_paths, based on Dijkstra’s algorithm.
• Several small memory leaks were removed, and a big one from the Spinglass community structure detection function
Bugs corrected in the C layer
• Several bugs were corrected in the (still experimental) C attribute handler.
• Pajek reader bug corrected, used to segfault if *Vertices was missing.
• Directedness is handled correctly when writing GML files. (But note that ‘correct’ conflicts the standard here.)
• Corrected a bug when calculating weighted, directed PageRank on an undirected graph. (Which does not make sense anyway.)
• Some code polish to make igraph compile with GCC 4.3
• Several bugs were fixed in the Reingold-Tilford layout to avoid edge crossings.
• A bug was fixed in the GraphML reader, when the value of a graph attribute was not specified.
• Fixed a bug in the graph isomorphism routine for small (3-4 vertices) graphs.
• Corrected the random sampling implementation (igraph_random_sample), now it always generates unique numbers. This affects the G(n,m) Erdos-Renyi generator, it always generates simple graphs now.
• The basic igraph constructor (igraph_empty_attrs, all functions are expected to call this internally) now checks whether the number of vertices is finite.
• The LGL, NCOL and Pajek graph readers handle errors properly now.
• The non-symmetric ARPACK solver returns results in a consistent form now.
• The fast greedy community detection routine now checks that the graph is simple.
• The LGL and NCOL parsers were corrected to work with all kinds of end-of-line encodings.
• Hub & authority score calculations initialize ARPACK parameters now.x
• Fixed a bug in the Walktrap community detection routine, when applied to unconnected graphs. | {"url":"https://igraph.org/2008/07/14/igraph-0.5.1-c.html","timestamp":"2024-11-08T20:39:19Z","content_type":"text/html","content_length":"13758","record_id":"<urn:uuid:5c1f0fcf-9196-4a53-883d-6efa7cf49fba>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00408.warc.gz"} |
Finding a dense-core in jellyfish graphs
The connectivity of the Internet crucially depends on the relationships between thousands of Autonomous Systems (ASes) that exchange routing information using the Border Gateway Protocol (BGP). These
relationships can be modeled as a graph, called the AS-graph, in which the vertices model the ASes, and the edges model the peering arrangements between the ASes. Based on topological studies, it is
widely believed that the Internet graph contains a central dense-core: Informally, this is a small set of high-degree, tightly interconnected ASes that participate in a large fraction of end-to-end
routes. Finding this densecore is a very important practical task when analyzing the Internet's topology. In this work we introduce a randomized sublinear algorithm that finds a densecore of the
AS-graph. We mathematically prove the correctness of our algorithm, bound the density of the core it returns, and analyze its running time. We also implemented our algorithm and tested it on real
AS-graph data. Our results show that the core discovered by our algorithm is nearly identical to the cores found by existing algorithms - at a fraction of the running time.
Original language English
Title of host publication Algorithms and Models for the Web-Graph - 5th International Workshop, WAW 2007, Proceedings
Pages 29-40
Number of pages 12
State Published - 2007
Externally published Yes
Event 5th Workshop on Algorithms and Models for the Web-Graph, WAW 2007 - San Diego, CA, United States
Duration: 11 Dec 2007 → 12 Dec 2007
Publication series
Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume 4863 LNCS
ISSN (Print) 0302-9743
ISSN (Electronic) 1611-3349
Conference 5th Workshop on Algorithms and Models for the Web-Graph, WAW 2007
Country/Territory United States
City San Diego, CA
Period 11/12/07 → 12/12/07
• Time Complexity
• Edge Density
• IEEE INFOCOM
• Sparse Graph
• Border Gateway Protocol
Dive into the research topics of 'Finding a dense-core in jellyfish graphs'. Together they form a unique fingerprint. | {"url":"https://cris.ariel.ac.il/en/publications/finding-a-dense-core-in-jellyfish-graphs-5","timestamp":"2024-11-04T18:09:40Z","content_type":"text/html","content_length":"56846","record_id":"<urn:uuid:bdf7fd49-873b-40dd-b19f-377f98c2bcb9>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00278.warc.gz"} |
Wind Turbines Just Keep Getting Bigger, But There’s a Limit
Illustration: Greg Mably
Wind turbines have certainly grown up. When the Danish firm Vestas began the trend toward gigantism, in 1981, its three-blade machines were capable of a mere 55 kilowatts. That figure rose to 500 kW
in 1995, reached 2 MW in 1999, and today stands at 5.6 MW. In 2021, MHI Vestas Offshore Wind’s V164 will rise 105 meters high at the hub, swing 80-meter blades, and generate up to 10 MW, making it
the first commercially available double-digit turbine ever. Not to be left behind, General Electric’s Renewable Energy is developing a 12-MW machine with a 260-meter tower and 107-meter blades, also
rolling out by 2021.
That is clearly pushing the envelope, although it must be noted that still larger designs have been considered. In 2011, the UpWind project released what it called a predesign of a 20-MW offshore
machine with a rotor diameter of 252 meters (three times the wingspan of an Airbus A380) and a hub diameter of 6 meters. So far, the limit of the largest conceptual designs stands at 50 MW, with
height exceeding 300 meters and with 200-meter blades that could flex (much like palm fronds) in furious winds.
To imply, as an enthusiastic promoter did, that building such a structure would pose no fundamental technical problems because it stands no higher than the Eiffel tower, constructed 130 years ago, is
to choose an inappropriate comparison. If the constructible height of an artifact were the determinant of wind-turbine design then we might as well refer to the Burj Khalifa in Dubai, a skyscraper
that topped 800 meters in 2010, or to the Jeddah Tower, which will reach 1,000 meters in 2021. Erecting a tall tower is no great problem; it’s quite another proposition, however, to engineer a tall
tower that can support a massive nacelle and rotating blades for many years of safe operation.
Larger turbines must face the inescapable effects of scaling. Turbine power increases with the square of the radius swept by its blades: A turbine with blades twice as long would, theoretically, be
four times as powerful. But the expansion of the surface swept by the rotor puts a greater strain on the entire assembly, and because blade mass should (at first glance) increase as a cube of blade
length, larger designs should be extraordinarily heavy. In reality, designs using lightweight synthetic materials and balsa can keep the actual exponent to as little as 2.3.
Even so, the mass (and hence the cost) adds up. Each of the three blades of Vestas’s 10-MW machine will weigh 35 metric tons, and the nacelle will come to nearly 400 tons. GE’s record-breaking design
will have blades of 55 tons, a nacelle of 600 tons, and a tower of 2,550 tons. Merely transporting such long and massive blades is an unusual challenge, although it could be made easier by using a
segmented design.
Exploring likely limits of commercial capacity is more useful than forecasting specific maxima for given dates. Available wind turbine power [PDF] is equal to half the density of the air (which is
1.23 kilograms per cubic meter) times the area swept by the blades (pi times the radius squared) times the cube of wind velocity. Assuming a wind velocity of 12 meters per second and an
energy-conversion coefficient of 0.4, then a 100-MW turbine would require rotors nearly 550 meters in diameter.
To predict when we’ll get such a machine, just answer this question: When will we be able to produce 275-meter blades of plastic composites and balsa, figure out their transport and their coupling to
nacelles hanging 300 meters above the ground, ensure their survival in cyclonic winds, and guarantee their reliable operation for at least 15 or 20 years? Not soon.
This article appears in the November 2019 print issue as “Wind Turbines: How Big?”
#Energy #Energy/renewables | {"url":"https://manidin.com/blogs/news/wind-turbines-just-keep-getting-bigger-but-there-s-a-limit","timestamp":"2024-11-13T21:09:42Z","content_type":"text/html","content_length":"67241","record_id":"<urn:uuid:2d960ea8-53a0-449f-b008-679aa2d9170a>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00763.warc.gz"} |
Proof Theory: History And Philosophical Significance
Proof Theory: History And Philosophical Significance
by Vincent F. Hendricks / 2000 / English / DjVu
3.2 MB Download
hiS volume in the Synthese Library Series is the result of a conference T held at the University of Roskilde, Denmark, October 31st-November 1st, 1997. The aim was to provide a forum within which
philosophers, math ematicians, logicians and historians of mathematics could exchange ideas pertaining to the historical and philosophical development of proof theory. Hence the conference was called
Proof Theory: History and Philosophical Significance. To quote from the conference abstract: Proof theory was developed as part of Hilberts Programme. According to Hilberts Programme one could
provide mathematics with a firm and se cure foundation by formalizing all of mathematics and subsequently prove consistency of these formal systems by finitistic means. Hence proof theory was
developed as a formal tool through which this goal should be fulfilled. It is well known that Hilbert's Programme in its original form was unfeasible mainly due to Gtldel's incompleteness theorems.
Additionally it proved impossible to formalize all of mathematics and impossible to even prove the consistency of relatively simple formalized fragments of mathematics by finitistic methods. In spite
of these problems, Gentzen showed that by extending Hilbert's proof theory it would be possible to prove the consistency of interesting formal systems, perhaps not by finitis tic methods but still by
methods of minimal strength. This generalization of Hilbert's original programme has fueled modern proof theory which is a rich part of mathematical logic with many significant implications for the
philosophy of mathematics. | {"url":"https://onlybooks.org/proof-theory-history-and-philosophical-significance","timestamp":"2024-11-10T05:19:48Z","content_type":"text/html","content_length":"150604","record_id":"<urn:uuid:7031de42-f66e-4d85-9e94-c31fd705966f>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00890.warc.gz"} |
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This online calculator converts miles per gallon to liters per 100 kilometers
Convert moles to liters and liters to moles
This online calculator converts moles to liters of gas and liters of gas to moles at STP (standard temperature and pressure).
Molar volume
This calculator calculates the molar volume of an ideal gas at different conditions (temperature and pressure) | {"url":"https://planetcalc.com/search/?tag=3557","timestamp":"2024-11-03T18:30:57Z","content_type":"text/html","content_length":"17402","record_id":"<urn:uuid:5b949304-b1a2-4f57-8606-f7cb968b849c>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00482.warc.gz"} |
Resulting 4339 - math word problem (4339)
Resulting 4339
Arrange fractions in ascending order: 7/9, 5/6, 2/3, 11/12, 3/4; Write the resulting order as a 5-digit number, digit = order.
Correct answer:
Did you find an error or inaccuracy? Feel free to
write us
. Thank you!
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fraction calculator
You need to know the following knowledge to solve this word math problem:
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Expository Notes
These are expository notes (often short) which I typed for various reasons over the years. (An important exception are the videos of Professor Illusie's lectures which I am parking here for the
moment, until a better spot is found.) For some of these notes I cannot remember the context. Most were as supplements to talks I gave at the institutes I worked in. Some were supplements to courses
I taught, and some supplements to courses or seminars others were running, which I attended out of interest and participated in. On a few occasions they were answers to questions students asked, and
which I did not have time to answer in person. As is often the case, when typing up answers to simple queries, one gets carried away and writes more than one means to. There is no grand theme. And
typos abound.
Luc Illusie's Lectures
February 7
Back to the Top.
Seminar Notes
Berkovich Spaces
Learning seminar run by Sukhendu Mehrotra at CMI, Jan-April 2020.
Visualizing ultrametrics. See Figure 2 of Jan E.Holly's article Pictures of Ultrametric Spaces, the p-adic Numbers, and Valued Fields, American Mathematical Monthly, October 2001.
Grothendieck Duality
Lectures given at CMI, Monsoon 2019.
Picture giving some idea of the Lipman-Kunz trace. See Overview.
Picard Schemes
These are notes from a seminar I ran on the subject in 2008-09 at East Carolina University and later at the Chennai Mathematical Institute in 2009-10. I gave all the lectures. I haven't uploaded all
the notes I have because they seem a bit too raw and need polishing. I am especially grateful to Sasha Shlapentokh and M.S. Ravi for asking me to run such a seminar and being a loyal audience. I have
to revisit this and polish up the notes. In particular the last lecture in which the Picard scheme was constructed. There are a lot of overlaps with the course I gave on torsors at CMI.
Picture giving descent for the surjective image of a faithfully flat pull-back. See Picard 5.
Basic Algebraic Geometry
Some of these are very basic.
• Upper and lower star. The adjointness of (-)^* and (-)[*].
• Duality for finite maps. A word of caution. What I have called f^!G is really H^0(f^!G)
• Projection Formula. Includes a proof of a special case where the projection formula applies to non locally free sheaves too.
• Spectral Sequences. For people who wish to learn to use double complex spectral sequences. The examples are from Algebraic Geometry.
• Degree of a line bundle on a singular curve. This note is a supplement to a course given by T.R. Ramadas at the Chennai Mathematical Institute. As usual it started innocently enough, but I got
carried away and did more than I meant to after proving the basic result that was needed for the course.
• Multiplicity and intersection number on a curve. Supplement to a course given by T.R. Ramadas at the Chennai Mathematical Institute (see above).
• Self-intersection, noether normalisation. This note is a supplement to a course given by T.R. Ramadas (see above). It proves that if we have an ample line bundle L on a d-dimensional projective
variety V, the self-intersection number ∫[V] c[1]^d(L) is positive. The idea was to do it without appealing to transcendental methods, and without using Bertini's theorem. Then things took on a
life of their own and along the way I gave a proof of the noether normalisation theorem (for projective varieties over an infinite field), as well as some comments (hopefully illuminating) on
étale maps.
• Self duality of elliptic curves. A proof that an elliptic curve (over a possibly non-algebraically closed field) is its own Jacobian.
Back to the Top
Weyl Character formula
I cannot remember the context, but I gave two talks on the subject when I worked at the Harish-Chandra Research Institute (then called the Mehta Research Institute). This must have been 1997. I
followed the material in V.S. Varadarajan's "An Introduction to Harmonic Analysis on Semisimple Lie Groups".
This and That
Random stuff
• Harmonic Series
A short proof of the divergence of the harmonic series. Modification of a proof by Shamik Banerjee in a Facebook maths forum.
• Characteristic Polynomials (ODEs and Recurrence Relations)
A unified way of looking at two elementary results concernings linear ODEs and linear recurrence relations
Back to the Top | {"url":"https://www.cmi.ac.in/~pramath/random.html","timestamp":"2024-11-12T12:10:37Z","content_type":"text/html","content_length":"14475","record_id":"<urn:uuid:fff11cee-9ff4-4618-8fe1-3b4c194d5c9c>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00628.warc.gz"} |
Northburn Primary School
Key Stage 1 Art and Design
National Curriculum England 2014
Year 1
• To use a range of materials creatively to design and make products
• To use drawing, painting and sculpture to develop and share their ideas, experiences and imagination
• To develop a wide range of art and design techniques in using colour, pattern, texture, line, shape, form and space
• About the work of a range of artists, craft makers and designers, describing the differences and similarities between different practices and disciplines, and making links to their own work
Year 2
• To use a range of materials creatively to design and make products
• To use drawing, painting and sculpture to develop and share their ideas, experiences and imagination
• To develop a wide range of art and design techniques in using colour, pattern, texture, line, shape, form and space
• About the work of a range of artists, craft makers and designers, describing the differences and similarities between different practices and disciplines, and making links to their own work
Key Stage 1 Computing
National Curriculum England 2014
Year 1
• 1. Understand what algorithms are, how they are implemented as programs on digital devices, and that programs execute by following precise and unambiguous instructions.
• 2. Create and debug simple programs.
• 3. Use logical reasoning to predict the behaviour of simple programs.
□ 4. Use technology purposefully to create, organise, store, manipulate and retrieve digital content.
• 5. Recognise common uses of information technology beyond school
• 6. Use technology safely and respectfully, keeping personal information private; identify where to go for help and support when they have concerns about content or contact on the internet or
other online technologies
Year 2
• 1. Understand what algorithms are, how they are implemented as programs on digital devices, and that programs execute by following precise and unambiguous instructions.
• 2. Create and debug simple programs.
• 3. Use logical reasoning to predict the behaviour of simple programs.
• 4. Use technology purposefully to create, organise, store, manipulate and retrieve digital content.
• 5. Recognise common uses of information technology beyond school.
• 6. Use technology safely and respectfully, keeping personal information private; identify where to go for help and support when they have concerns about content or contact on the internet or
other online technologies.
Key Stage 1 Design and Technology
National Curriculum England 2014
Cooking and Nutrition
Year 1
• Use the basic principles of a healthy and varied diet to prepare dishes
• Understand where food comes from
Year 2
• Use the basic principles of a healthy and varied diet to prepare dishes
• Understand where food comes from
Year 1
• Design purposeful, functional, appealing products for themselves and other users based on design criteria
• Generate, develop, model and communicate their ideas through talking, drawing, templates, mock-ups and, where appropriate, information and communication technology
Year 2
• Design purposeful, functional, appealing products for themselves and other users based on design criteria
• Generate, develop, model and communicate their ideas through talking, drawing, templates, mock-ups and, where appropriate, information and communication technology
Year 1
• Explore and evaluate a range of existing products
• Evaluate their ideas and products against design criteria
Year 2
• Explore and evaluate a range of existing products
• Evaluate their ideas and products against design criteria
Year 1
• Select from and use a range of tools and equipment to perform practical tasks [for example, cutting, shaping, joining and finishing]
• Select from and use a wide range of materials and components, including construction materials, textiles and ingredients, according to their characteristics
Year 2
• Select from and use a range of tools and equipment to perform practical tasks [for example, cutting, shaping, joining and finishing]
• Select from and use a wide range of materials and components, including construction materials, textiles and ingredients, according to their characteristics
Technical Knowledge
Year 1
• Build structures, exploring how they can be made stronger, stiffer and more stable
• Explore and use mechanisms [for example, levers, sliders, wheels and axles] in their products
Year 2
• Build structures, exploring how they can be made stronger, stiffer and more stable
• Explore and use mechanisms [for example, levers, sliders, wheels and axles] in their products
Key Stage 1 Geography
National Curriculum England 2014
Geographical skills and fieldwork
Year 1
• Use world maps, atlases and globes to identify the United Kingdom and its countries, as well as the countries, continents and oceans studied at this key stage
• Use simple compass directions (North, South, East and West) and locational and directional language [for example, near and far; left and right], to describe the location of features and routes on
a map
• Use aerial photographs and plan perspectives to recognise landmarks and basic human and physical features; devise a simple map; and use and construct basic symbols in a key
• Use simple fieldwork and observational skills to study the geography of their school and its grounds and the key human and physical features of its surrounding environment.
Year 2
• Use world maps, atlases and globes to identify the United Kingdom and its countries, as well as the countries, continents and oceans studied at this key stage
• Use simple compass directions (North, South, East and West) and locational and directional language [for example, near and far; left and right], to describe the location of features and routes on
a map
• Use aerial photographs and plan perspectives to recognise landmarks and basic human and physical features; devise a simple map; and use and construct basic symbols in a key
• Use simple fieldwork and observational skills to study the geography of their school and its grounds and the key human and physical features of its surrounding environment.
Human and physical geography
Year 1
• Identify seasonal and daily weather patterns in the United Kingdom and the location of hot and cold areas of the world in relation to the Equator and the North and South Poles
• Use basic geographical vocabulary to refer to:
□ Key physical features, including: beach, cliff, coast, forest, hill, mountain, sea, ocean, river, soil, valley, vegetation, season and weather
□ Key human features, including: city, town, village, factory, farm, house, office, port, harbour and shop
Year 2
• Identify seasonal and daily weather patterns in the United Kingdom and the location of hot and cold areas of the world in relation to the Equator and the North and South Poles
• Use basic geographical vocabulary to refer to:
□ Key physical features, including: beach, cliff, coast, forest, hill, mountain, sea, ocean, river, soil, valley, vegetation, season and weather
□ Key human features, including: city, town, village, factory, farm, house, office, port, harbour and shop
Locational knowledge
Year 1
• Name and locate the world’s 7 continents and 5 oceans
• Name, locate and identify characteristics of the 4 countries and capital cities of the United Kingdom and its surrounding seas
Year 2
• Name and locate the world’s 7 continents and 5 oceans
• Name, locate and identify characteristics of the 4 countries and capital cities of the United Kingdom and its surrounding seas
Place knowledge
Year 1
• Understand geographical similarities and differences through studying the human and physical geography of a small area of the United Kingdom, and of a small area in a contrasting non-European
Year 2
• Understand geographical similarities and differences through studying the human and physical geography of a small area of the United Kingdom, and of a small area in a contrasting non-European
Key Stage 1 History
National Curriculum England 2014
Year 1
• Changes within living memory – where appropriate, these should be used to reveal aspects of change in national life
• Events beyond living memory that are significant nationally or globally [for example, the Great Fire of London, the first aeroplane flight or events commemorated through festivals or
• The lives of significant individuals in the past who have contributed to national and international achievements, some should be used to compare aspects of life in different periods [for example,
Elizabeth I and Queen Victoria, Christopher Columbus and Neil Armstrong, William Caxton and Tim Berners-Lee, Pieter Bruegel the Elder and LS Lowry, Rosa Parks and Emily Davison, Mary Seacole and/
or Florence Nightingale and Edith Cavell]
• Significant historical events, people and places in their own locality
Year 2
• Changes within living memory – where appropriate, these should be used to reveal aspects of change in national life
• Events beyond living memory that are significant nationally or globally [for example, the Great Fire of London, the first aeroplane flight or events commemorated through festivals or
• The lives of significant individuals in the past who have contributed to national and international achievements, some should be used to compare aspects of life in different periods [for example,
Elizabeth I and Queen Victoria, Christopher Columbus and Neil Armstrong, William Caxton and Tim Berners-Lee, Pieter Bruegel the Elder and LS Lowry, Rosa Parks and Emily Davison, Mary Seacole and/
or Florence Nightingale and Edith Cavell]
• Significant historical events, people and places in their own locality
Key Stage 1 Mathematics
National Curriculum England 2014 - NAHT Assessment Framework
Geometry - position and direction
Year 1
• Describe position, direction and movement
• Describe whole, half, quarter and three-quarter turns
Year 2
• Order and arrange combinations of mathematical objects in patterns and sequences
• Uses mathematical vocabulary to describe position, direction and movement, including movement in a straight line
• Distinguishes between rotation as a turn and in terms of right angles for quarter, half and three-quarter turns (clockwise and anti-clockwise)
Geometry - properties of shapes
Year 1
• Recognise and name common 2-D and 3-D shapes, including:
□ 2-D shapes e.g. rectangles (including squares), circles and triangles.(WRM: Recognise and name 2-D shapes).
□ 3-D shapes e.g. cuboids (including cubes), pyramids and spheres.(WRM: Recognise and name 3-D shapes).
□ WRM: Patterns with 2-D and 3-D shapes.
• 1G-1: Recognise common 2D and 3D shapes presented in different orientations, and know that rectangles, triangles, cuboids and pyramids are not always similar to one another. (RtP)
• 1G-2: Compose 2D and 3D shapes from smaller shapes to match an example, including manipulating shapes to place them in particular orientations. (RtP)
Year 2
• Identify and describe the properties of 2-D shapes, including the number of sides.
□ WRM: Count sides on 2-D shapes.
□ WRM: Count vertices on 2-D shapes.
• Identify and describe the properties of 2-D shapes using line symmetry in a vertical line.
□ WRM: Lines of symmetry on shapes.
□ WRM: Use lines of symmetry to complete shapes.
• Identify and describe the properties of 3-D shapes, including the number of edges, vertices and faces.
□ WRM: Count faces on 3-D shapes.
□ WRM: Count edges on 3-D shapes.
□ WRM: Count vertices on 3-D shapes.
• Identify 2-D shapes on the surface of 3-D shapes, [for example, a circle on a cylinder and a triangle on a pyramid].
• Compare and sort common 2-D shapes.
• Compare and sort common 3-D shapes.
• Compare and sort everyday objects.
• 2G-1: Use precise language to describe the properties of 2D and 3D shapes, and compare shapes by reasoning about similarities and differences in properties. (RtP)
• WRM: Recognise 2-D and 3-D shapes.
• WRM: Make patterns with 2-D and 3-D shapes.
Year 1
• Compare, describe and solve practical problems for:
□ Lengths and heights e.g. long/short, longer/shorter, tall/short, double/half.
☆ WRM: Compare lengths and heights.
□ Mass/weight e.g. heavy/light, heavier than, lighter than.
□ Capacity and volume e.g. full/empty, more than, less than, half, half full, quarter.
□ Time e.g. quicker, slower, earlier, later
• Measure and begin to record the following:
☆ WRM: Measure length using objects.
☆ WRM: Measure length in centimetres.
□ Time (hours, minutes, seconds)
□ Recognise and know the value of different denominations of coins and notes
□ Sequence events in chronological order using language [for example, before and after, next, first, today, yesterday, tomorrow, morning, afternoon and evening]
• Recognise and use language relating to dates, including days of the week, weeks, months and years
• Tells the time to the hour and half past the hour and draw the hands on a clock face to show these times
Year 2
• Choose and use appropriate standard units to estimate and measure length/height in any direction (m/cm);to the nearest appropriate unit, using rulers.
□ WRM: Measure in centimetres.
• Choose and use appropriate standard units to estimate and measure mass (kg/g); to the nearest appropriate unit, using scales
□ WRM: Measure in kilograms.
□ WRM: Four operations with mass.
• Choose and use appropriate standard units to estimate and measure temperature (°c); to the nearest appropriate unit, using thermometers.
• Choose and use appropriate standard units to estimate and measure capacity (litres/ml) to the nearest appropriate unit, using measuring vessels.
□ WRM: Measure in millilitres.
□ WRM: Four operations with volume and capacity.
• Compare and order lengths and record the results using >, < and =.
□ WRM: Compare lengths and heights.
□ WRM: Order lengths and heights.
• Compare and order mass, and record the results using >, < and =
• Compare and order volume/capacity and record the results using >, < and =
□ WRM: Compare volume and capacity.
• Recognise and use symbols for pounds (£) and pence (p); combine amounts to make a particular value.
□ WRM: Count money - pence.
□ WRM: Count money - pounds (notes and coins).
□ WRM: Count money - pounds and pence.
□ WRM: Choose notes and coins.
□ WRM: Make the same amount.
□ WRM: Compare amounts of money.
• Find different combinations of coins that equal the same amounts of money.
• Solves simple problems in a practical context involving addition and subtraction of money of the same unit, including giving change.
□ WRM: Calculate with money.
• Compare and sequence intervals of time
• Tell and write the time to five minutes, including quarter past/to the hour and draw the hands on a clock face to show these times
• Know the number of minutes in an hour and the number of hours in a day
• WRM: Four operations with lengths and heights.
Number - addition and subtraction
Year 1
• Read, write and interpret mathematical statements involving addition (+), subtraction (−) and equals (=) signs.
□ WRM: Write number sentences.
□ WRM: Fact families - the eight facts.
□ WRM: Add or subtract 1 or 2.
• Represent and use number bonds and related subtraction facts within 20.
□ WRM: Fact families - addition facts.
□ WRM: Number bonds within 10.
□ WRM: Systematic number bonds within 10.
□ WRM: Addition - add together.
□ WRM: Addition - add more.
□ WRM: Find and make number bonds to 20.
• Add and subtract 1-digit and 2-digit numbers to 20, including zero.
□ WRM: Add by counting on within 20.
□ WRM: Add ones using number bonds.
□ WRM: Subtract ones using number bonds.
□ WRM: Subtraction-counting back.
□ WRM: Subtraction-finding the difference.
• Solve one-step problems that involve addition and subtraction, using concrete objects and pictorial representations, and missing number problems such as 7 = ? – 9.
□ WRM: Subtraction - find a part.
□ WRM: Subtraction - take away/cross out (How many left?)
□ WRM: Take away (How many left?)
□ WRM: Subtraction on a number line.
□ WRM: Missing number problems.
• 1NF-1: Develop fluency in addition and subtraction facts within 10. (RtP)
• 1NF-2: Count forwards and backwards in multiples of 2, 5 and 10, up to 10 multiples, beginning with any multiple, and count forwards and backwards through the odd numbers. (RtP)
• 1AS-1: Compose numbers to 10 from 2 parts, and partition numbers to 10 into parts, including recognising odd and even numbers. (RtP)
• 1AS-2: Read, write and interpret equations containing addition (+), subtraction (-) and equals (=) symbols, and relate additive expressions and equations to real-life contexts. (RtP)
• WRM: Introduce parts and wholes.
Year 2
• Solve addition problems by applying their increasing knowledge of mental and written methods.
• Solves subtraction problems by recalling and using addition and subtraction facts to 20 and 100.
• Add and subtract numbers using concrete objects, pictorial representations, and mentally, including:
□ A two-digit number and 1s.
□ A two-digit number and 10s.
□ Adding 3 one-digit numbers.
□ WRM: Fact families - addition and subtraction bonds within 20.
□ WRM: Bonds to 100 (tens).
□ WRM: Add and subtract 1s.
□ WRM: Add three 1-digit numbers.
□ WRM: Subtract a 1-digit number from a 2-digit number (across a 10).
□ WRM: Add and subtract 10s.
□ WRM: Add two 2-digit numbers (not across a 10).
□ WRM: Add two 2-digit numbers (across a 10).
□ WRM: Subtract two 2-digit numbers (not across a 10).
□ WRM: Subtract two 2-digit numbers (across a 10).
• Show that addition of 2 numbers can be done in any order (commutative) and subtraction of 1 number from another cannot.
• Recognise and use the inverse relationship between addition and subtraction and use this to check calculations and solve missing number problems.
□ WRM: Mixed addition and subtraction.
□ WRM: Compare number sentences.
□ WRM: Missing number problems.
• 2NF-1: Secure fluency in addition and subtraction facts within 10, through continued practice. (RtP)
• 2AS-1: Add and subtract across 10, for example: 8 + 5 = 13 13 - 5 = 8 (RtP)
• 2AS-2: Recognise the subtraction structure of 'difference' and answer questions of the form, "How many more...?" (RtP)
• 2AS-3: Add and subtract within 100 by applying related one-digit addition and subtraction facts: add and subtract only ones or only tens to/from a two-digit number. (RtP)
• 2AS-4: Add and subtract within 100 by applying related one-digit addition and subtraction facts: add and subtract any 2 two-digit numbers. (RtP)
Number - fractions
Year 1
• Recognises, finds and names a half as one of two equal parts of an object, shape or quantity
• Recognise, find and name a quarter as 1 of 4 equal parts of an object, shape or quantity
Year 2
• Recognises, find, name and write fractions ⅓, ¼,2/4and ¾
• Recognises, find, name and write fractions ⅓, ¼,2/4and ¾ of a shape
• Recognises, find, name and write fractions ⅓, ¼,2/4and ¾ of a set of objects or quantity
• Recognises, find, name and write fractions ⅓, ¼,2/4and ¾ of a length
• Write simple fractions, for example of ½ of 6 = 3 and recognise the equivalence of2/4and ½
Number - multiplication and division
Year 1
• Solve one-step problems involving multiplication, by calculating the answer using concrete objects, pictorial representations and arrays with the support of the teacher
• Solve one-step problems involving division, by calculating the answer using concrete objects, pictorial representations and arrays with the support of the teacher
Year 2
• Recall and use multiplication and division facts for the 2, 5 and 10 multiplication tables, including recognising odd and even numbers.
□ WRM: Doubling and halving.
□ WRM: Odd and even numbers.
□ WRM: The 5 and 10 times-tables.
• Calculate mathematical statements for multiplication and division within the multiplication tables and write them using the multiplication (x), division (÷)and equals (=) signs.
□ WRM: Recognise equal groups.
□ WRM: Introduce the multiplication symbol.
□ WRM: Multiplication sentences.
□ WRM: Make equal groups-grouping.
□ WRM: Make equal groups-sharing.
• Show that multiplication of 2 numbers can be done in any order (commutative) and division of 1 number by another cannot
• Solves problems involving multiplication using appropriate methods including, (using materials, arrays, repeated addition, mental methods, and multiplication facts), including problems in
• Solves problems involving division using appropriate methods including, (using materials, arrays, repeated addition, mental methods, and division facts), including problems in contexts
• 2MD-1: Recognise repeated addition contexts, representing them with multiplication equations and calculating the product, within the 2, 5 and 10 multiplication tables. (RtP)
• 2MD-2: Relate grouping problems where the number of groups is unknown to multiplication equations with a missing factor, and to division equations (quotitive division). (RtP)
Number - number and place value
Year 1
• Count to and across 100, forwards and backwards, beginning with 0 or 1, or from any given number.
□ WRM: Count on from any number.
□ WRM: Count backwards within 10.
□ WRM: Understand 11, 12 and 13.
□ WRM: Understand 14, 15 and 16.
□ WRM: Understand 17, 18 and 19.
□ WRM: Count from 20 to 50.
• Count, read and write numbers to 100 in numerals; count in multiples of 2s, 5s and 10s.
□ Read numbers 1-20 in numerals.
□ Read numbers 1-20 in words.
□ WRM: Recognise numbers as words.
□ Write numbers 1-20 in numerals.
□ Write numbers 1-20 in words.
□ WRM: Count by making groups of tens.
□ WRM: Groups of tens and ones.
□ WRM: Partition into tens and ones.
• Given a number, identifies 1 more and 1 less.
• Identify and represent numbers using objects and pictorial representations including the number line.
□ WRM: Count objects from a larger group.
□ WRM: Order objects and numbers.
□ WRM: The number line to 20.
□ WRM: Use a number line to 20.
□ WRM: Estimate on a number line to 20.
□ WRM: The number line to 50.
□ WRM: Estimate on a number line to 50.
• Use the language of: equal to, more than, less than (fewer), most, least.
□ WRM: Compare groups by matching.
□ WRM: Less than, greater than, equal to.
□ WRM: Compare numbers to 20.
□ WRM: Order numbers to 20.
• 1NPV-1: Count within 100, forwards and backwards, starting with any number. (RtP)
• 1NPV-2: Reason about the location of numbers to 20 within the linear number system, including comparing using < > and =. (RtP)
Year 2
• Count in steps of two, three, and five from 0, and in 10s from any number, forward and backward.
□ WRM: Count in 2s, 5s and 10s.
• Recognise the place value of each digit in a two-digit number (10s, 1s).
□ WRM: Recognise tens and ones.
• Identify, represent and estimate numbers using different representations, including the number line.
□ WRM: Use a place value chart.
□ WRM: Partition numbers to 100.
□ WRM: Flexibly partition numbers to 100.
□ WRM: 10s on the number line to 100.
□ WRM: 10s and 1s on the number line to 100.
□ WRM: Estimate numbers on a number line.
• Compare and order numbers from 0 up to 100.
□ WRM: Order objects and numbers.
• Use < > and = signs correctly.
