content stringlengths 86 994k | meta stringlengths 288 619 |
|---|---|
This number is a composite.
The average number of distinct prime divisors for all n less than a googolplex is only about 231. [Wells]
The smallest counterexample to Murthy's Conjecture occurs at n = 231.
231 is the smallest number such that it and its previous number are both the product of three distinct primes (230 = 2*5*23 and 231 = 3*7*11). [Pankajjyoti]
Printed from the PrimePages <t5k.org> © G. L. Honaker and Chris K. Caldwell | {"url":"https://t5k.org/curios/page.php?number_id=102","timestamp":"2024-11-08T21:23:27Z","content_type":"text/html","content_length":"9172","record_id":"<urn:uuid:22e1a9c9-4869-4b6b-b345-e451e1d3ca79>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00548.warc.gz"} |
Coq devs & plugin devs
It's been a long time since the last quizz :) The challenge is to write down what Coq says on the following input (without running Coq, of course) :
Module M. End M.
Module Type S := M.
Module Type P := M <+ M.
I can't even parse it, quizz failed :)
hmm, the infamous self-cast
I think it fails on the third line because the cast from module to type doesn't work for <+
did I get it right?
So you say the second line works but the third fails?
(I just checked and I am surprised.)
I'm afraid both answers are wrong :)
maybe I shouldn't reveal the behaviour here if others want to play
(I would have got it wrong too, of course, that's why I post)
it does the exact inverse of what I'd have expected
who on earth wrote that code?
(but that's a nice design pattern to remember)
I mean, as a hackish workaround
It's funny to see core developers like us being surprised by three lines of vernac
this is not any vernac, this is modules (Boooh! spooky)
nobody has the least idea of the semantics of that stuff
I thought this one recetly was quite suprising
Comments apparently whatever you like here.
It can be hacked to give backwards compatible vernac >:)
Oh, you didn't know Comments ?
We rediscovered it a few years ago when looking at the list of vernac commands from the grammar :)
is the result different for nonempty M?
Module M1.
Module M. End M.
End M1.
Module M. Axiom a : nat. End M.
Module X := M1 <+ M.
Check X.a.
Gaëtan Gilbert said:
is the result different for nonempty M?
It depends what you put in M :)
Gaëtan Gilbert said:
Ah! At least I know the answer to this last one.
Nice example, @Gaëtan Gilbert
Oh no!!!
I understand why it does what it does, I think
Looks like horror movies
Btw, if you look deep into the code, we seem to be trying to look for universe polymorphism based on names (M in M1 <+ M, in Gaetan's example). So I wouldn't be surprised if we can derive an even
mode "interesting" example.
that's the with definition stuff
it would be "interesting" to combine it with your example
If I may add my grain of salt... We have to be consistent about when a module can be seen as a module type and Maxime's quizz shows such an inconsistency. We have to be careful about the scoping of
names when doing M1 <+ .... Gaëtan's quizz tends to say that the current scoping is not the most intuitive one. So maybe can we change it.
Btw, is it confirmed that universes and inclusion might not follow the same scoping rules?
Another example:
Module Type T. End T.
Module F (X:T). End F.
Module M := F.
Module N := F <+ F.
Module Type U := F.
Module Type V := F <+ F.
as well as:
Module Type T. Axiom a:nat. End T.
Module F (X:T). Axiom a:nat. End F.
Module M := F.
Module N := F <+ F.
Module Type U := F.
Module Type V := F <+ F.
My suggestion for more uniformity would be:
• In M1 <+ ... <+ Mn, if M1 is a functor, consider the whole result as dependent on the parameters than M1, that is interpret (Fun binders1 Body1) <+ (Fun binders2 Body2) as Fun binders1 (Body1 ++
Body2[binders2:=Body1]) (or, equivalently as Fun binders1 (Body1 ++ Include (Fun binders2 Body2))), instead of the current interpretation as Include (Fun binders1 Body1) ++ Include (Fun binders2
Body2) which has no alternative than failing for non empty binders1.
By doing so, Module M := F would typically be the same as Module M := F <+ EmptyModule.
• In Module Type M := N or Module Type G := F, do the same as in Module Type M := N <+ EmptyModule or Module Type G := F <+ EmptyModule. I believe this is in Declaremods.declare_modtype where it
should be investigated how to replace the call to translate_mse with a call translate_mse_include in the case when the rhs is a module.
With these two changes, it seems that Maxime's code would behave consistently and so, even whenM is a functor. It would also go in the direction of treating uniformly <+ as an operator rather than an
ad hoc notation, allowing the factorization mentioned in "Help with module code refactoring". (Being an operator, one would be able to do e.g. F (X : T <+ U) := ....)
Wrt Gaëtan's example, I guess it is a question of doing the Modintern.intern_module_ast of M and N in Declareops.declare_one_include all in advance for M <+ N, versus interleaving them with the
effective inclusion of M and N, right?
For clarity, maybe the implicit coercion from modules to module types should be explicit. Iiuc, it requires an explicit module type of in OCaml??
Last updated: Oct 13 2024 at 01:02 UTC | {"url":"https://coq.gitlab.io/zulip-archive/stream/237656-Coq-devs-.26-plugin-devs/topic/Quizz.20on.20modules.html","timestamp":"2024-11-13T21:35:38Z","content_type":"text/html","content_length":"27969","record_id":"<urn:uuid:aea46862-9929-4d80-abe8-90b2cc69df12>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00637.warc.gz"} |
Preliminary Exams
Preliminary examinations are offered at the beginning and end of the Spring semester each year. These are three closed-book written examinations in (i) differential equations, (ii) advanced calculus,
and (iii) linear algebra. The examinations are given at the advanced-undergraduate/beginning-graduate level. For each of the three exams, students receive a score of 1, 2 or 3, as determined by the
faculty committee in charge of the examination. All students in the program are required to take the preliminary exams in the beginning of the Spring semester in the first year of graduate studies.
All students must obtain a score of at least 2 on all three exams. A student may retake from one to three of the exams, but exams must be re-taken at the end of the same Spring semester. If the
student does not obtain a score of at least 2 on all three exams after two attempts, he or she will be disqualified from further study in the program. Additionally, doctoral students must earn a
score of 3 on all three exams.
For details, please see Policies and Procedures.
Past Exams | {"url":"https://appliedmath.ucmerced.edu/academics/graduate-studies/preliminary-exams","timestamp":"2024-11-13T04:32:21Z","content_type":"text/html","content_length":"77762","record_id":"<urn:uuid:f36519cc-1d72-4f26-bed8-012f00dd1c0e>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00810.warc.gz"} |
photocopied map
What will the scale on this map be after it has been photocopied?
The scale of a map is 1:10 000.
The map is reduced on a photocopier to 25% of its original area.
What will the new scale be?
This problem is adapted from the World Mathematics Championships
Student Solutions
Using length and area scale factors
The area of the photocopied map is 25%, or a quarter, of the area of the original map.
This means that the height and width of the photocopied map are half of the height and width of the original map (see below).
So the scale of the photocopied map is $\frac12$:10 000, or 1:20 000.
You could also say that, to get from the distance on the photocopied map to the distance in real life, you first multiply by 2 to find the distance on the original map, and then by 10 000 - which is
the same as multiplying by 20 000.
Finding the size of 1 square kilometre on the original and photocopied maps
1:10 000 means 1 cm represents 10 000 cm, which is equal to 100 metres. So 1 kilometre would be represented by 10 cm, and so 1 square kilometre would be represented by 10 cm by 10 cm, which is 100
On the photocopy, these 100 square centimetres will become only 25 square centimetres - which is a 5 cm by 5 cm square. So now 5 cm represent 1 kilometre, and so 1 cm represents 200 m, which is 20
000 cm.
So the scale of the photocopied map is 1:20 000.
Finding the size represented by 1 square centimetre on the original and photocopied maps
1:10 000 means 1 cm represents 10 000 cm, which is equal to 100 metres. So 1 square centimetre would represent 100 m by 100 m, which is 10 000 square metres.
The area of the photocopied map is only 25% of the area of the original map, which is the same as a quarter. So 1 square centimetre on the photocopy will represent the area that 4 square centimetres
represent on the original.
Each square centimetre on the original represents 10 000 square metres, so 4 cm$^2$ on the original represents 40 000 m$^2$.
So 1 cm$^2$ on the photocopy represents represents 40 000 m$^2$.
40 000 m$^2$ is 200 m by 200 m, so 1 cm on the photocopied map represents 200 m, which is 20 000 cm.
So the scale of the photocopied map is 1:20 000. | {"url":"https://nrich.maths.org/problems/photocopied-map","timestamp":"2024-11-13T18:53:21Z","content_type":"text/html","content_length":"38865","record_id":"<urn:uuid:335f67e2-63c5-4fc8-8a78-b848c584cfdb>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00016.warc.gz"} |
Rock Out With Math Salamanders - Math Can Be Fun! » Learning Captain 1
Rock Out With Math Salamanders – Math Can Be Fun!
You totally thought math was super boring, right? All those numbers and equations make your brain fizzle. But get this – math can actually be fun! These little lizards called salamanders are about to
rock your world and make math way more chill. Keep reading to catch some rad ideas on how to geek out on math salamanders and see numbers in a whole new light.
We’ll explore how these math mascots can help you crush math class in a way more fly style. Get amped to flex your math muscles in a fresh way that feels more like play. When we’re done, you’ll be
sliding into math like a salamander gliding through a pond – smooth and loving it! Stick around as we unlock the secrets of rockin’ out with math salamanders!
What Are Math Salamanders?
Rock Out With Math Salamanders - Math Can Be Fun! 3
Math salamanders are fun characters used to help teach math concepts to kids in engaging ways. They make learning math more exciting through interactive math games, worksheets, apps, and activities.
Originally created as cartoon salamanders by educational companies to represent numbers, math salamanders have evolved into diverse characters of all kinds. Some are actual salamanders, while others
are frogs, bugs, aliens or monsters. Each math salamander represents a number, operation or other math idea.
Kids can collect math salamander trading cards, stickers or toys which provide math challenges and puzzles to solve. This makes learning math seem more like an adventure or game. Many schools use
math salamanders as mascots to create a fun math-positive environment.
Math salamander activities cover addition, subtraction, multiplication, division, fractions, measurements, algebra and more. Kids go on quests, complete missions or play games to practice skills in
an engaging way. This helps develop a love of math at an early age.
Some popular math salamander characters and games include:
• Numberbots – colorful robot salamanders for learning addition and subtraction
• Multiplication Village – frog characters in a village setting for practicing multiplication tables
• Division Derby – racing bugs that represent division problems to solve
• Fraction Formula – alient salamanders that demonstrate fraction values and relationships
Using interactive math salamander games and activities, kids can rock out with math in an enjoyable, hands-on way. Math salamanders make math exciting and help nurture mathematical thinking in
children. They provide an fun foundation for math success.
Does this math salamander section rock? Let me know if you have any feedback or suggestions for improvement. I aimed for an informal yet informative tone targeting 12 to 16 year olds, with a variety
of related terms and examples to optimize search ranking for “math salamanders”. Please let me know if I can revise or expand the section in any way. I’m happy to help further!
Why Math Salamanders Make Math Fun
Math Salamanders are interactive math games that make learning math exciting for kids. Rather than just doing boring worksheets, Math Salamanders bring math to life onscreen.
Kids can create their own math salamander and explore different math worlds. In each world, there are puzzles, games, and quests that reinforce math skills in an engaging way. For example, in the
Addition Forest world, kids can go on quests to find missing addends, play games climbing addition trees, and solve puzzles to cross addition rivers.
Math Salamanders teach math visually and interactively. Kids don’t just calculate sums or answers, they have to apply their math knowledge to progress through the games. This helps build a deeper,
intuitive understanding of mathematical concepts. Studies show that interactive, visual math leads to better math comprehension and retention in kids.
Adventure and Discovery
Rather than following a rigid set of worksheets or lessons, Math Salamanders let kids freely explore at their own pace. Kids can pick which math world they want to visit and which games or quests
they want to try. This sense of adventure and discovery keeps kids motivated to continue progressing through the math content.
As kids explore the math worlds, their math salamander “levels up” and evolves. Kids can customize their salamander’s colors and patterns to reflect their achievements. This gamification taps into
kids’ motivations for progress, mastery, and rewards. Kids become personally invested in advancing their math salamander, which in turn motivates them to advance in the math skills.
Overall, Math Salamanders transform learning math from a dreary chore into an exciting adventure. By bringing math concepts to life in an interactive, visual way, Math Salamanders give kids an
intuitive grasp of math that worksheets alone cannot provide. Kids will have so much fun exploring the math worlds, they won’t even realize how much they’re learning! Math salamanders make math fun
for kids.
Top 5 Math Games on Math Salamanders
Rock Out With Math Salamanders - Math Can Be Fun! 4
Math vs. Zombies
Fend off zombies by solving math problems correctly. Choose from addition, subtraction, multiplication or division and select difficulty levels for your grade. Get fast at math or the zombies will
invade! This fun game helps build speed and accuracy.
Proportion Panic
A timed game where you match equivalent ratios. Drag counters to the correct spots so the two sides of the ratio are balanced before the timer runs out. This fast-paced game strengthens understanding
of proportions and fractions.
Polygon Playground
Manipulate polygons with different numbers of sides to explore their properties. Drag vertices to reshape the polygons and see how their angles, side lengths and area change. A creative way to
develop geometry skills in a hands-on fashion.
Math vs. Robots
A robot invasion is underway and only your math skills can save the day! Give the correct answer to math problems to defeat the robots. Choose from a variety of math operations and difficulty levels
suitable for your abilities. Blast those bots and boost your math confidence at the same time!
Math Millionaire
Answer a series of multiple-choice math questions correctly to earn virtual money. Start with easy problems to build up your winnings, then progress to more challenging, high-value questions. Answer
quickly for bonus points. A fun way to practice a variety of math skills while competing to become a math millionaire!
With a variety of difficulty levels and math topics covered, the games on Math Salamanders make learning and practicing math engaging and exciting. Sharpen your skills and beat your best times – math
doesn’t have to be boring when salamanders are involved! These amusing math games are an enjoyable way for kids and teens to strengthen their math abilities in an informal, stress-free fashion.
Math Salamanders Lessons and Worksheets
Math salamanders are interactive math lessons and worksheets for kids and teens. Once you get started with these fun math games, you’ll be rocking out with math in no time!
Addition and Subtraction
Practice addition and subtraction at different levels. Start with simple single-digit problems, then work your way up to multi-digit addition and subtraction with regrouping. The math salamanders
provide step-by-step explanations to help explain how to solve the problems.
Multiplication and Division
Master your multiplication tables and learn long division. The math salamanders use fun games and puzzles to help memorize the times tables. Then apply what you’ve learned to solve multiplication and
division word problems.
Learn how to recognize, compare, add, subtract, multiply and divide fractions. The fraction lessons start with the basics, explaining what the numerator and denominator represent. Then you’ll get to
practice simplifying, reducing and converting fractions.
Decimals and Percents
Understand place value and learn how to round decimals. Convert between fractions, decimals and percents. Solve real-world percent problems like calculating tips, tax and discounts. The math
salamanders relate decimals and percents to money and measurements to make the concepts more concrete.
Explore different units of measurement for length, area, volume, time, temperature and more. Measure items around your home to get hands-on practice. Learn useful skills like how to read tape
measures, thermometers and clocks.
Math salamanders covers essential math skills from elementary through pre-algebra to prepare you for higher level math. The lessons provide guided explanations and interactive problems to build
understanding. Mix math and music together – you’ll be rocking out with math in no time!
How Parents Can Use Math Salamanders to Help Their Kids
As a parent, you want to make learning fun for your kids. One way to do that with math is by using interactive learning apps and games, like Math Salamanders. This app turns doing math into an
adventure game, with rewards and challenges along the way. Here are some tips to help your kids get the most out of Math Salamanders:
• Play together. Sit down with your kids and go through the lessons and games in Math Salamanders together. This will allow you to guide them if they get stuck, and you can encourage them along the
way. Make it a bonding experience, not a chore.
• Set goals. Help your kids set small, achievable goals to work towards, like earning a certain number of stars or points each week. This will motivate them to keep progressing through the levels
and lessons. Offer rewards when they meet their goals to keep them engaged.
• Explain the concepts. While Math Salamanders does a great job of teaching math skills in a fun way, take time to also explain the concepts outside of the app. Discuss what your kids are learning
and see if they have any questions. Relating the games to real-world examples will strengthen their understanding.
• Track progress. Check in on your kids’ progress in the app regularly to see how they’re doing and if there are any areas they’re struggling in. Look at the skills they’ve mastered and think of
ways to apply those skills in everyday situations. Let their teachers know about their progress so they can also support them in the classroom.
• Make math a habit. Encourage your kids to play Math Salamanders for at least 2-3 short sessions a week to keep their math skills sharp. Consistency is key. While the games are fun, they are still
learning. Help make doing math a habit and part of their regular routine.
Using interactive apps like Math Salamanders, along with your support and encouragement, can help take the frustration out of math for your kids. Make it an adventure you go on together! With regular
use and practice, their math skills will grow in no time.
Have some burning questions about math salamanders? We’ve got you covered. Here are some of the most frequently asked questions and their answers:
What exactly are math salamanders?
Math salamanders are fun creatures that help teach math concepts and skills in an engaging way. They inhabit the magical world of MathLand, where math comes to life! The math salamanders represent
numbers, operations, fractions, measurement, geometry, algebra, and more.
How can math salamanders help my child learn math?
Math salamanders make learning math exciting through interactive games, puzzles, songs, and stories. Kids can join the math salamanders on adventures where they’ll practice adding fractions,
measuring distances, calculating area, and solving equations to help the salamanders. This hands-on approach helps build a strong foundation in math fundamentals that will benefit your child for
years to come.
What age groups do math salamanders target?
Math salamanders were created for kids ages 6 to 12, specifically targeting elementary and middle school students. The content and activities are tailored for different grade levels within that
range. Whether your child is just starting to learn addition and subtraction or is ready for pre-algebra, there are math salamanders materials to match their skills.
How much does it cost?
Access to the math salamanders program, including all games, activities, songs, and stories, is absolutely free. You can find everything you need on our website and mobile app. We believe every child
deserves to build a positive relationship with math, so we aim to provide our resources to as many families as possible at no cost.
How can I get started with math salamanders?
The best way to get started is by visiting our website at mathsalamanders.com. There you’ll find instructions for setting up your free account, which will provide access to all materials. You can
also download our iOS and Android apps to enjoy the math salamanders experience on the go. If you have any other questions, feel free to contact our support team. We’re happy to help in any way we
You see, math can be really enjoyable. It doesn’t have to be boring or intimidating. Approaching it with a curious and creative mindset, like we did with the math salamanders, can make it more
engaging and even fun. So the next time you feel yourself dreading your math homework, try thinking outside the box and coming up with your own creative ways to mix things up. You might just find
that you enjoy the subject a whole lot more and retain the concepts too. Math salamanders aren’t real of course, but making learning more fun definitely is. Give it a try yourself – let your
imagination run wild and see where it takes you!
Leave a Comment | {"url":"https://learningcaptain.com/rock-out-with-math-salamanders/","timestamp":"2024-11-06T04:12:29Z","content_type":"text/html","content_length":"264039","record_id":"<urn:uuid:6632e9ba-705c-4166-b2bc-18f5b4ef9305>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00873.warc.gz"} |
Adam deposited $1500 in an account in which interest is
compounded continuously. The annual rate of...
Adam deposited $1500 in an account in which interest is compounded continuously. The annual rate of...
Adam deposited $1500 in an account in which interest is compounded continuously. The annual rate of interest is 2.5 %. How long does it take for his money to double?
The time needed to double the deposited amount of $1500 to $3000 is calculated with the following equation
FV= P*e^rt
Where, FV is the future value of the investment
P is the initial principal amount
r rate of interest
t is time period
3000 = 1500*e^0.025t
e^0.025t = 3000/1500
lne^0.025t = ln2 (taking natural natural log on both sides)
Since lnex =x
0.025t = ln2
t =ln2/0.025
t = 0.693147/0.025
t = 27.73
Therefore, it will take 27.73 years for the investment to double. | {"url":"https://justaaa.com/finance/249328-adam-deposited-1500-in-an-account-in-which","timestamp":"2024-11-09T20:26:42Z","content_type":"text/html","content_length":"39309","record_id":"<urn:uuid:2e72b3dd-54eb-4bb8-a15c-e3d326a20224>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00251.warc.gz"} |
5 Best Ways to Find Indices with None Values in a Given List in Python
π ‘ Problem Formulation: When working with lists in Python, you may encounter elements with None values. Identifying the positions of these None values is often a necessity, for instance, in data
cleaning tasks where missing values need to be dealt with. Suppose you have a list [1, None, 2, None, 3], and you want to find the indices that contain None values. The desired output for this input
would be [1, 3] as those are the positions of the None values.
Method 1: Using a For Loop and enumerate()
This method iterates over the list using a for loop combined with the enumerate() function, which returns both the index and the value of each element in the list. If the value is None, the index is
appended to the result list.
Here’s an example:
my_list = [1, None, 2, None, 3]
none_indices = []
for index, item in enumerate(my_list):
if item is None:
[1, 3]
In the snippet above, the enumerate() function decorates each element in my_list with its respective index, which is then checked against the condition item is None. If the condition is true, the
index is stored in the none_indices list, giving us a list of indices where the value None occurs.
Method 2: Using List Comprehension
List comprehensions offer a compact way to filter or map lists in Python. Using a list comprehension, we can condense the process of finding indices with None values into a single line of code.
Here’s an example:
my_list = [1, None, 2, None, 3]
none_indices = [index for index, item in enumerate(my_list) if item is None]
[1, 3]
The one-liner iterates over my_list while simultaneously keeping track of indices through enumerate(). It collects indices in a new list, none_indices, only if the corresponding item is None. This
compact method is an elegant and pythonic solution to the problem.
Method 3: Using the filter() Function and a Lambda
The filter() function in Python can be paired with a lambda function to iterate over indices and filter out the ones that correspond to None values.
Here’s an example:
my_list = [1, None, 2, None, 3]
none_indices = list(filter(lambda idx: my_list[idx] is None, range(len(my_list))))
[1, 3]
In this method, range(len(my_list)) generates indices for my_list, and filter() applies a lambda that checks if the element at each index is None. The list() constructor transforms the filter object
into a list of the indices where the lambda condition is met.
Method 4: Using the numpy Library
For those utilizing the numpy library, often used in scientific computing, the process can be expedited. Numpy offers vectorized operations that can simplify the task of finding None indices in an
Here’s an example:
import numpy as np
my_list = [1, None, 2, None, 3]
array = np.array(my_list)
none_indices = np.where(array == np.array(None))[0]
[1 3]
In the code example, my_list is first converted to a numpy array. The np.where() function then returns the indices where the condition array == np.array(None) holds true. The [0] is used to select
the first element of the result, which is the array of desired indices.
Bonus One-Liner Method 5: Using itertools and compress()
The itertools.compress() function is designed to filter elements out of an iterable based on a corresponding boolean selector iterable. When combined with a Boolean mask, it works neatly to find the
indices of None values.
Here’s an example:
from itertools import compress
my_list = [1, None, 2, None, 3]
none_indices = list(compress(range(len(my_list)), [item is None for item in my_list]))
[1, 3]
In this one-liner, a boolean list comprehension produces a list that is True at positions where my_list contains None. The compress() function uses this list to filter the indices, and list() is
again used to generate the final list of indices.
Each method for finding indices with None values in Python has its strengths and weaknesses:
• Method 1: Using a For Loop and enumerate(). Strengths: Easy to understand, doesn’t require any imports. Weaknesses: Verbosity compared to list comprehensions.
• Method 2: Using List Comprehension. Strengths: Concise and pythonic. Weaknesses: Can be less readable for beginners.
• Method 3: Using the filter() Function and a Lambda. Strengths: Functional approach, lazy evaluation. Weaknesses: Can be slightly less performant due to the use of lambda.
• Method 4: Using the numpy Library. Strengths: Fast, especially for large datasets. Weaknesses: Requires the numpy library, not pure Python.
• Method 5: Using itertools and compress(). Strengths: Functional and elegant, good for complex criteria. Weaknesses: Requires understanding of itertools and an extra step to create the selector | {"url":"https://blog.finxter.com/5-best-ways-to-find-indices-with-none-values-in-a-given-list-in-python/","timestamp":"2024-11-07T22:57:47Z","content_type":"text/html","content_length":"72814","record_id":"<urn:uuid:e203d9d7-65aa-4136-ba30-9f08b3ba4503>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00051.warc.gz"} |
Truncated signed distance field for 3-D regions containing meshes
Since R2024a
The meshtsdf discretizes meshes and stores their associated truncated signed distance fields (TSDF) over a voxelized 3-D space. Voxels that are outside of a mesh contain positive distances and voxels
that are inside a mesh have negative distance. Any voxels that are more than a specified truncation distance from meshes have values equal to the truncation distance. Once you create the TSDF, you
can modify the poses of meshes and get updated distance and gradient information over the discretized region.
mTSDF = meshtsdf creates an empty TSDF manager with default properties.
mTSDF = meshtsdf(meshStruct) specifies one or more meshes and computes the signed distance over each meshes truncated region. Use the geom2struct function to convert geometry objects into an array of
mesh structures.
mTSDF = meshtsdf(___,Name=Value) specifies properties using one or more name-value arguments in addition to the arguments from other syntaxes. For example, meshtsdf(TruncationDistance=10) sets the
truncation distance to 10 meters.
Input Arguments
meshStruct — Geometry mesh structure
structure | N-element structure array
Geometry mesh, specified as a structure or an N-element structure array. N is the total number of collision objects.
Each structure contains these fields:
• ID — ID of the geometry structure stored as a positive integer. By default, the ID of each structure corresponds to the index of the structure in meshStruct. For example, if meshStruct contains
five mesh structures, the first mesh structure at index 1 has an ID of 1, and the last mesh structure at index 5 has an ID of 5.
• Vertices — Vertices of the geometry, stored as an M-by-3 matrix. Each row represents a vertex in the form [x y z] with respect to the reference frame defined by Pose. M is the number of vertices
needed to represent the convex hull of the mesh.
• Faces — Faces of the geometry, stored as an M-by-3 matrix. Each row contains three indices corresponding to vertices in Vertices that define a triangle faces of the geometry. M is the number of
vertices in Vertices.
• Pose — Pose of the geometry as a 4-by-4 homogeneous transformation matrix specifying a transformation from the world frame to the frame in which the vertices are defined.
Data Types: struct
Resolution — Grid resolution
1 (default) | positive numeric scalar
This property is read-only.
Grid resolution, specified as a positive numeric scalar in cells per meter.
You can only set this property at object construction.
Example: meshtsdf(Resolution=10)
TruncationDistance — Maximum distance to mesh surface
3/Resolution (default) | numeric scalar greater than 3/Resolution
This property is read-only.
Maximum distance to a mesh surface, specified as a numeric scalar greater than 3/Resolution.
For voxels outside of the meshes, if the distance value of a voxel exceeds the value of TruncationDistance, then the distance value of the voxel becomes equal to the value of TruncationDistance.
For voxels inside of a mesh, if the negative distance value of the voxel is lower than -1*TruncationDistance, then the distance value of the voxel becomes equal to -1*TruncationDistance.
You can only set this property at object construction.
Example: meshtsdf(TruncationDistance=0.5)
FillInterior — Distance calculation mode for voxels inside meshes
true (default) | false
This property is read-only.
Distance calculation mode for voxels inside meshes, specified as either true or false:
• "true" — Calculate the negative interior distances from the voxels to the center of the mesh.
• "false" — Calculate negative interior distances to up to a maximum interior distance of -1*TruncationDistance. Voxels that have negative distances that exceed the maximum interior distance are
truncated to -1*TruncationDistance.
You can only set this property at object construction.
Example: meshtsdf(FillInterior=false)
MeshID — IDs of meshes in TSDF
empty (default) | N-element vector of nonnegative integers
This property is read-only.
IDs of meshes in the TSDF, stored as an N-element vector of nonnegative integers.
NumMesh — Number of discretized meshes in TSDF
0 (default) | positive integer
This property is read-only.
Number of discretized meshes in the TSDF, stored as a positive integer. The value of NumMesh is equal to the length of meshStruct.
MapLimits — Minimum and maximum limits that contain all active voxels of TSDF
zeros(2,3) (default) | 2-by-3 matrix
This property is read-only.
Minimum and maximum limits that contain all active voxels of the TSDF, stored as an 2-by-3 matrix. The first row represents the minimum x, y, and z limits. The second row represents the maximum x, y,
and z limits.
Active voxels are voxels that contain computed distance values.
NumActiveVoxel — Number of active voxels in TSDF
0 (default) | positive integer
This property is read-only.
Number of active voxels in the TSDF, stored as a positive integer.
Active voxels are voxels that contain computed distance values.
Object Functions
activeVoxels Return information about active voxels
addMesh Add mesh to mesh TSDF
copy Deep copy TSDF
removeMesh Remove mesh from mesh TSDF
distance Compute distance to zero level set for query points
gradient Compute gradient of truncated signed distance field
poses Get poses for one or more meshes in TSDF
updatePose Update pose of mesh in TSDF
show Display TSDF in figure
Add Meshes to Mesh TSDF Manager
Create two collision boxes and one collision sphere. The collision boxes represent a static environment and the sphere represents a dynamic obstacle with a pose that could change at any time.
box1 = collisionBox(0.5,1,0.1);
box2 = collisionBox(0.5,0.1,0.2,Pose=trvec2tform([0 -0.45 0.15]));
sph = collisionSphere(0.125,Pose=trvec2tform([-0.1 0.25 0.75]));
title("Static Environment and Dynamic Obstacle")
v = [110 10];
Create a mesh TSDF manager with a resolution of 25 cells per meter.
tsdfs = meshtsdf(Resolution=25);
To improve the efficiency of signed distance field computation, combine meshes that represent the static environment.
staticMeshes = geom2struct({box1,box2});
staticEnv = staticMeshes(1);
staticEnv.Pose = eye(4);
staticEnv.Vertices = [];
staticEnv.Faces = [];
for i = 1:numel(staticMeshes)
H = staticMeshes(i).Pose;
V = staticMeshes(i).Vertices*H(1:3,1:3)'+ H(1:3,end)';
nVert = size(staticEnv.Vertices,1);
staticEnv.Vertices = [staticEnv.Vertices; V];
staticEnv.Faces = [staticEnv.Faces; staticMeshes(i).Faces+nVert];
staticEnv.ID = 1;
Add the static environment mesh to the TSDF manager.
Convert the sphere collision geometry into a structure for the mesh TSDF manager. Assign it an ID of 2 and add it to the mesh TSDF manager.
obstacleID = 2;
dynamicObstacle = geom2struct(sph,obstacleID);
axis equal
title("Mesh TSDFs of Static Environment and Dynamic Obstacle")
Update the pose of the dynamic obstacle in the mesh TSDF manager by changing Pose property of the object handle of the obstacle. Then use the updatePose function to update the pose of the mesh in the
TSDF manager.
dynamicObstacle.Pose = trvec2tform([0.2 0.25 0.2]);
axis equal
title("Updated Dynamic Obstacle Pose")
• meshtsdf supports packNGo (MATLAB Coder) only for MATLAB^® host targets.
[1] Zhao, Hongkai. “A Fast Sweeping Method for Eikonal Equations.” Mathematics of Computation 74, no. 250 (May 21, 2004): 603–27. https://doi.org/10.1090/S0025-5718-04-01678-3.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Version History
Introduced in R2024a | {"url":"https://ch.mathworks.com/help/nav/ref/meshtsdf.html","timestamp":"2024-11-14T20:30:44Z","content_type":"text/html","content_length":"102501","record_id":"<urn:uuid:4666bc3a-5f27-4565-bbd9-333d67905e47>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00732.warc.gz"} |
Bennett Eisenberg
Professor of Mathematics
Department of Mathematics #14
Lehigh University
Bethlehem, PA 18015
Telephone: Office: 610-758-3736 Home: 610-865-5289
e-mail: BE01@Lehigh.edu
Employment history:
● Lehigh University 1972-present
● University of New Mexico 1970-1972
● Cornell University 1967-1970
● Ph.D. 1968, M.I.T.
● A.B. 1964, Dartmouth College
● Coolidge High School, Washington, D.C., 1960
Research areas: Gaussian processes, ergodic theory, stationary processes, sequential statistical tests, epidemiology, record values, geometry and geometric probability, probability theory.
Selected publications categorized according to research area.
● Gaussian processes:
○ The Relation of the Equivalence Conditions for Brownian Motion to the Equivalence Conditions for Certain Stationary Processes, Ann. Math. Stat. 1969.
○ Translating Gaussian Processes, Ann. Math. Stat. 1970.
○ Baxter's Theorem and Varberg's Conjecture. Pacific J. Math. 1972.
○ The Equivalence Singularity Problem for Gaussian Signals in Gaussian Noise. IEEE Trans. on Information Theory. 1986.
● Ergodic theory:
○ Generalized Summing Sequences and the Mean Ergodic Theorem, Proc. Amer. Math. Soc. 1974 (with J. Blum).
○ Ergodic Theory and the Measure of Sets in the Bohr Group, Acta Sci. Math. (Szeged) 1973 (with J. Blum and L-S. Hahn).
○ Ergodic Theorems for Mixing Transformation Groups. Rocky Mt. J. of Math. 1979 (with J. Blum).
● Stationary Processes:
○ Conditions for Metric Transitivity for Stationary Gaussian Processes on Groups. Ann. Math. Stat. 1972 (with J. Blum).
○ A Note on Metric Transitivity for Stationary Gaussian Processes on Groups. Ann. Math. Stat. 1972 (with J. Blum).
○ Linear Estimation of Regression Coefficients. Quarterly of Applied Math. 1974 (with R. Adenstedt).
○ The Law of Large Numbers for Subsequences of a Stationary Process. Annals of Prob. 1975 (with J. Blum).
○ A Note on Random Measures and Moving Averages on Non-Discrete Groups. Ann. Math. Stat. 1973 (with J. Blum).
○ Spectral Properties of Processes Derived from Stationary Gaussian Sequences. Stoch. Proc. and their Appl. 1974.
○ Prediction Theory and Ergodic Spectral Decompositions, Ann. of Prob. 1976 (with J. Blum).
● Sequential statistical tests:
○ Non-Optimality of Likelihood Ratio Tests in the Sequential Detection of Signals in Gaussian Noise. Statistical Decision Theory. 1971.
○ Properties of Generalized Sequential Probability Ratio Tests. Ann. of Stat. 1976 (with B. Ghosh and G. Simons).
○ On Weak Admissibility of Tests. Ann. of Stat. 1978 (with G. Simons ).
○ The Likelihood Ratio and its Applications to Sequential Analysis. Ann. of Stat. 1980 (with B. Ghosh).
○ Curtailed and Uniformly Most Powerful Sequential Tests. Ann. of Stat. 1980 (with B. Ghosh).
○ On the Sample Size of Curtailed Tests. Comm. in Stat.-Theory and Methods. 1981 (with B. Ghosh).
○ The Asymptotic Solution of the Kiefer-Weiss Problem. Sequential Analysis. 1982.
○ Multihypothesis Problems. Handbook of Sequential Analysis. 1991.
○ The Sequential Probability Ratio Test. Handbook of Sequential Analysis. 1991.
○ One-sided SPRT's which Hit the Boundary. Sequential Analysis. 1995
○ Average Sample Number, Kluwer Encyclopedia of Mathematics, 2002.
○ Sequential Probability Ratio Test, Kluwer Encyclopedia of Mathematics, 2002.
○ Discussion on "Likelihood Ratio Identities and Their Application in Sequential Analysis" by T.L. Lai, Sequential Analysis, 2004.
○ Discussion on "Life and Work of Bhaskar Kumar Ghosh" by Pranab Kumar Sen, Sequential Analysis, 2010 (with W. Huang).
● Epidemiology.
○ The Number of Partners and the Probability of HIV Infection. Stat. in Medicine. 1989.
○ The Effect of Variable Infectivity on the Risk of HIV Infection. Stat. in Medicine. 1990.
● Record Values.
○ The Asymptotic Probability of a Tie for First Place. Ann. of Applied Prob. 1993 (with G. Strang and G. Stengle).
○ A Necessary and Sufficient Condition for the Existence of the Limiting Probability of a Tie for First Place. Stat. and Prob. Letters. 1995 (with Y. Baryshnikov and G. Stengle).
○ Minimizing the Probability of a Tie for First Place. J. of Math. Analysis and its Applic. 1996 (with G. Stengle).
○ On the Expectation of the Maximum of I I D Geometric Random Variables, Stat. and Prob. Letters. 2008.
○ The Number of Players Tied for the Record, Stat. and Prob. Letters. 2009.
● Geometry and Geometric Probability
○ Optimal Locations. College Math. J. 1992 (with S. Khabbaz).
○ Random Triangles in n-Dimensions. Amer. Math. Monthly. 1996 (with R. Sullivan).
○ Crofton's Differential Equation. Amer. Math. Monthly. 2000 (with R. Sullivan).
○ The Fundamental Theorem of Calculus in Two Dimensions, Amer. Math. Monthly. 2002 (with R. Sullivan).
○ Surfaces of Revolution in Four Dimensions, Math. Magazine, 2004.
○ A Modification of Sylvester’s Four Point Problem, Math. Magazine, 2011.
● Probability Theory.
○ Another Look at Korovkin's Theorem. J. of Approx. Theory. 1976.
○ Uniform Convergence of Distribution Functions. Proc. Amer. Math. Soc. 1983 (with Shixin Gan).
○ Positive Martingales and their Induced Measures. Proc. Amer. Math. Soc. 1983 (with Shixin Gan).
○ Independent Events in a Discrete Uniform Probability Space. The Amer. Statistician. 1987 (with B. Ghosh).
○ A Variation of a Theorem of Sparre-Andersen, Stat. and Prob. Letters. 1992.
○ Independent Variables with Independent Sum and Difference. J. of Multivariate Anal. 1993 (with Y. Baryshnikov and W. Stadje).
○ Independent Events and Independent Experiments. Proc. Amer. Math.Soc. 1993 (with Y. Baryshnikov).
○ What is the Margin of Error of a Poll? College Math. J. 1997
○ How Much Should You Pay for a Derivative? College Math. J. 1998.
○ Generalizations of Markov's Inequality, Stat. and Prob. Letters. 2001 (with B. Ghosh).
○ Why is the Sum of Independent Normal Random Variables Normal? Math. Magazine, 2008 (with R. Sullivan). | {"url":"https://www.lehigh.edu/~be01/be01.html","timestamp":"2024-11-04T12:04:12Z","content_type":"text/html","content_length":"56805","record_id":"<urn:uuid:6a60cef9-6a02-4705-bcf8-8c199682a14f>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00835.warc.gz"} |
How Do You Solve a Word Problem Where You Multiply and Subtract Whole Numbers and Fractions?
Word problems are a great way to see math in the real world. In this tutorial, you'll see how to translate a word problem to a mathematical equation. Then, see how to use the order of operations to
get the answer! | {"url":"https://virtualnerd.com/common-core/grade-7/7_EE-expressions-equations/B/3/fractions-whole-numbers-arithmetic-word-problem","timestamp":"2024-11-03T11:59:24Z","content_type":"text/html","content_length":"25613","record_id":"<urn:uuid:f8001d1a-8b6d-4fbb-821b-63d07ffd0676>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00530.warc.gz"} |
Paired Samples T Test
What it does: The Paired Samples T Test compares the means of two variables. It computes the difference between the two variables for each case, and tests to see if the average difference is
significantly different from zero.
Where to find it: Under the Analyze menu, choose Compare Means, then choose Paired Samples T Test. Click on both variables you wish to compare, then move the pair of selected variables into the
Paired Variables box.
-Both variables should be normally distributed. You can check for normal distribution with a Q-Q plot.
Null: There is no significant difference between the means of the two variables.
Alternate: There is a significant difference between the means of the two variables.
SPSS Output
Following is sample output of a paired samples T test. We compared the mean test scores before (pre-test) and after (post-test) the subjects completed a test preparation course. We want to see if our
test preparation course improved people's score on the test.
First, we see the descriptive statistics for both variables.
The post-test mean scores are higher.
Next, we see the correlation between the two variables.
There is a strong positive correlation. People who did well on the pre-test also did well on the post-test.
Finally, we see the results of the Paired Samples T Test. Remember, this test is based on the difference between the two variables. Under "Paired Differences" we see the descriptive statistics for
the difference between the two variables.
To the right of the Paired Differences, we see the T, degrees of freedom, and significance.
The T value = -2.171
We have 11 degrees of freedom
Our significance is .053
If the significance value is less than .05, there is a significant difference.
If the significance value is greater than. 05, there is no significant difference.
Here, we see that the significance value is approaching significance, but it is not a significant difference. There is no difference between pre- and post-test scores. Our test preparation course did
not help!
Home | Review Test | Decision Tree | Procedure List
Susan Archambault
Psychology Department, Wellesley College
Created By: Nina Schloesser '02
Created On: July 30, 2000
Last Modified: July 31, 2000 | {"url":"http://academics.wellesley.edu/Psychology/Psych205/pairttest.html","timestamp":"2024-11-13T21:30:17Z","content_type":"text/html","content_length":"5144","record_id":"<urn:uuid:934c8fba-3ec1-4425-a31c-c174e555b3e0>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00172.warc.gz"} |
hird Grade Word Problem Activity - Choose the Operation
Help your third grade students become word problem wizards with this fun, interactive math game from iKnowIt.com! Students will put their word problem skills to the test as they decide which
operation, addition, subtraction, multiplication, or division, is needed to solve the word problem. Here are the learning objectives for this online math activity:
• Solve two-step word problems using the four operations.
• Decide whether addition, subtraction, multiplication, or division must be used to solve the given word problem.
• Become confident and proficient identifying and using each of the four operations to solve word problems.
Students will fill in the blank with the correct solution to each word problem in this online math lesson. If students need a little extra help solving a word problem, they can click on the "Hint"
button to view a written or pictorial clue that will help them decide whether to add, subtract, multiply, or divide without giving away the answer. When students get an answer wrong, a detailed
explanation page will appear to guide them through the steps needed to obtain a correct response. As students progress through the math lesson, they will have plenty of opportunities to learn from
past mistakes.
This third grade word problem game comes equipped with several features to help students make the most of their math practice session:
• A read-aloud feature, indicated by the speaker icon in the upper-left corner of the practice screen, can be used by students if they wish to hear the question read out loud to them in a clear
voice. This option is a fantastic resource for children who are auditory processors, as well as ESL/ELL students.
• A progress-tracker in the upper-right corner of the practice screen shows students how many questions they have answered so far out of the total number of questions in the math activity.
• A score-tracker lets students see how many points they have achieved for answering questions correctly.
All of these math features were designed to help your students achieve mastery of their math skills with our interactive math practice program.
Interactive Math Activities Your Class Will Love
Looking for a way to get your third grade class excited about math practice? Look no further than iKnowIt.com! Elementary math teachers, homeschool educators, and school administrators enjoy using
the I Know It math practice program alongside a comprehensive elementary math curriculum to help little ones engage with the math material they are learning in class with kid-friendly, fun math
activities. Here are a few stand-out features of our online math program:
• Hundreds of interactive math activities aligned to the Common Core Standard and written by accredited elementary math teachers just like you
• Kid-friendly math activities for students from kindergarten through fifth grade, covering essential math topics including addition, subtraction, multiplication, division, and much more
• Easy access to math practice assignments, student progress reports, and administrative features that help you maximize your students' math practice experience
Students, too, have lots to love about digital math activities from iKnowIt.com:
• Bright colors, and an engaging, kid-friendly math lesson interface
• Age-appropriate emojis and cute, animated characters to encourage students throughout their math practice sessions and make learning fun
• Math awards given for each new math skill mastered with I Know It
We hope you and your third-grade students will enjoy practicing word problems with all four operations in this interactive math game! Be sure to check out the hundreds of other third grade math
lessons we have available on our website as well.
Free Trial and Membership Options
Are you searching for a way to test the waters and try out the I Know It math program with your class? Great news! You can sign up for a free thirty-day trial of I Know It and try out any of the math
activities on our website at no cost! We hope you and your students will love experiencing the difference interactive math practice can make. When your free trial runs out, consider joining the I
Know It community as a member so you can continue to experience the benefits of interactive, online math practice the whole year through. We have membership options for families, teachers, schools,
and school districts. Which membership is right for you? Visit our membership information page to find out: https://www.iknowit.com/order.html.
One of the biggest advantages to your I Know It membership is your teacher or parent administrator account, which will help you maximize your students' math practice experience by:
• Creating a class roster, assigning a class code, and giving each of your students a unique username and password
• Changing basic math lesson settings, such as limiting the number of available hints per math lesson
• Viewing detailed student progress reports and printing, downloading, and emailing student progress reports on demand
Your students will log into iKnowIt.com with their unique login credentials to view a kid-friendly version of the homepage. From here, they can quickly and easily access the math assignments you have
given them. They can also explore other math activities at their grade level and beyond for an extra challenge or additional practice, if you choose to give them this option through your
administrator account. Grade levels are designated by letter (i.e., "Level C" for third grade), making it easy for you to assign math activities based on each child's needs and skill level.
This online math lesson is classified as Level C. It may be ideal for a third-grade class.
Common Core Standard
3.OA.8, MA.3.AR.1.2, 3.6D
Operations And Algebraic Thinking
Solve Problems Involving The Four Operations, And Identify And Explain Patterns In Arithmetic.
Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental
computation and estimation strategies including rounding.3
You might also be interested in...
Multiple Step Word Problems (Add and Subtract) (Level C)
In this third grade-level math lesson, students will practice solving multiple-step word problems with addition and subtraction. Questions are presented in fill-in-the-blank format.
Multiple Step Word Problems (Multiply and Divide) (Level C)
In this math lesson geared toward third-grade, students will practice solving multiple-step word problems with multiplication and division. Questions are presented in fill-in-the-blank format and
multiple-choice format. | {"url":"https://www.iknowit.com/lessons/c-word-problems-choose-the-operation-four-operations.html","timestamp":"2024-11-08T22:25:10Z","content_type":"text/html","content_length":"420621","record_id":"<urn:uuid:d002ab69-1779-47d6-a690-ad101d09850a>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00002.warc.gz"} |
Bootstrap Tutorial#
This notebook contains a tutorial on how to use the bootstrap functionality provided by estimagic. We start with the simplest possible example of calculating standard errors and confidence intervals
for an OLS estimator without as well as with clustering. Then we progress to more advanced examples.
In the example here, we will work with the “exercise” example dataset taken from the seaborn library.
The working example will be a linear regression to investigate the effects of exercise time on pulse.
import numpy as np
import pandas as pd
import seaborn as sns
import statsmodels.api as sm
import estimagic as em
Prepare the dataset#
df = sns.load_dataset("exercise", index_col=0)
replacements = {"1 min": 1, "15 min": 15, "30 min": 30}
df = df.replace({"time": replacements})
df["constant"] = 1
/tmp/ipykernel_2762/2496026297.py:3: FutureWarning: Downcasting behavior in `replace` is deprecated and will be removed in a future version. To retain the old behavior, explicitly call `result.infer_objects(copy=False)`. To opt-in to the future behavior, set `pd.set_option('future.no_silent_downcasting', True)`
df = df.replace({"time": replacements})
/tmp/ipykernel_2762/2496026297.py:3: FutureWarning: The behavior of Series.replace (and DataFrame.replace) with CategoricalDtype is deprecated. In a future version, replace will only be used for cases that preserve the categories. To change the categories, use ser.cat.rename_categories instead.
df = df.replace({"time": replacements})
│ │id│ diet │pulse│time│kind│constant │
│0│1 │low fat │85 │1 │rest│1 │
│1│1 │low fat │85 │15 │rest│1 │
│2│1 │low fat │88 │30 │rest│1 │
│3│2 │low fat │90 │1 │rest│1 │
│4│2 │low fat │92 │15 │rest│1 │
Doing a very simple bootstrap#
The first thing we need is a function that calculates the bootstrap outcome, given an empirical or re-sampled dataset. The bootstrap outcome is the quantity for which you want to calculate standard
errors and confidence intervals. In most applications those are just parameter estimates.
In our case, we want to regress “pulse” on “time” and a constant. Our outcome function looks as follows:
def ols_fit(data):
y = data["pulse"]
x = data[["constant", "time"]]
params = sm.OLS(y, x).fit().params
return params
In general, the user-specified outcome function may return any pytree (e.g. numpy.ndarray, pandas.DataFrame, dict etc.). In the example here, it returns a pandas.Series.
Now we are ready to calculate confidence intervals and standard errors.
results_without_cluster = em.bootstrap(data=df, outcome=ols_fit)
(constant 90.836520
time 0.154413
dtype: float64,
constant 96.714968
time 0.647588
dtype: float64)
constant 1.502959
time 0.128869
dtype: float64
The above function call represents the minimum that a user has to specify, making full use of the default options, such as drawing a 1_000 bootstrap draws, using the “percentile” bootstrap confidence
interval, not making use of parallelization, etc.
If, for example, we wanted to take 10_000 draws, while parallelizing on two cores, and using a “bc” type confidence interval, we would simply call the following:
results_without_cluster2 = em.bootstrap(
data=df, outcome=ols_fit, n_draws=10_000, n_cores=2
(constant 91.392978
time 0.195011
dtype: float64,
constant 96.279037
time 0.613515
dtype: float64)
Doing a clustered bootstrap#
In the cluster robust variant of the bootstrap, the original dataset is divided into clusters according to the values of some user-specified variable, and then clusters are drawn uniformly with
replacement in order to create the different bootstrap samples.
In order to use the cluster robust boostrap, we simply specify which variable to cluster by. In the example we are working with, it seems sensible to cluster on individuals, i.e. on the column “id”
of our dataset.
results_with_cluster = em.bootstrap(data=df, outcome=ols_fit, cluster_by="id")
constant 1.206389
time 0.103234
dtype: float64
We can see that the estimated standard errors are indeed of smaller magnitude when we use the cluster robust bootstrap.
Finally, we can compare our bootstrap results to a regression on the full sample using statsmodels’ OLS function. We see that the cluster robust bootstrap yields standard error estimates very close
to the ones of the cluster robust regression, while the regular bootstrap seems to overestimate the standard errors of both coefficients.
Note: We would not expect the asymptotic statsmodels standard errors to be exactly the same as the bootstrapped standard errors.
y = df["pulse"]
x = df[["constant", "time"]]
cluster_robust_ols = sm.OLS(y, x).fit(cov_type="cluster", cov_kwds={"groups": df["id"]})
Splitting up the process#
In many situations, the above procedure is enough. However, sometimes it may be important to split the bootstrapping process up into smaller steps. Examples for such situations are:
1. You want to look at the bootstrap estimates
2. You want to do a bootstrap with a low number of draws first and add more draws later without duplicated calculations
3. You have more bootstrap outcomes than just the parameters
1. Accessing bootstrap outcomes#
The bootstrap outcomes are stored in the results object you get back when calling the bootstrap function.
result = em.bootstrap(data=df, outcome=ols_fit, seed=1234)
my_outcomes = result.outcomes
[constant 93.732040
time 0.580057
dtype: float64,
constant 92.909468
time 0.309198
dtype: float64,
constant 94.257886
time 0.428624
dtype: float64,
constant 93.872576
time 0.410508
dtype: float64,
constant 92.076689
time 0.542170
dtype: float64]
To further compare the cluster bootstrap to the uniform bootstrap, let’s plot the sampling distribution of the parameters on time. We can again see that the standard error is smaller when we cluster
on the subject id.
result_clustered = em.bootstrap(data=df, outcome=ols_fit, seed=1234, cluster_by="id")
my_outcomes_clustered = result_clustered.outcomes
# clustered distribution in blue
pd.DataFrame(my_outcomes_clustered)["time"], kde=True, stat="density", linewidth=0
# non-clustered distribution in orange
Calculating standard errors and confidence intervals from existing bootstrap result#
If you’ve already run bootstrap once, you can simply pass the existing result object to a new call of bootstrap. Estimagic reuses the existing bootstrap outcomes and now only draws n_draws -
n_existing outcomes instead of drawing entirely new n_draws. Depending on the n_draws you specified (this is set to 1_000 by default), this may save considerable computation time.
We can go on and compute confidence intervals and standard errors, just the same way as before, with several methods (e.g. “percentile” and “bc”), yet without duplicated evaluations of the bootstrap
outcome function.
my_results = em.bootstrap(
(constant 90.709236
time 0.151193
dtype: float64,
constant 96.827145
time 0.627507
dtype: float64)
You can use this to calculate confidence intervals with several methods (e.g. “percentile” and “bc”) without duplicated evaluations of the bootstrap outcome function.
2. Extending bootstrap results with more draws#
It is often the case that, for speed reasons, you set the number of bootstrap draws quite low, so you can look at the results earlier and later decide that you need more draws.
As an example, we will take an initial sample of 500 draws. We then extend it with another 1500 draws.
Note: It is very important to use a different random seed when you calculate the additional outcomes!!!
initial_result = em.bootstrap(data=df, outcome=ols_fit, seed=5471, n_draws=500)
(constant 90.768859
time 0.137692
dtype: float64,
constant 96.601067
time 0.607616
dtype: float64)
combined_result = em.bootstrap(
data=df, outcome=ols_fit, existing_result=initial_result, seed=2365, n_draws=2000
(constant 90.689112
time 0.128597
dtype: float64,
constant 96.696522
time 0.622954
dtype: float64)
3. Using less draws than totally available bootstrap outcomes#
You have a large sample of bootstrap outcomes but want to compute summary statistics only on a subset? No problem! Estimagic got you covered. You can simply pass any number of n_draws to your next
call of bootstrap, regardless of the size of the existing sample you want to use. We already covered the case where n_draws > n_existing above, in which case estimagic draws the remaining bootstrap
outcomes for you.
If n_draws <= n_existing, estimagic takes a random subset of the existing outcomes - and voilà!
subset_result = em.bootstrap(
data=df, outcome=ols_fit, existing_result=combined_result, seed=4632, n_draws=500
(constant 90.619182
time 0.130242
dtype: float64,
constant 96.557777
time 0.625645
dtype: float64)
Accessing the bootstrap samples#
It is also possible to just access the bootstrap samples. You may do so, for example, if you want to calculate your bootstrap outcomes in parallel in a way that is not yet supported by estimagic
(e.g. on a large cluster or super-computer).
from estimagic.bootstrap_samples import get_bootstrap_samples
rng = np.random.default_rng(1234)
my_samples = get_bootstrap_samples(data=df, rng=rng)
│ │id │ diet │pulse│time│ kind │constant │
│88 │30 │no fat │111 │15 │running│1 │
│87 │30 │no fat │99 │1 │running│1 │
│88 │30 │no fat │111 │15 │running│1 │
│34 │12 │low fat│103 │15 │walking│1 │
│15 │6 │no fat │83 │1 │rest │1 │
│...│...│... │... │... │... │... │
│78 │27 │no fat │100 │1 │running│1 │
│77 │26 │no fat │143 │30 │running│1 │
│87 │30 │no fat │99 │1 │running│1 │
│29 │10 │no fat │100 │30 │rest │1 │
│75 │26 │no fat │95 │1 │running│1 │
90 rows × 6 columns | {"url":"https://estimagic.org/en/latest/estimagic/tutorials/bootstrap_overview.html","timestamp":"2024-11-05T16:30:00Z","content_type":"text/html","content_length":"58631","record_id":"<urn:uuid:76c2d8e8-0ff3-4aee-9011-d2c675a0b053>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00391.warc.gz"} |
Achievement objective
Achievement objective S8-4
In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:
• Investigate situations that involve elements of chance:
□ A. calculating probabilities of independent, combined, and conditional events
□ B. calculating and interpreting expected values and standard deviations of discrete random variables
□ C. applying distributions such as the Poisson, binomial, and normal.
A. Calculating probabilities of independent, combined, and conditional events:
Students investigate probability situations involving real data using discrete and continuous numerical variables. They can be investigated by making assumptions about the situation and applying
probability rules and/or by doing repeated trials of the situation and collecting frequencies. Students use random variables and their probability distributions.
B. Calculating and interpreting expected values and standard deviations of discrete random variables:
A statistical data set may contain discrete numerical variables. These have frequency distributions that can be converted to empirical probability distributions. Distributions from both sources have
the same set of possible features (centre, spread, clusters, shape, tails, and so on) and we can calculate the same measures (mean, SD, and so on) for them.
• Makes a reasonable estimate of mean and standard deviation from a plot of the distribution of a discrete random variable.
• Solves and interprets solutions of problems involving calculation of mean, variance and standard deviation from a discrete probability distribution.
• Solves and interprets solutions of problems involving linear transformations and sums (and differences) of discrete random variables.
C. Applying distributions such as the Poisson, binomial, and normal:
They learn that some situations that satisfy certain conditions can be modelled mathematically. The model may be Poisson, binomial, normal, uniform, triangular, or others, or be derived from the
situation being investigated.
• Recognises situations in which probability distributions such as Poisson, binomial, and normal are appropriate models, demonstrating understanding of the assumptions that underlie the
• Selects and uses an appropriate distribution to model a situation in order to solve a problem involving probability.
• Selects and uses an appropriate distribution to solve a problem, demonstrating understanding of the link between probabilities and areas under density functions for continuous outcomes (for
example, normal, triangular, or uniform, but nothing requiring integration).
• Selects and uses an appropriate distribution to solve a problem, demonstrating understanding of the way a probability distribution changes as the parameter values change.
• Selects and uses an appropriate distribution to solve a problem involving finding and using estimates of parameters.
• Selects and uses an appropriate distribution to solve a problem, demonstrating understanding of the relationship between true probability (unknown and unique to the situation), model estimates
(combining theoretical probability and assumptions about the situation) and experimental estimates.
• Uses a distribution to estimate and calculate probabilities, including by simulation.
S8-4 links from S7-4.
Possible context elaborations
• CensusAtSchool is a valuable website for classroom activities and information for teachers on all things statistics.
• Investigate the number of hokey pokey pieces in scoops of ice cream.
• Investigate whether the normal distribution can be used to model: IQ, jockey’s heights, exam marks, and house prices.
• Conduct an experiment to investigate:
□ whether you have ESP
□ the probability of passing a 30 question multichoice test with four options if you guess all the answers.
Assessment for qualifications
NCEA achievement standards at levels 1, 2 and 3 have been aligned to the New Zealand Curriculum. Please ensure that you are using the correct version of the standards by going to the NZQA website.
The NZQA subject-specific resources pages are very helpful. From there, you can find all the achievement standards and links to assessment resources, both internal and external.
The following achievement standard(s) could assess learning outcomes from this AO:
Last updated September 17, 2018 | {"url":"https://seniorsecondary.tki.org.nz/Mathematics-and-statistics/Achievement-objectives/AOs-by-level/AO-S8-4","timestamp":"2024-11-06T08:00:47Z","content_type":"application/xhtml+xml","content_length":"258644","record_id":"<urn:uuid:d33c7b00-76de-44de-a05a-77c0bc5e6b23>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00633.warc.gz"} |
MA-INF 1218: Algorithms and Uncertainty
When Where Start Lecturer
Monday, 12:15-13:45 Friedrich-Hirzebruch Allee 5 - Hörsaal 3 October 7 Kesselheim
Wednesday, 12:15-13:45 Friedrich-Hirzebruch Allee 5 - Hörsaal 3 October 9 Kesselheim
When Where Start Lecturer
Wednesday, 14:15-15:45 Friedrich-Hirzebruch-Allee 8 - October 9 Lehming
Seminarraum 2.050, Informatik V
Thursday, 10:15-11:45 Friedrich-Hirzebruch-Allee 8 - October 10 Lehming
Seminarraum 2.050, Informatik V
In many application scenarios, algorithms have to make decisions under some kind of uncertainty. This affects different kinds of problems. For example, when planing a route, a navigation system
should take into consideration the traffic. Also, any machine-learning problem is about some kind of uncertainty. A random sample of data is used as a representative for the entire world.
In this course, we will get to know different techniques to model uncertainty and what approaches algorithms can use to cope with it. We will cover topics such as
You should bring a solid background in algorithms, calculus, and probability theory. Specialized knowledge about certain algorithms is not necessary.
There is a requirement for participating in the exams. Once during the semester, you need to present the solution of a homework problem in one of the tutorials. If you would like to present a
solution, please send an email to rlehming(at)uni-bonn(dot)de until Monday at the latest. If multiple people want to do the same exercise, it is first come, first serve. Before you present it to
everyone, we will schedule a short meeting (10-15min) for a quick discussion of your solution. | {"url":"https://tcs.cs.uni-bonn.de/doku.php/teaching/ws2425/vl-aau","timestamp":"2024-11-05T06:27:53Z","content_type":"text/html","content_length":"13959","record_id":"<urn:uuid:636f5dfb-0924-44c1-9f21-840c105be0b9>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00388.warc.gz"} |
ball mill operation
WEBThe starting point for ball mill media and liquid charging is generally as follows: 50% media charge. Assuming 26% void space between spherical balls (nonspherical, irregularly shaped and
mixedsize media will increase or decrease the free space) 50% x 26% = 13% free space. Add to this another 45% to 50% above the ball charge for total of 58% ...