• Read and write numbers to at least 100 in numerals and in words.
□ WRM: Write numbers to 100 in words.
□ WRM: Write numbers to 100 in expanded form.
• Use place value and number facts to solve problems.
• WRM: Count objects to 100 by making 10s.
• 2NPV-1: Recognise the place value of each digit in two-digit numbers, and compose and decompose two-digit numbers using standard and non-standard partitioning. (RtP)
• 2NPV-2: Reason about the location of any two-digit number in the linear number system, including identifying the previous and next multiple of 10. (RtP)
Year 2
• Interpret and construct simple pictograms
• Interpret and construct simple tally charts
• Interpret and construct simple block diagrams
• Interpret and construct simple tables
• Ask and answer simple questions by counting the number of objects in each category and sorting the categories by quantity
• Asks and answers questions about totalling and comparing categorical data
Key Stage 1 Music
National Curriculum England 2014
Year 1
• Use their voices expressively and creatively by singing songs and speaking chants and rhymes
• Play tuned and untuned instruments musically
• Listen with concentration and understanding to a range of high-quality live and recorded music
• Experiment with, create, select and combine sounds using the interrelated dimensions of music
Year 2
• Use their voices expressively and creatively by singing songs and speaking chants and rhymes
• Play tuned and untuned instruments musically
• Listen with concentration and understanding to a range of high-quality live and recorded music
• Experiment with, create, select and combine sounds using the interrelated dimensions of music
Key Stage 1 Physical Education
National Curriculum England 2014
Year 1
• Master basic movements including running, jumping, throwing and catching, as well as developing balance, agility and co-ordination, and begin to apply these in a range of activities
• Participate in team games, developing simple tactics for attacking and defending
• Perform dances using simple movement patterns
• Swimming and Water Safety:
□ All schools must provide swimming instruction either in key stage 1 or key stage 2. In particular, pupils should be taught to:
☆ Swim competently, confidently and proficiently over a distance of at least 25 metres
☆ Use a range of strokes effectively [for example, front crawl, backstroke and breaststroke]
☆ Perform safe self-rescue in different water-based situations
Year 2
• Master basic movements including running, jumping, throwing and catching, as well as developing balance, agility and co-ordination, and begin to apply these in a range of activities
• Participate in team games, developing simple tactics for attacking and defending
• Perform dances using simple movement patterns
• Swimming and Water Safety:
□ All schools must provide swimming instruction either in key stage 1 or key stage 2. In particular, pupils should be taught to:
☆ Swim competently, confidently and proficiently over a distance of at least 25 metres
☆ Use a range of strokes effectively [for example, front crawl, backstroke and breaststroke]
☆ Perform safe self-rescue in different water-based situations
Key Stage 1 Reading
National Curriculum England 2014
Year 1
• Develop pleasure in reading, motivation to read, vocabulary and understanding by:
□ Listening to and discussing a wide range of poems, stories and non-fiction at a level beyond that at which they can read independently
□ Being encouraged to link what they read or hear to their own experiences
□ Becoming very familiar with key stories, fairy stories and traditional tales, retelling them and considering their particular characteristics
□ Recognising and joining in with predictable phrases
□ Learning to appreciate rhymes and poems, and to recite some by heart
□ Discussing word meanings, linking new meanings to those already known
• Understand both the books they can already read accurately and fluently and those they listen to by:
□ Drawing on what they already know or on background information and vocabulary provided by the teacher
□ Checking that the text makes sense to them as they read, and , correcting inaccurate reading
□ Discussing the significance of the title and events
□ Making inferences on the basis of what is being said and done
□ Predicting what might happen on the basis of what has been read so far
• Participate in discussion about what is read to them, taking turns and listening to what others say
• Explain clearly their understanding of what is read to them
Year 2
• Develop pleasure in reading, motivation to read, vocabulary and understanding by:
□ Listening to, discussing and expressing views about a wide range of contemporary and classic poetry, stories and non-fiction at a level beyond that at which they can read independently
□ Discussing the sequence of events in books and how items of information are related
□ Becoming increasingly familiar with and retelling a wider range of stories, fairy stories and traditional tales
□ Being introduced to non-fiction books that are structured in different ways
□ Recognising simple recurring literary language in stories and poetry
□ Discussing and clarifying the meanings of words, linking new meanings to known vocabulary
□ Discussing their favourite words and phrases
□ Continuing to build up a repertoire of poems learnt by heart, appreciating these and reciting some, with appropriate intonation to make the meaning clear
• Understand both the books that they can already read accurately and fluently and those that they listen to by:
□ Drawing on what they already know or on background information and vocabulary provided by the teacher
□ Checking that the text makes sense to them as they read, and correcting inaccurate reading
□ Making inferences on the basis of what is being said and done
□ Answering and asking questions
□ Predicting what might happen on the basis of what has been read so far
• Participate in discussion about books, poems and other works that are read to them and those that they can read for themselves, taking turns and listening to what others say
• Explain and discuss their understanding of books, poems and other material, both those that they listen to and those that they read for themselves
Word reading
Year 1
• Apply phonic knowledge and skills as the route to decode words
• Respond speedily with the correct sound to graphemes (letters or groups of letters) for all 40+ phonemes, including, where applicable, alternative sounds for graphemes
• Read accurately by blending sounds in unfamiliar words containing GPCs that have been taught
• Read common exception words, noting unusual correspondences between spelling and sound and where these occur in the word
• Read words containing taught GPCs and –s, –es, –ing, –ed, –er and –est endings
• Read other words of more than one syllable that contain taught GPCs
• Read words with contractions [for example, i’m, i’ll, we’ll], and understand that the apostrophe represents the omitted letter(s)
• Read books aloud, accurately, that are consistent with their developing phonic knowledge and that do not require them to use other strategies to work out words
• Reread these books to build up their fluency and confidence in word reading
Year 2
• Continue to apply phonic knowledge and skills as the route to decode words until automatic decoding has become embedded and reading is fluent
• Read accurately by blending the sounds in words that contain the graphemes taught so far, especially recognising alternative sounds for graphemes
• Read accurately words of two or more syllables that contain the same graphemes as above
• Read words containing common suffixes
• Read further common exception words, noting unusual correspondences between spelling and sound and where these occur in the word
• Read most words quickly and accurately, without overt sounding and blending, when they have been frequently encountered
• Read aloud books closely matched to their improving phonic knowledge, sounding out unfamiliar words accurately, automatically and without undue hesitation
• Reread these books to build up their fluency and confidence in word reading
Key Stage 1 Religious Education
KS1 RE
Year 1
• Unit 1.1 God: Identify what a parable is.
• Unit 1.1 God: Tell the story of the Lost Son from the Bible simply and recognise link with the Christian idea of God as a forgiving Father.
• Unit 1.1 God: Give clear, simple accounts of what the story means to Christians.
• Unit 1.1 God: Give at least two examples of a way in which Christians show their belief in God as loving and forgiving.
• Unit 1.1 God: Give an example of how Christians put their beliefs into practice in worship.
• Unit 1.1 God: Think, talk and ask questions about whether they can learn anything from the story for themselves, exploring different ideas.
• Unit 1.1 God: Give a reason for the ideas they have and the connections they make.
• Unit 1.2 Creation: Retell the story of creation from Genesis 1:1-2:3 simply.
• Unit 1.2 Creation: Recognise that Creation is the beginning of the 'big story' of the Bible.
• Unit 1.2 Creation: Say what the story tells Christians about God, Creation and the world.
• Unit 1.2 Creation: Give at least one example of what Christians do to say 'thank you' to God for Creation.
• Unit 1.2 Creation: Think, talk and ask questions about living in an amazing world.
• Unit 1.2 Creation: Give a reason for the ideas they have and the connections they make between the Jewish/Christian Creation story and the world they live in.
• Unit 1.3 Incarnation: Recognise that stories of Jesus' life come from the Gospels.
• Unit 1.3 Incarnation: Give a clear, simple account of the story of Jesus' birth and why Jesus is important to Christians.
• Unit 1.3 Incarnation: Give examples of ways in which Christians use the story of the Nativity to guide their beliefs and actions at Christmas.
• Unit 1.3 Incarnation: Think, talk and ask questions about Christmas for people who are Christians and for people who are not.
• Unit 1.3 Incarnation: Decide what they personally have to be thankful for, giving a reason for their ideas.
• Unit 1.4 Gospel: Tell stories from the Bible and recognise a link with the concept of 'Gospel' or 'good news'.
• Unit 1.4 Gospel: Give clear, simple accounts of what Bible texts (such as the story of Matthew the tax collector) means to Christians.
• Unit 1.4 Gospel: Recognise that Jesus gives instructions to people about how to behave.
• Unit 1.4 Gospel: Give at least two examples of ways in which Christians follow the teachings studied about forgiveness and peace, and bringing good news to the friendless.
• Unit 1.4 Gospel: Give at least two examples of ways in which Christians put these beliefs into practice in the Church community and their own lives (for example: charity, confession).
• Unit 1.4 Gospel: Think, talk and ask questions about whether Jesus' 'good news' is only good news for Christians, or if there are things for anyone to learn about how to live, giving a good
reason for their ideas.
• Unit 1.5 Salvation: Recognise that incarnation and salvation are part of a 'big story' of the Bible.
• Unit 1.5 Salvation: Tell stories of Holy Week and Easter from the Bible and recognise a link with the idea of salvation (Jesus rescuing people).
• Unit 1.5 Salvation: Give at least three examples of how Christians show their beliefs about Jesus' death and resurrection in church worship at Easter.
• Unit 1.5 Salvation: Think, talk and ask questions about whether the story of Easter only has something to say to Christians, or if it has anything to say to pupils about sadness, hope or heaven,
exploring different ideas and giving a good reason for their ideas.
• Unit 1.6 Muslim: Recognise the words of the Shahadah and that it is very important to Muslims.
• Unit 1.6 Muslim: Identify some of the key Muslim beliefs about God found in the Shahadah and the 99 names of Allah, and give a simple description of what some of them mean.
• Unit 1.6 Muslim: Give examples of how stories about the Prophet show what Muslims believe about Muhammad.
• Unit 1.6 Muslim: Give examples of how Muslims use the Shahadah to show what matters to them.
• Unit 1.6 Muslim: Give examples of how Muslims use stories about the Prophet to guide their beliefs and actions (e.g. care for creation, fast in Ramadan).
• Unit 1.6 Muslim: Give examples of how Muslims put their beliefs about prayer into action.
• Unit 1.6 Muslim: Think, talk about and ask questions about Muslim beliefs and ways of living.
• Unit 1.6 Muslim: Talk about what they think is good for Muslims about prayer, respect, celebration and self-control, giving a good reason for their ideas.
• Unit 1.6 Muslim: Give a good reason for their ideas about whether prayer, respect, celebration and self-control have something to say to them too.
• Unit 1.7 Jewish: Recognise the words of the Shema as a Jewish prayer.
• Unit 1.7 Jewish: Retell simply some stories used in Jewish celebrations (e.g. Chanukah).
• Unit 1.7 Jewish: Give examples of how the stories used in celebrations (e.g. Shabbat, Chanukah) remind Jews about what God is like.
• Unit 1.7 Jewish: Give examples of how Jewish people celebrate special times (e.g. Shabbat, Sukkot, Chanukah).
• Unit 1.7 Jewish: Make links between Jewish ideas of God found in the stories and how people live.
• Unit 1.7 Jewish: Give an example of how some Jewish people might remember God in different ways (e.g. mezuzah, on Shabbat).
• Unit 1.7 Jewish: Talk about what they think is good about reflecting, thanking, praising and remembering for Jewish people, giving a good reason for their ideas.
• Unit 1.7 Jewish: Give a good reason for their ideas about whether reflecting, thanking, praising and remembering have something to say to them too.
• Unit 1.8: Recognise that there are special places where people go to worship, and talk about what people do there.
• Unit 1.8: Identify at least three objects used in worship in two religions and give a simple account of how they are used and something about what they mean.
• Unit 1.8: Identify a belief about worship and a belief about God, connecting these beliefs simply to a place of worship.
• Unit 1.8: Give examples of stories, objects, symbols and actions used in churches, mosques and/or synagogues which show what people believe.
• Unit 1.8: Give simple examples of how people worship at a church, mosque or synagogue.
• Unit 1.8: Talk about why some people like to belong to a sacred building or a community.
• Unit 1.8: Think, talk and ask good questions about what happens in a church, mosque or synagogue, saying what they think about these questions, giving good reasons for their ideas.
• Unit 1.8: Talk about what makes some places special to people, and what the difference is between religious and non-religious special places.
• Unit 1.9: Identify a story or text that says something about each person being unique and valuable.
• Unit 1.9: Give an example of a key belief some people find in one of these stories (e.g. that God loves all people).
• Unit 1.9: Give a clear, simple account of what Genesis 1 tells Christians and Jews about the natural world.
• Unit 1.9: Give an example of how people show that they care for others (e.g. by giving to charity), making a link to one of the stories.
• Unit 1.9: Give examples of how Christians and Jews can show care for the natural earth.
• Unit 1.9: Say why Christians and Jews might look after the natural world.
• Unit 1.9: Think, talk and ask questions about what difference believing in God makes how people treat each other and the natural world.
• Unit 1.9: Give good reasons why everyone (religious and non-religious) should care for others and look after the natural world.
• Unit 1.10: Recognise that loving others is important in lots of communities.
• Unit 1.10: Say simply what Jesus and one other religious leader taught about loving other people.
• Unit 1.10: Give an account of what happens at a traditional Christian and Jewish or Muslim ceremony, and suggest what the actions and symbols mean.
• Unit 1.10: Identify at least two ways people show they love each other and belong to each other when they get married (Christian and/or Jewish and non-religious).
• Unit 1.10: Give examples of ways in which people express their identity and belonging within faith communities, responding sensitively to differences.
• Unit 1.10: Talk about what they think is good about being in a community, for people in faith communities and for themselves, giving a good reason for their ideas.
Year 2
• Unit 1.1 God: Identify what a parable is.
• Unit 1.1 God: Tell the story of the Lost Son from the Bible simply and recognise link with the Christian idea of God as a forgiving Father.
• Unit 1.1 God: Give clear, simple accounts of what the story means to Christians.
• Unit 1.1 God: Give at least two examples of a way in which Christians show their belief in God as loving and forgiving.
• Unit 1.1 God: Give an example of how Christians put their beliefs into practice in worship.
• Unit 1.1 God: Think, talk and ask questions about whether they can learn anything from the story for themselves, exploring different ideas.
• Unit 1.1 God: Give a reason for the ideas they have and the connections they make.
• Unit 1.2 Creation: Retell the story of creation from Genesis 1:1-2:3 simply.
• Unit 1.2 Creation: Recognise that Creation is the beginning of the 'big story' of the Bible.
• Unit 1.2 Creation: Say what the story tells Christians about God, Creation and the world.
• Unit 1.2 Creation: Give at least one example of what Christians do to say 'thank you' to God for Creation.
• Unit 1.2 Creation: Think, talk and ask questions about living in an amazing world.
• Unit 1.2 Creation: Give a reason for the ideas they have and the connections they make between the Jewish/Christian Creation story and the world they live in.
• Unit 1.3 Incarnation: Recognise that stories of Jesus' life come from the Gospels.
• Unit 1.3 Incarnation: Give a clear, simple account of the story of Jesus' birth and why Jesus is important to Christians.
• Unit 1.3 Incarnation: Give examples of ways in which Christians use the story of the Nativity to guide their beliefs and actions at Christmas.
• Unit 1.3 Incarnation: Think, talk and ask questions about Christmas for people who are Christians and for people who are not.
• Unit 1.3 Incarnation: Decide what they personally have to be thankful for, giving a reason for their ideas.
• Unit 1.4 Gospel: Tell stories from the Bible and recognise a link with the concept of 'Gospel' or 'good news'.
• Unit 1.4 Gospel: Give clear, simple accounts of what Bible texts (such as the story of Matthew the tax collector) means to Christians.
• Unit 1.4 Gospel: Recognise that Jesus gives instructions to people about how to behave.
• Unit 1.4 Gospel: Give at least two examples of ways in which Christians follow the teachings studied about forgiveness and peace, and bringing good news to the friendless.
• Unit 1.4 Gospel: Give at least two examples of ways in which Christians put these beliefs into practice in the Church community and their own lives (for example: charity, confession).
• Unit 1.4 Gospel: Think, talk and ask questions about whether Jesus' 'good news' is only good news for Christians, or if there are things for anyone to learn about how to live, giving a good
reason for their ideas.
• Unit 1.5 Salvation: Recognise that incarnation and salvation are part of a 'big story' of the Bible.
• Unit 1.5 Salvation: Tell stories of Holy Week and Easter from the Bible and recognise a link with the idea of salvation (Jesus rescuing people).
• Unit 1.5 Salvation: Give at least three examples of how Christians show their beliefs about Jesus' death and resurrection in church worship at Easter.
• Unit 1.5 Salvation: Think, talk and ask questions about whether the story of Easter only has something to say to Christians, or if it has anything to say to pupils about sadness, hope or heaven,
exploring different ideas and giving a good reason for their ideas.
• Unit 1.6 Muslim: Recognise the words of the Shahadah and that it is very important to Muslims.
• Unit 1.6 Muslim: Identify some of the key Muslim beliefs about God found in the Shahadah and the 99 names of Allah, and give a simple description of what some of them mean.
• Unit 1.6 Muslim: Give examples of how stories about the Prophet show what Muslims believe about Muhammad.
• Unit 1.6 Muslim: Give examples of how Muslims use the Shahadah to show what matters to them.
• Unit 1.6 Muslim: Give examples of how Muslims use stories about the Prophet to guide their beliefs and actions (e.g. care for creation, fast in Ramadan).
• Unit 1.6 Muslim: Give examples of how Muslims put their beliefs about prayer into action.
• Unit 1.6 Muslim: Think, talk about and ask questions about Muslim beliefs and ways of living.
• Unit 1.6 Muslim: Talk about what they think is good for Muslims about prayer, respect, celebration and self-control, giving a good reason for their ideas.
• Unit 1.6 Muslim: Give a good reason for their ideas about whether prayer, respect, celebration and self-control have something to say to them too.
• Unit 1.7 Jewish: Recognise the words of the Shema as a Jewish prayer.
• Unit 1.7 Jewish: Retell simply some stories used in Jewish celebrations (e.g. Chanukah).
• Unit 1.7 Jewish: Give examples of how the stories used in celebrations (e.g. Shabbat, Chanukah) remind Jews about what God is like.
• Unit 1.7 Jewish: Give examples of how Jewish people celebrate special times (e.g. Shabbat, Sukkot, Chanukah).
• Unit 1.7 Jewish: Make links between Jewish ideas of God found in the stories and how people live.
• Unit 1.7 Jewish: Give an example of how some Jewish people might remember God in different ways (e.g. mezuzah, on Shabbat).
• Unit 1.7 Jewish: Talk about what they think is good about reflecting, thanking, praising and remembering for Jewish people, giving a good reason for their ideas.
• Unit 1.7 Jewish: Give a good reason for their ideas about whether reflecting, thanking, praising and remembering have something to say to them too.
• Unit 1.8: Recognise that there are special places where people go to worship, and talk about what people do there.
• Unit 1.8: Identify at least three objects used in worship in two religions and give a simple account of how they are used and something about what they mean.
• Unit 1.8: Identify a belief about worship and a belief about God, connecting these beliefs simply to a place of worship.
• Unit 1.8: Give examples of stories, objects, symbols and actions used in churches, mosques and/or synagogues which show what people believe.
• Unit 1.8: Give simple examples of how people worship at a church, mosque or synagogue.
• Unit 1.8: Talk about why some people like to belong to a sacred building or a community.
• Unit 1.8: Think, talk and ask good questions about what happens in a church, mosque or synagogue, saying what they think about these questions, giving good reasons for their ideas.
• Unit 1.8: Talk about what makes some places special to people, and what the difference is between religious and non-religious special places.
• Unit 1.9: Identify a story or text that says something about each person being unique and valuable.
• Unit 1.9: Give an example of a key belief some people find in one of these stories (e.g. that God loves all people).
• Unit 1.9: Give a clear, simple account of what Genesis 1 tells Christians and Jews about the natural world.
• Unit 1.9: Give an example of how people show that they care for others (e.g. by giving to charity), making a link to one of the stories.
• Unit 1.9: Give examples of how Christians and Jews can show care for the natural earth.
• Unit 1.9: Say why Christians and Jews might look after the natural world.
• Unit 1.9: Think, talk and ask questions about what difference believing in God makes how people treat each other and the natural world.
• Unit 1.9: Give good reasons why everyone (religious and non-religious) should care for others and look after the natural world.
• Unit 1.10: Recognise that loving others is important in lots of communities.
• Unit 1.10: Say simply what Jesus and one other religious leader taught about loving other people.
• Unit 1.10: Give an account of what happens at a traditional Christian and Jewish or Muslim ceremony, and suggest what the actions and symbols mean.
• Unit 1.10: Identify at least two ways people show they love each other and belong to each other when they get married (Christian and/or Jewish and non-religious).
• Unit 1.10: Give examples of ways in which people express their identity and belonging within faith communities, responding sensitively to differences.
• Unit 1.10: Talk about what they think is good about being in a community, for people in faith communities and for themselves, giving a good reason for their ideas.
Key Stage 1 RSE
Year 1
• Understands who is in their family and how other families are different/similar.
• Recognises what they like about their friends and what their friends like about them.
• Understands how to make someone feel good about themselves and why you shouldn't tease people.
□ Knows the names of their own body parts and begin to name opposite sex body parts.
☆ Know which parts of their body are private and know when it is ok and not ok to let someone touch me.
☆ Understand how to say no if I dont want to be touched and I know who to tell if someone wants to touch my private parts.
☆ I know who I can ask if I need to know something and I know who I can go to if I am worried about something.
Year 2
• Identify and name biological terms for male and female sex parts • can label the male and female sex parts with confidence
• Understand that the male and female sex parts are related to reproduction.
• Can identify key stages in the human life cycle • understand some ways they have changed since they were babies • understand that all living things including humans start life as babies.
• Understand that we all have different needs and require different types of care • identify ways we show care towards each other • understand the links between needs, caring and changes throughout
the life cycle
• Can describe different types of family • identify what is special and different about their home life • understand families care for each other in a variety of ways
Key Stage 1 Science
National Curriculum England 2014 - NAHT Assessment Framework
Animals, including humans
Year 1
• Identify and name a variety of common animals including fish, amphibians, reptiles, birds and mammals.
• Identify and name a variety of common animals that are carnivores, herbivores and omnivores.
• Describe and compare the structure of a variety of common animals (fish, amphibians, reptiles, birds and mammals including pets).
• Name, locate parts of the human body, including those related to the senses
• Say which part of the body is associated with each sense.
Year 2
• Describe the main changes as young animals , including humans, grow into adults
• Find out about and describe the basic needs of animals, including humans, for survival (water, food and air).
• Name and locate parts of the human body, including those related to the senses and describe the importance for humans of exercise, eating the right amounts of different types of food, and
• Identify and name a variety of common animals including fish, amphibians, reptiles, birds and mammals.
• Identify and name a variety of common animals that are carnivores, herbivores and omnivores.
• Describe and compare the structure of a variety of common animals (fish, amphibians, reptiles, birds and mammals including pets).
• Identify, name, draw and label the basic parts of the human body.
• Say which part of the body is associated with each sense.
Everyday materials
Year 1
• Distinguish between an object and the material from which it is made.
• Identify and group everyday materials
• Describe the properties everyday materials.
Year 2
• Distinguish between an object and the material from which it is made.
• Identify and group everyday materials
• Describe the properties of everyday materials
• Compare the sustainability of materials for different uses
Living things and their habitats
Year 1
• Describe and compare the observable features of animals from a range of groups
• Group animals according to what they eat
Year 2
• Identify whether things are alive, dead or have never lived
• Describe how animals get their food from other animals and/or from plants
□ Use simple food chains to describe relationships between animals and plants
• name different plants and how they are suited to their environment
• Name different animals and how they are suited to their environment
Year 1
• Describe the basic needs of plants for surivival
Year 2
• Observe and describe how seeds and bulbs grow into mature plants.
• Describe the basic needs of plants for survival:-
• Find out and describe how plants need water, light and a suitable temperature to grow and stay healthy.
□ The impact of changing water, light and temperature on a plants survival
Seasonal changes
Year 1
• Describe seasonal changes
Year 2
• Observe changes across the 4 seasons.
Uses of everyday materials
Year 1
• Identify and compare the suitability of a variety of everyday materials, including wood, metal, plastic, glass, brick, rock, paper and cardboard for particular uses.
• Find out how the shapes of solid objects made from some materials can be changed by squashing, bending, twisting and stretching.
Year 2
• Identify and compare the suitability of a variety of everyday materials, including wood, metal, plastic, glass, brick, rock, paper and cardboard for particular uses.
• Find out how the shapes of solid objects made from some materials can be changed by squashing, bending, twisting and stretching.
Working scientifically
Year 1
• Ask their own questions about what they notice
• Use different types of scientific enquiry to gather and record data, using simple equipment where appropriate, to answer questions:
□ a) observing changes over time
☆ noticing patterns patterns
☆ grouping and classifying things
☆ carrying out simple comparative tests
Year 2
• Ask their own questions about what they notice
□ use different types of scientific enquiry to gather and record data, using simple equipment where appropriate, to answer questions
☆ observing changes over time
☆ grouping and classifying things
☆ carrying out simple comparative tests
☆ finding thingd out using secondary sourcwes of information
• communicate their ideas, what they do and what they find out in a variety of ways
Key Stage 1 Writing
National Curriculum England 2014
Year 1
• Write sentences by saying out loud what they are going to write about.
• Write sentences by composing sentences orally before writing them.
• Write sentences by re-reading what they have written to check that it makes sense.
• Write sentences by sequencing sentences to form short narratives.
• Discuss what they have written with the teacher or other pupils.
• Read their writing aloud, clearly enough to be heard by their peers and the teacher.
Year 2
• Develop positive attitudes towards and stamina for writing by writing narratives about personal experiences and those of others (real and fictional).
• Develop positive attitudes towards and stamina for writing by writing about real events.
• Develop positive attitudes towards and stamina for writing by writing poetry.
• Develop positive attitudes towards and stamina for writing by writing for different purposes.
• Consider what they are going to write before beginning by planning or saying out loud what they are going to write about.
• Consider what they are going to write before beginning by writing down ideas and / or key words, including new vocabulary.
• Consider what they are going to write before beginning by encapsulating what they want to say, sentence by sentence.
• Make simple additions, revisions and corrections to their own writing by re-reading to check that their reading makes sense and that verbs to indicate time are used correctly and consistently,
including verbs in the continuous form.
• Make simple additions, revisions and corrections to their own writing by proofreading to check for errors in spelling, grammar and punctuation (for example, ends of sentences punctuated
• Make simple additions, revisions and corrections to their own writing by evaluating their writing with the teacher and other pupils.
• Read aloud what they have written with appropriate intonation to make the meaning clear.
Year 1
• Sit correctly at a table, holding a pencil comfortably and correctly.
• Begin to form lower-case letters in the correct direction, starting and finishing in the right place.
• Understand which letters belong to which handwriting ‘families’ (ie letters that are formed in similar ways) and to practise these.
Year 2
• Form lower-case letters of the correct size relative to one another.
• Start using some of the diagonal and horizontal strokes needed to join letters and understand which letters, when adjacent to one another, are best left unjoined.
• Write capital letters and digits of the correct size, orientation and relationship to one another and to lower-case letters.
• Use spacing between words that reflects the size of the letters.
Transcription - Spelling
Year 1
• Spell words containing each of the 40+ phonemes already taught.
• Spell common exception words.
• Spell the days of the week.
• Name the letters of the alphabet in order.
• Use letter names to distinguish between alternative spellings of the same sound.
• Add prefixes and suffixes using the spelling rule for adding –s or –es as the plural marker for nouns and the third person singular marker for verbs.
• Use suffixes –ing, –ed, –er and –est where no change is needed in the spelling of root words (for example, helping, helped, helper, eating, quicker, quickest).
• Apply year 1 spelling rules.
• Write from memory simple sentences dictated by the teacher that include words using the GPCs and common exception words taught so far.
Year 2
• Spell by segmenting spoken words into phonemes and representing these by graphemes, spelling many correctly.
• Spell by learning new ways of spelling phonemes for which 1 or more spellings are already known, and learn some words with each spelling, including a few common homophones.
• Spell by learning to spell common exception words.
• Spell by learning to spell more words with contracted forms.
• Spell by learning the possessive apostrophe (singular) [for example, the girl’s book].
• Spell by distinguishing between homophones and near-homophones.
• Add suffixes to spell longer words including –ment, –ness, –ful, –less, –ly.
• Apply year 2 spelling rules.
• Write from memory simple sentences dictated by the teacher that include words using the GPCs, common exception words and punctuation taught so far.
Vocabulary, grammar and punctuation
Year 1
• Leave spaces between words.
• Join words and join clauses using ‘and’.
• Begin to punctuate sentences using a capital letter and a full stop, question mark or exclamation mark.
• Use a capital letter for names of people, places, the days of the week, and the personal pronoun ‘I'.
• Know the grammar for year 1 (prefix and suffix, joining clauses with and, short narratives, punctuation A . ? !, capital letters for names and personal pronoun I).
Year 2
• Use sentences with different forms: statement, question, exclamation, command.
• Use expanded noun phrases to describe and specify [for example, the blue butterfly].
• Use the present and past tenses correctly and consistently, including the progressive form.
• Use subordination (when, if, that, or because) and co-ordination (or, and, or but)
• Know the grammar for year 2 (suffixes to make adjectives -full, -less, -ly to turn adjectives into adverbs, subordination and so-ordination, expanded noun phrases, present and past tense, commas
in a list, apostrophes).
• Use and understand the year 2 grammatical terminology when discussing their writing.
Key Stage 2 Art and Design
National Curriculum England 2014
Year 3
• To create sketch books to record their observations and use them to review and revisit ideas
• To improve their mastery of art and design techniques, including drawing, painting and sculpture with a range of materials [for example, pencil, charcoal, paint, clay]
• About great artists, architects and designers in history
Year 4
• To create sketch books to record their observations and use them to review and revisit ideas
• To improve their mastery of art and design techniques, including drawing, painting and sculpture with a range of materials [for example, pencil, charcoal, paint, clay]
• About great artists, architects and designers in history
Year 5
• To create sketch books to record their observations and use them to review and revisit ideas
• To improve their mastery of art and design techniques, including drawing, painting and sculpture with a range of materials [for example, pencil, charcoal, paint, clay]
• About great artists, architects and designers in history
Year 6
• To create sketch books to record their observations and use them to review and revisit ideas
• To improve their mastery of art and design techniques, including drawing, painting and sculpture with a range of materials [for example, pencil, charcoal, paint, clay]
• About great artists, architects and designers in history
Key Stage 2 Computing
National Curriculum England 2014
Year 3
• 1. Design, write and debug programs that accomplish specific goals, including controlling or simulating physical systems; solve problems by decomposing them into smaller parts.
• 2. Use sequence, selection, and repetition in programs; work with variables and various forms of input and output.
• 3. Use logical reasoning to explain how some simple algorithms work and to detect and correct errors in algorithms and programs.
• 4. Understand computer networks, including the internet; how they can provide multiple services, such as the World Wide Web, and the opportunities they offer for communication and collaboration.