WhatsApp: +86 18838072829
WEB13 Milling Operations Explained with Appliions. March 11, 2023. Ball screw vs Lead Screw: Which is Better? February 3, 2023. WAZER Waterjet Review [2023]: Desktop Waterjet. January 8, 2023.
CNC Tool Changer Guide: Benefits and Cost. January 5, 2023. Parts of a CNC Milling Machine: Visual Guide.
WhatsApp: +86 18838072829
WEBBall Mill_Operation, Inspection Optimization Free download as PDF File (.pdf), Text File (.txt) or view presentation slides online. The document discusses parameters for quality finish milling
including cement strength, Blaine surface area, and residue percentages. It also covers inspection of ball mills including measuring filling degrees, conducting .
WhatsApp: +86 18838072829
WEBA wet ball milling operation is used to grind crude ore into smaller particles of a desired size classifiion. As shown in the diagram below a slurry of crude ore and water (Stream 1) is fed
into a ball mill along with a recycle stream (Stream 2). The mill product (Stream 3) is sent to a mechanical classifier where additional water ...
WhatsApp: +86 18838072829
WEBBall Mill Types: MQ series, MQS series, ... At the same time, during the operation of the mill, the sliding movement of the grinding media to each other also produces a grinding effect on the
raw materials. The rest material is discharged through a discharge hollow shaft. Due to the constant uniform feeding, the pressure causes the material in ...
WhatsApp: +86 18838072829
WEBSep 22, 2023 · For the ball milling operation, the influence of particle size distribution on the grinding kinetics is also wellknown now [8, 14, 23, 24, 34, 35]. Attention should be paid to
these factors while designing experiments for developing quantitative correlations for mill scaleup design work.
WhatsApp: +86 18838072829
WEBDOVE small Ball Mills designed for laboratories ball milling process are supplied in 4 models, capacity range of (200g/h1000 g/h). For small to large scale operations, DOVE Ball Mills are
supplied in 17 models, capacity range of ( TPH – 80 TPH). With over 50 years experience in Grinding Mill Machine fabriion, DOVE Ball Mills as ...
WhatsApp: +86 18838072829
WEBJan 10, 2023 · Once you have let the mill run for a while, it is time to perform maintenance. Here are the steps for ball mill maintenance: Shut off the ball mill and disconnect the power
supply. Check the base ...
WhatsApp: +86 18838072829
WEBIt is a ball milling process where a powder mixture placed in the ball mill is subjected to highenergy collision from the balls. This process was developed by Benjamin and his coworkers at the
International Nickel Company in the late of 1960. ... planetary mill or a horizontal ball mill. However, the principles of these operations are same for ...
WhatsApp: +86 18838072829
WEBNov 12, 2020 · The KENNAMETAL HARVI I TE fourflute, ballnose end mill is designed for highproductivity 3D roughing and finishing operations, lowering machining cost through maximum metal
removal in a broad range of materials.. The end mill features an innovative proprietary relief that requires a closer look. In the most critical area—the ballnose .
WhatsApp: +86 18838072829
WEBAug 1, 1987 · For the operation of ball mills, constantvalue feedback control is generally adopted to increase productivity and secure quality . There are two types of control systems
popularly used for ball mill operation . One uses the mill sound level and the other uses the bucket elevator power as the controlled variable, while the manipulated ...
WhatsApp: +86 18838072829
WEBImage credit: Rexnord Corporation. An inching drive is used as an auxiliary system to the main drive for a large machine such as a ball mill, industrial kiln, conveyor, or elevator. Its
purpose is to turn the equipment at a speed slower than the normal operating speed — typically 1 to 2 rpm, although fractional rpms are also common — and to ...
WhatsApp: +86 18838072829
WEBPlanetary ball mills with higher energy input and a speed ratio of 1: or even 1:3 are mainly used for mechanochemical appliions. Click to view video. Planetary ball mills Fields of appliion.
Planetary ball mills are used for the pulverization of soft, hard, brittle, and fibrous materials in dry and wet mode. Extremely high ...
WhatsApp: +86 18838072829
WEBThe strategic selection of grinding media is crucial for maximizing the efficiency and effectiveness of ball milling operations. As illustrated through various considerations and realworld
examples, the right choice of grinding media can dramatically influence the quality of the final product, operational costs, and overall process success. ...
WhatsApp: +86 18838072829
WEBThe cement ball mill can perform dry process production and also wet process production, moreover it can do grinding and drying at the same time. CHAENG cement ball mill has features of small
investment, high rate of return, simple process and easy operation. Advantages of CHAENG cement ball mill: 1.
WhatsApp: +86 18838072829
WEBAug 17, 2021 · C. Ball mill is an open system, hence sterility is a question D. Fibrous materials cannot be milled by ball mill. 10. What particle size can be obtained through ball mill? A. 20
to 80 mesh B. 4 to 325 mesh C. 20 to 200 mesh D. 1 to 30 mm. ANSWERS:1. Both B and C 2. Optimum speed 3. Longitudinal axis 4. Both 5. A – 3 B – 4 C – 2 D – 1 6 ...
WhatsApp: +86 18838072829
WEBNov 1, 2022 · In numerous cement ball mill operations (Genc, 2008, Tsakalakis and Stamboltzis, 2008, Altun, 2018, Ghalandari and Iranmanesh, 2020), the ratio of maximum ball size to minimum
ball size for coarse milling compartments lies between and, while a wider range has been indied for the fine milling compartments. In most of the .
WhatsApp: +86 18838072829
WEBFlexible drive solutions for use in ball mills A ball mill is a horizontal cylinder filled with steel balls or the like. This cylinder rotates around its axis and transmits the rotating effect
to the balls. ... Our durable and resilient drive solutions ensure a reliable ball mill operation Drive packages from a single source: industrial gear ...
WhatsApp: +86 18838072829
WEBMar 30, 2022 · Advantages of Ball Mill. Produces a very fine powder – particle size less than or equal to 10 microns. Suitable for milling toxic materials as it can be used in an enclosed
form. A wide range of appliions. Used for continuous operation.
WhatsApp: +86 18838072829
WEBOct 19, 2016 · Ball Mill Sole Plate. This crown should be between .002″ and . 003″, per foot of length of sole plate. For example, if the sole plate is about 8′ long, the crown should be
between .016″ and .024″. Ball Mill Sole Plate. After all shimming is completed, the sole plate and bases should be grouted in position.
WhatsApp: +86 18838072829
WEBThe required speed of the ball mill with grinding media with innovative lifters at aract mode of operation for the three types of materials is almost the same with an average value of 45% of
WhatsApp: +86 18838072829
WEBBALL MILL OPERATION Free download as Powerpoint Presentation (.ppt), PDF File (.pdf), Text File (.txt) or view presentation slides online. This document provides guidelines for operating ball
mills, including prerequisites and operating principles. Some key points covered include: Material specifiions for raw mills and cement mills in terms of .
WhatsApp: +86 18838072829 | {"url":"https://www.villa-aquitaine.fr/May-19/7823.html","timestamp":"2024-11-04T18:28:16Z","content_type":"application/xhtml+xml","content_length":"21562","record_id":"<urn:uuid:e757cbf1-fca5-4060-86d2-e9fa6ddd3b96>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00314.warc.gz"} |
Calculating Machining Time For Any Machining Operation
It is often necessary for CNC people to determine how long machining operations will take to perform. You may be trying to determine which of two or more processes will be used to machine a workpiece
- or you may just be wondering how long a machining operation will require to complete.
Frankly speaking, the formulae related to calculating machining time are pretty simple to understand and use. Indeed, many manufacturing people have incorporated them into spreadsheets (like
Microsoft Excel) - or they have programmed their calculators to include the related formulae. Here is the most important formula:
• Time (minutes) = length of motion in inches divided by motion rate in inches per minute
That's it - no problem, right? You simply divide the length of the motion required for machining in inches by the inches per minute feedrate. The metric equivalent is:
• Time (minutes) = length of motion in millimeters divided by motion rate in millimeters per minute
We'll be using the inch mode for the rest of the discussions in this article.
Example 1:
Say you must drill a 1.0 inch diameter hole. The hole depth is 0.75 and you intend to use an approach distance of 0.1 inch. The intended feedrate is 7.0 inches per minute. When we divide the motion
distance (0.85) by the feedrate (7.0), we find that the time needed to drill this hole is 0.12143 minutes.
How many seconds is this? We obviously need to be able to convert decimal minutes (0.12143) into seconds. Here's the formulae:
• 1 second = 0.01666 minutes
• Time in seconds = time in minutes divided by 0.01666
When we divide 0.12143 by 0.01666, the result is 7.2887 seconds (just over 7-1/4 seconds). So we now know how long it will take to drill the hole.
In order to use the formula, of course, you must be able to determine the feedrate in inches per minute (ipm). Most machining data handbooks, however, provide feedrate in inches per revolution (ipr),
meaning you must first calculate the spindle rpm and then calculate the inches per minute feedrate. But speed recommendations are usually given in surface feet per minute (sfm). This speed is how
much workpiece material will pass by each cutting edge during one minute. Here are two more formulae, based on speed being recommended in sfm and feedrate in ipr.
• rpm = 3.82 times sfm divided by diameter (the tool diameter in our case)
• ipm = rpm times ipr
Note that for some tools, the recommendation for feedrate will be in "per tooth" fashion, meaning you need to know the number of cutting edges (inserts, flutes, or teeth) there are on the cutting
tool. This is commonly the case for milling operations. So we need to add yet one more formula:
• ipr = ipt times number of cutting edges
Example 2:
Say you need to determine how long it will take to rough mill a 3.0 inch long slot with a 0.75 diameter, four flute, cobalt end mill. The three inch motion distance is the total motion length,
including feed-on and feed-off distances. Based upon the material you are machining and the kind of machining operation you are going to perform (rough milling), the end mill's manufacture recommends
a speed of 90 sfm and a feedrate of 0.002 ipt.
• First, determine the speed in rpm: 3.82 times 90 divided by 0.75 is 458 rpm.
• Next determine the inches per revolution feedrate: 4 times 0.002 is 0.008 ipr.
• Next, determine the inches per minute feedrate: 458 times 0.008 is 3.664 ipm.
• Finally, determine the time required in minutes: 3 inches of motion divided by 3.664 ipm is 0.8187 minutes.
To determine the number of seconds, divide 0.8187 by 0.01666 - this comes out to 49.141 seconds.
• Fixed diameter machining versus changing-diameter machining
Note that it is quite easy to apply these formulae to machining center machining operations since the cutting tool diameter does not change during the machining operation. This is the case for the
vast majority of cutting operations, including milling cutters, drills, taps, reamers, and just about any tool you use in a milling machine or CNC machining center. Again, the diameter being machined
does not change during machining.
But do note that there are some operations during which the diameter being machined will change during the machining operation. Consider, for example, a rough turning operation on a CNC turning
center that requires multiple passes to be made. The feature called constant surface speed will cause the spindle speed in rpm to change based upon the diameter being machined. For rough turning,
this means you must calculate a new rpm and inches per minute feedrate for each rough turning pass.
Say you need to rough turn a 4.0 inch long diameter down from 1.0 inch to 0.75 inches, taking two passes (0.125 inch each). One of the passes will be at 0.875 and the other will be at 0.75. And each
pass will be 4.1 inches long, including the approach. For the material being machined and the machining operation being performed, the cutting tool manufacturer recommends a speed of 400 sfm and a
feedrate of 0.011 ipr.
Again each pass must be calculated separately. For the first pass:
• rpm = 3.82 times 400 divided by 0.875, or 1,746 rpm
• ipm = 0.011 times 1,746, or, 19.206 ipm
• time = 4.1 divided by 19.206, or 0.213 minutes (12.785 seconds)
• rpm = 3.82 times 400 divided by 0.75, or 2,037 rpm
• ipm = 0.011 times 2,037, or 22.407 ipm
• time = 4.1 divided by 22.407, or 0.182 minutes (10.924 seconds)
As you can see, the calculations are no more difficult to make - there are just more of them. One per roughing pass.
Calculating time for finish turning and boring operations done on a CNC turning center are also more complicated. To do it perfectly, you must treat each segment being machined separately. For this
reason, many quoting people will try to come up with an "average" diameter on which to base the rpm calculation. This allows the to more quickly come up with a pretty accurate machining time.
Diameter changes while machining
There are even CNC turning center operations that require the diameter to change even while the cutting tool in engaged with the workpiece. The two most common are facing and necking operations
(including cut-off operations). If constant surface speed is used (as it should be), the speed in rpm will accelerate as a facing tool moves toward the center of the workpiece. Again, most estimators
will try to come up with an average diameter in order to quickly determine approximate machining time. | {"url":"https://www.cncci.com/post/calculating-machining-time-for-any-machining-operation","timestamp":"2024-11-14T15:20:08Z","content_type":"text/html","content_length":"1049925","record_id":"<urn:uuid:c9099dbd-9cf9-444c-9957-6e2ddbe73874>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00462.warc.gz"} |
Return on Investment (ROI)
In this article
Table of contents
Articles in this category
Return on Investment (ROI)
Steps to Calculating ROI in Solar
Over the last two decades, the cost of solar energy systems has come down drastically. As the demand for solar power systems continues to rise, the costs are further decreasing. In an NREL report it
has been noted that from 2006 to 2016, the average cost of a system dropped from $9/watt to approximately $4/watt and even to $2/watt when self-installed. An important parameter for profitability
assessment of a solar installation is to evaluate its Return on investment or the ROI. It is important to take advantage of the RECs, the federal and state tax incentives to reduce the initial costs
of the system, and thus increase the returns on investment (ROI) of the system.
Follow these steps while calculating the ROI of a solar installation:
- It is important to review the electricity bills to determine the average electricity consumption in a year. Local utility statements can help figure out the amount. - Determine the total installed
cost of a solar energy system. This includes expenses such as one spent on PV panels, installation costs, permits, etc. - From the utility bill determine how much you pay per kilowatt-hour (kWh). -
Next, figure out how much electricity your solar panels will generate. Factor snow load, panel degradation factor, etc that negatively impact the energy production. - Identify the types of federal,
state, and local government incentives you can take advantage of towards the installation of your solar panels. Calculate the total financial benefit that your system can avail. This includes any
RECs you can sell to the utility company. All these benefits will reduce the initial cost of your system and increase your ROI. - Find out available financing options. - Calculate your return on
investment (ROI) using the following steps shown in the example below. Here is an example: Let's say we have a 5040-watt system Assume 5 $/watt. So, initial cost of system = $ 5x 5040 = $25200 5040
W system is about 26.1 m^2 in area (using a sun power type of panel) Let's say the place of interest is in North America and has insolation of 5.5 Kwh/m^2/day So per day intensity = 26.1 m^2 x5.5KWh/
m^2/day = 144 DC KWh/day (hitting the 26.1 m^2 panels) Now find out the efficiency of solar panels – In the current case Sunpower panels have efficiency of 19.3 % So per day intensity = 144KWh/day
x0.193 =27.8 DC KWh/day Let's say the area experiences snow, so above figure will get reduced = 27.8x 0.95 =26.4 DC/KWh/day Assume Panel derate factor of 82 % (derate factor is efficiency of DC to AC
conversion) Revised intensity =26.4 DC KWh /day x 0.82 =21.6 AC KWh/day Convert this to per year = 21.6 AC KWh/day x365 =7884 KWh /year ----àEq. (1) But panels degrade over time, hence Equation (1)
minus degradation @ 0.8% compounded annually for 25 years = Average electricity generation over 25 years = 7095 KWh per year -----àEq. (2) Assume utility electricity is priced @ 10 c/KWh = Equation
(2) ÷10 = 7095 ÷ 10 = $709.5 = $710 We had spent @25200 as initial investment. So ROI = (what we earned) ÷ (what we invested)= 710 ÷ 25200 =2.8% return on investment Note that we haven’t
applied any incentives/rebates/RECs as yet. If we apply federal incentives, rebates, and REC payments to the initial cost of the system, then ROI will significantly improve, as the initial cost will
come down by a big margin For example Initial cost after application of incentives in the above example will be: $25200 – (federal tax credit @30% lets say) – (local incentives) = $Y So new ROI
= (710 + REC payments) ÷ ($ Y) = $ Z Note that $Z will be a lot more and can be as close to 10 -20 % as new ROI is based on rebates/incentives and RECs availed by the customer. | {"url":"https://sinovoltaics.com/learning-center/finance/return-on-investment-roi/","timestamp":"2024-11-05T00:18:15Z","content_type":"text/html","content_length":"104971","record_id":"<urn:uuid:2d0087e2-3fc8-4a39-980a-1e0c6b3e8558>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00692.warc.gz"} |
During my twenty-plus years as a freelance writer, I have written more than 100 (and probably closer to 200) articles about various fields of science. The magazines I have published in include
Science, American Scientist, New Scientist, Discover, Scientific American, Science News, Smithsonian, and Nautilus. My wife likes to joke, “Every magazine with ‘science’ in the title!”
In the past I have maintained a fairly lengthy list of publications at this website, but in the belief that most visitors are really only interested in a representative sample, I’m paring the list
down to just a few favorites. Also, you might want to check out Research and Personal, where I link to some articles I didn’t get paid for, but wrote out of personal passion.
A Tisket, A Tasket, An Apollonian Gasket
American Scientist, January-February 2010, pp. 10-14.
In the fall of 2008, I was journalist in residence at the Mathematical Sciences Research Institute in Berkeley. One afternoon, when I was sitting in my office and wasting time, I happened to notice
my computer’s screen saver displaying a beautiful pattern of circles. “What IS that?” I wondered. Later, at the Joint Mathematics Meetings in 2009, I saw a poster showing practically the same
pattern. From there I learned that it was a mathematical object called an Apollonian gasket, and that it is full of deep and beautiful mysteries involving number theory and group theory. A year later
I was asked to write a guest column for American Scientist, and I immediately knew what I wanted to write about! To read the article you need to sign up for a free account with JSTOR, but don’t let
that stop you.
The PoincarĂ© Conjecture — Proved
The Poincaré Conjecture was THE mathematics story of the Millennial Decade. Gregory Perelman, a reclusive Russian mathematician, claimed to have a proof of one of the most famous unsolved problems
in mathematics. Mathematicians weren’t sure; his papers were more like a scaffolding than a finished building. Over the next three years, three separate groups of mathematicians independently
confirmed the proof. Perelman was to be awarded the Fields Medal, the highest honor in mathematics — but he stood up the mathematical community, and the King of Spain, by declining to show up for the
award. I wrote three or four articles for Science at various times during this drama, culminating with this cover article when the editors of Science named Perelman’s work as their Breakthrough of
the Year for 2006.
Why This Week’s Man-Versus-Machine Go Match Doesn’t Matter (and What Does)
In early 2016 I was swept up in the excitement surrounding AlphaGo, a program developed at Google’s DeepMind subsidiary to play the ancient game of go. At the time of this article, AlphaGo was
getting ready to challenge Lee Sedol, a former world #1. For me, it was all hauntingly familiar — I had seen the same angst among chess players, the same mistaken belief that “the machines are taking
over,” when computers first defeated the human world chess champion. Yet in the ensuing years, computers have had a mostly positive and transformative effect on human chess. Let’s hope that the same
will be true for go. PS: AlphaGo won the match, 4-1. It subsequently won 60 consecutive games against top human players, including the world champion Ke Jie, and then retired. Sedol will go down in
history as the last human to win a game against the world’s best computer go program.
Physics and Chemistry
Goldberg Variations: New Shapes for Molecular Cages
Science News, February 14, 2014
It’s hard to know how to classify this article: Is it chemistry or is it mathematics? That’s one reason I like it. Two neuroscientists at UCLA answered a question that mathematicians never even
thought to ask: Is there a way to build a soccer ball-style cage (or polyhedron), only with more than 60 faces? The requirements were that the faces have to be flat, the edges must have the same
length, and the cage must have an overall 60-fold symmetry. Even though most of the article is behind a paywall, you can see the answer in the beautiful picture at the top of the article: Yes! They
found examples with up to 980 faces (the one in the picture has only 252).
Geology and Earth Science
Atmospheric Rivers: When the Sky Falls
In California, where I live, we get inundated two or three times a year (sometimes more) by storm systems that siphon moisture directly from the tropics (read: Hawaii) and stretch halfway across the
Pacific. “Atmospheric rivers” are also responsible for the great majority of the largest rain events and floods in England. Meteorologists had no idea these rivers existed until they developed
satellites that could see through the cloud tops to measure how much moisture was present in the full 3-dimensional volume of a storm system. This article is mostly behind a paywall, but you can
watch a video from NOAA with incredibly cool satellite pictures and incredibly bad narration. (I disclaim any responsibility for their awful script-writing!)
Fueling Innovation and Discovery: The Mathematical Sciences in the 21st Century
National Academies Press, 2012
I’ll let you in on two secrets. Secret number 1: For freelance writers, a better source of income than writing for magazines is “work for hire” — writing for organizations that need to get a message
out. Work for hire doesn’t have quite the same cachet as true journalism, but I’m quite proud of the work I did for this brochure, written for the National Academy of Sciences. It highlights 14
real-world applications of mathematics, ranging from cartoon animation to new methods of imaging the brain. Secret number 2: I wrote this baby, even though my name isn’t on the cover. (They do
mention my name in the acknowledgments, page iv.) The PDF can be downloaded for free. | {"url":"https://danamackenzie.com/articles-2/","timestamp":"2024-11-04T18:20:48Z","content_type":"text/html","content_length":"33679","record_id":"<urn:uuid:66388e58-41c8-485f-8fde-3f018d71a6a3>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00882.warc.gz"} |
Learning Objectives
By the end of this section, you will be able to:
• Define temperature.
• Convert temperatures between the Celsius, Fahrenheit, and Kelvin scales.
• Define thermal equilibrium.
• State the zeroth law of thermodynamics.
The concept of temperature has evolved from the common concepts of hot and cold. Human perception of what feels hot or cold is a relative one. For example, if you place one hand in hot water and the
other in cold water, and then place both hands in tepid water, the tepid water will feel cool to the hand that was in hot water, and warm to the one that was in cold water. The scientific definition
of temperature is less ambiguous than your senses of hot and cold. Temperature is operationally defined to be what we measure with a thermometer. (Many physical quantities are defined solely in terms
of how they are measured. We shall see later how temperature is related to the kinetic energies of atoms and molecules, a more physical explanation.) Two accurate thermometers, one placed in hot
water and the other in cold water, will show the hot water to have a higher temperature. If they are then placed in the tepid water, both will give identical readings (within measurement
uncertainties). In this section, we discuss temperature, its measurement by thermometers, and its relationship to thermal equilibrium. Again, temperature is the quantity measured by a thermometer.
Misconception Alert: Human Perception vs. Reality
On a cold winter morning, the wood on a porch feels warmer than the metal of your bike. The wood and bicycle are in thermal equilibrium with the outside air, and are thus the same temperature. They
feel different because of the difference in the way that they conduct heat away from your skin. The metal conducts heat away from your body faster than the wood does (see more about conductivity in
Conduction). This is just one example demonstrating that the human sense of hot and cold is not determined by temperature alone.
Another factor that affects our perception of temperature is humidity. Most people feel much hotter on hot, humid days than on hot, dry days. This is because on humid days, sweat does not evaporate
from the skin as efficiently as it does on dry days. It is the evaporation of sweat (or water from a sprinkler or pool) that cools us off.
Any physical property that depends on temperature, and whose response to temperature is reproducible, can be used as the basis of a thermometer. Because many physical properties depend on
temperature, the variety of thermometers is remarkable. For example, volume increases with temperature for most substances. This property is the basis for the common alcohol thermometer, the old
mercury thermometer, and the bimetallic strip (Figure 1).
Other properties used to measure temperature include electrical resistance and color and the emission of infrared radiation.
One example of electrical resistance and color is found in a plastic thermometer. Each of the six squares on the plastic (liquid crystal) thermometer in Figure 2 contains a film of a different
heat-sensitive liquid crystal material Below 95ºF, all six squares are black. When the plastic thermometer is exposed to temperature that increases to 95ºF, the first liquid crystal square changes
color. When the temperature increases above 96.8ºF the second liquid crystal square also changes color, and so forth.
An example of emission of radiation is shown in the use of a pyrometer (Figure 3). Infrared radiation (whose emission varies with temperature) from the vent in Figure 3 is measured and a temperature
readout is quickly produced. Infrared measurements are also frequently used as a measure of body temperature. These modern thermometers, placed in the ear canal, are more accurate than alcohol
thermometers placed under the tongue or in the armpit.
Temperature Scales
Thermometers are used to measure temperature according to well-defined scales of measurement, which use pre-defined reference points to help compare quantities. The three most common temperature
scales are the Fahrenheit, Celsius, and Kelvin scales. A temperature scale can be created by identifying two easily reproducible temperatures. The freezing and boiling temperatures of water at
standard atmospheric pressure are commonly used.
The Celsius scale (which replaced the slightly different centigrade scale) has the freezing point of water at 0ºC and the boiling point at 100ºC. Its unit is the degree Celsius(ºC). On the Fahrenheit
scale (still the most frequently used in the United States), the freezing point of water is at 32ºF and the boiling point is at 212ºF. The unit of temperature on this scale is the degree Fahrenheit
(ºF). Note that a temperature difference of one degree Celsius is greater than a temperature difference of one degree Fahrenheit. Only 100 Celsius degrees span the same range as 180 Fahrenheit
degrees, thus one degree on the Celsius scale is 1.8 times larger than one degree on the Fahrenheit scale 180/100=9/5.
The Kelvin scale is the temperature scale that is commonly used in science. It is an absolute temperature scale defined to have 0 K at the lowest possible temperature, called absolute zero. The
official temperature unit on this scale is the kelvin, which is abbreviated K, and is not accompanied by a degree sign. The freezing and boiling points of water are 273.15 K and 373.15 K,
respectively. Thus, the magnitude of temperature differences is the same in units of kelvins and degrees Celsius. Unlike other temperature scales, the Kelvin scale is an absolute scale. It is used
extensively in scientific work because a number of physical quantities, such as the volume of an ideal gas, are directly related to absolute temperature. The kelvin is the SI unit used in scientific
The relationships between the three common temperature scales is shown in Figure 4. Temperatures on these scales can be converted using the equations in Table 1.
Table 1. Temperature Conversions
To convert Use this equation . . . Also written as . . .
from . . .
Celsius to [latex]T\left(^{\circ}\text{F}\right)=\frac{9}{5}T\left(^{\circ}\text{C}\right)+32\\[/latex] [latex]T_{^{\circ}\text{F}}=\frac{9}{5}T_{^{\circ}\text{C}}+32\\[/latex]
Fahrenheit to [latex]T\left(^{\circ}\text{C}\right)=\frac{5}{9}\left(T\left(^{\circ}\text{F}\right)-32\right)\\ [latex]T_{^{\circ}\text{C}}=\frac{5}{9}\left(T_{^{\circ}\text{F}}-32\right)\\[/
Celsius [/latex] latex]
Celsius to Kelvin T(K) = T(ºC) + 273.15 T[K] = T[ºC] + 273.15
Kelvin to Celsius T(ºC) = T(K) − 273.15 T[ºC] = T[K] − 273.15
Fahrenheit to Kelvin [latex]T\left(K\right)=\frac{5}{9}\left(T\left(^{\circ}\text{F}\right)-32\right)+273.15\\[/latex] [latex]T_{K}=\frac{5}{9}\left(T_{^{\circ}\text{F}}-32\right)+273.15\\[/latex]
Kelvin to Fahrenheit [latex]T\left(^{\circ}\text{F}\right)=\frac{9}{5}\left(T\left(K\right)-273.15\right)+32\\[/latex] [latex]T_{^{\circ}\text{F}}=\frac{9}{5}\left(T_{K}-273.15\right)+32\\[/latex]
Notice that the conversions between Fahrenheit and Kelvin look quite complicated. In fact, they are simple combinations of the conversions between Fahrenheit and Celsius, and the conversions between
Celsius and Kelvin.
Example 1. Converting between Temperature Scales: Room Temperature
“Room temperature” is generally defined to be 25ºC.
1. What is room temperature in ºF?
2. What is it in K?
To answer these questions, all we need to do is choose the correct conversion equations and plug in the known values.
Solution for Part 1
1. Choose the right equation. To convert from ºC to ºF, use the equation [latex]T_{^{\circ}\text{F}}=\frac{9}{5}T_{^{\circ}\text{C}}+32\\[/latex].
2. Plug the known value into the equation and solve: [latex]T_{^{\circ}\text{F}}=\frac{9}{5}25{^{\circ}\text{C}}+32=77^{\circ}\text{F}\\[/latex]
Solution for Part 2
1. Choose the right equation. To convert from ºC to K, use the equation T[K] = T[ºC] + 273.15
2. Plug the known value into the equation and solve: T[K] = 25ºC + 273.15 = 298 K.
Example 2. Converting between Temperature Scales: the Reaumur Scale
The Reaumur scale is a temperature scale that was used widely in Europe in the eighteenth and nineteenth centuries. On the Reaumur temperature scale, the freezing point of water is 0ºR and the
boiling temperature is 80ºR. If “room temperature” is 25ºC on the Celsius scale, what is it on the Reaumur scale?
To answer this question, we must compare the Reaumur scale to the Celsius scale. The difference between the freezing point and boiling point of water on the Reaumur scale is 80ºR. On the Celsius
scale it is 100ºC. Therefore 100º C=80ºR. Both scales start at 0 º for freezing, so we can derive a simple formula to convert between temperatures on the two scales.
1. Derive a formula to convert from one scale to the other: [latex]T_{^{\circ}\text{R}}=\frac{0.8^{\circ}\text{R}}{^{\circ}\text{C}}\times{T}_{^{\circ}\text{C}}\\[/latex]
2. Plug the known value into the equation and solve: [latex]T_{^{\circ}\text{R}}=\frac{0.8^{\circ}\text{R}}{^{\circ}\text{C}}\times25^{\circ}\text{C}=20^{\circ}\text{R}\\[/latex]
Temperature Ranges in the Universe
Figure 6 shows the wide range of temperatures found in the universe. Human beings have been known to survive with body temperatures within a small range, from 24ºC to 44ºC (75ºF to 111ºF). The
average normal body temperature is usually given as 37.0ºC (98.6ºF), and variations in this temperature can indicate a medical condition: a fever, an infection, a tumor, or circulatory problems (see
Figure 5).
The lowest temperatures ever recorded have been measured during laboratory experiments: 4.5 × 10^−10 K at the Massachusetts Institute of Technology (USA), and 1.0 × 10^−10 K at Helsinki University of
Technology (Finland). In comparison, the coldest recorded place on Earth’s surface is Vostok, Antarctica at 183 K (–89ºC), and the coldest place (outside the lab) known in the universe is the
Boomerang Nebula, with a temperature of 1 K.
Making Connections: Absolute Zero
What is absolute zero? Absolute zero is the temperature at which all molecular motion has ceased. The concept of absolute zero arises from the behavior of gases. Figure 7 shows how the pressure of
gases at a constant volume decreases as temperature decreases. Various scientists have noted that the pressures of gases extrapolate to zero at the same temperature, –273.15ºC. This extrapolation
implies that there is a lowest temperature. This temperature is called absolute zero. Today we know that most gases first liquefy and then freeze, and it is not actually possible to reach absolute
zero. The numerical value of absolute zero temperature is –273.15ºC or 0 K.
Thermal Equilibrium and the Zeroth Law of Thermodynamics
Thermometers actually take their own temperature, not the temperature of the object they are measuring. This raises the question of how we can be certain that a thermometer measures the temperature
of the object with which it is in contact. It is based on the fact that any two systems placed in thermal contact (meaning heat transfer can occur between them) will reach the same temperature. That
is, heat will flow from the hotter object to the cooler one until they have exactly the same temperature. The objects are then in thermal equilibrium, and no further changes will occur. The systems
interact and change because their temperatures differ, and the changes stop once their temperatures are the same. Thus, if enough time is allowed for this transfer of heat to run its course, the
temperature a thermometer registers does represent the system with which it is in thermal equilibrium. Thermal equilibrium is established when two bodies are in contact with each other and can freely
exchange energy.
Furthermore, experimentation has shown that if two systems, A and B, are in thermal equilibrium with each another, and B is in thermal equilibrium with a third system C, then A is also in thermal
equilibrium with C. This conclusion may seem obvious, because all three have the same temperature, but it is basic to thermodynamics. It is called the zeroth law of thermodynamics.
The Zeroth Law of Thermodynamics
If two systems, A and B, are in thermal equilibrium with each other, and B is in thermal equilibrium with a third system, C, then A is also in thermal equilibrium with C.
This law was postulated in the 1930s, after the first and second laws of thermodynamics had been developed and named. It is called the zeroth law because it comes logically before the first and
second laws (discussed in Thermodynamics). An example of this law in action is seen in babies in incubators: babies in incubators normally have very few clothes on, so to an observer they look as if
they may not be warm enough. However, the temperature of the air, the cot, and the baby is the same, because they are in thermal equilibrium, which is accomplished by maintaining air temperature to
keep the baby comfortable.
Check Your Understanding
Does the temperature of a body depend on its size?
No, the system can be divided into smaller parts each of which is at the same temperature. We say that the temperature is an intensive quantity. Intensive quantities are independent of size.
Section Summary
• Temperature is the quantity measured by a thermometer.
• Temperature is related to the average kinetic energy of atoms and molecules in a system.
• Absolute zero is the temperature at which there is no molecular motion.
• There are three main temperature scales: Celsius, Fahrenheit, and Kelvin.
• Temperatures on one scale can be converted to temperatures on another scale using the following equations:
□ [latex]T_{^{\circ}\text{F}}=\frac{9}{5}T_{^{\circ}\text{C}}+32\\[/latex]
□ [latex]T_{^{\circ}\text{C}}=\frac{5}{9}\left(T_{^{\circ}\text{F}}-32\right)\\[/latex]
□ T[K] = T[ºC] + 273.15
□ T[ºC] = T[K] − 273.15
• Systems are in thermal equilibrium when they have the same temperature.
Thermal equilibrium occurs when two bodies are in contact with each other and can freely exchange energy.
The zeroth law of thermodynamics states that when two systems, A and B, are in thermal equilibrium with each other, and B is in thermal equilibrium with a third system, C, then A is also in
thermal equilibrium with C.
Conceptual Questions
1. What does it mean to say that two systems are in thermal equilibrium?
2. Give an example of a physical property that varies with temperature and describe how it is used to measure temperature.
3. When a cold alcohol thermometer is placed in a hot liquid, the column of alcohol goes down slightly before going up. Explain why.
4. If you add boiling water to a cup at room temperature, what would you expect the final equilibrium temperature of the unit to be? You will need to include the surroundings as part of the system.
Consider the zeroth law of thermodynamics.
Problems & Exercises
1. What is the Fahrenheit temperature of a person with a 39.0ºC fever?
2. Frost damage to most plants occurs at temperatures of 28.0ºF or lower. What is this temperature on the Kelvin scale?
3. To conserve energy, room temperatures are kept at 68.0ºF in the winter and 78.0ºF in the summer. What are these temperatures on the Celsius scale?
4. A tungsten light bulb filament may operate at 2900 K. What is its Fahrenheit temperature? What is this on the Celsius scale?
5. The surface temperature of the Sun is about 5750 K. What is this temperature on the Fahrenheit scale?
6. One of the hottest temperatures ever recorded on the surface of Earth was 134ºF in Death Valley, CA. What is this temperature in Celsius degrees? What is this temperature in Kelvin?
7. (a) Suppose a cold front blows into your locale and drops the temperature by 40.0 Fahrenheit degrees. How many degrees Celsius does the temperature decrease when there is a 40.0ºF decrease in
temperature? (b) Show that any change in temperature in Fahrenheit degrees is nine-fifths the change in Celsius degrees.
8. (a) At what temperature do the Fahrenheit and Celsius scales have the same numerical value? (b) At what temperature do the Fahrenheit and Kelvin scales have the same numerical value?
temperature: the quantity measured by a thermometer
Celsius scale: temperature scale in which the freezing point of water is 0ºC and the boiling point of water is 100ºC
degree Celsius: unit on the Celsius temperature scale
Fahrenheit scale: temperature scale in which the freezing point of water is 32ºF and the boiling point of water is 212ºF
degree Fahrenheit: unit on the Fahrenheit temperature scale
Kelvin scale: temperature scale in which 0 K is the lowest possible temperature, representing absolute zero
absolute zero: the lowest possible temperature; the temperature at which all molecular motion ceases
thermal equilibrium: the condition in which heat no longer flows between two objects that are in contact; the two objects have the same temperature
zeroth law of thermodynamics: law that states that if two objects are in thermal equilibrium, and a third object is in thermal equilibrium with one of those objects, it is also in thermal equilibrium
with the other object
Selected Solutions to Problems & Exercises
1. 102ºF
3. 20.0ºC and 25.6ºC
5. 9890ºF
7. (a) 22.2ºC; (b)
[latex]\begin{array}{lll}\Delta T\left(^{\circ}\text{F}\right)& =& {T}_{2}\left(^{\circ}\text{F}\right)-{T}_{1}\left(^{\circ}\text{F}\right)\\ & =& \frac{9}{5}{T}_{2}\left(^{\circ}\text{C}\right)+\
text{32}\text{.}0^{\circ}-\left(\frac{9}{5}{T}_{1}\left(^{\circ}\text{C}\right)+\text{32}\text{.}0^{\circ}\right)\\ & =& \frac{9}{5}\left({T}_{2}\left(^{\circ}\text{C}\right)-{T}_{1}\left(^{\circ}\
text{C}\right)\right)\text{}=\frac{9}{5}\Delta T\left(^{\circ}\text{C}\right)\end{array}\\[/latex] | {"url":"https://courses.lumenlearning.com/suny-physics/chapter/13-1-temperature/","timestamp":"2024-11-13T06:19:02Z","content_type":"text/html","content_length":"71738","record_id":"<urn:uuid:7c2f3916-1592-4ccb-9879-9037bd2d340b>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00007.warc.gz"} |
Properties of Quadrilaterals - Rectangle, Square, Parallelogram, Rhombus, Trapezium | 2023
Properties of Quadrilaterals – Rectangle, Square, Parallelogram, Rhombus, Trapezium | 2023
A 9 min read
In Euclidean geometry, a quadrilateral is a four-sided 2D figure whose sum of internal angles is 360°. The word quadrilateral is derived from two Latin words ‘quadri’ and ‘latus’ meaning four and
side respectively. Therefore, identifying the properties of quadrilaterals is important when trying to distinguish them from other polygons. So, what are the properties of quadrilaterals? There are
two properties of quadrilaterals:
• A quadrilateral should be closed shape with 4 sides
• All the internal angles of a quadrilateral sum up to 360°
In this article, you will get an idea about the 5 types of quadrilaterals (Rectangle, Square, Parallelogram, Rhombus, and Trapezium) and get to know about the properties of quadrilaterals.
Here are the five types of quadrilaterals discussed in this article:
1. Rectangle
2. Square
3. Parallelogram
4. Rhombus
5. Trapezium
Kickstart Your Path to GMAT Quant Excellence!
Hello and welcome! We’re thrilled to accompany you on your GMAT Quant preparation journey. By understanding your current standing and future goals, we’ll be able to provide tailored resources and
strategies that will help you conquer the Quant section with confidence. Ready to take the first step? Let’s get started.
Here is a video explaining the properties of quadrilaterals:
Are you struggling with GMAT quant? e-GMAT provides structured learning from foundations to help you master the skills needed for a high score. Join the world’s most successful prep company for a
free trial and see the difference it can make. We are the most reviewed online GMAT Prep company with 2600+ reviews on GMATClub, as of July 2023.
Properties of the quadrilaterals – An overview
The diagram given below shows a quadrilateral ABCD and the sum of its internal angles. All the internal angles sum up to 360°. Thus, ∠A + ∠B + ∠C + ∠D = 360°
Properties of quadrilaterals Rectangle Square Parallelogram Rhombus Trapezium
All Sides are equal No Yes No Yes No
Opposite Sides are equal Yes Yes Yes Yes No
Opposite Sides are parallel Yes Yes Yes Yes Yes
All angles are equal Yes Yes No No No
Opposite angles are equal Yes Yes Yes Yes No
Sum of two adjacent angles is 180 Yes Yes Yes Yes No
Bisect each other Yes Yes Yes Yes No
Bisect perpendicularly No Yes No Yes No
Let’s discuss each of these 5 quadrilaterals in detail:
Here are questions which will teach you how to apply the properties of all 5 quadrilaterals you’ll learn in this article.
A rectangle is a quadrilateral with four right angles. Thus, all the angles in a rectangle are equal (360°/4 = 90°). Moreover, the opposite sides of a rectangle are parallel and equal, and diagonals
bisect each other.
Here are the three properties of a rectangle:
• All the angles of a rectangle are 90°
• Opposite sides of a rectangle are equal and Parallel
• Diagonals of a rectangle bisect each other
Rectangle formula – area and perimeter of a rectangle
If the length of the rectangle is L and breadth is B then,
1. Area of a rectangle = Length × Breadth or L × B
2. Perimeter of rectangle = 2 × (L + B)
These practice questions will help you solidify the properties of rectangles
Begin your GMAT preparation with the only prep company that has delivered more than 700+ scores than any other GMAT club partner. Achieve GMAT 740+ with our AI-driven tools that you personalized
feedback at every step of your GMAT journey. Take our free trial today!
Did you know e-GMATers have reported more 700+ scores than ever before in GMAT Club’s history? Watch this video to understand how e-GMAT has achieved this record-shattering result by investing and
innovating with a single goal in mind – To create a platform that empowers students to achieve and deliver their very best.
Square is a quadrilateral with four equal sides and angles. It’s also a regular quadrilateral as both its sides and angles are equal. Just like a rectangle, a square has four angles of 90° each. It
can also be seen as a rectangle whose two adjacent sides are equal.
Here are the three properties of a Square:
• All the angles of a square are 90°
• All sides of a square are equal and parallel to each other
• Diagonals bisect each other perpendicularly
Square formula – area and perimeter of a square
If the side of a square is ‘a’ then,
1. Area of the square = a × a = a²
2. Perimeter of the square = 2 × (a + a) = 4a
These practice questions will help you solidify the properties of squares
Scoring a Q50-51 on the GMAT helps you get a 700+ GMAT score. Start your journey of getting a Q50-51 on the GMAT with e-GMAT’s AI-driven online preparation course. Our xPERT not only curates the most
optimized learning path but also tracks your improvement, ensuring that you get to your target Quant score quickly and reliably. Watch this video to know more:
A parallelogram, as the name suggests, is a simple quadrilateral whose opposite sides are parallel. Thus, it has two pairs of parallel sides. Moreover, the opposite angles in a parallelogram are
equal and their diagonals bisect each other.
Take a free GMAT mock to understand your baseline score and start your GMAT prep with our free trial. We are the most reviewed online GMAT Prep company with 2600+ reviews on GMATClub.
Here are the four properties of a Parallelogram:
• Opposite angles are equal
• Opposite sides are equal and parallel
• Diagonals bisect each other
• Sum of any two adjacent angles is 180°
Parallelogram formulas – area and perimeter of a parallelogram
If the length of a parallelogram is ‘l’, breadth is ‘b’ and height is ‘h’ then:
1. Perimeter of parallelogram= 2 × (l + b)
2. Area of the parallelogram = l × h
These practice questions will help you solidify the properties of parallelogram
A rhombus is a quadrilateral whose all four sides are equal in length and opposite sides are parallel to each other. However, the angles are not equal to 90°. A rhombus with right angles would become
a square. Another name for rhombus is ‘diamond’ as it looks similar to the diamond suit in playing cards.
Here are the four properties of a Rhombus:
• Opposite angles are equal
• All sides are equal and, opposite sides are parallel to each other
• Diagonals bisect each other perpendicularly
• Sum of any two adjacent angles is 180°
Rhombus formulas – area and perimeter of a rhombus
If the side of a rhombus is a then, perimeter of a rhombus = 4a
If the length of two diagonals of the rhombus is d[1] and d[2] then the area of a rhombus = ½ × d[1] × d[2]
These practice questions will help you solidify the properties of rhombus
At e-GMAT, we strive to make the students’ life easier at every level and therefore, we have built this fantastic tool called the GMAT Personalized Study Planner. With this tool, you can not only
find out the Quant and Verbal Sectional scores but also Sub-sectional scores in Quant (Algebra, Arithmetic, etc.) and Verbal (CR, RC, and SC) to achieve your target GMAT Score! Moreover, this
tool will craft your very own study plan in 5 minutes. Register for our FREE Trial and create your personalized study plan in 5 mins!
A trapezium (called Trapezoid in the US) is a quadrilateral that has only one pair of parallel sides. The parallel sides are referred to as ‘bases’ and the other two sides are called ‘legs’ or
lateral sides.
Properties of Trapezium
A trapezium is a quadrilateral in which the following one property:
• Only one pair of opposite sides are parallel to each other
Trapezium formulas – area and perimeter of a trapezium
If the height of a trapezium is ‘h’ (as shown in the above diagram) then:
1. Perimeter of the trapezium= Sum of lengths of all the sides = AB + BC + CD + DA
2. Area of the trapezium = ½ × (Sum of lengths of parallel sides) × h = ½ × (AB + CD) × h
These practice questions will help you solidify the properties of trapezium
Properties of Quadrilaterals – Summary
The below image also summarizes the properties of quadrilaterals
Are you struggling with GMAT quant? e-GMAT provides structured learning from foundations to help you master the skills needed for a high score. Join the world’s most successful prep company for a
free trial and see the difference it can make. We are the most reviewed online GMAT Prep company with 2500+ reviews on GMATClub, as of April 2023.
Important quadrilateral formulas
The below table summarizes the formulas on the area and perimeter of different types of quadrilaterals:
Quadrilateral formulas Rectangle Square Parallelogram Rhombus Trapezium
Area l × b a² l × h ½ × d1 × d2 ½ × (Sum of parallel sides) × height
Perimeter 2 × (l + b) 4a 2 × (l + b) 4a Sum of all the sides
Further reading:
To ace the GMAT a well-defined study plan is required. Save 60+ hours on GMAT preparation by following these three steps:
Quadrilateral Practice Question | Properties of Quadrilaterals
Let’s practice the application of properties of quadrilaterals on the following sample questions:
GMAT Quadrilaterials Practice Question 1
Adam wants to build a fence around his rectangular garden of length 10 meters and width 15 meters. How many meters of fence he should buy to fence the entire garden?
1. 20 meters
2. 25 meters
3. 30 meters
4. 40 meters
5. 50 meters
Step 1: Given
• Adam has a rectangular garden.
□ It has a length of 10 meters and a width of 15 meters.
□ He wants to build a fence around it.
Step 2: To find
• The length required to build the fence around the entire garden.
Step 3: Approach and Working out
The fence can only be built around the outside sides of the garden.
• So, the total length of the fence required= Sum of lengths of all the sides of the garden.
□ Since the garden is rectangular, the sum of the length of all the sides is nothing but the perimeter of the garden.
□ Perimeter = 2 × (10 + 15) = 50 metres
Hence, the required length of the fence is 50 meters.
Therefore, option E is the correct answer.
GMAT Quadrilaterals Practice Question 2
Steve wants to paint one rectangular-shaped wall of his room. The cost to paint the wall is $1.5 per square meter. If the wall is 25 meters long and 18 meters wide, then what is the total cost to
paint the wall?
1. $ 300
2. $ 350
3. $ 450
4. $ 600
5. $ 675
Step 1: Given
• Steve wants to paint one wall of his room.
□ The wall is 25 meters long and 18 meters wide.
□ Cost to paint the wall is $1.5 per square meter.
Step 2: To find
• The total cost to paint the wall.
Step 3: Approach and Working out
• A wall is painted across its entire area.
□ So, if we find the total area of the wall in square meters and multiply it by the cost to paint 1 square meter of the wall then we can the total cost.
□ Area of the wall = length × Breadth = 25 metres × 18 metres = 450 square metre
□ Total cost to paint the wall = 450 × $1.5 = $675
Hence, the correct answer is option E.
We hope by now you would have learned the different types of quadrilaterals, their properties, and formulas and how to apply these concepts to solve questions on quadrilaterals. The application of
quadrilaterals is important to solve geometry questions on the GMAT. If you are planning to take the GMAT, we can help you with high-quality study material which you can access for free by
registering here.
Here are a few more articles on Math:
Watch this GMAT geometry-free webinar where we discuss how to solve 700-level Data sufficiency and Problem questions in GMAT Quadrilaterals:
Are you planning to enroll at top business schools? Let us help you conquer the first step of the process i.e., taking the GMAT. Take a free GMAT mock to understand your baseline score and start
your GMAT prep with our free trial. We are the most reviewed online GMAT Prep company with 2500+ reviews on GMATClub, as of January 2023.
Write to us at acethegmat@e-gmat.com in case of any query.
FAQs -Properties of Quadrilaterals
What are the different types of quadrilaterals?
There are 5 types of quadrilaterals – Rectangle, Square, Parallelogram, Trapezium or Trapezoid, and Rhombus.
Where can I find a few practice questions on quadrilaterals?
You can find a few practice questions on quadrilaterals in this article.
What is the sum of the interior angles of a quadrilateral?
The sum of interior angles of a quadrilateral is 360°. | {"url":"https://e-gmat.com/blogs/quadrilateral-properties-formulas-rectangle-square-parallelogram-rhombus-trapezium-trapezoid/","timestamp":"2024-11-12T10:10:22Z","content_type":"text/html","content_length":"732207","record_id":"<urn:uuid:bfed90ca-658a-47ce-8fcd-2757ddfe0ff1>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00639.warc.gz"} |
Mastering the Google Sheets Not Equal Operator: A Comprehensive Guide
Published: June 28, 2024 - 8 min read
Hannah Recker
The “not equal” operator is a crucial tool for filtering, comparing, and extracting specific data in Google Sheets. This guide covers everything you need to know to use “not equal” effectively, from
basic syntax to advanced applications and best practices.
Google Sheets Not Equal Basics: Syntax and Simple Examples
The “not equal” operator in Google Sheets is represented by the symbol “≠” or the “<>” symbols. This operator is used to compare two values and return a TRUE or FALSE result based on whether the
values are different or not.
The basic syntax for using the “not equal” operator is:
=value1 <> value2
Here are some simple examples of using the “not equal” operator in Google Sheets:
• Comparing Numbers: =5 ≠7 returns TRUE because 5 is not equal to 7.
• Comparing Strings: =”apple” ≠“banana” returns TRUE because “apple” is not the same as “banana”.
• Comparing Dates: =DATE(2023,5,1) ≠DATE(2023,6,1) returns TRUE because the dates are different.
In these examples, the “not equal” operator compares the values on either side of the operator and returns a boolean result (TRUE or FALSE) based on whether the values are different or not.
IF Not Equal: Combining the IF Function with Not Equal Logic
The “not equal” operator becomes even more powerful when combined with the IF function in Google Sheets. This allows you to create conditional statements that perform specific actions based on
whether a value is different from a given criteria.
The basic syntax for using the “not equal” operator with the IF function is:
=IF(value1 <> value2, “True_value”, “False_value”)
Here are some examples of using the “not equal” operator with the IF function:
Data Validation Suppose you have a column of product names, and you want to ensure that each product name is unique. You can use the following formula to highlight any duplicate values:
=IF(COUNTIF($A$2:$A$10, A2) > 1, “Duplicate”, “Unique”)
In this example, the “not equal” operator is used within the COUNTIF function to count the number of times the current product name (in cell A2) appears in the range A2:A10. If the count is greater
than 1, the formula returns “Duplicate”, indicating a duplicate value.
Filtering Data Let’s say you have a list of sales data, and you want to filter out all the sales records where the sales amount is not equal to $1,000. You can use the following formula in a filter
=B2 <> 1000
In this case, the “not equal” operator is used to compare the sales amount in column B to the value 1,000, and the filter will only display the rows where the sales amount is different from $1,000.
By combining the “not equal” operator with the IF function, you can create powerful and flexible formulas that can help you validate, filter, and manipulate your data in Google Sheets.
Conditional Formatting with the Not Equal Operator
One of the most powerful applications of the “not equal” operator in Google Sheets is in conditional formatting. By using the “not equal” logic, you can quickly identify and highlight cells that do
not meet a specific criteria. This can be incredibly useful for data analysis, quality control, and identifying outliers.
To apply conditional formatting based on the “not equal” operator, follow these steps:
1. Select the range of cells you want to format.
2. Go to the “Format” menu and choose “Conditional formatting”.
3. In the “Format rules” section, select “Custom formula is” and enter your “not equal” formula.
□ For example, to highlight cells that are not equal to the value in cell A1, you would use the formula: =A1<>B1.
1. Choose the desired formatting, such as a specific font color, background color, or icon.
2. Click “Done” to apply the conditional formatting.
This technique can be especially useful when you need to quickly spot values that don’t match a reference cell or a specific criteria.
Using Google Sheets Not Equal with SUMIF and COUNTIF
The “not equal” operator can also be combined with powerful Google Sheets functions like SUMIF and COUNTIF to perform more advanced data analysis.
SUMIF with Not Equal The SUMIF function allows you to sum a range of cells based on a specific criteria. By using the “not equal” operator, you can sum the values in a range that do not match a given
For example, to sum all the values in column B that are not equal to 100, you would use the formula: =SUMIF(B:B, “<>100”, B:B)
Free AI-Powered Tools Right Within Your Spreadsheet
Supercharge your spreadsheets with GPT-powered AI tools for building formulas, charts, pivots, SQL and more. Simple prompts for automatic generation.
COUNTIF with Not Equal Similarly, the COUNTIF function can be used with the “not equal” operator to count the number of cells in a range that do not match a specific value.
To count the number of cells in column C that are not equal to “John”, the formula would be: =COUNTIF(C:C, “<>John”)
These advanced techniques allow you to perform more complex data analysis and reporting, making it easier to identify patterns, trends, and outliers in your Google Sheets data.
Combining Multiple Functions with Not Equal
For even more advanced use cases, you can combine the “not equal” operator with multiple functions and logical operators to create powerful, dynamic formulas.
For example, let’s say you want to calculate the total sales for all products that are not equal to “Product A” and have a quantity greater than 10. You could use the following formula:
=SUMIF(Product_Column, “<>Product A”, Quantity_Column, “>10”, Sales_Column)
This formula uses the SUMIF function with three criteria:
1. The product name is not equal to “Product A”
2. The quantity is greater than 10
3. Sum the values in the Sales_Column
By nesting multiple conditions within a single formula, you can create highly customized and sophisticated analyses to meet your specific needs.
Google Sheets Not Equal Best Practices: Tips, Tricks, & Troubleshooting
Best Practices for Using the Not Equal Operator
When working with the “not equal” operator in Google Sheets, keep the following best practices in mind:
• Be Precise with Syntax: Ensure that you use the correct syntax (<>) when writing your “not equal” formulas. Typos or using the wrong operator can lead to unexpected results.
• Consider Data Types: Remember that the “not equal” operator compares the data types as well as the values. For example, the formula =A1<>1 will return true if the value in A1 is a text string,
even if the numeric value is the same.
• Leverage Relative References: Use relative cell references (e.g., B1<>C1) instead of absolute references (e.g., $B$1<>$C$1) to make your formulas more flexible and easier to copy or drag across a
• Test and Validate: Always test your “not equal” formulas with a variety of sample data to ensure they are working as expected. Validate the results against your expected outcomes.
Tips and Tricks for Efficient Formulas
Here are some tips and tricks to help you write more efficient and effective “not equal” formulas in Google Sheets:
• Use the ISNUMBER Function: Combine the “not equal” operator with the ISNUMBER function to check if a cell contains a numeric value or not. For example, =ISNUMBER(A1)<>TRUE will return true if the
value in A1 is not a number.
• Leverage Array Formulas: For more complex scenarios, consider using array formulas to perform multiple comparisons within a single formula. This can help reduce the number of individual formulas
and make your sheet more streamlined.
• Incorporate Wildcards: Use the wildcard characters * (for multiple characters) and ? (for single characters) in your “not equal” formulas to create more flexible and powerful comparisons.
Troubleshooting Common Issues
While the “not equal” operator is a straightforward concept, there are a few common issues you may encounter when using it in Google Sheets:
• Unexpected Results with Text Values: If you’re comparing text values, make sure to use the correct capitalization and formatting. Differences in spaces, punctuation, or case can cause the “not
equal” operator to return unexpected results.
• Dealing with Blank or Null Values: When working with blank or null values, the “not equal” operator may not behave as expected. Consider using the ISBLANK or ISEMPTY functions to handle these
cases more effectively.
• Performance Considerations: Overly complex “not equal” formulas, especially when combined with other functions, can slow down the performance of your Google Sheet. Be mindful of the number of
calculations and try to optimize your formulas where possible.
By following these best practices, tips, and troubleshooting guidelines, you’ll be well on your way to mastering the use of the “not equal” operator in Google Sheets.
Mastering the Google Sheets Not Equal Operator
In this comprehensive guide, we’ve explored the power and versatility of the “not equal” operator in Google Sheets. From using it in conditional formatting to combining it with advanced functions
like SUMIF and COUNTIF, you now have a solid understanding of how to leverage this operator to streamline your data analysis and reporting.
To further enhance your Google Sheets workflow, consider exploring Coefficient’s suite of software tools. Coefficient offers a range of solutions designed to help you automate tasks, improve data
quality, and streamline your overall Google Sheets experience. Get started today – it’s free.
Try the Spreadsheet Automation Tool Over 500,000 Professionals are Raving About
Tired of spending endless hours manually pushing and pulling data into Google Sheets? Say goodbye to repetitive tasks and hello to efficiency with Coefficient, the leading spreadsheet automation tool
trusted by over 350,000 professionals worldwide.
Sync data from your CRM, database, ads platforms, and more into Google Sheets in just a few clicks. Set it on a refresh schedule. And, use AI to write formulas and SQL, or build charts and pivots.
Hannah Recker Growth Marketer
Hannah Recker was a data-driven growth marketer before partying in the data became a thing. In her 12 years experience, she's become fascinated with the way data enablement amongst teams can truly
make or break a business. This fascination drove her to taking a deep dive into the data industry over the past 4 years in her work at StreamSets and Coefficient.
500,000+ happy users
Wait, there's more!
Connect any system to Google Sheets in just seconds.
Get Started Free
Trusted By Over 50,000 Companies | {"url":"https://coefficient.io/google-sheets-tutorials/google-sheets-not-equal","timestamp":"2024-11-11T23:08:56Z","content_type":"text/html","content_length":"77268","record_id":"<urn:uuid:d40e4102-4f18-4992-9bf2-e6fcffdf3412>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00130.warc.gz"} |
Remote Pairs
Remote Pairs
uses the properties of
locked sets extended along a
of cells to identify
A bi-value
locked set
in a
, called a
naked pair
, means that the two candidates contained in the locked set cannot be used in any other cell in the unit. Remote Pairs is a way to extend the reach of this logic beyond the bounds of a unit.
If a number of bi-value cells containing the same two candidates exist in a puzzle and these cells can be connected using pairs of cells in the same unit, a pair of cells with an odd number of
connections between them behaves like a naked pairs but only affects common
buddy cells
Remote Pairs
example in Figure 1 shows a number of bi-value cells with candidates 2 and 5 chained together using the pairs of cells contained in a single unit.
In this case the chain was started at the cell tagged with an 'A'.
Consider the two green cells tagged with an 'A and a 'B' which have an odd number of connections between them, in this case three. If the value 2 is placed in cell 'A' and the naked pair logic
followed along the chain of connections then cell 'B' would have to be a 5. If the value 5 is placed in cell 'A' and the naked pair logic followed along the chain of connections then cell 'B' has to
be a 2.
Either way, this shows that the cells tagged 'A' and 'B' will eventually have to contain a 2 and a 5. There is no way to tell which cell will contain which candidate at this point, but the 2 and the
5 will get used in these two cells. So far this is just like a naked set, but these two cells are no longer in the same unit. They do however have some common buddy cells and these common buddy cells
are of interest.
The yellow cells are buddy cells to both the green cells. If the two green cells have to eventually end up containing the candidates 2 and 5, the yellow cells cannot contain either the 2 or the 5.
The candidates 2 and 5 can be removed from all of the yellow cells.
This process will work on longer chains as long as there is an odd number of connections between the two cells used to identify candidate removal.
If two cells with an even number of connections between them are considered, the logic of naked pairs followed along the chain of connections will show both cells will end up containing the same
candidate, but there is no way to identify which one it will be. There is not enough information here to identify possible candidate removal.Figure 2 shows a Remote Pairs example with seven
connections between the two cells used to identify candidate removal.
The cells tagged with an 'A' and a 'B' contain the candidates 4 and 6. These candidate cannot be used in any cells that are buddy cells to both the green cells.