• 5. Use search technologies effectively, appreciate how results are selected and ranked, and be discerning in evaluating digital content
• 6. Select, use and combine a variety of software (including internet services) on a range of digital devices to design and create a range of programs, systems and content that accomplish given
goals, including collecting, analysing, evaluating and presenting data and information
• 7. Use technology safely, respectfully and responsibly; recognise acceptable/unacceptable behaviour; identify a range of ways to report concerns about content and contact
Year 4
• 1. Design, write and debug programs that accomplish specific goals, including controlling or simulating physical systems; solve problems by decomposing them into smaller parts
• 2. Use sequence, selection, and repetition in programs; work with variables and various forms of input and output
• 3. Use logical reasoning to explain how some simple algorithms work and to detect and correct errors in algorithms and programs
• 4. Understand computer networks, including the internet; how they can provide multiple services, such as the World Wide Web, and the opportunities they offer for communication and collaboration
• 5. Use search technologies effectively, appreciate how results are selected and ranked, and be discerning in evaluating digital content
• 6. Select, use and combine a variety of software (including internet services) on a range of digital devices to design and create a range of programs, systems and content that accomplish given
goals, including collecting, analysing, evaluating and presenting data and information
• 7. Use technology safely, respectfully and responsibly; recognise acceptable/unacceptable behaviour; identify a range of ways to report concerns about content and contact
Year 5
• 1. Design, write and debug programs that accomplish specific goals, including controlling or simulating physical systems; solve problems by decomposing them into smaller parts
• 2. Use sequence, selection, and repetition in programs; work with variables and various forms of input and output
• 3. Use logical reasoning to explain how some simple algorithms work and to detect and correct errors in algorithms and programs
• 4. Understand computer networks, including the internet; how they can provide multiple services, such as the World Wide Web, and the opportunities they offer for communication and collaboration
• 5. Use search technologies effectively, appreciate how results are selected and ranked, and be discerning in evaluating digital content
• 6. Select, use and combine a variety of software (including internet services) on a range of digital devices to design and create a range of programs, systems and content that accomplish given
goals, including collecting, analysing, evaluating and presenting data and information
• 7. Use technology safely, respectfully and responsibly; recognise acceptable/unacceptable behaviour; identify a range of ways to report concerns about content and contact
Year 6
• 1. Design, write and debug programs that accomplish specific goals, including controlling or simulating physical systems; solve problems by decomposing them into smaller parts
• 2. Use sequence, selection, and repetition in programs; work with variables and various forms of input and output
• 3. Use logical reasoning to explain how some simple algorithms work and to detect and correct errors in algorithms and programs
• 4. Understand computer networks, including the internet; how they can provide multiple services, such as the World Wide Web, and the opportunities they offer for communication and collaboration
• 5. Use search technologies effectively, appreciate how results are selected and ranked, and be discerning in evaluating digital content
• 6. Select, use and combine a variety of software (including internet services) on a range of digital devices to design and create a range of programs, systems and content that accomplish given
goals, including collecting, analysing, evaluating and presenting data and information
• 7. Use technology safely, respectfully and responsibly; recognise acceptable/unacceptable behaviour; identify a range of ways to report concerns about content and contact
Key Stage 2 Design and Technology
National Curriculum England 2014
Cooking and Nutrition
Year 3
• Understand and apply the principles of a healthy and varied diet
• Prepare and cook a variety of predominantly savoury dishes using a range of cooking techniques
• Understand seasonality, and know where and how a variety of ingredients are grown, reared, caught and processed
Year 4
• Understand and apply the principles of a healthy and varied diet
• Prepare and cook a variety of predominantly savoury dishes using a range of cooking techniques
• Understand seasonality, and know where and how a variety of ingredients are grown, reared, caught and processed
Year 5
• Understand and apply the principles of a healthy and varied diet
• Prepare and cook a variety of predominantly savoury dishes using a range of cooking techniques
• Understand seasonality, and know where and how a variety of ingredients are grown, reared, caught and processed
Year 6
• Understand and apply the principles of a healthy and varied diet
• Prepare and cook a variety of predominantly savoury dishes using a range of cooking techniques
• Understand seasonality, and know where and how a variety of ingredients are grown, reared, caught and processed
Year 3
• Use research and develop design criteria to inform the design of innovative, functional, appealing products that are fit for purpose, aimed at particular individuals or groups
• Generate, develop, model and communicate their ideas through discussion, annotated sketches, cross-sectional and exploded diagrams, prototypes, pattern pieces and computer-aided design
Year 4
• Use research and develop design criteria to inform the design of innovative, functional, appealing products that are fit for purpose, aimed at particular individuals or groups
• Generate, develop, model and communicate their ideas through discussion, annotated sketches, cross-sectional and exploded diagrams, prototypes, pattern pieces and computer-aided design
Year 5
• Use research and develop design criteria to inform the design of innovative, functional, appealing products that are fit for purpose, aimed at particular individuals or groups
• Generate, develop, model and communicate their ideas through discussion, annotated sketches, cross-sectional and exploded diagrams, prototypes, pattern pieces and computer-aided design
Year 6
• Use research and develop design criteria to inform the design of innovative, functional, appealing products that are fit for purpose, aimed at particular individuals or groups
• Generate, develop, model and communicate their ideas through discussion, annotated sketches, cross-sectional and exploded diagrams, prototypes, pattern pieces and computer-aided design
Year 3
• Investigate and analyse a range of existing products
• Evaluate their ideas and products against their own design criteria and consider the views of others to improve their work
• Understand how key events and individuals in design and technology have helped shape the world
Year 4
• Investigate and analyse a range of existing products
• Evaluate their ideas and products against their own design criteria and consider the views of others to improve their work
• Understand how key events and individuals in design and technology have helped shape the world
Year 5
• Investigate and analyse a range of existing products
• Evaluate their ideas and products against their own design criteria and consider the views of others to improve their work
• Understand how key events and individuals in design and technology have helped shape the world
Year 6
• Investigate and analyse a range of existing products
• Evaluate their ideas and products against their own design criteria and consider the views of others to improve their work
• Understand how key events and individuals in design and technology have helped shape the world
Year 3
• Select from and use a wider range of tools and equipment to perform practical tasks [for example, cutting, shaping, joining and finishing], accurately
• Select from and use a wider range of materials and components, including construction materials, textiles and ingredients, according to their functional properties and aesthetic qualities
Year 4
• Select from and use a wider range of tools and equipment to perform practical tasks [for example, cutting, shaping, joining and finishing], accurately
• Select from and use a wider range of materials and components, including construction materials, textiles and ingredients, according to their functional properties and aesthetic qualities
Year 5
• Select from and use a wider range of tools and equipment to perform practical tasks [for example, cutting, shaping, joining and finishing], accurately
• Select from and use a wider range of materials and components, including construction materials, textiles and ingredients, according to their functional properties and aesthetic qualities
Year 6
• Select from and use a wider range of tools and equipment to perform practical tasks [for example, cutting, shaping, joining and finishing], accurately
• Select from and use a wider range of materials and components, including construction materials, textiles and ingredients, according to their functional properties and aesthetic qualities
Technical Knowledge
Year 3
• Apply their understanding of how to strengthen, stiffen and reinforce more complex structures
• Understand and use mechanical systems in their products [for example, gears, pulleys, cams, levers and linkages]
• Understand and use electrical systems in their products [for example, series circuits incorporating switches, bulbs, buzzers and motors]
• Apply their understanding of computing to program, monitor and control their products
Year 4
• Apply their understanding of how to strengthen, stiffen and reinforce more complex structures
• Understand and use mechanical systems in their products [for example, gears, pulleys, cams, levers and linkages]
• Understand and use electrical systems in their products [for example, series circuits incorporating switches, bulbs, buzzers and motors]
• Apply their understanding of computing to program, monitor and control their products
Year 5
• Apply their understanding of how to strengthen, stiffen and reinforce more complex structures
• Understand and use mechanical systems in their products [for example, gears, pulleys, cams, levers and linkages]
• Understand and use electrical systems in their products [for example, series circuits incorporating switches, bulbs, buzzers and motors]
• Apply their understanding of computing to program, monitor and control their products
Year 6
• Apply their understanding of how to strengthen, stiffen and reinforce more complex structures
• Understand and use mechanical systems in their products [for example, gears, pulleys, cams, levers and linkages]
• Understand and use electrical systems in their products [for example, series circuits incorporating switches, bulbs, buzzers and motors]
• Apply their understanding of computing to program, monitor and control their products
Key Stage 2 Geography
National Curriculum England 2014
Geographical skills and fieldwork
Year 3
• Use maps, atlases, globes and digital/computer mapping to locate countries and describe features studied
• Use the eight points of a compass, four and six-figure grid references, symbols and key (including the use of Ordnance Survey maps) to build their knowledge of the United Kingdom and the wider
• Use fieldwork to observe, measure, record and present the human and physical features in the local area using a range of methods, including sketch maps, plans and graphs, and digital
Year 4
• Use maps, atlases, globes and digital/computer mapping to locate countries and describe features studied
• Use the eight points of a compass, four and six-figure grid references, symbols and key (including the use of Ordnance Survey maps) to build their knowledge of the United Kingdom and the wider
• Use fieldwork to observe, measure, record and present the human and physical features in the local area using a range of methods, including sketch maps, plans and graphs, and digital
Year 5
• Use maps, atlases, globes and digital/computer mapping to locate countries and describe features studied
• Use the eight points of a compass, four and six-figure grid references, symbols and key (including the use of Ordnance Survey maps) to build their knowledge of the United Kingdom and the wider
• Use fieldwork to observe, measure, record and present the human and physical features in the local area using a range of methods, including sketch maps, plans and graphs, and digital
Year 6
• Use maps, atlases, globes and digital/computer mapping to locate countries and describe features studied
• Use the eight points of a compass, four and six-figure grid references, symbols and key (including the use of Ordnance Survey maps) to build their knowledge of the United Kingdom and the wider
• Use fieldwork to observe, measure, record and present the human and physical features in the local area using a range of methods, including sketch maps, plans and graphs, and digital
Human and physical geography
Year 3
• Describe and understand key aspects of:
□ Physical geography, including: climate zones, biomes and vegetation belts, rivers, mountains, volcanoes and earthquakes, and the water cycle
□ Human geography, including: types of settlement and land use, economic activity including trade links, and the distribution of natural resources including energy, food, minerals and water
Year 4
• Describe and understand key aspects of:
□ Physical geography, including: climate zones, biomes and vegetation belts, rivers, mountains, volcanoes and earthquakes, and the water cycle
□ Human geography, including: types of settlement and land use, economic activity including trade links, and the distribution of natural resources including energy, food, minerals and water
Year 5
• Describe and understand key aspects of:
□ Physical geography, including: climate zones, biomes and vegetation belts, rivers, mountains, volcanoes and earthquakes, and the water cycle
□ Human geography, including: types of settlement and land use, economic activity including trade links, and the distribution of natural resources including energy, food, minerals and water
Year 6
• Describe and understand key aspects of:
□ Physical geography, including: climate zones, biomes and vegetation belts, rivers, mountains, volcanoes and earthquakes, and the water cycle
□ Human geography, including: types of settlement and land use, economic activity including trade links, and the distribution of natural resources including energy, food, minerals and water
Locational knowledge
Year 3
• Locate the world’s countries, using maps to focus on Europe (including the location of Russia) and North and South America, concentrating on their environmental regions, key physical and human
characteristics, countries, and major citie
• Name and locate counties and cities of the United Kingdom, geographical regions and their identifying human and physical characteristics, key topographical features (including hills, mountains,
coasts and rivers), and land-use patterns; and understand how some of these aspects have changed over time
• Identify the position and significance of latitude, longitude, Equator, Northern Hemisphere, Southern Hemisphere, the Tropics of Cancer and Capricorn, Arctic and Antarctic Circle, the Prime/
Greenwich Meridian and time zones (including day and night)
Year 4
• Locate the world’s countries, using maps to focus on Europe (including the location of Russia) and North and South America, concentrating on their environmental regions, key physical and human
characteristics, countries, and major citie
• Name and locate counties and cities of the United Kingdom, geographical regions and their identifying human and physical characteristics, key topographical features (including hills, mountains,
coasts and rivers), and land-use patterns; and understand how some of these aspects have changed over time
• Identify the position and significance of latitude, longitude, Equator, Northern Hemisphere, Southern Hemisphere, the Tropics of Cancer and Capricorn, Arctic and Antarctic Circle, the Prime/
Greenwich Meridian and time zones (including day and night)
Year 5
• Locate the world’s countries, using maps to focus on Europe (including the location of Russia) and North and South America, concentrating on their environmental regions, key physical and human
characteristics, countries, and major citie
• Name and locate counties and cities of the United Kingdom, geographical regions and their identifying human and physical characteristics, key topographical features (including hills, mountains,
coasts and rivers), and land-use patterns; and understand how some of these aspects have changed over time
• Identify the position and significance of latitude, longitude, Equator, Northern Hemisphere, Southern Hemisphere, the Tropics of Cancer and Capricorn, Arctic and Antarctic Circle, the Prime/
Greenwich Meridian and time zones (including day and night)
Year 6
• Locate the world’s countries, using maps to focus on Europe (including the location of Russia) and North and South America, concentrating on their environmental regions, key physical and human
characteristics, countries, and major citie
• Name and locate counties and cities of the United Kingdom, geographical regions and their identifying human and physical characteristics, key topographical features (including hills, mountains,
coasts and rivers), and land-use patterns; and understand how some of these aspects have changed over time
• Identify the position and significance of latitude, longitude, Equator, Northern Hemisphere, Southern Hemisphere, the Tropics of Cancer and Capricorn, Arctic and Antarctic Circle, the Prime/
Greenwich Meridian and time zones (including day and night)
Place knowledge
Year 3
• Understand geographical similarities and differences through the study of human and physical geography of a region of the United Kingdom, a region in a European country, and a region within North
or South America
Year 4
• Understand geographical similarities and differences through the study of human and physical geography of a region of the United Kingdom, a region in a European country, and a region within North
or South America
Year 5
• Understand geographical similarities and differences through the study of human and physical geography of a region of the United Kingdom, a region in a European country, and a region within North
or South America
Year 6
• Understand geographical similarities and differences through the study of human and physical geography of a region of the United Kingdom, a region in a European country, and a region within North
or South America
Key Stage 2 History
National Curriculum England 2014
Year 3
• Changes in Britain from the Stone Age to the Iron Age
• The Roman Empire and its impact on Britain
• Britain’s settlement by Anglo-Saxons and Scots
• The Viking and Anglo-Saxon struggle for the Kingdom of England to the time of Edward the Confessor
• A study of an aspect or theme in British history that extends pupils’ chronological knowledge beyond 1066
• The achievements of the earliest civilizations – an overview of where and when the first civilizations appeared and a depth study of one of the following: Ancient Sumer, The Indus Valley, Ancient
Egypt, The Shang Dynasty of Ancient China
• Ancient Greece – a study of Greek life and achievements and their influence on the western world
• A non-European society that provides contrasts with British history – one study chosen from: early Islamic civilization, including a study of Baghdad c. AD 900; Mayan civilization c. AD 900;
Benin (West Africa) c. AD 900-1300
Year 4
• Changes in Britain from the Stone Age to the Iron Age
• The Roman Empire and its impact on Britain
• Britain’s settlement by Anglo-Saxons and Scots
• The Viking and Anglo-Saxon struggle for the Kingdom of England to the time of Edward the Confessor
• A study of an aspect or theme in British history that extends pupils’ chronological knowledge beyond 1066
• The achievements of the earliest civilizations – an overview of where and when the first civilizations appeared and a depth study of one of the following: Ancient Sumer, The Indus Valley, Ancient
Egypt, The Shang Dynasty of Ancient China
• Ancient Greece – a study of Greek life and achievements and their influence on the western world
• A non-European society that provides contrasts with British history – one study chosen from: early Islamic civilization, including a study of Baghdad c. AD 900; Mayan civilization c. AD 900;
Benin (West Africa) c. AD 900-1300
Year 5
• Changes in Britain from the Stone Age to the Iron Age
• The Roman Empire and its impact on Britain
• Britain’s settlement by Anglo-Saxons and Scots
• The Viking and Anglo-Saxon struggle for the Kingdom of England to the time of Edward the Confessor
• A study of an aspect or theme in British history that extends pupils’ chronological knowledge beyond 1066
• The achievements of the earliest civilizations – an overview of where and when the first civilizations appeared and a depth study of one of the following: Ancient Sumer, The Indus Valley, Ancient
Egypt, The Shang Dynasty of Ancient China
• Ancient Greece – a study of Greek life and achievements and their influence on the western world
• A non-European society that provides contrasts with British history – one study chosen from: early Islamic civilization, including a study of Baghdad c. AD 900; Mayan civilization c. AD 900;
Benin (West Africa) c. AD 900-1300
Year 6
• Changes in Britain from the Stone Age to the Iron Age
• The Roman Empire and its impact on Britain
• Britain’s settlement by Anglo-Saxons and Scots
• The Viking and Anglo-Saxon struggle for the Kingdom of England to the time of Edward the Confessor
• A study of an aspect or theme in British history that extends pupils’ chronological knowledge beyond 1066
• The achievements of the earliest civilizations – an overview of where and when the first civilizations appeared and a depth study of one of the following: Ancient Sumer, The Indus Valley, Ancient
Egypt, The Shang Dynasty of Ancient China
• Ancient Greece – a study of Greek life and achievements and their influence on the western world
• A non-European society that provides contrasts with British history – one study chosen from: early Islamic civilization, including a study of Baghdad c. AD 900; Mayan civilization c. AD 900;
Benin (West Africa) c. AD 900-1300
Key Stage 2 Languages
National Curriculum England 2014
Foreign language
Year 3
• Listen attentively to spoken language and show understanding by joining in and responding
• Explore the patterns and sounds of language through songs and rhymes and link the spelling, sound and meaning of words
• Engage in conversations; ask and answer questions; express opinions and respond to those of others; seek clarification and help*
• Speak in sentences, using familiar vocabulary, phrases and basic language structures
• Develop accurate pronunciation and intonation so that others understand when they are reading aloud or using familiar words and phrases*
• Present ideas and information orally to a range of audiences*
• Read carefully and show understanding of words, phrases and simple writing
• Appreciate stories, songs, poems and rhymes in the language
• Broaden their vocabulary and develop their ability to understand new words that are introduced into familiar written material, including through using a dictionary
• Write phrases from memory, and adapt these to create new sentences, to express ideas clearly
• Describe people, places, things and actions orally* and in writing
• Understand basic grammar appropriate to the language being studied, including (where relevant): feminine, masculine and neuter forms and the conjugation of high-frequency verbs; key features and
patterns of the language; how to apply these, for instance, to build sentences; and how these differ from or are similar to english
Year 4
• Listen attentively to spoken language and show understanding by joining in and responding
• Explore the patterns and sounds of language through songs and rhymes and link the spelling, sound and meaning of words
• Engage in conversations; ask and answer questions; express opinions and respond to those of others; seek clarification and help*
• Speak in sentences, using familiar vocabulary, phrases and basic language structures
• Develop accurate pronunciation and intonation so that others understand when they are reading aloud or using familiar words and phrases*
• Present ideas and information orally to a range of audiences*
• Read carefully and show understanding of words, phrases and simple writing
• Appreciate stories, songs, poems and rhymes in the language
• Broaden their vocabulary and develop their ability to understand new words that are introduced into familiar written material, including through using a dictionary
• Write phrases from memory, and adapt these to create new sentences, to express ideas clearly
• Describe people, places, things and actions orally* and in writing
• Understand basic grammar appropriate to the language being studied, including (where relevant): feminine, masculine and neuter forms and the conjugation of high-frequency verbs; key features and
patterns of the language; how to apply these, for instance, to build sentences; and how these differ from or are similar to english
Year 5
• Listen attentively to spoken language and show understanding by joining in and responding
• Explore the patterns and sounds of language through songs and rhymes and link the spelling, sound and meaning of words
• Engage in conversations; ask and answer questions; express opinions and respond to those of others; seek clarification and help*
• Speak in sentences, using familiar vocabulary, phrases and basic language structures
• Develop accurate pronunciation and intonation so that others understand when they are reading aloud or using familiar words and phrases*
• Present ideas and information orally to a range of audiences*
• Read carefully and show understanding of words, phrases and simple writing
• Appreciate stories, songs, poems and rhymes in the language
• Broaden their vocabulary and develop their ability to understand new words that are introduced into familiar written material, including through using a dictionary
• Write phrases from memory, and adapt these to create new sentences, to express ideas clearly
• Describe people, places, things and actions orally* and in writing
• Understand basic grammar appropriate to the language being studied, including (where relevant): feminine, masculine and neuter forms and the conjugation of high-frequency verbs; key features and
patterns of the language; how to apply these, for instance, to build sentences; and how these differ from or are similar to english
Year 6
• Listen attentively to spoken language and show understanding by joining in and responding
• Explore the patterns and sounds of language through songs and rhymes and link the spelling, sound and meaning of words
• Engage in conversations; ask and answer questions; express opinions and respond to those of others; seek clarification and help*
• Speak in sentences, using familiar vocabulary, phrases and basic language structures
• Develop accurate pronunciation and intonation so that others understand when they are reading aloud or using familiar words and phrases*
• Present ideas and information orally to a range of audiences*
• Read carefully and show understanding of words, phrases and simple writing
• Appreciate stories, songs, poems and rhymes in the language
• Broaden their vocabulary and develop their ability to understand new words that are introduced into familiar written material, including through using a dictionary
• Write phrases from memory, and adapt these to create new sentences, to express ideas clearly
• Describe people, places, things and actions orally* and in writing
• Understand basic grammar appropriate to the language being studied, including (where relevant): feminine, masculine and neuter forms and the conjugation of high-frequency verbs; key features and
patterns of the language; how to apply these, for instance, to build sentences; and how these differ from or are similar to english
Key Stage 2 Mathematics
National Curriculum England 2014 - NAHT Assessment Framework
Year 6
• Generate and describe linear number sequences
• Express missing number problems algebraically
• Find numbers that satisfy an equation with an unknown
• Find pairs of numbers that satisfy an equation with 2 unknowns
• Enumerate possibilities of combinations of 2 variables
Geometry - position and direction
Year 4
• Describe positions on a 2-D grid as coordinates in the first quadrant
• Describe movements between positions as translations of a given unit to the left/right and up/down
• Plots specified points and draw sides to complete a given polygon
• 4G-1: Draw polygons, specified by coordinates in the first quadrant, and translate within the first quadrant. (RtP)
Year 5
• Identify, describe and represent the position of a shape following a reflection, using the appropriate language, and know that the shape has not changed
• Identify, describe and represent the position of a shape following a translation, using the appropriate language, and know that the shape has not changed
Year 6
• Describe positions on the full coordinate grid (all 4 quadrants).
□ WRM: Describe positions on the full coordinate grid (all four quadrants).
□ WRM: Solve problems with coordinates.
• Draw and translate simple shapes on the coordinate plane, and reflect them in the axes.
Geometry - properties of shapes
Year 3
• Draw 2-D shapes and make 3-D shapes using modelling materials
• Recognise 3-d shapes in different orientations and describe them
• Recognise angles as a property of shape or a description of a turn
• Identify horizontal and vertical lines and pairs of perpendicular and parallel lines
• 3G-1: Recognise right angles as a property of a shape or a description of a turn, and identify right angles in 2D shapes presented in different orientations. (RtP)
• 3G-2: Draw polygons by joining marked points, and identify parallel and perpendicular sides. (RtP)
Year 4
• Compares and classifies geometric shapes
• Compares and classifies quadrilaterals based on their properties and sizes
• compares and classifies triangles, based on their properties and sizes
• Identify acute and obtuse angles
• compare and order angles up to 2 right angles by size
• Identify lines of symmetry in two dimensional shapes presented in different orientations
• Complete a simple symmetric figure with respect to a specific line of symmetry
• 4G-2: Identify regular polygons, including equilateral triangles and squares, as those in which the side-lengths are equal and the angles are equal. Find the perimeter of regular and irregular
polygons. (RtP)
• 4G-3: Identify line symmetry in 2D shapes presented in different orientations. Reflect shapes in a line of symmetry and complete a symmetric figure or pattern with respect to a specified line of
symmetry. (RtP)
Year 5
• Identify 3-D shapes, including cubes and other cuboids, from 2-D representations
• Know angles are measured in degrees: estimate and compare acute, obtuse and reflex angles
• Draws given angles, and measure them in degrees (°)
□ Angles at a point and 1 whole turn (total 360°)
□ Angles at a point on a straight line and half a turn (total 180°)
□ Use the properties of rectangles to deduce related facts and find missing lengths and angles
• Distinguishes between regular and irregular polygons based on reasoning about equal sides and angles
• 5G-1: Compare angles, estimate and measure angles in degrees and draw angles of a given size. (RtP)
• 5G-2: Compare areas and calculate the area of rectangles (including squares) using standard units. (RtP)
Year 6
• Draw 2-D shapes using given dimensions and angles.
□ WRM: Draw shapes accurately.
• Recognise, describe and build simple 3-D shapes, including making nets.
• Compare and classify geometric shapes based on their properties and sizes and find unknown angles in any triangles, quadrilaterals and regular polygons.
□ WRM: Angles in a triangle.
□ WRM: Angles in a triangle-special cases.
□ WRM: Angles in a triangle-missing angles.
□ WRM: Angles in a quadrilateral.
• Illustrate and name parts of circles, including radius, diameter and circumference and know that the diameter is twice the radius.
• Recognise angles where they meet at a point, are on a straight line, or are vertically opposite, and find missing angles.
□ WRM: Measure and classify angles.
□ WRM: Vertically opposite angles.
• 6G-1: Draw, compose and decompose shapes according to given properties, including dimensions, angles and area, and solve related problems. (RtP)
Year 3
• Measure, compare, add and subtract: lengths (m/cm/mm); mass (kg/g); volume/capacity (l/ml).
□ WRM: Measure in metres and centimetres.
□ WRM: Measure in millimetres.
□ WRM: Measure in centimetres and millimetres.
□ WRM: Metres, centimetres and millimetres.
□ WRM: Equivalent lengths (metres and centimetres).
□ WRM: Equivalent lengths (centimetres and millimetres).
□ WRM: Calculate perimeter.
□ WRM: Measure mass in grams.
□ WRM: Measure mass in kilograms and grams.
□ WRM: Equivalent masses (kilograms and grams).
□ WRM: Add and subtract mass.
□ WRM: Measure capacity and volume in millilitres.
□ WRM: Measure capacity and volume in litres and millilitres.
□ WRM: Equivalent capacities and volumes (litres and millilitres).
□ WRM: Compare capacity and volume.
□ WRM: Add and subtract capacity and volume.
• Measure the perimeter of simple 2-D shapes
• Adds and subtracts amounts of money to give change, using both £ and p in practical contexts
• Tells and writes the time from an analogue clock, including using roman numerals from I to XII, and 12-hour and 24-hour clocks
• Tells and writes the time from an analogue clock and 12-hour and 24-hour clocks
• Estimate and read time with increasing accuracy to the nearest minute; record and compare time in terms of seconds, minutes and hours; use vocabulary such as o’clock, am/pm, morning, afternoon,
noon and midnight
• Know the number of seconds in a minute and the number of days in each month, year and leap year
• Compare durations of events [for example, to calculate the time taken by particular events or tasks]
• Identifies right angles, recognises that two right angles make a half-turn, three make three quarters of a turn and four a complete turn; identifies whether angles are greater than or less than a
right angle
Year 4
• Convert between different units of measure [for example, kilometre to metre; hour to minute].
□ WRM: Measure in kilometres and metres.
□ WRM: Equivalent lengths (kilometres and metres).
• Measure and calculate the perimeter of a rectilinear figure (including squares) in centimetres and metres.
□ WRM: Perimeter on a grid.
□ WRM: Perimeter of a rectangle.
□ WRM: Perimeter of rectilinear shapes.
□ WRM: Find missing lengths in rectilinear shapes.
□ WRM: Calculate the perimeter of rectilinear shapes.
□ WRM: Perimeter of regular polygons.
□ WRM: Perimeter of polygons.
• Find the area of rectilinear shapes by counting squares.
• Estimate, compare and calculate different measures, including money in pounds and pence
• Read, write and convert time between analogue and digital 12- and 24-hour clocks
• Solve problems involving converting from hours to minutes, minutes to seconds, years to months, weeks to days
Year 5
• Converts between different units of metric measure;
☆ centimetre and millimeter
• Understand and use approximate equivalences between metric units and common imperial units such as inches, pounds and pints
• Measure and calculate the perimeter of composite rectilinear shapes in centimetres and metres.
□ WRM: Perimeter of rectangles.
□ WRM: Perimeter of rectilinear shapes.
□ WRM: Perimeter of polygons.
• Calculate and compare the area of rectangles (including squares), including using standard units, square centimetres (cm²) and square metres (m²) and estimate the area of irregular shapes.
□ WRM: Area of compound shapes.
• Estimate volume [for example, using 1 cm³ blocks to build cuboids (including cubes)] and capacity [for example, using water]
• Solve problems involving converting between units of time
• Use all four operations to solve problems involving measure [for example, length, mass, volume, money] using decimal notation, including scaling
Year 6
• Solve problems involving the calculation and conversion of units of measure, using decimal notation up to 3 decimal places where appropriate.
• Use, read, write and convert between standard units, converting measurements of length, mass, volume and time from a smaller unit of measure to a larger unit, and vice versa, using decimal
notation to up to 3 decimal places.
□ WRM: Convert metric measures.
□ WRM: Calculate with metric measures.
□ WRM: Miles and kilometres.
• Uses, reads and writes units of time.
• Recognise that shapes with the same areas can have different perimeters and vice versa
• Recognise when it is possible to use formulae for area and volume of shapes
• Calculate the area of parallelograms and triangles
• Calculate, estimate and compare volume of cubes and cuboids using standard units, including cubic centimetres (cm³) and cubic metres (m³), and extending to other units [for example, mm³ and km³]
Number - addition and subtraction
Year 3
• Add and subtract numbers mentally including:
□ A three-digit number and ones.
□ A three-digit number and tens.
□ A three-digit number and hundreds.
□ WRM: Apply number bonds within 10.
□ WRM: Add and subtract 1s.
□ WRM: Add and subtract 10s.
□ WRM: Add and subtract 100s.
• Add numbers with up to 3 digits, using formal written methods of columnar addition.
□ WRM: Add 2-digit and 3-digit numbers.
• Subtract numbers with up to 3 digits, using:
□ formal written methods of columnar subtraction.
□ WRM: Subtract a 2-digit number from a 3-digit number.
• Estimate the answer to a calculation and use inverse operations to check answers.
• Solve problems, including missing number problems, using number facts, place value, and more complex addition and subtraction.
• 3NF-1: Secure fluency in addition and subtraction facts that bridge 10, through continued practice. (RtP)
• 3AS-1: Calculate complements to 100, for example: 46 + ? = 100 (RtP)
• 3AS-2: Add and subtract up to three-digit numbers using columnar methods. (RtP)
• 3AS-3: Manipulate the additive relationship: understand the inverse relationship between addition and subtraction, and how both relate to the part-part-whole structure. Understand and use the
commutative property of addition, and understand the related property for subtraction. (RtP)
□ WRM: Add 10s across a 100.
□ WRM: Subtract 1s across a 10.
□ WRM: Subtract 10s across a 100.
□ WRM: Add two numbers (no exchange).
□ WRM: Subtract two numbers (no exchange).
□ WRM: Add two numbers (across a 10).
□ WRM: Add two numbers (across a 100).
□ WRM: Subtract two numbers (across a 10).
□ WRM: Subtract two numbers (across a 100).
Year 4
• Add numbers up to 4 digits using the formal written methods of columnar addition where appropriate.
□ WRM: Add up to two 4-digit numbers - no exchange.
□ WRM: Add two 4-digit numbers - one exchange.
□ WRM: Add two 4-digit numbers - more than one exchange.
• Subtract numbers with up to 4 digits using the formal written methods of columnar subtraction where appropriate.
□ WRM: Subtract two 4-digit numbers - no exchange.
□ WRM: Subtract two 4-digit numbers - one exchange.
□ WRM: Subtract two 4-digit numbers - more than one exchange.
□ WRM: Efficient subtraction.
• Estimate and use inverse operations to check answers to a calculation.
• Solve addition and subtraction two-step problems in context, deciding which operations and methods to use and why.
□ WRM: Checking strategies.
• WRM: Add and subtract 1s, 100s and 1,000s.
Year 5
• Add whole numbers with more than 4 digits, including using formal written methods (columnar addition).
□ WRM: Add whole numbers with more than four digits.
• Subtract whole numbers with more than 4 digits, including using formal written methods (columnar subtraction).
□ WRM: Subtract whole numbers with more than four digits.
• Use rounding to check answers to calculations and determine, in the context of a problem, levels of accuracy.
□ WRM: Round to check answers.
• Solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why.
□ WRM: Multi-step addition and subtraction problems.
□ Subtract numbers mentally with increasingly large numbers (eg 12,462 - 2,300 = 10,162).
□ Add numbers mentally with increasingly large numbers (eg 12,462 - 2,300 = 10,162).
• WRM: Inverse operations (addition and subtraction).
• WRM: Compare calculations.
• WRM: Find missing numbers.
Number - addition, subtraction, multiplication and division
Year 6
• Multiply multi-digit numbers up to four digits by a two-digit whole number using the formal written method of long multiplication.
□ WRM: Multiply up to a 4-digit number by a 2-digit number.
• Divide numbers up to 4 digits by a two-digit whole number using the formal written method of long division, and interpret remainders as whole number remainders, fractions, or by rounding, as
appropriate for the context.
□ WRM: Introduction to long division.
□ WRM: Long division with remainders.
• Divide numbers up to four digits by a two-digit number using the formal written method of short division where appropriate, interpreting remainders according to the context.
□ WRM: Division using factors.
• Perform mental calculations, including with mixed operations and large numbers.
□ WRM: Mental calculations and estimation.
• Identify common factors, common multiples and prime numbers.
• Use knowledge of the order of operations to carry out calculations involving the 4 operations.
□ WRM: Order of operations.
• Solve problems involving:
☆ multiplication(WRM: Solve problems with multiplication).
☆ division(WRM: Solve problems with division).
• Use estimation to check answers to calculations and determines, in the context of a problem, an appropriate degree of accuracy.
• 6AS/MD-1: Understand that 2 numbers can be related additively or multiplicatively, and quantify additive and multiplicative relationships (multiplicative relationships restricted to
multiplication by a whole number). (RtP)
• 6AS/MD-2: Use a given additive or multiplicative calculation to derive or complete a related calculation, using arithmetic properties, inverse operations, and place-value understanding. (RtP)
• 6AS/MD-3: Solve problems involving ratio relationships. (RtP)
• 6AS/MD-4: Solve problems with 2 unknowns. (RtP)
• WRM: Rules of divisibility.
• WRM: Square and cube numbers.
• WRM: Solve multi-step problems.
□ Solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why
☆ WRM: Add and subtract integers.
• WRM: Reason from known facts.
Number - fractions
Year 3
• Counts up and down in tenths
• Recognise that tenths arise from dividing an object into 10 equal parts and in dividing one-digit numbers or quantities by 10
• Recognise, find and write fractions of a discrete set of objects: unit fractions and non-unit fractions with small denominators.
□ WRM: Understand the denominators of unit fractions.
□ WRM: Understand the numerators of non-unit fractions.
□ WRM: Understand the whole.
• Recognise and use fractions as numbers: unit fractions and non-unit fractions with small denominators.
□ WRM: Fractions and scales.
□ WRM: Fractions on a number line.
□ WRM: Count in fractions on a number line.
• Recognise and show, using diagrams, equivalent fractions with small denominators.
□ WRM: Equivalent fractions on a number line.
□ WRM: Equivalent fractions as bar models.
• Add and subtract fractions with the same denominator within one whole [for example,5/7+1/7=6/7]
• Compare and order unit fractions, and fractions with the same denominators.
□ WRM: Compare and order unit fractions.
□ WRM: Compare and order non-unit fractions.
• Solve problems that involve all of the above
• 3F-1: Interpret and write proper fractions to represent 1 or several parts of a whole that is divided into equal parts. (RtP)
• 3F-2: Find unit fractions of quantities using known division facts (multiplication tables fluency). (RtP)
• 3F-3: Reason about the location of any fraction within 1 in the linear number system. (RtP)
• 3F-4: Add and subtract fractions with the same denominator, within 1. (RtP)
Year 4
• Recognise and show, using diagrams, families of common equivalent fractions
□ WRM: Equivalent fractions on a number line.
□ WRM: Equivalent fraction families.
• Counts up and down in hundredths; recognise that hundredths arise when dividing an object by 100 and dividing tenths by 10.
□ WRM: Hundredths as fractions.
• Solve problems involving increasingly harder fractions to calculate quantities
• Solve problems involving increasingly harder fractions to divide quantities
• Solve problems involving increasingly harder fractions to calculate quantities, and fractions to divide quantities, including non-unit fractions where the answer is a whole number
• Add and subtract fractions with the same denominator.
□ WRM: Add two or more fractions.
□ WRM: Add fractions and mixed numbers.
□ WRM: Subtract two fractions.
□ WRM: Subtract from whole amounts.
□ WRM: Subtract from mixed numbers.
• Recognise and write decimal equivalents of any number of tenths or hundreds.
□ WRM: Tenths on a place value chart.
□ WRM: Tenths on a number line.
□ WRM: Hundredths as decimals.
□ WRM: Hundredths on a place value chart.
• Recognise and write decimal equivalents to ¼, ½ and ¾
• Find the effect of dividing a one- or two-digit number by 10 and 100, identifying the value of the digits in the answer as ones, tenths and hundredths
□ WRM: Divide a 1-digit number by 10.