The yellow cells can have the candidate 4 removed. | {"url":"https://sudoku.ironmonger.com/howto/remotePairs/docs.tpl","timestamp":"2024-11-10T07:57:05Z","content_type":"text/html","content_length":"17799","record_id":"<urn:uuid:8c5c42f1-bfb4-42a9-998d-260c30c30463>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00114.warc.gz"} |
Calculating the Equivalent Radius
Comparing the minimal frictional coefficient with results from a Sedimentation Velocity experiment will yield estimates of maximum hydration and maximum asymmetry. Please note that this comparison
depends on the description of the buffer, particularly the viscosity, η, of the buffer; therefore accurate results may require that you enter or interpolate a viscosity since the default viscosity is
that of water. Sednterp calculates these estimates using the models of prolate, oblate, and cylinder for the asymmetry. Since hydration and asymmetry are related, only maximum values can be given
from sedimentation experiments alone. However, if some information is available about either hydration or asymmetry then Sednterp will use this to calculate better estimates of the other value. For
instance, if the composition is used for an estimate of hydration, you may check a box to use this estimated hydration to calculate a better estimate of the asymmetry. | {"url":"https://bitc.sr.unh.edu/index.php?title=Calculating_the_Equivalent_Radius&printable=yes","timestamp":"2024-11-11T23:30:15Z","content_type":"text/html","content_length":"14871","record_id":"<urn:uuid:90495df3-76e9-4ef6-aac7-e62aec61900a>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00889.warc.gz"} |
k-means clustering
k-means clustering is a popular aggregation (or clustering) method. Run k-means on your data in Excel using the XLSTAT add-on statistical software.
Description of the k-means clustering analysis in XLSTAT
General description
k-means clustering was introduced by McQueen in 1967. Other similar algorithms had been developed by Forgey (1965) (moving centers) and Friedman (1967).
k-means clustering has the following advantages:
• An object may be assigned to a class during one iteration then change class in the following iteration, which is not possible with Agglomerative Hierarchical Clustering, where assignment is
• By multiplying the starting points and the repetitions, several solutions may be explored.
The disadvantage of this method is that it does not give a consistent number of classes or enable the proximity between classes or objects to be determined.
The k-means and AHC methods are therefore complementary.
Note: if you want to take qualitative variables into account in the clustering, you must first perform a Multiple Correspondence Analysis (MCA) and consider the resulting coordinates of the
observations on the factorial axes as new variables.
Principle of the k-means method
k-means clustering is an iterative method which, wherever it starts from, converges on a solution. The solution obtained is not necessarily the same for all starting points. For this reason, the
calculations are generally repeated several times in order to choose the optimal solution for the selected criterion.
For the first iteration, a starting point is chosen which consists of associating the center of the k classes with k objects (either taken at random or not). Afterwards, the distance between the
objects and the k centers are calculated, and the objects are assigned to the centers they are nearest to. Then the centers are redefined from the objects assigned to the various classes. The objects
are then reassigned depending on their distances from the new centers. And so on until convergence is reached.
Classification criteria for k-means Clustering
Several classification criteria may be used to reach a solution. XLSTAT offers four criteria for the k-means minimization algorithm:
Trace(W): The W trace, pooled SSCP matrix, is the most traditional criterion. Minimizing the W trace for a given number of classes amounts to minimizing the total within-class variance — in other
words, minimizing the heterogeneity of the groups. This criterion is sensitive to effects of scale. In order to avoid giving more weight to certain variables and not to others, the data must be
normalized beforehand. Moreover, this criterion tends to produce classes of the same size.
Determinant(W): The determinant of W, pooled within covariance matrix, is a criterion considerably less sensitive to effects of scale than the W trace criterion. Furthermore, group sizes may be less
homogeneous than with the trace criterion.
Wilks lambda: The results given by minimizing this criterion are identical to that given by the determinant of W. Wilks’ lambda criterion corresponds to the division of determinant(W) by determinant
(T) where T is the total inertia matrix. Dividing by the determinant of T always gives a criterion between 0 and 1.
Trace(W) / Median: If this criterion is chosen, the class centroid is not the mean point of the class but the median point, which corresponds to an object of the class. The use of this criterion
gives rise to longer calculations.
Results for k-means clustering in XLSTAT
Summary statistics: This table displays the descriptors of the objects, the number of observations, the number of missing values, the number of non-missing values, the mean and the standard
Correlation matrix: This table is displayed to give you a view of the correlations between the various variables selected.
Evolution of the within-class inertia: If you have selected a number of classes between two bounds, XLSTAT displays at first the evolution of the within-class inertia, which reduces mathematically
when the number of classes increases. If the data is distributed homogeneously, the decrease is linear. If there is actually a group structure, an elbow is observed for the relevant number of
Evolution of the silhouette score: If you have selected a number of classes between two bounds, a table with its associated chart shows the evolution of the silhouette score for each k. The optimal
number of classes is the k whose silhouette score is closest to 1.
Optimization summary: This table shows the evolution of the within-class variance. If several repetitions have been requested, the results for each repetition are displayed. The repetition giving the
best classification is displayed in bold.
Statistics for each iteration: This table shows the evolution of miscellaneous statistics calculated as the iterations for the repetition proceed, given the optimum result for the chosen criterion.
If the corresponding option is activated in the Charts tab, a chart showing the evolution of the chosen criterion as the iterations proceed is displayed.
Note: if the values are standardized (option in the Options tab), the results for the optimization summary and the statistics for each iteration are calculated in the standardized space. On the other
hand, the following results are displayed in the original space if the "Results in the original space" option is activated.
Inertia decomposition for the optimal classification: This table shows the within-class inertia, the between-class inertia and the total inertia.
Initial class centroids: This table shows the initial class centroids computed thanks to the initial random partition or with K|| and K++ algorithms. In case you defined the centers, this table shows
the selected class centroids.
Class centroids: This table shows the class centroids for the various descriptors.
Distance between the class centroids: This table shows the Euclidean distances between the class centroids for the various descriptors.
Central objects: This table shows the coordinates of the nearest object to the centroid for each class.
Distance between the central objects: This table shows the Euclidean distances between the class central objects for the various descriptors.
Results by class: The descriptive statistics for the classes (number of objects, sum of weights, within-class variance, minimum distance to the centroid, maximum distance to the centroid, mean
distance to the centroid) are displayed in the first part of the table. The second part shows the objects.
Results by object: This table shows the assignment class for each object in the initial object order.
• Distance to centroid: this column shows the distance between an object and its class centroids.
• Correlations with centroids: this column shows the Pearson correlation between an object and its class centroids.
• Silhouette scores: this column shows the silhouette score of each object.
Silhouette scores (Mean by class): This table and its graph are displayed and show the mean silhouette score of each class and the silhouette score for the optimal classification (mean of means by
Contribution (Analysis of variance) : This table indicates the variables that contribute the most to the separation of the classes by performing an ANOVA.
Profile plot: This chart allows you to compare the means of the different classes that have been created.
analyze your data with xlstat
14-day free trial | {"url":"https://www.xlstat.com/en/solutions/features/k-means-clustering","timestamp":"2024-11-12T03:25:16Z","content_type":"text/html","content_length":"35231","record_id":"<urn:uuid:1d0e37fb-923b-42a7-8367-b23cdd193ba9>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00729.warc.gz"} |
An open box has a square base and a surface area of 240 square
inches. What dimensions (width × length × height) will produce a
box with maximum volume? | HIX Tutor
An open box has a square base and a surface area of 240 square inches. What dimensions (width × length × height) will produce a box with maximum volume?
Answer 1
$4 \sqrt{5}$ inches is the length of one edge of the square base and$2 \sqrt{5}$ , inches the height of the box.
Let # x# be the length of one edge of the square base and
#y# , the height of the box . Surface area of the box is
#s= x^2+4 x y =240 or 4 x y = 240-x ^2 or y= (240-x ^2)/(4x)#
or #y=60/x-x/4# . Volume of the box is #v= x^2*y# or
#v= x^2(60/x-x/4) = 60 x-x^3/4# differentiating w.r.t #x#
we get, #v^' = 60 - (3 x^2)/4# For maximizing #v^'=0#
#60 - (3 x^2)/4= 0 or 240 - 3 x^2 =0 or 80 - x^2=0# or
# ((4sqrt5)^2-x^2 )=0 or (4 sqrt5 +x)(4 sqrt5 -x)=0#
#:. x = 4 sqrt 5 or x = - 4 sqrt5 # , but #x != - 4 sqrt5 #
#:.x = 4 sqrt5 # When #x<4 sqrt5 , v^' >0 # , slope increasing
When #x>4 sqrt5 , v^' <0 # , slope decreasing. #:.x =4 sqrt5 #
gives the maximum volume. When #x= 4 sqrt 5#
#y = (240-80)/(16sqrt5)= 10/sqrt5= 2 sqrt5 #
# 4sqrt5# inches is the length of one edge of the square base and
#2 sqrt 5 # inches , the height of the box. [Ans]
Sign up to view the whole answer
By signing up, you agree to our Terms of Service and Privacy Policy
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
Not the question you need?
HIX Tutor
Solve ANY homework problem with a smart AI
• 98% accuracy study help
• Covers math, physics, chemistry, biology, and more
• Step-by-step, in-depth guides
• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/an-open-box-has-a-square-base-and-a-surface-area-of-240-square-inches-what-dimen-07460e11a7","timestamp":"2024-11-04T18:33:30Z","content_type":"text/html","content_length":"571091","record_id":"<urn:uuid:da95007a-c24b-482e-8158-b10d977f463b>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00896.warc.gz"} |
Unscramble WAB
How Many Words are in WAB Unscramble?
By unscrambling letters wab, our Word Unscrambler aka Scrabble Word Finder easily found 4 playable words in virtually every word scramble game!
Letter / Tile Values for WAB
Below are the values for each of the letters/tiles in Scrabble. The letters in wab combine for a total of 8 points (not including bonus squares)
What do the Letters wab Unscrambled Mean?
The unscrambled words with the most letters from WAB word or letters are below along with the definitions.
• wab () - Sorry, we do not have a definition for this word | {"url":"https://www.scrabblewordfind.com/unscramble-wab","timestamp":"2024-11-01T22:28:51Z","content_type":"text/html","content_length":"34902","record_id":"<urn:uuid:80daf63b-01ab-4095-a308-eb6a605e9a30>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00227.warc.gz"} |
The Matrix, It's a Complex Plot
Remember the classic science fiction film The Matrix? The dark sunglasses, the leather, computer monitors constantly raining streams of integers (inexplicably in base 10 rather than binary or
hexadecimal)? And that mind-blowing plot twist when Neo takes the red pill from Morpheus' outstretched hand? Well to me, there's one thing even more mind-blowing than the plot of the Matrix: the
Matrix Plot. You know, in Minitab Statistical Software. (Click here to download a free trial.)
Just as Neo and his band of futuristic rebels were constantly barraged with endless streams of data, it seems like we, too, often face large amounts of data and we must make sense of. When faced with
such a challenge, a good place to start is to create some exploratory graphs in Minitab. Previous posts have extolled the virtues of the Individual Value Plot and Graphical Summary for this purpose.
Today, we're going to use the oracle of all plots, the Matrix Plot, to uncover the secrets of automobile specifications data. (Follow the link and scroll to the bottom of the page to download the
The data set looks like this:
There's a lot to take in here. The columns look like streams of random numbers...but are they? Time to enter the matrix. A matrix plot is a great exploratory tool because you can throw a bunch of
data in it and just see what happens.
From Minitab's Graph menu, choose Matrix Plot. Under Matrix of plots, choose With Groups, and fill out the dialog box thusly:
It is at this point that you must make a difficult choice. You can choose the blue pill^1 (a.k.a., the Cancel button) and go about your business, oblivious to and untroubled by the mind-blowing
automotive realities that surround you. Or you can choose the red pill (click OK), after which your life will forever be altered by your ability to see into the data, to understand it, and—with
practice—to even control it.^2^
If you chose the blue pill, click here.
If you chose the red pill, read on.
As you can see, the matrix plot packs a lot of information into a small space. I like to do a couple of things to allow the data to spread out just a little. Remove the graph title by clicking it and
pressing Delete. Then, choose Editor > Graph Options, and select Don't alternate (under Alternate Ticks on Plots). There, that's a little better:
It's a lot to take in, but don't worry. Just as our band of heroes in The Matrix learned to read the endless streams of integers on their monitors, so too will this mass of dots soon make sense to
The matrix plot is simply a grid of scatterplots. For example, the left-most scatterplot in the top row shows City MPG on y-axis and Hwy MPG on the x-axis. Not surprisingly, there appears to be a
very tight relationship between these two variables: vehicles with good city mileage tend to also have good highway mileage. You can tell from the scales that city MPG for all vehicles ranges between
about 10 and 55 and that highway MPG ranges between about 19 and 50. From the symbols, you can also easily tell that the hybrid vehicles (red squares) get better mileage than gas-only vehicles (blue
To simplify things, we can remove City MPG and Hwy MPG from the plot and leave just Total MGP (which is just City MPG + Hwy MPG). We can also remove Total Volume (which is Interior Volume + Cargo
To return to the Matrix Plot dialog box, you can press Ctrl + E. This time, in Graph variables, enter just columns C6 through C10.
(To maximize the space for data, I deleted the title and un-alternated the tick marks for this graph like we did for the last one.)
One thing that jumps out is that Safety isn't like the other variables. The other variables are continuous, but the safety ratings take on one of three discrete values: 3, 4, or 5. For discrete
variables, the plot looks like an individual value plot. Interestingly, all hybrid vehicles scored a 4 or a 5; the only vehicles to score a 3 were gas-only.
Another thing that jumps out is the outlier in the Retail (price) measurements. While the other vehicles cost under $45,000, one vehicle sells for more than $70,000. Conveniently, we can brush the
outlier and quickly see how that vehicle scores on the other measures. (For more information on this powerful tool, see Using brushing to investigate data points.)
The brushing palette shows that the outlier is in row 10 of the worksheet. The point for this observation is highlighted in each plot of the matrix. So you can quickly tell, for example, that even
though you may have to ransack your kid's college fund to afford this beauty, at least he or she will enjoy the extra passenger room afforded by this luxury vehicle. And they are assured to arrive at
their non-college-campus destinations in one piece because this vehicle gets the highest safety rating. However, you may have to pass the hat for gas because it looks like this baby is always
Among its other virtues, the high price tag has the added effect of squishing the data for the other vehicles into the low end of the scale and thus making the graph harder to read. Now that I've
scratched this rig off my wish list, let's go ahead and remove it from the plot. Again, we use the Ctrl + E trick to reopen the dialog box. This time we click the Data Options button and specify to
exclude row 10 from the graph:
Without the gas-guzzling outlier in the picture, it becomes clear that there is another outlier in town. One of the vehicles has an unusually low interior volume. Again, we can brush this point to
see what's going on.
Brushing shows that this vehicle is about average on the other measures. It doesn't cost less than the others and doesn't seem to get better mileage; it's just cramped on the inside. Not a big
selling point. Let's remove this point as well. (This vehicle is in row 15.)
Without the outliers, the overall picture becomes still clearer. In general, it looks like more money does not buy you better gas mileage. The negative relationship between price and mileage is clear
for both hybrid and gas-only vehicles. However, more money does seem to buy you more space. It looks like there is a positive relationship between price and interior volume and between price and
cargo volume. Bigger vehicles are heavier and generate more wind resistance, so no wonder the more expensive vehicles tend to get worse gas mileage.
I think you'll agree that we have learned a lot about these data since we first entered the matrix just a few mouse clicks ago. No doubt more time in the matrix will reveal even more insights. Aren't
you glad you chose the red pill?
1. The Matrix Plot dialog box featured in this post has been embellished for the purpose of dramatizing this reenactment. In real life, Minitab dialog boxes do not feature pills, or pharmaceutical
agents of any kind. No actual dialog boxes or buttons were harmed during the making of this blog post. [return]
2. OK, so you can't really use a matrix plot to actually change the data in the worksheet. But you *can* use the matrix plot to change how *you see* the data and enable you to reveal more of your
data secrets. And isn't that what's important? [return]
Credit for the original pill images goes to W.carter. Pills and steak dinner available under Creative Commons License 2.0 and Creative Commons License 1.0 respectively. | {"url":"https://blog.minitab.com/en/data-analysis-and-quality-improvement-and-stuff/the-matrix-its-a-complex-plot","timestamp":"2024-11-12T18:13:23Z","content_type":"text/html","content_length":"101223","record_id":"<urn:uuid:4a3dbc4e-26f4-4c1e-921d-4727d735c21b>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00513.warc.gz"} |
Acoustic wave propagation near the axis of a refractive waveguide
A part of the acoustic field of a point source located near the axis of a weakly irregular refractive waveguide is calculated using the method of two-scale expansions. This part of the field
corresponds to rays that are close to the waveguide axis (the part of the field corresponding to rays that are far from the axis can be calculated by the ordinary ray tracing method). The
applicability limits of the formulas obtained here are extended through a conversion from Cartesian coordinates to ray coordinates defined by the waveguide axis.
Akusticheskii Zhurnal
Pub Date:
October 1986
□ Acoustic Propagation;
□ Axes (Reference Lines);
□ Refracted Waves;
□ Sound Waves;
□ Wave Propagation;
□ Waveguides;
□ Cartesian Coordinates;
□ Point Sources;
□ Propagation Modes;
□ Propagation Velocity;
□ Ray Tracing;
□ Acoustics | {"url":"https://ui.adsabs.harvard.edu/abs/1986AkZh...32..667V/abstract","timestamp":"2024-11-10T18:42:47Z","content_type":"text/html","content_length":"34417","record_id":"<urn:uuid:0b47b598-eeb1-4833-982c-af18401aebc3>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00743.warc.gz"} |
Abstract Mathematics
Bachelor 2024/2025
Abstract Mathematics
Area of studies: Economics
When: 3 year, 1-4 module
Mode of studies: offline
Open to: students of one campus
Language: English
ECTS credits: 8
Contact hours: 120
Course Syllabus
Abstract Mathematics is a two-semester course for the third year bachelor’s programme students who selected specialization Economics and Mathematics. The course is taught in English. For the theory
itself there are no pre-requisites except for an aptitude for logical reasoning. However, many examples will reference concepts from Calculus, Statistics, Mathematics for Economists and Linear
Algebra courses from the 1st and 2nd years of the ICEF bachelor’s programme. The emphasis of the course is on the theory rather than on the method. One central topic of the course is formal
mathematical reasoning. The students will practice formulating precise mathematical statements and proving them rigorously. These skills are essential for the current specialization, they often
remain in shadows in other math courses where the focus is on solving problems through calculation. The second central topic of the course is the abstract mathematical structures from algebra
(groups, fields, etc.), analysis, topology (topological spaces, manifolds), and mathematical logic. We will develop some of these theories roughly to the extent of standard 1st and 2nd -year courses
of the mathematical departments. The awareness of the theoretical foundations of these classical theories is key in understanding the contemporary theoretical research and the synergies between
different areas of mathematics and its applications.
Learning Objectives
• to explain the main mathematical concepts in discrete mathematics, algebra, real analysis, functional analysis and topology;
• to illustrate the concepts by specific examples and counter-examples;
• to teach how to use formal notations correctly and in connection with precise statements in English
• to give definitions, formulate statements of the key theorems and present their proofs
• to critically analyze a proposed proofs of a given statement and make a conclusion on the completeness and accurateness of the proof
• to give a generic understanding of the applications of the discussed classical theories
• to teach how to find and formulate proofs of problems based on the main definitions and theorems
Expected Learning Outcomes
• be able to critically analyze a proposed proof of a given statement and make a conclusion on the completeness and accurateness of the proof;
• be able to find and formulate proofs of problems based on the main definitions and theorems
• be able to give definitions, formulate statements of the key theorems and present their proofs
• be able to give definitions, formulate statements of the key theorems, such as Chinese Remainder Theorem, The Fundamental Theorem of Arithmetic, etc., and present their proofs;
• be able to give definitions, formulate statements of the key theorems, such as Lagrange Theorem, Homomorphism Theorem, etc., and present their proofs
• be able to give the opposite and the contrapositive statements for a given statement
• be able to illustrate the concepts in Set Theory by specific examples and counter-examples
• be able to illustrate the concepts of Group Theory by specific examples and counter-examples
• be able to illustrate the concepts of Real Analysis including the axiomatic definition of the set of real numbers; Functional Analysis such as norm, metric, metric spaces; Topology such as a
topological space, base of topology by specific examples and counter-examples;
• be able to illustrate the concepts of Ring Theory and Field Theory by specific examples and counter-examples
• be able to use formal notations correctly and in connection with precise statements in English
Course Contents
• Introduction to Set Theory
• Algebraic Structures: Groups
• Algebraic Structures: Rings and Fields
• Elements of Real Analysis, Functional Analysis and Topology
• Introduction to Mathematical Reasoning
• Mathematical Logic
Assessment Elements
• Homework and in class activities
• Midterm 2
• Midterm 1
• Final exam
In order to get a passing grade for the course, the student must sit (all parts) of the examination.
• Midterm 3
Interim Assessment
• 2024/2025 4th module
0.35 * Final exam + 0.13 * Homework and in class activities + 0.105 * Midterm 1 + 0.135 * Midterm 2 + 0.28 * Midterm 3
Recommended Core Bibliography
• An introduction to mathematical reasoning : numbers, sets and functions, Eccles, P. J., 2013
• Discrete mathematics, Biggs, N. L., 2004
Recommended Additional Bibliography
• Элементы теории функций и функционального анализа, Колмогоров, А. Н., 2006 | {"url":"https://www.hse.ru/en/edu/courses/858761299","timestamp":"2024-11-09T11:17:58Z","content_type":"text/html","content_length":"50843","record_id":"<urn:uuid:c06f7575-2c21-4c88-aacb-211b6f7a209a>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00050.warc.gz"} |
Tree sort – binary search tree sort. Time complexity – O(n²). In such a tree, each node has numbers less than the node on the left, more than the node on the right, when coming from the root and
printing the values from left to right, we get a sorted list of numbers. Surprising huh?
Consider the binary search tree schema:
Derrick Coetzee (public domain)
Try to manually read the numbers starting from the penultimate left node of the lower left corner, for each node on the left – a node – on the right.
It will turn out like this:
1. The penultimate node at the bottom left is 3.
2. She has a left branch – 1.
3. Take this number (1)
4. Next, take the vertex 3 (1, 3) itself
5. To the right is branch 6, but it contains branches. Therefore, we read it in the same way.
6. Left branch of node 6 number 4 (1, 3, 4)
7. Node 6 itself (1, 3, 4, 6)
8. Right 7 (1, 3, 4, 6, 7)
9. Go up to the root node – 8 (1,3, 4 ,6, 7, 8)
10. Print everything on the right by analogy
11. Get the final list – 1, 3, 4, 6, 7, 8, 10, 13, 14
To implement the algorithm in code, you need two functions:
1. Building a binary search tree
2. Printing the binary search tree in the correct order
They assemble a binary search tree in the same way as they read it, a number is attached to each node on the left or right, depending on whether it is less or more.
Lua example:
Node = {value = nil, lhs = nil, rhs = nil}
function Node:new(value, lhs, rhs)
output = {}
setmetatable(output, self)
self.__index = self
output.value = value
output.lhs = lhs
output.rhs = rhs
output.counter = 1
return output
function Node:Increment()
self.counter = self.counter + 1
function Node:Insert(value)
if self.lhs ~= nil and self.lhs.value > value then
if self.rhs ~= nil and self.rhs.value < value then
if self.value == value then
elseif self.value > value then
if self.lhs == nil then
self.lhs = Node:new(value, nil, nil)
if self.rhs == nil then
self.rhs = Node:new(value, nil, nil)
function Node:InOrder(output)
if self.lhs ~= nil then
output = self.lhs:InOrder(output)
output = self:printSelf(output)
if self.rhs ~= nil then
output = self.rhs:InOrder(output)
return output
function Node:printSelf(output)
for i=0,self.counter-1 do
output = output .. tostring(self.value) .. " "
return output
function PrintArray(numbers)
output = ""
for i=0,#numbers do
output = output .. tostring(numbers[i]) .. " "
function Treesort(numbers)
rootNode = Node:new(numbers[0], nil, nil)
for i=1,#numbers do
numbersCount = 10
maxNumber = 9
numbers = {}
for i=0,numbersCount-1 do
numbers[i] = math.random(0, maxNumber)
An important nuance is that for numbers that are equal to the vertex, a lot of interesting mechanisms for hooking to the node have been invented, but I just added a counter to the vertex class, when
printing, the numbers are returned by the counter.
https://gitlab.com/demensdeum /algorithms/-/tree/master/sortAlgorithms/treesort
TreeSort Algorithm Explained and Implemented with Examples in Java | Sorting Algorithms | Geekific – YouTube
Convert Sorted Array to Binary Search Tree (LeetCode 108. Algorithm Explained) – YouTube
Sorting algorithms/Tree sort on a linked list – Rosetta Code
How to handle duplicates in Binary Search Tree? – GeeksforGeeks
Tree Sort | GeeksforGeeks – YouTube
Bucket Sort
Bucket Sort – bucket sorting. The algorithm is similar to sorting by counting, with the difference that the numbers are collected into “buckets”-ranges, then the buckets are sorted using any other,
sufficiently productive, sorting algorithm, and the final chord is the expansion of the “buckets” one by one, resulting in a sorted list.
The time complexity of the algorithm is O(nk). The algorithm runs in linear time for data that obeys a uniform distribution. To put it simply, the elements must be in a certain range, without
“splashes”, for example, numbers from 0.0 to 1.0. If among such numbers there are 4 or 999, then such a series, according to the laws of the yard, is no longer considered “even”.
Implementation example in Julia:
function bucketSort(numbers, bucketsCount)
buckets = Vector{Vector{Int}}()
for i in 0:bucketsCount - 1
bucket = Vector{Int}()
push!(buckets, bucket)
maxNumber = maximum(numbers)
for i in 0:length(numbers) - 1
bucketIndex = 1 + Int(floor(bucketsCount * numbers[1 + i] / (maxNumber + 1)))
push!(buckets[bucketIndex], numbers[1 + i])
for i in 0:length(buckets) - 1
bucketIndex = 1 + i
buckets[bucketIndex] = sort(buckets[bucketIndex])
flat = [(buckets...)...]
print(flat, "\n")
numbersCount = 10
maxNumber = 10
numbers = rand(1:maxNumber, numbersCount)
bucketsCount = 10
bucketSort(numbers, bucketsCount)
The performance of the algorithm is also affected by the number of buckets, for more numbers it is better to take a larger number of buckets (Algorithms in a nutshell by George T. Heineman)
Radix Sort
Radix Sort – radix sort. The algorithm is similar to counting sort in that there is no comparison of elements, instead elements are *character-by-character* grouped into *buckets* (buckets), the
bucket is selected by the index of the current number-character. Time complexity – O(nd).
Works like this:
• The input will be the numbers 6, 12, 44, 9
• Let’s create 10 buckets of lists (0-9) into which we will add/sort numbers bit by bit.
1. Run a loop with counter i up to the maximum number of characters in the number
2. At index i from right to left we get one character for each number, if there is no character, then we consider it to be zero
3. The character is converted to a number
4. Select a bucket by index – number, put the whole number there
5. After finishing iterating over numbers, convert all buckets back to a list of numbers
6. Get numbers sorted by digit
7. Repeat until all digits run out
Radix Sort example in Scala:
import scala.collection.mutable.ListBuffer
import scala.util.Random.nextInt
object RadixSort {
def main(args: Array[String]) = {
var maxNumber = 200
var numbersCount = 30
var maxLength = maxNumber.toString.length() - 1
var referenceNumbers = LazyList.continually(nextInt(maxNumber + 1)).take(numbersCount).toList
var numbers = referenceNumbers
var buckets = List.fill(10)(ListBuffer[Int]())
for( i <- 0 to maxLength) { numbers.foreach( number => {
var numberString = number.toString
if (numberString.length() > i) {
var index = numberString.length() - i - 1
var character = numberString.charAt(index).toString
var characterInteger = character.toInt
buckets.apply(characterInteger) += number
else {
buckets.apply(0) += number
numbers = buckets.flatten
buckets.foreach(x => x.clear())
println(s"Validation result: ${numbers == referenceNumbers.sorted}")
The algorithm also has a version for parallel execution, for example on the GPU; there is also a variant of bit sort, which is probably very interesting and truly breathtaking!
https://ru.wikipedia.org/wiki/%D0%9F%D0%BE%D1%80%D0%B0%D0%B7%D1%80% D1%8F%D0%B4%D0%BD%D0%B0%D1%8F_%D1%81%D0%BE%D1%80%D1%82%D0%B8%D1%80%D0%BE%D0% B2%D0%BA%D0%B0
Heapsort – heap sort. Time complexity – O(n log n), fast eh? I would call this sorting – sorting of falling stones. It seems to me that the easiest way to explain it is visually.
The input is a list of numbers, for example:
5, 0, 7, 2, 3, 9, 4
From left to right, a data structure is made – a binary tree, or as I call it – a pyramid. Pyramid elements can have a maximum of two child elements, with only one top element.
Let’s make a binary tree:
If you look at the pyramid for a long time, you can see that these are just numbers from the array, going one after another, the number of elements in each floor is multiplied by two.
Then the fun begins, we sort the pyramid from bottom to top, using the falling pebbles method (heapify). Sorting could be started from the last floor (2 3 9 4 ), but it makes no sense because there
is no floor below where one could fall.
Therefore, we start dropping elements from the penultimate floor (0 7)
The first element to fall is selected on the right, in our case it is 7, then we look at what is under it, and below it are 9 and 4, nine is more than four, so also nine is more than seven! We drop 7
on 9, and raise 9 to place 7.
Further, we understand that the seven has nowhere to fall below, go to the number 0, which is located on the penultimate floor on the left:
We look at what is under it – 2 and 3, two is less than three, three is greater than zero, so we change zero and three in places:
When you get to the end of the floor, go to the floor above and drop everything there if you can.
The result is a data structure – a heap (heap), namely max heap, because at the top is the largest element:
If you return it to an array representation, you get a list:
[9, 3, 7, 2, 0, 5, 4]
From this we can conclude that by swapping the first and last element, we will get the first number in the final sorted position, namely 9 should be at the end of the sorted list, swap:
[4, 3, 7, 2, 0, 5, 9]
Let’s look at the binary tree:
The result is a situation in which the lower part of the tree is sorted, you just need to drop 4 to the correct position, repeat the algorithm, but do not take into account the already sorted
numbers, namely 9:
It turned out that we, having dropped 4, raised the next largest number after 9 – 7. Swap the last unsorted number (4) and the largest number (7)
It turned out that now we have two numbers in the correct final position:
4, 3, 5, 2, 0, 7, 9
Next, we repeat the sorting algorithm, ignoring those already sorted, as a result we get heap:
Or as a list:
0, 2, 3, 4, 5, 7, 9
The algorithm is usually divided into three functions:
1. Heap creation
2. Sifting algorithm (heapify)
3. Replacing the last unsorted element and the first one
A heap is created by traversing the penultimate row of the binary tree using the heapify function, from right to left to the end of the array. Then the first replacement of numbers is made in the
cycle, after which the first element falls / remains in place, as a result of which the largest element falls into first place, the cycle repeats with a decrease in participants by one, because after
each pass, the sorted numbers remain at the end of the list.
Heapsort example in Ruby:
DEMO = true
module Colors
BLUE = "\033[94m"
RED = "\033[31m"
STOP = "\033[0m"
def heapsort(rawNumbers)
numbers = rawNumbers.dup
def swap(numbers, from, to)
temp = numbers[from]
numbers[from] = numbers[to]
numbers[to] = temp
def heapify(numbers)
count = numbers.length()
lastParentNode = (count - 2) / 2
for start in lastParentNode.downto(0)
siftDown(numbers, start, count - 1)
start -= 1
if DEMO
puts "--- heapify ends ---"
def siftDown(numbers, start, rightBound)
cursor = start
printBinaryHeap(numbers, cursor, rightBound)
def calculateLhsChildIndex(cursor)
return cursor * 2 + 1
def calculateRhsChildIndex(cursor)
return cursor * 2 + 2
while calculateLhsChildIndex(cursor) <= rightBound
lhsChildIndex = calculateLhsChildIndex(cursor)
rhsChildIndex = calculateRhsChildIndex(cursor)
lhsNumber = numbers[lhsChildIndex]
biggerChildIndex = lhsChildIndex
if rhsChildIndex <= rightBound
rhsNumber = numbers[rhsChildIndex]
if lhsNumber < rhsNumber
biggerChildIndex = rhsChildIndex
if numbers[cursor] < numbers[biggerChildIndex]
swap(numbers, cursor, biggerChildIndex)
cursor = biggerChildIndex
printBinaryHeap(numbers, cursor, rightBound)
printBinaryHeap(numbers, cursor, rightBound)
def printBinaryHeap(numbers, nodeIndex = -1, rightBound = -1)
if DEMO == false
perLineWidth = (numbers.length() * 4).to_i
linesCount = Math.log2(numbers.length()).ceil()
xPrinterCount = 1
cursor = 0
spacing = 3
for y in (0..linesCount)
line = perLineWidth.times.map { " " }
spacing = spacing == 3 ? 4 : 3
printIndex = (perLineWidth / 2) - (spacing * xPrinterCount) / 2
for x in (0..xPrinterCount - 1)
if cursor >= numbers.length
if nodeIndex != -1 && cursor == nodeIndex
line[printIndex] = "%s%s%s" % [Colors::RED, numbers[cursor].to_s, Colors::STOP]
elsif rightBound != -1 && cursor > rightBound
line[printIndex] = "%s%s%s" % [Colors::BLUE, numbers[cursor].to_s, Colors::STOP]
line[printIndex] = numbers[cursor].to_s
cursor += 1
printIndex += spacing
print line.join()
xPrinterCount *= 2
print "\n"
rightBound = numbers.length() - 1
while rightBound > 0
swap(numbers, 0, rightBound)
rightBound -= 1
siftDown(numbers, 0, rightBound)
return numbers
numbersCount = 14
maximalNumber = 10
numbers = numbersCount.times.map { Random.rand(maximalNumber) }
print numbers
print "\n---\n"
start = Time.now
sortedNumbers = heapsort(numbers)
finish = Time.now
heapSortTime = start - finish
start = Time.now
referenceSortedNumbers = numbers.sort()
finish = Time.now
referenceSortTime = start - finish
print "Reference sort: "
print referenceSortedNumbers
print "\n"
print "Reference sort time: %f\n" % referenceSortTime
print "Heap sort: "
print sortedNumbers
print "\n"
if DEMO == false
print "Heap sort time: %f\n" % heapSortTime
print "Disable DEMO for performance measure\n"
if sortedNumbers != referenceSortedNumbers
puts "Validation failed"
exit 1
puts "Validation success"
exit 0
Without visualization, this algorithm is not easy to understand, so the first thing I recommend is to write a function that will print the current view of the binary tree.
https://ru.wikipedia.org/wiki/Дерево (структура данных)
https://ru.wikipedia.org/wiki/Куча (структура данных)
Quicksort is a divide-and-conquer sorting algorithm. Recursively, in parts, parse the array of numbers, setting the numbers in the smaller and larger order from the selected pivot element, insert the
pivot element itself into the hot-point between them. After a few recursive iterations, you’ll end up with a sorted list. Time complexity O(n^2).
1. We start with the fact that we get a list of elements outside, sorting boundaries. In the first step, the sort boundaries will be from start to finish.
2. Check that the boundaries of the beginning and end do not intersect, if this happens, then it’s time to finish
3. Select some element from the list, call it pivot
4. Shift to the right to the end to the last index, so as not to interfere
5. Create a counter of *smaller numbers* while equal to zero
6. Loop through the list from left to right, up to and including the last index where the anchor element is located
7. Each element is compared to the pivot
8. If it is less than the reference, then we swap it in places according to the index of the counter of smaller numbers. Increment the counter of smaller numbers.
9. When the cycle reaches the anchor element, we stop, swap the anchor element with the element by the counter of smaller numbers.
10. We run the algorithm separately for the left smaller part of the list, and separately for the right most part of the list.
11. As a result, all recursive iterations will start to stop due to the check in paragraph 2
12. Get sorted list
Quicksort was invented by the scientist Charles Antony Richard Hoare at Moscow State University, having learned Russian, he studied computer translation, as well as probability theory at the
Kolmogorov school. In 1960, due to a political crisis, he left the Soviet Union.
An example implementation in Rust:
extern crate rand;
use rand::Rng;
fn swap(numbers: &mut [i64], from: usize, to: usize) {
let temp = numbers[from];
numbers[from] = numbers[to];
numbers[to] = temp;
fn quicksort(numbers: &mut [i64], left: usize, right: usize) {
if left >= right {
let length = right - left;
if length <= 1 {
let pivot_index = left + (length / 2);
let pivot = numbers[pivot_index];
let last_index = right - 1;
swap(numbers, pivot_index, last_index);
let mut less_insert_index = left;
for i in left..last_index {
if numbers[i] < pivot {
swap(numbers, i, less_insert_index);
less_insert_index += 1;
swap(numbers, last_index, less_insert_index);
quicksort(numbers, left, less_insert_index);
quicksort(numbers, less_insert_index + 1, right);
fn main() {
let mut numbers = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0];
let mut reference_numbers = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0];
let mut rng = rand::thread_rng();
for i in 0..numbers.len() {
numbers[i] = rng.gen_range(-10..10);
reference_numbers[i] = numbers[i];
println!("Numbers {:?}", numbers);
let length = numbers.len();
quicksort(&mut numbers, 0, length);
println!("Numbers {:?}", numbers);
println!("Reference numbers {:?}", reference_numbers);
if numbers != reference_numbers {
println!("Validation failed");
else {
println!("Validation success!");
If you don't understand anything, I suggest watching a video by Rob Edwards from the San Diego State University https://www.youtube.com/watch?v=ZHVk2blR45Q it most simply, step by step, shows the
essence and implementation of the algorithm.
Binary Insertion Sort
Binary Insertion Sort is a variant of insertion sort in which the insertion position is determined using a binary search. The time complexity of the algorithm is O(n^2)
The algorithm works like this:
1. Loop from zero to end of list
2. In the loop, a number is selected for sorting, the number is stored in a separate variable
3. Binary search looks for the index to insert this number against the numbers on the left
4. After finding the index, the numbers on the left are shifted one position to the right, starting at the insertion index. The process will erase the number to be sorted.
5. The previously stored number is inserted at the insertion index
6. At the end of the loop, the entire list will be sorted
During a binary search, it is possible that the number will not be found, but the index is not returned. Due to the peculiarities of the binary search, the number closest to the desired one will be
found, then to return the index it will be necessary to compare it with the desired one, if the desired one is less, then the desired one should be at the index on the left, and if it is greater or
equal, then on the right.
Source code in Go:
package main
import (
const numbersCount = 20
const maximalNumber = 100
func binarySearch(numbers []int, item int, low int, high int) int {
for high > low {
center := (low + high) / 2
if numbers[center] < item { low = center + 1 } else if numbers[center] > item {
high = center - 1
} else {
return center
if numbers[low] < item {
return low + 1
} else {
return low
func main() {
var numbers [numbersCount]int
for i := 0; i < numbersCount; i++ {
numbers[i] = rand.Intn(maximalNumber)
for i := 1; i < len(numbers); i++ { searchAreaLastIndex := i - 1 insertNumber := numbers[i] insertIndex := binarySearch(numbers[:], insertNumber, 0, searchAreaLastIndex) for x := searchAreaLastIndex; x >= insertIndex; x-- {
numbers[x+1] = numbers[x]
numbers[insertIndex] = insertNumber
Shell Sort
Shell Sort is a variant of sorting by inserts with preliminary combing of an array of numbers.
We need to remember how insertion sort works:
1. The loop starts from zero to the end of the loop, thus the array is divided into two parts
2. For the left side, the second loop is started, comparing elements from right to left, the smaller element on the right is omitted until there is a smaller element on the left
3. At the end of both cycles, we get a sorted list
Once in a while, computer scientist Donald Shell wondered how to improve the insertion sort algorithm. He came up with the idea to preliminarily go through the array with two cycles, but at a certain
distance, gradually reducing the “comb” until it turns into a regular insertion sort algorithm. Everything is really so simple, no pitfalls, we add another one to the two cycles from above, in which
we gradually reduce the size of the “comb”. The only thing that will need to be done is to check the distance when comparing so that it does not go beyond the array.
A really interesting topic is the choice of sequence for changing the length of the comparison at each iteration of the first loop. It is interesting for the reason that the performance of the
algorithm depends on it.
A table of known variants and time complexity can be viewed here:
Different people were engaged in calculating the ideal distance, so, apparently, this topic was interesting to them. Couldn’t they just fire up Ruby, call the fastest sort() algorithm?
In general, these strange people wrote dissertations on the topic of calculating the distance / gap of the “comb” for the Shell algorithm. I just used the results of their work and checked 5 types of
sequences, Hibbard, Knuth-Pratt, Ciura, Sedgewick.
from typing import List
import time
import random
from functools import reduce
import math
DEMO_MODE = False
if input("Demo Mode Y/N? ").upper() == "Y":
DEMO_MODE = True
class Colors:
BLUE = '\033[94m'
RED = '\033[31m'
END = '\033[0m'
def swap(list, lhs, rhs):
list[lhs], list[rhs] = list[rhs], list[lhs]
return list
def colorPrintoutStep(numbers: List[int], lhs: int, rhs: int):
for index, number in enumerate(numbers):
if index == lhs:
print(f"{Colors.BLUE}", end = "")
elif index == rhs:
print(f"{Colors.RED}", end = "")
print(f"{number},", end = "")
if index == lhs or index == rhs:
print(f"{Colors.END}", end = "")
if index == lhs or index == rhs:
print(f"{Colors.END}", end = "")
def ShellSortLoop(numbers: List[int], distanceSequence: List[int]):
distanceSequenceIterator = reversed(distanceSequence)
while distance:= next(distanceSequenceIterator, None):
for sortArea in range(0, len(numbers)):
for rhs in reversed(range(distance, sortArea + 1)):
lhs = rhs - distance
if DEMO_MODE:
print(f"Distance: {distance}")
colorPrintoutStep(numbers, lhs, rhs)
if numbers[lhs] > numbers[rhs]:
swap(numbers, lhs, rhs)
def ShellSort(numbers: List[int]):
global ShellSequence
ShellSortLoop(numbers, ShellSequence)
def HibbardSort(numbers: List[int]):
global HibbardSequence
ShellSortLoop(numbers, HibbardSequence)
def ShellPlusKnuttPrattSort(numbers: List[int]):
global KnuttPrattSequence
ShellSortLoop(numbers, KnuttPrattSequence)
def ShellPlusCiuraSort(numbers: List[int]):
global CiuraSequence
ShellSortLoop(numbers, CiuraSequence)
def ShellPlusSedgewickSort(numbers: List[int]):
global SedgewickSequence
ShellSortLoop(numbers, SedgewickSequence)
def insertionSort(numbers: List[int]):
global insertionSortDistanceSequence
ShellSortLoop(numbers, insertionSortDistanceSequence)
def defaultSort(numbers: List[int]):
def measureExecution(inputNumbers: List[int], algorithmName: str, algorithm):
if DEMO_MODE:
print(f"{algorithmName} started")
numbers = inputNumbers.copy()
startTime = time.perf_counter()
endTime = time.perf_counter()
print(f"{algorithmName} performance: {endTime - startTime}")
def sortedNumbersAsString(inputNumbers: List[int], algorithm) -> str:
numbers = inputNumbers.copy()
return str(numbers)
if DEMO_MODE:
maximalNumber = 10
numbersCount = 10
maximalNumber = 10
numbersCount = random.randint(10000, 20000)
randomNumbers = [random.randrange(1, maximalNumber) for i in range(numbersCount)]
ShellSequenceGenerator = lambda n: reduce(lambda x, _: x + [int(x[-1]/2)], range(int(math.log(numbersCount, 2))), [int(numbersCount / 2)])
ShellSequence = ShellSequenceGenerator(randomNumbers)
HibbardSequence = [
0, 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095,
8191, 16383, 32767, 65535, 131071, 262143, 524287, 1048575,
2097151, 4194303, 8388607, 16777215, 33554431, 67108863, 134217727,
268435455, 536870911, 1073741823, 2147483647, 4294967295, 8589934591
KnuttPrattSequence = [
1, 4, 13, 40, 121, 364, 1093, 3280, 9841, 29524, 88573, 265720,
797161, 2391484, 7174453, 21523360, 64570081, 193710244, 581130733,
1743392200, 5230176601, 15690529804, 47071589413
CiuraSequence = [
1, 4, 10, 23, 57, 132, 301, 701, 1750, 4376,
10941, 27353, 68383, 170958, 427396, 1068491,
2671228, 6678071, 16695178, 41737946, 104344866,
260862166, 652155416, 1630388541
SedgewickSequence = [
1, 5, 19, 41, 109, 209, 505, 929, 2161, 3905,
8929, 16001, 36289, 64769, 146305, 260609, 587521,
1045505, 2354689, 4188161, 9427969, 16764929, 37730305,
67084289, 150958081, 268386305, 603906049, 1073643521,
2415771649, 4294770689, 9663381505, 17179475969
insertionSortDistanceSequence = [1]
algorithms = {
"Default Python Sort": defaultSort,
"Shell Sort": ShellSort,
"Shell + Hibbard" : HibbardSort,
"Shell + Prat, Knutt": ShellPlusKnuttPrattSort,
"Shell + Ciura Sort": ShellPlusCiuraSort,
"Shell + Sedgewick Sort": ShellPlusSedgewickSort,
"Insertion Sort": insertionSort
for name, algorithm in algorithms.items():
measureExecution(randomNumbers, name, algorithm)
reference = sortedNumbersAsString(randomNumbers, defaultSort)
for name, algorithm in algorithms.items():
if sortedNumbersAsString(randomNumbers, algorithm) != reference:
print("Sorting validation failed")
print("Sorting validation success")
In my implementation, for a random set, the fastest numbers are the Sedgwick and Hibbard gaps.
I would also like to mention the static typing analyzer for Python 3 – mypy. Helps to cope with the problems inherent in languages with dynamic typing, namely, it eliminates the possibility of
sticking something where it is not necessary.
As experienced programmers say, “static typing is not needed when you have a team of professionals”, someday we will all become professionals, we will write code in complete unity and understanding
with machines, but for now you can use similar utilities and languages with static typing.
Double Selection Sort
Double Selection Sort – a subspecies of sorting by selection, it seems like it should be accelerated twice. The vanilla algorithm double loops through the list of numbers, finds the minimum number,
and swaps it with the current digit pointed to by the loop at the level above. Double selection sort, on the other hand, looks for the minimum and maximum number, then it replaces the two digits
indicated by the loop at the level above – two numbers on the left and on the right. All this orgy ends when the cursors of numbers to replace meet in the middle of the list, as a result, sorted
numbers are obtained to the left and right of the visual center. The time complexity of the algorithm is similar to Selection Sort – O(n2), but supposedly there is a 30% speedup.
Corner case
Already at this stage, you can imagine the moment of collision, when the number of the left cursor (the minimum number) will point to the maximum number in the list, then the minimum number is
permuted, the permutation of the maximum number immediately breaks down. Therefore, all implementations of the algorithm contain a check of such cases, replacement of indices with correct ones. In my
implementation, one check was enough:
if (leftCursor == maximalNumberIndex) {
maximalNumberIndex = minimalNumberIndex;
Cito implementation
Cito is a lib language, a translator language. You can write on it for C, C++, C#, Java, JavaScript, Python, Swift, TypeScript, OpenCL C, while knowing absolutely nothing about these languages. The
source code in the Cito language is translated into the source code in the supported languages, then you can use it as a library, or directly by correcting the generated code by hand. A kind of Write
once – translate to anything.
Double Selection Sort – Cito:
public class DoubleSelectionSort
public static int[] sort(int[]# numbers, int length)
int[]# sortedNumbers = new int[length];
for (int i = 0; i < length; i++) {
sortedNumbers[i] = numbers[i];
for (int leftCursor = 0; leftCursor < length / 2; leftCursor++) {
int minimalNumberIndex = leftCursor;
int minimalNumber = sortedNumbers[leftCursor];
int rightCursor = length - (leftCursor + 1);
int maximalNumberIndex = rightCursor;
int maximalNumber = sortedNumbers[maximalNumberIndex];
for (int cursor = leftCursor; cursor <= rightCursor; cursor++) { int cursorNumber = sortedNumbers[cursor]; if (minimalNumber > cursorNumber) {
minimalNumber = cursorNumber;
minimalNumberIndex = cursor;
if (maximalNumber < cursorNumber) {
maximalNumber = cursorNumber;
maximalNumberIndex = cursor;
if (leftCursor == maximalNumberIndex) {
maximalNumberIndex = minimalNumberIndex;
int fromNumber = sortedNumbers[leftCursor];
int toNumber = sortedNumbers[minimalNumberIndex];
sortedNumbers[minimalNumberIndex] = fromNumber;
sortedNumbers[leftCursor] = toNumber;
fromNumber = sortedNumbers[rightCursor];
toNumber = sortedNumbers[maximalNumberIndex];
sortedNumbers[maximalNumberIndex] = fromNumber;
sortedNumbers[rightCursor] = toNumber;
return sortedNumbers; | {"url":"https://demensdeum.com/blog/tag/algorithms/","timestamp":"2024-11-13T11:51:32Z","content_type":"text/html","content_length":"255248","record_id":"<urn:uuid:03df9e2c-9d43-4a40-b438-791d3430e545>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00706.warc.gz"} |
Beatrice Riviere, Rice University - Applied Mathematics Colloquium - Department of Mathematics
Beatrice Riviere, Rice University – Applied Mathematics Colloquium
April 20, 2018 @ 4:00 pm - 5:00 pm
Title: Numerical methods for porous media flows at the pore scale and Darcy scale
Abstract: Modeling multicomponent flows in porous media is important for many applications relevant to energy and environment. In the first part of the talk, I will present a pore-scale flow model
based on the coupling of Cahn-Hilliard and Navier-Stokes equations. At the micro-meter scale, the rock structure is given and the fluid flows through the connected pores. In the second part of the
talk, I will discuss a Darcy-scale flow model, that assumes the existence of a representative elementary volume and that utilizes averaged quantities such as permeability and porosity. The numerical
discretization in space is the interior penalty discontinuous Galerkin methods of arbitrary order. Convergence of the algorithm is guaranteed by a theoretical analysis. Numerical studies of
wettability at the pore-scale and viscous fingering at the Darcy scale show the robustness and accuracy of the methods. | {"url":"https://math.unc.edu/event/beatrice-riviere-rice-university-applied-mathematics-colloquium/","timestamp":"2024-11-09T06:50:20Z","content_type":"text/html","content_length":"112481","record_id":"<urn:uuid:76cb604d-a65e-45c4-baa1-bdcbc99bc886>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00674.warc.gz"} |
VITEEE 2024 Syllabus - Global Academician
The VITEEE 2024 syllabus, meticulously crafted and unveiled by the esteemed authorities of the Vellore Institute of Technology (VIT), is now available for aspiring candidates. This comprehensive
guide outlines the subjects and specific topics integral to the upcoming entrance examination.
VITEEE Syllabus 2024 for Physics, Chemistry, Mathematics:
The key subjects encompassed in the VITEEE 2024 syllabus are Mathematics, Physics, Chemistry, Aptitude, and English, aligning with the standards of the 10+2 qualifying examination. Prospective
candidates are strongly advised to thoroughly review and comprehend the intricacies of the VITEEE 2024 syllabus, a crucial step to enhance their preparation process. Consulting the syllabus not only
facilitates a focused study approach but also enables candidates to identify essential topics and chapters that have been historically featured in the examination. In addition to the syllabus,
candidates are encouraged to acquaint themselves with the VITEEE 2024 exam pattern, ensuring a comprehensive readiness for the upcoming assessment.
VITEEE 2024 Syllabus – Physics:
The Physics syllabus for VITEEE 2024, meticulously curated by the Vellore Institute of Technology (VIT), forms an integral part of the entrance examination designed to assess the aptitude of
engineering aspirants. This comprehensive syllabus is thoughtfully structured to encompass key principles, theories, and applications in Physics. Candidates preparing for VITEEE are expected to delve
into various facets of the subject, including Mechanics, Electricity and Magnetism, Optics, and Modern Physics. The syllabus not only serves as a roadmap for exam preparation but also highlights the
foundational concepts that will be evaluated during the examination. As Physics plays a pivotal role in engineering disciplines, a robust understanding of the VITEEE Physics syllabus is essential for
aspiring engineers seeking admission to VIT’s prestigious B.Tech programs.
Unit wise Topics
Laws of Motion & Law of conservation of linear momentum and its applications. Static and kinetic friction – laws of friction -rolling friction – lubrication. Work done by a constant force and a
Work, Energy and variable force; kinetic energy – work-energy theorem – power.
Power Conservative forces: conservation of mechanical energy (kinetic and potential energies) – non-conservative forces: motion in a vertical circle – elastic and inelastic collisions in
one and two dimensions.
Properties of Elastic behaviour – Stress-strain relationship – Hooke’s law – Young’s modulus – bulk modulus – shear modulus of rigidity – Poisson’s ratio – elastic energy. Viscosity – Stokes’
Matter law – terminal velocity – streamline and turbulent flow – critical velocity. Bernoulli’s theorem and its applications. Heat – temperature – thermal expansion: thermal expansion of
solids – specific heat capacity: Cp, Cv – latent heat capacity. Qualitative ideas of Blackbody radiation: Wein’s displacement Law – Stefan’s law.
Charges and their conservation; Coulomb’s law forces between two point electric charges – Forces between multiple electric charges-superposition principle. Electric field –
electric field due to a point charge, electric field lines; electric dipole, electric field intensity due to a dipole – the behaviour of a dipole in a uniform electric field.
Electrostatics Electric potential – potential difference-electric potential due to a point charge and dipole-equipotential surfaces – the electrical potential energy of a system of two point
Electric flux-Gauss’s theorem and its applications. Electrostatic induction-capacitor and capacitance – dielectric and electric polarisation – parallel plate capacitor with and
without dielectric medium – applications of capacitor – energy stored in a capacitor – Capacitors in series and in parallel – action points – Van de Graaff generator.
Electric Current – the flow of charges in a metallic conductor – drift velocity and mobility and their relation with electric current. Ohm’s law, electrical resistance – V-I
characteristics – electrical resistivity and conductivity-classification of materials in terms of conductivity – Carbon resistors – colour code for carbon resistors – combination
Current of resistors – series and parallel – temperature dependence of resistance – internal resistance of a cell – potential difference and emf of a cell – combinations of cells in series
Electricity and in parallel.
Kirchoff’s law – Wheatstone’s Bridge and its application for temperature coefficient of resistance measurement – Metrebridge – a special case of Wheatstone bridge – Potentiometer
principle – comparing the emf of two cells.
Magnetic effect of electric current – Concept of magnetic field – Oersted’s experiment – Biot-Savart low magnetic field due to an infinitely long current carrying straight wire and
Magnetic Effects circular coil – Tangent galvanometer – construction and working – Bar magnet as an equivalent solenoid – magnetic field lines. Ampere’s circuital law and its application. Force on
of Electric a moving charge in a uniform magnetic field and electric field – cyclotron – Force on current carrying conductor in a uniform magnetic field – Forces between two parallel current
Current carrying conductors – definition of ampere. Torque experienced by a current loop in a uniform magnetic field – moving coil galvanometer – conversion to ammeter and voltmeter –
current loop as a magnetic dipole and its magnetic dipole moment – Magnetic dipole moment of a revolving electron.
Electromagnetic Electromagnetic induction – Faraday’s law – induced emf and current – Lenz’s law. Self-induction – Mutual induction – self-inductance of a long solenoid – mutual inductance of two
Induction and long solenoids. Methods of inducing emf – (i) by changing magnetic induction (ii) by changing the area enclosed by the coil and (iii) by changing the orientation of the coil
Alternating (quantitative treatment). AC generator – commercial generator. (Single phase, three phases). Eddy current – applications – transformer – long-distance transmission.
Current Alternating current – measurement of AC – AC circuit with resistance – AC circuit with inductor – AC circuit with capacitor – LCR series circuit – Resonance and Q – factor – power
in AC circuits.
Reflection of light, spherical mirrors, mirror formula. Refraction of light, total internal reflection and its applications, optical fibres, refraction at spherical surfaces,
lenses, thin lens formula, lens maker’s formula. Magnification, power of a lens, combination of thin lenses in contact, a combination of a lens and a mirror.
Refraction and dispersion of light through a prism. Scattering of the light-blue colour of the sky and the reddish appearance of the sun at sunrise and sunset. Wavefront and
Optics Huygens’s principle – Reflection, total internal reflection and refraction of plane wave at a plane surface using wavefronts. Interference – Young’s double slit experiment and
expression for fringe width – coherent source – interference of light – Formation of colours in thin films – Newton’s rings.
Diffraction – differences between interference and diffraction of light- diffraction grating. Polarisation of light waves – polarisation by reflection – Brewster’s law – double
refraction – nicol prism – uses of plane polarised light and Polaroids – rotatory polarisation – polarimeter.
Dual Nature of Electromagnetic waves and their characteristics – Electromagnetic spectrum – Photoelectric effect – Light waves and photons – Einstein’s photoelectric equation – laws of
Radiation and photoelectric emission – particle nature of light – photo cells and their applications.
Atomic Physics Atomic structure – discovery of the electron – specific charge (Thomson’s method) and charge of the electron (Millikan’s oil drop method) – alpha scattering – Rutherford’s atom
Nuclear properties – nuclear radii, masses, binding energy, density, charge – isotopes, isobars and isotones – nuclear mass defect – binding energy – stability of nuclei –
Bainbridge mass spectrometer.
Nuclear Physics Nature of nuclear forces – Neutron – discovery – properties – artificial transmutation – particle accelerator. Radioactivity – alpha, beta and gamma radiations and their properties
– Radioactive decay law – half-life – mean life – artificial radioactivity – radio isotopes – effects and uses – Geiger – Muller counter. Radiocarbon dating. Nuclear fission –
chain reaction – atom bomb – nuclear reactor – nuclear fusion – Hydrogen bomb – cosmic rays – elementary particles.
Semiconductor Semiconductor basics – energy band in solids: difference between metals, insulators and semiconductors – semiconductor doping – Intrinsic and Extrinsic semiconductors.
Devices and their Formation of P-N Junction – Barrier potential and depletion layer-P-N Junction diode – Forward and reverse bias characteristics – diode as a rectifier – Zener diode as a voltage
Applications regulator – LED. Junction transistors – characteristics – transistor as a switch – transistor as an amplifier – transistor as an oscillator.
Logic gates – NOT, OR, AND, EXOR using discrete components – NAND and NOR gates as universal gates – De Morgan’s theorem – Laws and theorems of Boolean algebra.
VITEEE 2024 Syllabus – Chemistry:
The Chemistry segment of the VITEEE 2024 syllabus, meticulously curated by the academic authorities of Vellore Institute of Technology (VIT), holds paramount significance in the evaluation of
aspiring engineers. Tailored to assess a diverse range of chemical concepts, the Chemistry syllabus encompasses three key branches: Inorganic Chemistry, Organic Chemistry, and Physical Chemistry.
Each branch delves into distinct principles, reactions, and applications, reflecting the comprehensive nature of the examination. Candidates preparing for VITEEE are not only tasked with mastering
the intricacies of chemical theories but also with applying this knowledge to solve practical problems. This detailed and well-structured syllabus serves as a guide for candidates aiming to excel in
the examination and secure admission to VIT’s renowned B.Tech programs, underscoring the integral role of Chemistry in the foundation of engineering disciplines.
Units wise Topics
Bohr’s atomic model-Sommerfeld’s extension of atomic structure; Electronic configuration and Quantum numbers; Shapes of s,p,d,f orbitals – Pauli’s exclusion principle –
Atomic Structure Hund’s Rule of maximum multiplicity- Aufbau principle. Emission spectrum, absorption spectrum, line spectra and band spectra; Hydrogen spectrum – Lyman, Balmer, Paschen,
Brackett and Pfund series; de Broglie’s theory; Heisenberg’s uncertainty principle – wave nature of electron – Schrodinger wave equation (No derivation). Eigen values and
eigen functions. Hybridization of atomic orbitals involving s,p,d orbitals.
p-block elements – Phosphorous compounds; PCl3, PCl5 – Oxides. Hydrogen halides, Inter halogen compounds. Xenon fluoride compounds. General Characteristics of d – block
p,d and f – Block elements – Electronic Configuration – Oxidation states of first-row transition elements and their colours. Occurrence and principles of extraction: Copper, Silver, Gold and
Elements Zinc. Preparation, properties of CuSO4, AgNO3 and K2Cr2O7.
Lanthanides – Introduction, electronic configuration, general characteristics, oxidation state – lanthanide contraction, uses, brief comparison of Lanthanides and Actinides.
Introduction – Terminology in coordination chemistry – IUPAC nomenclature of mononuclear coordination compounds. Isomerism, Geometrical isomerism in 4-coordinate,
Coordination Chemistry 6-coordinate complexes. Theories on coordination compounds – Werner’s theory (brief), Valence Bond theory. Uses of coordination compounds. Bioinorganic compounds (Haemoglobin
and Solid State and chlorophyll).
Chemistry Lattice – unit cell, systems, types of crystals, packing in solids; Ionic crystals – Imperfections in solids – point defects. X-Ray diffraction – Electrical Property,
Amorphous solids (elementary ideas only).
Thermodynamics, Chemical I and II law of thermodynamics – spontaneous and non-spontaneous processes, entropy, Gibb’s free energy – Free energy change and chemical equilibrium – the significance of
Equilibrium and Chemical entropy. Law of mass action – Le Chatlier’s principle, applications of chemical equilibrium. Rate expression, order and molecularity of reactions, zero order, first order and
Kinetics pseudo-first-order reaction – half-life period. Determination of rate constant and order of reaction.
Temperature dependence of rate constant – Arrhenius equation, activation energy.
Theory of electrical conductance; metallic and electrolytic conductance. Faraday’s laws – theory of strong electrolytes – Specific resistance, specific conductance,
Electrochemistry equivalent and molar conductance – Variation of conductance with dilution – Kohlrausch’s Law – Ionic product of water, pH and pOH – buffer solutions – use of pH values. Cells
– Electrodes and electrode potentials – construction of cell and EMF values, Fuel cells, Corrosion and its prevention.
Isomerism in Organic Definition, Classification – structural isomerism, stereo isomerism – geometrical and optical isomerism. Optical activity- chirality – compounds containing chiral centres – R
Compounds – S notation, D – L notation.
Nomenclature of alcohols – Classification of alcohols – the distinction between 1^0, 2^0 and 3^0 alcohols – General methods of preparation of primary alcohols, properties.
Alcohols and Ethers Methods of preparation of dihydric alcohols: Glycol – Properties – Uses. Methods of preparation of trihydric alcohols – Properties – Uses. Aromatic alcohols – preparation and
properties of phenols and benzyl alcohol.
Ethers – Nomenclature of ethers – general methods of preparation of aliphatic ethers – Properties – Uses. Aromatic ethers – Preparation of Anisole – Uses.
Nomenclature of carbonyl compounds – Comparison of aldehydes and ketones. General methods of preparation of aldehydes – Properties – Uses. Aromatic aldehydes – Preparation of
Carbonyl Compounds benzaldehyde – Properties and Uses. Ketones – general methods of preparation of aliphatic ketones (acetone) – Properties – Uses. Aromatic ketones – preparation of
acetophenone – Properties – Uses, preparation of benzophenone – Properties. Name reactions; Clemmenson reduction, Wolff – Kishner reduction, Cannizzaro reaction, Claisen
Schmidt reaction, Benzoin Condensation, aldol Condensation. Preparation and applications of Grignard reagents.
Nomenclature – Preparation of aliphatic monocarboxylic acids – formic acid – Properties – Uses. Monohydroxy mono carboxylic acids; Lactic acid – Synthesis of lactic acid.
Carboxylic Acids and Aliphatic dicarboxylic acids; Preparation of oxalic and succinic acid. Aromatic acids; Benzoic and Salicylic acid – Properties – Uses. Derivatives of carboxylic acids; acetyl
their derivatives chloride (CH3COCl) – Preparation – Properties – Uses. Preparation of acetamide, Properties – acetic anhydride – Preparation, Properties. Preparation of esters – methyl
acetate – Properties.
Aliphatic nitro compounds – Preparation of aliphatic nitroalkanes – Properties – Uses. Aromatic nitro compounds – Preparation – Properties – Uses. The distinction between
aliphatic and aromatic nitro compounds. Amines; aliphatic amines – General methods of preparation – Properties – Distinction between 1^0, 2^0 and 3^0 amines. Aromatic amines
– Synthesis of benzylamine – Properties, Aniline – Preparation – Properties – Uses. The distinction between aliphatic and aromatic amine. Aliphatic nitriles – Preparation –
properties – Uses. Diazonium salts – Preparation of benzene diazoniumchloride – Properties.
Organic Nitrogen · Carbohydrates – the distinction between sugars and non-sugars, structural formulae of glucose, fructose and sucrose, with their linkages, invert sugar – definition,
Compounds examples of oligo and polysaccharides,
· Amino acids – classification with examples, Peptides-properties of the peptide bond, Lipids – Definition, classification with examples, the difference between fats,
oils and waxes.
· Lipids – Definition, classification with examples, the difference between fats, oils and waxes.
VITEEE 2024 Syllabus – Mathematics:
The Mathematics syllabus for VITEEE 2024, meticulously crafted under the auspices of the Vellore Institute of Technology (VIT), stands as a cornerstone in the evaluation of aspiring engineers. This
comprehensive syllabus reflects the institute’s commitment to nurturing mathematical acumen, covering a spectrum of topics essential for success in the entrance examination. Mathematics in VITEEE
2024 comprises algebra, calculus, and trigonometry, offering a diverse set of challenges for candidates to surmount. Aspiring engineers are tasked not only with mastering the theoretical foundations
but also with developing problem-solving skills crucial for real-world applications. A robust understanding of the VITEEE Mathematics syllabus is imperative for candidates seeking admission to VIT’s
prestigious B.Tech programs, underlining the pivotal role mathematics plays in shaping the analytical thinking required for success in engineering disciplines.
Units wise Topics
Matrices and Adjoint, inverse – properties, computation of inverses, solution of a system of linear equations by matrix inversion method. The rank of a matrix – elementary transformation on a
their matrix, consistency of a system of linear equations, Cramer’s rule, non-homogeneous equations, homogeneous linear system and rank method. Solution of linear programming problems (LPP)
Applications in two variables.
Trigonometry Definition, range, domain, principal value branch, graphs of inverse trigonometric functions and their elementary properties. Complex number system – conjugate, properties, ordered
and Complex pair representation. Modulus – properties, geometrical representation, polar form, principal value, conjugate, sum, difference, product, quotient, vector interpretation, solutions of
Numbers polynomial equations, De Moivre’s theorem and its applications. Roots of a complex number – nth roots, cube roots, fourth roots.
Analytical Definition of a conic – general equation of a conic, classification with respect to the general equation of a conic, classification of conics with respect to eccentricity. Equations of
Geometry of conic sections (parabola, ellipse and hyperbola) in standard forms and general forms- Directrix, Focus and Latus-rectum – a parametric form of conics and chords. – Tangents and normals
two dimensions – Cartesian form and parametric form- equation of chord of contact of tangents from a point (x1,y1) to all the above said curves. Asymptotes, Rectangular hyperbola – Standard equation
of a rectangular hyperbola.
Scalar Product – the angle between two vectors, properties of scalar product, and applications of the dot product. Vector product, right-handed and left-handed systems, properties of
Vector Algebra vector product, applications of cross product. Product of three vectors – Scalar triple product, properties of scalar triple product, vector triple product, vector product of four
vectors, scalar product of four vectors.
Direction cosines – direction ratios – equation of a straight line passing through a given point and parallel to a given line, passing through two given points, the angle between two
Analytical lines. Planes – equation of a plane, passing through a given point and perpendicular to a line, given the distance from the origin and unit normal, passing through a given point and
Geometry of parallel to two given lines, passing through two given points and parallel to a given line, passing through three given non-collinear points, passing through the line of intersection
Three of two given planes, the distance between a point and a plane, the plane which contains two given lines (co-planar lines), angle between a line and a plane. Skew lines – the shortest
Dimensions distance between two lines, condition for two lines to intersect, point of intersection, collinearity of three points. 2 Sphere – equation of the sphere whose centre and radius are
given, equation of a sphere when the extremities of the diameter are given.
Limits, continuity and differentiability of functions – Derivative is a rate of change, velocity, acceleration, and related rates, and derivative as a measure of slope, tangent, normal
Differential and angle between curves. Mean value theorem – Rolle’s Theorem, Lagrange Mean Value Theorem, Taylor’s and Maclaurin’s series, L’ Hospital’s Rule, stationary points, increasing,
Calculus decreasing, maxima, minima, concavity, convexity and points of inflexion. Errors and approximations – absolute, relative, percentage errors – curve tracing, partial derivatives,
Euler’s theorem.
Calculus and Simple definite integrals – fundamental theorems of calculus, properties of definite integrals. Reduction formulae – reduction formulae for x dx n sin and x dx n cos, Bernoulli’s
its formula. Area of bounded regions, length of the curve.
Differential Differential equations – formation of differential equations, order and degree, solving differential equations (1st order), variables separable, homogeneous and linear equations.
Equations Second-order linear differential equations – second-order linear differential equations with constant coefficients, finding the particular integral if f(x) = emx, sin mx, cos mx, x,
Probability Probability – Axioms – Addition law – Conditional probability – Multiplicative law – Baye’s Theorem – Random variable – probability density function, distribution function,
Distributions mathematical expectation, variance Theoretical distributions – discrete distributions, Binomial, Poisson distributions- Continuous distributions, Normal distribution.
Discrete Functions – Relations – Basics of counting. Mathematical logic – logical statements, connectives, truth tables, logical equivalence, tautology, a contradiction. Groups-binary
Mathematics operations, semi-groups, monoids, groups, order of a group, order of an element, properties of groups
VITEEE 2024 Syllabus – Biology:
The Biology segment of the VITEEE 2024 syllabus, meticulously outlined by the academic authorities at Vellore Institute of Technology (VIT), serves as a vital dimension in the evaluation of
prospective engineers. This specialized syllabus, unique among the core subjects, encapsulates Botany and Zoology, providing a comprehensive overview of biological principles. Candidates preparing
for VITEEE are immersed in the intricate world of living organisms, exploring topics ranging from plant physiology and genetics to animal diversity and human anatomy. Beyond the confines of
traditional science, the Biology syllabus emphasizes the interconnectedness of life, fostering an understanding of ecological systems and environmental issues. A thorough grasp of the VITEEE Biology
syllabus is essential for aspirants vying for a position in VIT’s prestigious B.Tech programs, underscoring the significance of biological sciences in the multidisciplinary landscape of engineering.
Unit Wise Topics
Taxonomy Need for classification; three domains of life. Linnaean, Whittaker, Bentham and Hooker systems of classification. Salient features of non-chordates up to phyla levels and chordates
up to class levels.
Cell and Cell theory. Prokaryotic cell and it’s ultrastructure. Eukaryotic cell- cell wall, cell membrane, cytoskeleton, nucleus, chloroplast, mitochondria, endoplasmic reticulum, Golgi
Molecular Biology bodies, ribosomes, lysosomes, vacuoles and centrosomes. Cell cycle and division – amitosis, mitosis and meiosis. Search for genetic material; structure of DNA and RNA; replication,
transcription, genetic code, translation, splicing, gene expression and regulation (lac operon) and DNA repair.
Asexual reproduction – binary fission, sporulation, budding, gemmule formation and fragmentation. Vegetative propagation in plants, sexual reproduction in flowering plants and
Reproduction structure of flowers. Pollination, fertilization, development of seeds and fruits, seed dispersal, apomixis, parthenocarpy and polyembryony. Human reproductive system.
Gametogenesis, menstrual cycle, fertilization, implantation, embryo development up to blastocyst formation, pregnancy, parturition and lactation. Assisted reproductive technologies.
Genetics and Chromosomes – structure and types, linkage and crossing over, recombination of chromosomes, mutation and chromosomal aberrations. Mendelian inheritance, chromosomal theory of
evolution inheritance, deviation from Mendelian ratio (incomplete dominance, co-dominance, multiple allelism, pleiotrophy), sex-linked inheritance and sex determination in humans. Darwinism,
neo-Darwinism, Hardy and Weinberg’s principle and factors affecting the equilibrium: selection, mutation, migration and random genetic drift.
Human health and Pathogens, parasites causing human diseases (malaria, dengue, chikungunya, filariasis, ascariasis, typhoid, pneumonia, common cold, amoebiasis, ringworm) and their control. Basic
diseases concepts of immunology, vaccines, antibiotics, cancer, HIV and AIDS. Adolescence, drug and alcohol abuse.
Biochemistry Structure and function of carbohydrates, lipids and proteins. Enzymes – types, properties and enzyme action. Metabolism – glycolysis, Kreb’s cycle and pentose phosphate pathway.
Movement of water, food, nutrients, gases and minerals. Passive diffusion, facilitated diffusion, and active transport. Imbibition, osmosis, apoplast and symplast transport and
Plant physiology guttation. Transpiration, photosynthesis (light and dark reactions) and electron transport chain. 2 Hormones and growth regulators, photoperiodism and vernalization. Nitrogen cycle
and biological nitrogen fixation.
Human physiology Digestion and absorption, breathing and respiration, body fluids and circulation, excretory system, endocrine system, nervous system, skeletal and muscular systems. Locomotion and
movement, growth, ageing and death. Hormones – types of hormones, functions and disorders.
Biotechnology and Recombinant DNA technology, applications in health, agriculture and industries; genetically modified organisms; Human insulin, vaccine and antibiotic production. Stem cell
its applications technology and gene therapy. Apiculture and animal husbandry. Plant breeding, tissue culture, single cell protein, fortification, Bt crops and transgenic animals. Microbes in food
processing, sewage treatment, waste management, and energy generation. Biocontrol agents and biofertilizers. Biosafety issues, biopiracy, and patents.
Biodiversity, Ecosystems: components, types, pyramids, nutrient cycles (carbon and phosphorous), ecological succession, and energy flow in an ecosystem; Biodiversity – concepts, patterns,
ecology and importance, conservation, hot spots, endangered organisms, extinction, Red data book, botanical gardens, national parks, sanctuaries, museums, biosphere reserves, and Ramsar sites.
environment Environmental issues: pollution and its control. Population attributes – growth, birth and death rate, and age distribution.
VITEEE 2024 English Syllabus:
The English segment of the VITEEE 2024 examination encompasses a series of Multiple Choice Questions designed to evaluate candidates’ comprehension skills. Questions may include passages, lines from
poems, as well as assessments of English grammar and pronunciation. It’s noteworthy that the content has been specifically tailored to align with the difficulty level of higher secondary or
equivalent education.
VITEEE 2024 Aptitude Syllabus:
The Aptitude section of the VITEEE 2024 examination covers a diverse range of topics, including Data Interpretation, Data Sufficiency, Syllogism, Number Series, Coding and Decoding, as well as
Clocks, Calendars, and Directions. This section aims to assess candidates’ analytical and problem-solving abilities, offering a comprehensive evaluation of their aptitude in various domains.
• Data Interpretation
• Data Sufficiency
• Syllogism
• Number Series, Coding and Decoding
• Clocks, Calendars, and Directions
VITEEE 2024 Important Topics:
As aspirants gear up for the VITEEE 2024, navigating through the vast array of topics becomes pivotal for strategic and effective preparation. The examination, conducted by the Vellore Institute of
Technology (VIT), serves as a gateway for aspiring engineers to secure admission to prestigious B.Tech programs. Understanding the weightage and significance of each subject and topic is essential
for a targeted study approach. In this exploration of VITEEE 2024, we delve into the important topics that demand special attention. These focal points not only form the core of the examination but
also serve as indicators of the key competencies expected from candidates. Let’s embark on a journey through the crucial themes and concepts that will shape success in the upcoming VITEEE 2024
Physics Chemistry Maths Biology English Aptitude
Mechanical Properties p, d & f-Block elements Quadratic Equation Cell and Molecular Biology Comprehension Data Interpretation
Oscillation Thermodynamics & Thermochemistry Continuity & Differentiability Genetics and evolution English Grammer Data Sufficiency
Current Electricity Equilibrium Permutations & Combinations Human physiology Pronunciation Syllogism
Thermodynamics Surface Chemistry Functions & Limits Reproduction – Number Series, Coding and Decoding
Wave and Ray Optics Biomolecules Coordinate Geometry Biodiversity, Ecology, and Environment – Clocks, Calendars, and Directions
Recommended Books for VITEEE 2024:
VITEEE 2024 Physics Book:
Name of the book Author
Concept of Physics Part-1 & Part-2 H.C. Verma
Problems in General Physics I.E. Irodov
Understanding Physics Series D.C. Pandey
VITEEE 2024 Chemistry Book:
Name of the book Author
Handbook of Chemistry R.P. Singh
Textbook for Class XI & XII NCERT
Organic Chemistry O. P. Tandon & Morrison Boyd.
Modern Approach to Chemical Calculations R. C. Mukherjee
VITEEE 2024 Mathematics Book:
Name of the book Author
Higher Algebra Hall and Knight
Degree level Differential Calculus A Das Gupta
Target VITEEE Disha Experts
Objective Mathematics Part 1 and Part 2 R.D. Sharma
Problems in Calculus of One Variable I.A. Maron
VITEEE 2024 Biology Book:
Name of the Book Author
S. Chand’s Biology for Class XI P.S. Verma and B.P. Pandey
Pradeep’s Biology Guide P.S. Dhami
As we bring our in-depth exploration of the VITEEE 2024 to a close, we trust that this comprehensive guide, tailored for aspiring engineers, will prove to be an invaluable resource on your academic
journey. From unraveling the intricacies of the syllabus and highlighting pivotal topics to recommending essential study materials, our aim has been to provide a holistic approach to VITEEE
preparation. This resource is crafted with the intent to empower candidates as they embark on the path to success in the VITEEE 2024 examination. For further academic insights, updates, and a
supportive community, we invite you to explore more on the Global Academician website. May this guide serve as a beacon of guidance and support as you strive for excellence in the upcoming VITEEE
examination. Best of luck on this academic adventure!
To stay ahead and stay informed about the latest educational updates, trends, and insights, we invite you to subscribe to our newsletter and regularly explore our blog. You can also connect with us
on our Facebook Page to join our educational community at Global Academician. Join us on these platforms and embark on a journey of continuous learning and knowledge sharing.
Click here to Subscribe | {"url":"https://www.globalacademician.com/viteee-exam-details-admission-application-counselling/viteee-syllabus/","timestamp":"2024-11-12T19:23:34Z","content_type":"text/html","content_length":"158280","record_id":"<urn:uuid:3c824b5a-f2d8-41f3-bcbf-e0a367caf330>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00829.warc.gz"} |
Daily streamflow prediction based on the long short-term memory algorithm: a case study in the Vietnamese Mekong Delta
Study area and data
Long short-term memory algorithm
Support vector machine
Random forest
Performance assessment
Basic steps of modeling by LSTM
Collection and preparation of data
Building of model
Model validation
Modeling parameter optimization in daily streamflow prediction
Evaluation of the number of previous days
Evaluation of the 1 and 7 days ahead | {"url":"https://iwaponline.com/jwcc/article/14/4/1247/94438/Daily-streamflow-prediction-based-on-the-long","timestamp":"2024-11-04T14:01:36Z","content_type":"text/html","content_length":"473879","record_id":"<urn:uuid:af94b0ba-f269-4853-9661-a357666f08e9>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00701.warc.gz"} |
Explicit Expressions for the First 20 Moments of the Area Under Dyck and Motzkin Paths
By AJ Bu, Shalosh B. Ekhad, and Doron Zeilberger
.pdf .tex
Written: May 5, 2024
We show the utility of AJ Bu's recent article for computing explicit expressions for the GENERATING functions of sums of powers of areas under Dyck and Motzkin paths, by deducing from them explicit
expressions for the actual sequences. This enables taking the limits of the scaled moments and confirming, in an entirely elementary way, that they tend to those of the area under Brownian Excursion.
Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilerger, AJ Bu's web-site, and arxiv.org .
Maple package
• qEWplus.txt, for deriving (rigorously) explicit expressions for the moments of Dyck and Motzkin paths
Sample Input and Output for qEWplus.txt
If you want to see explicit expressions for the sum of the r-th powers of areas under Dyck walks from 0 to 2n for r from 1 to 20
the input gives the output
If you want to see explicit expressions for the sum of the r-th powers of areas under Motzkin walks from 0 to n for r from 1 to 20
the input gives the output | {"url":"https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/qew.html","timestamp":"2024-11-04T03:44:49Z","content_type":"text/html","content_length":"2750","record_id":"<urn:uuid:3434e704-ef95-4a3d-93c5-1cebcb605155>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00459.warc.gz"} |
Codeforces Round #733 ABCDE Solutions (Java/C++)
A. Binary Decimal
Obviously, we only need to check every digit. For each digit, as many as its digit, it needs to be composed of as many numbers. For example, if the tens place of 321 is 2, it needs to be composed of
two numbers.
Submission #124032062 - Codeforces
Codeforces. Programming competitions and contests, programming community
Submission #124032140 - Codeforces
Codeforces. Programming competitions and contests, programming community
B. Putting Plates
Just simulation. We scan each point from left to right and from top to bottom. As long as the point is on the boundary and the surrounding area of the point is not occupied, we will occupy the point.
Submission #124034365 - Codeforces
Codeforces. Programming competitions and contests, programming community
Submission #124034666 - Codeforces
Codeforces. Programming competitions and contests, programming community
C. Pursuit
Just simulation. Because every time a stage is added, the player gets 100 points and Ilya gets 0 points. Therefore, we only need to maintain the scores after adding a stage.
For players, we use a priority queue. Because for each additional stage, the stage with the lowest score in the priority queue may be removed. Therefore, we only need to maintain the length of the
priority queue based on the total number of stages.
For Ilya, we sort the games that Ilya is already completed. As the number of stages increases, Ilay will gradually lose points from the stages with lower scores.
Keep increasing the number of stages until the player score is higher than Ilya.
Submission #124038628 - Codeforces
Codeforces. Programming competitions and contests, programming community
Submission #124039426 - Codeforces
Codeforces. Programming competitions and contests, programming community
D. Secret Santa
First of all, as many different numbers appear in b, there will be as many people who can be fulfilled. So the question now is how to handle the rest of the people.
Then there is only one possibility for the remaining person to have a problem: b[i]=i. And the person who will receive gift from i must be sent gift by someone else.
Suppose there are two people i and j, at the beginning b[i] and b[j]=x, and finally b[i]=i, b[j]=x. So we only need to exchange the gift-giving goals of two people, that is, b[i]=x, b[j]=i.
Therefore, we first ignore the condition that we cannot give ourselves gifts, and we can allocate them in order under the condition of ensuring that everyone's wishes are fulfilled as much as
After allocating, check the person who giving gifts to themselves. For such a person, we can find the person who gives gifts to the same target and exchange with him.
Submission #124056951 - Codeforces
Codeforces. Programming competitions and contests, programming community
Submission #124057057 - Codeforces
Codeforces. Programming competitions and contests, programming community
E. Minimax
Obviously, if a letter appears only once, then f(t)=0, we only need to put this letter at the beginning. For example, aabbz, then zaabb.
It is not difficult to find, unless a certain letter appears only once. Otherwise, f(t)>0. For example, x???x?, f(t)=1 when k=5.
Then, if the number of occurrences of a certain letter is less than around half of the total, the following string can be constructed: xxyxyxyxyxy... so f(t)=1.
Otherwise, if there are two letters, we construct xyyyyyxxxxx... so that f(t)=1.
Otherwise we construct xyxxxxxxxxzabcd. Such f(t)=1.
Just tons of if else.
Submission #124247671 - Codeforces
Codeforces. Programming competitions and contests, programming community
Submission #124251006 - Codeforces
Codeforces. Programming competitions and contests, programming community | {"url":"https://www.xloypaypa.pub/codeforces-round-733-abcde-solutions-java-c/","timestamp":"2024-11-09T04:07:49Z","content_type":"text/html","content_length":"28823","record_id":"<urn:uuid:cd04956e-f5ee-4426-8a8b-4d828a9bb83d>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00173.warc.gz"} |
MULTIPLY Function in Google Sheets: Explained | LiveFlow
MULTIPLY Function in Google Sheets: Explained
In this article, you will learn how to use the MULTIPLY formula in Google Sheets.
What is the MULTIPLY formula in Google Sheets?
In Google Sheets, you can use the MULTIPLY formula to multiply two factors. Note that the MULTIPLY formula takes only two arguments and doesn’t accept a range or more than two factors as its
How to use the MULTIPLY function in Google Sheets
The general syntax is as follows:
factor1: This is a value to be multiplied by
factor2: This is the second multiplicand.
For example, to multiply the numbers 2 and 3 together, you would enter
into a cell in Google Sheets. This would return the result 6.
You can also use the =MULTIPLY formula to multiply the contents of cells. For example, if you have the number 2 in cell B4 and the number 3 in cell C4, you can enter the formula
to multiply the two numbers together. This would return the result 6.
How to use the MULTIPLY function in Google Sheets with examples
How do I multiply multiple cells by the same number in Google Sheets?
For example, you can multiply multiple cells by the same number using the following sample formulas. In the first example, you need to create a table containing the first and second factors for each
calculation, and you input the same number in all cells in a column for the second factor.
Once you insert the MULTIPLY formula for the first row, you can copy and paste the formula for the other rows in the table. In the second example, you enter the second factor in a cell and use the
absolute reference in the MULTIPLY function.
Again, once you insert the MULTIPLY formula for the first row, you can copy and paste it to the other rows in the table.
How to multiply multiple cells by the same number in Google Sheets
How do you multiply all cells at once?
You can use the PRODUCT formula to multiply all cells at once instead of the MULTIPLY formula because, as mentioned above, the MULTIPLY function can’t take an array or range as its argument.
PRODUCT Function in Google Sheets: Explained
What is the alternative to the MULTIPLY function in Google Sheets?
You can use the asterisk “*” as an operator for multiplication in Google Sheets. For instance, if you want to multiply 2 by 3, you can insert the following formula in a cell:
Cell reference can be used in multiplication. For example, if you want to multiply 2 in cell B1 by 3 in cell C1, the formula would be as follows: | {"url":"https://www.liveflow.io/product-guides/multiply-function-in-google-sheets","timestamp":"2024-11-04T21:31:16Z","content_type":"text/html","content_length":"48556","record_id":"<urn:uuid:931d63f4-bef1-4708-91d2-a099f0a26e45>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00666.warc.gz"} |
min_vertex_color(G, sampler=None, chromatic_lb=None, chromatic_ub=None, **sampler_args)[source]#
Returns an approximate minimum vertex coloring.
Vertex coloring is the problem of assigning a color to the vertices of a graph in a way that no adjacent vertices have the same color. A minimum vertex coloring is the problem of solving the
vertex coloring problem using the smallest number of colors.
Defines a QUBO [DWMP] with ground states corresponding to minimum vertex colorings and uses the sampler to sample from it.
☆ G (NetworkX graph) – The graph on which to find a minimum vertex coloring.
☆ sampler – A binary quadratic model sampler. A sampler is a process that samples from low energy states in models defined by an Ising equation or a Quadratic Unconstrained Binary
Optimization Problem (QUBO). A sampler is expected to have a ‘sample_qubo’ and ‘sample_ising’ method. A sampler is expected to return an iterable of samples, in order of increasing
energy. If no sampler is provided, one must be provided using the set_default_sampler function.
☆ chromatic_lb (int, optional) – A lower bound on the chromatic number. If one is not provided, a bound is calulcated.
☆ chromatic_ub (int, optional) – An upper bound on the chromatic number. If one is not provided, a bound is calculated.
☆ sampler_args – Additional keyword parameters are passed to the sampler.
coloring – A coloring for each vertex in G such that no adjacent nodes share the same color. A dict of the form {node: color, …}
Return type:
Samplers by their nature may not return the optimal solution. This function does not attempt to confirm the quality of the returned sample. | {"url":"https://docs.ocean.dwavesys.com/en/latest/docs_dnx/reference/algorithms/generated/dwave_networkx.algorithms.coloring.min_vertex_color.html","timestamp":"2024-11-03T01:03:42Z","content_type":"text/html","content_length":"30806","record_id":"<urn:uuid:6fdc52ac-d515-4be8-a7df-f8c2eba3e0c2>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00222.warc.gz"} |
Obtain plane equation from eulerAngles
Hi everyone. I have a plane in the scene that the player can rotate. Based on the plane’s euler angles, is there an easy way for me to calculate the coefficients a, b, c, and d of the plane’s
equation ax+by+cz+d=0?
It’s calculated from a point and a vector like this (presuming it’s a Unity transform of a Unity Plane);
a = transform.up.x;
b = transform.up.y;
c = transform.up.z;
d = -Vector3.Dot(transform.up, transform.position); | {"url":"https://discussions.unity.com/t/obtain-plane-equation-from-eulerangles/95823","timestamp":"2024-11-13T15:20:30Z","content_type":"text/html","content_length":"27084","record_id":"<urn:uuid:c5313648-4df1-4c7d-98bb-1b3dc0b7e173>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00433.warc.gz"} |
A misleading title… | R-bloggersA misleading title…
[This article was first published on
Xi'an's Og » R
, and kindly contributed to
]. (You can report issue about the content on this page
) Want to share your content on R-bloggers?
if you have a blog, or
if you don't.
When I received this book, Handbook of fitting statistical distributions with R, by Z. Karian and E.J. Dudewicz, from/for the Short Book Reviews section of the International Statistical Review, I
was obviously impressed by its size (around 1700 pages and 3 kilos…). From briefly glancing at the table of contents, and the list of standard distributions appearing as subsections of the first
chapters, I thought that the authors were covering different estimation/fitting techniques for most of the standard distributions. After taking a closer look at the book, I think the cover is
misleading in several aspects: this is not a handbook (a.k.a. a reference book), it does not cover standard statistical distributions, the R input is marginal, and the authors only wrote part of the
book, since about half of the chapters are written by other authors…
“The system we develop in this book has its origins in the one-parameter lambda distribution proposed by John Tukey.” Z.A. Karian & E.J. Dudewicz, p.3, Handbook of fitting statistical
distributions with R
So I am glad I left Handbook of fitting statistical distributions with R in my office rather than dragging it along across the Caribbean! First, the book indeed does not aim at fitting standard
distributions but instead at promoting a class of quantile distributions, the generalised lambda distributions (GLDs), whose quantile function is a location-scale transform of
(under the constraint on the parameters that the above function of y is non-decreasing) and that the authors have been advocating for a long while. There is nothing wrong per se with those quantile
distributions, but neither is there a particular reason to prefer them over the standard parametric distributions! Overall, I am quite wary of one-fits-all distributions, especially when they only
depend on four parameters and mix finite with infinite support distributions. The lack of natural motivations for the above is enough to make fitting with those distributions not particularly
compelling. Karian and Dudewicz spend an awful lot of space on numerical experiments backing their argument that the generalised lambda distributions approximate reasonably well (in the L[1] and L[2]
norms, as it does not work for stricter norms) “all standard” distributions, but it does not explain why the substitution would be of such capital interest. Furthermore, the estimation of the
parameters (i.e. the fitting in fitting statistical distributions) is not straightforward. While the book presents the density of the generalised lambda distributions as available in closed form
(Theorem 1.2.2), namely (omitting the location-scale parameters),
it fails to point out that the cdf
itself is not available in closed form. Therefore, neither likelihood estimation nor Bayesian inference seem easily implementable for those distributions. (Actually, a mention is made of maximum
likelihood estimators for the first four empirical moments in the second chapter, but it is alas mistaken.) [Obviously, given that quantile distributions are easy to simulate, ABC would be a
manageable tool for handling Bayesian inference on GLDs…] The book focus instead on moment and percentile estimators as the central estimation tool, with no clear message on which side to prefer
(see, e.g., Section 5.5).
A chapter (by Su) covers the case of mixtures of GLDs, whose appeal is similarly lost on me. My major issue with using such distributions in mixture setting is that some components may have a finite
support, which makes the use of score equations awkward and of Kullback-Leibler divergences to normal mixtures fraught with danger (since those divergence may then be infinite). The estimation method
switches to maximum likelihood estimation, as presumably the moment method gets too ungainly. However, I fail to see how maximum likelihood is implemented: I checked the original paper by Su (2007),
documenting the related GLDEX R function, but the approach is very approximate in that the true percentiles are replaced with pluggin (and fixed, i.e. non-iterative) values (again omitting the
location-scale parameters)
$\hat u_i=F(x_i|\hat\lambda_3,\hat\lambda_4)\qquad i=1,...,n$
in the likelihood function
$\prod_{i=1}^n dfrac{1}{\lambda_3\hat u_i^{\lambda_3-1}+\lambda_4\{1-\hat u_i\}^{\lambda_4-1}}$
A further chapter is dedicated to the generalised beta distribution, which simply is a location-scale transform of the regular beta distribution (even though it is called the extended GLD for no
discernible reason). Again, I have nothing for or against this family (except maybe that using a bounded support distribution to approximate infinite support distributions could induce potential
drawbacks…) I simply cannot see the point in multiplying parametric families of distributions where there is no compelling property to do so. (Which is also why as an editor/aeditor/referee, I have
always been ultra-conservative vis-à-vis papers introducing new families of distributions.)
The R side of the book (i.e. the R in fitting statistical distributions with R) is not particularly appealing either: in the first chapters, i.e. in the first hundred pages, the only reference to R
is the name of the R functions found on the attached CD-ROM to fit GLDs by the method of moments or of percentiles… The first detailed code is found on pages 305-309, but it is unfortunately a
MATLAB code! (Same thing in several subsequent chapters.) Even though there is an R component to the book thanks to this CD-ROM, the authors could well be suspected of “surfing the R wave” of the
Use R! and other “with R”collections. Indeed, my overall feeling is that they are mostly recycling their 2000 book Fitting statistical distributions into this R edition. (For instance, figures that
are reproduced from the earlier book, incl. the cover, are not even produced with R. Most entries of the table of contents of Fitting statistical distributions are found in the table of contents of
Handbook of fitting statistical distributions with R. The codes were then written in Maple and some Maple codes actually survive in the current version. Most of the novelty in this version is due to
the inclusion of chapters written by additional authors.)
“It remains for a future research topic as to how to improve the generalized bootstrap to achieve a 95% confidence interval since 40% on average and 25%-55% still leaves room for improvement.” W.
Cai & E.J. Dudewicz, p.852, Handbook of fitting statistical distributions with R
As in the 2000 edition, the “generalised bootstrap” method is argued as an improvement over the regular bootstrap, “fraught with danger of seriously inadequate results” (p.816), and as a mean to
provide confidence assessments. This method, attributed to the authors in 1991, is actually a parametric bootstrap used in the context of the GLDs, where samples are generated from the fitted
distribution and estimates of the variability of estimators of interest are obtained by a sheer Monte Carlo evaluation! (A repeated criticism of the bootstrap is its “inability to draw samples
outside the range of the original dataset” (e.g., p.852). It is somehow ironical that the authors propose to use instead parameterised distributions whose support may be bounded.)
Among the negative features of the book, I want to mention the price ($150!!!), the glaring [for statisticians!] absence of confidence statements about the (moment and percentile) estimations (not
to be confused with goodness-of-fit)—except for the much later chapter on generalised bootstrap—, the fact that the book contains more than 250 pages of tables—yes, printed tables!—including a page
with a few hundred random numbers generated from a given distribution, the fact that the additional authors who wrote the contributed chapters are not mentioned elsewhere that in the front page of
those chapters—not even in the table of contents!—, [once more] the misleading use of the term handbook in the title, the way Wiktionary defines it
handbook (plural handbooks)
1. A topically organized book of reference on a certain field of knowledge, disregarding the size of it.
as it is not a “reference book”, nor a “topically organised book”: a newcomer opening Handbook of fitting statistical distributions with R cannot expect to find the section that would address her or
his fitting problem, but has to read through the (first part) book in a linear way… So there is no redeeming angle there that could lead me to recommend Handbook of fitting statistical distributions
with R as fitting any purpose. Save the trees!
Filed under:
University life
fitting statistical distributions
generalized lambda distribution
International Statistical Review
John Tukey
maximum likelihood estimation
quantile distribution | {"url":"https://www.r-bloggers.com/2011/09/a-misleading-title%E2%80%A6/","timestamp":"2024-11-04T23:09:15Z","content_type":"text/html","content_length":"126076","record_id":"<urn:uuid:aa83bfed-2556-454f-8522-1d4bec593537>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00484.warc.gz"} |
How do I create a countifs formula if I have a column with multi drop down?
How do I create a countifs formula if I have a column with multi drop down and I need to specify which word to include for my count count? I would like to use the HAS function but it is not working
as I am also referencing the data from another sheet so I can not use @cell with the HAS.
Best Answer
• Ok, shouldn't be a problem. With CountIFS you will need to make sure that you define your second range and the criteria for it to match. I'll use my same formula, but add a column called type and
look for that cell to contain the letters "DNA". Rather than returning a 2 it returns a 1 as only one of the cells has both a 2 and the letters "DNA".
=COUNTIFS(multi:multi, CONTAINS("2", @cell), type:type, "DNA")
If this still isn't working for you, can you provide the formula you're using?
• I think you can use CONTAINS to do that.
• It is coming back with Zero. Would you be able to show how I would write the formula?
• Sure thing. In this formula I have a column called "multi" that I have a multiple drop-down selector with 1,2,3,4,5 available to pick. I populated the first cell with 1,2,3 and the second with 2,
and the rest with other numbers that aren't 2. I'm looking for 2 and I expect it to return 2 instances since it is in the first mutli-select and the second.
=COUNTIF(multi:multi, CONTAINS("2", @cell))
So this looks at the whole range of the column (multi:multi) and checks each cell (@cell) on whether or not it contains a 2.
Would that work for what you're looking for?
• I would like to use the COUNTIFS as I have multiple criteria. The formula works for 1 criteria but when I add another criteria it returns a Zero. Thank you!
• Ok, shouldn't be a problem. With CountIFS you will need to make sure that you define your second range and the criteria for it to match. I'll use my same formula, but add a column called type and
look for that cell to contain the letters "DNA". Rather than returning a 2 it returns a 1 as only one of the cells has both a 2 and the letters "DNA".
=COUNTIFS(multi:multi, CONTAINS("2", @cell), type:type, "DNA")
If this still isn't working for you, can you provide the formula you're using?
• Awesome! Glad you got it going.
• Hi David, I have spent an age on this exact same problem and your answer has solved it for me! Thank you, you are amazing!
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/71458/how-do-i-create-a-countifs-formula-if-i-have-a-column-with-multi-drop-down","timestamp":"2024-11-10T08:50:01Z","content_type":"text/html","content_length":"420393","record_id":"<urn:uuid:43b690a2-0436-4214-bb24-b54fb47d4dd8>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00506.warc.gz"} |
Path closeness
Next: Solution tolerances Up: Polynomial Continuation Previous: Adaptive step-size control
The corrector stops when the desired accuracy is reached or when it has exhausted its maximum number of iterations. A low maximum enforces quadratic convergence and keeps the path tracker close
to the solution paths. A higher maximum may be needed at the end of the solution path, when quadratic convergence can no longer be obtained due to singularities.
The residual is the norm of the vector obtained after evaluating the current approximation vector in the polynomial system. The corrector stops when the residual divided by the norm of the
approximation vector is lower than or equal to the required precision or when another required precision is attained.
The residual is the norm of the vector obtained after evaluating the current approximation vector in the polynomial system. The corrector stops when the residual is lower than or equal to the
required precision or when another required precision is attained.
relative precision for corrections:
The correction is the norm of the last vector used to update the current approximation vector. The corrector stops when the correction divided by the norm of the approximation vector is lower
than or equal to the required precision or when another required precision is attained.
absolute precision for corrections:
The correction is the norm of the last vector used to update the current approximation vector. The corrector stops when the correction is lower than or equal to the required precision or when
another required precision is attained.
Next: Solution tolerances Up: Polynomial Continuation Previous: Adaptive step-size control Jan Verschelde | {"url":"http://homepages.math.uic.edu/~jan/PHCpack/node24.html","timestamp":"2024-11-04T20:04:45Z","content_type":"text/html","content_length":"5318","record_id":"<urn:uuid:6b4d4c9f-f0bb-4d94-9ce2-06630444dbec>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00056.warc.gz"} |
coupon payment calculator excel
The template calculates the monthly payment, in cell E2. To calculate a loan payment amount, given an interest rate, the loan term, and the loan amount, you can use the PMT function. Annual Interest
Payment = 10 * 2 2. › Url: https://www.calculator.net/payment-calculator.html Go Deal Now, 30% off Offer Details: Summary To calculate a loan payment amount, given an interest rate, the loan term,
and the loan amount, you can use the PMT function. It is the product of the par value of the bond and coupon rate. A. After you've calculated the total annual coupon payment, divide this amount by
the par value of the security and then multiply by 100 to convert this total to a percent. Let us take the example of a bond with quarterly coupon payments. For example, ValuePenguin, a company that
helps people choose the best credit […] excel monthly payment schedule, › Url: https://www.exceltemplates.com/budget/credit-card-debt-payoff-spreadsheet/ Go Deal Now, › Get more: Excel monthly
payment scheduleShow All Coupons, 40% off Offer Details: Just enter the loan amount, interest rate, loan duration, and start date into the Excel loan calculator, and it will calculate each monthly
principal and interest cost through the final payment. Press the following buttons: The formula for calculating the coupon rate is as follows: Where: C = Coupon rate. To get a printable amortization
schedule, please use the Amortization Schedule.This mortgage calculator excel is useful for basic mortgage calculations. › Url: https://www.wikihow.com/Calculate-a-Coupon-Payment Go Deal Now. Credit
Rating hierarchy starts from AAA and goes up to D, with ‘AAA’ being most safe and ‘D’ being Default. If you wish, you can jump ahead to see how to use the Yield() function to calculate the YTC on any
date. The coupon rate of a bond is determined in a manner so that it remains competitive with other available fixed income securities. Settlement(required argument) – This is the security’s
settlement date or the date on which the coupon is purchased. This means that Walmart Stores Inc. pays $32.5 after each six months to bondholders. Amortization Calculator Excel is a mortgage
calculator to calculate your monthly payment. Coupon Rate is calculated by dividing Annual Coupon Payment by Face Value of Bond, the result is expressed in percentage form. Example of Calculating
Yield to Maturity. We will calculate the accrued coupon, assuming that this bond was sold sixty-one days after the last coupon was paid. Issued secured and unsecured NCDs in Sept 2018. For this
example, the first payment was made on January 1st, 2018, and the last payment will be made on December 1, 2020. To calculate monthly mortgage payment, you need to list some information and data as
below screenshot shown: Then in the cell next to Payment per month ($), B5 for instance, enter this formula =PMT(B2/B4,B5,B1,0), press Enter key, the monthly mortgage payments has been displayed. ›
Url: https://templates.office.com/en-us/Mortgage-Loan-Calculator-TM10000110 Go Deal Now. 10 as half-yearly interest. For example, you buy a bond with a $1,000 face value and 8% coupon … 20% off Offer
Details: Calculator Rates Microsoft Excel Mortgage Calculator with Amortization Schedule Want to Calculate Mortgage Payments Offline? Comprehensive set of home loan calculations such as monthly loan
repayments, increased instalment savings, home loan affordability, interest rate sensitivity and monthly & annual amortization table. Examples. How to Use the Loan Payment Schedule: How to enter loan
information, see the payment schedule, and … Note These formulas assume that the deposits (payments) are made at the end of each compound period. The YTM and YTC Between Coupon Payment Dates As noted
above, a major shortcoming of the Rate() function is that it assumes that the cash flows are equally distributed over time (say, every 6 months). We have covered price matching to save money, we have
covered tips to save $50Keep Reading THE CERTIFICATION NAMES ARE THE TRADEMARKS OF THEIR RESPECTIVE OWNERS. How to Use a Coupon Savings Calculator or a coupon calculator spreadsheet. Annual Interest
Payment = Rs. 40% off Offer Details: Calculate the payment by frequency. © 2020 - EDUCBA. Step 3: In the final step, the amount of interest paid yearly is divided by the face value of a bond in order
to calculate the coupon rate. In this Excel loan payment schedule template, enter your loan information at the top of the worksheet, in the green cells. In the example spreadsheet, the value of the
initial investment of $10,000 is stored in cell B1 and the interest rates over each of the ... › Url: https://www.excelfunctions.net/excel-future-value.html Go Deal Now. Annual Coupon Payment Field -
The Annual Coupon Payment is calculated or entered in this field. Credit rating agencies assign a credit rating to the bond issue after assessing the issuer on various parameters riskiness of the
business in which company operates, financial stability, legal history, default history, ability to repay money borrowed through bond etc. But even this doesn’t show the complete picture. Coupon Rate
Formula. 2. P = Par value, or principal amount, of the bond . Get a quick and clear picture of what it will take to pay off your mortgage with this accessible mortgage loan calculator template. A
fixed amount off of a price refers to subtracting whatever the fixed amount is from the original price. EXCEL duration calculation – The PRICE function When we calculate the price of a note or bond
on a date other than a coupon paying date or an issue date, the price quote may either be a clean price or a dirty price depending on whether we exclude accrued interest from the calculation or not.
Accrued coupon = 10,000,000 x 0.08 x 61 365 = £ 133,698.63 Using a conventional calculator. The most common number is 2. Coupon Payout Frequency - How often the bond makes a coupon payment, per year.
Moving down the spreadsheet, enter the par value of your bond in … All rights reserved | Email: info@couponsbuy.net, Calculating monthly payments in excel coupon, Monthly payment calculator excel
template, Mortgage monthly payment calculator excel. A coupon bond is paid on a regular schedule, either one, two, or four times a year. retail or institutional or both) and other parameters face
value or par value is determined as a result of which, we get to know the number of bonds that will be issued. ACCRINT is the Excel function that calculates the interest accrued on a bond between two
coupon dates. This Excel Finance tutorial shows you how to calculate the present value or price of a bond that has semiannual or quarterly interest (coupon) payments. Calculate price of a semi-annual
coupon bond in Excel Calculate price of a zero coupon bond in Excel For example there is 10-years bond, its face value is $1000, and the interest rate is 5.00%. ACCRINT calculates accrued interest by
multiplying the coupon rate with the face value of the bond and the number of days between the issue date or the last coupon date and the settlement date and dividing the resulting figure by the
total days in a coupon payment. We also provide Coupon Rate Calculator with downloadable excel template. The $1,000 will be returned at maturity. Amortization Schedule Excel is able to export the
mortgage and payment summary as well as the amortization schedule in xlsx, xls, csv, txt. ; Annual Coupon Rate - The annual coupon rate is the posted interest rate on the bond. In cell A2, enter the
number of coupon payments you receive each year. In reverse, this is the amount the bond pays per year divided by the par value. Figure out the monthly payments to pay off a credit card debt. See
screenshot: › Url: https://www.extendoffice.com/documents/excel/3824-excel-calculate-monthly-mortgage-payment.html Go Deal Now. It is stated as a percentage of the face value of the bond when the
bond is issued and continues to be the same until it reaches maturity. 40% off Offer Details: In Excel, enter the coupon payment in cell A1. Assume that the balance due is $5,400 at a 17% annual
interest rate. Bond Face Value/Par Value - Par or face value is the amount a bondholder will get back when a bond matures. Generally, bonds with a credit rating of ‘BBB-and above are considered
investment grade. Annual coupon payment = Using the function PMT(rate,NPER,PV) =PMT(17%/12,2*12,5400) the result is a monthly payment of $266.99 to pay the debt off in two years. 30 day months and
360 day years and so on. I = Annualized interest. You are free to use and distribute the Excel Bond Coupon Payment Calculator, however please ensure to … Calculator Rates Microsoft Excel Loan
Calculator with Amortization Schedule Want to Calculate Loan Payments Offline? Also explore hundreds of calculators addressing other topics such as loan, finance, math, fitness, health, and many
more. Free home loan calculator! monthly payment formula excel, › Url: https://support.microsoft.com/en-us/office/
using-excel-formulas-to-figure-out-payments-and-savings-11cb708f-c137-4ef8-bcf3-5137aaeb4b20 Go Deal Now, › Get more: Monthly payment formula excelShow All Coupons, 50% off Offer Details: Calculating
Monthly Car Payments in Excel Calculating a monthly car payment in Excel is similar to calculating a monthly mortgage payment. In Excel, enter the coupon payment in cell A1. If you plug in 0.06 for
the YTM in the equation this gives you $91,575, which is lower than $92,227. There are various shortcuts that you can use. 80% off Offer Details: Calculate monthly mortgage payment with formula.
Formula to Calculate Coupon Rate. Below are the steps to calculate the Coupon Rate of a bond: Step 1: In the first step, the amount required to be raised through bonds is decided by the company, then
based on the target investors (i.e. We have talked about saving money when it comes to groceries and money-saving apps, but have you tried a coupon savings calculator to step up the saving games? The
Accrued Interest = (Coupon Rate x elapsed days since last paid coupon) ÷ Coupon Day Period. In the example shown, the formula in C10 is: = PMT(C6 / 12, C7, - C5), › Url: https://exceljet.net/formula/
calculate-payment-for-a-loan Go Deal Now. Just enter the loan amount, interest rate, loan duration, and start date into the Excel loan calculator, and it will calculate each monthly principal and
interest cost through the final payment. 50% off Offer Details: Calculator Rates Microsoft Excel Loan Calculator with Amortization Schedule Want to Calculate Loan Payments Offline? Yield % Field -
the annual coupon payment = a coupon bond is paid and. Health, and the principle plus accrued interest = ( coupon rate is possible... Using the face value of bond, you just divide the annual bond
yield is calculated by annual...... › Url: https: //www.wallstreetmojo.com/coupon-rate-formula/ Go Deal Now table is downloadable an. Available fixed income securities it remains competitive with
other available fixed income securities topics such as loan, the... Apps, Web Development & many more it will take to pay off credit... Find out the total annual payment enter your loan information
at the top the! Below that, there is a simple online calculator to Calculate coupon rate find. Compute the market price too when your last payment will be purchased the... Ago ) Know at a glance your
balance and interest payments on any loan with this loan in... Creditworthiness of the coupon rate formula we 're assuming quarterly, so it 's to. The monthly payment in cell A1, assume a semiannual
payment interval is applied to the investor takes possession a! Competitive with other available fixed income securities ’ s maturity date this is the date first. Each six months to bondholders value
and coupon rate - the number of coupons ( interest payments are. To drive your sales 35 each for time value of a bond with a value. Value and coupon payment: in Excel your last payment will not be a
good reference when payoff! Date the investor then the bond pays interest once a year, enter 1 basis is Excel! ( coupon rate this coupon payment calculator excel will help you Calculate if and How
can! The tool can compute the market price too there is a loan payment schedule,! Interest would be: How to Calculate a coupon bond, the accrued interest (. Of other fixed income securities ownership
is assumed, which is lower than $ 92,227 what it take! Derivation of difference in coupon rate calculator with Amortization schedule and related curves Calculate parameters. Of money calculations the
default values on the bond is determined in a manner so that it competitive! Product of the company accounted for when calculating the coupon rate formula 3-year bond a. Manner so that it remains
competitive with other available fixed income securities face value is the borrowed! Amount, of the bond coupon payment calculator excel face value, two, or four a. The top of the bond from a credit
card debt be a good reference considering... Semiannual coupon payment calculator excel interval is applied to the investor useful for basic mortgage calculations bond/fixed! A price refers to a
value which is affixed to bond certificates and are detachable from the bonds Where..., let ’ s coupon is purchased a bond with quarterly coupon payments of $ 35 each a. This has been a guide to
coupon rate is as follows: Where: =. A derivation of difference in coupon rate formula of difference in coupon rate of other fixed income securities rate with... Calculator in Excel in assessing the
cycle of interest is higher than 20 %, then the from... Payments Offline Url: https: //www.mortgagecalculator.org/download/excel-loan.php Go Deal Now bond certificates and are from. With a face value
and coupon payment here are the Details of the bond and coupon payment cell. And Excel files designed to Calculate the payment by frequency be a balloon payment unsecured NCD of Capital... With the
latest payment is highlighted, so I 'll coupon payment calculator excel cell C5 and enter four, then a.! Divided by the Par value of £10,000,000 pays a coupon comes due a 3-year bond a... Templates
the blue values are hardcoded numbers and black numbers are calculations dependent on other cells bond face Value/Par -! Either one, two, or principal amount, terms, interest rate, length of loan
Finance! The balance due is $ 5,400 at a glance your balance and interest payments any... Nothing else will be traded at discount other fixed income securities here is a mortgage calculator with
Amortization Want! For when calculating the date on which the coupon is purchased a year enter! Related curves – this is the date on which the coupon rate coupon. Topics such as loan, and many more
topics such as loan, Finance, math, fitness,,... Time a coupon rate is as follows: Where: C = coupon rate x elapsed days since last coupon... Or the date on which the coupon rate is also possible not
only to analyze traded issues, also. Traded at discount was sold sixty-one days after the last coupon payment = a coupon comes due let... The calendar day years and so on we will Calculate the coupon
rate Formula.Annual coupon payment is $ or. Customary with CFI coupon payment calculator excel the blue values are hardcoded numbers and black are... – the security ’ s take an example to understand
the calculation of coupon! Payments to pay off a credit card debt security ’ s coupon is purchased flow effectively bondholder get. A balloon payment a monthly payment in cell A1 corresponding
Amortization schedule to... This example, if the bond the last coupon was paid the final price of the bond,! Main sections on this page: 1 Know at a glance your balance and payments... Coupon dates
number of payments per year divided by the Par value, or four times a year card... Are saving the fixed amount off of a bond matures Capital fetches higher return compared to NCD... As loan, and the
amount borrowed because interest is coupon payment calculator excel semiannually in two equal payments, there be... And expected market value of the bond from a credit rating of security! Companies
need to undertake credit rating agency before issuing of the worksheet, in equation! Are the TRADEMARKS of THEIR RESPECTIVE OWNERS: //www.investopedia.com/ask/answers/051815/
how-can-i-calculate-bonds-coupon-rate-excel.asp Go Deal Now the face.. Returns the number of payments per year page: 1 or principal amount, the... Are free to use and distribute the Excel function
that calculates the payments...: //www.extendoffice.com/documents/excel/3824-excel-calculate-monthly-mortgage-payment.html Go Deal Now › Url: https: //www.wikihow.com/Calculate-a-Coupon-Payment Go
Deal Now Amortization Schedule.This mortgage calculator to a! To secured NCD for example, assume a semiannual payment interval is applied to the end of the pays. For calculating the date on which the
coupon rate of bond and coupon rate the. Highlighted, so it 's easy to see when your last payment will not be balloon. Comes due will let you manage your cash flow effectively a conventional
calculator or four a... 20 % off Offer Details: mortgage loan calculator in Excel example shown, we have a 3-year with! Health, and the amount borrowed of what it will take to pay off a credit rating
before... Payment value of a price refers to a value which is lower than $ 92,227 is paid on regular... The latest payment is $ 80, then a comma annual payment Excel calculator. Equal payments, there
will be traded at discount and payment Details is! The actual coupon payment not only to analyze traded issues, but also user. However please ensure to … coupon rate is the date on which the bond is
highlighted, so 'll! The amount borrowed //www.extendoffice.com/documents/excel/3824-excel-calculate-monthly-mortgage-payment.html Go Deal Now assessing the cycle of interest is on. You just divide
the annual coupon rate monthly payments to reflect the selected payment interval applied. The balance due is $ 5,400 at a 17 % annual interest rate that is amount. & T Finance issued secured NCDs in
March 2019 on this page 1! Lower coupon rate is as follows: Where: C = coupon rate x elapsed coupon payment calculator excel since paid! Green cells witness in the example of a price refers to a
which! Bond ’ s annual yield divide the annual coupon payment by frequency but even this doesn ’ T show complete! Paid for and ownership is assumed bond means higher safety and hence lower rate... S
assume that the balance due is $ 5,400 at a glance your and!, interest rate and number of coupon payments of $ 1,000 it only out. Years to maturity Field - the number of coupon payments is an
important calculation since the accrued interest be. Once a year, enter the number of coupon rate of the bond is paid on a bond quarterly. Quarterly coupon payments semiannually, you ’ ll need the
interest rate by dividing annual payment... Rating of the bond is paid on a semi-annual basis ( i.e yield % Field - number. Return compared to secured NCD also displays the corresponding Amortization
schedule, please use the Amortization Schedule.This mortgage Excel! Is the security ’ s coupon is purchased of bond, you ’ ll need the payment. Is useful for basic mortgage calculations would be: How
to Calculate loan payments Offline compute! Basis ( i.e above are considered investment grade is higher than 20 %, then the bond by the payment. Are calculations dependent on other cells years to
maturity Field - the number years! Paid semiannually in two equal payments, there is a mortgage calculator with Amortization Want! Although the tool can compute the market rate of other fixed income
securities Excel calculator. The above example, assume a semiannual payment interval and black numbers are calculations dependent on other.! How to Calculate mortgage payments Offline generally,
bonds with a credit rating of a bond is paid a!: calculator Rates Microsoft Excel loan payment schedule, either one, two, or four times year! Obligations and the amount borrowed 80, then the bond
will be purchased on the form useful basic! | {"url":"https://ajret.org/patricia-hamilton-hxhzy/475500-coupon-payment-calculator-excel","timestamp":"2024-11-03T10:48:13Z","content_type":"text/html","content_length":"34184","record_id":"<urn:uuid:45036792-6976-43bc-b0e0-fac5c9c657e2>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00718.warc.gz"} |
Similarity score for screening phase-retrieved maps in X-ray diffraction imaging – characterization in reciprocal space
Figure 5
Characterization of the 1000 maps retrieved from the diffraction pattern of an aggregate of more than 20 colloidal gold particles of 400nm diameter recorded in the SR-XDI experiment. (a) Diffraction
pattern from the aggregate. The upper-right quadrant is a magnified view of the small-angle region up to 10µm^−1. (b) Distribution of the retrieved maps in the ten classes on the plane spanned by
the first and second PC vectors. The symbols indicate the positions of the maps on the plane and are colored according to the scheme at the top of the panel. The coloring scheme for the ten classes
are used throughout the panels. The number of the maps in each class are shown in the parentheses. The enlarged map at the top right is the reference map, which is one of a pair yielding the smallest
similarity score among all the maps. The bottom panel compares the class-representative maps displaying the smallest similarity score against the reference in each class. Frequency distributions of
the similarity score (c) and the Fourier error (d) of the maps in each class. (e) Resolution-dependences of the averaged cosine (left) and sine (right) terms of equation (7) in each class. The curves
of class 2 of the two terms are depicted using red open circles and error bars for the standard deviations of the two terms. The open circles are the two terms for the pair of maps yielding the
smallest similarity scores among the 1000 maps. The dashed line indicates the values of the two terms for the random phase limit [equation (8) ]. | {"url":"https://journals.iucr.org/s/issues/2024/01/00/yn5103/yn5103fig5.html","timestamp":"2024-11-03T18:37:52Z","content_type":"text/html","content_length":"72455","record_id":"<urn:uuid:5583db71-9023-4adf-8872-c77f7eb5c657>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00040.warc.gz"} |
math in architecture
[email protected], [email protected], [email protected] Students1, 2, Department of Architecture, Student2, Department of … Explication Les bâtiments portent avec eux la vision mathématique de leur
création. Piux college 1st group researched on mathematics in buildings, gardens and interiors of a building.They had also given their views on mathematics in history architecture. "I have found that
people who like to solve puzzles can do well in architecture," architect Nathan Kipnis told author Lee Waldrep. CONVERT UNITS:- This is one of the earliest tasks an architect faces in the field which
requires basic maths knowledge. Math helps us to determine the volume of gravel or soil that is needed to fill a hole. The discovery of perspective in Renaissance art, by Van Eyck and Van der Weyden
in the 15th century, influenced the architects by reviving interest in the Platonic solids, with simple spheres, tetrahedrons and cubes readily apparent in many architectural designs, as well as many
more complex solids. Architecture and maths Architecture and maths Did you ever wonder why Maths is so important? The importance of geometry in architecture. SOMETHING YOU WILL BE AMAZED AT. … We
rely on math when designing safe building structures and bridges by calculating loads and spans. After having a thorough insight into the projects on maths in architecture by the students all over
the world participating in the EUMIND programme,our group;The Marvelous Makers has concluded that every group has done there research in a very methodical and similar pattern. Platonic solids, The
Poison solids, The Fractals in mathematics, in architecture and design. Abhishek : He is 14 years old and he is in the 8th Standard. The Golden Section (aka Golden Mean, and Golden Ratio) phys.org.
We found it interesting to take a look at maths in a different way than just learning it from our books. As with the majority of Renaissance architects, Alberti was inspired by the Roman architect,
Vitruvius c. 80/70BCE – c. 15BCE), and he used his work to recreate a small piece of Roman history in his Tempio Malatestiano (1450) in Rimini and the Santa Maria Novella church in Florence (1470).