□ WRM: Divide a 2-digit number by 10.
□ WRM: Divide a 1- or 2-digit number by 100.
• Rounds decimals with one decimal place to the nearest whole number
• Compare numbers with the same number of decimal places up to 2 decimal places
• Solves simple measure problems involving fractions and decimals to two decimal places
• Solves simple money problems involving fractions and decimals to two decimal places
• 4F-1: Reason about the location of mixed numbers in the linear number system. (RtP)
• 4F-2: Convert mixed numbers to improper fractions and vice versa. (RtP)
• 4F-3: Add and subtract improper and mixed fractions with the same denominator, including bridging whole numbers, for example:7/5 + 4/5 = 11/5 3 7/8 - 2/8 = 3 5/8 7 2/5 + 4/5 = 8 1/5 8 1/5 - 4/5 =
7 2/5 (RtP)
• WRM: Understand the whole.
• WRM: Partition a mixed number.
• WRM: Number lines with mixed numbers.
• WRM: Compare and order mixed numbers.
• WRM: Understand improper fractions.
• WRM: Convert mixed numbers to improper fractions.
• WRM: Convert improper fractions to mixed numbers.
• WRM: Tenths as fractions.
Year 5
• Compare and order fractions whose denominators are all multiples of the same number.
□ WRM: Compare fractions less than 1.
□ WRM: Order fractions less than 1.
□ WRM: Compare and order fractions greater than 1.
• Identify, name and write equivalent fractions of a given fraction, represented visually, including tenths and hundredths.
□ WRM: Find fractions equivalent to a unit fraction.
□ WRM: Find fractions equivalent to a non-unit fraction.
□ WRM: Recognise equivalent fractions.
• Recognise mixed numbers and improper fractions and convert from one form to the other and write mathematical statements > 1 as a mixed number.
□ WRM: Convert improper fractions to mixed numbers.
□ WRM: Convert mixed numbers to improper fractions.
• Write mathematical statements > 1 as a mixed number [for example, ⅖ + ⅘ =6/5= 1⅕ ]
• Add and subtract fractions with the same denominator, and denominators that are multiples of the same number.
□ WRM: Add and subtract fractions with the same denominator.
□ WRM: Add fractions within 1.
□ WRM: Add fractions with total greater than 1.
□ WRM: Add to a mixed number.
□ WRM: Add two mixed numbers.
□ WRM: Subtract from a mixed number.
□ WRM: Subtract from a mixed number – breaking the whole.
□ WRM: Subtract two mixed numbers.
• Multiply proper fractions and mixed numbers by whole numbers, supported by materials and diagrams
□ WRM: Multiply a unit fraction by an integer.
□ WRM: Multiply a non-unit fraction by an integer.
□ WRM: Multiply a mixed number by an integer.
□ WRM: Calculate a fraction of a quantity.
□ WRM: Fraction of an amount.
□ WRM: Use fractions as operators.
• Read and write decimal numbers as fractions e.g. 0.71 =71/100.
□ WRM: Equivalent fractions and decimals (tenths).
□ WRM: Equivalent fractions and decimals (hundredths).
□ WRM: Equivalent fractions and decimals.
• Recognise and use thousandths and relate them to tenths, hundredths and decimal equivalents.
□ WRM: Thousandths as fractions.
□ WRM: Thousandths as decimals.
• Round decimals with 2 decimal places to the nearest whole number and to 1 decimal place.
□ WRM: Round to the nearest whole number.
□ WRM: Round to 1 decimal place.
• Read, write, order and compare numbers with up to 3 decimal places.
□ WRM: Decimals up to 2 decimal places.
□ WRM: Thousandths on a place value chart.
□ WRM: Order and compare decimals (same number of decimal places).
□ WRM: Order and compare any decimals with up to 3 decimal places.
• Solve problems involving number up to 3 decimal places
• Recognise the per cent symbol (%) and understand that per cent relates to ‘number of parts per 100’, and write percentages as a fraction with denominator 100, and as a decimal fraction.
□ WRM: Understand percentages.
□ WRM: Percentages as fractions.
□ WRM: Percentages as decimals.
□ WRM: Equivalent fractions, decimals and percentages.
• Solves problems which require knowing percentage and decimal equivalents of ½, ¼, ⅕, ⅖, ⅘
• Solve problems which require knowing percentage and decimal equivalents of 1/2, 1/4, 1/5, 2/5, 4/5 and those fractions with a denominator of a multiple of 10 or 25
• 5F-1: Find non-unit fractions of quantities. (RtP)
• 5F-2: Find equivalent fractions and understand that they have the same value and the same position in the linear number system. (RtP)
• 5F-3: Recall decimal fraction equivalents for 1/2, 1/4, 1/5 and 1/10, and for multiples of these proper fractions. (RtP)
Year 6
• Use common factors to simplify fractions; use common multiples to express fractions in the same denomination.
□ WRM: Equivalent fractions and simplifying.
□ WRM: Equivalent fractions on a number line.
• Compare and order fractions, including fractions >1.
□ WRM: Compare and order (denominator).
□ WRM: Compare and order (numerator).
• Add and subtract fractions with different denominators and mixed numbers, using the concept of equivalent fractions.
□ WRM: Add and subtract simple fractions.
□ WRM: Add and subtract any two fractions.
□ WRM: Subtract mixed numbers.
□ WRM: Multi-step problems.
• Multiply simple pairs of proper fractions, writing the answer in its simplest form [for example, ¼ x ½ = ⅛].
□ WRM: Multiply fractions by integers.
□ WRM: Multiply fractions by fractions.
• Divide proper fractions by whole numbers [for example, ⅓ ÷ 2 = ⅙ ].
□ WRM: Divide a fraction by an integer.
□ WRM: Divide any fraction by an integer.
• Associate a fraction with division and calculate decimal fraction equivalents [for example, 0.375] for a simple fraction [for example, ⅜].
□ WRM: Fraction of an amount.
□ WRM: Fraction of an amount – find the whole.
• Identify the value of each digit in numbers given to 3 decimal places and multiply and divide numbers by 10, 100 and 1,000 giving answers up to 3 decimal places.
• Multiply one-digit numbers with up to 2 decimal places by whole numbers.
• Uses written division methods in cases where the answer has up to two decimal places.
• Solves problems which require answers to be rounded to specified degrees of accuracy.
• Recalls and uses equivalences between simple fractions, decimals and percentages, including in different contexts.
• 6F-1: Recognise when fractions can be simplified, and use common factors to simplify fractions. (RtP)
• 6F-2: Express fractions in a common denominator and use this to compare fractions that are similar in value. (RtP)
• 6F-3: Compare fractions with different denominators, including fractions greater than 1, using reasoning, and choose between reasoning and common denomination as a comparison strategy. (RtP)
• WRM: Mixed questions with fractions.
Number - multiplication and division
Year 3
• Recall and use multiplication and division facts for the 3, 4 and 8 multiplication tables.
□ WRM: Multiples of 5 and 10.
□ WRM: The 2, 4 and 8 times-tables.
• Write and calculate mathematical statements for multiplication and division using the multiplication tables that they know, including for two-digit numbers times one-digit numbers, using mental
and progressing to formal written methods.
□ WRM: Multiplication-equal groups.
□ WRM: Sharing and grouping.
□ WRM: Related calculations.
□ WRM: Reasoning about multiplication.
□ WRM: Multiply a 2-digit number by a 1-digit number (no exchange).
□ WRM: Multiply a 2-digit number by a 1-digit number (with exchange).
□ WRM: Divide a 2-digit number by a 1-digit number (no exchange).
□ WRM: Divide a 2-digit number by a 1-digit number (flexible partitioning).
□ WRM: Divide a 2-digit number by a 1-digit number (with remainders).
• Solve problems, including missing number problems, involving multiplication and division, including positive integer scaling problems and correspondence problems in which n objects are connected
to m objects.
□ WRM: Link multiplication and division.
• 3NF-2: Recall multiplication facts, and corresponding division facts, in the 10, 5, 2, 4 and 8 multiplication tables, and recognise products in these multiplication tables as multiples of the
corresponding number. (RtP)
• 3NF-3: Apply place-value knowledge to known additive and multiplicative number facts (scaling facts by 10), for example: 80 + 60 = 140 140 - 60 = 80 30 x 4 = 120 120 divided by 4 = 30. (RtP)
• 3MD-1: Apply known multiplication and division facts to solve contextual problems with different structures, including quotitive and partitive division. (RtP)
Year 4
• Recalls multiplication and division facts for multiplication tables up to 12 x 12
□ WRM: Multiply and divide by 6.
□ WRM: 6 times-table and division facts.
□ WRM: Multiply and divide by 9.
□ WRM: 9 times-table and division facts.
□ WRM: The 3, 6 and 9 times-tables.
□ WRM: Multiply and divide by 7.
□ WRM: 7 times-table and division facts.
□ WRM: 11 times-table and division facts.
□ WRM: 12 times-table and division facts.
• Use place value, known and derived facts to multiply and divide mentally, including: multiplying by 0 and 1; dividing by 1; multiplying together three numbers.
□ WRM: Multiply by 1 and 0.
□ WRM: Divide a number by 1 and itself.
□ WRM: Multiply three numbers.
□ WRM: Divide a 2-digit number by a 1-digit number (1)
□ WRM: Divide a 2-digit number by a 1-digit number (2)
□ WRM: Divide a 3-digit number by a 1-digit number.
• Recognise and use factor pairs and commutativity in mental calculations
□ WRM: Recognise and use factor pairs and commutativity in mental calculations.
• Multiply 2-digit and 3-digit numbers by a 1-digit number using formal written layout.
□ Multiply a two-digit by a one-digit number using formal written layout.
□ Multiply three-digit numbers by a one-digit number using formal written layout.
□ WRM: Multiply a 2-digit number by a 1-digit number.
□ WRM: Multiply a 3-digit number by a 1-digit number.
• Multiply three-digit numbers by a one-digit number using formal written layout
• Solve problems involving multiplying and adding, including using the distributive law to multiply two-digit numbers by 1 digit, integer scaling problems and harder correspondence problems such as
n objects are connected to m objects.
□ WRM: Related facts-multiplication and division.
□ WRM: Informal written methods for multiplication.
□ WRM: Correspondence problems.
□ WRM: Efficient multiplication.
• 4NF-1: Recall multiplication and division facts up to 12 x 12, and recognise products in multiplication tables as multiples of the corresponding number. (RtP)
• 4NF-2: Solve division problems, with two-digit dividends and one-digit divisors, that involve remainders, for example: 74 divided by 9 = 8 r 2 and interpret remainders appropriately according to
the context. (RtP)
• 4NF-3: Apply place-value knowledge to known additive and multiplicative number facts (scaling facts by 100), for example:8 + 6 = 14 and 14 - 6 = 8 so 800 + 600 = 1,400 1,400 - 600 = 800 3 x 4 =
12 and 12 divided by 4 = 3 so 300 x 4 = 1,200 1,200 divided by 4 = 300 (RtP)
• 4MD-1: Multiply and divide whole numbers by 10 and 100 (keeping to whole number quotients); understand this as equivalent to making a number 10 or 100 times the size. (RtP)
• 4MD-2: Manipulate multiplication and division equations, and understand and apply the commutative property of multiplication. (RtP)
• 4MD-3: Understand and apply the distributive property of multiplication. (RtP)
Year 5
• Identifies multiples and factors, including finding all factor pairs of a number, and common factors of two numbers.
• Establish whether a number up to 100 is prime and recall prime numbers up to 19.
□ Know and use the vocabulary of prime numbers, prime factors and composite (non-prime) numbers.
• Multiply numbers up to 4 digits by a one- or two-digit number using a formal written method, including long multiplication for two-digit numbers
□ WRM: Multiply up to a 4-digit number by a 1-digit number.
□ WRM: Multiply a 2-digit number by a 2-digit number (area model).
□ WRM: Multiply a 2-digit number by a 2-digit number.
□ WRM: Multiply a 3-digit number by a 2-digit number.
□ WRM: Multiply a 4-digit number by a 2-digit number.
□ WRM: Solve problems with multiplication.
• Multiply and divide numbers mentally, drawing upon known facts.
□ WRM: Multiples of 10, 100 and 1,000.
• Divide numbers up to 4 digits by a one-digit number using the formal written method of short division and interpret remainders appropriately for the context
□ WRM: Divide a 4-digit number by a 1-digit number.
□ WRM: Divide with remainders.
• Multiply and divide whole numbers and those involving decimals by 10, 100 and 1,000.
□ WRM: Multiply by 10, 100 and 1,000.
□ WRM: Divide by 10, 100 and 1,000.
• Recognise and use square numbers and cube numbers, and the notation for squared (²) and cubed (³).
• Solve problems involving multiplication and division, including using a knowledge of factors and multiples, squares and cubes.
□ WRM: Solve problems with multiplication and division.
• Solve problems involving;
• Solve problems involving addition, subtraction, multiplication and division and a combination of these, including understanding the meaning of the equals sign
• Solves problems involving multiplication and division, including scaling by simple fractions and problems involving simple rates
• 5MD-1: Multiply and divide numbers by 10 and 100; understand this as equivalent to making a number 10 or 100 times the size, or 1 tenth or 1 hundredth times the size. (RtP)
• 5MD-2: Find factors and multiples of positive whole numbers, including common factors and common multiples, and express a given number as a product of 2 or 3 factors. (RtP)
• 5MD-3: Multiply any whole number with up to 4 digits by any one-digit number using a formal written method. (RtP)
• 5MD-4: Divide a number with up to 4 digits by a one-digit number using a formal written method, and interpret remainders appropriately for the context. (RtP)
• 5NF-1: Secure fluency in multiplication table facts, and corresponding division facts, through continued practice. (RtP)
• 5NF-2: Apply place-value knowledge to know additive and multiplicative number facts (scaling facts by 1 tenth or 1 hundredth), for example: 8 + 6 = 14 0.8 + 0.6 = 1.4 0.08+ 0.06 = 0.14 3 x 4 = 12
0.3 x 4 = 1.2 0.03 x 4 = 0.12 (RtP) | {"url":"https://northburn-northumberland.frogos.net/app/curriculum/key%20stages","timestamp":"2024-11-13T21:26:20Z","content_type":"text/html","content_length":"1049491","record_id":"<urn:uuid:8e8671da-c32d-41c7-89e4-85f6c6deccd8>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00847.warc.gz"} |
CPH Theory
Godel and the End of Physics
This lecture is the intellectual property of Professor S.W.Hawking. You may not reproduce, edit, translate, distribute, publish or host this document in any way with out the permission of Professor
Note that there may be incorrect spellings, punctuation and/or grammar in this document. This is to allow correct pronunciation and timing by a speech synthesiser.
In this talk, I want to ask how far can we go in our search for understanding and knowledge. Will we ever find a complete form of the laws of nature? By a complete form, I mean a set of rules that in
principle at least enable us to predict the future to an arbitrary accuracy, knowing the state of the universe at one time. A qualitative understanding of the laws has been the aim of philosophers
and scientists, from Aristotle onwards. But it was Newton's Principia Mathematica in 1687, containing his theory of universal gravitation that made the laws quantitative and precise. This led to the
idea of scientific determinism, which seems first to have been expressed by Laplace. If at one time, one knew the positions and velocities of all the particles in the universe, the laws of science
should enable us to calculate their positions and velocities at any other time, past or future. The laws may or may not have been ordained by God, but scientific determinism asserts that he does not
intervene to break them.
At first, it seemed that these hopes for a complete determinism would be dashed by the discovery early in the 20th century; that events like the decay of radio active atoms seemed to take place at
random. It was as if God was playing dice, in Einstein's phrase. But science snatched victory from the jaws of defeat by moving the goal posts and redefining what is meant by a complete knowledge of
the universe. It was a stroke of brilliance whose philosophical implications have still not been fully appreciated. Much of the credit belongs to Paul Dirac, my predecessor but one in the Lucasian
chair, though it wasn't motorized in his time. Dirac showed how the work of Erwin Schrodinger and Werner Heisenberg could be combined in new picture of reality, called quantum theory. In quantum
theory, a particle is not characterized by two quantities, its position and its velocity, as in classical Newtonian theory. Instead it is described by a single quantity, the wave function. The size
of the wave function at a point, gives the probability that the particle will be found at that point, and the rate at which the wave function changes from point to point, gives the probability of
different velocities. One can have a wave function that is sharply peaked at a point. This corresponds to a state in which there is little uncertainty in the position of the particle. However, the
wave function varies rapidly, so there is a lot of uncertainty in the velocity. Similarly, a long chain of waves has a large uncertainty in position, but a small uncertainty in velocity. One can have
a well defined position, or a well defined velocity, but not both.
This would seem to make complete determinism impossible. If one can't accurately define both the positions and the velocities of particles at one time, how can one predict what they will be in the
future? It is like weather forecasting. The forecasters don't have an accurate knowledge of the atmosphere at one time. Just a few measurements at ground level and what can be learnt from satellite
photographs. That’s why weather forecasts are so unreliable. However, in quantum theory, it turns out one doesn't need to know both the positions and the velocities. If one knew the laws of physics
and the wave function at one time, then something called the Schrodinger equation would tell one how fast the wave function was changing with time. This would allow one to calculate the wave function
at any other time. One can therefore claim that there is still determinism but it is determinism on a reduced level. Instead of being able accurately to predict two quantities, position and velocity,
one can predict only a single quantity, the wave function. We have re-defined determinism to be just half of what Laplace thought it was. Some people have tried to connect the unpredictability of the
other half with consciousness, or the intervention of supernatural beings. But it is difficult to make either case for something that is completely random.
In order to calculate how the wave function develops in time, one needs the quantum laws that govern the universe. So how well do we know these laws? As Dirac remarked, Maxwell's equations of light
and the relativistic wave equation, which he was too modest to call the Dirac equation, govern most of physics and all of chemistry and biology. So in principle, we ought to be able to predict human
behavior, though I can't say I have had much success myself. The trouble is that the human brain contains far too many particles for us to be able to solve the equations. But it is comforting to
think we might be able to predict the nematode worm, even if we can't quite figure out humans. Quantum theory and the Maxwell and Dirac equations indeed govern much of our life, but there are two
important areas beyond their scope. One is the nuclear forces. The other is gravity. The nuclear forces are responsible for the Sun shining and the formation of the elements including the carbon and
oxygen of which we are made. And gravity caused the formation of stars and planets, and indeed, of the universe itself. So it is important to bring them into the scheme.
The so called weak nuclear forces have been unified with the Maxwell equations by Abdus Salam and Stephen Weinberg, in what is known as the Electro weak theory. The predictions of this theory have
been confirmed by experiment and the authors rewarded with Nobel Prizes. The remaining nuclear forces, the so called strong forces, have not yet been successfully unified with the electro weak forces
in an observationally tested scheme. Instead, they seem to be described by a similar but separate theory called QCD. It is not clear who, if anyone, should get a Nobel Prize for QCD, but David Gross
and Gerard ‘t Hooft share credit for showing the theory gets simpler at high energies. I had quite a job to get my speech synthesizer to pronounce Gerard's surname. It wasn't familiar with apostrophe
t. The electro weak theory and QCD together constitute the so called Standard Model of particle physics, which aims to describe everything except gravity.
The standard model seems to be adequate for all practical purposes, at least for the next hundred years. But practical or economic reasons have never been the driving force in our search for a
complete theory of the universe. No one working on the basic theory, from Galileo onward, has carried out their research to make money, though Dirac would have made a fortune if he had patented the
Dirac equation. He would have had a royalty on every television, walkman, video game and computer.
The real reason we are seeking a complete theory, is that we want to understand the universe and feel we are not just the victims of dark and mysterious forces. If we understand the universe, then we
control it, in a sense. The standard model is clearly unsatisfactory in this respect. First of all, it is ugly and ad hoc. The particles are grouped in an apparently arbitrary way, and the standard
model depends on 24 numbers whose values can not be deduced from first principles, but which have to be chosen to fit the observations. What understanding is there in that? Can it be Nature's last
word? The second failing of the standard model is that it does not include gravity. Instead, gravity has to be described by Einstein's General Theory of Relativity. General relativity is not a
quantum theory unlike the laws that govern everything else in the universe. Although it is not consistent to use the non quantum general relativity with the quantum standard model, this has no
practical significance at the present stage of the universe because gravitational fields are so weak. However, in the very early universe, gravitational fields would have been much stronger and
quantum gravity would have been significant. Indeed, we have evidence that quantum uncertainty in the early universe made some regions slightly more or less dense than the otherwise uniform
background. We can see this in small differences in the background of microwave radiation from different directions. The hotter, denser regions will condense out of the expansion as galaxies, stars
and planets. All the structures in the universe, including ourselves, can be traced back to quantum effects in the very early stages. It is therefore essential to have a fully consistent quantum
theory of gravity, if we are to understand the universe.
Constructing a quantum theory of gravity has been the outstanding problem in theoretical physics for the last 30 years. It is much, much more difficult than the quantum theories of the strong and
electro weak forces. These propagate in a fixed background of space and time. One can define the wave function and use the Schrodinger equation to evolve it in time. But according to general
relativity, gravity is space and time. So how can the wave function for gravity evolve in time? And anyway, what does one mean by the wave function for gravity? It turns out that, in a formal sense,
one can define a wave function and a Schrodinger like equation for gravity, but that they are of little use in actual calculations.
Instead, the usual approach is to regard the quantum spacetime as a small perturbation of some background spacetime; generally flat space. The perturbations can then be treated as quantum fields,
like the electro weak and QCD fields, propagating through the background spacetime. In calculations of perturbations, there is generally some quantity called the effective coupling which measures how
much of an extra perturbation a given perturbation generates. If the coupling is small, a small perturbation creates a smaller correction which gives an even smaller second correction, and so on.
Perturbation theory works and can be used to calculate to any degree of accuracy. An example is your bank account. The interest on the account is a small perturbation. A very small perturbation if
you are with one of the big banks. The interest is compound. That is, there is interest on the interest, and interest on the interest on the interest. However, the amounts are tiny. To a good
approximation, the money in your account is what you put there. On the other hand, if the coupling is high, a perturbation generates a larger perturbation which then generates an even larger
perturbation. An example would be borrowing money from loan sharks. The interest can be more than you borrowed, and then you pay interest on that. It is disastrous.
With gravity, the effective coupling is the energy or mass of the perturbation because this determines how much it warps spacetime, and so creates a further perturbation. However, in quantum theory,
quantities like the electric field or the geometry of spacetime don't have definite values, but have what are called quantum fluctuations. These fluctuations have energy. In fact, they have an
infinite amount of energy because there are fluctuations on all length scales, no matter how small. Thus treating quantum gravity as a perturbation of flat space doesn't work well because the
perturbations are strongly coupled.
Supergravity was invented in 1976 to solve, or at least improve, the energy problem. It is a combination of general relativity with other fields, such that each species of particle has a super
partner species. The energy of the quantum fluctuations of one partner is positive, and the other negative, so they tend to cancel. It was hoped the infinite positive and negative energies would
cancel completely, leaving only a finite remainder. In this case, a perturbation treatment would work because the effective coupling would be weak. However, in 1985, people suddenly lost confidence
that the infinities would cancel. This was not because anyone had shown that they definitely didn't cancel. It was reckoned it would take a good graduate student 300 years to do the calculation, and
how would one know they hadn't made a mistake on page two? Rather it was because Ed Witten declared that string theory was the true quantum theory of gravity, and supergravity was just an
approximation, valid when particle energies are low, which in practice, they always are. In string theory, gravity is not thought of as the warping of spacetime. Instead, it is given by string
diagrams; networks of pipes that represent little loops of string, propagating through flat spacetime. The effective coupling that gives the strength of the junctions where three pipes meet is not
the energy, as it is in supergravity. Instead it is given by what is called the dilaton; a field that has not been observed. If the dilaton had a low value, the effective coupling would be weak, and
string theory would be a good quantum theory. But it is no earthly use for practical purposes.
In the years since 1985, we have realized that both supergravity and string theory belong to a larger structure, known as M theory. Why it should be called M Theory is completely obscure. M theory is
not a theory in the usual sense. Rather it is a collection of theories that look very different but which describe the same physical situation. These theories are related by mappings or
correspondences called dualities, which imply that they are all reflections of the same underlying theory. Each theory in the collection works well in the limit, like low energy, or low dilaton, in
which its effective coupling is small, but breaks down when the coupling is large. This means that none of the theories can predict the future of the universe to arbitrary accuracy. For that, one
would need a single formulation of M-theory that would work in all situations.
Up to now, most people have implicitly assumed that there is an ultimate theory that we will eventually discover. Indeed, I myself have suggested we might find it quite soon. However, M-theory has
made me wonder if this is true. Maybe it is not possible to formulate the theory of the universe in a finite number of statements. This is very reminiscent of Godel's theorem. This says that any
finite system of axioms is not sufficient to prove every result in mathematics.
Godel's theorem is proved using statements that refer to themselves. Such statements can lead to paradoxes. An example is, this statement is false. If the statement is true, it is false. And if the
statement is false, it is true. Another example is, the barber of Corfu shaves every man who does not shave himself. Who shaves the barber? If he shaves himself, then he doesn't, and if he doesn't,
then he does. Godel went to great lengths to avoid such paradoxes by carefully distinguishing between mathematics, like 2+2 =4, and meta mathematics, or statements about mathematics, such as
mathematics is cool, or mathematics is consistent. That is why his paper is so difficult to read. But the idea is quite simple. First Godel showed that each mathematical formula, like 2+2=4, can be
given a unique number, the Godel number. The Godel number of 2+2=4, is *. Second, the meta mathematical statement, the sequence of formulas A, is a proof of the formula B, can be expressed as an
arithmetical relation between the Godel numbers for A- and B. Thus meta mathematics can be mapped into arithmetic, though I'm not sure how you translate the meta mathematical statement, 'mathematics
is cool'. Third and last, consider the self referring Godel statement, G. This is, the statement G can not be demonstrated from the axioms of mathematics. Suppose that G could be demonstrated. Then
the axioms must be inconsistent because one could both demonstrate G and show that it can not be demonstrated. On the other hand, if G can't be demonstrated, then G is true. By the mapping into
numbers, it corresponds to a true relation between numbers, but one which can not be deduced from the axioms. Thus mathematics is either inconsistent or incomplete. The smart money is on incomplete.
What is the relation between Godel’s theorem and whether we can formulate the theory of the universe in terms of a finite number of principles? One connection is obvious. According to the positivist
philosophy of science, a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted. One example might
be the Goldbach conjecture. Given an even number of wood blocks, can you always divide them into two piles, each of which can not be arranged in a rectangle? That is, it contains a prime number of
Although this is incompleteness of sort, it is not the kind of unpredictability I mean. Given a specific number of blocks, one can determine with a finite number of trials whether they can be divided
into two primes. But I think that quantum theory and gravity together, introduces a new element into the discussion that wasn't present with classical Newtonian theory. In the standard positivist
approach to the philosophy of science, physical theories live rent free in a Platonic heaven of ideal mathematical models. That is, a model can be arbitrarily detailed and can contain an arbitrary
amount of information without affecting the universes they describe. But we are not angels, who view the universe from the outside. Instead, we and our models are both part of the universe we are
describing. Thus a physical theory is self referencing, like in Godel’s theorem. One might therefore expect it to be either inconsistent or incomplete. The theories we have so far are both
inconsistent and incomplete.
Quantum gravity is essential to the argument. The information in the model can be represented by an arrangement of particles. According to quantum theory, a particle in a region of a given size has a
certain minimum amount of energy. Thus, as I said earlier, models don't live rent free. They cost energy. By Einstein’s famous equation, E = mc squared, energy is equivalent to mass. And mass causes
systems to collapse under gravity. It is like getting too many books together in a library. The floor would give way and create a black hole that would swallow the information. Remarkably enough,
Jacob Bekenstein and I found that the amount of information in a black hole is proportional to the area of the boundary of the hole, rather than the volume of the hole, as one might have expected.
The black hole limit on the concentration of information is fundamental, but it has not been properly incorporated into any of the formulations of M theory that we have so far. They all assume that
one can define the wave function at each point of space. But that would be an infinite density of information which is not allowed. On the other hand, if one can't define the wave function point
wise, one can't predict the future to arbitrary accuracy, even in the reduced determinism of quantum theory. What we need is a formulation of M theory that takes account of the black hole information
limit. But then our experience with supergravity and string theory, and the analogy of Godel’s theorem, suggest that even this formulation will be incomplete.
Some people will be very disappointed if there is not an ultimate theory that can be formulated as a finite number of principles. I used to belong to that camp, but I have changed my mind. I'm now
glad that our search for understanding will never come to an end, and that we will always have the challenge of new discovery. Without it, we would stagnate. Godel’s theorem ensured there would
always be a job for mathematicians. I think M theory will do the same for physicists. I'm sure Dirac would have approved.
Thank you for listening.
Source: Hawking Org
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Use Ranking System to Plan for Replacement
We are too focused on cost when it comes to thinking about equipment replacement. It certainly is important to know what it will cost to own and operate a unit in the year ahead and to make wise
replacement decisions when the old unit—the defender—is likely to cost more than the minimum lifecycle cost you can expect from a new unit—the challenger. The defender-versus-challenger replacement
theory is well accepted in practice, and many companies use this approach to plan replacements and manage fleet average age.
But there is more to it than cost. Other metrics such as age and utilization are important, and we frequently face the problem of balancing many factors when we try to identify or, at the very least,
rank units for replacement.
The American Public Works Association (APWA) proposes the use of a simple subjective points system to rank vehicles for replacement. The system asks users to award a certain number of points for
factors such as age, miles travelled, reliability, cost and condition, based on their assessment of each factor. The points are totaled, and the unit with the highest score “needs immediate
consideration” for replacement. It has been proven to work, but it is subjective.
The need to combine many factors when trying to identify units for replacement is a common problem. Building on the APWA system, let’s add a little technology and develop a spreadsheet that combines
subjectivity and analysis to give us a practical tool we can use to help in replacement decisions for large equipment groups such as pickups and tri-axle trucks.
The nearby spreadsheet table serves as an example of how this might work with a grouping of tri-axle trucks.
Columns C, D, E, H, I and J show we are looking at six factors: year of manufacture; miles travelled, life to date; miles travelled in the past 12 months; a subjective inspection score; repair labor
cost per mile; and repair parts cost per mile.
The unit numbers and the required data need to be entered into the green cells. Then calculate cost per mile for labor and parts by dividing columns F and G by column E to obtain the values in
columns I and J.
To set up the scoring system, award “points” to each unit under each of the six factors. Determine what value is “good” and worthy of 10 points and what value is “unacceptable” and not worthy of any
points. Rows 3 and 4 indicate those ranges. For example, a year of manufacture of 2013 or later is worth 10 points, and a year of manufacture of 1994 or earlier is worth zero. The process of deciding
what is “good” and “bad” takes some discipline. You need to quantify your expectations for each factor and decide where the boundaries lie. It is not easy, but it is necessary. It makes the process
repeatable, and once you have set the boundaries, the process is defendable.
Units will not be at the boundaries; some will be above expectations, some below expectations, and some in between. The next step is to decide how to apportion points regardless of where units fall.
If they fall above and below the boundary, values are easy: 10 points or zero points. If they fall in between boundary values, we apportion the points on a straight-line basis defined by the values
in rows 3 and 4.
The graph on the left shows how it is done for year of manufacture: zero points for units older than 1994, a straight-line proportion of the points between 1994 and 2013, and 10 points for units
newer than 2013. The graph on the right shows how it is done for labor cost per mile travelled during the last 12 months: 10 points for under $0.09 per mile, a straight-line proportion between $0.09
and $0.30 per mile, and zero if the cost is more than $0.30 per mile.
These six factors do not, or should not, count equally in determining the final score. A weighting factor applied to each of the six factors aids in calculating the final score. In row 5, 30 percent
of the weighting goes to factors that measure age (15 percent for age in years, column C, and 15 percent for age in miles, column D); 30 percent of the weight goes to utilization (miles in the last
12 months, column E); 30 percent of the weight goes to cost factors (cost per mile in the last 12 months for labor and parts, columns I and J); and 10 percent of the weight goes to the subjective
inspection (column H).
With all the data in place, do the mathematics to determine the final score. It is a little technical but not complicated. The final score is the weighted total of the points calculated for each of
the six factors considered.
Finally, use the analysis to help in your decisions. In this example, units in rows 9, 14 and 15 have low scores and are clearly candidates for replacement. Row 9 must go, the other two might depend
on available capex budget. The unit in row 7 is better than the boundary values for all factors except labor cost per mile. It is a keeper, as are the units in rows 11, 8 and 6.
Tools like this enable us to move away from a fixation about cost. Other factors influence our decisions. We need to know how to include them. | {"url":"https://www.constructionequipment.com/topical/executive-institute/article/10751710/use-ranking-system-to-plan-for-replacement","timestamp":"2024-11-05T00:59:16Z","content_type":"text/html","content_length":"262434","record_id":"<urn:uuid:fd315705-c43d-4a51-a3c5-32a37ef341d2>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00381.warc.gz"} |
Solving ratios and rates problems in 6th Grade: Tips, Examples, and Worksheets
• Are you a 6th-grader who wants to learn how to solve ratios and rates problems? Or are you a parent or teacher who wants to help your child or student master this vital math skill? If so, you've
come to the right place! In this article, we will provide fun tips, examples, and worksheets that your students can practice daily and are free to download.
Simplify learning ratios and rates: Download Free 6th Grade Worksheets PDF with step-by-step solutions.
Like fractions, this article will help 6^th graders learn how to simplify ratios and rates to their lowest forms using a simple step-by-step solution. You can download these free 6th Grade
Worksheets PDF from Mathskills4kids.com and start practicing now.
In this article, we'll explain what ratios and rates are, why they are important for 6th-graders, and how to write, represent, compare, and solve them in different ways.
We'll also provide tips, examples, and worksheets to practice and improve your ratios and rates skills. And as a bonus, we'll share some web links with more resources that you can use to learn more
about ratios and rates.
So, let's get started!
□ What are ratios and rates?
A ratio compares two quantities of the same kind by division. For example, if 12 boys and 18 girls are in a class, the ratio of boys to girls will be 12:18 or 12/18. A rate is a special ratio
that compares two quantities of different kinds by division. For example, if you can run 3 miles in 30 minutes, the rate of your speed is 3 miles per 30 minutes or 3/30.
□ Why are ratios and rates important for 6th-graders?