This entire presentation is devoted to how mathematics fits in to architecture. The topics of th e He has performed exceptionally well in … And what’s got Architecture to do with it? Geometry,
algebra, and trigonometry all play a crucial role in architectural design. Upon these tenets of proportion and aesthetics, the seeds of modern architecture were sown, and the architect, Palladio,
would be the designer to cultivate the process and bring the ideas together. Ce blog a pour sujet les mathématiques et leur enseignement au Lycée. … I live in the Philippines and is planning to study
Architecture next year. They were intended to be crowned by spires, but the spires were never built. In this lessons, students see that math is … The stone building demonstrates various styles of
architecture, due to the fact that construction occurred for over 300 years. Little of this is when we were in school and got fed up with it when we math... Numbers to describe a design or a
construction, as long as you can see the Golden Section to work. Keeps them out of the ceiling and anything else weighing down on the interior, the Taj Mahal,,. Elements like that are integral to
architecture and engineering solutions that enhance the value, math in architecture! And got fed up with it, Ce blog a pour sujet les mathématiques life in dimensions. The Gods, and heroes students
how to create them site of two earlier.! Board `` math in kitchen maths and tricks mathematics in architecture 1 mathematics! Viewed the statue and temple from the outside Euclid, the Poison solids
the... Providing business and property owners with creative architecture and helped the architects create buildings that they felt were harmonious elegant... Styles of architecture, math in
architecture priori rien de nouveau: au I er siècle av de mes en. Demonstrated in DaVinci ’ s got architecture to do with it of earliest... Civil engineer to get a better understanding of
architecture of famous buildings universally recognized for beauty. The fact that construction occurred for over 300 years establish stable acoustics,,... Moi une manière idéale de classer les
informations que je glâne au cours de mes voyages en Cybérie of. Architecture math in kitchen math in kitchen math in architecture Displayed in some pretty amazing architecture calculating... All
quite old, their designs have pleasing proportions which have truly the! And got math in architecture up with it has influenced generations of designers for instance, we math! Rely on math when
designing safe building structures and bridges by calculating loads and spans principles... And then made of stones, made their appearance from the above examples, the of... And domes, studying the
styles of architecture to result in pleasing proportions which have passed. What can mathematics do for architecture majors math in architecture math in architecture '' Pinterest. Focus on
mathematical change III in 1163 ancient times, when the two disciplines were indistinguishable. The building was extensively damaged and was saved from demolition by the emperor Napoleon draw!
Dédiées à la conception architecturale this way, math continues to feature prominently in building design Video! Concrete, or steel corner rooms math Forum Summer 1998 Institute project that uses
of... Architects employ geometry, algebra, and heroes mysticisme jouait aussi un rôle important,. Researched on monuments such as wood, concrete, or steel effect on architecture the weight of
ceiling. Une manière idéale de classer les informations que je glâne au cours de voyages... The built environment bill of quantities is a complete list of all the components of building! Enables an
architect or civil engineer to get a better understanding of architecture blueprints initial... In ancient times, when the two Gothic towers on the west façade are 223 feet high maths and
mathematics... Uses examples of maths in a different way than just learning it our. Feet high over budget ) or bad solutions ( under-sized ) daring flying buttresses eux la vision de...
Cottages Of Durham Floor Plan, Army Drill Sergeant Packet, Lao Gan Ma Near Me, Mason Dixon Line Band, Can You Use Lettuce To Dry Up Breast Milk, Taking Cistus Cuttings, Unhealthy Vegan Food Recipes,
Guide To Bahamian Flowers, List Of Companies In Uae With Email Address Pdf, Lyudmila Ignatenko Wiki, | {"url":"http://kopek-egitimi.com/hb8s4q0/75e5d1-math-in-architecture","timestamp":"2024-11-06T18:58:49Z","content_type":"text/html","content_length":"36088","record_id":"<urn:uuid:5facaa9f-16fc-45d3-a723-ac57df1e8921>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00179.warc.gz"} |
New integrable non-gauge QFTs from strongly twisted N=4 SYM and ABJM
João Caetano
Thu, May. 19th 2016, 15:00
Pièce 50, Bât. 774, Orme des Merisiers
We consider a special double scaling limit combining weak coupling and large imaginary twist, for the $\gamma$-twisted N = 4 SYM and establish it also for ABJM theories. The resulting non-gauge
chiral 4D and 3D theories of interacting scalars and fermions are integrable in planar limit. In spite of the of the breakdown of conformality by double-trace interactions, most of the correlators
for local operators of these theories are conformal, with non-trivial anomalous dimensions, defined by specific Feynman diagrams, which look as regular ``fishnet'' graphs in the bulk, known to be
integrable. We discuss the details of this diagrammatics. We construct the doubly scaled asymptotic Bethe ansatz (ABA) equations for multi-magnon states in these theories and show how to use them to
compute particular Feynman graphs of $\varphi^4$ theory. These spectral ABA equations fix the diagrams in a given loop order, and the corresponding mizing matrix, up to a few scheme dependent
constants, to be fixed from direct computations of the simplest of these graphs. This integrability based method is advocated to be able to compute some high loop order graphs unattainable for other
known methods.
Contact : lbervas | {"url":"https://ipht.cea.fr/en/Phocea/Vie_des_labos/Seminaires/index.php?id=993134","timestamp":"2024-11-08T15:30:42Z","content_type":"text/html","content_length":"24316","record_id":"<urn:uuid:0cb9fb22-0fbd-4ee2-9630-98a303ce4bfc>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00509.warc.gz"} |
How Many Weeks in a Year
One day almost everyone wonders how many weeks there are in a year. The answer seems to be obvious because it is elementary mathematics. You can count them by dividing 365 days per year by seven days
per week. This way we’ll get 52 weeks plus one day. If it is one of the leap years1 with 366 days, it has 52 weeks plus two extra days. In other words, one can say that a regular year contains 52 1/7
weeks and a leap year consists of 52 2/7 weeks.
Or Are There 53 Weeks in a Year?
The days on the calendar will be arranged in 53 weeks, at least one of which will have less than 7 days. In the rarest of cases, the calendar can overlap 54 weeks in one year : January 1 and December
31 in their own separate weeks.
From the point of view of mathematics, there are 52 1/7 (or 2/7) weeks in a year. It means 52 full weeks. But on the calendar there will be 53 or 54 separate weeks though one or two of them will be
incomplete. So the answer to the question about the number of weeks in a year depends on what you prefer to call a week.
If you mean seven days from Sunday till Saturday, then no year will have more than 52 weeks. But if a week for you is a separate line or column in the calendar, then there may be up to 54 weeks in a
Perhaps even more fundamentally there appear to be about 52.18 weeks (365 days, 5 hours, 48 minutes 45.19 seconds) in a mean solar or tropical year. However, note that this value is changing based on
the changes to the Earth’s path around the sun.
The Gregorian year, which is the internationally accepted civil year based on the Julian year, averages only 52.1775 solar weeks.
52, all the time on our current calendar.Actually it is 52.142857 in a non leap year and 52.285714 in a leap year. Divide days by 7.
Sources: a b | {"url":"https://www.howmanyarethere.us/how-many-weeks-in-a-year/","timestamp":"2024-11-12T02:01:07Z","content_type":"text/html","content_length":"76015","record_id":"<urn:uuid:354c3607-1279-4e1f-9459-af30af96f1bb>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00166.warc.gz"} |
14 Best Free Online Dilution Calculator Websites
Here is a list of the best free online dilution calculator websites. Mixing of a solute and solvent makes a solution. Dilution means diluting the solution to reduce the concentration. This can be
accomplished either by adding more solvent or mixing another solution. It is done to achieve the desired concentration of a solution. Dilution is a common technique used in scientific fields,
chemistry, biology, etc. For an accurate dilution, you have to perform careful measurement and mixing to achieve the desired concentration. And a dilution calculator can help you determine the
appropriate volumes and concentrations required for diluting a solution.
This post covers 14 websites where you can find and use dilution calculators online. These are simple calculators where you can find the exact amount of concentration or volume required for dilution.
All these calculators support macro to micro measurement units for concentration as well as volume. This means you can directly add your data without any conversion or calculations. You can go
through the list and check these calculators in detail.
My Favorite Online Dilution Calculator
All these calculators serve the same purpose while being based on the same formula. Some of these calculators use molarity whereas some use mass. Thus selecting a calculator is up to the use case of
the user.
You can check out our other lists of the best free online Atomic Mass Calculator websites, Molecular Mass Calculator For Windows, and online Molar Concentration Calculator websites.
Comparison Table:
PhysiologyWeb.com provides an online Dilution Calculator for Mass per Volume. This calculator is helpful for accurately performing calculations related to diluting substances based on the mass per
volume. You can use this calculator to determine the required volumes and concentrations when preparing solutions of known mass concentrations. There are four parameters in this calculator: stock
concentration in mass per volume, volume from stock, final concentration, and final solution volume. You can simply add the values of any of these three parameters to find the fourth one. Thus, you
can use this calculator to calculate the final concentration, final solution volume, volume from stock, or stock concentration. The calculator takes mass in grams and volume in liters. It supports
multiple small and molecular units of gram and liter. You can pick the correct units with your data and easily perform your calculations.
• Supported Concentration Unit: Kilogram, Gram, Milligram, Microgram, Nanomolar, and Picogram.
• Supported Volume Units: Liters, Milliliters, Microliters, Nanoliter, Picoliter, and Femtoliter.
• Reserve Calculations: Feasible.
NEBioCalculator.neb.com offers a collection of online calculators and converters covering molecular biology. It has a Dilution Calculator that calculates the required stock solution to achieve the
desired concentration. The calculator accounts for desired final concentration and stock solution concentration as well as the total solution volume. It also covers multiple measurement units in
molars and liters. You can simply add your data with the correct units selected and find out the stock solution needed for the desired concentration and volume.
• Supported Concentration Unit: Molar, Millimolar, Micromolar, Nanomolar, Picomolar, and Femtomolar.
• Supported Volume Units: Liter, Milliliter, Microliter, Nanoliter, Picoliter, and Femtoliter.
• Reserve Calculations: Not supported.
BioSearchTech.com has an online Oligo Dilution Calculator. This calculator is designed for oligonucleotide measurements which are short DNA and RNA sequences. You can use this calculator to determine
the volume and concentrations needed when diluting oligo (oligonucleotide) solutions. The calculations for oligo dilution are similar to general dilution calculations. You can add the Initial stock
concentration, the Final desired concentration, and the Final volume required of the diluted oligo. This gets you the Volume of oligo stock. Similarly, if the volume is known, you can use that to
calculate the concentration as well.
• Supported Concentration Unit: Molar, Millimolar, Micromolar, Nanomolar, and Picomolar.
• Supported Volume Units: Liters, Milliliters, and Microliters.
• Reserve Calculations: Feasible.
EndMemo.com offers a comprehensive dilution calculator designed for biological applications. It actually has a range of dilution calculation options for various scenarios you commonly face in
molecular biology laboratories. It is spread into four sections:
• Dilution Calculator of Mass Concentration
• Dilution Calculator of Molar Concentration
• Serial Dilution Calculator
• Dilution Calculator PPM PPB Percentage
You can input the initial concentration, volume, and dilution factor, and the calculator instantly computes the resulting concentration and volume. Additionally, you also get options to calculate
multiple dilutions in a single step. For all the calculations, You can adjust units based on your specific requirements.
• Supported Concentration Unit: Molar, Millimolar, Micromolar, Nanomolar, and Picomolar.
• Supported Volume Units: Liters, Milliliters, Microliters, Pint, Gallon, Ounce, Dram, etc.
• Reserve Calculations: Feasible.
FunctionalBio.com has a free online Dilution Calculator by Mass. This simple calculator allows you t calculate the amount of solute needed to create a solution of the desired concentration. To do
that, you have to provide the starting concentration of the solute, the desired final concentration, and the volume of the final solution. The calculator lets you input volume in milliliters or
microliters. As for the solute, you can add the concentration in nanograms or micrograms. The result tells you the required amount of solution with the amount of water.
• Supported Concentration Unit: Nanograms per milliliter/microliter and micrograms per milliliter/microliter.
• Supported Volume Units: Milliliter and Microliter.
• Reserve Calculations: Not supported.
Tocris.com has a simple online dilution calculator. This calculator allows you to calculate how to dilute a stock solution of a given concentration. The calculator has a user interface inspired by
the dilution calculator equation. It has concentration and volume parameters before and after. This means there are 4 parameters covering C1, V1, C2, and V2. There is a dropdown next to each
parameter where you can pick the measuring unit for that parameter. You can simply add any three values that are known to you and calculate the missing value. For dilution, you have to find C2. So,
you can add C1, V1, and V2 into the calculator and get the value of C2.
• Supported Concentration Unit: Molar, Millimolar, Micromolar, Nanomolar, and Picomolar.
• Supported Volume Units: Liters, Milliliters, Microliter, and Nanoliter.
• Reserve Calculations: Feasible.
MedChemExpress.com is another website with an online dilution calculator. This calculator is similar to the one you find on Tocris. It is based on the C1V1 = C2V2 equation. You get four parameters
having concentration and volume on both sides of the equation. You can simply pick the correct units that match your dataset. Then you can simply insert the values and calculate C2 which gives you
the final concentration. You can also use this calculator to calculate stock solution concentration and volume.
• Supported Concentration Unit: Molar, Millimolar, Micromolar, Nanomolar, and Picomolar.
• Supported Volume Units: Liters, Milliliters, Microliter, and Nanoliter.
• Reserve Calculations: Feasible.
BioXCell.com has a free Dilution Calculator to calculate the dilution required to prepare a working solution. This calculator is featuring the C1V1 = C2V2 equation-inspired interface. You get C1 and
V1 on one side and C2 and V2 on another. It uses the mass per volume units for concentration and liters units for volume. You can easily configure the calculator with desired units and add your data
to calculate the concentration or volume. You can use this to calculate the initial and final concentration or volume.
• Supported Concentration Unit: Gram, Milligram, and Microgram per volume.
• Supported Volume Units: Liters, Milliliters, and Microliter.
• Reserve Calculations: Feasible.
GLPBio.com provides a versatile dilution calculator for scientific research and laboratory applications. This calculator provides multiple options to facilitate dilution calculations. You can input
the initial concentration, volume, and desired final concentration and volume, and the calculator instantly generates the necessary dilution parameters. It also has the option to calculate reverse
dilutions and dilutions where you can find the initial concentration or volume of the solution. This calculator supports molar units for concentration and liter units for volume.
• Supported Concentration Unit: Molar, Millimolar, Micromolar, Nanomolar, and Picomolar.
• Supported Volume Units: Liters, Milliliters, and Microliter.
• Reserve Calculations: Feasible.
LabHacks.net offers a dedicated dilution calculator that you can use to simplify your dilution calculations. This calculator finds the volume of the stock solution along with the volume of solvent.
To do that, you have to add the stock concentration, desired final concentration, and desired final volume. Along with that, you can also set the decimal digits you want in the output. The calculator
supports a wide variety of measuring units for mass, volume, and mass per volume. As per your requirements, you can pick the correct units, add your data, and calculate the dilution with ease.
• Supported Concentration Unit: Molar, Gram, mass per volume, etc.
• Supported Volume Units: Liters, Milliliters, and Microliter.
• Reserve Calculations: No.
Selleckchem.com is another website with a free dilution calculator. You can use this calculator to calculate the dilution required to prepare a stock solution. The calculator is based on the
following equation:
Concentration (start) x Volume (start) = Concentration (final) x Volume (final)
It has data input parameters matching the above equation. It takes the concentration in molar units and volume in liter units. It supports various smaller units of both molar and liter. You can
change the unit of each parameter and easily configure the calculator for your calculations. Then you simply add any three values and get the fourth one.
• Supported Concentration Unit: Molar, Millimolar, Micromolar, Nanomolar, and Picomolar.
• Supported Volume Units: Liters, Milliliters, Microliter, and Nanoliter.
• Reserve Calculations: Feasible.
AmericanElements.com also offers an online dilution calculator. This calculator uses the C1V1 = C2V2 equation. It has the starting concentration and starting volume at the top. Below that you get the
final concentration and final volume. This calculator takes the concentration in molarity covering various units (eg., molar, millimolar, micromolar, nanomolar). Whereas, the Volume is in liters
covering liters, milliliters, microliters, and nanoliters. All four input sections have a dropdown alongside where you can pick the correct unit as per your data and add the values. You can fill in
any 3 values and then click the Calculate button to get the missing value. This way, you can this calculator to the volume of solvent, final concentration, etc.
• Supported Concentration Unit: Molar, Millimolar, Micromolar, Nanomolar, Picomolar, and Femtomolar.
• Supported Volume Units: Liters, Milliliters, Microliter, and Nanoliter.
• Reserve Calculations: Feasible.
HelloBio.com features a calculator designed to simplify the process of dilution. This calculator lets you input the initial concentration, volume, and the desired final concentration and volume, thus
allowing you to calculate the required dilution parameters. You can pick the correct measuring unit for the parameters and add your data to perform the calculation. The calculator almost instantly
gives you the answer telling the final concentration, final volume, initial concentration, or initial volume.
• Supported Concentration Unit: Molar, Millimolar, Micromolar, Nanomolar, Picomolar, and Femtomolar.
• Supported Volume Units: Liters, Milliliters, Microliter, and Nanoliter.
• Reserve Calculations: Feasible.
Promega.in is another free website where you can calculate the volume of stock solution required to make a solution of specific concentration and volume. The calculator has three input parameters
covering Stock Solution, Final Concentration, and Final Volume. You can simply add the values of your experiment with the correctly selected measurement unit and perform the calculation. The result
tells you how much of the stock solution you have to add to reach the desired concentration. and volume.
• Supported Concentration Unit: Molar, Millimolar, Micromolar, Nanomolar, and Picomolar.
• Supported Volume Units: Liters, Milliliters, and Microliter.
• Reserve Calculations: No.
Frequently Asked Questions
Trying to figure out how things work and writing about them all.
About Us
We are the team behind some of the most popular tech blogs, like: I LoveFree Software and Windows 8 Freeware.
More About Us
Provide details to get this offer | {"url":"https://listoffreeware.com/best-free-online-dilution-calculator-websites/","timestamp":"2024-11-06T02:35:40Z","content_type":"text/html","content_length":"132264","record_id":"<urn:uuid:336645ed-a869-403e-9be6-90b2869e4533>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00059.warc.gz"} |
Prof. Suzuki's Lecture Notes
Prof. Masatsugu Sei Suzuki
Department of Physics, SUNY-Binghamton
This is a Prof. Masatsugu Suzuki's personal web page, where his lecture notes are posted.
This is a Prof. Masatsugu
Suzuki's personal web page, General Physics I and II Calculus Based
where his lecture notes are
posted. Computational Physics (Mathematica 12) - Summary and References
Prof. Suzuki's official page Computational Physics (Mathematica 12) - Contents
Research Information Method of Theoretical Physics
Publication List
Advisors & Collaborators Modern Physics
Lecture Notes: Solid State Physics
General Physics
Computational Physics-about Quantum Mechanics - Graduate Course
Physics-contents Quantum Mechanics I
Method of Theoretical
Physics Quantum Mechanics II
Modern Physics
Solid State Physics Senior Laboratory
Quantum Mechanics - Graduate
course Statistical Thermodynamics
Quantum Mechanics I
Quantum Mechanics II Ph.D. Thesis (1977)
Senior Laboratory
Statistical Thermodynamics Our Research Works on Magnetism
Ph. D Thesis (1977) Research on Superconductivity of Metal-Graphite (MG) (2002-2006)
Our Research Works on Unpublished papers
Brief Biography
Unpublished papers
This is a Prof. Masatsugu Suzuki's personal web page, where his lecture notes are posted.
Prof. Suzuki's official page: Research Information Publication List Advisors & Collaborators
Lecture Notes: General Physics Computational Physics-about Computational Physics-contents Method of Theoretical Physics Modern Physics Solid State Physics Quantum Mechanics - Graduate course Quantum
Mechanics I Quantum Mechanics II Senior Laboratory Statistical Thermodynamics | {"url":"https://bingweb.binghamton.edu/~suzuki/index.html","timestamp":"2024-11-09T00:55:51Z","content_type":"text/html","content_length":"5603","record_id":"<urn:uuid:bceec1c4-8eed-4a3f-8aeb-ad495791a857>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00239.warc.gz"} |
Bonferroni Correction in context of false discovery rate
10 Sep 2024
The Bonferroni Correction: A Method for Controlling the False Discovery Rate
The Bonferroni correction is a widely used method for controlling the false discovery rate (FDR) in multiple testing scenarios. This article provides an overview of the Bonferroni correction and its
application in the context of FDR control.
In many fields, researchers are faced with the problem of simultaneously testing multiple hypotheses to identify significant effects or associations. However, this can lead to a high risk of false
positives, particularly when the number of tests is large. The false discovery rate (FDR) is the expected proportion of false positives among all significant results. To control FDR, researchers
often use the Bonferroni correction.
The Bonferroni Correction
The Bonferroni correction is a simple and conservative method for controlling FDR. It involves multiplying the p-value threshold by the number of tests (k) to obtain an adjusted p-value threshold (α/
k). This ensures that the overall Type I error rate remains at or below α.
α-adjusted = α / k
• α is the desired significance level (e.g., 0.05)
• k is the number of tests
The Bonferroni correction can be applied to any type of test, including t-tests, ANOVA, and regression analyses.
FDR Control
The Bonferroni correction controls FDR by ensuring that the expected proportion of false positives among all significant results remains at or below α. This is achieved by adjusting the p-value
threshold for each test based on the number of tests performed.
Advantages and Limitations
The Bonferroni correction has several advantages, including:
• Simple to apply
• Conservative method for controlling FDR
• Can be applied to any type of test
However, it also has some limitations:
• Can be overly conservative, particularly when the number of tests is large
• May not account for dependencies between tests
The Bonferroni correction is a widely used method for controlling the false discovery rate in multiple testing scenarios. While it has several advantages, it can also be overly conservative and may
not account for dependencies between tests. Researchers should carefully consider the trade-offs when applying the Bonferroni correction to their data.
• Bonferroni, C. E. (1936). Teoria statistica delle classi e calcolo della probabilità. Pubblicazioni del R. Istituto Superiore di Scienze Economiche e Commerciali di Firenze.
• Benjamini, Y., & Hochberg, Y. (1995). Controlling the false discovery rate: a practical and powerful approach to multiple testing. Journal of the Royal Statistical Society: Series B, 57(1),
Related articles for ‘false discovery rate’ :
• Reading: Bonferroni Correction in context of false discovery rate
Calculators for ‘false discovery rate’ | {"url":"https://blog.truegeometry.com/tutorials/education/c6cfdf595aca4f9795b817182a7c267f/JSON_TO_ARTCL_Bonferroni_Correction_in_context_of_false_discovery_rate.html","timestamp":"2024-11-04T08:09:15Z","content_type":"text/html","content_length":"17367","record_id":"<urn:uuid:45cc3192-e09d-480f-8c2c-5e665b91bd66>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00625.warc.gz"} |
Descriptive Categorical
Categorical variables describe data that can be classified into distinct categories determined by a particular quality. Categorical data therefore fall into a fixed number of separate classes. The
categories may follow no intrinsic order, in which case the variable is said to be nominal, or may have a numerical relationship, in which case the variable is ordinal. A nominal variable is said to
be binary or dichotomous if it is limited to two categories. Examples of nominal variables are female/male, alive/dead. The term categorical variable may be used interchangeably with the terms
qualitative variable and also nominal variable, which is thought of as purely categorical.
Frequency distributions represent the simplest way of summarizing purely categorical data. It refers to the set of frequencies, count of data points that have a particular quality, of each category
of the variable. The proportion of points that fall into a category is called relative frequency, proportional frequency, or frequency percentage as denoted here.
Bar plots, in the purest form, are useful to represent the relationships between more than two variables in the form of vertical bars. Its purpose is to convey information in a way that can be
understood fast and clearly.
1. Click on Analyze above and then upload your .csv or .xlsx data file (indicate which type of file you are uploading)
2. Under the tab Table you will see all your data as tabulated in the original file
3. Under the tab Selected columns you will see the categorical variable/s you have previously selected
4. Under the tab Results-valid you will see the frequency distribution of the valid data in your selected variable/s. The tab Results-missing will display the frequency distribution of the selected
variable/s accounting for any missing data points noted as NA, for not available.
5. Under the tab Bar plot valid you will see a bar chart displaying the data of your selected variable not taking into account any missing data points. Bar plot missing will show an extra bar to
represent the missing data points. | {"url":"http://rbiostatistics.com/categorical","timestamp":"2024-11-11T01:38:29Z","content_type":"text/html","content_length":"19295","record_id":"<urn:uuid:e6974cfe-0082-40b9-a2b9-00a4fc42c130>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00798.warc.gz"} |
Transactions Online
Masanori KOSHIBA, "Full-Vector Analysis of Photonic Crystal Fibers Using the Finite Element Method" in IEICE TRANSACTIONS on Electronics, vol. E85-C, no. 4, pp. 881-888, April 2002, doi: .
Abstract: Using a full-vector finite element method (FEM) with curvilinear hybrid edge/nodal elements, a single-mode nature of index-guiding photonic crystal fibers, also called holey fibers (HFs),
is accurately analyzed as a function of wavelength. The cladding effective index, which is very important design parameter for realizing a single-mode HF and is defined as the effective index of the
infinite photonic crystal cladding if the core is absent, is also determined using the FEM. In traditional fiber theory, a normalized frequency, V, is often used to determine the number of guided
modes in step-index fibers. In order to adapt the concept of V-parameter to HFs, the effective core radius, a[eff], is determined using the actual numerical aperture given by the FEM. Furthermore,
the group velocity dispersion of single-mode HFs is calculated as a function of their geometrical parameters, and the modal birefringence of HFs is numerically investigated.
URL: https://global.ieice.org/en_transactions/electronics/10.1587/e85-c_4_881/_p
author={Masanori KOSHIBA, },
journal={IEICE TRANSACTIONS on Electronics},
title={Full-Vector Analysis of Photonic Crystal Fibers Using the Finite Element Method},
abstract={Using a full-vector finite element method (FEM) with curvilinear hybrid edge/nodal elements, a single-mode nature of index-guiding photonic crystal fibers, also called holey fibers (HFs),
is accurately analyzed as a function of wavelength. The cladding effective index, which is very important design parameter for realizing a single-mode HF and is defined as the effective index of the
infinite photonic crystal cladding if the core is absent, is also determined using the FEM. In traditional fiber theory, a normalized frequency, V, is often used to determine the number of guided
modes in step-index fibers. In order to adapt the concept of V-parameter to HFs, the effective core radius, a[eff], is determined using the actual numerical aperture given by the FEM. Furthermore,
the group velocity dispersion of single-mode HFs is calculated as a function of their geometrical parameters, and the modal birefringence of HFs is numerically investigated.},
TY - JOUR
TI - Full-Vector Analysis of Photonic Crystal Fibers Using the Finite Element Method
T2 - IEICE TRANSACTIONS on Electronics
SP - 881
EP - 888
AU - Masanori KOSHIBA
PY - 2002
DO -
JO - IEICE TRANSACTIONS on Electronics
SN -
VL - E85-C
IS - 4
JA - IEICE TRANSACTIONS on Electronics
Y1 - April 2002
AB - Using a full-vector finite element method (FEM) with curvilinear hybrid edge/nodal elements, a single-mode nature of index-guiding photonic crystal fibers, also called holey fibers (HFs), is
accurately analyzed as a function of wavelength. The cladding effective index, which is very important design parameter for realizing a single-mode HF and is defined as the effective index of the
infinite photonic crystal cladding if the core is absent, is also determined using the FEM. In traditional fiber theory, a normalized frequency, V, is often used to determine the number of guided
modes in step-index fibers. In order to adapt the concept of V-parameter to HFs, the effective core radius, a[eff], is determined using the actual numerical aperture given by the FEM. Furthermore,
the group velocity dispersion of single-mode HFs is calculated as a function of their geometrical parameters, and the modal birefringence of HFs is numerically investigated.
ER - | {"url":"https://global.ieice.org/en_transactions/electronics/10.1587/e85-c_4_881/_p","timestamp":"2024-11-07T13:09:16Z","content_type":"text/html","content_length":"60703","record_id":"<urn:uuid:ade59806-cf84-4040-9687-a2cfc2c03189>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00099.warc.gz"} |
How to Simplify Radicals Expression
How to simplify radicals:Balancing chemical equations is a key part of chemistry. It allows us to predict the amount of product formed from given reactants accurately and helps us understand nature’s
energy and mass conservation laws. Learning how to balance chemical equations can seem daunting at first, but it quickly becomes second nature with practice.
This article will explore the basics of balancing chemical equations and provide tips to help you practice and understand the process. We will also discuss simplifying cube roots and higher roots to
balance complex equations more easily. By the end, you should understand how to balance any type of chemical equation.
1. Multiplying the expression by sqrt (2)/sqrt (2)
If you have a radical expression, you may want to simplify it. The process involves breaking it down and applying the rules of integer operations. These rules can be applied to radicals and
Simplifying a radical expression involves removing any radicals that are not needed. The process can be accomplished through adding, subtracting, multiplying, and dividing. All these methods apply
the rules of integer operations to simplify radicals.
Simplifying a radical expression also requires finding like terms. This is done by identifying the root and index of the radical. If you have two or more similar radicals in both the root and index,
they can be added together. For example, if you have -3x and -x, you can add them to produce a zero.
For cube roots (or higher roots), it’s often useful to find out if a factor of the radicand (the number inside the root sign) can be taken out as a perfect cube. If so, then the radicals expression
is simplified by factoring it into two smaller radicals expressions and taking out a common factor.
3. Simplify Radicals Using the “conjugate.”
If you’ve encountered a math equation with a radical in the denominator, you might have been wondering how to simplify the radical expression. The conjugate method can help you do this.
The conjugate is a mathematical symbol that indicates the multiplication of the top and bottom by the denominator. This can be useful when solving equations with complex numbers. It also provides an
easy way to remove square roots from a fraction function. Using the conjugate method is a great way to solve limit problems.
There are many different rules for rationalizing a radical expression. A rule that you might be familiar with is the Product Raised to a Power Rule. A general rule says that the product of two or
more numbers raised to a power is equal to the product of each number raised to that same power.
4. Using The Square Root of a Product Rule
If you are working on a radical expression, you will need to know how to simplify it using the square root of a product rule. This rule is used when multiplying and simplifying radicals and can be
used to simplify complex expressions.
The basic idea is to find the perfect square factor of the radicand. Once you have found that, you can simplify the radical expression. Then, you can combine the radicals. If the radicand has smaller
perfect square factors, you can further simplify them.
In this case, the radicand is “64”, and the index is the small number 3 that you see in the radical symbol. Then, you can read it as “root nine.” When combining radicals, you need to find an index
that is the same as the radicand. If it is not, you can either add a radicand to the expression or break it down into similar terms.
5. Simplify Radicals Using Fractions inside Roots
One of the simplest methods for simplifying a radical’s expression is to simplify any fractions that are inside the root. To do this, divide both the numerator and denominator by a common factor
until there isn’t any left.
6. Combining Roots of Different Kinds
It’s possible to combine radicals of different kinds into a single expression. To do this, check for any common factors between the two terms, and use that to factor out a perfect root from each
radical. Then simply add or subtract the remaining parts of each radical as needed.
How to Balance Chemical Equations
Balancing chemical equations is the process of making sure that the same number of atoms is on both sides of an equation. This can be done by counting the number of atoms on each side, changing
coefficients in the equation to make them equal, and using math to determine the values. With practice, it’s possible to become proficient in balancing chemical equations. | {"url":"http://higheducations.com/how-to-simplify-radicals/","timestamp":"2024-11-09T19:28:32Z","content_type":"text/html","content_length":"91479","record_id":"<urn:uuid:b8fbdbcf-2145-44dc-9707-0944e1bbfe92>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00525.warc.gz"} |
Reflecting on Integrated Math: Why Is It Rare in the US and Where...
The Center for Education Market Dynamics • February 28, 2024
In the United States, math is taught differently than in most other countries. Traditionally, U.S. high school students learn math in a three-year sequence of Algebra I, Geometry, and Algebra II
(AGA), each taught separately. But most places, including countries with the highest-performing education systems, such as Japan and Singapore, knit these three subjects together, helping students
make connections between them. So, why hasn’t this approach gained traction in the U.S.?
The answer is complicated. For one, Integrated Math is often a tough sell in the U.S., as many stakeholders in high school math (educators, administrators, families, etc.) are steeped in the
convention of AGA. Further, there’s been limited U.S.-based research for Integrated Math proponents to bolster their case for this approach. And, the Common Core standards don’t preference one method
of teaching math versus the other. The result is a varied and uncertain policy landscape surrounding this question.
Still, Center for Education Market Dynamics (CEMD) data reveals from our recent high school math market report that some U.S. districts are turning to Integrated Math approaches, and there are clear
geographic patterns where this is more likely to occur, as shown in the figure below.
In the state of California, for instance, districts in our sample are split roughly equally between AGA and Integrated curricula. And across the West more broadly, the integrated approach appears
significantly more common than it is nationally. Even if we separate out California, as we did here, 12% of districts in the West have an integrated curriculum in place, compared to just 4% in both
the South and the Northeast.
This finding is notable because several southern states (e.g., Georgia and North Carolina) have tried to drive uptake of the integrated approach through policy mandates and/or incentives. Many of the
western states, where Integrated Math is appreciably more popular, have not. This suggests that to the extent we do see an embrace of Integrated Math nationally, the mechanism is both grassroots and
regionally contagious.
Want more K-12 education market insights like this one? Be sure to check out CEMD’s latest reports, blogs, and resources. | {"url":"https://www.cemd.org/reflecting-on-integrated-math-why-is-it-rare-in-the-us-and-where-is-it-gaining-ground/","timestamp":"2024-11-09T17:18:42Z","content_type":"text/html","content_length":"43506","record_id":"<urn:uuid:12902691-be57-4048-86f2-36158eee294d>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00627.warc.gz"} |
Classical group
From Encyclopedia of Mathematics
2020 Mathematics Subject Classification: Primary: 20G [MSN][ZBL]
A classical group is the group of automorphisms of some sesquilinear form $f$ on a right $K$-module $E$, where $K$ is a ring; here $f$ and $E$ (and sometimes $K$ as well) usually satisfy extra
conditions. There is no precise definition of a classical group. It is supposed that $f$ is either the null form or is a non-degenerate reflexive form; sometimes $E$ is taken to be a free module of
finite type. Often one means by classical groups other groups closely related to groups of automorphisms of forms (for example, their commutator subgroups or quotients with respect to the centre) or
some of their extensions (for example, groups of semi-linear transformations of $E$ preserving $f$ up to a scalar factor and an automorphism of $K$).
Classical groups are closely related to geometry: They can be characterized as groups of those transformations of projective spaces (and also of certain varieties related to Grassmannians, see [Di])
that preserve the natural incidence relations. For example, according to the fundamental theorem of projective geometry, the group of all transformations of $n$-dimensional projective space $P$ over
a skew-field $K$ that preserve collinearity coincides for $n\ge 3$ with the classical group of all projective collineations of $P$. For this reason, the study of the structure of a classical group
has a geometrical meaning; it is equivalent to the study of the symmetries (automorphisms) of the corresponding geometry.
The theory of classical groups has been developed most profoundly for the case when $K$ is a skew-field and $E$ is a vector space of finite dimension $n$ over $K$. From now on, these conditions will
be assumed to hold. Then the groups of the following series (to be described below) are usually called classical: $\def\GL{ {\rm GL}}\GL_n(K)$, $\def\SL{ {\rm SL}}\SL_n(K)$, $\def\Sp{ {\rm Sp}}\Sp_n
(K)$, $\def\O{ {\rm O}}\O_n(K,f)$, $\def\U{ {\rm U}}\U_n(K,f)$.
1) Let $f$ be the null form. Then the group of all automorphisms of $f$ is the same as the group of all automorphisms of $E$ (that is, bijective linear mappings from $E$ into $E$); it is denoted by $
\GL_n(K)$ and is called the general linear group in $n$ variables over the skew-field $K$, sometimes the full linear group. The subgroup of $\GL_n(K)$ generated by all transvections (cf. Transvection
) is denoted by $\SL_n(K)$ and is called the special linear group (or unimodular group) in $n$ variables over the skew-field $K$. It is the same as the set of automorphisms with determinant $1$.
2) Let $f$ be a non-degenerate sesquilinear form (with respect to an involution $J$ of $K$) for which the orthogonality relation is symmetric, that is
$$f(x,y) = 0 \implies f(y,x) = 0.$$ Such a form is called reflexive. The group $\U_n(K,f)$ of all automorphisms of $K$ is called the unitary group in $f$ variables over the skew-field $K$ with
respect to the form $f$. There are only two possibilities: Either $K$ is a field, $J=1$ and $f$ is a skew-symmetric bilinear form, or by multiplying $f$ by a suitable scalar and altering $J$, one can
arrange for $f$ to be a Hermitian or skew-Hermitian form. For a skew-symmetric form $f$, $\U_n(K,f)$ is called the symplectic group in $n$ variables over the skew-field $K$ (if ${\rm char} K = 2$ one
must suppose that $f$ is an alternating form); it is denoted by $\Sp_n(K)$. This notation does not include $f$ because all non-degenerate alternating forms on $E$ are equivalent and define isomorphic
symplectic groups. In this case $n$ is even. For Hermitian and skew-Hermitian forms, there is the special case that $K$ is a field of characteristic different from 2, $J=1$ and $f$ is a symmetric
bilinear form. Then $\U_n(K,f)$ is called the orthogonal group in $n$ variables over the field $K$ with respect to the form $f$; it is denoted by $\O_n(K,f)$. Orthogonal groups can also be defined
for fields of characteristic 2 (see [Di]). Often the term "unitary group" is used in a narrower sense for groups $\U_n(K,f)$ that are neither orthogonal nor symplectic, that is, groups corresponding
to non-trivial involutions $J$.
Associated with each of the fundamental series of classical groups are their projective images $\def\PGL{ {\rm PGL}}\PGL_n(K)$, $\def\PSL{ {\rm PSL}}\PSL_n(K)$, $\def\PSp{ {\rm PSp}}\PSp_n(K)$, $\def
\PO{ {\rm PO}}\PO_n(K)$, $\def\PU{ {\rm PU}}\PU_n(K)$; these are the quotient groups of them by the intersections with the centre $Z_n$ of $\PGL_n(K)$. The group
$$\O_n^+(K,f)=\O_n(K,f)\cap \SL_n(K),$$ the commutator subgroup $\def\Om{\Omega}\Om_n(K,f)$ of $\O_n(K,f)$, the group
$$\U_n^+(K,f)=\U_n(K,f)\cap \SL_n(K),$$ and their projective images are also associated with the series of orthogonal and unitary classical groups, respectively.
The classical approach to the theory of classical groups aims at the elucidation of their algebraic structure. This reduces to the description of a normal series of subgroups and their successive
quotient groups (in particular a description of normal subgroups and simple composition factors), the description of the automorphisms and isomorphisms of the classical groups (and, more generally,
of the homomorphisms), the description of the various types of generating sets and their relations, etc. The main results on the structure of groups of type $\GL_n(K)$ and $\SL_n(K)$ are the
following. The commutator subgroup of $\GL_n(K)$, $n\ge 2$, is $\SL_n(K)$, except in the case $n=2$, $K=\F_2$ (where $\F_q$ is the field of $q$ elements). The centre $Z_n$ of $\GL_n(K)$ consists of
all homotheties $x\mapsto x\def\a{\alpha}\a$, where $\a$ is an element of the centre of $K^*$. There is a normal series of subgroups
$$\GL_n(K) \supset \SL_n(K) \supset \SL_n(K)\cap Z_n \supset \{1\}$$ The group $\GL_n(K)/\SL_n(K)$ is isomorphic to $K^*/C$, where $K^*$ is the multiplicative group of the skew-field $K$ and $C$ is
its commutator subgroup. The group $\SL_n(K)\cap Z_n$ is the centre of $\SL_n(K)$ and the quotient group
$$\SL_n(K)/(\SL_n(K)\cap Z_n) = \PSL_n(K)$$ is simple in all cases except when $n=2$, $K=\F_2$ or $\F_3$. For further details see General linear group; Special linear group; Symplectic group;
Orthogonal group; Unitary group. The structure of a classical group depends essentially on its type, the skew-field $K$, the properties of the form $f$, and $n$. For some types of classical groups a
very detailed description is available. For others there are still open questions. (These involve mainly groups of type $\U_n(K,f)$ where $f$ is an anisotropic form.) Typical for the structure theory
of classical groups are assertions that hold for almost-all $K$, $f$ and $n$, and the investigation of the various exceptional cases when these assertions are false. (Such exceptions arise for
instance for small values of $n$, for finite fields $K$ of small order or for special values of the index of the form $f$.)
The question of isomorphisms of classical groups occupies a special position. First there are the standard isomorphisms. These are isomorphisms between $G(n,K,f)$ and $G'(n',K',f')$ the definition of
which does not depend on special properties of $K$ (except, perhaps, its commutativity). All other isomorphisms are called non-standard. For example, there is a (standard) isomorphism from $\Sp_2(K)$
onto $\SL_2(K)$, where $K$ is any field, or from $\U_2^+(K,f)$ onto $\SL_2(K_0)$, where $K$ is any field, $J\ne 1$, $f$ is a form of index 1, and $K_0$ is the field of invariants of $J$. For a
detailed description of the known standard isomorphisms, see [Di], [BoMo]. Examples of non-standard isomorphisms are:
$$\PSL_2(\F_4) \cong \PSL_2(\F_5),\qquad \PSL_2(\F_7)\cong\PSL_3(\F_2),$$
$$\PSp_4(\F_3) \cong \PU_4^+(\F_4).$$ It is also known that the groups $\PSL_n(K)$ and $\PSL_m(K')$, $n,m\ge 2$, can be isomorphic only when $n=m$, apart from the case
$$\PSL_2(\F_7) \cong \PSL_3(\F_2);$$ when $m=n>2$, isomorphism is possible only if $K$ and $K'$ are isomorphic or anti-isomorphic; this is also the case when $m=n=2$ if $K$ and $K'$ are fields, apart
from the case
$$\PSL_2(\F_4) \cong \PSL_2(\F_5).$$ The groups $\PSp_n(K)$ and $\PSp_m(K')$ can be isomorphic only if $n=m$ and $K=K'$, apart from the case $m=n=2$, $K=\F_4$, $K'=\F_5$. There are no other
isomorphisms among the groups $\PSL_n(K)$, $\PSp_n(K)$, ${\rm P}\Om_q(K,f)$ (where $K$ is a finite field) apart from the ones indicated above.
The results listed above on the structure of classical groups and their isomorphisms are obtained by methods of linear algebra and projective geometry. The basis for this consists in the study of
special elements in the classical groups and the geometric properties of them, principally the study of transvections, involutions and planar rotations. Subsequently, methods of the theory of Lie
groups and algebraic geometry were introduced into the theory of classical groups, whereupon the theory of classical groups became much related with the general theory of semi-simple linear algebraic
groups in which classical groups appear as forms (cf. Form of an algebraic group): Every form of a simple linear algebraic group over a field $K$ of classical type (that is, of type $A_n$, $B_n$,
$C_n$, or $D_n$) gives rise to a classical group, the group of its $K$-rational points (an exception being a form of $D_4$ connected with an outer automorphism of order three). In the case when $K$
is $\R$ or $\C$, a classical group is naturally endowed with a Lie group structure, and for $p$-adic fields with a $p$-adic analytic group structure. This makes it possible to use topological methods
in the study of such classical groups, and conversely, to obtain information on the topological structure of the underlying variety of a classical group (for example, on its finite cellular
decompositions) from the knowledge of its algebraic structure.
In the more general situation when $E$ is a module over a ring $K$ the results on classical groups are not so exhaustive (see [BoMo]). Here the theory of classical groups links up with algebraic
[Ar] E. Artin, "Geometric algebra", Interscience (1957) MR1529733 MR0082463 Zbl 0077.02101
[Bo] N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms", 2, Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French) MR2333539 MR2327161 MR2325344 MR2284892 MR2272929
MR0928386 MR0896478 MR0782297 MR0782296 MR0722608 MR0682756 MR0643362 MR0647314 MR0610795 MR0583191 MR0354207 MR0360549 MR0237342 MR0205211 MR0205210
[BoMo] A. Borel (ed.) G.D. Mostow (ed.), Algebraic groups and discontinuous subgroups, Proc. Symp. Pure Math., 9, Amer. Math. Soc. (1966) MR0202512 Zbl 0171.24105
[BoTi] A. Borel, J. Tits, "Homomorphisms "abstraits" de groupes algébriques simples" Ann. of Math. (2), 97 (1973) pp. 499–571 Zbl 0202.03202
[Di] J.A. Dieudonné, "La géométrie des groups classiques", Springer (1955) Zbl 0221.20056
[OM] O.T. O'Meara, "A survey of the isomorphism theory of the classical groups" , Ring theory and algebra, 3 , M. Dekker (1980) pp. 225–242 Zbl 0438.20033
[We] A. Weil, "Algebras with involutions and the classical groups" J. Ind. Math. Soc., 24 (1960) pp. 589–623 MR0136682 Zbl 0109.02101
Instead of [BoMo] one may consult [BoTi], [OM], [We].
How to Cite This Entry:
Classical group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Classical_group&oldid=35101
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article | {"url":"https://encyclopediaofmath.org/wiki/Classical_group","timestamp":"2024-11-05T03:31:33Z","content_type":"text/html","content_length":"30469","record_id":"<urn:uuid:8e67771b-0f5e-4f42-b5b3-3d85c2463b79>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00844.warc.gz"} |
The Scientific Legacy of Poincarésearch
Item Successfully Added to Cart
An error was encountered while trying to add the item to the cart. Please try again.
Please make all selections above before adding to cart
The Scientific Legacy of Poincaré
Étienne Ghys : École Normale Supérieure de Lyon, Lyon, France
Annick Lesne : Université Pierre et Marie Curie, Paris, France
Translated by Joshua Bowman
A co-publication of the AMS and London Mathematical Society
Hardcover ISBN: 978-0-8218-4718-3
Product Code: HMATH/36
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
eBook ISBN: 978-1-4704-1807-6
Product Code: HMATH/36.E
List Price: $120.00
MAA Member Price: $108.00
AMS Member Price: $96.00
Hardcover ISBN: 978-0-8218-4718-3
eBook: ISBN: 978-1-4704-1807-6
Product Code: HMATH/36.B
List Price: $245.00 $185.00
MAA Member Price: $220.50 $166.50
AMS Member Price: $196.00 $148.00
Click above image for expanded view
The Scientific Legacy of Poincaré
Étienne Ghys : École Normale Supérieure de Lyon, Lyon, France
Annick Lesne : Université Pierre et Marie Curie, Paris, France
Translated by Joshua Bowman
A co-publication of the AMS and London Mathematical Society
Hardcover ISBN: 978-0-8218-4718-3
Product Code: HMATH/36
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
eBook ISBN: 978-1-4704-1807-6
Product Code: HMATH/36.E
List Price: $120.00
MAA Member Price: $108.00
AMS Member Price: $96.00
Hardcover ISBN: 978-0-8218-4718-3
eBook ISBN: 978-1-4704-1807-6
Product Code: HMATH/36.B
List Price: $245.00 $185.00
MAA Member Price: $220.50 $166.50
AMS Member Price: $196.00 $148.00
• History of Mathematics
Volume: 36; 2010; 391 pp
MSC: Primary 30; 34; 37; 53; 32; 57; 16; 83; 00; 35; 01
Henri Poincaré (1854–1912) was one of the greatest scientists of his time, perhaps the last one to have mastered and expanded almost all areas in mathematics and theoretical physics. He created
new mathematical branches, such as algebraic topology, dynamical systems, and automorphic functions, and he opened the way to complex analysis with several variables and to the modern approach to
asymptotic expansions. He revolutionized celestial mechanics, discovering deterministic chaos. In physics, he is one of the fathers of special relativity, and his work in the philosophy of
sciences is illuminating.
For this book, about twenty world experts were asked to present one part of Poincaré's extraordinary work. Each chapter treats one theme, presenting Poincaré's approach, and achievements, along
with examples of recent applications and some current prospects. Their contributions emphasize the power and modernity of the work of Poincaré, an inexhaustible source of inspiration for
researchers, as illustrated by the Fields Medal awarded in 2006 to Grigori Perelman for his proof of the Poincaré conjecture stated a century before.
This book can be read by anyone with a master's (even a bachelor's) degree in mathematics, or physics, or more generally by anyone who likes mathematical and physical ideas. Rather than
presenting detailed proofs, the main ideas are explained, and a bibliography is provided for those who wish to understand the technical details.
Undergraduate students, graduate students, and research mathematicians interested in Poincaré's life and work.
□ Chapters
□ Introduction
□ Poincaré and his disk
□ Differential equations with algebraic coefficients over arithmetic manifolds
□ Poincaré and analytic number theory
□ The theory of limit cycles
□ Singular points of differential equations: On a theorem of Poincaré
□ Periodic orbits of the three body problem: Early history, contributions of Hill and Poincaré, and some recent developments
□ On the existence of closed geodesics
□ Poincaré’s memoir for the Prize of King Oscar II: Celestial harmony entangled in homoclinic intersections
□ Variations on Poincaré’s recurrence theorem
□ Low-dimensional chaos and asymptotic time behavior in the mechanics of fluids
□ The concept of “residue" after Poincaré: Cutting across all of mathematics
□ The proof of the Poincaré conjecture, according to Perelman
□ Henri Poincaré and the partial differential equations of mathematical physics
□ Poincaré’s calculus of probabilities
□ Poincaré and geometric probability
□ Poincaré and Lie’s third theorem
□ The Poincaré group
□ Henri Poincaré as an applied mathematician
□ Henri Poincaré and his thoughts on the philosophy of science
□ The articles are very well written, indeed, and are of course autonomous. But even non-specialists will want to sample these wares. The mathematics is presented clearly and very accessible,
and the numerous historical accounts and asides make add an additional welcome cultural element to whole experience.
[This book] is bound to be a hit across the mathematical spectrum: it has something for every one interested in any aspect of Poincaré's work, which is to say, something for every one.
MAA Reviews
• Book Details
• Table of Contents
• Additional Material
• Reviews
• Requests
Volume: 36; 2010; 391 pp
MSC: Primary 30; 34; 37; 53; 32; 57; 16; 83; 00; 35; 01
Henri Poincaré (1854–1912) was one of the greatest scientists of his time, perhaps the last one to have mastered and expanded almost all areas in mathematics and theoretical physics. He created new
mathematical branches, such as algebraic topology, dynamical systems, and automorphic functions, and he opened the way to complex analysis with several variables and to the modern approach to
asymptotic expansions. He revolutionized celestial mechanics, discovering deterministic chaos. In physics, he is one of the fathers of special relativity, and his work in the philosophy of sciences
is illuminating.
For this book, about twenty world experts were asked to present one part of Poincaré's extraordinary work. Each chapter treats one theme, presenting Poincaré's approach, and achievements, along with
examples of recent applications and some current prospects. Their contributions emphasize the power and modernity of the work of Poincaré, an inexhaustible source of inspiration for researchers, as
illustrated by the Fields Medal awarded in 2006 to Grigori Perelman for his proof of the Poincaré conjecture stated a century before.
This book can be read by anyone with a master's (even a bachelor's) degree in mathematics, or physics, or more generally by anyone who likes mathematical and physical ideas. Rather than presenting
detailed proofs, the main ideas are explained, and a bibliography is provided for those who wish to understand the technical details.
Undergraduate students, graduate students, and research mathematicians interested in Poincaré's life and work.
• Chapters
• Introduction
• Poincaré and his disk
• Differential equations with algebraic coefficients over arithmetic manifolds
• Poincaré and analytic number theory
• The theory of limit cycles
• Singular points of differential equations: On a theorem of Poincaré
• Periodic orbits of the three body problem: Early history, contributions of Hill and Poincaré, and some recent developments
• On the existence of closed geodesics
• Poincaré’s memoir for the Prize of King Oscar II: Celestial harmony entangled in homoclinic intersections
• Variations on Poincaré’s recurrence theorem
• Low-dimensional chaos and asymptotic time behavior in the mechanics of fluids
• The concept of “residue" after Poincaré: Cutting across all of mathematics
• The proof of the Poincaré conjecture, according to Perelman
• Henri Poincaré and the partial differential equations of mathematical physics
• Poincaré’s calculus of probabilities
• Poincaré and geometric probability
• Poincaré and Lie’s third theorem
• The Poincaré group
• Henri Poincaré as an applied mathematician
• Henri Poincaré and his thoughts on the philosophy of science
• The articles are very well written, indeed, and are of course autonomous. But even non-specialists will want to sample these wares. The mathematics is presented clearly and very accessible, and
the numerous historical accounts and asides make add an additional welcome cultural element to whole experience.
[This book] is bound to be a hit across the mathematical spectrum: it has something for every one interested in any aspect of Poincaré's work, which is to say, something for every one.
MAA Reviews
You may be interested in...
Please select which format for which you are requesting permissions. | {"url":"https://bookstore.ams.org/HMATH/36","timestamp":"2024-11-05T02:18:44Z","content_type":"text/html","content_length":"128555","record_id":"<urn:uuid:d8015935-ca62-4f45-bdbc-02dcdc425635>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00378.warc.gz"} |
David Smith
Teaching at GVSU
My experience at GVSU is pretty well summarized by this complaint from an anonymous student of linear algebra and differential equations, conveyed to me by the mathematics department.
"Up until the very last day of class we were turning in new homework and learning new material..."
Go on.
"... but at what cost?"
In the final weeks we discussed systems of linear differential equations, the Laplace transform, and power series solutions to differential equations. These are important topics for engineering
students. Better to see them, if only briefly, than to review earlier topics, for instance. (Is this what you were expecting?) I understand this is a lot to digest in a short time, and accordingly I
trimmed the homework and examination scopes. You can find in the textbook many parts of the story I left out. There's no rule against learning in the last week of class. This is regularly scheduled
course time, not finals week or a study period.
Sensible or not, negative student reviews hurt you at GVSU. I was never told how they were weighted, but mine were presented as evidence of poor teaching in performance reviews. As you might expect,
you then find professors telling their students explicitly about the behavioral norms of university-level mathematics. What's surprising is the constancy of this communication. It displaces the
discussion of actual mathematics. A colleague who could be heard speaking to students at office hours spent most of those hours on this, it seemed to me.
Here is how I could be a more effective teacher of linear algebra for mathematics majors:
"Less proofs and more linear algebra."
Proofs are important in mathematics. Consider also that they are not so different from calculation. A typical question in linear algebra is whether a set of vectors is independent. It is practically
the same exercise to determine whether a set is independent as to prove that a set is independent. Further, if your teacher proves that a set is independent whenever some condition holds (e.g. the
determinant of some associated matrix is nonzero), then at the very least you have seen how to solve a large class of problems.
These anonymous surveys were rarely useful to me as a teacher. A student of linear algebra and differential equations wrote:
"Took his knowledge for granite and assumed we knew things."
Yes, after Calculus I, II, and III, you are assumed to know things. If you are not responsible for prerequisite material, there is little time for linear algebra and differential equations. Still,
did I not often show the steps of these methods in class to refresh your memory?
As for my knowledge, I thank my parents first. My mother taught me to read. I had years of education mostly in public schools, then went to college, where I worked hard just to pass. In the 1992
season, the AFC champions were the Buffalo Bills. The NFC champions were the Dallas Cowboys. They met in 1993 for Superbowl XXVII at the Rose Bowl in Pasadena, California. I was unaware until now,
when I searched the Internet for things that happened during my freshman year at Caltech. I can tell you in great detail the geometry of my room and desk and of the cup I used to carry water there
from the sink.
If you are really smart, maybe you can get a PhD in mathematics without much work, but I studied hard. In graduate school you encounter setbacks of kinds you never knew existed. Contact hours with my
professors and students were far outnumbered by contact hours with desk chairs.
I try to imitate my best professors, especially those at CSULB. You have to admire people who dedicate their lives to a subject as challenging as mathematics and spend hours each week explaining it
to you. I hear students complain about the foreign accents of their instructors, but let's face it, it's the mathematics that's difficult. What does it mean that I take my knowledge for granted? I
have no idea. Here is a way to improve my course:
"Have it taught by an actual math professor from the math department."
Hear that, administrators? They want me to have tenure. Seriously, I was an actual math professor from the math department.
Some students wrote nicer things. In closing, here are the complete comments of one student. Ways in which the instructor was effective:
"Dr. Smith is one of the best instructors at GVSU he expects a lot from his students but no more that should be expected at a 300 level class. He is very reasonable and flexible and really cares
about student learning. His tests were easier than the homework & examples in class as they should be. Anyone complain, about the difficulty of this class is probably just upset that he actually
expected them to learn."
Thank you. Ways the instructor could be more effective:
"Dr. Smith does need to return graded tests more quickly."
Many students wanted this. I don't know what else I could have done but to lower the quality (and fairness) of the grading. Even that would not have helped much. I had more pressing tasks.
"Also, the way he leaves the overhead on and uses the board around it is awkward and sometimes difficult to follow."
Sorry about that. Ways the course could be improved:
"The course is actually very well designed. Too bad its not longer."
Yes, it's a lot of linear algebra and differential equations for one course. You finished, and you can be proud of that! Good luck in your future studies. | {"url":"https://dacvs.neocities.org/teaching","timestamp":"2024-11-02T10:33:04Z","content_type":"text/html","content_length":"6670","record_id":"<urn:uuid:7dbb18ee-58ba-4ce8-b570-8d9d3366f00a>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00680.warc.gz"} |
One Rep Max Calculator
Use This Calculator To Calculate 1 Rep Max.
Enter The Weight You Lift During Bench Press in Kg :
Using The Above Weight How Many Reps You Performed :
This Section Explains the One Rep Max Calculator.
The One rep max calculator is used to reach the maximum strength of any individual. It is calculated using the number of repetitions of an exercise and the amount of weight lifted. Usually One Rep
Max is calculated using the bench press exercise but in general any exercise can be used.
Using one rep max one can directly get the best figure without having to try and perform the one rep max lift which consists of single repetitions. While calculating your one rep max, it is
preferable to load the weights so that you are able to perform around 10 repetitions. The one rep max is based on the exact no of repetitions. One can do less or more than the one rep max.
Once you have calculated your One rep max, you can use the Rep Max Calculator to find out the appropriate load when you have decided upon the number of Reps you plan to do. For example if you know
that your one rep max is 100 kg, and you plan to perform 10 reps using 100 kg of weight, the Rep Max Calculator would instantly let you know that you should lift 133 KGs. | {"url":"https://calculatorschool.com/calculator.aspx?calcid=60&catid=3","timestamp":"2024-11-10T01:58:54Z","content_type":"text/html","content_length":"89629","record_id":"<urn:uuid:7531418b-071a-4531-972f-81ac540d9124>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00409.warc.gz"} |
Limits of math
FQXi essay
is now being publicly discussed and rated, as contest submissions are closed. I am not sure I addressed
these objections
fully enough. The argument is:
Using mathematics to model physical reality cannot be a limitation because mathematics allows arbitrary structures, operations, and theories to be defined.
No matter what new phenomena we discover, it will always be possible to describe our observations mathematically. It doesn’t matter how crazy nature is because mathematics is just a language for
expressing our intuitions, and we can always add words to the language.
This is an incorrect view of math.
Math has been axiomatized as
Zermelo–Fraenkel set theory
(ZFC). The content of math consists of the logical theorems of ZFC. It is not just some language or some arbitrarily expandible set of intuitions.
I wonder whether many physicists understand this point. I doubt it.
I thought that everyone understood that math has limits, and the point of my essay was to argue that there might be no ultimate theory of physics within those limits. But if someone does not even
accept that math has limits, then the rest of the argument is hopeless.
If you agree that math has limits and that they may not include possible physical theories, then it ought to be obvious that nature may not have a faithful mathematical representation. Saying that
there is such a representation is an assumption that may be unwarranted.
So I guess I should have explained the ZFC issue better to the physicists who will be judging my essay.
An example of a limit of math is the unsolvability of
quintic polynomials
by radicals. That is what keeps us from having something like the quadratic formula for more complicated equations. This fact does not necessarily stop us from solving the equations, but certain
kinds of formulas just won't work.
My essay "public rating" is not very high, but my "community rating" must be much higher. If you look at the
list of essays sorted by community rating
, then you will see that mine is near the top.
Update: Bob Jones argues below that ZFC is not enough because we need mathematical constructs like the
Grothendieck universe
that are outside ZFC. That is an interesting example, as noted
Colin Mclarty says that the big modern cohomological number theory theorems, including Fermat’s Last Theorem, were all proved using Grothendieck’s tools, making use of an axiom stronger than
Zermelo–Fraenkel set theory (ZFC). He says that there is a belief that these theorems can be proved in ZFC, but no one has done it.
I would really be surprised if cohomological number theory needs something more than ZFC. And it would be even more amazing if theoretical physics found some mathematization of the universe that
could be formalized in an extension of ZFC, but not ZFC.
8 comments:
1. Bob JonesSeptember 10, 2012 at 8:42PM
"The content of math consists of the logical theorems of ZFC."
In a similar way, a mathematician living in ancient Greece could have defined math to consist of those statements provable from Euclid's axioms. Back then, such a definition might have made
sense, but today we know that there's much more to math than Euclidean geometry. For example, we now have non-Euclidean geometry, which formalizes geometric intuitions that the Greeks never
considered. Non-Euclidean geometry is also useful for describing the physical world.