Ratios and rates are important for 6th-graders because they help them understand the relationships between different quantities and measure how fast or slow something changes.
Ratios and rates also help them solve real-world problems involving proportions, fractions, decimals, percentages, unit conversions, and more.
In addition, ratios and rates are the foundation for learning more advanced math topics such as ratios and proportional relationships, unit rate, unit analysis, scaling, slope, and linear
□ How to write ratios and rates in different ways
There are three main ways to write ratios and rates: using words, colons (:), or fractions (/). For example, the ratio of boys to girls in a class can be written as:
☆ "12 boys to 18 girls" using words
☆ "12:18" using colons
☆ "12/18" using fractions
The rate of your speed can be written as:
☆ "3 miles per 30 minutes" using words
☆ "3:30" using colons
☆ "3/30" using fractions
We can also simplify ratios and rates by dividing both quantities by a common factor. For example, the ratio of boys to girls in a class can be simplified by dividing both numbers by 6:
☆ "12 boys to 18 girls" simplifies to "2 boys to 3 girls"
☆ "12:18" simplifies to "2:3"
☆ "12/18" simplifies to "2/3"
The rate of your speed can be simplified by dividing both numbers by 3:
☆ "3 miles per 30 minutes" simplifies to "1 mile per 10 minutes."
☆ "3:30" simplifies to "1:10"
☆ "3/30" simplifies to "1/10"
□ How to find equivalent ratios and simplify rates
Equivalent ratios are ratios that have the same value when simplified. For example, the ratios 2:3, 4:6, and 6:9 are equivalent because they all simplify to 2/3. We can multiply or divide
both quantities by the same factor to find equivalent ratios. For example, to find an equivalent ratio to 2:3, we multiply both numbers by 2 to get 4:6 or divide both numbers by 2 to get
Simplifying rates is similar to simplifying ratios, except that we have to make sure that the units of the quantities are consistent. For example, to simplify the rate of 3 miles per 30
minutes, we will convert the units of distance and time to the same unit, such as miles per hour or minutes per mile. To do this, we can use unit conversions or unit fractions.
For example, to convert 3 miles per 30 minutes to miles per hour, we can multiply by the unit fraction 60 minutes/1 hour, which is equal to 1:
3 miles / 30 minutes x 60 minutes / 1 hour = 180 miles / 1800 minutes = 1 mile / 10 minutes = 6 miles / 60 minutes = 6 miles per hour
□ How to compare ratios and rates using cross-multiplication
Cross-multiplication is a method that helps you compare ratios and rates by finding a common denominator. To use cross-multiplication, we will set up a proportion, an equation that states
that two ratios or rates are equal. For example, if we want to compare the ratio of boys to girls in two classes, we can write a proportion like this:
12/18 = x/24,
Where x is the number of boys in the second class. Then, we can cross-multiply by multiplying the numerator of one ratio by the denominator of the other ratio and setting them equal. For
12 x 24 = 18 x (x)
Then, we can solve for x by dividing both sides by 18:
x = (12 x 24) / 18
x = 288 / 18
x = 16
So, the ratio of boys to girls in the second class is 16:24, equivalent to 2:3.
We can also use cross-multiplication to compare rates by ensuring consistent units. For example, if we want to compare the speed of two runners who run different distances at different times,
we can write a proportion like this:
3 miles / 30 minutes = x miles / 45 minutes
Where x is the distance run by the second runner. Then, we can cross-multiply and solve for x:
3 x 45 = 30 x x
x = (3 x 45) / 30
x = 135 / 30
x = 4.5
So, the second runner runs 4.5 miles in 45 minutes.
□ How to solve word problems involving ratios and rates
One of the most common applications in 6th Grade math is solving ratios and rates problems. Word problems are mathematical questions that use words and numbers to describe a real-life
situation. To solve word problems involving ratios and rates, we can follow these steps:
1. Read the problem carefully and identify the given information and the unknown quantity.
2. Write a ratio or a rate to represent the relationship between the given information and the unknown quantity.
3. Use equivalent ratios, cross-multiplication, or unit rates to find the value of the unknown quantity.
4. Check your answer by plugging it back into the ratio or rate and see if it makes sense in the problem context.
5. Write a complete sentence to answer the question.
Let's look at examples of applying these steps to different word problems involving ratios and rates.
□ 10 ratios and rates word problems with solutions for 6th-Grade
Here are 10-word problems selected from Mathskills4kids.com that your students can practice with to improve their ratios and rates skills in Grade 6. Please encourage them to solve these
problems independently, then check the solutions below.
1. In a class of 24 students, 18 students like chocolate ice cream, and the rest like vanilla ice cream. What is the ratio of students who like chocolate ice cream to vanilla ice cream?
Write your answer in the simplest form.
2. A recipe for lemonade calls for 6 cups of water and 1 cup of lemon juice. How many cups of lemon juice are needed to make 18 cups of lemonade?
3. A car travels 120 miles in 3 hours. What is the average speed of the car in miles per hour?
4. A bag of candy contains 12 red candies, 8 blue candies, and 10 green candies. What is the probability of picking a red candy from the bag?
5. A map has a scale of 1 inch: 50 miles. How many inches on the map represent a distance of 300 miles?
6. A printer can print 15 pages in 2 minutes. How long will it take to print 60 pages?
7. A bag of oranges weighs 12 pounds. If each orange weighs 0.25 pounds, how many oranges are in the bag?
8. A bicycle costs $180 and is on sale for 25% off. What is the sale price of the bicycle?
9. A cookie recipe makes 24 cookies and uses 2 cups of flour. How many cups of flour are needed to make 36 cookies?
10. A painter can paint a wall in 4 hours. How much of the wall can he paint in 1 hour?
1. The ratio of students who like chocolate ice cream to vanilla ice cream is 18:6, which can be simplified by dividing both terms by 6, giving 3:1.
2. We can use equivalent ratios to find the number of cups of lemon juice needed to make 18 cups of lemonade. If 6 cups of water and 1 cup of lemon juice make one batch of lemonade, then we
can multiply both terms by 3 to get another equivalent ratio: 18 cups of water and 3 cups of lemon juice make three batches of lemonade.
Therefore, we need 3 cups of lemon juice to make 18 cups of lemonade.
3. To find the average speed of the car in miles per hour, we can use a unit rate, which is a rate that compares a quantity to one unit of another quantity. In this case, we want to compare
miles to one hour, so we divide both terms by the time:
120 miles / 3 hours = 40 miles/hour
The average speed of the car is 40 miles per hour.
4. To find the probability of picking a red candy from the bag, we can use a ratio that compares the number of favorable outcomes to the total number of possible outcomes. In this case, the
number of favorable outcomes is the number of red candies, which is 12, and the total number of possible outcomes is the number of all candies, which is 12 + 8 + 10 = 30. The ratio is
12 / 30 = 0.4
The probability of picking a red candy from the bag is 0.4 or 40%.
5. We can use equivalent ratios again to find how many inches on the map represent a distance of 300 miles. If one inch on the map represents 50 miles in reality, then we can multiply both
terms by x to get another equivalent ratio:
x inches on the map represent 300 miles in reality.
Then, we can solve for x by cross-multiplying:
1 x 300 = (x) x 50
x = (1 x 300) /50
x = 6
Therefore, 6 inches on the map represents a distance of 300 miles.
6. We can use another unit rate to find how long it will take to print 60 pages. If the printer can print 15 pages in 2 minutes, then we can divide both terms by 15 to get a unit rate that
compares pages to one minute:
15 pages / 2 minutes = 1 page / (2/15) minutes
The printer can print one page in (2/15) minutes, equivalent to 0.133 minutes or 8 seconds. To print 60 pages, we can multiply both terms by 60:
1 page / (2/15) minutes = 60 pages / (120/15) minutes
The printer can print 60 pages in (120/15) minutes, equivalent to 8 minutes.
7. To find how many oranges are in the bag, we can use a rate that compares the bag's weight to the number of oranges. The rate is:
12 pounds / x oranges
To find x, we can use the fact that each orange weighs 0.25 pounds, which means that x oranges weigh x x 0.25 pounds. We can then set up an equation and solve for x:
12 pounds = x x 0.25 pounds
x = 12 / 0.25
x = 48
There are 48 oranges in the bag.
8. To find the sale price of the bicycle, we can use a percentage, which is a ratio that compares a part to a whole and is expressed as a fraction of 100. In this case, the part is the
discount amount, and the whole is the original price of the bicycle. The percentage is:
25% = 25 / 100 = 0.25
To find the amount of discount, we can multiply the percentage by the original price:
0.25 x $180 = $45
The amount of discount is $45. To find the sale price, we can subtract the amount of discount from the original price:
$180 - $45 = $135
The sale price of the bicycle is $135.
9. To find how many cups of flour are needed to make 36 cookies, we can use a proportion, which is an equation that states that two ratios are equal. In this case, the two ratios are:
2 cups of flour / 24 cookies = x cups of flour / 36 cookies
To find x, we can cross-multiply and solve for x:
2 x 36 = (x) x 24
x = (2 x 36) /24
x = 3
We need 3 cups of flour to make 36 cookies.
10. We can use another proportion to find how much of the wall he can paint in one hour. The two ratios are:
1 wall / 4 hours = x wall / 1 hour
To find x, we can cross-multiply and solve for x:
1 x 1 = (x) x4
x = (1 x 1) /4
x = 0.25
He can paint 0.25 or one-fourth of the wall in one hour.
Bonus: More resources for solving ratios and rates problems in 6th Grade
If you want more practice for your 6^th graders on solving ratios and rates problems, then check out these web links:
Thank you for sharing the links of MathSkills4Kids.com with your loved ones. Your choice is greatly appreciated.
Conclusion: the benefits of mastering ratios and rates in everyday situations
Your 6th-grade students have learned much about ratios and rates in this article. They have seen how to write them in different ways, how to use tables, graphs, and diagrams to represent them, how to
find equivalent ratios and simplify rates, how to compare them using cross-multiplication, and how to solve word problems involving them. They have also practiced their skills with 10 ratios and
rates word problems with solutions for 6th grade.
But why are ratios and rates important to us? How can they help us in our daily life? Here are some examples of how we can use ratios and rates in everyday situations:
• Cooking: We can use ratios and rates to measure ingredients, adjust recipes, compare prices, and calculate nutritional values. For example, if we want to make a cake that serves 12 people, but
the recipe is for 8 people, we can use a ratio to determine how much each ingredient is needed.
• Sports: We can use ratios and rates to analyze performance, compare players, and predict outcomes. For example, if we want to compare two basketball players, we can use ratios to compare their
points, rebounds, assists, and other statistics.
• Art: We can use ratios and rates to create designs, patterns, and proportions. For example, if we want to draw a face, we can use ratios to divide the head into sections and place the features
• Science: We can use ratios and rates to explore nature, conduct experiments, and interpret data. For example, if we want to know how many birds are in a park, we can use a ratio to compare the
number of birds we see in a sample area to the total park area.
As we can see, ratios and rates are useful for many aspects of our lives. They help us make sense of the world around us and solve problems creatively.
By mastering ratios and rates in 6th grade, your students will be ready for more advanced math topics in the future. They will also develop critical thinking skills that benefit them in school and | {"url":"https://mathskills4kids.com/6th-grades-ratios-and-rates-worksheets-pdf-with-answers","timestamp":"2024-11-08T02:46:40Z","content_type":"text/html","content_length":"94522","record_id":"<urn:uuid:ca9fa5e5-961b-477e-8770-ae5a7cd5af66>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00429.warc.gz"} |
5. Raja wants to purchase a bike
2. Subtract the sum of 33452 and 46771 from 927521.
3. Subtract the sum of 19067 and 51237 from 100000.
4. Subtract the difference of 45637 and 22427 from 666667.
5. Raja wants to purchase a bike worth * 76100 and sells his old bike for 2270
how much more money is required to purchase a new bike.
6. Rashmi has* 200000. She buys a refrigerator for 35350, a television for
and a microwave for 33470. Find how much money is left with her.
7. Mr and Mrs Sharma work in a software MNC. Their total annual sa
927000. If Mr Sharma earns580726, find how much Mrs Sharma earns.
8. Population of a town is 72500. If there are 26180 males, 20000 females, ho
children are there in town?
Ajay buys a box that can hold 70000 marbles. He has 52170 grey n
2612 white marbles and 12112 red marbles. Will all the marb
he box?
tiple Choice Questions (MCOS)
Here's the answer hope it helps you. | {"url":"https://alumniagri.in/task/2-subtract-the-sum-of-33452-and-46771-from-927521-3-subtract-42344404","timestamp":"2024-11-05T16:01:05Z","content_type":"text/html","content_length":"25335","record_id":"<urn:uuid:225a87c9-4197-4109-bdb9-da2193f3e4d8>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00597.warc.gz"} |
The effects of the supply of credit on real estate prices ... - P.PDFKUL.COM
The effects of the supply of credit on real estate prices: Venezuela as a policy laboratory Claire A. Boeing-Reicher†
David Pinto‡
Kiel Institute for the World Economy (IfW) Kiel, Germany This Version: Mar. 10, 2016 Abstract We identify the effects of the supply of mortgage credit on house prices, using the politically-directed
credit-targeting regime of Venezuela as quasi-natural experiment. We find a large effect of the supply of housing credit on time path of house prices (or housing markups), with an elasticity of
housing markups with respect to credit of about 0.23 under our baseline specification, and similar results under a set of alternative specifications. These estimates are close to previous panel
estimates for the United States, which suggests that these estimates capture similar phenomena. Keywords: House prices, credit supply, Venezuela, credit targeting. We are grateful to Jakob De Haan,
Henning Weber, Maik Wolters, Ester Faia, Matthias Burgert, Harmen Lehment, Jean Imbs, and Peter Neary, and to seminar participants in Kiel, for their useful feedback. This paper expresses solely the
views of the authors and not of their respective institutions. JEL Codes: E51, E65, R31. Keywords: Hugo Ch´avez, credit supply, house prices, Venezuela. †
Corresponding author, Kiel Institute for the World Economy. Email: claire dot reicher at ifw-kiel dot de. ‡ Kiel Institute for the World Economy: Advanced Studies Program, 2013-2014.
The nationwide expansion of mortgage credit and the increase in house prices during the 2000s, and the subsequent crash, have provoked a wave of research on the causal nexus between the supply of
credit and asset prices. However, it is difficult to clearly state in econometric terms that the expansion in mortgage credit caused the increase in house prices, since it is conceivable that the
increase in house prices may have caused the expansion in credit, or something else caused both the increase in house prices and the expansion in credit. In fact, it is only recently that two panel
studies on U.S. metropolitan areas have managed to identify the causal effect of the supply of credit on house prices. To add to our knowledge on this subject, we offer some evidence on this effect
based on Venezuelan time-series data. We use Venezuelan data because in Venezuela, the supply of mortgage credit is politically determined, to such an extent that shocks to the supply of credit
provide a quasi-natural experiment. This allows us to treat Venezuela under the administration of former president Hugo Ch´avez as a policy laboratory. After doing this, we find similar results to
the panel studies–we find a large and persistent effect of the supply of mortgage credit on the housing markup (our preferred measure of house prices), with a medium-run (12-quarter) elasticity of
about 0.23.
Our results add to a growing set of findings which suggest that changes to the supply of housing credit have an important causal effect on house prices. To that end, there are two panel studies which
look at the effects of the supply of credit on regional house prices in the United States, and these studies also rely on a political identification strategy. The first panel study in question is
that of Favara and Imbs (2015), who use changes in branch banking regulations to identify exogenous movements in the supply of housing credit at the MSA level. Favara and Imbs find a large,
economically significant effect of the supply of credit on house prices. The medium-run elasticity that they estimate, holding income constant, is somewhat below 0.25, which is close to our estimate
of 0.23.1 The second panel study to examine this issue is that of Adelino et al. (2012), who estimate the effects of regional changes in the conforming loan limit of the Federal Housing Finance
We derive this figure from Favara and Imbs (2015), Table 5, second column. An elasticity of 0.25 is obtained by taking 0.134/(1 − 0.457).
(FHFA), which the authors take to represent a exogenous policy-driven shock to credit, on house prices at the metro area level. Adelino et al. find that an increase in the conforming loan limit leads
to a substantial increase in house prices, which in turn suggests that the supply of credit causally affects house prices. In light of this panel evidence for the United States, our results find that
the aggregate time-series evidence for Venezuela produces a similar set of patterns, which suggests that both sets of estimates capture basically similar phenomena.
Apart from the two politically-identified panel studies, most other studies on the link between the supply of credit and house prices take an observational approach. However, these studies also find
a correlation between the supply of credit and house prices, with some possibility of causation. One such study is the panel study of Agnello and Schuknecht (2009). Agnello and Schuknecht estimate a
panel probit model in order to see which variables can help to forecast boom and bust episodes in the housing market for 18 major industrialized economies. In their findings, they observe that lagged
credit growth helps to forecast booms and busts. Another such study is that of Goodhart and Hofmann (2008), who estimate a panel VAR model with fixed effects, to estimate the degree of comovement
among house prices, credit and money, using data for the 17 major industrialized economies. Based on these estimates, they conclude that money, credit, and house prices are linked to each other.
Exploring this link with respect to the recent crisis, Mian and Sufi (2009) also present results that provide evidence for such a relationship. This evidence comes from the fact that the US house
price boom from 2001 to 2005 coincided with an unusual increase in subprime lending which affected ZIP codes asymmetrically. It turns out that those ZIP codes which experienced a larger increase in
subprime lending also experienced larger increases in mortgage credit and also in house prices, with some hint of causality.2 Finally, based more on a descriptive approach, Borio and Lowe (2002)
observe that asset price cycles seem to comove with the credit cycle. Taken together, the literature suggests that there is a statistical relationship between the supply of credit and house prices,
and in conjunction with the identified panel studies,
We refer to results from the working paper version, since the published version omits this finding. Also note that Glaeser et al. (2010) find that none of this cross-sectional correlation seems to
come through changes in approval rates or loan-to-value ratios.
our work helps to further establish that this relationship also appears to be a causal one.
This set of empirical results also has implications for ongoing theoretical work which has sought to explain the 2000s boom and crash in house prices. To cite one example of such work, Justiniano et
al. (2014) set up a model of real estate prices which features a credit channel. Based on this model, Justiniano et al. argue that credit supply shocks may be a major driver of the boom and crash in
U.S. house prices. On the credit supply side, their model features a lending constraint following Kiyotaki and Moore (1997), and on the demand side, their model features a collateral constraint. A
credit supply shock works in the following manner. First, a loosening of the lending constraint leads to an expansion in the supply of housing credit, which then leads to an expansion in the demand
for housing. This expansion in demand drives house prices up, which further loosens the collateral constraint on the credit demand side. In fact, based on this feedback mechanism, this model
generates a stark prediction: a house price elasticity with respect to credit of about one, which in comparison with the data, overexplains the original runup in house prices. Nonetheless, the basic
qualitative predictions of their model are compatible with our esults, which suggests that changes in lending constraints may help to explain some of the empirical patterns that we see.
Identification: the economic and political environment in Venezuela during the credit targeting regime
Our results are based on an identification strategy that takes the supply of credit in Venezuela as an exogenous variable, which then affects the housing markup (our proxy for real estate prices). As
Figure 1 shows, between 1999 and 2008, the housing markup and the nominal supply of housing credit tended to move together. The housing markup serves as a proxy for the value of land, and this markup
is equal to a nominal house price index of the capital city Caracas, divided by a construction input price index.3 The nom-
The nominal house price index is the “Indicador Inmobiliario Consolidado (Inpi)”, which was calculated by the Central Bank of Venezuela for research and internal purposes until the first quarter of
2008. The construction input price index is the “´Indice de Precios de Insumos de la Construcci´on”, which is published on a monthly basis by the Central Bank of Venezuela.
inal supply of housing credit is equal to the aggregate supply of residential mortgages within the Venezuelan banking system.4 Examining these series, several episodes are worth noting. First of all,
during the 2002-2003 period of political instability, both the housing markup and the supply of housing credit fell together. Then, during the second half of 2003, both variables began to recover
from their depressed levels, and from 2005 onward (marked by the vertical line), both variables began to show a sustained sharp increase. Altogether, the visual evidence indicates that both variables
tended to fall and rise together. Based on our identification strategy, which takes credit as exogenous, we argue that the increase in the housing markup alongside the increase in credit supports the
idea that an increase in credit causes the housing markup to rise.
To support this identification strategy, we argue that the documentary evidence suggests that the supply of housing credit in Venezuela during the 2000s was driven by political decisions rather than
economic decisions and is hence exogenous to the housing markup. We argue that these conditions especially hold after 2005. After 2005, the sharp increase in the supply of housing credit results
directly from the Venezuelan government’s implementation of an expanded credit targeting regime. Under this credit targeting regime, the government forced private and public banks to direct ten
percent of their respective loan portfolios to housing credit in the form of mortgages, with an objective toward expanding home ownership among workers. Since banks on average had initially directed
only two percent of their loan portfolios to housing credit, in order to avoid shrinking their portfolios in other sectors, banks were forced to progressively increase their overall supply of
credit.5 In fact, the Central Bank of Venezuela points out in its 2005 Annual Economic Report that real credit supply for the purchase of houses increased by 157.5 percent in 2005 due to the
implementation of this credit target.6 The imposition of this credit target also coincided with the imposition of a lower interest rate for housing credit (shown in Figure 2), with an objective
toward limiting “usury”.7 Im-
These data were obtained from the Central Bank of Venezuela. For further information about these changes and other changes to the Venezuelan banking system, see Levy-Carciente et al. (2014). For
further information, see Banco Central de Venezuela, Informe Econ´omico 2005, page 179. See Gaceta Oficial No. 38.098: Ley Especial de Protecci´ on al Deudor Hipotecario de Vivienda (2005), for which
a rough translation would be, “The special law for the protection of residential mortgages”.
portantly, the credit targeting law itself states that its objectives mainly lie in expanding homeownership and in limiting usury, while that law does not make any reference to thencurrent economic
conditions. These stated objectives suggest that the credit targeting regime and its associated expansion of housing credit were implemented for exogenous, political reasons rather than as an
endogenous response to macroeconomic conditions.
There is more evidence that points toward a uniquely strong role for shocks to housing credit in driving the housing markup. Although Venezuela experienced a number of other shocks during the
1999-2008 period, none of these shocks seems to have coincided closely with the rise in housing markups. One set of shocks involves shocks to exchange rate policy. Venezuelan exchange rate policy may
conceivably have an effect on housing markups, since house prices in Venezuela are often tied to the US dollar to avoid issues associated with a high and variable inflation rate. To address this
possibility, Figure 3 displays the housing markup and the shadow and official exchange rates between the Venezuelan Bolivar (VEB) and the US dollar.8 The Venezuelan government implemented controls on
these exchange rates beginning in 2003, after which the official and shadow exchange rates diverged. The official exchange rate increased only gradually after 2003, while the shadow exchange rate
increased sharply beginning in late 2006, nearly two years after the sharp increase in the housing markup was well underway. The timing of this increase suggests that the rise in the housing markup
may have been driven by something other than shocks to exchange rate policy.
In addition to oil price shocks, another set of shocks to affect Venezuela during this period would be shocks to oil prices. Given that oil forms more than 90 percent of Venezuelan exports, changes
in oil prices may have affected the relative price of nontrad-
The discussion about expanding home ownership could be found in Articles 8 and 9 of this law, while the discussion about usury could be found in Article 11. Both exchange rate indicators are in
nominal terms. While the official exchange rate was published by the Central Bank of Venezuela, the shadow market exchange rate was published on private websites which were not officially recognized
by the Venezuelan government. Until 2010 the shadow market exchange rate was published on a daily basis at http://www.dolarparalelo.blogspot.com, while until February 2015 the shadow market exchange
rate was published on a daily basis at http://dolartoday.com. In February 2015 the shadow foreign exchange market was finally legalized by the government, and since then foreign exchange can be
traded freely at the Sistema Marginal de Divisas (SIMADI).
ables in general and the price of real estate in particular.9 To address this possibility, Figure 4 displays the housing markup and the real price of the Venezuelan oil basket in US dollars.10 A look
at this figure shows that the real price of the Venezuelan oil basket began to rise from 2001 onward–four years before the sharp increase in the real estate markup. The timing of the oil price
shocks, as with the exchange rate shocks, does not seem to coincide closely with the rise in housing markups.
Altogether, it appears that the timing of exchange rate shocks and oil price shocks does not coincide with the timing of the sharp increase in the housing markup, while the sharp increase in the
housing markup does seem to coincide with a sharp increase in the supply of housing credit and with a fall in interest rates, both of which are driven by the credit targeting regime. Furthermore,
this credit targeting regime appears to be driven by political events rather than by economic events, which suggests that it is reasonable to assume that the nominal supply of housing credit is
exogenous to the rest of the system. Using this exogeneity as an identifying assumption in a VAR, we go on to show that there does in fact seem to be a close relationship between the supply of
housing credit and the housing markup, and this relationship is robust to including these other confounding factors.
Housing credit and the housing markup: Evidence from a VAR
The VAR specification
The institutional setup of the credit targeting regime–in particular, the exogenous imposition of that regime for political reasons–makes it possible to identify the effects of credit supply shocks.
This identification scheme implies that a VAR model would capture
Given this reliance on oil, an increase in oil prices could result in the “Dutch Disease”, following the terminology of Corden and Neary (1982). 10 The real oil price is obtained by dividing the
nominal price of the Venezuelan oil basket in US dollars, which is published by the Venezuelan Ministry of Energy and Oil, by the US CPI, which is published by the US Bureau of Labor Statistics.
the effects of an exogenous shock to growth in the nominal supply of housing credit, or nominal credit growth, under the assumption that nominal credit growth in the current period is exogenous to
all of the other variables in the system. To estimate such a VAR, we use a Bayesian MCMC estimation procedure in order to generate exact credible intervals (analogous to confidence intervals) given
the data. Furthermore, the credit supply shock may have persistent–or even unit-root or explosive–effects on observables, which renders the use of asymptotic reasoning problematic.
In the baseline VAR setup, the observables yt are given by the two-by-one matrix: nominal housing credit growtht . yt = housing markup growtht
The elements of yt are equal to the change in log total nominal housing credit (nominal housing credit growth) and the change in the log housing markup (housing markup growth). The aggregates yt
evolve according to a VAR. The VAR specification itself takes the reduced form: yt =c+
P X
Ap yt−p +εt .
On the right-hand side of the VAR, c equals a 2 by 1 matrix containing a set of intercepts; Ap equals a 2 by 2 matrix comprising the coefficients on lagged endogenous variables at lag p; and the
residuals εt are a 2 by 1 vector of innovations which is i.i.d across time and multivariate normal with a mean of zero and a covariance matrix of Σ. There are T observations in total. Taking the
inclusion of P lagged endogenous variables on the right-hand side into account, there remain T − P usable observations in calculating the VAR coefficients.
The estimation procedure
To estimate the VAR, it is necessary to first set the lag length P . In the baseline setup, the Akaike and Schwarz information criteria both point toward a value of P equal to one, and so the
estimated VAR contains one lag. We estimate this VAR using an MCMC algorithm, where each MCMC draw is indexed by (i). For notational simplicity, the matrix A denotes all of the coefficients in the
VAR stacked into a column matrix; X denotes 7
the stacked right-hand side variables of the VAR (including a column of ones); and Y denotes the stacked left-hand side variables of the VAR. Furthermore, we set our priors such that the matrix A is
multivariate normal with a zero mean and zero precision, and Σ is Inverse Wishart, equivalent to having observed zero observations, with a product of residuals equal to a matrix of zeroes.
To initialize the MCMC, we first set A(1) and Σ(1) to the observed coefficients and to the observed covariance matrix of VAR residuals, which are calculated using the usual OLS formulae. Then, for
each iteration (i) from 2 through 50,001, we draw a set of coefficients A(i) given Y , X, and Σ(i−1) , from a posterior distribution given by A(i) ∼ N ((X ′ X)−1 (X ′ Y ), Σ(i−1) ⊗ (X ′ X)−1 ). Next,
we generate a set of VAR residuals ε(i) of dimension 2 by T −P , given A(i) , Y , and X. Based on these residuals, we draw a covariance matrix Σ(i) , from a posterior distribution given by Σ(i) ∼ IW
(ε(i) (ε(i) )′ , T −P ).11
Once the VAR coefficients for iteration (i) are recovered, we then apply a set of Cholesky identifying assumptions in order to recover impulse responses. We generate these impulse responses for
iteration (i) by shocking the first element of εt and tracing through its contemporaneous and dynamic effects. We accomplish this by representing the innovations to equation (2) as εt = C (i) ηt ,
where ηt is a matrix of shocks with an identity covariance matrix. By assuming that the credit supply shock can affect the other variables contemporaneously but not vice versa, we equivalently assume
that C (i) is given by a lower-triangular matrix which we in turn derive using the Cholesky decomposition of Σ(i) . By setting the first element of ηt to one at the initial period (which represents a
one-standard-deviation credit supply shock), premultiplying by C (i) gives εt conditional on that shock, which allows us to iterate through equation (2) to map out the impulse response for later
After engaging in these steps for a given iteration (i), we store the impulse responses in memory, and then we move on to the next iteration (i + 1) and repeat the whole process.
The notation used here for the Inverse Wishart distribution takes as arguments the sum of squared residuals and number of observations.
After the final iteration, we discard the first 10,000 iterations as burn-in, which leaves the remaining 40,001 iterations to calculate exact posterior medians, credible intervals, and the posterior
distributions of impulse responses.
Baseline VAR results: The increase in housing markups after a housing credit supply shock
Baseline results: the effects of a nominal credit supply shock
Figure 5 shows a set of posterior median impulse responses of yt to a one standard deviation housing credit supply shock, along with a set of credible intervals. The lefthand panels of Figure 5
display the quarterly responses of the nominal housing credit growth rate and the housing markup growth rate to a one-standard-deviation nominal housing credit supply shock, while the right-hand
panels show the cumulative responses of housing credit and the housing markup. Table 1, part A, displays some posterior statistics with respect to these impulse responses. Altogether, these impulse
responses show that housing credit supply shocks appear to drive up the housing markup, which is directly in line with the visual evidence from Figure 1. A nominal housing credit supply shock on
impact triggers a 5.40 percent posterior median increase in the nominal housing credit supply and a 0.63 percent posterior median increase in the housing markup growth rate (both in log terms), with
a 92.52 percent probability that the latter increase is above zero. After twelve quarters the housing credit supply shock leads on average to a 12.34 percent posterior median cumulative increase in
the total housing credit supply and a 2.82 percent posterior median cumulative increase in the housing markup growth rate, with a 97.82 percent probability that the latter effect is above zero. This
response is economically significant. These numbers imply that the medium-run elasticity of the housing markup with respect to the supply of housing credit is equal to approximately 0.23.
Furthermore, a variance decomposition (using OLS estimates of the VAR system) suggests that within a horizon of 12 quarters, between 18 and 19 percent of the variance in the housing markup growth
rate can be explained by the housing credit supply shock. Altogether, it appears that credit supply shocks are an economically significant factor behind fluctuations in the housing markup in
An overidentifying restriction: the effects of a residual housing markup shock
Under the identifying assumptions used to identify the effects of a credit supply shock, the supply of credit should be driven by the credit supply shock alone, and residual shocks to the housing
markup (shocks to the second element of ηt ) should therefore have no effect on the supply of credit. A nonzero effect of such a shock on the credit supply would mean that there is some feedback
process running from the housing markup to the housing credit supply, which would put the identifying assumptions underlying the VAR into doubt. To check that the housing markup does not drive future
changes to the housing credit supply, Figure 6 and Table 1, part B, display impulse response paths and posterior statistics with respect to the housing credit growth rate and housing markup growth
rate after a residual shock to the housing markup. It turns out that a residual shock to the housing markup in fact has little effect on the supply of housing credit. The posterior credible intervals
for these responses encompass zero, and furthermore, the estimated responses of the housing credit supply to a housing markup shock are small and economically insignificant. Furthermore, a variance
decomposition shows that shocks to the housing markup only explain about 6 percent of the variance of housing credit growth. Taken together, this evidence is in line with the narrative evidence and
with the identifying assumption that a residual shock to the housing markup has no effect on the supply of housing credit in later periods.
3.4 3.4.1
Robustness: results based on alternative observables Real instead of nominal housing credit supply shocks
One possible objection to our analysis might take the form that we combine real and nominal variables in an unusual way. To address this objection, we show that the baseline VAR results are robust to
looking at real housing credit growth instead of nominal housing credit growth. To do this, we replace nominal housing credit growth in equation (2) with real housing credit growth, such that:
yt =
real housing credit growtht housing markup growtht
and then proceed as before. Here, we obtain real credit by deflating nominal housing credit by the Venezuelan (Caracas) consumer price index, which serves as a proxy for the level of consumer prices
in all of Venezuela.
As seen in Figure 7 and Table 2, part A, a real housing credit supply shock on impact triggers a 14.12 percent posterior median increase in the real housing credit growth rate and a 0.67 percent
posterior median increase in the housing markup growth rate, with a 93.78 percent probability that this increase is above zero. After twelve quarters the real housing credit supply shock leads to a
34.58 percent cumulative increase in the real housing credit supply and a 3.03 percent posterior cumulative increase in the housing markup growth rate, with a 98.37 percent probability that this
effect is above zero. As with a nominal credit supply shock, the effects of a real credit supply shock are statistically distinguishable from zero and economically significant. Furthermore, within a
horizon of 12 quarters, between 20 and 21 percent of the variance in housing markup growth can be explained by the housing credit supply shock. Altogether, these results show that the baseline
results are not sensitive to taking nominal as opposed to real housing credit, and in fact, the baseline results could be viewed as somewhat conservative. 3.4.2
Real house prices instead of housing markups
The baseline VAR results are also robust to looking at an alternative measure of house prices, which involves replacing the housing markup in equation (2) with real house price growth, such that:
nominal housing credit growtht , yt = real house price growtht
and then proceeding as before. Here, real house prices are obtained by deflating nominal house prices by the Venezuelan (Caracas) consumer price index.
As seen in Figure 8 and Table 2, part B, a nominal housing credit supply shock on impact triggers a 5.47 percent posterior median increase in nominal housing credit growth and a 0.33 percent
posterior median increase in real house price growth, with an 81.11 percent probability that this increase is above zero. After twelve quarters the nominal housing credit supply shock leads to a
12.36 percent cumulative increase in the 11
nominal credit supply and a 1.65 percent posterior cumulative increase in the real house price, with a 94.48 percent probability that this effect is above zero. These effects are (borderline)
statistically distinguishable from zero and economically significant, though less significant than under the housing markup measure. After 12 quarters, about 10.5 percent of the variance in house
price growth can be explained by the housing credit supply shock. Based on these results, it appears our results are robust to choosing a different indicator of house prices.