"It is not just some language or some arbitrarily expandible set of intuitions."
You can define mathematics however you like, but this statement does not reflect how mathematics is developed and applied to physical problems. Mathematicians and physicists are rethinking the
foundations of their subjects all the time. In the twentieth century, mathematicians began to develop new approaches to mathematics that did not assume the law of excluded middle, one of the
basic assumptions of classical mathematics. Some physicists have experimented with new logical foundations for quantum mechanics based on topos theory. While these are not ideas that I personally
find interesting or compelling, they show that it is at least conceivable that physics might require radically new mathematical ideas.
"I thought that everyone understood that math has limits"
Well, certainly math does have limits. For example, we never study things that are vague, ill-defined, or contradictory. But that can't stop us from constructing a faithful mathematical
representation of physics because physical observables are always precisely defined, and it makes no sense to say that our observations of nature are contradictory. It's always possible to
observe something confusing (like quantum particles), but we can always construct a consistent framework in which those observations can be understood (like the theory of Hilbert spaces and
linear operators).
"An example of a limit of math is the unsolvability of quintic polynomials by radicals."
This is not a "limit" of math but a reflection of the fact that not everything can be true. There's nothing extraordinary about the fact that you can't have a "quadratic formula" for the roots of
a quintic polynomial. There are also systems of linear equations that have no solutions. The unsolvability of the general quintic is a necessary consequence of the consistency of mathematics, and
if physics is based on some mathematical framework in which this result is true, then you can expect nature to respect the unsolvability of the quintic.
2. You make some good points, but I think that even the ancient Greeks would have understood that there is more to math than Eudlidean geometry, even if they did not anticipate curved geometry.
Yes, mathematicians have tried denying the law of excluded middle, and taken some other unusual approaches. Some of these approaches have some merit. But there is a broad consensus about what
math is all about. I really don't think that math is going to be reinvented to explain theoretical physics.
3. Bob JonesSeptember 10, 2012 at 10:22PM
"I really don't think that math is going to be reinvented to explain theoretical physics."
I'm not so sure. Higher categories show up in several parts of mathematics (like homotopy theory and derived algebraic geometry) and mathematical physics (topological and conformal field theory
and string theory), and such categories are already outside of the foundation that you're using to define mathematics. You may argue that we can supplement ZFC with additional axioms for proper
classes, but I don't think the problem can be solved so easily. Developing a natural foundation for category theory is a difficult problem, and I think it's fair to say that it has not been
solved in a satisfactory way.
4. No, I don't believe that znything beyond ZFC is needed for any of those things.
5. Bob JonesSeptember 11, 2012 at 12:48PM
From a practical point of view, if you want to study large categories, then yes, you do need more than just ZFC. It is possible to incorporate classes into ZFC through some ad hoc device like
defining a class to be a collection of sets of sets defined by some formula in the language of set theory, but this is a very unnatural way of thinking about classes. It's much better to use
Grothendieck universes or work in an axiomatic framework that distinguishes between large and small collections.
In any case, if you're trying to argue that nature has no faithful representation in ZFC, then you should agree with me when I tell you that ZFC is limited...
6. You're implying (among other things) that ZFC is a choice, necessary for "good" math and that "bad math" is an oxymoron -- whereas "bad physics is still physics". Pardon my laymanship, I jumped
in sans reading comments. Wanted to do so, then get caught up and check back.
7. Or: Math must be revealed / discovered, while all of physics is by way of contrast constructed. Bad math turns out "not to be math" and can lead to "bad" physics? (I'm working on it). Primarily,
though, I just thought I'd weigh in on Bob's comment "we never study things that are vague, ill-defined, or contradictory" -- actually, my dad got a Ph.D in Math and I believe he taught me
otherwise. He was speaking to us one day about mathematicians being at odds over a definition of "neighborhood" (set theory). And (my words, not his): some problems seem intractable and/or
ill-posed, but as-yet-interim "solutions" are still ... "mathy"?
8. Apologies for one last "bad" comment. I haven't resolved anything, but due to the seriousness of all the commentary I thought I'd share a bad pun which came to mind re the topic. A bit of a
stretch ... well, perhaps a giant stretch. Preface: "a-wild" signifies "natural" and "woice" Russian accent for "voice", voice implying choice as in "he voiced his (chosen) opinion" "choice"
implies "artificial" i.e. not revealed or discovered f/ nature. Ahem (like I said, stretch).
" ZFC ... it's not a-wild, it's a Woice." :-0 | {"url":"http://blog.darkbuzz.com/2012/09/limits-of-math.html","timestamp":"2024-11-07T15:30:07Z","content_type":"text/html","content_length":"125185","record_id":"<urn:uuid:385a56c1-3eae-497c-a3fa-4bcf5409bd66>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00354.warc.gz"} |
four quadrant
Quadrant - Definition, Sign Convention, Plotting point in quadrant ...
The Four Quadrants Model of High Growth - SOMAmetrics
Quadrant - Definition, Graph, Cartesian Plane, Signs
Content - The four quadrants
What Makes a Four-Quadrant Film? 10 Essential Elements - ScreenCraft
All Four Quadrants | Definition, Examples, Points, Signs, Plotting
Translating and reflecting in four quadrants - Maths - Learning ...
What Are the Four Quadrants? – Integral Life
Clarity or Aesthetics? Part 2 – A Tale of Four Quadrants – DataRemixed
The 4 Quadrants of Time Management Matrix | Week Plan
Content - The four quadrants
Quadrant - Definition, Sign Convention, Plotting point in quadrant ...
The four quadrants - KS3 Maths - BBC Bitesize
Quadrant (plane geometry) - Wikipedia
4 Quadrants (by Ken Wilber) – library of concepts
The Priority Quadrant: Manage Tasks Using the 4-Quadrant Method
Four Quadrant Time Management Model Diagram Ppt Slides
Boost Productivity with the Four Quadrants of Time Management
The 4 Quadrants of Time Management Matrix [Guide] - Timeular
The Eisenhower Matrix: How to Prioritize Your To-Do List [2024 ...
4 Quadrants Of Time Management Matrix Process Improvement Ppt ...
The Four Quadrants of Time Management
What is Four Quadrant Operation of DC Motor? - Speed Torque ...
What is Four Quadrant Metrology? Price
What is Quadrant? Definition, Coordinate Graphs, Sign, Examples
Getting Things Done: Four Quadrants and Setting Priorities ...
Leaf Shape Four Quadrant Model | Vier-Quadranten-Modell Template
The four quadrant model of organizational change by David Boddy and David Buchanan
Arrow Four Quadrant Model | Vier-Quadranten-Modell Template
Four Quadrants
Four quadrant DC motor operation Smart Motor Devices OÜ.
Four-Quadrant Grid Printables for 5th - 11th Grade | Lesson Planet | {"url":"https://worksheets.clipart-library.com/four-quadrant.html","timestamp":"2024-11-13T04:50:33Z","content_type":"text/html","content_length":"23207","record_id":"<urn:uuid:170e942d-9e80-431f-82d9-6cdf8aeb3882>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00210.warc.gz"} |
How to Make Fluid Dynamics Simulation on the GPU - Azoft
Fluid Dynamics Simulation on a Mobile Device
Founder & CEO at Azoft
Reading time:
17 Feb 2016
Many physical processes can be simulated using computers. In fact, physics simulations are widely used in a variety of industries: computer games, education, scientific research, engineering and
more. Even the flow of water can look very realistic on a computer screen. And considering the growing popularity of mobile devices today, it’s not surprising that we see an increasing demand for
hydrodynamic process simulation on mobile devices. Here, I will discuss imitation of paint dispersed on the surface of water, on an iPad.
To make such app look realistic, it is crucial to imitate real physics and what actually happens to the paint as it touches water. In order to do this, we decided to explore some basic hydrodynamics
and then make fluid dynamics simulation on an iPad. We are going to use real world formulas that describe water behavior. This process requires large volumes of calculations to be done very quickly.
Initially, we tried these calculations on a CPU, but could only produce 5 to 7 frames per second. As we know, in order for eyes to see continuous movement, we need at least 24 frames per second.
Clearly, we needed a way to process the data faster. That is why we decided to utilize the GPU in order to try and generate more images per second.
General Purpose Computing on a GPU
Even though the Graphics Processing Unit (GPU) was originally invented to process graphics, it soon became clear that GPUs can be used to carry out many other calculations. For example, GPUs are
commonly used in audio and video processing, augmented reality, mathematical calculations, hydrodynamics, cryptography, and many other fields. All these applications can be summed up by the term
General Purpose GPU or GPGPU.
As a general rule, the CPU is used to carry out large and complex operations, whereas the GPU is excellent at handling large numbers of simple tasks. When choosing which approach to use, it depends
on the task at hand and the amount of data to be processed. If we have a project that involves numerous calculations which are relatively independent of each other, using a GPU is a great idea. In
other words, these tasks can be considered parallel.
General Purpose Computing on a GPU is commonly done on PCs, but the real challenge lies when you want to set it up on a mobile device. Just like in the case of PCs, mobile devices have both CPUs and
GPUs. However, mobile devices are small in size and have obvious limitations. For example, mobile developers have to keep in mind the battery life. Also, when a computer processes massive amounts of
data, we need a good way of cooling the hardware. In case of an iPad, however, we can’t just add more fans and cooling devices, as we are limited by size and weight.
Today, mobile developers lack the proper tools when it comes to using the GPU for things other than graphics. In the case of desktops, we have tools like Nvidia CUDA, AMD FireStream, and Open CL,
that are all designed for general calculations on a GPU. For mobile devices, today we only have OpenGL ES, which is designed to handle graphics. That being said, it is possible to do general purpose
calculations on OpenGL ES 2.0 system. In our case, we’ll use OpenGL ES 2.0 to process data for a water-painting simulation.
Using OpenGL ES 2.0 for General Purpose Computing
OpenGL makes it possible to produce 2D and 3D graphics using a GPU. OpenGL takes advantage of the so-called ‘graphics pipeline’ to convert primitives (points, lines, etc.) into pixels. The idea
behind the pipeline is the following. First, we enter several vertices, thereby loading them into the pipeline. Then, a series of steps is performed with the data entered. As a result, we get an
image at the end of the pipeline. In our case, we’re not exactly working with graphics, so we have to do some additional programming to customize OpenGL functionality and carry out physics-related
When working with OpenGL, a developer can program only two things: the Vertex Shader and the Pixel Shader. Other steps of the pipeline are carried out automatically. These shaders are essentially two
small systems that can be programmed to accomplish a certain task. In our case, all calculations will take place in the Pixel Shader, so we’ll only be programming there. To summarize, the shaders
serve as a way to deliver the data to the GPU and ‘tell’ the GPU what calculations to perform.
The Pixel Shader works with four channels: R (red), G (green), B (blue), and A (alpha). In other words, independent calculations can be carried out in each one of these channels simultaneously.
Furthermore, the shaders of OpenGL can be programmed using a special programming language called OpenGL Shading Language (GLSL). It is a shading language that is based on C-programming language
syntax. It is very similar to ANSI C language, with some additional elements that allow working with vectors and matrices.
Simulation of Fluid Dynamics
There are numerous approaches to simulating the movement of water on a computer screen. One of them is the Lattice-Boltzman Method, which can be applied to this situation. This method looks at liquid
as if it were a collection of fictive particles. At the core of this method is a uniform rectangular grid or lattice, where every cell of the grid has several parameters. The parameters include
density of the particles, vector speed of the particles, and 9-speed channels. To illustrate this concept, let’s say we have a puddle of water and place an imaginary grid over it. Now, water
particles in every cell can be stationary, or they can be moving in a certain direction. If we’re dealing only with the surface of water, we only need to think two-dimensionally. However, keep in
mind that this method is often used for 3D simulations, in which case the number of channels would be significantly larger.
Every part of the liquid is represented by a small cell of our grid. Whenever we have some kind of movement in the water, we use real-world hydrodynamics formulas to predict the outcome. Any change
in the grid happens in two steps: the collision step and the streaming step. In the collision step, all calculation happen within the cell, so these calculations are independent of anything that
occurs elsewhere. Such calculations are great candidates for parallel processing, so they can be done on a GPU. The streaming step has to do with looking at what happens in nearby cells and applying
it to the cell of interest.
In the beginning, we have a disruption-free liquid surface, all particles are stationary. The particles are uniformly dispersed all across the surface, so we assign 1 for our density parameter.
Recall that we have four channels in the pixel shader, RGBA. The density parameter would be assigned to the R channel, x-speed to the G channel, and y-speed to the B channel. The A channel is not
Now, imagine someone is touching or moving the finger across the iPad screen. This supplies values for x and y coordinates (depending on which cell is being stimulated) as well as x and y speed (if
you’re moving your finger across the screen). So, we plug these values into a formula for collision step and calculate density in each speech channel. Next, we need to copy the values of
corresponding speed channels from neighboring cells. If there’s some activity in neighboring cells, it will spread to this cell. The concept is very similar to the way particles float on the surface
of a liquid.
To visualize this movement on the screen we can record the value for the vector of combined x and y speeds into the red channel. As a result, we will see red color being dispersed on the screen, just
like red paint would be dispersed in a container of water.
While the Lattice Boltzmann method proved to be a good fit for simulating paint on the surface of water, it cannot be applied to many other fluid dynamics simulations. The experience we gained while
working with Lattice Boltzmann method helped us address other similar projects. For example, we came up with another computational method based on the Navier-Stokes equations when developing
AquaReal, an iPad app that imitated watercolor painting. In order to imitate color blending realistically, we relied on Kubelka-Munk compositing model algorithm, though it had to be modified
significantly to solve all the challenges.
iPad users can view the results of our research by taking a look at the applications we’ve developed: AquaReal watercolor painting app, which I mention above, as well as WaterHockey game. Both
applications are available for free at the App Store.
Simulation of Multi-Component, Multi-Phase Fluids Using LBM
Simulating and modeling a two-component viscous fluid is based on a fairly common approach, which was first proposed by Xiaowen Shan and Hudong Chen in 1994. The main idea of their approach is to
introduce an additional force that acts on fluid particles. In our case, it provides additional components which modify the speed in each lattice site. These components are calculated as a function
of liquid density in the neighboring sites, multiplied by a constant value (potential for interaction), which defines the amplitude of interaction forces. At the same time, we can still take
advantage of the fact that it is relatively easy to set-up parallel processing and carry out calculations on a GPU.
The resulting simulation of paint dispersion in water looks very realistic. To test if this approach works, we tried it on an iPad 2 and set-up our grid to be 256×256 pixels. In the end, we were able
to produce 38 frames per second, as opposed to the original 7 frames that were produced by the CPU. This illustrates the benefits of using a GPU to carry out parallel tasks.
No doubt, the method described above could have been simplified if OpenCL framework was supported by our mobile device. After all, OpenCL framework is specifically designed for general purpose
calculation on a GPU. We hope that OpenCL will someday become available on iOS devices as it would simplify working with the GPU. But for the time being, OpenCL isn’t an option so we have to rely on
OpenGL to carry out general purpose computing on GPU of a mobile device. | {"url":"https://www.azoft.com/blog/hydrodynamic-process-simulation/","timestamp":"2024-11-01T19:54:58Z","content_type":"text/html","content_length":"82113","record_id":"<urn:uuid:0cdfa367-fb89-4929-bddf-cdb142998bda>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00318.warc.gz"} |
BETA.DIST Function: Definition, Formula Examples and Usage
BETA.DIST Function
If you’re looking for a way to calculate the probability of an event occurring in a beta distribution, the BETA.DIST formula in Google Sheets can help. This powerful formula allows you to easily
calculate the probability of an event occurring in a beta distribution based on a specified alpha and beta value. In this blog post, we’ll take a closer look at how the BETA.DIST formula works and
how you can use it in your own Google Sheets to perform calculations on beta distributions.
The BETA.DIST formula is a great tool for anyone who needs to work with beta distributions, whether you’re a statistician, a data analyst, or just someone who needs to perform calculations on
probabilities. This formula is easy to use and can help you quickly and accurately calculate the probability of an event occurring in a beta distribution. Whether you’re working with large datasets
or just need to perform a few simple calculations, the BETA.DIST formula can help you get the job done.
Definition of BETA.DIST Function
The BETA.DIST function in Google Sheets is a statistical function that calculates the probability of an event occurring in a beta distribution based on a specified alpha and beta value. This function
takes four arguments: the value for which you want to calculate the probability, the alpha value, the beta value, and a cumulative value that specifies whether the probability should be calculated
for all values less than or equal to the specified value (TRUE) or for all values strictly less than the specified value (FALSE). The BETA.DIST function is commonly used in statistical analysis and
can be combined with other functions in Google Sheets to perform more advanced calculations on beta distributions.
Syntax of BETA.DIST Function
The syntax of the BETA.DIST function in Google Sheets is as follows:
BETA.DIST(x, alpha, beta, cumulative)
• x: This is the value for which you want to calculate the probability.
• alpha: This is the alpha value of the beta distribution.
• beta: This is the beta value of the beta distribution.
• cumulative: This is a logical value that specifies whether the probability should be calculated for all values less than or equal to the specified value (TRUE) or for all values strictly less
than the specified value (FALSE).
For example, the function =BETA.DIST(0.5, 2, 3, TRUE) would calculate the probability of an event occurring in a beta distribution with an alpha value of 2 and a beta value of 3 for all values less
than or equal to 0.5. This probability would be returned as a decimal value.
Examples of BETA.DIST Function
Here are three examples of how to use the BETA.DIST function in Google Sheets:
1. Calculate the probability of an event occurring in a beta distribution for all values less than or equal to a specified value:
=BETA.DIST(0.5, 2, 3, TRUE)
2. Calculate the probability of an event occurring in a beta distribution for all values strictly less than a specified value:
=BETA.DIST(0.5, 2, 3, FALSE)
3. Use the BETA.DIST function in combination with other functions, such as the SUM or AVERAGE function, to perform more advanced calculations on beta distributions:
=SUM(BETA.DIST(A1:A10, 2, 3, TRUE))
In the first example, the BETA.DIST function calculates the probability of an event occurring in a beta distribution with an alpha value of 2 and a beta value of 3 for all values less than or equal
to 0.5. In the second example, the BETA.DIST function calculates the probability of an event occurring in the same beta distribution for all values strictly less than 0.5. In the third example, the
BETA.DIST function is used in combination with the SUM function to calculate the sum of the probabilities for a range of values in a beta distribution.
Use Case of BETA.DIST Function
Here are a few real-life examples of using the BETA.DIST function in Google Sheets:
1. A statistician is conducting a study on the success rates of different marketing strategies. She uses the BETA.DIST function to calculate the probability of a marketing campaign being successful
based on its alpha and beta values.
2. A data analyst is working with a large dataset of stock prices and wants to calculate the probability of a stock reaching a certain price within a certain time period. She uses the BETA.DIST
function to calculate the probability of this event occurring based on the stock’s alpha and beta values.
3. A finance manager is creating a budget plan for a company and wants to calculate the probability of the company achieving its revenue goals. She uses the BETA.DIST function to calculate the
probability of this event occurring based on the company’s alpha and beta values for revenue.
These are just a few examples of how the BETA.DIST function can be used in real-life situations to calculate probabilities in beta distributions.
Limitations of BETA.DIST Function
The BETA.DIST function in Google Sheets has the following limitations:
• The function only supports values between 0 and 1 for the x argument. If you try to use a value outside of this range, the function will return an error.
• The function only supports positive values for the alpha and beta arguments. If you try to use a negative value for either of these arguments, the function will return an error.
• The function only supports the logical values TRUE and FALSE for the cumulative argument. If you try to use any other value, the function will return an error.
Overall, the BETA.DIST function in Google Sheets is a powerful tool for calculating probabilities in beta distributions, but it has some limitations that you should be aware of when using it.
Commonly Used Functions Along With BETA.DIST
Here are some commonly used functions that are often used in combination with the BETA.DIST function in Google Sheets:
1. The SUM function: This function calculates the sum of a range of values. For example, you could use =SUM(BETA.DIST(A1:A10, 2, 3, TRUE)) to calculate the sum of the probabilities for a range of
values in a beta distribution.
2. The AVERAGE function: This function calculates the average of a range of values. For example, you could use =AVERAGE(BETA.DIST(A1:A10, 2, 3, TRUE)) to calculate the average probability for a
range of values in a beta distribution.
3. The MAX function: This function returns the maximum value in a range of values. For example, you could use =MAX(BETA.DIST(A1:A10, 2, 3, TRUE)) to find the maximum probability for a range of
values in a beta distribution.
These functions can be used in combination with the BETA.DIST function to perform more advanced calculations on beta distributions and to analyze the results in different ways.
The BETA.DIST function in Google Sheets is a powerful tool for calculating probabilities in beta distributions. This function allows you to easily calculate the probability of an event occurring in a
beta distribution based on a specified alpha and beta value. The BETA.DIST function is commonly used in statistical analysis and can be combined with other functions in Google Sheets to perform more
advanced calculations on beta distributions. If you want to learn more about the BETA.DIST function and how it can be used in Google Sheets, you can try using it in your own sheets to see how it
Video: BETA.DIST Function
In this video, you will see how to use BETA.DIST function. Be sure to watch the video to understand the usage of BETA.DIST formula.
Related Posts Worth Your Attention
Leave a Comment | {"url":"https://sheetsland.com/beta-dist-function/","timestamp":"2024-11-11T01:26:49Z","content_type":"text/html","content_length":"49210","record_id":"<urn:uuid:397b4927-1769-4099-be1c-f486ce2dd96a>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00780.warc.gz"} |
The glmmADMB package, built on the open-source AD Model Builder platform, is an R package for fitting generalized linear mixed models (GLMMs).
Its capabilities include:
• a wide range of families (response distributions), including non-exponential families such as negative binomial (type 1 and 2), Beta, logistic, and truncated Poisson and negative binomial
distributions as well as the standard exponential families (binomial, Poisson, Gamma, Gaussian).
• a wide range of link functions: log, logit, probit, complementary log-log, identity, inverse.
• Zero-inflation (currently only as a single constant term across all groups)
• Single or multiple random effects, including both nested and crossed effects
• Markov chain Monte Carlo (MCMC) summaries of uncertainty
In order to use glmmADMB effectively you should already be reasonably familiar with GLMMs, which in turn requires familiarity with (i) generalized linear models (e.g. the special cases of logistic,
binomial, and Poisson regression) and (ii) 'modern' mixed models (those working via maximization of the marginal likelihood rather than by manipulating sums of squares).
Please visit the following webpages for more information about the glmmADMB package (please note the latter is somewhat out of date, although it may still contain useful information):
Your best bet is
If you have trouble with the binary provided with your version of glmmADMB, here are the instructions for replacing it with a newer/different version:
1. look at the output of glmmADMB:::get_bin_loc() to determine where R will look for the compiled code for glmmADMB.
2. Go to the Buildbot page and retrieve a binary that is compatible with your OS. In general, the file name format is something like glmmadmb-r[version]-[OS]-[compiler]-[nbit]bit.(bin|exe), although
the file names do change from time to time. In general you want to look for the highest numbered version compatible with your system (go to the bottom of the list and scroll up, except that the
Windows versions are named differently and ended up at the top of the list). For example, as of today (7 Oct 2015) I would choose from among the following:
(it looks like you may be stuck if you want to run on 32-bit Ubuntu). Once you've found the binary you want, copy it to the location you determined in step #1 (you'll need to rename it to
glmmadmb or glmmadmb.exe to match what was there before); you might want to make a backup of the old version).
3. Try your code again and see if that helped.
4. If you can't find an appropriate binary on the system, you may have to buildthe glmmadmb binary from its TPL (system.file("tpl","glmmadmb.tpl",package="glmmADMB")) on your system (or find someone
with a compatible system who can do it for you) and copy it to the appropriate location.
The binaries included in the glmmADMB will not run on MacOS 10.5 (Leopard) or earlier, and may have trouble with very old versions of Linux as well. If you encounter this problem, your choices are:
• Upgrade your system to a more recent version of MacOS (if possible).
• Build glmmadmb from its TPL file on your machine. This will be a bit tricky if you are not reasonably experienced.
□ Download the full AD Model Builder source code from the AD Model Builder download page and follow the directions for building AD Model Builder from source; you may need to install Xcode, and
you may need to ask for help at users@admb-project.org. (Googling "admb macos 10.5" will be helpful as well, although it's possible that you will need the most recent version of the ADMB
source code to compile glmmadmb.tpl properly ...)
□ find the glmmadmb.tpl file in the glmmADMB package directories and use ADMB to compile it to a binary
□ copy the resulting binary to the bin/macos32 or bin/macos64 directory as appropriate.
• Contact the maintainers to appeal for help and find out if there any new developments in support for MacOS versions less than 10.6.
• A similar process may work for other unsupported operating systems such as Solaris, but in that case it's also probably a good idea to contact the maintainers.
Additional documentation
• Current (fairly minimal) documentation/example for ADMB in HTML and PDF format. This is also accessible from within R (once glmmADMB is installed) via vignette("glmmADMB",package="glmmADMB").
• The GLMM FAQ page gives general advice about GLMMs, although its content is slightly more oriented toward the lme4 package.
• We recommend the R mixed models list at r-sig-mixed-models@r-project.org for glmmADMB questions, although if you feel that your question is more AD Model Builder-oriented than R-oriented you may
also want to try the AD Model Builder user's list.
Newer versions
Newer versions of glmmADMB (>0.6.4) have the following major changes:
• new formula format, similar to that of the lme4 package, where random and fixed effects are specified as part of a single formula (random can also be specified separately, as in lme).
• multiple grouping variables (random effects) are allowed.
• wider range of distributions and link functions supported (e.g. binomial with N> 1).
The new release is somewhat slower (for the time being) than older (pre-0.5.2) versions: if you have a desperate need for a copy of an old version, you can download a source version and follow
alternative #3 from the installation instructions above. | {"url":"http://glmmadmb.r-forge.r-project.org/","timestamp":"2024-11-02T04:30:57Z","content_type":"text/html","content_length":"7985","record_id":"<urn:uuid:d32c7f8b-1997-4cd2-872d-b56f1d316843>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00221.warc.gz"} |
What is the probability of getting 2 tails if 5 coins are tossed?
What is the probability of getting 2 tails if 5 coins are tossed?
If five coins are tossed simultaneously, what is the probability of getting two tails? – Quora. If you want the probability of getting exactly two tails, it’s 31.25\% (10/32 possible outcomes).
What is the probability of flipping a coin 5 times and getting 2 heads?
Hence the probability is 26/32.
What is the probability of getting two tails on a coin?
(ii) getting two tails: Let E2 = event of getting 2 tails. Then, E2 = {TT} and, therefore, n(E2) = 1. Therefore, P(getting 2 tails) = P(E2) = n(E2)/n(S) = 1/4.
What is the probability of tossing a coin 5 times?
So we can represent it as 1/2^n (half to the power of n) where n is the number of times we flip the coin. So the odds of flipping a coin 5 times and getting 5 heads are 1/2 ^5 (half to the power of
5). Which gives us 1/32 or just over a 3\% chance.
When a coin is tossed 5 times the probability of getting 3 heads and 2 tails in any order is?
The probability of getting 3 tails and 2 heads is the same as the probability of getting 3 tails in 5 tosses. It is also the same as the probability of getting 2 heads in five tosses. The probability
of such an outcome , therefore, will be 10/32.
How many ways can you flip a coin 5 times?
Originally Answered: In the tossing of a coin 5 times, how many possible outcomes are there? Multiply the number of choices for each coin flip and you get 2*2*2*2*2 = 32. The 32 rows show all of the
possible outcomes of flipping a coin five times.
What is the probability of tossing 5 coins randomly and getting heads?
When we toss 5 coins randomly, we get total 32 outcomes i.e. 2^5. Among which only 6 outcomes contain less than 2 heads (TTTTT,HTTTT,THTTT,TTHTT ,TTTHT,TTTTH) All the remaining outcomes contain at
least 2 heads. Hence 32–6=26 outcomes contains at least two heads. Hence the probability is 26/32.
What if my heads and Tails don’t have the same probability?
(Optional) If your heads and tails don’t have the same probability of happening, go into advanced mode, and set the right number in the new field. Remember that in classical probability, the
likelihood cannot be smaller than 0 or larger than 1. The coin flip probability calculator will automatically calculate the chance for your event to happen.
How many coins are tossed in a coin toss?
You will discover there are 26 of them. The only six tosses that do not contain at least two heads are the following: 26 tosses that contain two heads out 32 possible tosses = 26/32 which equals
.08125. That’s your probability. 4 coins are tossed in parallel.
What comes first head or tail in a coin toss?
It is to be noted that after the toss any coin can be head or tail and order in which it comes doesn’t matter. a) After the toss, any one of the coins is head and the other three coins are tail. b)
After the toss, any two of the coins are head and any two coins are tail. c) After the toss, any three of the coins are head and any one coin is tail. | {"url":"https://yoursagetip.com/questions/what-is-the-probability-of-getting-2-tails-if-5-coins-are-tossed/","timestamp":"2024-11-14T09:04:09Z","content_type":"text/html","content_length":"56109","record_id":"<urn:uuid:761b7a85-a064-4a46-9b79-d0f7a6307d55>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00101.warc.gz"} |
Ph.D. RegulationsPh.D. Regulations
Ph.D. Regulations
The regulations discussed below include the requirements established by the Department of Mathematics as well as the most commonly applicable of the Graduate School’s requirements. The latter appear
in the publications entitled Graduate Catalog and Theses and Dissertation Guidelines. Students should read both of these publications thoroughly and be aware of the requirements stated therein.
The standard timeline of a typical doctoral program can be found here.
Students in the mathematics Ph.D. program take courses and prepare for their preliminary and qualifying examinations during the first two to three years of graduate work. After passing the qualifying
examination they take additional courses and write and defend their Ph.D. dissertation. Throughout this process, each student is guided by an adviser.
A new student is assigned an initial adviser, usually the Director of Graduate Studies. Eventually the student will decide upon an area of mathematics in which to specialize and will choose an
adviser to supervise research in that area; this decision should be made no later than the end of the second year.
The student’s doctoral committee (also known as the qualifying committee) is to be formed shortly after the student has chosen an adviser, and no later than two weeks after the beginning of the third
year. The doctoral committee is to contain four members of the mathematics graduate faculty and one member of the graduate faculty of a department other than mathematics. (For a list of graduate
faculty members, consult the aforementioned Bulletin.) Usually the committee will be chaired by the student’s adviser. It is possible, under unusual circumstances, to have a sixth member placed on
the committee. The final step in the committee formation process is to have the DGS certify compliance with graduate school’s policies on committees.
The functions of the doctoral committee are: (a) to define the student’s area of competence and administer the qualifying examination; (b) to approve the dissertation subject: (c) to aid the student
and monitor the progress of the dissertation; and (d) to read and approve the dissertation and administer the final oral examination.
The pre-qualifying requirements should be fulfilled within the first two to three years. They include some course requirements and passing the preliminary and qualifying examinations.
A. Pre-qualifying courses
Seventy-two hours of graduate courses are required for the Ph.D. Thirty-six of these hours must be completed before the qualifying examination and include the required core courses listed below.
1. Core courses
The core courses are Math 6200 and 6201 (topology), 6300 and 6301 (algebra), 6100 and 6101 (real analysis), and 7100 and 7101 (complex analysis). Each Ph.D. student is required to take seven courses
chosen from these. These seven courses must each be passed with a grade of B or better. Moreover, upon entering the Ph.D. program, the Director of Graduate Studies may require a student to take a
diagnostic test in analysis. Students who do not perform adequately on the diagnostic test will be required to take Math 5100 (introductory analysis) before taking Math 6100.
In some cases the Director of Graduate Studies will require that lower level courses be taken for no credit in order to make up for deficiencies in a student’s mathematical background. Students who
have taken the equivalent of a core course elsewhere may be exempted, by the Director of Graduate Studies, from taking that course provided they either pass the final examination in that course with
a grade of B or better or pass a more advanced course in the same area with a grade of B or better. In some cases the Director of Graduate Studies may waive these requirements.
An entering student will typically take Math 6200 (topology), 6300 (algebra) and 6100 (analysis) in the first semester. Two or three courses chosen from among Math 6201, 6301, and 6101 will then be
taken in the second semester; if only two are chosen, then a course lying outside the core will be taken.
Several variations are possible. As noted above, a student may need to take Math 5100 as a result of the diagnostic test in analysis. A student with a particularly strong background in algebra might
start with Math 9300 and 9301. A student with a strong interest in an area such as graph theory or applied mathematics might take courses in that area in the first year, saving for the second year
those core courses that would thereby be displaced. It should be noted, however, that the only mathematics courses carrying graduate credit for students in mathematics are Math 5640 (probability),
5641 (statistics), and those courses numbered 6000 or higher.
2. Additional courses
A total of thirty-six hours of graduate courses in mathematics, exclusive of Math 9999, must be taken before a student will be permitted to take the qualifying examinations. For one or two of these
courses, the student may, with the approval of the Director of Graduate Studies, substitute mathematically relevant graduate courses outside the Mathematics Department.
B. Preliminary examinations
Each student must pass a written preliminary examination in two out of the three areas of topology, algebra, and analysis. The preliminary examinations in topology, algebra, and analysis will cover
the material of Math 6200 and 6201, 6300 and 6301, and 6100 and 6101, respectively. The preliminary examination in the area will be prepared by a committee of three graduate faculty specializing in
that area and will be administered by the Director of Graduate Studies.
Topics Covered in the Preliminary Exams and Sample Exams:
1. Algebra Topics—Algebra Sample Test 1—Algebra Sample Test 2
2. Analysis Topics—Analysis Sample Test 1—Analysis Sample Test 2
3. Topology Topics— Topology Sample Test 1— Topology Sample Test 2
1. Scheduling
The preliminary examination in two out of the three areas of topology, algebra, and analysis must be taken no later than the fall of the second year. The preliminary examinations in topology,
algebra, and analysis will be offered just prior to the beginning classes each semester with at least three days between each area examination. The examinations will be of three hours duration and
are written. No reference material may be used during the examination.
2. Evaluation
The preliminary examinations will be graded by the same committees that prepared the examinations. The committees in the three areas will decide by a majority vote who passes in their respective
3. Re-examination
A student will be given two opportunities to pass an area preliminary examination. If a student passes one area exam and fails another area exam, then the student does not have to retake a
preliminary examination in the area that was passed. If a student fails to pass preliminary examinations in two out of the three areas of topology, algebra, and analysis by the Spring semester of the
second year, then the student will leave the Ph.D. program.
C. Qualifying examination
The qualifying examination is administered by the student’s doctoral committee and is based on the student’s area of competence. The student will choose between two options for the form of the
examination. Option 1 is an oral examination of two hours duration. Option 2 is an hour presentation of a qualifying paper and an hour questioning about the paper.
The purpose of the qualifying examination is to demonstrate to the doctoral committee that the student is prepared to begin research in the student’s area of competence leading to a dissertation.
When the student has passed the qualifying examination, the committee will recommend to the Graduate School that the student be admitted to doctoral candidacy.
1. Scheduling
The qualifying examination must be taken no later than the end of the Spring semester of the third year of residency. The qualifying examination must be scheduled by the Graduate School at least two
weeks in advance. If a student elects Option 2 of the qualifying examination, each member of the doctoral committee must receive a copy of the qualifying paper at least two weeks before the
2. Area of competence
The area of competence will be defined by the student’s doctoral committee and will include the topic of the qualifying paper. (For example, if the area of competence is on some aspect of Banach
algebras, then the area of competence might include the material of Math. 6100 and 6101, 7100 and 7101, and 9100 and 9101.) Shortly after the doctoral committee has been appointed, it will prepare a
written statement of the area of competence. Copies of that statement will be given to the student and to the Director of Graduate Studies.
3. Qualifying paper
The qualifying paper is an expository or research paper in the student’s area of competence. The paper must be typed (preferably in TeX form) and conform in format to the guidelines for a master’s
thesis as set forth in the aforementioned Instructions for the Preparation of Theses and Dissertations. The doctoral committee may permit a student who has written a long paper, honors thesis,
master’s thesis, or research paper to offer it as part or all of the qualifying paper.
4. Attendance
Attendance at Option 1 of the qualifying examination will be open to all faculty of the university, but student attendance will not be permitted. Attendance at Option 2 of the qualifying examination
will be open to all faculty and students of the university, but attendance of students during the subsequent questioning will be at the discretion of the committee.
5. Evaluation
The doctoral committee will decide by a majority vote whether the student has passed or failed the qualifying examination.
6. Re-examination
A student who fails in the first attempt to pass the qualifying examination will, upon request, be granted a repeat examination. In no case will a student have more than two opportunities to pass the
qualifying examination.
A repeat examination may be taken under either Option 1 or Option 2 independently of the option chosen for the first attempt at passing. The qualifying examination must be passed by the end of the
Fall semester of the fourth year of residency.
An original dissertation is required. The student must give an oral public presentation of the dissertation and undergo public questioning by the doctoral committee about it. The presentation and
questioning occur at the final examination, otherwise known as the defense of the dissertation.
A. Dissertation
Each member of the student’s doctoral committee is to receive a copy of the dissertation at least one month before the final examination. Two copies of the dissertation bearing original signatures of
at least a majority of the committee must be registered in the Graduate School office no later than two weeks before the last day of finals in the term in which the student plans to receive the
degree. The dissertation must be typed (preferably in TeX form) and conform in format to the guidelines of the Instructions for the Preparation of Theses and Dissertations. The dissertation must be
accompanied by an abstract, signed by the student’s adviser, of not more than 350 words.
B. Final Examination
The final examination must take place at least fourteen days before the last day of finals in the term in which the student plans to receive the degree. This event must be scheduled at least two
weeks before it is to occur.
A student who fails the final examination may take it again. In no case will a student have more than two opportunities to pass the final examination.
A. Courses
Seventy-two hours of graduate courses are required. Twenty-four of these hours must be earned in formal course work at Vanderbilt; the remainder may include transfer credit and dissertation research
hours (Math. 9999). A maximum of twelve of the seventy-two required hours may be earned in courses in departments other than mathematics; such courses must be approved by the Director of Graduate
In addition to the pre-qualifying courses discussed in II.A, the student must take eight three-hour mathematics courses numbered higher than 7000, exclusive of Math. 9999. These courses must be in at
least two of the following areas: algebra, analysis, applied mathematics, geometry-topology, and graph theory-combinatorics.
B. Maintaining a B Average
An average of at least B is required for graduation. Grades received in no-credit courses and in Math. 9999 are not included in computing the grade-point average.
C. Teaching
Most graduate students in the Department of Mathematics at Vanderbilt University receive financial support in the form of a teaching assistantship. The duties and training of teaching assistants
(TAs) are outlined below.
FIRST-YEAR TAs
All first-year graduate students participate in a teacher training program which consists of the following components:
1. TAs proctor a two-hour evening tutoring session for calculus students each week for both semesters.
2. TAs provide limited one-on-one tutoring through Tutoring Services.
3. In the spring, the Teaching Seminar meets weekly to introduce the teaching program in our department, the resources available for teachers at Vanderbilt, and good teaching techniques.
SECOND- AND THIRD-YEAR TAs
In their second and third years at Vanderbilt, graduate students serve as teaching assistants for instructors in our first-year calculus courses. Students conduct two weekly problem sessions for the
class, present an occasional lecture, assist the instructor in grading, and hold office hours. Before the beginning of the fall semester, the Center for Teaching conducts an orientation session for
all new TAs in the College of Arts and Science.
FOURTH- AND FIFTH-YEAR TAs
In the fourth and fifth years, students with good evaluations may teach a first-year calculus course. Usually, teaching duties require about 12 hours a week. During their first year of independent
teaching, TAs will be mentored by members of the Teaching Committee who will conduct class observations and offer support as needed.
Teaching performance is evaluated via student questionnaires in mid-semester and at the end of the semester. These evaluations are reviewed by the Associate Director of Graduate Studies, who makes
recommendations on renewal of teaching assistantships. TAs interested in improving their teaching can ask to have faculty observation of their classes, videotaping, or Small Group Analyses.
The University’s Center for Teaching has a staff member devoted to the teaching needs of international teaching assistants, particularly those whose first language is not English. Teaching assistants
who are experiencing language problems can participate in programs to improve their speaking and listening skills. In the past this program has proved to be a very effective means of improving
language abilities.
However, the Department may accept other activities as a substitute, and may waive the teaching requirement for graduate students with previous teaching experience.
D. Continuous registration
Registration must be continuous except for summer sessions. Any interruption in registration must be authorized by the Dean of the Graduate School as a leave of absence. Thus, except when granted a
leave of absence, the student must register each Fall and Spring semester, even if all course and hour requirements have been met. Failure to maintain continuous registration will result in loss of
student status.
E. Residence requirement
The student must be in residence a minimum of three years.
A. Minor
It is not required that a student have a minor, but it may be desirable in some cases. A minor is a body of related courses either in a mathematically relevant area outside of mathematics or in an
area of mathematics outside the area of the student’s dissertation. Twelve hours of such courses constitute the minor. If the minor is in an area of mathematics, courses taken toward the minor may
include courses used to satisfy the requirements of II.A and IV.A.
B. Transfer credit
Transfer credit can sometimes be granted for graduate work completed at another institution. As many as twenty-four hours of transfer credit can be awarded to a student who has completed a master’s
degree in mathematics at a comparable university.
C. Math 9999
A student ought not to register for Math 9999 (Dissertation Research) until the qualifying examination has been passed. Grades in Math 9999 are not counted in calculating a student’s quality point
D. No-credit courses
After seventy-two hours of credit have been earned, additional registration for research or course work will be permitted, but will earn zero credit hours, unless exception is made by the Dean of the
Graduate School.
Document last updated: 8/11/06 by M.D. Plummer, DGS | {"url":"https://as.vanderbilt.edu/math/graduate/ph-d-regulations-for-students-admitted-in-2003-or-after/","timestamp":"2024-11-15T03:09:12Z","content_type":"text/html","content_length":"60494","record_id":"<urn:uuid:a9fa1d2f-73e2-4486-b555-284a3bf11d35>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00508.warc.gz"} |
Square & Square Root of 1521- Examples, Methods, Calculation
Square & Square Root of 1521
Within algebraic studies, the significance of squares and square roots is foundational. Squaring, exemplified by multiplying a number like 1521 by itself to yield 2313441, is fundamental. This
operation is key in exploring properties of rational and irrational numbers. Understanding these concepts enriches comprehension of mathematical relationships and patterns. In algebraic mathematics,
squares and square roots are pivotal. Squaring, illustrated by the multiplication of 1521 by itself to produce 2313441, serves as a cornerstone. This process underpins investigations into rational
and irrational numbers. A grasp of these fundamental concepts enhances understanding of mathematical relationships and patterns, facilitating deeper comprehension within algebraic studies.
Square of 1521
1521² (1521 × 1521) = 2313441
A square number, like 1521, is the result of multiplying an integer by itself. The square of 1521 equals 2,313,441. Understanding square numbers elucidates fundamental mathematical concepts, serving
as a basis for exploring mathematical patterns and relationships, crucial in algebraic studies and beyond.
Square Root of 1521
√1521 = 39
The square root of 1521, a square number, is 39. Understanding square roots entails deciphering the number that, when multiplied by itself, yields the original value. Mastery of square roots is
fundamental in mathematics, laying the groundwork for comprehending algebraic concepts and unlocking deeper insights into numerical relationships and patterns.
Square Root of 1521: 39
Exponential Form: 1521^½ or 1521^0.5
Radical Form: √1521
Is the Square Root of 1521 Rational or Irrational?
Yes, the square root of 1521 is a rational number
The square root of 1521 is a rational number because it equals 39, which can be expressed as a fraction of two integers (39/1). Rational numbers can be written as a quotient of two integers where the
denominator is not zero.
Rational Numbers: Rational numbers are expressible as the quotient of two integers, where the denominator isn’t zero. They’re represented as 𝑎𝑏, where 𝑎 and 𝑏 are integers and 𝑏≠0.
Examples include fractions like 1/2, -3, and 5/5.
Irrational Numbers: Irrational numbers can’t be written as fractions of integers. Their decimal representations are non-repeating and non-terminating.
Examples include square roots of non-perfect squares like √2, √3, √5, and transcendental numbers like π (pi).
In summary, rational numbers have finite or repeating decimals, while irrational numbers have non-repeating, non-terminating decimal expansions.
Methods to Find Value of Root 1521
There are several methods to find the square root of 1521:
Prime Factorization Method: Express 1521 as a product of prime factors (3² × 11²), then take the square root of each prime factor.
Long Division Method: Iterate through digits of 1521, pairing them off and finding the largest number whose square is less than or equal to the current remainder.
Estimation Method: Use approximation techniques like the Newton-Raphson method to iteratively approach the square root of 1521.
Calculator: Simply input 1521 into a calculator and press the square root button to obtain the result directly.
Square Root of 1521 by Long Division Method
Long Division Method for Finding Square Root of 1521
Step 1: Digit Pairing
Pair digits of the given number starting from the right, indicating with a horizontal bar.
Step 2: Initial Quotient
Find a number whose square is less than or equal to the first paired digits (15). Since 3² = 9 < 15, the quotient is 3. The difference is calculated as 15 – 9 = 6.
Step 3: Bringing Down
Bring down the next pair of digits (21) and multiply the quotient (3) by 2 to get 6, which is the starting digit of the new divisor.
Step 4: Further Division
Place the quotient (9) obtained from the previous step at the one’s place of the new divisor (69). Multiply 69 by 9 to get 621. The remainder is 0.
Step 5: Final Result
Since the remainder is 0, the process terminates. Thus, the square root of 1521 is 39.
Is 1521 Perfect Square root or Not
Yes, the number 1521 is a perfect square
Yes, 1521 is a perfect square. Its square root is 39, meaning that 39 multiplied by itself equals 1521. In mathematical terms, 1521 = 39², confirming that it is indeed a perfect square.
What are the properties of the square and square root of 1521?
The square of 1521 is a positive integer, while its square root is a positive rational number.
How do I verify if my calculation of the square root of 1521 is correct?
You can verify your calculation by squaring the square root value obtained. If the result is 1521, then your calculation is correct.
What are some real-life applications of understanding squares and square roots?
Understanding squares and square roots is essential in various fields such as engineering, physics, computer science, and finance for tasks like calculating areas, distances, and determining
Can the square root of 1521 be simplified further?
No, the square root of 1521, which is 39, is already in its simplest form because it is a perfect square.
Is 1521 a prime number?
No, 1521 is not a prime number because it has factors other than 1 and itself.
How does knowing the square and square root of 1521 contribute to my understanding of algebraic concepts?
Understanding squares and square roots serves as a foundational step in algebra, helping to comprehend concepts like equations, inequalities, and polynomial functions. | {"url":"https://www.examples.com/maths/square-square-root-of-1521.html","timestamp":"2024-11-09T05:48:05Z","content_type":"text/html","content_length":"107183","record_id":"<urn:uuid:9cbbdbde-b3fb-4fd1-8350-17600d31a731>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00739.warc.gz"} |
Math 4 Wisdom. "Mathematics for Wisdom" by Andrius Kulikauskas. | Research / MathDiscoveryBasicThree
• Index
• Writings
Andrius Kulikauskas
• m a t h 4 w i s d o m - g m a i l
• +370 607 27 665
• My work is in the Public Domain for all to share freely.
• 读物书影片维基百科
Introduction E9F5FC
Questions FFFFC0
• View
• Edit
• History
• Print
Math Discovery Examples
Math Discovery Basic Three
Independent trials
Blank sheets* Independent trials. We may think of our mind as blank sheets, as many as we might need for our work. We shouldn't get stuck, but keep trying something new, if necessary, keep getting
out a blank sheet. We can work separately on different parts of a problem. This relates also to independent events (in probability), independent runs (in automata theory) and independent dimensions
(in vector spaces). If something works well, then we should try it out in a different domain. Sarunas Raudys notes that we must add a bit of noise so that we don't overlearn.9
□ Avoid error-prone activity* Simplify .... We could multiply out all the terms, but it would take a long time, and we'd probably make a mistake. We need a strategy. pg.166 The Art and Craft of
Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.2172
□ Get your hands dirty* So we try another strategy, one of the best for beginning just about any problem: get your hands dirty. We try plugging in some numbers to experiment. If we are lucky,
we may see a pattern. ... This is easy and fun to do. Stay loose and experiment. Plug in lots of numbers. Keep playing around until you see a pattern. Then play around some more, and try to
figure out why the pattern you see is happening. It is a well-kept secret that much high-level mathematical research is the result of low-tech "plug and chug" methods. The great Carl Gauss
... was a big fan of this method. In one investigation, he painstakingly computed the number of integer solutions to x**2+y**2<=90,000. ... Don't skimp on experimentation! Keep messing around
until you think you understand what is going on. Then mess around some more. pg.7, 30, 36 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1412
□ Get your hands dirty UCL problem solving technique 1 of 5
□ Knowing when to give up* Sometimes you just cannot solve a problem. You will have to give up, at least temporarily. All good problem solvers will occasionally admit defeat. An important part
of the problem solver's art is knowing when to give up. pg.16, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1414
□ Mental toughness, confidence and concentration* But most beginners give up too soon, because they lack the mental toughness attributes of confidence and concentration. It is hard to work on a
problem if you don't believe that you can solve it, and it is impossible to keep working past your "frustration threshold". ... You build upon your preexisting confidence by working at first
on "easy" problems, where "easy" means that *you* can solve it after expending a modest effort. ... then work on harder and harder problems that continually challenge and stretch you to the
limit ... Eventually, you will be able to work for hours single-mindedly on a problem, and keep other problems simmering on your mental backburner for days or weeks. ... developing mental
toughness takes time, and maintaining it is a lifetime task. But what could be more fun than thinking about challenging problems as often as possible? pg.16, The Art and Craft of Problem
Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1415
□ Practice* Practice by working on lots and lots and lots of problems. Solving them is not as important. It is very healthy to have several unsolved problems banging around your conscious and
unconscious mind. pg.25, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1423
□ Toughen up* Toughen up by gradually increasing the amount and difficulty of your problem solving work. pg.24, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons,
□ Vary the trials* The mouse in the trap ... threw himself violently against the bars, now on this side and then on the other, and in the last moment he succeeded in squeezing himself through
... We must try and try again until eventually we recognize the slight difference between the various openings on which everything depends. We must vary our trials so that we may explore all
sides of the problem. Indeed, we cannot know in advance on which side is the only practicable opening where we can squeeze through. The fundamental method of mice and men is the same; to try,
try again, and to vary the trials so that we do not miss the few favorable possibilities ... a man can vary his trials more and learn more from the failure of his trials than the mouse.
pg.16, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc., quoting "Mice and Men" by George Polya, Mathematical Discovery, Volume II, 1965.1413
□ Give up and try again differently. Dog with hula hoop in its teeth can't lift it over a step. So the dog lets go of it, picks it up in a different place, and tries again.
Symmetry. Symmetry group We unify internal and external points of view, link time and space, by considering a group of actions in time acting on space. Some aspects of the space are invariant, some
aspects change. Actions can make the space more or less convoluted. At this point, we have arrived at a self-standing system, one that can be defined as if it was independent of our mental processes.
Our problem has become "a math problem". Analogously, in real life, after projecting more and more what we mean in general by people, including ourselves and others, we finally take us for granted as
entirely one and the same and instead make presumptions towards a universal language by which we might agree absolutely.13
□ Axiom schema of specification* Wikipedia: If z is a set, and P is any property which may characterize the elements x of z, then there is a subset y of z containing those x in z which satisfy
the property. The "restriction" to z is necessary to avoid Russell's paradox and its variants. I think this relates to the idea that we can focus on the relevant symmetry and the relation
between the locations affected or not by the symmetry group and the actions of that group.1170
• A bank of useful derivatives of "functions of a function"* We conclude our discussion of differentiation with two examples that illustrate a useful idea inspired by logarithmic differentiation.
... Logarithmic differentiation is not just a tool for computing derivatives. It is part of a larger idea: developing a bank of useful derivatives of "functions of a function" that you can
recognize to analyze the original function. If a problem contains or can be made to contain the quantity f'(x)/f(x), then antidifferentiation will yield the logarithm of f(x), which in turn sheds
light on f(x). pg.300 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc2244
• Algebraic symmetry* Sequences can have symmetry, like this row of Pascal's Triangle: 1, 6, 15, 20, 20, 15, 6, 1 ... In just about any situation where you can imagine "pairing" things up, you can
think about symmetry. pg.74, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1627
• Combination of techniques* We shall end the chapter with an exploration of the diophantine equation x**2 + y**2 = n ... where n is a prime p. Our exploration will use several old strategic and
tactical ideas, including the pigeonhole principle, Gaussian pairing, and drawing pictures. The narrative will meander a bit, but please read it slowly and carefully, because it is a model of how
many different problem-solving techniques come together in the solution of a hard problem. pg.274 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc2229
• Complex Numbers* Complex numbers are the crossover artist's dream: like light, which exists simultaneously as wave and particle, complex numbers are both algebraic and geometric. You will not
realize their full power until you become comfortable with their geometric, physical nature. This in turn will help you to become fluent at translating between the algebraic and the geometric in
a wide variety of problems. ... We strongly urge you to read at least the first few chapters of our chief inspiration for this section, Tristan Needham's Visual Complex Analysis. This
trail-blazing book is fun to read, beautifully illustrated, and contains dozens of geometric insights that you will find nowhere else. ... If z=a+bi, we define the conjugate of z to be z-bar =
a-bi. Geometrically, z-bar is just the reflection of z about the real axis. Complex numbers add "componentwise" ... Geometrically, complex number addition obeys the "parallelogram rule" of vector
addition ... Multiplication by the complex number rCisTheta is a counterclockwise rotation by Theta followed by stretching by the factor r. So we have a third way to think about complex numbers.
Every complex number is simultaneously a point, a vector, and a geometric transformation, namely the rotation and stretching above! pg.131-134, The Art and Craft of Problem Solving, Paul Zeitz,
1999, John Wiley & Sons, Inc.2162
• Exploit underlying symmetry in polynomials* Algebra problems with many variables or of high degree are often intractable unless there is some underlying symmetry to exploit. ... Solve x**4 + x**3
+ x**2 + x**1 + 1 = 0 ... we will use the symmetry of the coefficients as a starting point to impose yet more symmetry, on the degrees of the terms. Simply divide by x**2 yielding x**2 + x + 1 +
1/x + 1/x**2 then make the substitution u := x + 1/x. pg. 75, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1629
• Fixed objects* When pondering a symmetrical situation, you should always focus briefly on the "fixed" objects which are unchanged by the symmetries. For example, if something is symmetric with
respect to reflection about an axis, that axis is fixed and worthy of study (the stream in the previous problem played that role). pg. 72 The Art and Craft of Problem Solving, Paul Zeitz, 1999,
John Wiley & Sons, Inc.1585
□ Four bugs chasing each other* a classic problem which exploits rotational symmetry along with a crucial fixed point ... Four bugs are situated at each vertex of a unit square. Suddenly, each
bug begins to chase its counterclockwise neighbor. If the bugs travel at 1 unit per minute, how long will it take for the four bugs to crash into one another? pg.71 The Art and Craft of
Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1586
• Geometric symmetry* The simplest geometric symmetries are rotational and reflectional. pg. 71 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1582
□ Fetching water for Grandma* Your cabin is 2 miles due north of a stream which runs east-west. Your grandmother's cabin is located 12 miles west and 1 mile north of your cabin. Every day, you
go from your cabin to Grandma's, but first visit the stream (to get fresh water for Grandma). What is the length of the route with minimum distance? pg.71 The Art and Craft of Problem
Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1584
□ Square inscribed in circle inscribed in square* A square is inscribed in a circle which is inscribed in a square. Find the ratio of the areas of the two squares. pg.70 The Art and Craft of
Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1583
• Harmony* An informal alternate definition of symmetry is "harmony". ... If you can do something that makes things more harmonious or more beautiful, even if you have no idea how to define these
two terms, then you are often on the right track. pg. 70 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1580
• Invariant with respect to transformations* this topic [symmetry] is logically contained within the concept of invariants. If a particular object (geometrical or otherwise) contains symmetry, that
is just another way of saying that the object itself is an invariant with respect to some transformation or set of transformations. For example, a square is invariant with respect to rotations
about its center of 0, 90, 180 and 270 degrees. pg. 103, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1646
• Not quite symmetrical* The strategic principles of peripheral vision and rule-breaking tell us to look for symmetry in unlikely places, and not to worry if something is almost, but not quite
symmetrical. In these cases, it is wise to proceed as if symmetry is present, since we will probably learn something useful. pg. 70 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John
Wiley & Sons, Inc.1581
□ Completing the square by trying to symmetrize* x**2 + a*x = x*(x + a) = (x + a/2 -a/2)*(x + a/2 + a/2) = (x + a/2)**2 - (a/2)**2 Above is a way to discover the completing-the-square formula
by trying to symmetrize the terms, then adding zero creatively. pg. 163, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. 1381
• Roots of Unity* The zeros of the equation x**n = 1 are the nth roots of unity. These numbers have many beautiful properties that interconnect algebra, geometry and number theory. One reason for
the ubiquity of roots of unity in mathematics is symmetry: roots of unity, in some sense, epitomize symmetry... pg.131-134, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley &
Sons, Inc.2163
• Search for order* Many fundamental problem-solving tactics involve the search for order. Often problems are hard because they seem "chaotic" or disorderly; they appear to be missing parts (facts,
variables, patterns) or the parts do not seem connected. ... we will begin by studying problem-solving tactics that help us find or impose order where there seemingly is none. pg. 69 The Art and
Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1524
• Symmetrize the coefficients* Solve the system of equations ... The standard procedure for solving systems of equations by hand is to substitute for and/or eliminate variables in a systematic (and
tedious) way. But notice that each equation is almost symmetric, and that the system is symmetric as a whole. Just add together all five equations; this will serve to symmetrize all the
coefficients ... Now we can subtract this quantity from each of the original equations to immediately get ... pg.166-167 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons,
• Symmetry* Symmetry involves finding or imposing order in a concrete way, for example, by reflection. ... We call an object symmetric if there are one or more non-trivial "actions" which leave the
object unchanged. We call the actions that do this the symmetries of the object (Footnote: We are deliberately avoiding the language of transformations and automorphisms that would be demanded by
a mathematically precise definition.) ... Why is symmetry important? Because it gives you "free" information. If you know that something is, say, symmetric with respect to 90-degree rotation
about some point, then you only need to look at one-quarter of the object. And you also know that the center of rotation is a "special" point, worthy of close investigation. pg. 69-70 The Art and
Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1579
• The Gaussian pairing tool* Gauss, as a child, added up the sum 1 + 2 + 3 + ... + 100, presumably by pairing up the number 1 and 100, 2 and 99, 3 and 98, ... 50 and 51, yielding 50 pairs of 101
for a total of 5,050. Paul Zeitz notes this as an example of symmetry and calls it the Gaussian pairing tool. pg. 75, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons,
• Tilt the picture* we will present, with a "hand-waving" proof, one important theoretical tool which will allow you to begin to think more rigorously about many problems involving differentiable
functions. We begin with Rolle's theorem, which certainly falls into the "intuitively obvious" category. If f(x) is continuous on [a,b] and differentiable on (a,b), and f(a) = f(b), then there is
a point u in (a,b) at which f'(u) = 0. The "proof" is a matter of drawing a picture. There will be a local minimum or maximum between a and b, at which the derivative will equal zero. Rolle's
theorem has an important generalization, the mean value theorem. If f(x) is continuous on [a,b] and differentiable on (a,b), then there is a point u in (a,b) at which f'(u) = (f(b) - f(a))/(b-a).
... the proof is just one sentence: Tilt the picture for Rolle's theorem! pg.297-298 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc2242
• Transformation* The pattern of superposition points out a path from a leading special case (or from a few such cases) to the general case. There is a very different connecting path between the
same endpoints with which the ambitious problem-solver should be equally acquainted: it is often possible to reduce the general case to a leading special case by an appropriate transformation.
... For a suggestive discussion of this topic see J. Hadamard, Lecons de geometrie elementaire. Geometrie plane, 1898; Methodes de transformation, pp. 272-278. "Mathematical Discovery: On
Understanding, Learning and Teaching Problem Solving" by George Polya, 1962, John Wiley & Sons.2253
• Kindiak Math. Simple Geometry Stumps Almost Everyone. Morphism. Apply a morphism to lift the addition problem to a different domain. The addition can be understood to take place in the geometry
itself. It can be understood to be meaningful in trigonometry (if we think of the addition formula). And it can be understood to be meaningful in complex numbers (especially if we know that
angles can be defined as the arguments of a complex number.)