3.5 3.5.1
Robustness: results based on expanded specifications Taking the interest rate into account
As discussed in Section 2, the imposition of the credit targeting regime coincided with the imposition of a preferential interest rate for housing credit; and in fact, it is conceivable that this is
one mechanism through which the credit targeting regime operated. To investigate the sensitivity of our results to movements in interest rates, we also estimate a specification which includes the
interest rate on housing credit into the VAR, ordered second. In this setup, the vector yt consists of the following observables, such that:
nominal housing credit growtht yt = change in nominal interest ratet , housing markup growtht
where the change in the nominal interest rate is given by changes to the quarterly nominal lending rate until 2004 and the preferential nominal interest rate for residential mortgages from 2005
onward. As before, we utilize a one-lag specification based on the Akaike and Schwarz Information Criteria.
The impulse response functions in Figure 9 and the results in Table 3, part A, show that adding the change in the nominal interest rate to the system does not significantly alter the results of the
baseline model specification. On impact, a nominal housing credit shock triggers a 5.49 percent median increase in the housing credit growth rate, a fall in the nominal interest rate of about 0.90
percent, and a 0.54 percent posterior median increase in the housing markup growth rate, with an 88.45 percent probability that this
increase is above zero. After twelve quarters, the nominal housing credit supply shock leads on average to an 11.47 percent posterior median cumulative increase in the total housing credit supply, a
2.71 percent posterior cumulative increase in the nominal interest rate, and a 2.24 percent posterior cumulative increase in the housing markup growth rate, with a 94.00 percent probability that this
effect is above zero. Over a 12 quarter horizon, between 22 and 23 percent of the variance in changes in interest rates and between 16 and 17 percent of the variance in the housing markup growth rate
can be explained by the nominal housing credit supply shock. These results are broadly in line with those obtained in the baseline model, and these results suggest that the nominal housing credit
supply shock may reduce interest rates at the outset, with an ambiguous effect in later periods.
Turning to the issue of identification, under the identifying assumptions used to identify the effects of a nominal housing credit supply shock, the supply of housing credit should be driven by the
housing credit supply shock alone, and residual shocks to the absolute change in the nominal interest rate (shocks to the second element of ηt ) should therefore have no effect either on the nominal
supply of housing credit or on the housing markup. Nonzero effects of such a shock on the housing credit supply or on the housing markup would put the identifying assumptions underlying the
three-variable VAR into doubt.
To check that the absolute change in the nominal interest rate does not drive future changes in the credit supply, Figure 10 and Table 3, part B, display paths and statistics with respect to the
impulse responses to the nominal housing credit growth rate, housing markup growth rate, and the absolute change in the nominal interest rate after a residual shock to the nominal interest rate. It
turns out that a residual shock to the nominal interest rate in fact has little effect on the nominal supply of housing credit, and only a small effect on the housing markup. The posterior credible
intervals for these responses encompass zero, and furthermore, the estimated responses of the nominal housing credit supply to an interest rate shock are small and negative over the course of 12
quarters, and economically insignificant. In addition, a variance decomposition shows that shocks to the nominal interest rate only explain between 2 and 3 percent of the variance of nominal
housing credit growth. Taken together, this evidence is in line with the narrative evidence and with the identifying assumption that a residual shock to the interest rate has no effect on the supply
of housing credit in later periods. 3.5.2
Taking oil prices and exchange rates into account
Finally, we see what happens when we add real oil price growth and exchange rate growth to our system. Given the narrative evidence in Section 2, we do not expect much to happen to the effects of
credit supply shocks when we include these two variables. This is indeed the case. To see this, in the expanded system, the vector yt now consists of the following observables, such that: nominal
housing credit growtht oil price growtht , yt = exchange rate growtht housing markup growtht
As before, we utilize a one-lag specification based on the Akaike and Schwarz Information Criteria.
The impulse response functions in Figure 11 and the statistics in Table 4, part A, show that adding these auxiliary variables does not significantly change the results relative to the baseline model
specification. On impact, a nominal housing credit shock now triggers a 5.51 percent median increase in housing credit growth, a 2.05 percent median decrease in oil prices, a 1.48 percent decrease in
the nominal exchange rate, and a 0.60 percent posterior median increase in the housing markup growth rate, with an 88.74 percent probability that this last increase is above zero. After twelve
quarters, the nominal housing credit supply shock leads on average to a 12.17 percent posterior cumulative increase in the total housing credit supply, a 1.82 percent posterior cumulative decrease in
oil prices, a 1.79 percent posterior decrease in the nominal exchange rate, and a 2.49 percent posterior cumulative increase in the housing markup, with a 94.18 percent probability that this last
effect is above zero. Over a 12 quarter horizon, between 14 and 15 percent of the variance in the housing markup growth rate can be explained by the nominal housing credit supply shock. These results
indicate that 14
the results from the baseline model are not affected by taking oil prices and exchange rates into account, which is also in line with the narrative evidence presented in Section 2.
Interestingly, these results also point toward a moderate probability (100 percent minus 3.66 percent, or 96.34 percent) of a negative relationship between innovations to credit growth and oil price
growth in the very short run, while the probability of this relationship falls in the long run. While the probability of this relationship in the very short run does not quite reach the 97.5 percent
threshold that would be applied in a two-sided hypothesis test, and such a test would in any case require a Bonferroni-like correction for multiple comparisons, these results point toward that
possibility that political authorities in Venezuela tended to slightly increase the supply of credit whenever the oil price fell. To the extent that this is a true pattern in the data, and not an
artifact of multiple comparisons, it turns out that this pattern would result in this robustness check yielding somewhat too conservative of a result in the short run.
To investigate this issue and other issues related to the identification of this model, it is useful to look at the effects of the other shocks. Under the identifying assumptions used to identify the
effects of a nominal housing credit supply shock, the supply of housing credit should be driven by the housing credit supply shock alone, and residual shocks to the real oil price growth rate (shocks
to the second element of ηt ) as well as residual shocks to the exchange rate (shocks to the third element of ηt ) should therefore have no effect either on the nominal supply of housing credit or on
the housing markup. Furthermore, looking at the effects of these shocks is likely to reveal the extent and direction of any bias that might result from a correlation between credit supply shocks and
oil price shocks.
First, to check that the real oil price growth rate does not drive future changes in the supply of housing credit, Figure 12 and Table 4, part B, display median impulse response paths and statistics
with respect to the impulse responses after a residual shock to the real oil price growth rate. It turns out that a residual shock to the real oil price growth rate has at most a minor effect on the
nominal supply of housing credit, and only a small positive effect on the housing markup. After twelve quarters, the residual real oil price shock leads on average to a 4.29 percent posterior
cumulative increase in the total housing
credit supply, with a 91.28 percent probability that this effect is above zero. Over the course of twelve quarters, a residual real oil price shock triggers a 1.14 percent median increase in the
housing markup, with an 79.68 percent probability that this increase is above zero. Furthermore, the variance decomposition shows that shocks to the real oil price growth rate only explain about 8
percent of the variance of nominal housing credit growth and between 2 and 3 percent of the variance of housing markup growth over a 12-quarter horizon. Taken together, this evidence is in line with
the narrative evidence and with the identifying assumption that a residual shock to the real oil price growth rate has only a negligible effect on the supply of housing credit in later periods.
Furthermore, this set of impulse responses suggests that the estimates of the very short-run effects of a housing credit supply shock are somewhat conservative, to the extent that these shocks are
actually related to decreases in oil prices.
Then, to check that the shadow exchange rate does not drive future changes in the supply of housing credit, Figure 13 and Table 4, part C, display median paths and statistics with respect to the
impulse responses after a residual shock to the shadow exchange rate. It turns out that a residual shock to the shadow exchange rate growth in fact has little effect on the nominal supply of housing
credit, and only a small effect on the housing markup. The posterior credible intervals for these responses encompass zero, and furthermore, the estimated responses of the nominal supply of housing
credit to a shadow exchange rate growth shock are small and economically insignificant. In addition, a variance decomposition shows that shocks to the shadow exchange rate only explain less than 5
percent of the variance of nominal housing credit growth. Taken together, this evidence is in line with the narrative evidence and with the identifying assumption that a residual shock to the shadow
exchange rate has no effect on the supply of housing credit in later periods.
Altogether, the narrative evidence and quantitative evidence from a VAR for Venezuela both point toward a strong effect of the supply of housing credit on real estate markups, or on real estate
prices more generally. Based on exogenous, politically-driven movements
in the supply of housing credit that were implemented through a credit-targeting regime, we find that increases in the supply of housing credit appear to have resulted in large, robust increases in
the housing markup, with a medium-run elasticity of the housing markup with respect to the supply of credit of about 0.23. Furthermore, we argue that these credit supply shocks have effects which are
broadly compatible with the theoretical literature on the credit channel. The key to these results lies in the fact that the supply of credit in Venezuela is politically determined, which allows us
to treat Venezuela as a policy laboratory and to conduct a quasi-natural experiment to see what happens after a shock to the supply of housing credit.
These results suggest that the findings reported by observational studies on this topic might represent, in large part, a causal relationship, rather than mere correlation. In interpreting these
results in such a manner, it is worth pointing out that the fact that Venezuela can serve as a policy laboratory in the first place implies that Venezuela is not completely representative with
respect to the experiences of other countries. Much of the Venezuelan economy is subject to state intervention, and the economy itself relies heavily on oil and natural resources. In these respects,
Venezuela is different from a large, diversified economy such as the United States or most of Europe. As a result of these differences, our exact estimates should be treated with a degree of caution.
Nonetheless, we believe that some of the same mechanisms that operate within the Venezuelan economy might operate within other economies, particularly with respect to the demand for real estate by
the household sector. Furthermore, our aggregate time-series results are very close to those from the panel study of Favara and Imbs (2015). This similarity suggests that both sets of results seem to
possibly capture a similar set of phenomena.
Taking into account these considerations, we believe that our results may help to shape ongoing work on the effects of changes in the supply of credit, and to provide a quantitative target for
researchers to match. Additionally, within the macro literature, future work might help to examine the spillovers between the channels identified by Justiniano et al. (2014) and the real economy,
particularly within credit-sensitive sectors such as the construction sector. Given the basic similarities between our empricial results and the theoretical results of Justiniano et al., our results
also suggest that future
work might further uncover the degree to which monetary, fiscal, and regulatory policymakers might wish to directly monitor, or respond to, shocks to the supply of credit. Altogether, the evidence
points toward exogenous fluctuations in the supply of credit as having important effects on real estate prices, and given recent experience, these effects are are of substantial interest to
policymakers and researchers.
References Adelino, M., A. Schoar, and F. Severino (2012), “Credit Supply and House Prices: Evidence from Mortgage Market Segmentation”, NBER Working Paper 17832. Agnello, L., and L. Schuknecht
(2009), “Booms and Busts in Housing Markets: Determinants and Implications”, ECB Working Paper 1071. Banco Central de Venezuela (2005), Informe Econ´omico 2005. Borio, C., and P. Lowe (2002), “Asset
Prices, Financial and Monetary Stability: Exploring the Nexus”, BIS Working Paper Series 114. Corden, W., and P. Neary (1982), “Booming Sector and De-Industrialization in a Small Open Economy”, The
Economic Journal 92(268), pp. 825-848. Favara, G., and J. Imbs (2015), “Credit Supply and the Price of Housing”, The American Economic Review 105(3), pp. 958-992. Goodhart, C., and B. Hofmann (2008),
“House Prices, Money, Credit and the Macroeconomy”, ECB Working Paper 888. Glaeser, E., J. Gottlieb, and J. Gyourko (2010), “Can Cheap Credit Explain the Housing Boom?”, NBER Working Paper 16230.
Justiniano A., G. Primiceri G., and A. Tambalotti (2014), “Credit Supply and the Housing Boom”, Federal Reserve Bank of Chicago Working Paper 2014-21. Kiyotaki N., and J. Moore (1997), “Credit
Chains”, Journal of Political Economy 105(2), pp. 211-248. 18
Levy-Carciente S., D. Kennet, A. Avakian, H. Stanley, and H. Shlomo (2014), “Dynamical Macro-Prudential Stress Testing Using Network Theory”, Working Paper. Mian A., and A. Sufi (2009), “The
Consequences of Mortgage Credit Expansion: Evidence from the 2007 Mortgage Default Crisis”, Quarterly Journal of Economics 124(4), pp. 1449-1496. Rep´ ublica Bolivariana de Venezuela: “Ley Especial
de Protecci´on al Deudor Hipotecario de Vivienda” (2005), Gaceta Oficial 38.098.
Table 1: Two-variable baseline model: Cumulative responses to shocks: Results on impact (t=0) and after three years (t=12) Credit Growth Median [P>0] (Var. Decomp. %)
Housing Markup Growth Median [P>0] (Var. Decomp. %)
0.0063 [92.52] 0.0282 [97.82] (18.46)
0.0342 [89.10] (6.12)
A. Nominal credit supply shock (Baseline)
B. Residual markup shock (Baseline)
Part A displays the quarterly posterior median responses of nominal housing credit growth and housing markup growth to a one-standard-deviation nominal housing credit supply shock. The values in
square brackets below the median response outcomes show the probability of the shock’s effect being above zero. The values in parentheses display how much of the variance in the housing markup growth
rate can be explained by the nominal housing credit supply shock (variance decomposition). Part B displays the same statistics as described above, but for the responses to a one-standard-deviation
residual housing markup shock.
Table 2: Two-variable model with alternative indicators: Cumulative responses to shocks: Results on impact (t=0) and after three years (t=12) Credit Growth Median [P>0] (Var. Decomp. %)
Housing Markup / House Price Growth Median [P>0] (Var. Decomp. %)
0.0067 [93.78]
0.0303 [98.37] (20.60)
0.0033 [81.11] 0.0165 [94.48] (10.45)
A. Real credit supply shock (Alternative credit indicator: real credit)
B. Nominal credit supply shock (Alternative house price indicator: real house prices)
Part A displays the quarterly posterior median responses of real housing credit growth and housing markup growth to a one-standard-deviation real housing credit supply shock. Real housing credit is
obtained by deflating nominal housing credit with the consumer price index (cpi) of the Venezuelan capital city Caracas. The values in square brackets below the median response outcomes show the
probability of the shock’s effect being above zero. The values in parentheses display how much of the variance in the housing markup growth rate can be explained by the real housing credit supply
shock (variance decomposition). Part B displays the same statistics described as above, but for the nominal housing credit and house price growth responses to a one-standard-deviation nominal housing
credit supply shock.
Table 3: Larger system with interest rate: Cumulative responses to shocks: Results on impact (t=0) and after three years (t=12)
Credit Growth Median [P>0] (Var. Decomp.%)
Markup Growth Median [P>0] (Var. Decomp. %)
Change in Interest Rate Median [P>0] (Var. Decomp. %)
0.0054 [88.45]
-0.0090 [12.88]
0.0224 [94.00] (16.24)
0.0271 [97.20] (22.55)
-0.0213 [13.48] (2.79)
0.0425 [55.52] -0.0093 [16.82] (4.64)
A. Nominal credit supply shock (System with interest rate)
B. Residual interest rate shock (System with interest rate)
Part A displays the quarterly posterior median responses of nominal housing credit growth, housing markup growth, and the change in the nominal interest rate to a onestandard-deviation nominal
housing credit supply shock. The values in square brackets below the median response outcomes show the probability of the shock’s effect being above zero. The values in parentheses display how much
of the variance in the housing markup growth rate can be explained by the nominal housing credit supply shock (variance decomposition). Part B displays the same statistics described as above, but for
the responses to a one-standard-deviation residual nominal interest rate shock.
Table 4: Larger system with oil prices and exchange rate: Cumulative responses to shocks: Results on impact (t=0) and after three years (t=12) Markup Growth Median [P>0] (Var. Decomp. %)
Oil P. Growth Median [P>0] (Var. Decomp. %)
FX Rate Growth Median [P>0] (Var. Decomp. %)
A. Nominal credit supply shock (System with oil price, FX rate) t=0
0.0060 [88.74] 0.0249 [94.18] (14.33)
-0.0205 [3.66] -0.0182 [27.83] (10.71)
-0.0148 [7.47] -0.0179 [25.92] (6.65)
0 [0] 0.0429 [91.28] (8.10)
0.0029 [73.40] 0.0114 [79.68] (2.61)
-0.0011 [45.42] -0.0197 [18.73] (4.57)
Credit Growth Median [P>0] (Var. Decomp.%)
B. Residual oil price shock (System with oil price, FX rate) t=0 t=12
The table displays the quarterly posterior median responses of nominal housing credit growth, housing markup growth, real oil price growth, and the FX rate growth rate to a one-standard-deviation
nominal housing credit supply shock (Part A) and to a one-standarddeviation residual oil price shock (Part B). The values in square brackets below the median response outcomes show the probability of
the shocks’ effect being above zero. The values in parentheses display how much of the variance in the housing credit growth rate, the housing markup growth rate, the real oil price growth rate, and
the FX rate growth rate can be explained by the shocks.
Table 4 (continued): Larger system with oil prices and exchange rate: Cumulative responses to the shocks: Results on impact (t=0) and after three years (t=12)
C. Residual exchange rate shock (System with oil price, FX rate) t=0 t=12
Credit Growth Median [P>0] (Var. Decomp.%)
Markup Growth Median [P>0] (Var. Decomp. %)
Oil P. Growth Median [P>0] (Var. Decomp. %)
FX Rate Growth Median [P>0] (Var. Decomp. %)
0 [0] -0.0333 [16.75] (4.14)
0.0022 [69.21] -0.0142 [14.91] (9.23)
0 [0] 0.0025 [55.10] (1.67)
0.0521 0.0678
The table displays the quarterly posterior median responses of nominal housing credit growth, housing markup growth, oil price growth, and the FX rate growth rate to a one-standard-deviation shadow
FX rate shock. The values in square brackets below the median response outcomes show the probability of the shock’s effect being above zero. The values in parentheses display how much of the variance
in the housing credit growth rate, the housing markup growth rate, the real oil price growth rate, and the FX rate growth rate can be explained by the shock.
Figure 1: Housing markup and nominal supply of housing credit
The housing markup index (continuous line) serves as a proxy for the value of land. This index is obtained by dividing a nominal house price index by a construction input price index. The nominal
house price index is called “Indicador Inmobilirario Consolidado (Inpi)” and is estimated by the Central Bank of Venezuela. The construction input price index is called ´Indice de Precios de Insumos
de la Construcci´on and is published on a monthly basis by the Central Bank of Venezuela. The dotted line shows data published by the Central Bank of Venezuela on the aggregate nominal credit supply
for housing mortgages. The vertical line highlights the implementation of a credit target for housing mortgages during the first quarter of 2005. Source: Central Bank of Venezuela, and authors’
Figure 2: Housing markup and interest rate to be charged on housing credit
The housing markup index (continuous line) serves as a proxy for the value of land. This index is obtained by dividing a nominal house price index by a construction input price index. The nominal
house price index is called “Indicador Inmobilirario Consolidado (Inpi)” and is estimated by the Central Bank of Venezuela. The construction input price index is called ´Indice de Precios de Insumos
de la Construcci´on and is published on a monthly basis by the Central Bank of Venezuela. The dotted line shows the nominal market interest rate for housing mortgages until 2004. From the first
quarter of 2005 onward, the dotted line shows the preferential interest rate for housing mortgages within the credit targeting framework. This preferential interest rate is obtained by applying a
predefined haircut to the average nominal market interest rate. Source: Central Bank of Venezuela, Venezuelan Bank Supervision Authority (Sudeban), and authors’ calculations.
Figure 3: Housing markup and shadow and official exchange rates between the Venezuelan Bolivar (VEB) and the US dollar
The housing markup index (continuous line) serves as a proxy for the value of land. This index is obtained by dividing a nominal house price index by a construction input price index. The nominal
house price index is called “Indicador Inmobilirario Consolidado (Inpi)” and is estimated by the Central Bank of Venezuela. The construction input price index is called “´Indice de Precios de Insumos
de la Construcci´on” and is published on a monthly basis by the Central Bank of Venezuela. The fine dotted line shows the evolution of the official exchange rate and the bold dotted line shows the
evolution of the shadow exchange rate. Since 2003 an exchange rate control has been implemented in Venezuela. Under these controls, the shadow market exchange rate could be seen as a proxy to the
free market rate. The official exchange rate is published by the Central Bank of Venezuela. For the shadow exchange rate there are specialized websites which publish the shadow exchange rate on a
daily basis, based on information from the shadow market. Source: Central Bank of Venezuela, http://www.dolarparalelo.blogspot.com, http://dolartoday.com, and authors’ calculations.
Figure 4: Housing markup and real price of the Venezuelan oil basket in US dollars
The housing markup index (continuous line) serves as a proxy for the value of land. This index is obtained by dividing a nominal house price index by a construction input price index. The nominal
house price index is called “Indicador Inmobilirario Consolidado (Inpi)” and is estimated by the Central Bank of Venezuela. The construction input price index is called ´Indice de Precios de Insumos
de la Construcci´on and is published on a monthly basis by the Central Bank of Venezuela. The dotted line shows the evolution of the real oil price. The real oil price is obtained by deflating the
nominal price of the Venezuelan oil basket in US dollars, which is published by the Venezuelan Ministry of Energy and Oil on a monthly basis, by the US consumer price index. The US consumer price
index is obtained from the US Bureau of Labor Statistics (BLS). Source: Central Bank of Venezuela, US Bureau of Labor Statistics, Venezuelan Ministry of Energy and Oil, and authors’ calculations.
Figure 5: Baseline model: Responses to a nominal housing credit supply shock Housing credit (log change)
Housing credit (log change), cumulative
0.05 0.25 0.04 0.2 0.03 0.15 0.02 0.1
0 −0.01 0
Housing markup (log change)
x 10
Housing markup (log change), cumulative 0.12
0.1 15 0.08 10
0.02 0 0
−5 0
−0.02 12 0
The left-hand panels display the quarterly responses of the nominal housing credit growth rate and the housing markup growth rate to a one-standard-deviation nominal credit supply shock, while the
right-hand panels show the cumulative responses of the nominal housing credit growth rate and the housing markup growth rate.
Figure 6: Baseline model: Responses to a residual housing markup shock Housing credit (log change)
Housing credit (log change), cumulative
0.03 0.15
0.025 0.02
0.1 0.015 0.01 0.05 0.005 0
−0.005 −0.01 0
−0.05 0 12
Housing markup (log change)
Housing markup (log change), cumulative
0.1 0.09
0.025 0.08 0.02
0.05 0.04
0.005 0.03 0 −0.005 0
0.02 2
0.01 0
The left-hand panels display the quarterly responses of the housing markup growth rate and the nominal housing credit growth rate to a one-standard-deviation residual housing markup shock, while the
right-hand panels show the cumulative responses of the housing markup growth rate and the nominal housing credit growth rate.
Figure 7: Alternative observables (real housing credit): Responses to a real housing credit supply shock Real housing credit (log change), cumulative
Real housing credit (log change) 1.4
1.2 0.15 1 0.1
0.4 0 0.2 −0.05 0
Housing markup (log change)
x 10
Housing markup (log change), cumulative 0.14 0.12
15 0.1 0.08
0.06 5
0.04 0.02
0 0 −5 0
−0.02 0
The left-hand panels display the quarterly responses of the real housing credit growth rate and housing markup growth rate to a one-standard-deviation housing credit shock, while the right-hand
panels show the cumulative responses. Real housing credit is obtained by the deflating nominal housing credit by the consumer price index of the capital city of Caracas.
Figure 8: Alternative observables (real house prices): Responses to a nominal housing credit supply shock Housing credit (log change)
Housing credit (log change), cumulative
0.05 0.25 0.04 0.2 0.03 0.15 0.02 0.1
0 −0.01 0
Real house price (log change)
x 10
Real house price (log change), cumulative 0.08
8 0.05 6 0.04 4 0.03 2 0.02
0 −2
−6 0
−0.01 0
The left-hand panels display the quarterly responses of the nominal housing credit growth rate and real house price index growth rate to a one-standard-deviation housing credit shock, while the
right-hand panels show the cumulative responses. The real house price index is obtained by the deflating the nominal house price index with the consumer price index of the capital city of Caracas.
Figure 9: System with interest rates (absolute change): Responses to a nominal housing credit supply shock Housing credit (log change)
Housing credit (log change), cumulative
0.05 0.25 0.04 0.2 0.03 0.15 0.02 0.1
0 −0.01 0
Housing markup (log change)
Housing markup (log change), cumulative
0.08 0.015 0.06 0.01 0.04 0.005 0.02 0
−0.005 0
−0.02 12 0
The left-hand panels display the quarterly responses of the nominal housing credit growth rate and the housing markup growth rate to a one-standard-deviation nominal credit supply shock, while the
right-hand panels show the cumulative responses of these objects.
Figure 9 (continued): System with interest rates (absolute change): Response to a nominal housing credit supply shock Change in nominal interest rate
Change in nominal interest rate, cumulative
−0.03 0
−0.04 0 12
The left-hand panel displays the quarterly response of the nominal interest rate to a onestandard-deviation nominal credit supply shock, while the right-hand panel shows the cumulative response of
this object. The nominal interest rate is the market rate through 2004 and the preferential interest rate beginning in 2005.
Figure 10: System with interest rates (absolute change): Responses to a residual interest rate shock Housing credit (log change)
Housing credit (log change), cumulative
0 −0.02 −0.005 −0.04 −0.01 −0.06 −0.015 −0.08
−0.025 −0.03 0
−0.12 0
Housing markup (log change)
Housing markup (log change), cumulative
0 0 −0.01 −0.005 −0.02 −0.01 −0.03 −0.015
−0.02 0
−0.05 0
The left-hand panels display the quarterly responses of the nominal housing credit growth rate and the housing markup growth rate to a one-standard-deviation residual nominal interest rate shock,
while the right-hand panels show the cumulative responses of these objects. The nominal interest rate is the market rate through 2004 and the preferential interest rate beginning in 2005.
Figure 10 (continued): System with interest rates (absolute change): Responses to a residual interest rate shock
Change in nominal interest rate
Change in nominal interest rate, cumulative
0.055 0.05
0.04 0.02 0.035 0.01 0.03 0
−0.02 −0.03 0
0.015 2
0.01 0
The left-hand panel displays the quarterly response of the nominal interest rate to a one-standard-deviation residual nominal interest rate shock, while the right-hand panel shows the cumulative
response of this object. The nominal interest rate is the market rate through 2004 and the preferential interest rate beginning in 2005.
Figure 11: System with exchange rates and oil prices: Responses to a nominal housing credit supply shock Housing credit (log change), cumulative
Housing credit (log change) 0.08
0.45 0.4
0.3 0.04 0.25 0.03 0.2 0.02
−0.01 0
Housing markup (log change)
x 10
Housing markup (log change), cumulative 0.16 0.14
0.12 0.1
10 0.08 0.06 5 0.04 0.02
0 −5 0
−0.02 12 0
The left-hand panels display the quarterly responses of the nominal housing credit growth rate and the housing markup growth rate to a one-standard-deviation nominal credit supply shock, while the
right-hand panels show the cumulative responses of the nominal housing credit growth rate and the housing markup growth rate.
Figure 11 (continued): System with exchange rates and oil prices: Responses to a nominal housing credit supply shock
Real oil price (log change)
Real oil price (log change), cumulative
0.01 0.05 0 0
−0.03 −0.1 −0.04
−0.05 0
−0.15 0 12
Shadow exchange rate (log change), cumulative
Shadow exchange rate (log change) 0.15
0.05 0 0 −0.01 −0.05 −0.02 −0.1 −0.03
−0.04 0
−0.2 0
The left-hand panels display the quarterly responses of the real oil price growth rate and the shadow exchange rate growth rate to a one-standard-deviation nominal credit supply shock, while the
right-hand panels show the cumulative responses of the real oil price growth rate and the shadow exchange rate growth rate.
Figure 12: System with exchange rates and oil prices: Responses to a residual oil price shock Housing credit (log change), cumulative
Housing credit (log change) 0.3
0.04 0.035
0.25 0.03 0.2
0.025 0.02
0.015 0.1
0.01 0.005
0 0 −0.005 −0.01 0
−0.05 0 12
Housing markup (log change)
Housing markup (log change), cumulative
0.06 0.005 0.04 0 0.02 −0.005
−0.01 0
−0.02 0 12
The left-hand panels display the quarterly responses of the nominal housing credit growth rate and the housing markup growth rate to a one-standard-deviation residual real oil price shock, while the
right-hand panels show the cumulative responses of the nominal housing credit growth rate and the housing markup growth rate.
Figure 12 (continued): System with exchange rates and oil prices: Responses to a residual oil price shock Real oil price (log change), cumulative
Real oil price (log change) 0.08
0.04 0.12 0.03 0.1 0.02 0.08
−0.01 −0.02 0
0.02 0 12
Shadow exchange rate (log change) 0.02
Shadow exchange rate (log change), cumulative 0.04 0.02
0.01 0 0
−0.02 −0.04
−0.01 −0.06 −0.02
−0.08 −0.1
−0.03 −0.12 −0.04 0
−0.14 12 0
The left-hand panels display the quarterly responses of the real oil price growth rate and the shadow exchange rate growth rate to a one-standard-deviation residual real oil price shock, while the
right-hand panels show the cumulative responses of the real oil price growth rate and the shadow exchange rate growth rate.
Figure 13: System with exchange rates and oil prices: Responses to a residual exchange rate shock Housing credit (log change)
Housing credit (log change), cumulative
0.01 0.05 0.005 0
0 −0.005
−0.01 −0.1
−0.015 −0.02
−0.025 −0.2 −0.03 −0.035 0
−0.25 0 12
Housing markup (log change)
Housing markup (log change), cumulative
0.005 −0.02 0 −0.04 −0.005 −0.06 −0.01 −0.08
−0.02 0
−0.1 0
The left-hand panels display the quarterly responses of the nominal housing credit growth rate and the housing markup growth rate to a one-standard-deviation residual shadow exchange rate shock,
while the right-hand panels show the cumulative responses of the nominal housing credit growth rate and the housing markup growth rate.
Figure 13 (continued): System with exchange rates and oil prices: Responses to a residual exchange rate shock Real oil price (log change), cumulative
Real oil price (log change) 0.06
0.02 0.02 0 0.01 −0.02 0 −0.04 −0.01
−0.02 0
−0.08 0 12
Shadow exchange rate (log change), cumulative
Shadow exchange rate (log change) 0.07
0.18 0.16
0.14 0.04 0.12 0.03 0.1 0.02 0.08 0.01
0 −0.01 0
0.02 0
The left-hand panels display the quarterly responses of the real oil price growth rate and the shadow exchange rate growth rate to a one-standard-deviation residual shadow exchange rate shock, while
the right-hand panels show the cumulative responses of the real oil price growth rate and the shadow exchange rate growth rate. | {"url":"https://p.pdfkul.com/the-effects-of-the-supply-of-credit-on-real-estate-prices-_5ad22c907f8b9a177a8b4573.html","timestamp":"2024-11-14T05:05:12Z","content_type":"text/html","content_length":"122663","record_id":"<urn:uuid:6c0ae2ca-24d7-4fea-822a-a83d0c63aa87>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00510.warc.gz"} |
Discrete Fourier Transform | Brilliant Math & Science Wiki
Discrete Fourier Transform
The discrete Fourier transform (DFT) is a method for converting a sequence of \(N\) complex numbers \( x_0,x_1,\ldots,x_{N-1}\) to a new sequence of \(N\) complex numbers,
\[ X_k = \sum_{n=0}^{N-1} x_n e^{-2\pi i kn/N}, \]
for \( 0 \le k \le N-1.\)
The \(x_i\) are thought of as the values of a function, or signal, at equally spaced times \(t=0,1,\ldots,N-1.\) The output \(X_k\) is a complex number which encodes the amplitude and phase of a
sinusoidal wave with frequency \(\frac kN\) cycles per time unit. \(\big(\)This comes from Euler's formula: \( e^{2\pi i kn/N} = \cos(2\pi kn/N) + i\sin(2\pi kn/N).\big)\) The effect of computing the
\(X_k\) is to find the coefficients of an approximation of the signal by a linear combination of such waves. Since each wave has an integer number of cycles per \(N\) time units, the approximation
will be periodic with period \(N.\) This approximation is given by the inverse Fourier transform
\[ x_n = \frac1{N} \sum_{k=0}^{N-1} X_k e^{2\pi ikn/N}. \]
The DFT is useful in many applications, including the simple signal spectral analysis outlined above. Knowing how a signal can be expressed as a combination of waves allows for manipulation of that
signal and comparisons of different signals:
• Digital files (jpg, mp3, etc.) can be shrunk by eliminating contributions from the least important waves in the combination.
• Different sound files can be compared by comparing the coefficients \(X_k\) of the DFT.
• Radio waves can be filtered to avoid "noise" and listen to the important components of the signal.
Other applications of the DFT arise because it can be computed very efficiently by the fast Fourier transform (FFT) algorithm. For example, the DFT is used in state-of-the-art algorithms for
multiplying polynomials and large integers together; instead of working with polynomial multiplication directly, it turns out to be faster to compute the DFT of the polynomial functions and convert
the problem of multiplying polynomials to an analogous problem involving their DFTs.
Let \( x_0 = 1,\) \( x_1 = x_2 = \cdots =x_{N-1} = 0.\) Then the DFT of the \(x_n\) is \[ X_k = \sum_{n=0}^{N-1} x_n e^{-2\pi i k n/N} = 1. \] So this gives an expression of \( x_n\) as \[ x_n =
\frac1{N} \sum_{k=0}^{N-1} e^{2\pi i kn /N}. \] Check: When \( n=0,\) this returns \(x_0 = \frac1{N}\cdot N = 1,\) and when \(n \ne 0\) this returns \[ x_n &= \frac1{N} \sum_{k=0}^{N-1} \big(e^{2
\pi in/N}\big)^k \\\\ &= \frac1{N}\frac{\big(e^{2\pi in/N}\big)^N-1}{e^{2\pi in/N}-1} \\\\ &= \frac1{N}\frac{e^{2\pi i n} - 1}{e^{2\pi in/N}-1} \\\\ &= 0. \]
So the DFT gives a breakdown of a "spike" into a sum of waves (equally weighted in this case), which all peak at \(t=0,\) but interfere with each other and cancel out perfectly at other integer time
values \( < N.\)
If the spike occurs at a different time, the coefficients change:
Find the DFT of \((x_0,x_1,x_2,x_3) = (0,1,0,0).\)
In this case, \[ X_k = \sum_{n=0}^3 x_n e^{-2\pi i kn/4} = e^{-2\pi i k/4}. \] So \( X_0 = 1, X_1 = -i, X_2 = -1, X_3 = i.\) The answer is \( (1,-i,-1,i).\)
Proceeding as in the previous example, this gives the expansion \[ x_n = 1 - i e^{2\pi i n/4} - e^{4\pi i n/4} + i e^{6\pi i n/4}. \] Converting the complex coefficients to complex exponentials
gives \[ x_n &= 1 + e^{6\pi i/4} e^{2\pi i n/4} + e^{4\pi i/4} e^{4\pi i n/4} + e^{2\pi i/4} e^{6 \pi i n/4} \\ &= 1 + e^{2\pi i(n+3)/4} + e^{2\pi i (2n+2)/4} + e^{2\pi i (3n+1)/4}. \] This is an
example of phase shifting occurring in the sum. Taking the real parts of both sides gives a sum of cosine waves: \[ x_n = 1 + \cos(2\pi n/4 + 3\pi/2) + \cos(4\pi n/4 + \pi) + \cos(6\pi n/4 + \pi/
2), \] where the addition of \(3\pi/2, \pi, \pi/2\) has the effect of shifting the waves forward by \( 270^\circ, 180^\circ, 90^\circ,\) respectively. \(_\square\)
Orthogonality and the Inverse Transform
Why is the formula for the inverse transform true? Substitute the formula for \( X_k\) into the formula for \( x_n\): \[ \sum_{k=0}^{N-1} X_k e^{-2\pi ikn/N} &= \frac1{N} \sum_{k=0}^{N-1} \sum_{m=0}^
{N-1} x_m e^{2\pi i k m/N} e^{-2\pi ikn/N} \\ &= \frac1{N} \sum_{k=0}^{N-1} \sum_{m=0}^{N-1} x_m e^{2\pi i k(m-n)/N} \\ &= \frac1{N} \sum_{m=0}^{N-1} x_m \sum_{k=0}^{N-1} e^{2\pi i k(m-n)/N}. \] When
\(m \ne n,\) the inner sum is \(0\) by the formula for a geometric series (as in the first example in the previous section). When \(m=n,\) the inner sum is \( N.\) So the entire sum is \( \frac1{N} \
cdot x_n \cdot N = x_n,\) as desired.
Another way to think of this argument is that it is a consequence of orthogonality with respect to the complex dot product. Consider the complex vectors \[ v_k = \left( 1, \omega_N^k, \omega_N^{2k},
\ldots, \omega_N^{k(N-1)} \right), \] where \(\omega_N = e^{2\pi i/N}.\) It is not hard to check, by an argument similar to the one given above, that these complex vectors are orthogonal with respect
to the complex dot product (or inner product): \( v_k \cdot v_\ell = N\) if \(k=\ell\), otherwise \(0.\) \(\big(\)Remember that the complex dot product of two vectors \( (x_i)\) and \( (y_i)\) is \(\
sum x_i{\overline{y_i}},\) where the bar denotes complex conjugation.\(\big)\) There are \(N\) of these vectors, so they form an orthogonal basis of \( {\mathbb C}^N.\)
The DFT formula for \( X_k \) is simply that \(X_k = x \cdot v_k,\) where \(x\) is the vector \( (x_0,x_1,\ldots,x_{N-1}).\) The inverse formula recovers \(x\) in terms of \(X\), by writing \(x\)
using the standard formula for expressing any vector as a linear combination of orthogonal basis vectors: \[ x = \sum_{k=0}^{N-1} \frac{x \cdot v_k}{v_k \cdot v_k} v_k = \frac1{N} \sum_{k=0}^{N-1}
X_k v_k. \]
Convolution and Polynomial Multiplication
As a sample application of the DFT, consider polynomial multiplication.
Given two polynomials \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0\) and \(g(x) = b_mx^m + b_{m-1}x^{m-1} + \cdots + b_1 x + b_0,\) call the coefficient vectors \( (a_0,a_1,\cdots)\) and
\( (b_0,b_1,\cdots)\) \(\bf a\) and \( \bf b\), respectively. The product \( f(x)g(x)\) will have degree \(n+m\) with the coefficients given by the convolution vector \( {\bf a} * {\bf b},\) with
\[ ({\bf a} * {\bf b})_k = \sum_{c=0}^k a_c b_{k-c}. \]
It is convenient to extend the vectors \( \bf a\) and \( \bf b\) to a common space \( {\mathbb C}^N\) by padding them with extra \(0\)s: take a value of \(N\) larger than \(m+n,\) and let \( a_x =
b_y = 0\) if \(x\) or \(y\) is larger than \(n\) or \(m\), respectively. (For FFT applications it is often best to let \(N\) be a power of 2.) Then the beautiful fact about convolution and the DFT is
The DFT of \( {\bf a} * {\bf b} \) is the componentwise product of the DFT of \( \bf a \) and the DFT of \( \bf b\).
The proof of this fact is straightforward and can be found in most standard references.
So multiplying \(f(x)\) and \(g(x)\) can be accomplished by padding the coefficient vectors, computing their DFTs, multiplying the DFTs, and applying the inverse DFT to the result.
Find \( (1+x)\big(1+x+x^2\big)\) using the DFT.
Pad the coordinate vectors to \( (1,1,0,0) \) and \( (1,1,1,0).\) The DFT of \((1,1,0,0)\) is \( (2,1-i, 0, 1+i) \) \(\big(\)this follows from the two examples above, since the DFT is additive
and we know the DFTs of \( (1,0,0,0) \) and \( (0,1,0,0)\big),\) and the DFT of \( (1,1,1,0) \) is \( (3,-i,1,i).\) The coordinatewise product of these two is \( (6,-1-i,0,-1+i),\) and the
inverse DFT of this vector is \( (1,2,2,1).\) So the product is \( 1+2x+2x^2+x^3.\) \(_\square\)
This may seem like a roundabout way to accomplish a simple polynomial multiplication, but in fact it is quite efficient due to the existence of a fast Fourier transform (FFT). The point is that a
normal polynomial multiplication requires \( O(N^2)\) multiplications of integers, while the coordinatewise multiplication in this algorithm requires only \( O(N)\) multiplications. The FFT algorithm
is \( O(N \log N),\) so the polynomial multiplication algorithm which uses the FFT is \( O(N \log N)\) as well.
Multiplying large integers is done in the same way: an integer like \( 1408\) can be viewed as the evaluation of the polynomial \( 8 + 4x^2+x^3\) at \(x=10,\) so multiplying integers can be reduced
to multiplying polynomials (and then evaluating at 10, or whatever base is most convenient). | {"url":"https://brilliant.org/wiki/discrete-fourier-transform/","timestamp":"2024-11-05T15:14:04Z","content_type":"text/html","content_length":"51018","record_id":"<urn:uuid:741069d7-4e24-4a35-8cc0-3c158b379d73>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00459.warc.gz"} |
How do you calculate work done in joules?
How do you calculate work done in joules?
Work can be calculated with the equation: Work = Force × Distance. The SI unit for work is the joule (J), or Newton • meter (N • m). One joule equals the amount of work that is done when 1 N of force
moves an object over a distance of 1 m.
How do u calculate force?
Learning the Formula. Multiply mass times acceleration. The force (F) required to move an object of mass (m) with an acceleration (a) is given by the formula F = m x a. So, force = mass multiplied by
What is the work done formula?
Scientifically Work done formula will be given as, W = F * d: In this case, the force exerting on the block is constant, but the direction of force and direction of displacement influenced by this
force is different.
How do I calculate potential energy?
Simplified, this formula can be written as: Potential Energy = mgh, where m is the mass, measured in kilograms; g is the acceleration due to gravity (9.8 m/s^2 at the surface of the Earth); and h is
the height, measured in meters.
How do you calculate joules in electricity?
Electric energy = P × t = V × I × t = I2 × R × t = V2t / R.
1. S.I unit of electric energy is joule (denoted by J), where 1joule = 1watt × 1 second = 1volt × 1ampere × 1second.
2. Commercial unit of electric energy is kilowatt-hour (kWh), where 1kWh = 1000 Wh = 3.6 ×106J = one unit of electric energy consumed.
How do you calculate watts from Joules?
Watts are defined as 1 Watt = 1 Joule per second (1W = 1 J/s) which means that 1 kW = 1000 J/s. A Watt is the amount of energy (in Joules) that an electrical device (such as a light) is burning per
second that it’s running. So a 60W bulb is burning 60 Joules of energy every second you have it turned on.
What is 50n in KG?
5.0985810649 kgf
Newton to Kilogram-force Conversion Table
Newton [N] Kilogram-force [kgf]
20 N 2.039432426 kgf
50 N 5.0985810649 kgf
100 N 10.1971621298 kgf
1000 N 101.9716212978 kgf
How do you convert kilograms to Newtons?
N is the force in Newton. Kg is the mass in kilograms….Kg and Newton.
Kg to Newton 1 kg = 9.81 N
Newton to kg 1N = 0.10197 kg
What is the formula of power Class 10?
P = E/t: This formula is also called the mechanical power equation. Here E stands for energy in joule and t stands for time in seconds. This formula states that the consumption of energy per unit of
time is called power.
How do you calculate work energy and power?
We define the capacity to do the work as energy. Power is the work done per unit of time….Overview of Work, Energy and Power.
What is Work, Energy and Power?
Formula The formula for power is P = W/t
Unit The SI unit of power is Watt (W).
What is potential energy class 10?
Potential energy is the energy possessed by a body because of its position relative to other bodies. Gravitational potential energy is the energy held by a body when it is at some height relative to
the zero potential level ground.
What is potential difference class 10th?
Answer: Potential difference between any two points is defined as the amount of work done in moving a unit charge from one point to another. dv=dwdq.
Is WS-10B engine used in J-10C?
The J-10C uses a diverterless air intake. In 2021, China begun retrofitting J-10 fighter jets with WS-10B engine. Chinese broadcaster CCTV has shown a J-10C fighter with Peoples Liberation Army Air
Force emblem fitted with WS-10B engine.
Why was the J-10 so difficult to develop?
The Gulf War renewed interest and brought adequate resourcing. Unlike earlier programs, the J-10 avoided crippling requirement creep. Technical development was slow and difficult. The J-10
represented a higher level of complexity than earlier generations of Chinese aircraft.
What kind of radar does the J-10 use?
According to Chengdu Aircraft Industry Corporation officials the J-10 uses a multi-mode fire-control radar designed in China. The radar has a mechanically scanned planar array antenna and is capable
of tracking 10 targets.
Is this the WS-10B Taihang Engine on the J-10C Vigorous Dragon?
According to the images posted by China National Radio of a PLAAF live-firing exercise at an unspecified location in May 2021, J-10C Vigorous Dragons were equipped with distinctive exhaust nozzles of
the WS-10B Taihang turbofan engine. This marks the first time the WS-10 has been officially seen on an operational J-10. | {"url":"https://heimduo.org/how-do-you-calculate-work-done-in-joules/","timestamp":"2024-11-09T03:50:34Z","content_type":"text/html","content_length":"139767","record_id":"<urn:uuid:012c5b4f-6efa-495b-9f46-4335898ef8c3>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00746.warc.gz"} |
Master the Slitherlink Puzzle: Tips, Tricks, and Strategies
Mastering the Slitherlink Puzzle – A Guide to Solving the Challenging Brainteaser
Ready to embark on an exciting mental challenge? Look no further than the captivating world of Slitherlink! This intricate puzzle, often referred to as the Loop Link or Fence Puzzle, will put your
logical thinking and problem-solving abilities to the test. With its grid of interconnected lines and numbers, Slitherlink offers a unique and stimulating experience that will keep you engaged for
Slitherlink revolves around creating a loop of lines that traverse the grid, adhering to specific rules. The goal is to enclose each numbered cell within the loop, while ensuring that the number
reflects the total number of adjacent lines that are part of the loop. Sounds simple, right? Well, things quickly become more complex as you progress to larger grids and encounter various additional
puzzle constraints.
Immerse yourself in the challenging world of Slitherlink, and let your mind unravel the mysteries hidden within each puzzle. Sharpen your analytical thinking, as well as your ability to deduce
patterns and logical connections. With each solved puzzle, you’ll gain a deeper understanding of the strategies and techniques that will allow you to conquer even the most mind-boggling of
Slitherlink grids.
Understand the Basics of Slitherlink Puzzle
In this section, we will delve into the fundamental concepts of the slitherlink puzzle. To successfully navigate this perplexing game, it is essential to grasp the puzzle’s core mechanics and
familiarize yourself with key terms like numbers, loops, links, and fences. By understanding these basics, you will be well-equipped to approach more complex challenges with confidence and ease.
Numbers play a crucial role in the slitherlink puzzle. These digits are strategically placed throughout the grid, acting as clues to guide you towards the solution. Each number indicates the exact
number of fence segments that need to be drawn around it. By analyzing these numbers and their relationships with neighboring digits, you can start to unravel the paths and connections that will form
the completed loop.
Loops are the central component of the slitherlink puzzle. They represent the continuous closed lines that wind their way across the grid, connecting the numbered cells. Your goal is to create a
single loop that respects all the given clues while efficiently covering as much of the grid as possible. Paying attention to the loop’s shape and considering potential branching points are key
strategies to overcome challenges as they arise.
Links in the slitherlink puzzle refer to the connections or bridges that form between two adjacent cells. These links are essential for creating a continuous loop and ensuring that all segments of
fence are placed correctly. Careful analysis of the numbers and the existing loop is required to determine where these links should be placed. Recognizing when and how to establish links is a
critical skill that will aid you in advancing through the puzzle with precision.
Fences are the segments that make up the loop, forming a barrier between cells or enclosing certain areas. Creating an unbroken fence around each numbered cell is necessary to satisfy the given clue
and complete the puzzle. However, not all cells will have fences surrounding them, and determining which cells require fences and which do not is a key aspect of solving the slitherlink puzzle.
By gaining a solid understanding of these basic elements of the slitherlink puzzle, you will be well on your way to honing your skills and tackling more complex challenges. Remember, the key to
success is to approach each puzzle with patience, logical thinking, and a keen eye for detail. Happy solving!
Start with the Easy Puzzles
If you are new to the intriguing world of Slitherlink puzzles, it is recommended to begin with the easier ones. Starting with these less challenging puzzles will allow you to grasp the basic concepts
and strategies of solving Slitherlink puzzles without feeling overwhelmed.
When you encounter an easy Slitherlink puzzle, the main objective is to create a closed loop by connecting the dots on the grid. Each number on the grid represents how many of its adjacent edges are
part of the loop. By strategically linking the dots and forming the loop, you can solve the puzzle successfully.
As you tackle the easy Slitherlink puzzles, pay attention to the numbers on the grid. They serve as hints and guidelines for determining the correct link between the dots. Start by identifying the
numbers with only one adjacent edge and their corresponding dots. This will help you establish crucial links and reinforce the loop.
In addition, focus on finding the areas of the grid where the loop can be extended without violating the constraints imposed by the numbers. By gradually expanding the loop and making logical
deductions, you will progress towards completing the puzzle.
• Observe the numbers carefully to identify the dots with multiple adjacent edges.
• Consider the possible combinations of linking the dots, ensuring that the loop remains continuous and without any self-intersections.
• Use logical reasoning to eliminate incorrect links and narrow down the possibilities.
• Make use of the spaces within the grid to create separate loops that will aid in solving the puzzle.
Remember, starting with the easy puzzles will familiarize you with the fundamental techniques required to solve more challenging Slitherlink puzzles in the future. So, take your time, sharpen your
skills, and enjoy the journey of becoming a Slitherlink master!
Use the Given Numbers Wisely
When solving a slitherlink puzzle, the given numbers play a crucial role in guiding you towards the correct solution. It is essential to use these numbers wisely in order to create valid links and
loops within the puzzle grid. By carefully analyzing the given numbers, you can strategically place the necessary fences to connect the dots and form a coherent puzzle solution.
Utilize the Provided Clues:
Each number in a slitherlink puzzle represents the number of adjacent edges that need to be included in the loop. By taking into account these clues, you can make informed decisions on where to place
fences and where to leave open spaces. Remember, a number signifies a link that must be included, so it is important to find the right connections based on the given numbers.
Consider the Possibilities:
Keep in mind that there can be multiple valid solutions for a slitherlink puzzle. As you progress, you may encounter situations where different interpretations of the given numbers can lead to
distinct configurations. It is crucial to explore all possible options and visualize the potential links and loops that can be formed to determine the most logical and efficient solution.
Using the given numbers wisely is the key to successfully solving a slitherlink puzzle. By thoroughly assessing the clues, considering all possible outcomes, and strategically placing fences, you can
uncover the correct solution and master this challenging puzzle.
Look for Patterns and Connections
Discovering patterns and connections within the Slitherlink puzzle is crucial to mastering this challenging and engaging game. By identifying the relationships between the fences, numbers, and
possible links, you can solve the puzzle with greater efficiency and accuracy.
Identifying Number Patterns
One of the first steps in solving a Slitherlink puzzle is to carefully analyze the numbers provided. Look for patterns in the distribution of the numbers as well as their relationships to the
neighboring fences. Identifying patterns such as consecutive numbers, ascending or descending sequences, or symmetry can provide valuable clues towards solving the puzzle.
Exploring Link Connections
Examining the potential links between the numbers and the surrounding fences is another crucial strategy. As you progress in the puzzle, you will notice that certain numbers can only be connected to
specific fences, while others offer multiple possibilities. By analyzing the connections and ruling out the impossible links, you can narrow down the potential solutions and ultimately reach the
correct solution.
Pattern Description
Consecutive Numbers Numbers that appear consecutively suggest that the fences adjacent to them must form a continuous line.
Symmetry If the puzzle exhibits symmetry, the fences and numbers on one side can reflect identically on the other side, aiding in solving the puzzle.
Ascending/Descending Sequences Numbers arranged in ascending or descending order indicate a specific pattern in the link connections.
By carefully observing and analyzing these patterns and connections, you can elevate your Slitherlink puzzle-solving skills to the next level. Keep practicing and hone your ability to identify these
crucial elements within the game, and soon enough, you’ll become a true Slitherlink master!
Break the Puzzle into Smaller Sections
When tackling a Slitherlink puzzle, one valuable strategy is to break the puzzle into smaller sections. By doing so, you can focus on solving one section at a time, making the overall puzzle more
Instead of trying to solve the entire puzzle in one go, start by identifying specific boundaries or ‘fences’ within the grid. These fences define the loops that need to be created, and by breaking
the puzzle into smaller sections, you can better visualize where these loops should go.
Begin by examining the numbers provided in the puzzle. Each number represents the cumulative count of links that need to be connected around it. By identifying these numbers and their corresponding
areas, you can start dividing the puzzle into achievable sections.
Divide and Conquer
Once you have identified the areas that need to be connected, focus on one section at a time. Look for clues such as nearby numbers or existing links, which can help guide your decisions. By
concentrating your efforts on smaller sections, you can gradually build the loops and connect the numbers, reducing the complexity of the puzzle as a whole.
Developing a System
As you break the puzzle into smaller sections, it’s essential to develop a systematic approach. Create a mental or physical map of the puzzle and mark the areas you have solved to avoid confusion.
Additionally, consider keeping track of the links you have made to prevent errors and ensure a logical progression through the puzzle.
Breaking the Slitherlink puzzle into smaller sections not only increases your chances of success but also enhances your problem-solving skills. With a clear focus on manageable areas, you can
strategically link numbers, solve loops, and ultimately conquer the entire puzzle.
Loop the Loop Puzzle: Connecting the Dots
Are you ready to embark on a new challenge? Introducing the loop the loop puzzle, a captivating game that will put your strategic thinking to the test. In this engaging brain teaser, your goal is to
connect the dots in a loop by drawing lines according to a set of rules and numbers. No worries, it’s not just any ordinary puzzle – it’s a thrilling experience that will keep you engaged for hours.
To solve this intriguing puzzle, you’ll need to use your logical reasoning skills and attention to detail. Each puzzle consists of a grid with various numbered cells. Your task is to create a loop by
connecting adjacent dots in a way that satisfies the provided numeric clues. The challenge lies in figuring out the exact path of the loop, ensuring it doesn’t intersect or overlap with itself.
The link between the numbers and the loop is the key to solving these puzzles. The numbers indicate how many of the surrounding edges of a cell should be part of the loop. For example, a cell with
the number 2 means that two of its edges should be included in the loop. By strategically analyzing the numbers and their positioning, you’ll be able to deduce the path of the loop, gradually filling
in the grid and completing the puzzle.
Get ready to exercise your mind with the loop the loop puzzle. With its challenging gameplay and intricate designs, it’s an addictive game that will keep you coming back for more. So, sharpen your
strategic thinking skills, flex your mental muscles, and dive into the world of loop the loop puzzles. Happy connecting!
Begin by Connecting the Dots
When starting a slitherlink puzzle, one of the first steps is to connect the dots. This simple act sets the foundation for solving the puzzle and reveals the hidden paths that form the loop. By
strategically linking adjacent dots, you can start to uncover the dynamics of the puzzle.
Remember, the goal of a slitherlink puzzle is to create a single loop that passes through all the dots. Each number inside a cell represents how many of its surrounding borders are part of the loop.
By carefully connecting the dots and considering the numbers, you can gradually construct an interconnected web of fences.
One strategy is to start by connecting dots that have the lowest numbers adjacent to them. These dots are more likely to have a smaller number of fences connected to them, allowing for a more
straightforward path to be established. Following this approach helps to build a preliminary structure and provides a starting point for further deductions.
Another tactic is to pay attention to the relationship between neighboring cells. If two adjacent cells have the same number, it often means that they share the same number of fences. By connecting
the dots between these cells, you can eliminate the possibility of additional fences and create a more efficient loop.
Furthermore, be aware of dots that are located near the border of the puzzle grid. These dots often have fewer adjacent cells, limiting the number of fences they can be connected to. By identifying
these dots and linking them appropriately, you can reduce the number of possibilities and make progress towards solving the puzzle.
Begin by connecting the dots – it may seem like a simple step, but it forms the basis for solving the slitherlink puzzle. By carefully considering the numbers, the relationships between cells, and
the position of dots, you can gradually uncover the correct path for the loop. So grab a pen and start connecting the dots to tackle this captivating puzzle!
Look for Symmetry and Alignment
When solving a Slitherlink puzzle, one useful strategy is to look for symmetry and alignment in the placement of the numbers and the connectivity of the links.
Symmetry can often provide clues about the correct placement of numbers and the arrangement of the fences. For example, if you notice that there is a symmetrical pattern emerging on one side of the
puzzle, you can use that information to deduce the placement of numbers on the other side. Additionally, symmetry can help you identify potential loops and eliminate incorrect possibilities.
Alignment refers to the positioning of numbers and links in a straight line or in a pattern. When you find alignment in a Slitherlink puzzle, it can be a valuable hint for solving the puzzle. For
instance, if you see two numbers in a row with a link connecting them, it indicates that the adjacent fences must connect to those numbers. Similarly, if you notice a vertical alignment of numbers in
a column, it implies that the corresponding fences must be linked accordingly.
By paying attention to symmetry and alignment, you can gain insights into the overall structure of the puzzle and make progress in solving the Slitherlink. These observations help you form a logical
approach towards identifying correct number placement and determining the correct links to create the desired loop.
Use Trial and Error
When attempting to solve a Slitherlink puzzle, one effective strategy to consider is employing the method of trial and error. This approach involves systematically trying different combinations of
fences and numbers to determine the correct placement for each element within the puzzle grid. By using trial and error, solvers can explore various possibilities and gradually narrow down the
potential solutions until they reach the correct configuration of the loop and its connecting links.
To begin with, it is essential to identify the clues provided by the numbers in the puzzle. These numbers indicate the exact number of fences that should surround them. By analyzing the adjacent
cells and examining the layout of the overall puzzle, you can strategically deduce where the fences should be placed. It is important to remember that each number should have the exact number of
fences surrounding it, and any excess or fewer fences will result in an incorrect solution.
As you progress through the Slitherlink puzzle, you may encounter situations where there are conflicting or ambiguous clues. In such cases, trial and error becomes particularly useful. By making
educated guesses and trying different combinations of fences and numbers, you can test the validity of your assumptions and eliminate incorrect possibilities. It is crucial to keep track of your
trials and remember to go back and undo any incorrect placements to maintain the integrity of your puzzle-solving progress.
During the trial and error process, it can be helpful to focus on areas that provide the most significant impact on the puzzle’s overall solution. Identifying critical points where changes in the
loop and link placements can lead to drastic alterations in the rest of the puzzle will help guide your trial and error efforts effectively. By strategically selecting specific locations to test
different configurations, you can efficiently narrow down the correct solution.
While using trial and error, it is essential to approach the puzzle-solving process with patience and persistence. Solving a Slitherlink puzzle requires careful analysis and a keen eye for patterns
and logical deductions. By combining trial and error with other strategies and techniques, such as identifying forced placements or making implications from adjacent clues, you can enhance your
ability to master even the most challenging Slitherlink puzzles.
Number Link Puzzle: Connect the Numbers
In this section, we will explore the fascinating world of the Number Link Puzzle, a captivating game that challenges your ability to connect numbers by creating loops with fences. Just like in the
renowned Slitherlink puzzle, you’ll need to use your logical thinking and problem-solving skills to successfully complete each puzzle and create a continuous loop that connects all the given numbers.
The Fundamentals of the Number Link Puzzle
The Number Link Puzzle is a grid-based game where you are presented with various numbers scattered across the board. Your goal is to connect these numbers by drawing lines, or fences, in a way that
forms a single loop. Each number represents an endpoint for the loop, and you must ensure that all endpoints are connected in the final solution.
A key challenge in the Number Link Puzzle is that the fences cannot cross or overlap with each other. Additionally, the loop cannot contain any branches or dead ends. It must be a continuous path
that connects all the numbers without any interruptions or breaks.
Strategies for Solving the Number Link Puzzle
To successfully solve the Number Link Puzzle, it is important to approach it with a systematic strategy. Here are a few tips and techniques that can help you navigate through the puzzles:
• Start with the obvious: Look for numbers that have only one possible connection and draw the corresponding fence. This can help you establish anchor points in the puzzle.
• Eliminate possibilities: Identify numbers that have limited options for connections. By deducing the available paths, you can eliminate incorrect options and narrow down the possible solutions.
• Make use of intersections: When multiple fences intersect, consider the possible outcomes and choose the option that leads to the most logical and consistent solution.
• Create sections: Divide the puzzle into manageable sections by isolating groups of numbers. Focus on solving each section individually, ensuring that the loop connects all the endpoints within
that specific group.
By adopting these strategies and practicing regularly, you’ll enhance your skills in solving the Number Link Puzzle and conquer even the most challenging grids. Remember to approach each puzzle with
patience and a methodical approach, and soon you’ll be unraveling the mysteries of this captivating game.
Start with the Number 1
The first step to solving a slitherlink puzzle is to locate the number 1. This initial number serves as the starting point for building the puzzle’s loop of fences.
Begin by scanning the puzzle grid for the number 1, which represents the total number of fence segments required to surround the adjacent squares. Once you have identified the number 1, you can
proceed to link the fences in a way that creates a closed loop.
Keep in mind that each fence segment can only be connected to two other fence segments. As you start establishing the connections from the number 1, look for adjacent squares with higher numbers, as
they will indicate the number of fence segments that need to be connected to those particular squares.
By starting with the number 1 and expanding from there, you can gradually form the loop of fences that satisfies the number clues provided in the puzzle. This approach allows you to strategically
plan the connections and carefully link the fences, ensuring a successful and logical solution.
Link the fences Create a loop Surround adjacent squares
Connect fence segments Establish connections Identify higher numbers
Form a closed loop Plan strategically Ensure a logical solution
Avoid Creating Separate Loops
When solving a Slitherlink puzzle, it is crucial to prevent the formation of multiple independent loops. Instead, aim to create a single continuous loop that connects all the given numbers in the
grid. By avoiding separate loops, you can ensure a more logical and coherent solution.
Understand the Concept of Loops
In Slitherlink, loops are formed by connecting adjacent cells with fences. The numbers in the grid indicate how many sides of the cell should be surrounded by fences. It is essential to grasp the
concept of loops as it lays the foundation for solving the puzzle efficiently. The goal is to connect all the numbers to form one loop without any isolated or disconnected areas.
Identify Possible Loop Divisions
During the solving process, it is common to encounter opportunities where separate loops may be inadvertently created. These situations often arise when connecting cells with different numbers or
when trying to resolve conflicting clues. By carefully considering the potential consequences of each move, you can avoid dividing the main loop and maintain the integrity of your solution.
Fences Puzzle: Connect the Islands
In the realm of number-based logic puzzles, one that stands out is the Fences Puzzle, also known as “Connect the Islands.” This captivating game challenges you to create a loop by connecting numbered
islands, while following specific rules.
Similar to the Slitherlink puzzle, the Fences Puzzle requires you to form a continuous loop. However, instead of drawing lines to create a loop around the entire grid, in this puzzle, you must
connect the numbered islands with fences to form a closed loop.
The objective is to create a loop that passes through each numbered island, fulfilling the required number of connected fences. Each number on an island indicates how many fence segments should be
connected to it. Additionally, the loop cannot intersect itself, and it must form a single closed loop without any branches or dead ends.
To solve the Fences Puzzle efficiently, it’s important to analyze the given clues and visualize potential connections. Start by identifying islands with the highest number of fences connected to
them, as they will play a crucial role in forming the loop. The loop must pass by these islands to fulfill their fence requirements.
While solving the puzzle, remember that the loop can only make right-angled turns and cannot cross over itself. Utilize deductive reasoning to eliminate potential connections and determine the
correct path for the loop. Be mindful of islands with neighboring fences already connected, as they limit the available options for the loop.
Logic and strategic thinking are key to mastering the Fences Puzzle. Carefully consider the constraints of each island when connecting the fences, and continue to analyze the grid to make informed
decisions. With practice and perseverance, you’ll become adept at navigating the intricate web of fences to create a complete loop and conquer this challenging puzzle. | {"url":"https://triogical.com/blog/mastering-the-slitherlink-puzzle-a-guide-to-solving-the-challenging-brainteaser","timestamp":"2024-11-12T06:12:54Z","content_type":"text/html","content_length":"67933","record_id":"<urn:uuid:6256df93-48b0-4bed-98bf-2614aec9ad59>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00458.warc.gz"} |
Worksheet Multiplication By 2
Math, especially multiplication, forms the foundation of various scholastic techniques and real-world applications. Yet, for several students, mastering multiplication can position a challenge. To
resolve this difficulty, educators and parents have accepted an effective tool: Worksheet Multiplication By 2.
Intro to Worksheet Multiplication By 2
Worksheet Multiplication By 2
Worksheet Multiplication By 2 -
Here is 2 times table worksheets PDF a simple an enjoyable set of x2 times table suitable for your kids or students Multiplying by 2 activities is giving to help children to practice their
multiplication skills Times Table worksheets 1 times table worksheets 2 times table worksheets 3 times table worksheets 4 times table worksheets
Welcome to The Multiplying 2 Digit by 2 Digit Numbers A Math Worksheet from the Long Multiplication Worksheets Page at Math Drills This math worksheet was created or last revised on 2021 02 17 and
has been viewed 8 714 times this week and 10 384 times this month
Value of Multiplication Method Recognizing multiplication is critical, laying a solid structure for advanced mathematical principles. Worksheet Multiplication By 2 supply structured and targeted
method, cultivating a much deeper understanding of this essential arithmetic operation.
Advancement of Worksheet Multiplication By 2
Multiplication Worksheets For 2nd Graders Free With No Login MathWorksheets
Multiplication Worksheets For 2nd Graders Free With No Login MathWorksheets
Our multiplication worksheets start with the basic multiplication facts and progress to multiplying large numbers in columns We emphasize mental multiplication exercises to improve numeracy skills
Choose your grade topic Grade 2 multiplication worksheets Grade 3 multiplication worksheets Grade 4 mental multiplication worksheets
Multiplying by two is an important math skill Kids in third grade or anyone starting to learn their multiplication facts will get targeted multiplication practice with this worksheet First math
students will practice a series of one digit multiplication problems and then they will fill in a simple multiplication chart for the number 2
From conventional pen-and-paper workouts to digitized interactive layouts, Worksheet Multiplication By 2 have advanced, dealing with varied learning styles and choices.
Types of Worksheet Multiplication By 2
Standard Multiplication Sheets Simple workouts focusing on multiplication tables, helping learners develop a solid math base.
Word Issue Worksheets
Real-life situations incorporated right into troubles, boosting important reasoning and application skills.
Timed Multiplication Drills Examinations made to enhance speed and precision, helping in rapid mental mathematics.
Benefits of Using Worksheet Multiplication By 2
The Multiplying 2 Digit by 2 Digit Numbers Multiplication worksheets Multiplication Math
The Multiplying 2 Digit by 2 Digit Numbers Multiplication worksheets Multiplication Math
Multiplication by 2s Print worksheets for teaching students to multiply single digit numbers by the number 2 This page has printable flash cards practice worksheets math sliders timed tests puzzles
and task cards Multiply By 2s Only Learn to Multiply by 2s FREE
Worksheets Multiply by 2 s Multiply by 2 s Lunita2 Member for 3 years 5 months Age 7 9 Level Grade 2 3 Language English en ID 149870 03 05 2020 Country code US Country United States School subject
Math 1061955 Main content Multiplication 2013181 One digit multiplication by 2 Share Print Worksheet Finish
Enhanced Mathematical Abilities
Consistent method hones multiplication efficiency, improving total mathematics capacities.
Enhanced Problem-Solving Abilities
Word troubles in worksheets create analytical thinking and method application.
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Worksheets accommodate specific knowing rates, fostering a comfortable and versatile knowing environment.
Just How to Produce Engaging Worksheet Multiplication By 2
Incorporating Visuals and Shades Vivid visuals and shades record attention, making worksheets visually appealing and engaging.
Consisting Of Real-Life Scenarios
Connecting multiplication to day-to-day scenarios includes significance and practicality to workouts.