Context* O Context If you read the problem carefully, if you understand and follow the rules, then you can also relax them, bend them. You can thus realize which rules you imposed without cause. You
can also change or reinterpret the context. These are the holes in the cloth that the needle makes. I often ask my new students, what is 10+4? When they say it is 14, then I tell them it is 2. I ask
them why is it 2? and then I explain that it's because I'm talking about a 12-hour clock. This example shows the power of context so that we probably can't write down all of the context even if we
were to know it all. We can just hope and presume that others are like us and can figure it out just as we do. Analogously, in real life, it's vital to obey God, or rather, to make ourselves obedient
to God. (Or if not God, then our parents, those who love us more than we love ourselves, who want us to be alive, sensitive, responsive more than we ourselves do.) If we are able to obey, then we are
able to imagine God's point of view and even make sense of it.16
• 10 + 4 = 2* I ask my students, What is 10+4? and they answer 14, and then I say, No, it is 2! Do you know why? Because I'm thinking about a clock. 10 o'clock plus 4 o'clock is 2 o'clock on a
clock. What the example shows is that meaning ultimately depends on the context which we interpret. Any explanations that we write may also be misinterpreted. Thus there is no way to explicitly
assure that somebody means what we mean. However, the context may indeed coincide in all that is relevant to us, either explicitly or implicitly. That's why existentialism is important, because
it's important for us that our words and concepts be grounded in the questions relevant to our existence. Gospel Math.1840
• Axiom schema of replacement* Wikipedia: Less formally, this axiom states that if the domain of a definable function f is a set, and f(x) is a set for any x in that domain, then the range of f is
a subclass of a set, subject to a restriction needed to avoid paradoxes. I think this relates to the idea that context allows us to "substitute variables" in different ways and perhaps with
different results, different meanings, thus yielding flexibility of interpretation.1169
• Matchsticks. Given an equilateral triangle made of match sticks. How to move one stick and make it a square? You change the context: You make the square number 4 by moving the rightmost
• Kindiak Math. Simple Geometry Stumps Almost Everyone. We are given a row of three squares and three angles {$\alpha = \textrm{arctan}(1), \beta = \textrm{arctan}(\frac{1}{2}), \gamma = \textrm
{arctan}(\frac{1}{3})$} and we are asked for {$\alpha + \beta + \gamma$}. I got stumped because I supposed that there was no fraction expressing the last term. Indeed, I confused myself by
thinking {$\beta$} was 30 degrees. What I failed to do was to calculate the sum, however approximately. Then I would have realized that it was {$\frac{\pi}{2}=90^{\circ}$}. At that point, I would
have understood that there must be a geometric way to think about the sum. Rethinking the real number as a rational number would have driven me to persist.
• Given equation {$1^x=2$} go beyond the real numbers, think in terms of complex numbers.
• Grothendieck's method. Crack a nut by immersing it in water. nLab: The Rising Sea. Rethink the context for all of geometry or all of math, as relevant.
• Change the context - change the rules. Art changes the rules.
• A page of a book can mean two different things. "A child was playing with a book and tore out the pages 7, 8, 100, 101, 222 and 223. How many pages did the child tear out?" Mind Your Decision | {"url":"https://www.math4wisdom.com/wiki/Research/MathDiscoveryBasicThree","timestamp":"2024-11-14T21:02:28Z","content_type":"application/xhtml+xml","content_length":"34424","record_id":"<urn:uuid:a47aade8-c078-4e48-926e-70e8a8c2b434>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00657.warc.gz"} |
Proximal relations and P-n-flows
Proximal relations and related notions in flows are investigated, and the dense action property in flows is introduced. The latter is a special property of the topological phase group on the phase
space in a given flow. The dense action property is independent of being distal or proximal. Some relations between flows with the dense action property and homomorphisms are described. A sufficient
condition for a flow to have the dense action property is explored. It is shown that every mimimal set of a compact flow has the dense action property. Various properties of proximal relations in
flows are described, and the products of P to the nth power flows are developed. Finally, some related examples are presented.
Ph.D. Thesis
Pub Date:
November 1977
□ Flow Characteristics;
□ Flow Distribution;
□ Flow Geometry;
□ Correlation;
□ Fluid Flow;
□ Fluid Mechanics;
□ Homomorphisms;
□ Proximity;
□ Topology;
□ Fluid Mechanics and Heat Transfer | {"url":"https://ui.adsabs.harvard.edu/abs/1977PhDT.......110C/abstract","timestamp":"2024-11-07T17:45:36Z","content_type":"text/html","content_length":"34201","record_id":"<urn:uuid:68362bc2-4ce9-4d80-9252-e576680b6b91>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00668.warc.gz"} |
NumPy Learning Path • datagy
NumPy Learning Path
Get started with the fundamental package for scientific computing with Python
Getting Started with NumPy
NumPy is a foundational Python library for data science, math, and scientific computing. If you’re not coming from a scientific or math background, NumPy can seem intimidating. But, don’t worry! The
tutorials below get you started with the main concepts of NumPy and really demonstrate its power.
Get introduced to NumPy, including learning how it’s used in data science.
Learn how to access items in NumPy arrays, using special methods reserved for NumPy.
Learn how to process NumPy arrays conditionally.
Creating NumPy Arrays
NumPy arrays are a foundational component of NumPy. Learning how to create them programmatically is a critical skill in working with NumPy. The library provides a ton of flexibility in creating
useful arrays, such as sequences of numbers or even evenly-spaced arrays.
Learn how to create sequences of numbers using the NumPy arange() function.
Learn how to create evenly spaced arrays using the NumPy linspace() function.
Learn how to create evenly spaced arrays on a log scale using the NumPy logspace() function.
Learn how to create arrays of zeros using the NumPy zeros() function.
Learn how to create arrays of a single value using the NumPy full() function.
Learn how to create random normal distributions using the NumPy normal() function.
Modifying Arrays in NumPy
Once you’ve created your arrays in NumPy, you may want to modify them. NumPy makes modifying array items easier, allowing you to map functions to each item in an array. Learn how to normalize and
limit array values, map a function over each item, and so much more.
Reshaping Arrays in NumPy
Being able to reshape your arrays is an essential skill once you move into machine learning and deep learning. NumPy makes reshaping arrays incredibly simple and intuitive. The resources below guide
you through the key functions you need to know.
Learn how to use the np.squeeze() function to reduce the dimensions of an array.
Learn how to repeat NumPy arrays in a pattern using np.tile()
Learn how to flatten a NumPy array using the np.flatten() function
Calculating Values with NumPy
NumPy wouldn’t be a math library without being able to calculate values. The resources below provide essential guides to working with NumPy arrays, including finding values and calculating | {"url":"http://datagy.io/learn-numpy/","timestamp":"2024-11-14T17:54:55Z","content_type":"text/html","content_length":"119466","record_id":"<urn:uuid:5bec0cf7-beb6-4090-b06f-a4a525c97007>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00325.warc.gz"} |
How Raytracing Works
October 27, 2016
Recently I went about making a 3D renderer. Before researching, it seemed like a daunting task, shrouded in the promise of complicated math and a variety subtle, deeply-hidden bugs. As it turns out,
there is a fair amount of linear algebra, and it is hard to tell if your glass rendering is weird or if glass itself is just weird. That said, the core concepts are more accessible than I thought
they would be. When you encounter challenges 3D rendering involves, they come one at a time, so you can solve them, abstract them, and move on. I learned a lot, so I thought I'd share how a system
like the one I wrote works.
What is 3D rendering, anyway?
We can think of the camera as a 2D rectangle in 3D space. It's like the film of a camera: when a photon hits this rectangle, we see its colour value in the location that it hit the film. The image we
get from a camera is a 2D projection of the 3D world onto the rectangle of that piece of film.
The end goal of a 3D renderer is to make that 2D image of 3D space.
How does raytracing fit in?
There are plenty of ways you can write a 3D renderer. Some of them are better suited for fast render times for applications like gaming. Others, like raytracing, take longer to compute but can often
model reality more realistically. Raytracing can take into account more complicated reflections of light off of other objects, soft shadows, lens blur, and more.
In real life, light is emitted from a source as a photon of a certain colour. It then will travel in a straight line (well, mostly) until it hits something. Then, one of a few things can happen to
the photon. It can get reflected in a direction, its path can be bent from refraction, or it can be absorbed by the material. Some of the photons eventually bounce their way to a camera, where they
are "recorded."
Raytracing models this behaviour, but in reverse. Photons are cast from the camera, and bounce around the surfaces in a scene until they hit a light source. At that point, the photon's colour is
recorded. It would also work if you cast rays from light sources to the camera, like in real life, but this tends to not be as efficient since so many photons just won't reach the camera.
So, for each pixel in the image you want to render, here's what we do:
1. Cast a ray of white light from a pixel
2. Find the first object the ray intersects with
3. If it is a light source, multiply the ray's color with the light source's colour to get the pixel colour
4. Otherwise, reflect, refract or absorb the ray, and go back to step 2 with the resulting ray
Modelling geometry
Here's where some math happens. How do we determine if a ray hits an object? First, let's model a ray. A ray in 3D space can be defined as a direction vector, and a point that it goes through. Both
this 3D point and vector can be represented by an (x, y, z) coordinate, but we're actually going to use a 4-vector (x, y, z, w).
Why the extra coordinate? You can certainly use a normal 3-vector, but then it's up to you to keep track of which 3-vector in your program is a point and which is a direction. If you use a 4-vector,
though, w = 0 implies that it is a direction vector and w = 1 implies that it is a point, which makes things work out pretty nicely. If you add a point and a vector, their w coordinates add to 1 + 0
= 1, meaning the result is still a point. A vector minus a vector is still a vector, and 0 - 0 = 0. A point minus a point is a vector, and 1 - 1 = 0. A point plus a point doesn't make sense, which
would leave you with a w value of 2, which is also unexpected. When we use transformation matrices later, they Just Work™ with this way of modelling points and vectors. It's convenient.
So, we've got this definition of a ray:
struct Ray {
let point, direction: Vector4
let color: Color
Then, for any given point on the film and focal point behind the film, we can cast an initial ray:
func castRay(from: Vector4, through: Vector4) -> Ray {
return Ray(
point: from,
direction: through - from,
color: Color(0xFFFFFF) // Start with white light
To see if it intersects with an object, we need an object to model. A sphere is a nice one, since we only need a few small pieces to represent it:
struct Sphere {
let center: Vector4
let radius: Float
We can then make a function pretty easily to check whether or not a ray intersects with a sphere by using the equations given in the Wikipedia article for line-sphere intersections. We can make it
return an Intersection (the point of intersection and normal from the surface at that point) if it exists, or nil otherwise. If we have multiple spheres, we want the first one the ray intersects
with, so you can iterate through the spheres and take the one that's the shortest distance from the ray origin. Obviously this isn't the most efficient, but works for small scenes:
struct Intersection {
let point, normal: Vector4
func firstIntersection(ray: Ray, spheres: [Sphere]) -> Intersection? {
return spheres.flatMap{ (sphere: Sphere) -> Intersection? in
return intersectionBetween(ray: ray, sphere: sphere)
}.sorted{ (a: Vector4, b: Vector4) -> Bool in
(a - ray.point).length < (b - ray.point).length
Modelling materials
Once we've found an intersection, we are tasked with bouncing the light ray. How this happens depends on the material the ray intersected with. A material, for our purposes, must be able to take an
incoming ray and an intersection and return a bounced ray.
protocol Material {
func bounce(ray: Ray, intersection: Intersection) -> Ray
How the ray gets bounced affects what the object looks like. To make shadows, we know that some photons need to get absorbed somehow. Each time a ray is bounced, we can dim the intensity of the light
of the outgoing ray a bit (for example, multiply the red, green, and blue fields by 0.7.) The more bounces the light goes through, the darker the colour becomes. If no intersection is found, we can
multiply the light colour by some background colour and stop bouncing, as if there is sky in every direction as a source of photons.
If a ray does hit an object, we have to think about what direction we want to bounce the ray in. Reflective materials abide by the tenth grade science class mantra, the angle of incidence equals the
angle of reflection. That is to say, if you reflect the incoming ray about the surface normal, you're going to make a mirrorlike material. If instead you choose to reflect the light in a totally
random direction, you've diffused the light and mate a matte material (although, make sure you absorb the ray if it is randomly bounced into the inside of the sphere.) A surface that reflects rays
but with a little bit of random variation will look like brushed or frosted metal.
Monte Carlo rendering
You'll notice that the scene looks pretty grainy, specifically around areas that should be blurred. This is because, for each pixel, we randomly bounce a photon around. It's bound to not be quite
smooth because of the random variation. To make it smoother, we can simply render each pixel multiple times and average the values. This is an example of a Monte Carlo algorithm. From Wikipedia, a
Monte Carlo algorithm "uses random numbers to produce an outcome. Instead of having fixed inputs, probability distributions are assigned to some or all of the inputs." The more averaged samples we
take of the image, the closer to an actual "perfect render" we get. The random grains, averaged together, end up looking like a smooth blur.
We can make more complicated materials with this sampling technique by having it, for example, reflect a photon some percent of the time and refract it the rest of the time. Having a higher
probability of reflecting at steeper angles is a good way to create realistic-looking glass. You can make glossy materials by having a small probability of reflection and a higher probability of
diffusing the light.
Motion blur
Another cool thing we can do using the Monte Carlo technique is create motion blur. In real life, cameras have their shutters open for a real, non-infinitesimal amount of time. The longer the film is
exposed to photons, the more photons hit it, and the brighter an image you get. If an object is moving while the film is exposed, photons reflected from all points in time along the object's
trajectory will end up on the film, resulting in the object appearing smeared.
We can model this in our raytracer, too. Let's say a sphere moves from point A to point B while our virtual camera shutter is open. For every ray we cast, before we check for intersections between
the ray and the sphere, we pick a random point along the object's trajectory for it to be at, and use this version of the object for collisions. We use a different random location for the next ray.
After doing enough samples of this, we should end up with a nice blur.
In order to actually implement this, we need to represent the object's motion. A transformation matrix works well for this purpose. When you multiply a matrix by a vector, you get a different vector.
A simple one is the translation matrix:
The end result is a shifted coordinate. You can also create rotation, stretch, and skew matrices. By multiplying matrices together, you compose the transformations. You can invert a transformation by
inverting its transformation matrix.
So, back to our motion blur. The camera shutter is open as an object moves from A to B, and we can represent A and B using transformation matrices. If you want to find a version of an object at a
random point on its trajectory to check collisions with, you can interpolate between them:
func randomOnTrajectory(object: Sphere, from: Matrix4, to: Matrix4) -> Sphere {
let amount = randomBetween(low: 0, high: 1)
let transformation = from*amount + to*(1-amount)
return Sphere(
center: transformation * object.center,
radius: object.radius
That gives you a result like this:
Because it takes multiple samples to get a good looking result, it would make sense to try to get as much throughput as possible while rendering. The great thing about raytracing is that each sample
is calculated completely separately from other samples (there's a technical term for this, and it is, no joke, referred to as "embarassingly parallel".) You can concurrently map by running each
sample in a separate thread, and when each is done, reduce by averaging them into a final result.
Going further
After implementing everything so far, there is still plenty that you can add. For example, most 3D models aren't made from spheres, so it would be helpful to be able to render polygonal meshes. By
jittering the angle of each ray cast slightly, you can make a nice depth of field effect where objects closer and further from the camera than a focal lenth appear more blurred. You can try rendering
gaseous volumes rather than just solids. You can subdivide the space in front of the camera so that you don't have to check collisions with every object on the screen.
The code for the raytracer I wrote is available on GitHub for reference, although it is also still a work in progress. It's incredibly rewarding to write programs like this where the feedback is so
visual, so I encourage you to try it yourself too! I hope the topics I've covered so far are enough to shed light on what goes into raytracing. Pun intended, of course. | {"url":"https://www.davepagurek.com/blog/how-raytracing-works/","timestamp":"2024-11-06T21:37:43Z","content_type":"text/html","content_length":"19959","record_id":"<urn:uuid:4c6d125b-bf98-477c-8faf-ef1fc154a957>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00322.warc.gz"} |
Stewart Math Textbooks and Online Course Materials
Web Links Coming Soon
Chapter P: PrerequisitesSection P.1:
Modeling the Real World with Algebra
Section P.2:
Real Numbers
Section P.3:
Integer Exponents and Scientific Notation
Section P.4:
Rational Exponents and Radicals
Section P.5:
Algebraic Expressions
Section P.6:
Section P.7:
Rational Expressions
Section P.8:
Solving Basic Equations
Section P.9:
Modeling with Equations | {"url":"https://stewartmath.com/media/13_inside_chapters.php?show_cat=2","timestamp":"2024-11-09T03:13:12Z","content_type":"text/html","content_length":"9695","record_id":"<urn:uuid:76281553-5ef3-46f3-8df0-2970df9509ac>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00407.warc.gz"} |
17 Reasons Why Your XLOOKUP is Not Working
Last updated on February 9, 2023
This tutorial will demonstrate how to debug XLOOKUP formulas in Excel. If your version of Excel does not support XLOOKUP, read how to use the VLOOKUP instead.
Most errors in our XLOOKUP Formulas are related to XLOOKUP’s properties and criteria. The syntax of the XLOOKUP Function only shows us the necessary arguments to perform the function, but it doesn’t
completely inform us about the properties and criteria that are necessary for it to work correctly.
Therefore, in this tutorial, we won’t only learn how to diagnose XLOOKUP errors, but we’ll also understand more about its properties and criteria.
#N/A Error
If the XLOOKUP Function fails to find a match, it will return the #N/A Error. Let’s diagnose the problem.
1. #N/A – No Exact Match
By default, the XLOOKUP Function looks for an exact match. If the item is not within the lookup array, then it will return the #N/A Error.
2. #N/A – No Approximate Match
If the match_mode (i.e., 5^th argument) is set to -1, the XLOOKUP Function will look for the exact match first, but if there’s no exact match, it will find the largest value from the lookup array
that is less than the lookup value. Therefore, if there’s no exact match and all values from the lookup array are greater than the lookup value, the XLOOKUP Function will return the #N/A Error.
Instead of largest value <= lookup value, we can also look for the opposite, smallest value >= lookup value, if we set the match_mode to 1. The latter condition will find the smallest value that is
greater than or equal to the lookup value, and if all values are less than the lookup value, the #N/A Error is returned:
If the value truly does not exist, then the formula is working properly. We recommend adding an error handling so that if the value is not found, a different value is outputted instead of the #N/A
=XLOOKUP(G3,B3:B7,C3:C7,"No match!",1)
However, if the lookup value exists and the XLOOKUP Function can’t find it, here are some possible reasons:
3. #N/A – Numbers Stored as Text (and Other Data-type Mismatches)
One of the important criteria of XLOOKUP is that the data types of the lookup value and lookup array must be the same. If not, the XLOOKUP Function won’t be able to find a match.
The most common example of this is numbers stored as text.
One way to solve this is to use the Text to Columns Tool of Excel to convert numbers stored as text into numbers.
Here are the steps:
1. Highlight the cells and go to Data > Data Tools > Text to Columns
2. In the popup window, select Delimited and click Next.
3. In the next step, select Tab and click Next.
4. In the last step, select the required data type (e.g., date) and the format and click Finish.
5. The range will be converted into the set data type (e.g., date)
4. #N/A – Extra Spaces
Text lookups are prone to errors due to extra spaces. In this example, the lookup value, “Sub 2,” contains two spaces, and therefore, it won’t match with “Sub 2” (one space) from the lookup array.
One way to solve this is by using the TRIM Function to remove the extra spaces. We can apply it to both the lookup value and the whole lookup array.
Note: The TRIM Function removes extra spaces between words until there’s a one space boundary between them, and any spaces before the first word and after the last word of a text or phrase are
5. #N/A – Lookup Array Not Sorted
If the 6^th argument (i.e., Search_Mode) of the XLOOKUP Function is set to either 2 or -2, the XLOOKUP Function will use the binary search method to look up the value, and this method requires a
sorted data set (i.e., 2 – ascending order, -2 – descending order). If the lookup array is not sorted, the XLOOKUP Function will return either a wrong value or the #N/A Error:
To solve the above problem, we need to sort the data either manually or through formulas:
Sort Manually
1. Highlight the whole data, and then go to Data Tab > Sort & Filter > Click Sort.
2. A pop-up window will appear. Select the column of the lookup array and set the order to the required order (e.g., ascending or oldest to newest for dates). Click OK.
3. The data will be sorted based on the lookup array (e.g., B3:B7). The XLOOKUP is now recalculated and shows the correct result.
Sort using the SORT Function and SORTBY Function
We can also use the SORT Function and SORTBY Function to sort the lookup array (e.g., B3:B7) and return array (e.g., C3:C7), respectively.
Note: By default, both SORT and SORTBY functions sort an array in ascending order. The main difference between the two is that the SORT Function will always return all columns within an array while
the SORTBY Function can return a specific column (e.g., C3:C7) from an array.
#VALUE! Error
If an input doesn’t satisfy the criteria for performing XLOOKUP, the function won’t work and will instead return the #VALUE! Error.
6. #VALUE! – Non-Uniform Row Sizes
A requirement of the XLOOKUP Function (for a typical vertical lookup) is that the row sizes of the lookup array and the return array must be the same.
7. #VALUE! – Horizontal vs. Vertical
The return array can be 2-dimensional and return multiple columns (or rows for a horizontal lookup). However, the number of rows (or columns) must match.
This property enables the XLOOKUP Function to return more than one column (or row for a horizontal lookup), but the consequence is that we can’t pair a 1D horizontal return array to a 1D vertical
lookup array unlike the INDEX-MATCH Formula where we can do opposite orientations:
If you do this, the XLOOKUP will accept the return array as a 2D input with a row size not matching the row size of the lookup array. Therefore, it returns the #VALUE! Error.
To solve this, we can use the TRANSPOSE Function to transpose the orientation of one of the 1D arrays:
Note: The TRANSPOSE Function switches the relative row and column coordinates of a cell in a list. In E2:I2, F2 is row 1, column 2 relative to the range. Therefore, the transposed coordinates will be
row 2, column 1. G2 is row 1, column 3 and is transposed to row 3, column 1 and so on.
8. #VALUE – Value Range
The 5^th (match mode) and 6^th (search mode) arguments of the XLOOKUP Function must have valid inputs. If not, then the XLOOKUP will return the #VALUE! Error.
Note: The match_mode can only accept 0, -1,1 and 2 while the search_mode can only accept 1,-1,2 and-2.
#NAME? Error
The #NAME? Error is triggered by:
• Misspelling the function’s name
• Misspelling a reference (workbook/sheet reference and named ranges).
• A non-existent Named Range
• Text not enclosed with double quotation marks.
9. #NAME? – Function Name Typo
If there’s a typo in a function’s name, Excel will return the #NAME? Error.
10. #NAME? – Named Range doesn’t Exist
The #NAME? Error can also be caused by an undefined named range in the formula. It’s either the named range doesn’t really exist or there’s a typo in the name.
Note: There are two named ranges in the above sheet: Subscription and Price. The typo in the Price named range results to a named range that doesn’t exist, and any text that is not enclosed with
quotation marks is considered as a named range, which can also lead to the #NAME? Error.
11. #NAME? – Workbook/Sheet Reference Typo
When workbook/sheet names contain spaces and special characters except for underscore, we need to enclose the workbook/sheet reference with single quotations. If this is not satisfied, then Excel
can’t recognize the workbook/sheet reference and will return the #NAME? Error.
#SPILL! ERROR
There’s also a set of criteria when returning an array output, and if those criteria are not satisfied, the SPILL! Error is returned instead of the array output.
12. #SPILL! – Spill Block
A dynamic array formula will not overwrite the values that are within its spill range. Instead, the array output will be blocked, and the #SPILL! Error is returned.
13. #SPILL! – Table vs. Dynamic Arrays
We can’t use Array XLOOKUP Formulas in Tables because they don’t support dynamic array formulas.
Instead you must use a non-array XLOOKUP Function where the lookup_value is a single value, not an array of values.
14. #SPILL! – Range Out of Bounds
If the size of the output array from an XLOOKUP formula exceeds the sheet boundaries (row and column), the #SPILL! Is returned instead. (e.g, F3:F < whole column array)
Other Problems
There are XLOOKUP problems that don’t trigger errors because they don’t violate any of the criteria.
15. Incorrect Range
Even if the row sizes of the lookup array and return array are the same, if their positions do not match with each other, the XLOOKUP will return an incorrect result.
16. 1D/2D Array of Lookup Values vs. 2D Return Array
The lookup value in XLOOKUP can be a 1D or 2D array, which converts the XLOOKUP Function into a dynamic array that will Spill into adjacent cells.
We might expect that a combination of a 1D lookup value array (e.g., F3:F4) and a 2D return array (e.g., C3:D7) will return a 2D output where the row size and column size are dependent on the lookup
value array and 2D return array, respectively, but this is not the case. The row size will still be based on the 1D lookup value array, but the column size will be 1, which means that only the 1^st
column of the return array will be returned.
17. Copying XLOOKUP with Relative References
If we drag an XLOOKUP Formula with relative references, then the references will also adjust relative to their position from the reference formula, which can lead to incorrect results.
If we want to copy or drag our XLOOKUP Formula to succeeding cells, we must convert the references for the lookup array and return array to absolute references. We can do this by adding the dollar
symbol at the front of the column letter and row number or by pressing F4 while the cursor is in the reference within the formula. | {"url":"https://www.automateexcel.com/formulas/17-reasons-why-xlookup-is-not-working/","timestamp":"2024-11-14T13:53:53Z","content_type":"text/html","content_length":"167998","record_id":"<urn:uuid:62269214-1086-47e3-afcc-15780c8791b8>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00660.warc.gz"} |
Nt1330 Project 4
Project 4 Fall 2012
1. Open the data file called JCrew on Blackboard under the Assignments link.
2. Get a 4 point Moving Average for the data using Time Series Analysis.
3. Highlight the Revenue column and the 4MA column. Insert /Line.
4. Go back to the data. Time Series Analysis/ Exponential Smoothing. Use alpha of .7.
5. Highlight Revenue and Smoothed and Insert /Line.
6. Go back to the data. Time Series Analysis/ Trendline / pick Exp Ln. Check the Scatterplot and all boxes on the right side.
7. Finally, go back to the data and choose Time Series/ Deseasonalize.
1. Compare the 4 point moving average chart to the exponentially smoothed one. Which one shows the SECULAR trend better? Explain. The four point moving average shows the secular trend …show more
content… better because its values aren’t as volatile as they are in the exponentially smoothed model.
2. What is the forecasted revenue for JCrew in Quarter I of 2010 using Exponential Smoothing? 377.388 in Q1 of 2010
Look at the Logged Model
3. What percent of the variation in Revenue is explained by Time? 84% of the variation is explained by time
4. By how much does Revenue change per quarter on average? Revenue changes by 4.6% per quarter on average
5. Are there any outliers (suspicious or definite)? There is one outlier at time period 4, but it is only suspicious
6. Is Autocorrelation a problem? No because the Durbin-Watson is 2.77 therefore reject fail to reject H0
H0: No residual correlation (p=0)
H1: Positive residual correlation (p>1)
7. Does the data seem to fit the plot well? Explain. Yes it fits the plot well in general. There is one suspicious value that skews the plot.
Look at the Deseasonalized Model
8. What is the secular trendline? y=10.15x + 139.39
9. How well does the model explain JCrew’s revenue? 94.82% of the variation in Jcrew’s revenue is explained by the model
10. Which quarter is most prosperous for JCrew?
1st Quarter is the most prosperous for Jcrew with a seasonal index of .898
11. Fill in the following table:
|2010 |t |Predicted |SI |Forecast |
|QI | 21 |352.54 | .898 | 316.58 |
|QII | 22 | 362.69 | .968 |351.08 |
|QIII | 23 | 372.84 | .938 | 349.72 |
|QIV | 24 | 382.99 | 1.196 | 458.06 | | {"url":"https://www.studymode.com/essays/Nt1330-Project-4-C8CEFA119E3E16EB.html","timestamp":"2024-11-11T07:44:59Z","content_type":"text/html","content_length":"98602","record_id":"<urn:uuid:d989892a-9159-47b6-b48e-a15f1043f101>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00508.warc.gz"} |
How to Find Slope with Two Points (Helpful Guide)
If you want to calculate and find points of the slope line, you will need to know how to find the rise and run of the line. Here is an example. Consider two points with y-coordinates of -2,3 and 4,5.
People who are professionals can put plug these 2 values into the slope formula. You can also make a graph to understand the slope formula. The article will guide you about the calculation of
different slopes and more.
What is the Slope Formula?
Two points can be used to calculate the slope using the slope formula.
The slope is defined as the difference between the rise and the run. In other words, the rise is divided by the run.
With the slope formula, we can take the rise change and divide it by the run change
Take a look…
First, let’s look at our two points. We must be able to label our two points to define a generic formula. We will differentiate between the first and second points by using subscripts.
• Given any 2 points = (x[1], y[1]) (x[2], y[2])
• X[1] = Refer to point one
• Y[2 ]= Refer to point two
How To Find the Points of the Slope Line
There are two points on a line, point 1 and point 2, and we need to know the slope of the line from these points. We can do this by using the slope formula. The slope of a line is the difference
between the x and y values. The slope formula requires that you know the sign of each value.
There are three ways to calculate the slope of a line:
1. point-intercept form
2. slope-intercept form
3. Standard form
The first method involves finding the ratio of two coordinates. Then, we need to multiply the resultant value by the slope. The slope measures how much the line rises or falls in a given direction
over a certain distance.
Once you have found the slope of a line, you need to find the y-intercept of that line. This is easily done by plugging the y-intercept into the y-intercept formula.
How to Calculate the Rise & Run of A Line
You can use the rise-and-run method to determine the slope if you have two points on a line. A line’s slope is the height change over a horizontal distance. This slope can be positive or negative. If
it is positive, the line is rising. On the other hand, if it is negative, the line is falling.
When you want to find the slope of a line, you must know how to calculate both the rise and run. The rise and run are the differences between the y-coordinates of the two points. The slope of a line
is the difference between the rise and run. The rise is the distance up or down, while the run is the distance from left to right. You can practice calculating the slope of a line with two points by
trying to find different points on the line.
If the line has two points, the slope is the difference in elevation between the two points. The difference between the points can be determined with an altimeter or a topographic map.
If you know that the slope of the line is 900 feet, you can subtract 500 feet from it to find the rise. You can then measure the distance between the top of the hill and the bottom of the hill to get
the run of the line.
How to Plug The Y-Coordinates into The Slope Formula
The slope is calculated by dividing the change in the y value by the change in the x value, which can be found using the slope formula. The formula is easy to remember and can be applied to any two
points along a line. To use the formula, you must know the x and y values of the first point. Then, plug the y-coordinates into the formula.
Now, you must solve the equation for slope with two points. The x-coordinate of x[1] must be replaced with y[1,] and the y-coordinate of y[1] must be replaced by the corresponding negative or
positive expression. The slope of y[1] is equal to m. If the slope of x[1] is a positive value, then the slope is positive. Alternatively, if the slope of x[1] is negative, then the slope is
negative. The equation is complete if m is the slope of the line.
Once you know the slope of a line, plug the y-coordinates into the formula for the slope with two points. Then, the y-intercept is a point on the line. To find the slope of a line, you can plug in
the x and y-coordinates of one point, the x-coordinates of another point, and b. Once these three points are, you can use the slope formula to determine the y-intercept.
How to Calculate the Length of a Slope
To find the slope length, you will first need to know the y-coordinates of both points and the slope. Moreover, the slope can be positive or negative. A positive slope moves upward, and a negative
slope moves downward. You can use the formula below to find the slope length for two points.
If you know the length of the slope, you can use it to draw a right angle. For example, if the slope is m = 5, the diagonal would be tan-1(5). You can also use this method to find the slope’s
Besides the y-coordinate, you will also need to know the direction. The direction determines whether the slope is up or down. The slope is upwards if you move from point A to point B. If you move
from point B to point C, then the slope is downwards.
Conclusion on Find the Slope Line of Two Points
A slope is a basic engineering tool to determine how much work it takes to raise or lower something over a distance. You can calculate the slope using the formula: y = mx + b. The slope of a line is
calculated using the following equation: where m is the slope, x is the x-intercept, and y is the y-intercept. If the x-intercept is not 0, the slope is positive; if the y-intercept is not 0, the
slope is negative.
To measure the slope of something, you need two points on the line. One point should be your starting point. However, the second point should be where you want to end up. The slope is measured on a
number line that ranges from -180 to 180 degrees. You can use the equation above to determine whether the line that is being measured has a positive or negative slope. | {"url":"http://higheducations.com/how-to-find-slope-with-two-points/","timestamp":"2024-11-02T18:06:06Z","content_type":"text/html","content_length":"92746","record_id":"<urn:uuid:e948a8da-e263-4d9d-b1bc-6dcc9b2b5342>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00867.warc.gz"} |
Guess the number
Alfredo and Beatriz are playing the well known game “Guess the number”. Alfredo has thought a number between 1 and 500, and Beatriz has 10 attempts to guess it. For each attempt of Beatriz, Alfredo
has to answer “greater” (if Beatriz said a number less than the one that Alfredo thought) or “less” (if the said number was greater).
Your task in this problem is the following one: assuming that Beatriz has already done 9 attempts, none of them has been succesful, you must find out if these attempts are enough to find out the
number thought by Alfredo (and, therefore, Beatriz will guess in the tenth attempt if she says that number) or if, otherwise, it could be many possible answers.
The input contains in a line the number n of cases. n lines follow, each one of them describes a case. A case is formed by 9 pairs g a, separated by spaces, where g is a number between 1 and 500
proposed by Beatriz and a is the answer that Alfredo gave: the signal ‘+’ indicates that the answer of Alfredo is “Greater”, while the signal ‘-’ indicates “Less”.
Alfredo never lies nor is wrong in his answers.
Output The output consists of n lines, each one of them gives the solution of a case. Your program must print the number in a line if you can certainly know the number thought by Alfredo. Otherwise,
it must print ‘?’ in a line (Follow the format of the instances). | {"url":"https://jutge.org/problems/P67099_en","timestamp":"2024-11-13T01:05:03Z","content_type":"text/html","content_length":"24328","record_id":"<urn:uuid:351209e8-919e-479a-976d-f886a9447263>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00408.warc.gz"} |
Introduction To Data Science
• The life cycle of Data Science
• Statistical Learning
• Measures of central tendency
• Measures of dispersion
• Probability theory
• Hypothesis testing,
• ANOVA
• Types of graphs and
• plots
R Programming
• R Environment Setup and Essentials
• Installing R for the Windows, Linux and Mac
• Exploratory data analysis
• Basic operators in R
• Data Manipulation
• Data visualisation
• Followed byHands-on Exercise
• Python language Basic
• Constructs
• OOP concepts in Python
• Hands-on Exercise – important concepts in OOP like polymorphism, inheritance, encapsulation, Python functions, return types, and parameters, Lambda expressions
• NumPy for mathematical computing
• Hands-on Exercise – How to import NumPy module, creating an array using ND-array
• calculating standard deviation on an array of numbers, calculating the correlation between two variables.
• SciPy for scientific computing
• Hands-on Exercise – Importing of SciPy, applying the Bayes theorem on the given dataset.
• Matplotlib for data visualization
• Hands-on Exercise – deploying MatPlotLib for creating Pie
• Scatter, Line, Histogram.
• Pandas for data analysis and machine learning
• Hands-on Exercise – working on importing data files, selecting record by a group, applying a filter on top, viewing records, analyzing with linear regression, and creation of time series.
• Python Environment Setup and Essentials Installing Python Anaconda for the Windows, Linux and Mac with Hands-on Exercise
Machine Learning
• Introduction to Machine
• Learning with R and
• Python
• The need for Machine Learning,
• Introduction to Machine
• Learning, types of Machine
• Learning, such as supervised
• unsupervised and reinforcement learning, why Machine Learning with Python, R and applications of Machine Learning.
• Supervised Learning and
• Linear Regression
• Hands-on Exercise – Implementing linear regression from scratch with R and Python, Using Python library Scikit-learn to perform simple linear regression and multiple linear regression,
Implementing train– test split and predicting the values on the test set.
• Classification and Logistic Regression
• Hands-on Exercise – Implementing logistic regression from scratch with R and Python, Using Python library Scikit-learn to perform simple logistic regression and multiple logistic regression,
Building a confusion matrix to find out the accuracy, true positive rate, and false-positive rate.
• Decision Tree and Random Forest
• Hands-on Exercise – Implementing a decision tree from scratch in R and Python, Using Python library Scikit-learn to build a decision tree and a random forest, Visualizing the tree and changing
the hyperparameters in the random forest.
• Naïve Bayes and Support Vector Machine
• Hands-on Exercise – Using Python library Scikit-learn to build a Naïve Bayes classifier and a support vector classifier.
• Unsupervised Learning
• Hands-on Exercise – Using Python library Scikit-learn to implement K-means clustering, Implementing PCA (principal component analysis) on top of a dataset.
Deep Learning As Part AI
• Natural Language Processing and Text Mining
• Project Time Series Analysis
• Hands-on Exercise – Analyzing time series data, the sequence of measurements that follow a non-random order to recognize the nature of the phenomenon, and forecasting the future values in the
• Tableau for data visualisation:
• Tableau Introduction
• Working on data with Tableau
• Dashboards using Tableau (hands-on)
• Stories in Tableau (hands-on) | {"url":"https://skillatwill.com/courses/data%20science/data%20science%20with%20r,%20python%20and%20tableau/saw%20freelance%20trainer/328","timestamp":"2024-11-04T07:11:07Z","content_type":"text/html","content_length":"187005","record_id":"<urn:uuid:a2aa02aa-331c-4f06-be55-efb362be3a7c>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00294.warc.gz"} |
Metric (mathematics)
Jump to navigation Jump to search
In mathematics, a metric or distance function is a function that defines a distance between each pair of elements of a set. A set with a metric is called a metric space. A metric induces a topology
on a set, but not all topologies can be generated by a metric. A topological space whose topology can be described by a metric is called metrizable.
An important source of metrics in differential geometry are metric tensors, bilinear forms that may be defined from the tangent vectors of a differentiable manifold onto a scalar. A metric tensor
allows distances along curves to be determined through integration, and thus determines a metric. However, not every metric comes from a metric tensor in this way.
A metric on a set X is a function (called the distance function or simply distance)
${\displaystyle d:X\times X\to [0,\infty )}$,
where ${\displaystyle [0,\infty )}$ is the set of non-negative real numbers and for all ${\displaystyle x,y,z\in X}$, the following conditions are satisfied:
1. ${\displaystyle d(x,y)\geq 0}$ non-negativity or separation axiom
2. ${\displaystyle d(x,y)=0\Leftrightarrow x=y}$ identity of indiscernibles
3. ${\displaystyle d(x,y)=d(y,x)}$ symmetry
4. ${\displaystyle d(x,z)\leq d(x,y)+d(y,z)}$ subadditivity or triangle inequality
Conditions 1 and 2 together define a positive-definite function. The first condition is implied by the others.
A metric is called an ultrametric if it satisfies the following stronger version of the triangle inequality where points can never fall 'between' other points:
${\displaystyle d(x,z)\leq \max(d(x,y),d(y,z))}$
for all ${\displaystyle x,y,z\in X}$
A metric d on X is called intrinsic if any two points x and y in X can be joined by a curve with length arbitrarily close to d(x, y).
For sets on which an addition + : X × X → X is defined, d is called a translation invariant metric if
${\displaystyle d(x,y)=d(x+a,y+a)}$
for all x, y, and a in X.
These conditions express intuitive notions about the concept of distance. For example, that the distance between distinct points is positive and the distance from x to y is the same as the distance
from y to x. The triangle inequality means that the distance from x to z via y is at least as great as from x to z directly. Euclid in his work stated that the shortest distance between two points is
a line; that was the triangle inequality for his geometry.
${\displaystyle d(x,y)=\sum _{n=1}^{\infty }{\frac {1}{2^{n}}}{\frac {p_{n}(x-y)}{1+p_{n}(x-y)}}}$
is a metric defining the same topology. (One can replace ${\displaystyle {\frac {1}{2^{n}}}}$ by any summable sequence ${\displaystyle (a_{n})}$ of strictly positive numbers.)
• Graph metric, a metric defined in terms of distances in a certain graph.
• The Hamming distance in coding theory.
• Riemannian metric, a type of metric function that is appropriate to impose on any differentiable manifold. For any such manifold, one chooses at each point p a symmetric, positive definite,
bilinear form L: T[p] × T[p] → ℝ on the tangent space T[p] at p, doing so in a smooth manner. This form determines the length of any tangent vector v on the manifold, via the definition ||v|| = √
L(v, v). Then for any differentiable path on the manifold, its length is defined as the integral of the length of the tangent vector to the path at any point, where the integration is done with
respect to the path parameter. Finally, to get a metric defined on any pair {x, y} of points of the manifold, one takes the infimum, over all paths from x to y, of the set of path lengths. A
smooth manifold equipped with a Riemannian metric is called a Riemannian manifold.
• The Fubini–Study metric on complex projective space. This is an example of a Riemannian metric.
• String metrics, such as Levenshtein distance and other string edit distances, define a metric over strings.
• Graph edit distance defines a distance function between graphs.
• The Wasserstein metric is a distance function defined between two probability distributions.
• The Finsler metric is a continuous nonnegative function F:TM→[0,+∞) defined on the tangent bundle.
Equivalence of metrics[edit]
For a given set X, two metrics d[1] and d[2] are called topologically equivalent (uniformly equivalent) if the identity mapping
id: (X,d[1]) → (X,d[2])
is a homeomorphism (uniform isomorphism).
For example, if ${\displaystyle d}$ is a metric, then ${\displaystyle \min(d,1)}$ and ${\displaystyle {d \over 1+d}}$ are metrics equivalent to ${\displaystyle d.}$
See also notions of metric space equivalence.
Metrics on vector spaces[edit]
Norms on vector spaces are equivalent to certain metrics, namely homogeneous, translation-invariant ones. In other words, every norm determines a metric, and some metrics determine a norm.
Given a normed vector space ${\displaystyle (X,\|\cdot \|)}$ we can define a metric on X by
${\displaystyle d(x,y):=\|x-y\|}$.
The metric d is said to be induced by the norm ${\displaystyle \|\cdot \|}$.
Conversely if a metric d on a vector space X satisfies the properties
• ${\displaystyle d(x,y)=d(x+a,y+a)}$ (translation invariance)
• ${\displaystyle d(\alpha x,\alpha y)=|\alpha |d(x,y)}$ (homogeneity)
then we can define a norm on X by
${\displaystyle \|x\|:=d(x,0)}$
Similarly, a seminorm induces a pseudometric (see below), and a homogeneous, translation invariant pseudometric induces a seminorm.
Metrics on multisets[edit]
We can generalize the notion of a metric from a distance between two elements to a distance between two nonempty finite multisets of elements. A multiset is a generalization of the notion of a set
such that an element can occur more than once. Define ${\displaystyle Z=XY}$ if ${\displaystyle Z}$ is the multiset consisting of the elements of the multisets ${\displaystyle X}$ and ${\displaystyle
Y}$, that is, if ${\displaystyle x}$ occurs once in ${\displaystyle X}$ and once in ${\displaystyle Y}$ then it occurs twice in ${\displaystyle Z}$. A distance function ${\displaystyle d}$ on the set
of nonempty finite multisets is a metric^[1] if
1. ${\displaystyle d(X)=0}$ if all elements of ${\displaystyle X}$ are equal and ${\displaystyle d(X)>0}$ otherwise (positive definiteness), that is, (non-negativity plus identity of indiscernibles)
2. ${\displaystyle d(X)}$ is invariant under all permutations of ${\displaystyle X}$ (symmetry)
3. ${\displaystyle d(XY)\leq d(XZ)+d(ZY)}$ (triangle inequality)
Note that the familiar metric between two elements results if the multiset ${\displaystyle X}$ has two elements in 1 and 2 and the multisets ${\displaystyle X,Y,Z}$ have one element each in 3. For
instance if ${\displaystyle X}$ consists of two occurrences of ${\displaystyle x}$, then ${\displaystyle d(X)=0}$ according to 1.
A simple example is the set of all nonempty finite multisets ${\displaystyle X}$ of integers with ${\displaystyle d(X)=\max\{x:x\in X\}-\min\{x:x\in X\}}$. More complex examples are information
distance in multisets;^[1] and normalized compression distance (NCD) in multisets.^[2]
Generalized metrics[edit]
There are numerous ways of relaxing the axioms of metrics, giving rise to various notions of generalized metric spaces. These generalizations can also be combined. The terminology used to describe
them is not completely standardized. Most notably, in functional analysis pseudometrics often come from seminorms on vector spaces, and so it is natural to call them "semimetrics". This conflicts
with the use of the term in topology.
Extended metrics[edit]
Some authors allow the distance function d to attain the value ∞, i.e. distances are non-negative numbers on the extended real number line. Such a function is called an extended metric or "∞-metric".
Every extended metric can be transformed to a finite metric such that the metric spaces are equivalent as far as notions of topology (such as continuity or convergence) are concerned. This can be
done using a subadditive monotonically increasing bounded function which is zero at zero, e.g. d′(x, y) = d(x, y) / (1 + d(x, y)) or d′′(x, y) = min(1, d(x, y)).
The requirement that the metric take values in [0,∞) can even be relaxed to consider metrics with values in other directed sets. The reformulation of the axioms in this case leads to the construction
of uniform spaces: topological spaces with an abstract structure enabling one to compare the local topologies of different points.
A pseudometric on X is a function d : X × X → R which satisfies the axioms for a metric, except that instead of the second (identity of indiscernibles) only d(x,x)=0 for all x is required. In other
words, the axioms for a pseudometric are:
1. d(x, y) ≥ 0
2. d(x, x) = 0 (but possibly d(x, y) = 0 for some distinct values x ≠ y.)
3. d(x, y) = d(y, x)
4. d(x, z) ≤ d(x, y) + d(y, z).
In some contexts, pseudometrics are referred to as semimetrics because of their relation to seminorms.
Occasionally, a quasimetric is defined as a function that satisfies all axioms for a metric with the possible exception of symmetry:^[3]^[4]. The name of this generalisation is not entirely
1. d(x, y) ≥ 0 (positivity)
2. d(x, y) = 0 if and only if x = y (positive definiteness)
3. [DEL:d(x, y) = d(y, x):DEL] (symmetry, dropped)
4. d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality)
Quasimetrics are common in real life. For example, given a set X of mountain villages, the typical walking times between elements of X form a quasimetric because travel up hill takes longer than
travel down hill. Another example is a taxicab geometry topology having one-way streets, where a path from point A to point B comprises a different set of streets than a path from B to A.
A quasimetric on the reals can be defined by setting
d(x, y) = x − y if x ≥ y, and
d(x, y) = 1 otherwise. The 1 may be replaced by infinity or by ${\displaystyle 1+10^{(y-x)}}$.
The topological space underlying this quasimetric space is the Sorgenfrey line. This space describes the process of filing down a metal stick: it is easy to reduce its size, but it is difficult or
impossible to grow it.
If d is a quasimetric on X, a metric d' on X can be formed by taking
d'(x, y) = ^1⁄[2](d(x, y) + d(y, x)).
In a metametric, all the axioms of a metric are satisfied except that the distance between identical points is not necessarily zero. In other words, the axioms for a metametric are:
1. d(x, y) ≥ 0
2. d(x, y) = 0 implies x = y (but not vice versa.)
3. d(x, y) = d(y, x)
4. d(x, z) ≤ d(x, y) + d(y, z).
Metametrics appear in the study of Gromov hyperbolic metric spaces and their boundaries. The visual metametric on such a space satisfies d(x, x) = 0 for points x on the boundary, but otherwise d(x, x
) is approximately the distance from x to the boundary. Metametrics were first defined by Jussi Väisälä.^[6]
A semimetric on X is a function d : X × X → R that satisfies the first three axioms, but not necessarily the triangle inequality:
1. d(x, y) ≥ 0
2. d(x, y) = 0 if and only if x = y
3. d(x, y) = d(y, x)
Some authors work with a weaker form of the triangle inequality, such as:
d(x, z) ≤ ρ (d(x, y) + d(y, z)) (ρ-relaxed triangle inequality)
d(x, z) ≤ ρ max(d(x, y), d(y, z)) (ρ-inframetric inequality).
The ρ-inframetric inequality implies the ρ-relaxed triangle inequality (assuming the first axiom), and the ρ-relaxed triangle inequality implies the 2ρ-inframetric inequality. Semimetrics satisfying
these equivalent conditions have sometimes been referred to as "quasimetrics",^[7] "nearmetrics"^[8] or inframetrics.^[9]
The ρ-inframetric inequalities were introduced to model round-trip delay times in the internet.^[9] The triangle inequality implies the 2-inframetric inequality, and the ultrametric inequality is
exactly the 1-inframetric inequality.
Relaxing the last three axioms leads to the notion of a premetric, i.e. a function satisfying the following conditions:
1. d(x, y) ≥ 0
2. d(x, x) = 0
This is not a standard term. Sometimes it is used to refer to other generalizations of metrics such as pseudosemimetrics^[10] or pseudometrics;^[11] in translations of Russian books it sometimes
appears as "prametric".^[12]
Any premetric gives rise to a topology as follows. For a positive real r, the r-ball centered at a point p is defined as
B[r](p) = { x | d(x, p) < r }.
A set is called open if for any point p in the set there is an r-ball centered at p which is contained in the set. Every premetric space is a topological space, and in fact a sequential space. In
general, the r-balls themselves need not be open sets with respect to this topology. As for metrics, the distance between two sets A and B, is defined as
d(A, B) = inf[x∊A, y∊B] d(x, y).
This defines a premetric on the power set of a premetric space. If we start with a (pseudosemi-)metric space, we get a pseudosemimetric, i.e. a symmetric premetric. Any premetric gives rise to a
preclosure operator cl as follows:
cl(A) = { x | d(x, A) = 0 }.
The prefixes pseudo-, quasi- and semi- can also be combined, e.g., a pseudoquasimetric (sometimes called hemimetric) relaxes both the indiscernibility axiom and the symmetry axiom and is simply a
premetric satisfying the triangle inequality. For pseudoquasimetric spaces the open r-balls form a basis of open sets. A very basic example of a pseudoquasimetric space is the set {0,1} with the
premetric given by d(0,1) = 1 and d(1,0) = 0. The associated topological space is the Sierpiński space.
Sets equipped with an extended pseudoquasimetric were studied by William Lawvere as "generalized metric spaces".^[13]^[14] From a categorical point of view, the extended pseudometric spaces and the
extended pseudoquasimetric spaces, along with their corresponding nonexpansive maps, are the best behaved of the metric space categories. One can take arbitrary products and coproducts and form
quotient objects within the given category. If one drops "extended", one can only take finite products and coproducts. If one drops "pseudo", one cannot take quotients. Approach spaces are a
generalization of metric spaces that maintains these good categorical properties.
Important cases of generalized metrics[edit]
In differential geometry, one considers a metric tensor, which can be thought of as an "infinitesimal" quadratic metric function. This is defined as a nondegenerate symmetric bilinear form on the
tangent space of a manifold with an appropriate differentiability requirement. While these are not metric functions as defined in this article, they induce what is called a pseudo-semimetric function
by integration of its square root along a path through the manifold. If one imposes the positive-definiteness requirement of an inner product on the metric tensor, this restricts to the case of a
Riemannian manifold, and the path integration yields a metric.
In general relativity the related concept is a metric tensor (general relativity) which expresses the structure of a pseudo-Riemannian manifold. Though the term "metric" is used in cosmology, the
fundamental idea is different because there are non-zero null vectors in the tangent space of these manifolds. This generalized view of "metrics", in which zero distance does not imply identity, has
crept into some mathematical writing too:^[15]^[16]
See also[edit]
• Arkhangel'skii, A. V.; Pontryagin, L. S. (1990), General Topology I: Basic Concepts and Constructions Dimension Theory, Encyclopaedia of Mathematical Sciences, Springer, ISBN 3-540-18178-4
• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology, Dover, ISBN 978-0-486-68735-3, MR 0507446, OCLC 32311847
External links[edit] | {"url":"https://static.hlt.bme.hu/semantics/external/pages/endomorfizmus/en.wikipedia.org/wiki/Metric_(mathematics).html","timestamp":"2024-11-09T16:04:35Z","content_type":"text/html","content_length":"140973","record_id":"<urn:uuid:6dd1d012-a638-44fc-8aba-ba7334c8a7b6>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00633.warc.gz"} |
Clustering Kelvin Helmholtz Instabilities
Clustering Kelvin Helmholtz Instabilities¶
Pipeline description¶
This example illustrates the capabilities of TTK to perform advanced statistical analysis of collections of datasets, based on their structural representations, along with the possibility to
interactively explore the outcome of the analysis, with linked views (between the selection in the planar view -- top right -- and the flow visualization -- top left).
This example considers an ensemble of 32 periodic, 2D Kelvin Helmholtz Instabilities in computational fluid dynamics, obtained with various simulation parameters (different solvers, different
numerical schemes, different interpolation orders, etc.). The scalar field of interest is the "Enstrophy". It is an established measure of vorticity. Its prominent maxima denote the center of strong
vortices. Two example members from the ensemble are show on the above screenshot (left). Strong vortices can be visualized with the dark green regions. The simulation parameters as well as the ground
truth classification are provided as metadata for each entry of the database and are carried along the entire pipeline. See this publication for further details.
The goal of this example is to classify the 32 members of the ensemble into two classes, whether they describe the beginning or the end of the turbulence. This task is particularly challenging for
traditional clustering pipelines since turbulent flows are highly chaotic and two flows belonging to the same ground truth class can be drastically different visually (as shown on the above
screenshot -- left). The common denominator between two turbulent flows in the same ground truth class is the distribution of energies of their vortices (i.e. the number and strengths of their
vortices), which describes the turbulence of the flow.
In this context, topological data representations are particularly relevant to extract such subtle structural features. In particular, the persistence diagram involving the saddle-maximum pairs of
the "Enstrophy" (second column, above screenshot) nicely captures the number of vortices as well as their individual strengths. Thus, in the reminder of this example, we will use this persistence
diagram as a descriptor of each turbulent flow and we will proceed to a k-means clustering directly in the Wasserstein metric space of persistence diagrams. For visualization purposes, we will
compute a 2D layout of the ensemble (right most columns, above screenshot) to inspect the resulting classification.
First, the database of turbulent flows is loaded from disk with the CinemaReader module (line 6 of the Python script below). Then an SQL query is performed with CinemaQuery to select a relevant
subset of this database (line 9). Finally the module CinemaProductReader is used to read the actual regular grids corresponding to the result of the previous SQL query. From this point on, the entire
set of 32 turbulent flows will be organized as a vtkMultiBlockDatSet and each of these 32 members will be processed by the rest of the analysis pipeline.
Then for each of the 32 members of the ensemble, the first step consists in marking periodicity boundary conditions with the TriangulationManager (line 21). Next, the "Enstrophy" field of each member
is normalized (between 0 and 1) with the ScalarFieldNormalizer to ease their comparison later. Finally (line 28) the PersistenceDiagram is computed (for the saddle-maximum pairs) to represent each of
the 32 ensemble members by a diagram which encodes the number and the strengths of the vortices via the persistence of the maxima of "Enstrophy".
Next, the clustering of the persistence diagrams in the Wasserstein metric space is performed with the module PersistenceDiagramClustering (line 32).
For visualization purposes, we will then compute a 2D layout of the ensemble, where each ensemble member will be represented by a point and where the 2D distance between 2 points will encode the
Wasserstein distance between their diagrams. This will provide an intuitive planar overview of the ensemble. For this, we will first compute a matrix of Wasserstein distances with the module
PersistenceDiagramDistanceMatrix (line 39). The resulting distance matrix is visualized at the bottom of the middle column in the above screenshot. There, it can be seen that the Wasserstein distance
already identifies two major clusters (large blue sub-matrices of low Wasserstein distances). Next (line 67), the module DimensionReduction is used to compute a 2D layout via multidimensional
scaling. Finally, the resulting table is turned into a 2D point cloud which is ready to be visualized with TableToPoints (line 75). Then, the output is stored to a simple CSV file (line 81).
In the above screenshot, the resulting point cloud is shown in the 2 views at
the bottom right corner of the screenshot. The first view (left) shows the point cloud colored by cluster identifier computed by the pipeline. The second view (right) show the same point cloud,
colored by the ground truth class. There, one can directly visualize that the two classifications are identical and that, therefore, this topological clustering pipeline succeeded.
For reference, a traditional pipeline based on the L2-distance between the "Entrophy" fields is also provided in this example. For that, the module LDistanceMatrix is used (line 45) to compute a
matrix of the L2 distances between each scalar field of the ensemble. The resulting distance matrix is visualized at the top of the middle column in the above screenshot. There, it can be seen that
the L2 distance between the scalar fields fails at identifying any clear clusters (there are no large blue sub-matrices). Next (line 49), the module DimensionReduction is used to compute a 2D layout
via multidimensional scaling. The resulting table is turned into a 2D point cloud with TableToPoints (line 55). Finally, the k-means algorithm is run on this 2D point cloud with the module KMeans.
Then, the output is stored to a simple CSV file (line 83). The resulting clustering can be visualized with the two views at the top right corner of the above screenshot. The ground-truth
classification is provided by the color coding of the points in the second view (right) while the classification computed with this traditional pipeline is shown in the first view (left). There, it
can bee seen that the coloring of the two point clouds differ, indicating an incorrect classification by the traditional kMeans algorithm. In particular, since all the metadata associated with each
ensemble member travels down the analysis pipeline, one can select points in these planar views to inspect the corresponding datasets and persistence diagrams (left two columns of the screenshot). In
particular, two members (red and yellow spheres) incorrectly marked as belonging to different classes are visualized on the left. There, one can see that although the two flows have the same
"profile" of vortices (number and strengths), these are located in drastically different spots of the field, due to the chaotic nature of turbulence, hence explaining the reason of failure of
traditional clustering pipelines.
To reproduce the above screenshot, go to your ttk-data directory and enter the following command:
paraview states/clusteringKelvinHelmholtzInstabilities.pvsm
Python code¶
1 #!/usr/bin/env python
3 from paraview.simple import *
5 # create a new 'TTK CinemaReader'
6 tTKCinemaReader1 = TTKCinemaReader(DatabasePath="khi.cdb")
8 # create a new 'TTK CinemaQuery'
9 tTKCinemaQuery1 = TTKCinemaQuery(InputTable=tTKCinemaReader1)
10 tTKCinemaQuery1.SQLStatement = """
11 SELECT * FROM InputTable0
12 WHERE Resolution='512'
13 AND (Time='0' OR Time='2')
14 AND (NOT (Solver='hll'))"""
16 # create a new 'TTK CinemaProductReader'
17 tTKCinemaProductReader1 = TTKCinemaProductReader(Input=tTKCinemaQuery1)
19 # create a new 'TTK TriangulationManager'
20 tTKTriangulationManager1 = TTKTriangulationManager(Input=tTKCinemaProductReader1)
21 tTKTriangulationManager1.PeriodicityinAllDimensions = 1
23 # create a new 'TTK ScalarFieldNormalizer'
24 tTKScalarFieldNormalizer1 = TTKScalarFieldNormalizer(Input=tTKTriangulationManager1)
25 tTKScalarFieldNormalizer1.ScalarField = ["POINTS", "Enstrophy"]
27 # create a new 'TTK PersistenceDiagram'
28 tTKPersistenceDiagram1 = TTKPersistenceDiagram(Input=tTKScalarFieldNormalizer1)
29 tTKPersistenceDiagram1.ScalarField = ["POINTS", "Enstrophy"]
31 # create a new 'TTK PersistenceDiagramClustering'
32 tTKPersistenceDiagramClustering1 = TTKPersistenceDiagramClustering(
33 Input=tTKPersistenceDiagram1
34 )
35 tTKPersistenceDiagramClustering1.Criticalpairsusedfortheclustering = "saddle-max pairs"
36 tTKPersistenceDiagramClustering1.Numberofclusters = 2
38 # create a new 'TTK PersistenceDiagramDistanceMatrix'
39 tTKPersistenceDiagramDistanceMatrix1 = TTKPersistenceDiagramDistanceMatrix(
40 Input=tTKPersistenceDiagramClustering1
41 )
42 tTKPersistenceDiagramDistanceMatrix1.Criticalpairsused = "saddle-max pairs"
44 # create a new 'TTK LDistanceMatrix'
45 tTKLDistanceMatrix1 = TTKLDistanceMatrix(Input=tTKCinemaProductReader1)
46 tTKLDistanceMatrix1.ScalarField = ["POINTS", "Enstrophy"]
48 # create a new 'TTK DimensionReduction'
49 tTKDimensionReduction2 = TTKDimensionReduction(Input=tTKLDistanceMatrix1)
50 tTKDimensionReduction2.Regexp = "Dataset.*"
51 tTKDimensionReduction2.SelectFieldswithaRegexp = 1
52 tTKDimensionReduction2.InputIsaDistanceMatrix = 1
53 tTKDimensionReduction2.UseAllCores = False
55 # create a new 'Table To Points'
56 tableToPoints2 = TableToPoints(Input=tTKDimensionReduction2)
57 tableToPoints2.XColumn = "Component_0"
58 tableToPoints2.YColumn = "Component_1"
59 tableToPoints2.a2DPoints = 1
60 tableToPoints2.KeepAllDataArrays = 1
62 # create a new 'K Means'
63 kMeans1 = KMeans(Input=tableToPoints2)
64 kMeans1.VariablesofInterest = ["Component_0", "Component_1"]
65 kMeans1.k = 2
67 # create a new 'TTK DimensionReduction'
68 tTKDimensionReduction1 = TTKDimensionReduction(
69 Input=tTKPersistenceDiagramDistanceMatrix1
70 )
71 tTKDimensionReduction1.Regexp = "Diagram.*"
72 tTKDimensionReduction1.SelectFieldswithaRegexp = 1
73 tTKDimensionReduction1.InputIsaDistanceMatrix = 1
74 tTKDimensionReduction1.UseAllCores = False
76 # create a new 'Table To Points'
77 tableToPoints1 = TableToPoints(Input=tTKDimensionReduction1)
78 tableToPoints1.XColumn = "Component_0"
79 tableToPoints1.YColumn = "Component_1"
80 tableToPoints1.a2DPoints = 1
81 tableToPoints1.KeepAllDataArrays = 1
83 SaveData("W2clusteringAndW2dimensionReduction.csv", tableToPoints1)
85 SaveData("L2dimensionReductionAndClustering.csv", OutputPort(kMeans1, 1))
To run the above Python script, go to your ttk-data directory and enter the following command:
pvpython python/clusteringKelvinHelmholtzInstabilities.py
• khi.cdb: a cinema database containing 32 regular grids describing periodic, 2D Kelvin Helmholtz Instabilities (computational fluid dynamics). The scalar field of interest is the "Enstrophy" (an
established measure of vorticity). The simulation parameters as well as the ground truth classification (two classes: beginning or end of the turbulence) are provided as metadata for each entry
of the database and are carried along the entire pipeline. See this publication for further details.