Customizing Worksheets to Various Skill Degrees Tailoring worksheets based upon differing effectiveness degrees makes certain comprehensive knowing. Interactive and Online Multiplication Resources
Digital Multiplication Equipment and Games Technology-based resources provide interactive discovering experiences, making multiplication interesting and satisfying. Interactive Internet Sites and
Applications Online systems supply varied and easily accessible multiplication method, supplementing standard worksheets. Customizing Worksheets for Different Discovering Styles Aesthetic Students
Aesthetic help and diagrams aid comprehension for students inclined toward visual learning. Auditory Learners Verbal multiplication problems or mnemonics deal with learners that realize principles
through acoustic ways. Kinesthetic Learners Hands-on activities and manipulatives sustain kinesthetic learners in understanding multiplication. Tips for Effective Implementation in Learning
Uniformity in Practice Normal technique reinforces multiplication abilities, promoting retention and fluency. Stabilizing Rep and Selection A mix of repetitive exercises and varied issue layouts
preserves interest and comprehension. Offering Useful Comments Comments help in determining locations of enhancement, motivating continued progression. Challenges in Multiplication Method and
Solutions Motivation and Engagement Hurdles Boring drills can bring about uninterest; innovative techniques can reignite inspiration. Getting Over Worry of Math Adverse assumptions around math can
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Multiplying 1 To 12 by 2 A
Fun Double Digit Multiplication Worksheets Best Kids Worksheets
Check more of Worksheet Multiplication By 2 below
Printable 2 Times Table Worksheets Activity Shelter
2 Digit By 2 Digit Multiplication Worksheets With Answers Free Printable
Multiplication Sheets 4th Grade
Double Digit Multiplication Worksheets 99Worksheets
Free Multiplication Worksheet 1s And 2s Free4Classrooms
FREE PRINTABLE MULTIPLICATION WORKSHEETS WonkyWonderful
Multiplying 2 Digit by 2 Digit Numbers A Math Drills
Welcome to The Multiplying 2 Digit by 2 Digit Numbers A Math Worksheet from the Long Multiplication Worksheets Page at Math Drills This math worksheet was created or last revised on 2021 02 17 and
has been viewed 8 714 times this week and 10 384 times this month
Multiplication 2 Digits Times 2 Digits Super Teacher Worksheets
Shape Multiplication 2 Digit by 2 Digit Numbers At the top of this worksheet students are shown a dozen shapes with two digit numbers in them They multiply similar shapes For example Find the product
of the numbers in the hexagons 4th through 6th Grades View PDF Find the Errors 2 Digit by 2 Digit Numbers
Welcome to The Multiplying 2 Digit by 2 Digit Numbers A Math Worksheet from the Long Multiplication Worksheets Page at Math Drills This math worksheet was created or last revised on 2021 02 17 and
has been viewed 8 714 times this week and 10 384 times this month
Shape Multiplication 2 Digit by 2 Digit Numbers At the top of this worksheet students are shown a dozen shapes with two digit numbers in them They multiply similar shapes For example Find the product
of the numbers in the hexagons 4th through 6th Grades View PDF Find the Errors 2 Digit by 2 Digit Numbers
Double Digit Multiplication Worksheets 99Worksheets
2 Digit By 2 Digit Multiplication Worksheets With Answers Free Printable
Free Multiplication Worksheet 1s And 2s Free4Classrooms
FREE PRINTABLE MULTIPLICATION WORKSHEETS WonkyWonderful
2 Digit Multiplication Worksheet
2 By 2 Digit Multiplication Worksheets Free Printable
2 By 2 Digit Multiplication Worksheets Free Printable
2 Digit by 2 Digit Multiplication Worksheets
FAQs (Frequently Asked Questions).
Are Worksheet Multiplication By 2 ideal for every age teams?
Yes, worksheets can be tailored to various age and skill levels, making them versatile for numerous students.
How commonly should students practice using Worksheet Multiplication By 2?
Consistent technique is essential. Normal sessions, preferably a couple of times a week, can yield significant renovation.
Can worksheets alone improve mathematics skills?
Worksheets are an useful tool however ought to be supplemented with diverse understanding methods for thorough skill growth.
Exist on-line platforms offering free Worksheet Multiplication By 2?
Yes, numerous academic internet sites supply free access to a vast array of Worksheet Multiplication By 2.
Just how can parents support their kids's multiplication technique in your home?
Encouraging consistent technique, offering aid, and producing a positive knowing atmosphere are beneficial steps. | {"url":"https://crown-darts.com/en/worksheet-multiplication-by-2.html","timestamp":"2024-11-12T22:42:19Z","content_type":"text/html","content_length":"28185","record_id":"<urn:uuid:25124a06-a172-4ef4-8996-183f39985878>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00089.warc.gz"} |
ProCal Uncertainty Rounding Rules
There are multiple interpretations of the GUM with regard to rounding, with the guidance in the GUM (https://www.bipm.org/en/committees/jc/jcgm/publications) stating :
7.2.6 The numerical values of the estimate y and its standard uncertainty uc(y) or expanded uncertainty U should not be given with an excessive number of digits. It usually suffices to quote uc(y)
and U [as well as the standard uncertainties u(xi) of the input estimates xi] to at most two significant digits, although in some cases it may be necessary to retain additional digits to avoid
round‑off errors in subsequent calculations.
In reporting final results, it may sometimes be appropriate to round uncertainties up rather than to the nearest digit. For example, uc(y) = 10,47 mΩ might be rounded up to 11 mΩ. However, common
sense should prevail and a value such as u(xi) = 28,05 kHz should be rounded down to 28 kHz. Output and input estimates should be rounded to be consistent with their uncertainties; for example, if y
= 10,057 62 Ω with uc(y) = 27 mΩ, y should be rounded to 10,058 Ω. Correlation coefficients should be given with three‑digit accuracy if their absolute values are near unity.
ProCal as of V6.8.91 has been updated so that the below rules are always followed.
• Intermediary calculations will be calculated with no rounding to a high precision.
• Expanded Uncertainties will always be rounded UP to the least significant digit of the resolution. I.e. for a resolution of 0.001V, and an uncertainty of 0.00101V, the reported uncertainty will
be rounded up to 0.002V (2uV)
• Uncertainties will be reported and rounded up to 2 Significant Places, based on the resolution and the calculated uncertainty. I.e., an uncertainty of 587.2 kOhms will be rounded UP to 590 kOhms
This is taking the most risk-averse approach to reporting and calculating the uncertainty. Rounding UP is always used in any event where the reporting resolution causes the expanded uncertainty to be
rounded for display.
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Matrix Theory 1.5 Cramer Theorem
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Joules to Volts Calculator - calculator
Joules to Volts Calculator
Joules to Volts Calculator to easily convert energy in joules to voltage in volts. Joules to Volts, Energy Conversion, Voltage Calculator, Physics Calculator
Related Calculator-
Calculate Joules to Volts
What is Joules to Volts?
Joules and Volts are both units used in the realm of electricity and energy. Joules measure energy, while Volts measure electrical potential difference. Converting Joules to Volts involves
understanding the relationship between energy and potential difference.
What is a Joules to Volts Calculator?
A Joules to Volts Calculator is an online tool designed to help you convert energy (measured in Joules) to electrical potential difference (measured in Volts). It simplifies the conversion process by
using a straightforward formula to deliver quick and accurate results.
How to Use the Joules to Volts Calculator
To use a Joules to Volts Calculator, follow these simple steps:
1. Enter the value of energy in Joules.
2. Click on the 'Calculate' button.
3. View the result, which will be displayed in Volts.
Most calculators offer a user-friendly interface, making it easy to perform these conversions without needing complex calculations yourself.
The Formula for Joules to Volts Conversion
The formula used by the Joules to Volts Calculator is:
Voltage (V) = Energy (J) / Charge (C)
In this formula, Voltage (V) is the electrical potential difference in Volts, Energy (J) is the energy in Joules, and Charge (C) is the charge in Coulombs. Ensure you have the charge value to perform
the conversion accurately.
Advantages and Disadvantages of Joules to Volts Calculator
• Quick and Accurate: Provides instant results with precise calculations.
• User-Friendly: Easy to use with a straightforward interface.
• Convenient: Accessible online from any device with an internet connection.
• Dependence on Input: Requires accurate charge values for correct results.
• Limited Scope: Only useful for converting between Joules and Volts if the charge is known.
Additional Information
Some advanced calculators may also allow you to input different units or perform additional calculations related to electrical energy. It’s important to verify that the tool you use provides
accurate results and is suitable for your specific needs.
Frequently Asked Questions
How do I convert Joules to Volts manually?
To convert Joules to Volts manually, use the formula: Voltage (V) = Energy (J) / Charge (C). You need to know the amount of charge in Coulombs to perform the conversion.
Can I use the Joules to Volts Calculator for all types of energy conversions?
No, the Joules to Volts Calculator is specifically designed for converting Joules to Volts. For other types of energy conversions, different calculators or formulas are needed.
Is the Joules to Volts Calculator accurate?
Yes, the calculator is accurate as long as the input values are correct. Ensure you enter the correct charge value for precise results. | {"url":"https://calculatordna.com/joules-to-volts-calculator/","timestamp":"2024-11-11T19:29:19Z","content_type":"text/html","content_length":"94554","record_id":"<urn:uuid:7b86c0a8-2c28-45c4-b727-f457f9dcc0a2>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00060.warc.gz"} |
Powerdomains and hyperspaces IV: theories
Last time, we concluded with a mysterious observation. There is a theory, that of unital inflationary topological semi-lattices, which plays a fundamental role in the study of the Hoare powerspace.
On the one hand, H(X) is the free sober such thing. On the other hand, the algebras of the H monad are exactly those things that are sober. We shall investigate that by looking at theories
themselves, and show how those constructions arise from a logical perspective.
Terms, inequalities, theories
In the rest, we shall consider a language of first-order terms over a given signature Σ. A signature is just a set of so-called function symbols, together with their arities, that is, the number of
arguments they take. For unital inflationary topological semi-lattices, the signature would contain two symbols, ∨ of arity 2, and 0 of arity 0.
We also consider a set V of so-called variables.
A term over the signature Σ and the set V of variables is a finite tree whose vertices are:
• either variables from V, and have no direct successor (those are leaves),
• or are labeled with function symbols f from Σ, and have n direct successors, where n is the arity of f.
Such terms are usually written as strings, such as f(g(a, x), a). The latter denotes a tree whose root is labeled f, of arity 2. Its two direct successors are the trees g(a, x) and a. The first
one is itself a tree whose root is labeled g, of arity 2, with two direct successors a and x. I am assuming that x is a variable here. The constant a is a tree labeled a, of arity 0, and has no
direct successor, just like variables. (We should write a(), but a is more readable.)
For unital inflationary topological semi-lattices, we would have terms of the form ∨(∨(a, b), c), for example. We find it more practical to write s ∨ t instead of ∨(s, t).
The type of terms over a signature is easily encoded in a type of computer data, provided the signature is finite. For example, one can specify the set of terms over the signature ∨, 0 as follows in
type 'a SUITopSLatTerm = Var of 'a
| Zero
| Vee of 'a SUITopSlatTerm * 'a SUITopSLatTerm;;
Here Zero denotes 0, Vee (s, t) denotes s ∨ t, and Var(x) denotes the variable x.
We specify a theory by listing inequalities, of the form s ≤ t, for pairs of terms s and t. We also take s=t to denote the pair of inequalities s ≤ t and t ≤ s. Variables are universally
quantified. That means that, if we write, say, x ≤ x ∨ y as inequality, where x and y are variables, then we mean that s should be below s ∨ t for all values s and t. Oh, oops, I haven’t mentioned
that inequalities should have a meaning.
Algebras, models
Meaning is given by what logicians call a model. A Σ-algebra M is just a set D (the support of M), together with an n-ary function M[f] from D^n to D for each function symbol f in Σ. An environment
ρ is a map from V to D, serving to interpret the variables. Then one can interpret each term t in D, modulo ρ, as an element M[t] ρ of D defined recursively by:
• M[x] ρ = ρ (x) for every variable x;
• M[f(t[1], …, t[n])] ρ = M[f] (M [t[1]] ρ, …, M [t[n]] ρ).
A model of T, also known as a T-algebra, is a Σ-algebra that satisfies all inequalities of T, in the sense that, for each equality s=t in the theory T, the model satisfies M[s] ρ = M[t] ρ for every
environment ρ. (What about inequalities? Allow me to only consider equalities here, for simplicity. I will deal with inequalities below, in the ‘The order-theoretic case’ section. The following
example will use one inequality, so you’ll have to overlook the simplification for a second.)
For example, H(X) is a model of the theory of unital inflationary topological semi-lattices, given by the following inequalities:
• (unit) x ∨ 0 = x
• (associativity) (x ∨ y) ∨ z = x ∨ (y ∨ z)
• (commutativity) x ∨ y = y ∨ x
• (idempotence) x ∨ x = x
• (inflationary) x ≤ x ∨ y
where x, y, z are distinct variables. The model is given by interpreting ∨ as union and 0 as the empty set, that is, by defining M[∨] as union and M[0] as the empty set.
If you prefer to reason categorically, given a signature Σ, there is a category of “sets with a function for each function symbol in Σ”. (I am ignoring arities for the sake of readability.) This
is the category Alg[Σ] of all Σ-algebras.
The forgetful functor to Set has a left adjoint Term[Σ], which maps every set V (yes, any set can play the role of a set of variables) to the set of terms over Σ and V. The latter is a very simple
kind of Σ-algebra: for each function symbol f, there is a map M[f], which maps every tuple of terms t[1], …, t[n] to the term f (t[1], …, t[n]). So much for terms.
The unit of the adjunction, η : V → Term[Σ](V), maps every variable x to x itself, seen as a term. (In our OCaml example, this map sends every variable x to Var(x) instead.)
Since Term[Σ](V) is the free Σ-algebra over the set V, for each set map ρ : V → D, there is a unique map of Σ-algebras, from Term[Σ](V) to D, that gives back ρ when precomposed with η : V → Term[Σ](V
). This is the interpretation map, the one that sends every term t to M[t] ρ.
Building the free sober T-algebra: first ideas
All right, now take a fixed theory T. There is a known construction of a free continuous domain T-algebra over any continuous dcpo X [1, Section 6.1.2]. (A domain T-algebra is just a T-algebra,
except the support D is a dcpo and the functions M[f] are required to be Scott-continuous.)
The rough idea is not that complicated, but I have always been puzzled by how subtle it could be, and by the fact that this only worked for continuous dcpos.
This works as follows. We start from Term[Σ](X)—think of that as a set of trees whose leaves are elements of X instead of variables. You may need to pause on that. There is nothing special about
variables. The set of variables can be anything we wish, and in particular X. Have you digested this? If so, then we can go on.
We declare two terms s, t equivalent if and only if one can deduce the two inequalities s ≤ t and t ≤ s from the inequalities of the theory T, by the usual rules (I will be more precise below). Then
we consider the quotient of Term[Σ](X) by this equivalence, and consider that quotient as an abstract basis. To this end, we need to define an interpolative relation ≺ on that set (see Lemma 5.1.32
in the book). We declare that (the equivalence class of) s directly approximates (the equivalence class of) t if and only if one can reach s, starting from t, by first going to a term u ≤ t, then
replacing all the variables in u (which are just values in X) by values that are way-below in X, obtaining a new term v, then checking that s ≤ v. The interpolative relation ≺ is the transitive
closure of that relation of direct approximation [1, Lemma 6.1.5]. Finally, the rounded ideal completion of that abstract basis will give you the free object in the category of continuous T
-algebras, over X [1, Theorem 6.1.6].
As for many things domain-theoretic, I had the feeling that this can be made clearer and more general by looking at the problem in a topological way instead. My initial guess was to put a topology
directly on Term[Σ](X), satisfying a few properties, and to take the sobrification of the resulting space. The properties that I had imagined were:
• η : X → Term[Σ](X), which maps every variable to itself as a term, is continuous;
• for each n-ary function symbol f, the map that sends every tuple of terms t[1], …, t[n] to the term f (t[1], …, t[n]), is continuous (we shall simply call f that map);
• every open U is upward-closed, in the sense that for every term s in U and for every term t such that we can deduce s ≤ t from the theory T, then t is in U.
The resulting topology would fail very badly to be T[0], because of the latter item. But sobrification, which in particular equates all the points that are in the same open subsets, would take care
of that.
This approach does not work out of the box. We would need something like the coarsest topology satisfying the above. However, the requirement for f to be continuous is hard to enforce. Requiring f
to be separately continuous instead would probably give us a topology, but that is not quite the same thing.
Instead, we shall look at a simpler case first (the order-theoretic case). We shall find our way through successive generalizations, the algebraic case, then the general topological case and finally
the sober case.
If you ever get lost, keep in mind the following:
The free (ordered, or topological) T-algebra on X will always be a set of trees with function symbols from the signature Σ, and variables from X, suitably quotiented.
The only problem will be to find the right topology to put on this set of trees. The discussion above shows that this is the real problem.
The order-theoretic case
Instead of working in the category of topological case, we first deal with the simpler case of quasi-orderings. The situation is as follows. Consider a quasi-ordered set X, and try to find the free
quasi-order T-algebra on X. A quasi-order T-algebra is a T-algebra with a quasi-ordering ≤ such that:
• for every inequality s ≤ t in the theory T, the model satisfies M[s] ρ ≤ M[t] ρ for every environment ρ;
• for every function symbol f, M[f] is monotonic.
An order T-algebra is defined similarly, except that ≤ is required to be an ordering.
Given a quasi-ordered set X, we extend the quasi-ordering ≤ on X to one on Term[Σ](X) as the smallest relation ≤^* that satisfies the following. We write ≤^* for the extension, so as to distinguish
it clearly from ≤.
• (Extension) for all x, y in X, if x ≤ y in X, then x ≤^* y as terms, in Term[Σ](X);
• (Reflexivity) s ≤^* s for every s in Term[Σ](X);
• (Theory) given any inequality s ≤ t in T, for every environment ρ : V → Term[Σ](X), sρ ≤^* tρ; sρ denotes the result of replacing each variable x in s by the corresponding term ρ(x), and
similarly for tρ (if you don’t mind being puzzled for a second, note that sρ is just the meaning [s]ρ of s in environment ρ, considering Term[Σ](X) itself as a T-algebra);
• (Transitivity) if s ≤^* t and t ≤^* u, then s ≤^* u;
• (Application) if s[1] ≤^* t[1], …, s[n] ≤^* t[n], then f(s[1], …, s[n]) ≤^* f(t[1], …, t[n]), for every f in Σ, of arity n.
With the quasi-ordering ≤^*, Term[Σ](X) is a quasi-order-T-algebra. The map η : X → Term[Σ](X), which maps every variable x to itself as a term, is monotonic, by (Extension).
Finally, for every monotonic map β from X to the support D of a quasi-order-T-algebra M, β extends to a unique monotonic morphism Φ of quasi-order-T-algebras from Term[Σ](X) to M. For each term t, Φ(
t) is defined by induction on t:
• for every x in X, Φ(x)=β(x);
• Φ(f (t[1], …, t[n])) = M[f] (Φ(t[1]), …, Φ(t[n])).
In other words, Term[Σ](X), with the quasi-ordering ≤^*, is the free quasi-order-T-algebra over the quasi-ordered set X, ≤. Good.
The quasi-ordering ≤^* is not in general an ordering, even if X is a poset. For example, if the theory T contains the axiom (associativity) (x ∨ y) ∨ z = x ∨ (y ∨ z), then the terms (s ∨ t) ∨ u and s
∨ (t ∨ u) will both be below each other, for all terms s, t, and u.
However, every quasi-ordering induces an equivalence relation: define s=^*t if and only if both s ≤^* t and t ≤^* s. For equivalence classes [s] and [t] (of s and t, respectively, under that
equivalence relation), we define [s] ≤^* [t] if and only if s ≤^* t. This defines an ordering (not just a quasi-ordering), and the quotient poset Term[Σ](X)/=^* is then the free order-T-algebra over
the poset X.
The algebraic case
Now consider an algebraic dcpo X. Recall that a domain T-algebra is an order T-algebra in which the support D is a dcpo and where the functions M[f] are Scott-continuous. We build the free domain T
-algebra above X as follows. We first extract the poset K of finite elements of X. We then build the free quasi-order T-algebra on K. This is Term[Σ](K), with the quasi-ordering ≤^*, as we have just
seen. (We could also take the free order T-algebra on K, Term[Σ](K)/=^*, but this really does not matter.) To obtain a dcpo from that, we build its ideal completion I(Term[Σ](K)).
The point of that construction is that finding continuous operations on an algebraic dcpo is exactly the same thing as finding monotonic operations on their poset of finite elements. Therefore,
finding the free domain T-algebra on an algebraic domain will reduce to the problem of finding the free (quasi-)order T-algebra on a poset, a problem we have just solved.
The ideal completion is dealt with in Definition 5.1.45, and Proposition 5.1.46 tells us that the ideal completion I(B) of a poset B is an algebraic dcpo, whose finite elements are isomorphic to B.
When B is merely quasi-ordered, the same results hold, except the finite elements of I(B) are isomorphic to the quotient of B by the equivalence relation induced by the quasi-ordering.
In particular, I(Term[Σ](K)) is an algebraic dcpo. There are a few properties we must check:
• η : X → I(Term[Σ](K)) is continuous: instead of defining it and checking that it is continuous afterwards, we can just build it so that it is continuous by definition. There is a map that sends
each x in K to x itself, as an element of Term[Σ](K). Compose that with the inclusion of Term[Σ](K) into I(Term[Σ](K)) (which maps every term t to ↓t), so that we obtain a monotonic map from K to
I(Term[Σ](K)). This has a unique continuous extension to X (see Exercise 5.1.62), and we call it η.
• Each function f in Σ is continuous, namely, Scott-continuous. Call n the arity of f. We proceed as for η. First, f defines a monotonic map from Term[Σ](K)^n to Term[Σ](K), by rule (Application).
We obtain a monotonic map from Term[Σ](K)^n to I(Term[Σ](K)) by composing with the inclusion of Term[Σ](K) into I(Term[Σ](K)). This extends to a unique continuous map from I(Term[Σ](K)^n) to I(
Term[Σ](K)). This is almost what we want, except that we need the domain of the map to be [I(Term[Σ](K))]^n, not I(Term[Σ](K)^n). We are saved by the fact that ideal completions commute with
finite products, up to natural isomorphism. You can check it by hand. Alternatively, use the following roundabout argument: ideal completion is sobrification (see comment after Exercise
8.2.48), and sobrification preserves product up to natural isomorphism (Theorem 8.4.8).
• I(Term[Σ](K)) satisfies all the inequalities of the theory T. Given any term t in Term[Σ](V), the meaning map [t], which maps every environment ρ : V → Term[Σ](K) to [t]ρ, is monotonic, as a
composition of monotonic maps. Again, it extends to a unique continuous map, again written [t], from extended environments ρ’ : V → I(Term[Σ](K)) to I(Term[Σ](K)). Since it is unique, it
coincides with the unique continuous map defined by [x]ρ’ = ρ'(x), [f(t[1], …, t[n])]ρ’ = f ([t[1]]ρ’, …, [t[n]]ρ’), where the effect of f on the ideal completion was found in the last bulleted
item. To check that [s]ρ’ ≤ [t]ρ’ for every extended environment, it is enough to check that [s]ρ ≤ [t]ρ for every environment with values in Term[Σ](K). But we know that already, from the
previous section on the ordered case.
Finally, for every continuous map β from X to the support D of a domain T-algebra M, β restricts to a continuous map from K to M. We know from the previous section that β extends to a unique
monotonic morphism Φ of quasi-order-T-algebras from Term[Σ](K) to M. Since I is left adjoint to the forgetful functor from dcpos to orders (Exercise 5.5.3), Φ extends to a unique continuous map
from I(Term[Σ](K)) to M. The same uniqueness arguments as above show that Φ commutes with application of f, for every function symbol f in Σ, so Φ is a continuous morphism of T-algebras. Finally, it
coincides with β not only on K, but on the whole of X, again by uniqueness of extensions.
We conclude that:
Theorem 1. Given any algebraic dcpo X, there is a free domain T-algebra on X, and it is algebraic as a dcpo.
This free domain T-algebra is the ideal completion I(Term[Σ](K)) of the quasi-order T-algebra of terms built from function symbols in Σ and from “variables” taken from the set K of finite elements of
This is important per se. The continuous case can be obtained by realizing that, if X is continuous, then it is a retract of some algebraic dcpo Y (see Theorem 5.1.48). Then build the free domain
T-algebra on Y, and construct the free domain T-algebra on X as a suitable retract of the former. (Hint: both I and Term[Σ] are functors.) This shows that every continuous dcpo has a free domain T
-algebra on it, and that it is continuous. The construction from [1, Section 6.1.2] builds the same object, up to natural isomorphism, except in a more concrete way.
The topological case
Let us jump to the whole category of all T[0] topological spaces. We wish to build the free T[0] topological T-algebra on X. A T[0] topological T-algebra is, as you can expect, a T-algebra M whose
support is a T[0] topological space, and on which every function M[f] is continuous.
To do so, we reduce to the previous case (again). Explicitly, we embed X in a much larger space, which will happen to be an algebraic dcpo. Then we apply the previous construction to that large
space, and carve out the needed bit from it.
To do so, let X be a T[0] space. We can embed it into an algebraic dcpo Y. The typical construction is to take the powerset P(O(X)) of the set O(X) of open subsets of X for Y; the embedding i : X → Y
maps each point x in X to its set of open neighborhoods. (See Proposition 9.3.5.) Note that Y is also a complete lattice.
In the sequel, I will forget about i and simply consider that X is a subspace of an algebraic complete lattice Y. I will then build the free T[0] topological T-algebra on X as a subspace A of the
free domain T-algebra on the much larger space Y.
In the previous section, we have seen that there is a free domain T-algebra on Y. This is I(Term[Σ](K)), where K is the set of finite elements of Y, namely the set of finite sets of open subsets of X
… but reasoning at that level of detail would get us lost. Simply call T(Y) the free domain T-algebra on Y, for the rest of this post.
We again forget about embeddings, and consider that Y itself appears as a subset of T(Y). If we were to be formal, that would be a topological embedding (not a T-algebra embedding). With those
conventions, note that X is then an even smaller subspace of T(Y).
There is a subspace A of T(Y), built as the smallest subset that contains X and that is closed under application of function symbols f in Σ. (The effect of those function symbols is taken to be their
effect in T(Y).) As such, A is not only a subspace, but also a sub-T-algebra.
It is now fairly easy to see that A is the free T[0] topological T-algebra on X. For that, consider any continuous map β from X to (the underlying topological space) of a topological T-algebra M,
with support D. To this end, we notice that β extends to a continuous map from Y to D, which then extends to a unique domain T-algebra morphism from T(Y) to M. Restrict the latter to A, obtaining a
map β’. As the restriction of a continuous map, β’ is continuous. It is also a morphism of T-algebras. The fact that β’ is unique is obvious: we must have β'(x)=β(x) for every x in X, and β'(f(t[1],
…, t[n])) = M[f] (β’ (t[1]), …, β’ (t[n])), and this suffices to define β’ uniquely on A.
Theorem 2. For every T[0] topological space X, there is a free T[0] topological T-algebra on X.
Although it might look very abstract, that free T[0] topological T-algebra is just the set Term[Σ](X) of trees with “variables” taken from X, suitably quotiented and topologized. The reason for all
the complication with ideal completions, embeddings into powersets and what have you is just to define the topology on that set of trees.
Let us return for a moment to the theory of unital inflationary topological semi-lattices. The terms are built from the constant 0, the binary symbol ∨ and variables. One must quotient those terms,
considering ∨ as associative, commutative and idempotent, with 0 as neutral element. A moment’s reflection should convince you that this quotient is isomorphic to some set of finite subsets of X. For
example, the equivalence class of the term x ∨ (y ∨ z) should look like the finite set {x, y, z}. However, the (inflationary) inequality also forces some other identifications, such as {x} = {x, y}
if x ≤ y. The complex construction we have described above serves to put the right topology on this set of finite subsets. Eventually, a canonical representative of a finite set will be its
downward-closure (e.g., ↓{x, y, z}): the free T[0] topological T-algebra on X, where T is the theory of unital inflationary semi-lattices, is the set of finitary closed subsets of X, with some
topology. I’ll let you check that this is the lower Vietoris topology. But please! do not try to check it directly. The only nice way to do so that I know of is to check that the set of finitary
closed subsets of X with the lower Vietoris topology is the unique (up to natural isomorphism) T[0] topological T-algebra on X; so it must coincide with what our tree-based construction yields, up to
natural isomorphism.
Sober T-algebras
We had seen in part I that H(X) is the free sober T-algebra, for the theory T of unital inflationary semi-lattices. The free T[0] topological T-algebra is a bit disappointing, as it contains only
finitary closed sets, and free sober T-algebras look more promising.
So let us look for free sober T-algebras in general. Sober T-algebras are just T[0] topological T-algebras whose support is sober.
Looking back at the construction of the previous section, we realize that A is not necessarily sober. (The space of finitary closed subsets of a space, for one, is not sober, as its sobrification is
in fact the whole of H(X).) But it sobrification S(A) will be. We claim that S(A) will be the sought after free sober T-algebra.
The topological embedding of A into T(Y) lifts to a topological embedding of S(A) into S(T(Y)) (Lemma 8.4.11). Furthermore, since T(Y) is an algebraic dcpo, hence sober, the latter is just T(Y). All
this allows us to see S(A) as a sober subspace of T(Y).
Any continuous map from A^n to a sober space extends to a unique continuous map from S(A^n) to the same sober space (Lemma 8.2.44). Since S(A^n) is naturally isomorphic to [S(A)]^n (Theorem 8.4.8),
we obtain that S(A) is also a Σ-algebra where the application of each function symbol is continuous. Inequalities are preserved, too, because sobrification preserves order, namely if f ≤ g as
continuous maps, then S(f) ≤ S(g). (Exercise: use Lemma 8.2.42 for a definition of S(f).) Therefore S(A) is a sober T-algebra.
We have almost finished to prove that S(A) is the free sober T-algebra on X. Consider any continuous map β from X to (the underlying topological space) of a sober T-algebra M, with support D. This
extends to a unique continuous morphism of T-algebras β’ from A to M, by the results of the previous section. In turn, β’ extends to a unique continuous map from S(A) to M, since M is sober.
Using arguments similar to some that we have already seen (uniqueness of extensions), that extension also commutes with applications of function symbols from Σ. Therefore it is also a morphism of T
-algebras. We have proved:
Theorem 3. For every T[0] topological space X, there is a free sober T-algebra on X.
Whew. Done. Of course, this free sober T-algebra will be naturally isomorphic to H(X) in case T is the theory of unital inflationary semilattices, by general category-theoretic arguments.
In general, it is very hard to find a concrete description of free sober T-algebras. There are a few cases where this can be done. We have seen the case of the Hoare powerspace. I’ll talk some day
about the Smyth powerspace, where the (inflationary) axiom is replaced by a similar one with the order reversed, and which, in nice cases, yields the powerspace of compact saturated subsets of X.
The case of the Plotkin powerdomain is a notoriously tough one. One that has occupied me for some time is Daniele Varacca’s domains of indexed valuations [2]. That one is given by an inequational
theory, but we have no idea what the free sober T-algebra in that case looks like, concretely.
— Jean Goubault-Larrecq (June 30th, 2015)
1. Samson Abramsky and Achim Jung. Domain Theory. Handbook of Logic in Computer Science, Oxford University Press, 1994, pages 1-168.
2. Daniele Varacca. Probability, Nondeterminism and Concurrency: Two Denotational Models for Probabilistic Computation. PhD Thesis, Dept. of Computer Science, U. Aarhus, 2003. BRICS Dissertation
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Ampere unit
Ampere definition
Ampere or amp (symbol: A) is the unit of electrical current.
The Ampere unit is named after Andre-Marie Ampere, from France.
One Ampere is defined as the current that flows with electric charge of one Coulomb per second.
1 A = 1 C/s
Ampere meter or ammeter is an electrical instrument that is used to measure electrical current in amperes.
When we want to measure the electrical current on the load, the ampere-meter is connected in series to the load.
The resistance of the ampere-meter is near zero, so it will not affect the measured circuit.
Table of ampere unit prefixes
name symbol conversion example
microampere (microamps) μA 1μA = 10^-6A I[ ]= 50μA
milliampere (milliamps) mA 1mA = 10^-3A I[ ]= 3mA
ampere (amps) A - I[ ]= 10A
kiloampere (kiloamps) kA 1kA = 10^3A I[ ]= 2kA
How to convert amps to microamps (μA)
The current I in microamperes (μA) is equal to the current I in amperes (A) divided by 1000000:
I[(μA)] = I[(A)] / 1000000
How to convert amps to milliamps (mA)
The current I in milliamperes (mA) is equal to the current I in amperes (A) divided by 1000:
I[(mA)] = I[(A)] / 1000
How to convert amps to kiloamps (kA)
The current I in kiloamperes (mA) is equal to the current I in amperes (A) times 1000:
I[(kA)] = I[(A)] ⋅ 1000
How to convert amps to watts (W)
The power P in watts (W) is equal to the current I in amps (A) times the voltage V in volts (V):
P[(W)] = I[(A)] ⋅ V[(V)]
How to convert amps to volts (V)
The voltage V in volts (V) is equal to the power P in watts (W) divided by the current I in amperes (A):
V[(V)] = P[(W)] / I[(A)]
The voltage V in volts (V) is equal to the current I in amperes (A) times the resistance R in ohms (Ω):
V[(V)] = I[(A)] ⋅ R[(Ω)]
How to convert amps to ohms (Ω)
The resistance R in ohms (Ω) is equal to the voltage V in volts (V) divided by the current I in amperes (A):
R[(Ω)] = V[(V)] / I[(A)]
How to convert amps to kilowatts (kW)
The power P in kilowatts (kW) is equal to the current I in amps (A) times the voltage V in volts (V) divided by 1000:
P[(kW)] = I[(A)] ⋅ V[(V)] / 1000
How to convert amps to kilovolt-ampere (kVA)
The apparent power S in kilovolt-amps (kVA) is equal to RMS current I[RMS] in amps (A), times the RMS voltage V[RMS] in volts (V), divided by 1000:
S[(kVA)] = I[RMS(A)] ⋅ V[RMS][(V)] / 1000
How to convert amps to coulombs (C)
The electric charge Q in coulombs (C) is equal to the current I in amps (A), times the time of current flow t in seconds (s):
Q[(C)] = I[(A)] ⋅ t[(s)]
See also | {"url":"https://jobsvacancy.in/electric/ampere.html","timestamp":"2024-11-03T23:28:32Z","content_type":"text/html","content_length":"9684","record_id":"<urn:uuid:6d3d9d09-5721-4c9e-a48e-69da68786bed>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00428.warc.gz"} |