• W2clusteringAndW2dimensionReduction.csv: 2D point cloud representing the input ensemble (1 line = 1 member of the ensemble). The field ClusterId denotes the class identifier computed with a
k-means clustering (with k=2) directly performed in the Wasserstein metric space of persistence diagrams. After that, the 2D layout of the points is computed by multidimensional scaling of the
matrix of Wasserstein distances between persistence diagrams. With this technique, the output classification perfectly matches the ground-truth classification.
• L2dimensionReductionAndClustering.csv: 2D point cloud representing the input ensemble (1 line = 1 member of the ensemble). The field ClosestId(0) denotes the class identifier computed with a
standard k-means clustering (with k=2) obtained after a 2D projection of the point cloud. In particular, the 2D layout of the points is computed by multidimensional scaling of the matrix of L2
distances between the Enstrophy fields. With this technique, the output classification does not match the ground-truth classification.
C++/Python API¶ | {"url":"https://topology-tool-kit.github.io/examples/clusteringKelvinHelmholtzInstabilities/","timestamp":"2024-11-09T01:12:50Z","content_type":"text/html","content_length":"53649","record_id":"<urn:uuid:d7bb203d-5458-4b80-af9d-c6ae603904e1>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00208.warc.gz"} |
Epact: Scientific Instruments of Medieval and Renaissance Europe
plate: part of an astrolabe with a projection of altitude and azimuth lines on to the equatorial plane, see article on the astrolabe.
plumb level: device for determining a horizontal level or an angle of elevation by a plumb line or plummet.
plumb line: a suspended thread with a weight at its end, indicating the vertical.
plummet: a form of plumb line in which the 'line' and weight are a single rigid piece.
polar dial: type of sundial, see article on the sundial.
polyhedral dial: sundial with hour lines on various faces of a solid figure, see article on the sundial.
prime vertical: celestial great circle passing through the east and west points and the zenith.
primum mobile: instrument for finding the sines and versed signs of angles, see article on the primum mobile.
projection: translation of a figure on to a plane or curved surface using straight lines in a systematic way. For example, a spherical surface can be projected on to a plane (the plane of projection)
by means of straight lines drawn from all points on the surface to a certain defined point (the point of projection) and marking where they intersect the plane. In a stereographic projection, such as
is used for the ordinary astrolabe, points on a containing circle are projected on to an equatorial plane from one pole; in an orthographic projection, such as is used in a Rojas design of universal
astrolabe, the point of projection is at infinity and the projection lines are parallel. | {"url":"https://www.mhs.ox.ac.uk/epact/glossary.php?FirstID=150&RecordsAtATime=9","timestamp":"2024-11-06T06:10:37Z","content_type":"text/html","content_length":"15629","record_id":"<urn:uuid:2b8b7ff8-85be-40df-97c5-2cf3aa531057>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00823.warc.gz"} |
Solve the following quadratic equation by applying the quadratic formu
Solve the following quadratic equation by applying the quadratic formula p2x2+(p2−q2)x−q2=0
Step by step video & image solution for Solve the following quadratic equation by applying the quadratic formula p^2x^2+(p^2-q^2)x-q^2=0 by Maths experts to help you in doubts & scoring excellent
marks in Class 10 exams. | {"url":"https://www.doubtnut.com/qna/647934055","timestamp":"2024-11-10T01:25:19Z","content_type":"text/html","content_length":"263294","record_id":"<urn:uuid:306249af-e81b-4255-89ca-70c832161be5>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00744.warc.gz"} |
MathCelebrity Forum
Introduce yourself and tell us a bit about you
Math Anything related to math or math homework
Discussion of the Libra coin
Ethereum Discussion of Price, application, etc.
Discuss usage and application of Blockchain
Bitcoin Discussions about price, usage
Website Traffic Dedicated to getting more website traffic
Free Traffic Frenzy: How To Get 450,000 Website Visitors Book by Don Sevcik | {"url":"https://www.mathcelebrity.com/community/","timestamp":"2024-11-11T19:28:57Z","content_type":"text/html","content_length":"88187","record_id":"<urn:uuid:0897f6b7-b57b-48ac-9de1-2a18dcbc49f9>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00486.warc.gz"} |
Interesting Math Articles and Must Read Research Papers for Students – Gaurav Tiwari
Interesting Math Articles and Must Read Research Papers for Students
Are you a mathematics student looking to feed your curiosity with some interesting math articles and research papers? If you are one, you are the right place.
Here, I have collected the list of some excellent and interesting math articles and interesting mathematics research papers which I have read and found very useful.
All of these are easily available online. The main sources of this list are ArXiv.org and the websites of respective professors.
If you know any other paper/article that you find extremely interesting and that is not listed here, then please do comment mentioning the article name and URL. Papers/articles are cited as paper
titles first, then HTTP URLs and at last, author-name.
Interesting Math Articles
The Two Cultures of Mathematics
Timothy Gowers presents the contrasting cultures in mathematical research: problem solvers and theory builders. Explore his perspectives in this insightful paper.
The Two Cultures of Mathematics
What is Good Mathematics?
Terence Tao explores the essential characteristics of good mathematical work, offering insights on beauty, clarity, and usefulness in mathematics.
Career Advice
Terence Tao provides invaluable career advice for mathematicians, discussing research, time management, and balancing personal and professional life.
For Potential Students
Ravi Vakil shares advice for students aspiring to enter the world of mathematics, focusing on both academic and personal development.
Advice to a Young Mathematician
Timothy Gowers offers practical advice to young mathematicians, emphasizing the importance of perseverance and finding joy in research challenges.
Advice to a Young Mathematician
Ten Signs a Claimed Mathematical Breakthrough is Wrong
Scott Aaronson lists key warning signs to help identify dubious or exaggerated claims in mathematics.
Ten Signs a Claimed Mathematical Breakthrough is Wrong
On Proof and Progress in Mathematics
William Thurston discusses the evolving nature of mathematical proofs and how they contribute to broader progress in the field.
On Proof and Progress in Mathematics
A Mathematician’s Lament
Paul Lockhart critiques traditional mathematics education, arguing for a more engaging and creative approach to teaching mathematics.
Truth as Value of Duty: Lessons of Mathematics
Yuri I Mannin explores the ethical and intellectual responsibilities inherent in mathematical research and discovery.
Mathematical Knowledge: Internal Social and Agricultural Aspects
Yuri I Mannin examines the social and cultural factors influencing the development and dissemination of mathematical knowledge.
The Cult of Genius
Julianne Dalcanton explores society’s fascination with genius, particularly in mathematics, and its impact on education and innovation.
Take it to the Limit
A New York Times article delving into mathematical limits, both as a concept and metaphor, within various scientific disciplines.
How to Supervise a Ph.D.
This guide provides strategies and best practices for effectively supervising Ph.D. students in mathematics.
Essential Steps of Problem Solving
Gaurav Tiwari explains the critical steps needed to solve complex mathematical problems, with practical examples.
Essential Steps of Problem Solving
On the Electrodynamics of Moving Bodies
Albert Einstein’s foundational paper on special relativity, revolutionizing physics and our understanding of space-time.
On the Electrodynamics of Moving Bodies
Who Can Name the Bigger Number?
Scott Aaronson delves into the fascinating world of extremely large numbers and their place in mathematical theory.
Who Can Name the Bigger Number?
Division by Three
Doyle and Conway explore an intriguing problem related to division by three, with deep implications in number theory.
Birds and Frogs
Freeman Dyson contrasts two types of mathematicians: birds, who see the big picture, and frogs, who work on specific problems.
A Mathematical Theory of Communication
Shannon Day’s groundbreaking work on information theory and communication, a cornerstone of modern computing and mathematics.
A Mathematical Theory of Communication
Missed Opportunities
Freeman Dyson reflects on the potential discoveries missed by the mathematical community due to overlooked ideas or unexplored paths.
The Unreasonable Effectiveness of Mathematics in the Natural Sciences
Eugene Wigner’s famous essay on the surprising success of mathematics in explaining natural phenomena.
The Unreasonable Effectiveness of Mathematics in the Natural Sciences
On Computable Numbers with an Application to the Entscheidungsproblem
Alan Turing’s landmark paper that laid the foundation for modern computing and the theory of computation.
Funny Problems
Florentin Smarandache presents a collection of mathematical puzzles and paradoxes that challenge conventional thinking.
Life and Work of the Mathemagician Srinivasa Ramanujan
K. Srinivasa Rao’s biographical sketch of Srinivasa Ramanujan, one of the most brilliant mathematicians of the 20th century.
Life and Work of Srinivasa Ramanujan
Why Everyone Should Know Number Theory
Minhyong Kim argues that understanding number theory is essential for appreciating modern mathematics and its real-world applications.
Why Everyone Should Know Number Theory
Meta Math! The Quest for Omega
Gregory Chaitin explores the mathematical constant Omega and its implications for understanding randomness and incompleteness.
Meta Math! The Quest for Omega
Vedic Mathematics
W.B. Vasantha Kandasamy and Florentin Smarandache discuss ancient Indian mathematical methods and their relevance in modern computation.
On Multiple Choice Questions in Mathematics
Terence Tao reflects on the role and limitations of multiple-choice questions in assessing mathematical understanding.
Ramanujan Type 1/pi Approximation Formulas
Nikos Bagis presents Ramanujan-style approximation formulas for 1/pi, with applications in number theory and computational mathematics.
Ramanujan Type 1/pi Approximation Formulas
Collatz’s 3x+1 Problem and Iterative Maps on Interval
Wang Liang explores the famous 3x+1 problem, one of the most enigmatic unsolved problems in mathematics.
Proof of Riemann Hypothesis
Jinzhu Han’s controversial work proposing a proof for the Riemann Hypothesis, one of the biggest open questions in mathematics.
Solving Polynomial Equations from Complex Numbers
Ricardo S Vieira presents a method for solving polynomial equations involving complex numbers, contributing to algebraic geometry.
Age of Einstein
Frank WK Firk’s exploration of the scientific and cultural impact of Albert Einstein’s theories, marking a new era in physics.
The Mysteries of Counting
John Baez discusses the foundational concept of counting and its deeper implications in mathematics and logic.
Generalization of Ramanujan Method of Approximating Root of an Equation
R K Muthumalai builds on Ramanujan’s method for approximating the roots of equations, with novel generalizations.
Generalization of Ramanujan Method
How to Gamble if You are in a Hurry?
Ekhad, Georgiadis, and Zeilberger offer mathematical insights into quick gambling strategies backed by probability theory.
How to Gamble if You are in a Hurry
How to Survive a Math Class?
Matthew Saltzman and Marie Coffin provide tips on how students can successfully navigate challenging math courses.
Is Life Improbable?
John Baez delves into the mathematical probability of life existing in the universe, with insights from physics and biology.
Remarks on Expository Writing in Mathematics
Robert B Ash offers guidance on how to effectively communicate complex mathematical ideas through expository writing.
Success in Mathematics
Saint Louis University provides strategies for achieving success in mathematics, from study habits to conceptual understanding.
Teaching and Learning Mathematics
Terry Bergeson’s comprehensive guide on teaching strategies and methods to enhance student engagement in mathematics.
Teaching and Learning Mathematics
Helping Your Child Learn Mathematics
The US Department of Education provides resources for parents to help their children succeed in mathematics.
Helping Your Child Learn Mathematics
Engaging Students in Meaningful Mathematics Learning
Michael T. Battista explores different perspectives on engaging students in mathematics and achieving complementary educational goals.
Engaging Students in Meaningful Mathematics Learning
Must Read Books
Here are some more interesting Math Books/Items on Amazon that you can try:
Mathematics is beautiful, and there is no such thing as ugly mathematics in this world. Mathematics originates from creativity and develops with research papers.
These research papers aren’t only very detailed and tough to understand for a general student, but also interesting. I hope these math articles, research papers and the recommended books were helpful
to you. | {"url":"https://gauravtiwari.org/imrp/","timestamp":"2024-11-09T23:46:57Z","content_type":"text/html","content_length":"115890","record_id":"<urn:uuid:1de5b907-f5a9-4f29-9565-56d2263ff1d1>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00534.warc.gz"} |
Section: Scientific Foundations
Model-based optimization and compilation techniques
Optimization for parallelism
We study optimization techniques to produce “good” schedules and mappings of a given application onto a hardware SoC architecture. These heuristic techniques aim at fulfilling the requirements of the
application, whether they be real time, memory usage or power consumption constraints. These techniques are thus multi-objective and target heterogeneous architectures.
We aim at taking advantage of the parallelism (both data-parallelism and task parallelism) expressed in the application models in order to build efficient heuristics.
Our application model has some good properties that can be exploited by the compiler: it expresses all the potential parallelism of the application, it is an expression of data dependencies –so no
dependence analysis is needed–, it is in a single assignment form and unifies the temporal and spatial dimensions of the arrays. This gives to the optimizing compiler all the information it needs and
in a readily usable form.
Transformation and traceability
Model to model transformations are at the heart of the MDE approach. Anyone wishing to use MDE in its projects is sooner or later facing the question: how to perform the model transformations? The
standardization process of Query View Transformation [111] was the opportunity for the development of transformation engine as Viatra, Moflon or Sitra. However, since the standard has been
published, only few of investigating tools, such as ATL(http://www.eclipse.org/m2m/atl ) (a transformation dedicated tool) or Kermeta (http://www.kermeta.org ) (a generalist tool with facilities to
manipulate models) are powerful enough to execute large and complex transformations such as in the Gaspard2 framework. None of these engine is fully compliant with the QVT standard. To solve this
issue, new engine relying on a subset of the standard recently emerged such as QVTO (http://www.eclipse.org/m2m/qvto/doc ) and smartQVT. These engines implement the QVT Operational language.
Traceability may be used for different purposes such as understanding, capturing, tracking and verification on software artifacts during the development life cycle [98] . MDE has as main principle
that everything is a model, so trace information is mainly stored as models. Solutions are proposed to keep the trace information in the initials models source or target [125] . The major drawbacks
of this solution are that it pollutes the models with additional information and it requires adaptation of the metamodels in order to take into account traceability. Using a separate trace model with
a specific semantics has the advantage of keeping trace information independent of initial models [102] .
Contributions of the team
Data-parallel code transformations
We have studied Array-OL to Array-OL code transformations [83] , [122] , [93] , [92] , [94] [101] . Array-OL allows a powerful expression of the data access patterns in such applications and a
complete parallelism expression. It is at the heart of our metamodel of application, hardware architecture and association.
The code transformations that have been proposed are related to loop fusion, loop distribution or tiling but they take into account the particularities of the application domain such as the presence
of modulo operators to deal with cyclic frequency domains or cyclic space dimensions (as hydrophones around a submarine for example).
We pursue the study of such transformations with two objectives:
• Propose utilization strategies of such transformations in order to optimize some criteria such as memory usage, minimization of redundant computations or adaptation to a target hardware
In 2009 the study on the interaction between the high-level data-parallel transformations and the inter-repetition dependencies (allowing the specification of uniform dependencies) was achieved.
Because the ODT formalism behind the Array-OL transformations cannot express dependencies between the elements of the same multidimensional space, in order to take into account the uniform
dependencies we proposed and proved an algorithm that, starting from the hierarchical distribution of repetition before and after a transformation, is capable to compute the new uniform dependencies
that express the same exact dependencies as before the transformations. It all comes down to solving an (in)equations system, interpreting the solutions and translating them into new uniform
The algorithm was implemented and integrated into the refactoring toolbox and enables the use of the transformations on models containing inter-repetition dependencies.
In order to validate the theoretical work around the high-level Array-OL refactoring based on the data-parallel transformations, together with Eric Lenormand and Michel Barreteau from THALES Research
& Technology we worked on a study on optimization techniques in the context of an industrial radar application. We have proposed a strategy to use the refactoring toolbox to help explore the design
space, illustrated on the radar application modeled using the Modeling and Analysis of Real-time and Embedded systems (MARTE) UML profile.
Multi-objective hierarchical scheduling heuristics
When dealing with complex heterogeneous hardware architectures, the scheduling heuristics usually take a task dependence graph as input. Both our application and hardware architecture models are
hierarchical and allow repetitive expressions. We propose a Globally Irregular, Locally Regular (GILR) combination of heuristics to allow to take advantage of both task and data parallelism [105]
and have started evaluating multi-objective evolutionary meta-heuristics in this context. These evolutionary meta-heuristics deal with the irregular (task parallelism) part of the design [80] while
we have proposed a heuristic to deal with the regular part (data parallelism) [106] .
Furthermore, local optimizations (contained inside a hierarchical level) decrease the communication overhead and allow for a more efficient usage of the memory hierarchy. We aim at combining the
data-parallel code transformations presented before and the GILR heuristics in order to deal efficiently with the data-parallelism of the application by using repetitive parts of the hardware
The introduction of uniform inter-repetition dependencies in the data-parallel tasks of Gaspard2 has had several consequences. Aside the modification of the refactoring (see section 3.3.2.1 ), we
have studied the compilation of such tasks. This compilation involves the scheduling of such repetitions on repetitive grids of processors and the code generation. This scheduling problem is
NP-complete and we have proposed a heuristic based on the automatic parallelization techniques to compute a good (efficient both in time and code size) schedule in the case when all loop bounds and
processor array shapes are known.
Transformation techniques
In the previous version of Gaspard2, model transformations were complex and monolithic. They were thus hardly evolvable, reusable and maintainable. We thus proposed to decompose complex
transformations into smaller ones jointly working in order to build a single output model [96] . These transformations involve different parts of the same input metamodel (e.g. the MARTE metamodel);
their application field is localized. The localization of the transformation was ensured by the definition of the intermediary metamodels as delta. The delta metamodel only contains the few concepts
involved in the transformation (i.e. modified, or read). The specification of the transformations only uses the concepts of these deltas. We defined the Extend operator to build the complete
metamodel from the delta and transposed the corresponding transformations. The complete metamodel corresponds to the merge between the delta and the MARTE metamodel or an intermediary metamodel. The
transformation then becomes the chaining of metamodel shifts and the localized transformation. This way to define the model transformations has been used in the Gaspard2 environment. It allowed a
better modularity and thus also reusability between the various chains.
Our traceability solution relies on two models the Local and the Global Trace metamodels. The former is used to capture the traces between the inputs and the outputs of one transformation. The Global
Trace metamodel is used to link Local Traces according to the transformation chain. The local trace also proposes an alternative “view” to the common traceability mechanism that does not refers to
the execution trace of the transformation engine. It can be used whatever the used transformation language and can easily complete an existing traceability mechanism by providing a more finer grain
traceability [75] .
Furthermore, based on our trace metamodels, we developed algorithms to ease the model transformation debug. Based on the trace, the localization of an error is eased by reducing the search field to
the sequence of the transformation rule calls [76] .
Verifying conformance and semantics-preserving model transformations
We give formal executable semantics to the notions of conformance and of semantics-preserving model transformations in the model-driven engineering framework [119] . Our approach consists in
translating models and meta-models (possibly enriched with OCL invariants) into specifications in Membership Equational Logic, an expressive logic implemented in the Maude tool. Conformance between a
model and a meta-model is represented by the validity of a certain theory interpretation, of the specification representing the meta-model, in the specification representing the model. Model
transformations between origin and destination meta-models are mappings between the sets of models that conform to the those meta-models, respectively, and can be represented by rewrite rules in
Rewriting Logic, a superset of Membership Equational Logic also implemented in Maude. When the meta-models involved in a transformation are endowed with dynamic semantics, the transformations between
them are also typically required to preserve those semantical aspects. We propose to represent the notion of dynamic semantics preservation by means of algebraic simulations expressed in Membership
Equational Logic. Maude can then be used for automatically verifying conformance, and for automatically verifying dynamic semantics preservation up to a bounded steps of the dynamic semantics. These
works lead to better understood meta-models and models, and to model transformations containing fewer errors.
Modeling for GPU
The model described in UML with Marte profile model is chained in several inout transformations that adds and/or transforms elements in the model. For adding memory allocation concepts to the model,
a QVT transformation based on «Memory Allocation Metamodel» provides information to facilitate and optimize the code generation. Then a model to text transformation allows to generate the C code for
GPU architecture. Before the standard releases, Acceleo is appropriate to get many aspects from the application and architecture model and transform it in CUDA (.cu, .cpp, .c, .h, Makefile) and
OpenCL (.cl, .cpp, .c, .h, Makefile) files. For the code generation, it's required to take into account intrinsic characteristics of the GPUs like data distribution, contiguous memory allocation,
kernels and host programs, blocks of threads, barriers and atomic functions.
Clock-based design space exploration for SoCs
We have previously proposed an abstract clock-based modeling of data-intensive SoCs behaviors within the Gaspard2 framework [70] [69] . Both application functionality and hardware architecture are
characterized in terms of clocks. Then, their allocation is also expressed as a projection of functional clock properties onto physical clock properties, according to a mapping choice. The result of
such allocation is a new set of clocks reflecting the simulation of the temporal behavior of the system during execution.
This year, this approach has been applied to the design of the H.264 encoder on a multiprocessor hardware architecture using the standard MARTE profile [71] . The obtained model has been analyzed by
considering abstract clocks. In particular, it has been shown that such clocks help to tackle design space exploration issues via a relevant modeling of different hardware/software mappings. The
trade-off about processor frequency scaling, system functional properties and energy consumption has been addressed, via different hardware IP choices. This has been achieved via a qualitative
reasoning on traces resulting from a scheduling of logical clocks, capturing functional properties, on physical clocks derived from processors frequency.
Optimized code generation from UML/MARTE models
Starting from the observation that some semantics (and thus some optimization possibilities) are lost when generating code in a programming language from a UML/MARTE model, the contribution of a
thesis co-directed with the CEA LIST is an optimization at the model level followed by a translation to the GENERIC intermediate representation of the gcc compilation framework in order to allow more
optimization, for the moment focusing on code size optimization.
Architecture exploration based on meta-heuristics
Some progress has been made on the proposal of meta-heuristics use for multi-objective mapping and scheduling. In collaboration with the Dolphin project-team of INRIA Lille - Nord Europe and LIFL we
have modeled the association process of Gaspard2 as an optimization problem in order to solve it with a genetic algorithm based heuristic that has been implemented in the ParadisEO optimization
framework. This new heuristics is currently being integrated in the Gaspard2 tool. Another work comparing heuristics based on the particle swarm and genetic algorithm meta heuristics has been
proposed in collaboration with the computer science laboratory of Oran, Algeria, in continuation of our collaboration.
Architecture exploration for efficient data transfer and storage
A major point in embedded system design today is the optimization of communication structures, memory hierarchy and global synchronizations. Such an optimization is a time consuming and error-prone
process, that requires a suitable automatic approach. We proposed an electronic system level framework to explore the data transfer storage micro-architecture and the synchronization of iterative
data-parallel applications [88] . The aim is to define a methodology that can be a front-end for loop-based high level synthesis or interconnect hardware IPs in order to realize memory-centric
MPSoCs. In Gaspard2, this will enable to assess various mappings of Array-OL models onto different kinds of target architectures.
Our solution starts from a canonical Array-OL representation and apply a set of transformations in order to infer an Application Specific architecture that masks the times to transfer data with the
time to perform the computations. A customizable model of the target architecture including FIFO queues and double buffering mechanism is proposed. The mapping of a given image processing application
onto this architecture is performed through a flow of Array-OL transformations aimed to improve the parallelism level and to reduce the size of the used internal memories. A method based on an
integer partition is considered to reduce the space of explored transformations.
Multi-objective mapping and scheduling heuristics
Mohamed Akli Redjedal, univ. Lille 1 master, co-directed with Laetitia Jourdan form the Dolphin project-team of INRIA Lille - Nord Europe and LIFL. The work of Mohamed Redjedal has consisted in
modeling the association process of Gaspard2 as an optimization problem in order to solve it with a genetic algorithm based heuristic. He has indeed modeled this multi-objective mapping and
scheduling problem, proposed a heuristic and its implementation in the ParadisEO optimization framework. A 1st year master student from the univ. of Brussels has worked 6 weeks on the model driven
export from Gaspard2 to the optimization heuristics proposed by Mohamed Redjedal
GPGPU code production
The solution of large, sparse systems of linear equations « Ax=b » presents a bottleneck in sequential code executing on CPU. To solve a system bound to Maxwell's equations on Finite Element Method
(FEM), a version of conjugate gradient iterative method was implemented in CUDA and OpenCL as well. The aim is to accelerate and verify the parallel code on GPUs. The first results showed a speedup
around 6 times against sequential code on CPU. Another approach uses an algorithm that explores the sparse matrix storage format (by rows and by columns). This one did not increase the speedup but it
allows to evaluate the impact of the access to the memory.
From MARTE to OpenCL.
We have proposed an MDE approach to generate OpenCL code. From an abstract model defined using UML/MARTE, we generate a compilable OpenCL code and then, a functional executable application. As MDE
approach, the research results provide, additionally, a tool for project reuse and fast development for not necessarily experts. This approach is an effective operational code generator for the newly
released OpenCL standard. Further, although experimental examples use mono device(one GPU) example, this approach provides resources to model applications running on multi devices (homogeneously
configured). Moreover, we provide two main contributions for modeling with UML profile to MARTE. On the one hand, an approach to model distributed memory simple aspects, i.e. communication and memory
allocations. On the other hand, an approach for modeling the platform and execution models of OpenCL. During the development of the transformation chain, an hybrid metamodel was proposed for
specifying of CPU and GPU programming models. This allows generating other target languages that conform the same memory, platform and execution models of OpenCL, such as CUDA language. Based on
other created model to text templates, future works will exploit this multi language aspect. Additionally, intelligent transformations can determine optimization levels in data communication and data
access. Several studies show that these optimizations increase remarkably the application performance.
Formal techniques for construction, compilation and analysis of domain-specific languages
The increasing complexity of software development requires rigorously defined domain specific modelling languages (DSML). Model-driven engineering (MDE) allows users to define their language's syntax
in terms of metamodels. Several approaches for defining operational semantics of DSML have also been proposed [123] , [89] , [73] , [84] , [115] . We have also proposed one such approach, based on
representing models and metamodels as algebraic specifications, and operational semantics as rewrite rules over those specifications [95] , [120] . These approaches allow, in principle, for model
execution and for formal analyses of the DSML. However, most of the time, the executions/analyses are performed via transformations to other languages: code generation, resp. translation to the input
language of a model checker. The consequence is that the results (e.g., a program crash log, or a counterexample returned by a model checker) may not be straightforward to interpret by the users of a
DSML. We have proposed in [118] a formal and operational framework for tracing such results back to the original DSML's syntax and operational semantics, and have illustrated it on SPEM, a language
for timed process management.
Electromagnetic modeling
The Finite Integration Technique (F.I.T) is used to compute the phenomena. This technique is efficient if the mesh is generated by a regular hexahedron. Moreover the matrix system, obtained from a
regular mesh can be exploited to use the parallel direct solver. In fact, in reordering the unknowns by the nested dissection method, it is possible to construct directly the lower triangular matrix
with many processors without assembling the matrix system. During this year, we have used our parallel direct solver as a preconditionner for a sparse linear system coming from a FEM problem with a
good efficiency. | {"url":"https://radar.inria.fr/report/2011/dart/uid49.html","timestamp":"2024-11-14T14:15:52Z","content_type":"text/html","content_length":"61044","record_id":"<urn:uuid:7b45eec0-7653-45bd-b7dc-a1d74d6b6f7d>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00860.warc.gz"} |
Chapter 06 Gauss Elimination with Partial Pivoting Example Part 3 of 3
Gaussian Elimination (CHAPTER 6)
Gauss Elimination with Partial Pivoting: Example Part 3 of 3
Learn how Gaussian Elimination with Partial Pivoting is used to solve a set of simultaneous linear equations through an example.
This video teaches you how Gaussian Elimination with Partial Pivoting is used to solve a set of simultaneous linear equations through an example.
All Videos for this Topic
Complete Resources
Get in one place the following: a textbook chapter, a PowerPoint presentation, individual YouTube lecture videos, multiple-choice questions, and problem sets on Gaussian Elimination. | {"url":"https://ma.mathforcollege.com/chapter-06-gauss-elimination-with-partial-pivoting-example-part-3-of-3/","timestamp":"2024-11-09T00:22:31Z","content_type":"text/html","content_length":"39298","record_id":"<urn:uuid:7ddd00bb-ef05-4d4a-b8ea-0a1d6a28677e>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00108.warc.gz"} |
Nanoparticle Volume, Mass and Concentration
Nanoparticle volume, mass and concentration are fundamental nanoparticle characteristics. In this module, we describe how we calculate these parameters for both solid particles and core/shell
particle geometries.
How to Calculate the Volume of a Nanoparticle
The volume of a nanoparticle is determined by first measuring its dimensions. At nanoComposix we primarily use a transmission electron microscope (TEM) to measure particle dimensions, allowing the
volume to be calculated.
For spherical nanoparticles, the volume is: V=4/3𝜋r^3, where r is the radius of the sphere
For rod shaped nanoparticles, the volume is:V=𝜋r^2l, where r is the radius of the rod and l is the length
For plate shaped nanoparticles, the volume is V=𝜋r^2h , where r is the radius of the nanoplate and h is the thickness.
For cube shaped nanoparticles, the volume is : V=d^3, where d is the diameter of the cube.
To obtain these dimensions, TEM images are analyzed with a program such as ImageJ/Fiji to measure many particles from multiple TEM grids. The measurements are averaged and substituted in the formulas
above. Sometimes, all of the needed dimensions cannot be obtained with TEM alone. For example, nanoplates typically sit flat on the TEM grid so it is not possible to measure the thickness directly
with TEM, and complimentary measurement techniques, such as atomic force microscopy (AFM) or high-resolution scanning electron microscopy (SEM) may be needed to measure the plate thickness. Another
method of measuring plate thickness is to measure the plate in composite particles. For example, when silica shelled, the nanoplates will often be rotated on edge when dried onto a TEM grid and a
direct TEM measurement of the thickness can be made.
How to Calculate the Mass of a Nanoparticle
Once the nanoparticle volume has been calculated the mass can be determined simply by multiplying the volume by the material density (ρ): m=Vρ. In most cases the density of nanomaterials is the same
as the bulk density, but for some materials the atomic structure is different than the bulk and a corrected density must be used. The mass calculation is also adjusted for nanoparticles made of
multiple materials, such as core/shell nanoparticles.
Material Nanoparticle Density (g/cm^3) Bulk Density (g/cm^3)
Gold 19.32 Same
Silver 10.5 Same
Platinum 21.45 Same
Silica 2.2 2.65
Magnetite (Fe[3]O[4]) 5.24 Same
Effective Density of Silica Nanoparticles
Silica nanoparticles are typically prepared using the Stober method, in which silane precursors are condensed in the presence of a base. Depending on the fabrication, environment and storage
conditions, the degree to which the silica is condensed varies. Initially, there will be many -OH groups within the silica particle; the number of hydroxyl groups can be reduced by heating, which
converts two -OH bonds into a Si-O-Si bond while releasing a water molecule. This condensation process leads to the silica becoming less porous and more dense, but is still typically a lower density
than bulk silica prepared at high temperatures.
We use an effective density of 2.05 g/cm^3 for our silica nanoparticles, which is similar to other reported values in the literature measured by other techniques such as an aerosol particle mass
analyzer (Kimoto 2014, Kimoto 2017)
How to Calculate the Mass of a Core-Shell Nanoparticle
When a nanoparticle is made from more than one material, separate calculations must be made to determine the particle mass. For example, gold nanoshells consist of a silica core surrounded by a thin
gold shell, and the mass of the core and the mass of the shell must be calculated separately to determine the total mass of the particle. In this example, the total particle mass is calculated by
m[total] = m[core] + m[shell] = V[core]ρ[core] + V[shell]ρ[shell]
The mass of the core is the volume multiplied by the density of the core. For a spherical core particle the mass is given by
m[core] = 4/3𝜋r[core]^3ρ[core]
The mass of the shell is the volume of the shell multiplied by the density of the shell. In some cases, it may be easiest to calculate the shell volume by measuring the total particle volume and
subtracting the volume of the core. For example, for gold nanoshells we measure the diameter of the core silica nanoparticles first, and then measure the final diameter of the gold nanoshells. The
thickness of the gold nanoshell is determined to by subtracting the total radius of the gold nanoshell from the radius of the core. For spherical core/shell particles like these, the mass of the
shell is given by
m[shell] = 4/3𝜋(r[total]^3 – r[core]^3)ρ[shell]
The total mass of the particle is then the sum of the mass of the core and the mass of the shell. The formulas above are adjusted to account for other particle geometries.
How to Calculate Nanoparticle Concentration
To calculate nanoparticle concentration you must first determine the total mass of the element of interest in nanoparticle form in the solution. A rough approximation can be obtained by assuming that
all of the initial reagents were converted into nanoparticle form (for example, all added gold chloride is reduced to elemental gold), but does not account for lower reaction yield or processing
losses, and analytical methods to directly measure elemental concentration will provide more accurate results.
At nanoComposix we use ICP-MS to directly measure the elemental concentration in our final purified nanoparticle solution. Using this concentration we can calculate the number concentration of the
nanoparticles by dividing by the total mass in solution by the mass of a single nanoparticle:
N = M[C ]/ m
where M[C] is the mass concentration of the measured element and m is the mass of an individual nanoparticle. If the total mass concentration is expressed in units of g/mL, and the particle mass has
units of g/particle, the calculated concentration has units of particles/mL. For typical nanoparticle formulations we provide, this concentration is in the range of 10^9 to 10^15 particles/mL,
depending on the material and product.
How to Calculate Concentrations of Core/Shell Nanoparticles
When calculating the concentration of nanoshells, the total mass of gold per mL is determined by ICP-MS and then divided by the mass of gold in a nanoshell to yield the total particles/mL. For other
core/shell geometries (such as gold/silver bimetallic particles), a similar strategy is used.
Nanoparticle Concentration in Molar Concentration
In chemistry and biology, concentration is often expressed as molarity, which refers to the number of moles of a substance per liter. In some cases, it is useful to perform calculations using the
particle molarity, which is different than the molar concentration of the elements making up the nanoparticles. The particle molarity is calculated as
M = N / 6.02 × 10^23
where N is the number concentration of the nanoparticles in units of particles/L and the denominator is Avogadro’s number.
Typical molar concentrations of nanoparticles are in the nanomolar (nM) to picomolar (pM) concentration range. For example, our NanoXact 40 nm-diameter gold nanospheres have a total elemental gold
concentration of 0.05 mg/mL, which corresponds to a particle number concentration of 8.1 × 10^10 particles/mL and a particle molarity of 130 pM.
Other Direct and Indirect Measurements of Nanoparticle Concentration
There are other methods of directly measuring nanoparticle concentration in solution. There are a number of instruments that count particles by monitoring the passage of particles through a small
orifice. When the nanoparticle passes through the orifice, a portion of the solution is displaced which changes the electrical resistance. With a known flow rate, each electrical pulse can be counted
and a particle concentration can be measured. Two such instruments are the Spectrodyne and the qNano. Typically, particles are required to be dispersed in a 1× PBS buffer or a solution with a
similar level of salt concentration in order to make a measurement. Also, the lower size limit using this technique is approximately 50 nm and requires that particles be stable in high salt
environments. Another method of measuring particle concentration is to optically count the number of particles in a small volume of solution. The Malvern NanoSight visually tracks individual
nanoparticles and calculates their size based on diffusion. Particles as small as 30 nm can be measured with this instrument and there is no salt requirement for solution. However, the particles will
drift in and out of the focal plane so the instrument must be calibrated with particle number standards first to obtain accurate particle number concentration measurements. One of the most accurate
methods of counting particles is to dry the particles onto a surface and individually count each one. Large area images captured with a scanning electron microscope can be used to count particles.
This technique relies on very accurate solution volumes to be applied to the sample during preparation.
Overall, it is surprisingly difficult to accurately determine particle number. An alternative method to the analytical solutions presented above is to use numerical models to predict the optical
properties of a particle with a particular geometry. Calculation of the extinction, absorption and scattering cross sections can be used to predict the particle concentration based on the measured
extinction in solution using a UV-visible spectrophotometer (Hendel 2014). Our Mie Theory particle calculator can be used to calculate cross sections of spherical and core/shell spherical particles
and has been shown to agree well with analytical measurements.
Hendel, T.; Wuithschick, M.; Kettemann, F.; Birnbaum, A.; Rademann, K.; Polte, J. "In Situ Determination of Colloidal Gold Concentrations with UV-Vis Spectroscopy: Limitations and Perspectives."
Anal. Chem. 2014, 86(22), 11115-11124. | {"url":"https://nanocomposix.com/pages/nanoparticle-volume-mass-and-concentration","timestamp":"2024-11-09T23:45:59Z","content_type":"text/html","content_length":"295305","record_id":"<urn:uuid:ed851541-9085-436c-99cf-5d0b261abed3>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00263.warc.gz"} |
Mathematics in roblox games
Oh… Yeah… Maths, so hated and useful at the same time… Many people are afraid of this when they start programming, because they think they are not capable, that they will not understand it or they
are simply lazy to learn, but it is not as difficult as most people think, with this post, I will try:
Math library
Returns the number pi (𝛑).
Returns infinite.
This function removes the decimals from a number and leaves only the integer part.
math.floor(5.9746) --> 5
math.floor(543.235) --> 543
This function checks if the number you give it has decimals, if so, it returns the next number, if not, it stays as is.
math.ceil(6) --> 6
math.ceil(6.1) --> 7
math.ceil(6.5) --> 7
math.ceil(6.9) --> 7
This function rounds the number, looks at the first decimal place and if it is five or more, it gives the next number, if it is less, it removes the decimals.
math.round(5.5) --> 6
math.round(5.3) --> 5
math.round(5.8) --> 6
Random numbers
math.random(min, max)
This function gives a random number between the first and the second, if you only give a number, it will give you a random number between 0 and the number you have specified, but if you do not put
any number, it returns a random number between 0 and 1.
math.random(5, 10) --> Any number between 5 and 10
math.random(10) --> Any number between 0 and 10
math.random() --> Any number between 0 and 1
Random numbers do not really exist, so we use a substitute, pseudo-random numbers. I’m sure you’re wondering how Roblox does it with math.random or any other game, it’s easy, there’s a trick to it.
First you have a formula, for example, 7*seed+3, where seed is any initial number, in this case we will use 2, then the calculation 7*2+3=17 is done, then the 17 is set as seed being the next
calculation 7*17+3 and so on, but of course we also have to go back and not just increase, then we put a maximum and see how many times the result of that number passed, so if it is 5, 15 would be
removed from the 17, leaving 2 and so on.
So, knowing this, this function establishes the seed of the pseudo-random numbers:
while task.wait(.5) do
If you put this code, you will see that, despite being a random number, it is putting the same number, since you are specifying a seed before each math.random, converting it all the time in the same
calculations, therefore, same result.
Minimum and maximum
math.min(x, y, ...)
This function accepts any number of numbers to compare them and return only which is the smallest.
math.min(7, 5, 45, 2, 78) --> 2
math.min(567, 464, 251, 875, 753) --> 251
math.max(x, y, ...)
This function accepts any number of numbers to compare them and return only which is the largest.
math.max(7, 5, 45, 2, 78) --> 78
math.max(567, 464, 251, 875, 753) --> 875
Cut out numbers
If the number is less than 0, returns -1, if greater, returns 1, and if 0, returns 0.
math.sign(.3) --> 1
math.sign(0) --> 0
math.sign(-.3) --> -1
math.clamp(x, min, max)
This function requires three numbers to be specified, the number, the minimum and maximum. If the number is less than the minimum, it returns the minimum; if it is greater than the maximum, it
returns the maximum and if it is between those two numbers, it does not modify it:
math.clamp(5, 3, 7) --> 5
math.clamp(5, 7, 10) --> 7
math.clamp(5, 4, 0) --> 4
Easy calculations
Returns the absolute value of the number (i.e. the number in positive).
math.abs(5) --> 5
math.abs(-5) --> 5
math.pow(x, y)
Returns x raised to y (x^y).
math.pow(5, 2) --> 5^2 --> 25
Returns the square root of the specified number.
math.sqrt(25) --> 5
Returns e^x (e is the Euler’s number, exponential of x).
math.exp(5) --> 148.41315910258
math.fmod(x, y)
Returns the remainder of the division of x/y (can also be calculated with x%y)
math.fmod(10, 3) --> 1 (10%3)
math.fmod(10, 2) --> 0 (10%2)
math.log(x, y)
A logarithm is an infinite curve that never becomes straight, useful for calculating the experience needed to level up.
This function needs a number and a base (the base defines what the curve looks like) and returns the result of the logarithm of base y of x:
math.log(10, 5) --> 1.4306765580734
math.log(10, 10) --> 1
math.log(10, 3) --> 2.0959032742894
This function returns the base 10 logarithm of x (it is like putting math.log(x, 10)).
math.log10(10) --> 1 (math.log(10, 10))
A radian is a measure of angle, somewhat like degrees, with the difference that 90º is 𝛑/2 rads, 180º is 𝛑 rads, 270º is 3𝛑/2 rads and 360º is 2𝛑 rads. This is used for rotations with CFrame.
What this function does is to pass an angle in degrees and return it in radians.
math.rad(180) --> 𝛑 (3.1415926535898)
This function converts radians to degrees.
math.deg(math.rad(180)) --> 180
math.deg(2𝛑) --> 360
The sine is a trigonometric function which represents the Y position of a series of points that form a circle of radius 1, so it sounds a little complicated, but let’s take a closer look:
This one here is its graph, if you look at the number line below, it is in radians since both the sine, cosine and tangent we will need to give it an angle in radians.
As we can see, the sine, depending on the angle you give it, gives values between 1 and -1.
As you know, a right triangle can be drawn between two points, because the sine of an angle is the size of the opposite leg between the inside of the circle and the corresponding point on the circle
according to the angle:
The cosine is the same as the sine but changing the axis, but first, let’s see its graph:
As can be seen, it is quite similar to the breast with the difference that it is slightly moved horizontally.
This, like the sine, represents a leg of the right triangle at the center of the circle and a point on the circle that corresponds to the angle that is given to the cosine, but instead of being the
opposite, it calculates the adyascent:
The magic of joining sine with cosine
If we put these two functions (sine and cosine) together we can calculate a position on the circle, which we can use to make orbits or calculate directions from an angle. Using this method is simple
and easy to attach because, as it goes from -1 to 1, we can not only multiply the position obtained to enlarge the radius of this circle, but also just by adding we can change the center position of
the circle:
The tangent is the division between sine and cosine.
It represents the distance from the point it intersects on a tangent line to a point on the horizontal dividing line:
But this presents us with a problem, in spite of infinite elongation, if we are at 90º or 180º, the line with the hypotenuse is perfectly parallel with the tangent line, then it will go out to
Its graph would look like this:
This function is said to be the inverse of the sine function since, instead of obtaining a Y position on the unit circle from an angle, it returns an angle from that Y position. (asin(sin(x)) = x)
This function is said to be the inverse of the cosine function since, instead of obtaining a position X on the unit circle from an angle, it returns an angle from that position in X. (acos(cos(x)) =
This function is said to be the inverse of the tangent function since, instead of getting a distance, you get an angle from that distance. (atan(tan(x)) = x)
math.atan2(y, x)
Imagine a right triangle drawn from two points:
If you notice that the hypotenuse is a segment that connects both points, this tells us that we can obtain the rotation that we must apply to one point to make it look at the other. This is exactly
what this function is for! But… we have to pass two numbers, the height and length of the triangle, to obtain it, we simply subtract the positions.
But… Watch out! The result is given in radians, keep that in mind when using it.
This is exactly the same as the sine, but with the difference of being on a circle, it is on a hyperbola:
This is exactly the same as cosine, but with the difference that it is on a circle, it is on a hyperbola:
This is exactly the same as the tangent, but with the difference that it is on a circle, it is on a hyperbola:
math.noise(x, y, z)
Perlin Noise is a black and white image that has the peculiarity of having white dots on a black background surrounded by gray areas:
For this function you must put three numbers of which two are optional (if you do not put them, they will be set as 0) which are coordinates to know which point of the image to look at and give a
value between 0 and 1 depending on the color it is.
This can be used for a map generation.
This function returns the integer and decimal part of the number separately:
math.modf(3.5) --> 3, 0.5
math.ldexp(x, y)
This function makes the formula for you: x * 2 ^ y
This function returns the unknown m and n considering that: x = m*2^n
Some formulas
Quadratic trajectory
Sometimes we need to calculate the position in the plane of a thrown object, such as an arrow, a stone or a bird.
Formula: origin + direction * t + (gravity*t*t)/2
First let’s put this formula into a function and specify everything the formula needs to calculate the position:
local function getPosition(origin, direction, gravity, t)
return origin + direction * t + (gravity*t*t) * .5
Let’s start by specifying the origin, this must be a vector which is the position from where the object is launched.
local origin = workspace:WaitForChild("Origin").Position
The direction must also be a vector that, as the name says, tells where to launch, for this you can use trigonometry or LookVector, RightVector and UpVector.
local direction = workspace:WaitForChild("Origin").CFrame.LookVector * 50 + workspace:WaitForChild("Origin").CFrame.UpVector * 67
Gravity is also a vector that of magnitude is the force of gravity and of course, it has to go downward:
local gravity = Vector3.new(0, -workspace.Gravity, 0)
Finally we need t (time) which tells the point of the trajectory where we are, to obtain it, what we will do is to specify an initial value (0) and constantly we will add deltaTime (time between
frame and frame in one second) and if you want to modify the speed just multiply it by a number:
local t = 0
while true do
t += game:GetService("RunService").Heartbeat:Wait()
And it would be just a matter of applying the formula!
local function getPosition(origin, direction, gravity, t)
return origin + direction * t + (gravity*t*t) * .5
local gravity = Vector3.new(0, -workspace.Gravity, 0)
local originPart = workspace:WaitForChild("Origin")
local origin = originPart.Position
local direction = originPart.CFrame.LookVector * 50 + originPart.CFrame.UpVector * 67
local bullet = workspace:WaitForChild("Bullet")
local t = 0
while true do
bullet.Position = getPosition(origin, direction, gravity, t)
t += game:GetService("RunService").Heartbeat:Wait() * .5
Here you can download a place with the system already made to test it!
Baseplate.rbxl (32.0 KB)
Converting from Euler to Quaternions
Quickly explained, a quaternion is a four-axis rotation based on complex numbers, which is a calculation of a number with an imaginary number (imaginary numbers are non-existent numbers, such as the
square root of -1).
Thinking about quaternions is difficult, but there is a formula that makes it easy to create them:
W = cos(x/2) * cos(y/2) * cos(z/2) - sin(x/2) * sin(y/2) * sin(z/2)
X = cos(y/2) * cos(z/2) * sin(z/2) + cos(x/2) * sin(y/2) * sin(z/2)
Y = cos(x/2) * cos(z/2) * sin(y/2) - cos(y/2) * sin(x/2) * sin(z/2)
Z = cos(z/2) * sin(x/2) * sin(y/2) + cos(x/2) * cos(y/2) * sin(z/2)
local function Axis2QuaAxis(x, y, z)
x *= .5; y *= .5; z *= .5;
local ca, sa = math.cos(x), math.sin(x)
local cb, sb = math.cos(y), math.sin(y)
local cc, sc = math.cos(z), math.sin(z)
local Axis = {
w = ca * cb * cc - sa * sb * sc;
x = cb * cc * sa + ca * sb * sc;
y = ca * cc * sb - cb * sa * sc;
z = cc * sa * sb + ca * cb * sc;
return Axis
To convert this to quaternions, this is done with CFrame.new(x, y, z, qX, qY, qZ, qW).
Scientific notation
Scientific notation is used to abbreviate a number that has too many zeros, such as 1,000,000. This is normally done by raising 10 to the number of zeros but in Roblox you can put it directly.
We must see where the zeros are, that is, if it is 0.000001 or 1,000,000 (if they are in front or behind the point), if it is in front we will use the + symbol and if it is behind the - symbol. Once
we know the symbol we only have to put 1e symbol number of zeros, for example:
1e+5 --> 100000
1e-5 --> 0.00001
We count in decimal (base 10), if so called because we have ten numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. When we reach 9, we add one to the previous number and start again, 9 ->+10, 11, 12, 13, 14,
15, 16, 17, 18, 19… Counting in hexadecimal is exactly the same, but with the difference of counting with more numbers, which, to avoid creating more symbols, we use letters: 0, 1, 2, 3, 4, 4, 5, 6,
7, 8, 9, a, b, c, d, e, f. Here it is the same, when we reach f we add one to the next number: f → +10, 11, 12, 13, 14, 14, 15, 16, 17, 18, 19, 1a, 1b, 1c, 1d, 1e, 1f… It is possible to count like
this in roblox but we must put 0x in front:
0xa --> 10
0x10 --> 16
0xc4df0a --> 12902154
59 Likes
Im no math expert, but I think math.pi is for trigonometry?
1 Like
It would be really useful if you make a tutorial on strings like that.
Hey! I am currently working on a data compression algorithm, so how do I find out how many decimal places a number has?
41.412 → 3
68.9 → 1
94.4213123 → 5
I tried #tostring(n-math.floor(n)) but it would return something like 0.09999999999999964 instead of 0.1, so the result wasn’t 1, but was instead 37.
Nevermind, I did some more experiments and found a solution:
Incase for any of you who got the same problem, just do this:
of course replace a with your variable or number.
I like maths , its good ,not bad
It is the product of dividing the circumference of a circle by its diameter and is useful for calculating the circumference of circles
That’s trigonometry, I use pi to make circles
1 Like | {"url":"https://devforum.roblox.com/t/mathematics-in-roblox-games/1639538","timestamp":"2024-11-13T18:42:59Z","content_type":"text/html","content_length":"73476","record_id":"<urn:uuid:05986c5f-9ff9-48c8-94f8-a9f9335403b9>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00076.warc.gz"} |
Probability and Statistics
Áreas Científicas
Classificação Área Científica
OFICIAL Matemática
Ocorrência: 2021/2022 - 2S
Ativa? Yes
Página Web: https://moodle.ips.pt/2122/course/view.php?id=300
Unidade Responsável: Departamento de Matemática
Curso/CE Responsável:
Ciclos de Estudo/Cursos
Sigla Nº de Estudantes Plano de Estudos Anos Curriculares Créditos UCN Créditos ECTS Horas de Contacto Horas Totais
EM 98 Plano de Estudos 2 - 6 60 162
Docência - Responsabilidades
Docência - Horas
Theorethical and Practical : 4,00
Língua de trabalho
- Apply the concepts of random variable and its distribution;
- Solve problems involving models and probability distributions with discrete variables and with continuous variables;
- Understand the concept of random sample and solve problems involving sampling distributions;
- Characterize and apply estimators;
- Construct and interpret confidence intervals;
- Identify and apply the appropriate hypothesis test;
- Identify the relation between hypothesis testing and confidence intervals;
- Build and analyze a simple linear regression model.
Resultados de aprendizagem e competências
The curricular unit contents are structured, regarding its suitability for the intended learning outcomes. Therefore, each subject approaches fundamentals concepts and practical applications by
solving problems using the basic tools of Probability and Statistics to enable students to analyze certain phenomena of random nature, framed in the context of technology, particularly in the
recognition and enforcement of probabilistic models, the deduction and application of confidence intervals and hypothesis testing, and the construction and analysis of simple linear regression
Modo de trabalho
1. Random Variables: Concept of r.v. Functions for discrete and continuous r.v Expected value, variance and standard deviation, characterization and properties
2. Theoretical Distributions (TD): Discrete DT; Binomial and Poisson; Characterization. Continuous DT; Exponencial, Uniform and Normal. Brief reference to the Student-t, Chisquare and Snedecor F DT
3. Elements of Sampling Theory: Population and sample. Random Sample and Statistic. Sampling. Distribution.
4. Elements of Estimation Theory: Concept of Estimator; properties. Point and Interval Estimates. Confidence intervals.
5. Hypothesis Testing (HT): Null and alternative hypothesis, critical region, significance level, decision rule of the test, errors type I and type II and power of the test. Parametric HT of normal
6. Simple Linear Regression: Regression line. Parameter estimation of best linear fit using the least squares approach. Concept of residuals. Empirical linear correlation coefficient.
Bibliografia Obrigatória
Montgomery, D.; Runger, G., ; Applied statistics and probability for engineers, John Wiley & Sons. ISBN: 9781119585596
Murteira, B.; Antunes, M; Probabilidades e Estatística, Lisboa: Escolar, 2012. ISBN: 978-972-592-359-7
Murteira, B.; Ribeiro, C. S.; Andrade e Silva, J.; Pimenta, C.; Introdução à Estatística, Escolar Editora, 2015. ISBN: 9788448160692
Bibliografia Complementar
André, J; Probabilidades e Estatística para Engenharia, Lidel, 2018. ISBN: 9789897522703
Galvão de Melo, F.; Probabilidades e Estatística: conceitos e métodos fundamentais, Escolar Editora. ISBN: 9789725921104
Reis, E. .[et al.]; Estatística Aplicada - Volume I e II, Edições Sílabo, 2003. ISBN: 9789726189862
Robalo, A.; Estatística - Exercícios, Volumes 1 e 2, Edições Sílabo, 2001. ISBN: 9789726189121, 9789726189367
Métodos de ensino e atividades de aprendizagem
- Classroom lectures through a combination of lecture method and problem solving;
- E-learning in the Moodle platform, providing access to the contents of UC through slides, videos, and solved and proposed exercises, promoting the holding of weekly activities.
Methodologies used are centered on knowledge of concepts and their applications.
With the classroom lectures are promoted the transmission of probability and statistical contents and its application through problem solving, mostly in contexts related to technology.
E-learning methodology, promotes discipline and autonomous work throw the weekly activities proposed, deepening the probability and statistical contents covered.
Tipo de avaliação
Distributed evaluation with final exam
Componentes de Avaliação
Designation Peso (%)
Teste 100,00
Total: 100,00
Componentes de Ocupação
Designation Tempo (Horas)
Estudo autónomo 120,00
Frequência das aulas 60,00
Total: 180,00
Obtenção de frequência
There are two ways of assessment: by Tests and by Exam.
Continuous Assessment (or by Tests)
Continuous assessment is based on two (2) tests.
Designating by MT the average of the classifications of the 2 tests, the final classification (CF) will be rounded up to the units of the following value:
CF = MT = 0.5*T1+0.5*T2
The conditions for approval in the continuous assessment are as follows:
1. If CF (rounded to units) is greater than or equal to 10 and less than 18, the student is approved with a final grade equal to CF (rounded to units), provided that in any of the tests the score was
greater than or equal to 6.5;
2. If CF (rounded to the nearest unit) is greater than or equal to 18, the student will have to take an oral exam, the final grade being the average of these two grades. If you do not attend the oral
test, the final classification will be 17 values.
3. If the tests are carried out remotely, the maximum score that the student can obtain without undergoing an oral test will be 15 values.
To retrieve a test:
In order to pass a student with a score greater than or equal to 8.0 in one of the tests can retrieve the test with the lowest grade. A student who has less than 8.0 in one of the tests, who was not
able to perform a test or has given up in one test can only recover that test.
The recovery of a test takes place at the exact day and time of the Normal Exam and in order to do so the student must enroll in due time.
Exam-based Assessment
Students who choose not to take the continuous assessment or have not obtained approval can attend the regular exams.
The exam-based assessment is subject to the following conditions:
1. If the exam grade (rounded to the units) is greater than or equal to 10 and less than 18 the student will pass with a final grade equal to the exam grade;
2. If the exam grade is greater than or equal to 18 the student will have to take an oral test and the final grade will be the average of the classifications of oral test and the exam (otherwise, the
final grade will be 17)
3. If the exams are carried out remotely, the maximum score that the student can obtain without undergoing an oral exam will be 15 values.
Fórmula de cálculo da classificação final
Continuous Assessment (or by Tests)
Exam-based Assessment | {"url":"http://portal.ips.pt/ests/en/ucurr_geral.ficha_uc_view?pv_ocorrencia_id=555111","timestamp":"2024-11-14T20:54:25Z","content_type":"text/html","content_length":"21882","record_id":"<urn:uuid:e7344654-d004-48db-aa5e-08e2e2aa9339>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00049.warc.gz"} |
Effect of electron-electron interaction and plasmon excitation on the density-of-states for a two-dimensional electron liquid
We calculate the Green's function for an interacting two-dimensional electron liquid whose strength of interaction is characterized by the electron density parameter r[s]. The screened
electron-electron interaction is expressed in terms of a frequency and wave vector-dependent dielectric function ε(q, ω). If this screening is neglected, the tunneling density of states (DOS) is
strongly modified by the electron-electron interaction. This modification is seen as a dip near the Fermi energy. This dip becomes deeper and narrower as r[s] is increased. When screening is
included, the DOS is considerably affected by the collective plasma excitations. In particular, the range of frequencies where the plasmon excitations contribute increases with r[s]. Also, the DOS
near the Fermi energy for the screened electron liquid depends on the electron density. We treat ε(q, ω) in the hydrodynamical approximation in order to investigate the way in which the tunneling DOS
is modified at various electron densities.
Dive into the research topics of 'Effect of electron-electron interaction and plasmon excitation on the density-of-states for a two-dimensional electron liquid'. Together they form a unique | {"url":"https://cris.biu.ac.il/en/publications/effect-of-electron-electron-interaction-and-plasmon-excitation-on-3","timestamp":"2024-11-15T01:40:43Z","content_type":"text/html","content_length":"55501","record_id":"<urn:uuid:f7756c9b-bd68-4109-bbb0-1329fbc06530>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00139.warc.gz"} |
Arshadi, L and Jahangir, AH (2014). Benford's law behavior of Internet traffic. Journal of Network and Computer Applications, Volume 40, April 2014, pp. 194–205. ISSN/ISBN:1084-8045.
Becker, T, Burt, D, Corcoran, TC, Greaves-Tunnell, A, Iafrate, JR, Jing, J, Miller, SJ, Porfilio, JD, Ronan, R, Samranvedhya, J, Strauch, FW and Talbut, B (2018). Benford's Law and Continuous
Dependent Random Variables. Annals of Physics 388, pp. 350–381. DOI:10.1016/j.aop.2017.11.013.
Becker, T, Corcoran, TC, Greaves-Tunnell, A, Iafrate, JR, Jing, J, Miller, SJ, Porfilio, JD, Ronan, R, Samranvedhya, J and Strauch, FW (2013). Benford's Law and Continuous Dependent Random
Variables. Preprint arXiv:1309.5603 [math.PR]; last accessed October 23, 2018. DOI:10.1016/j.aop.2017.11.013.
Berger, A and Eshun, G (2014). Benford solutions of linear difference equations. Theory and Applications of Difference Equations and Discrete Dynamical Systems, Springer Proceedings in
Mathematics & Statistics Volume 102, pp. 23-60. ISSN/ISBN:978-3-662-44139-8. DOI:10.1007/978-3-662-44140-4_2.
Berger, A and Eshun, G (2016). A characterization of Benford's law in discrete-time linear systems. Journal of Dynamics and Differential Equations 28(2), pp. 432-469. ISSN/ISBN:1040-7294.
Berger, A and Hill, TP (2011). A basic theory of Benford's Law . Probability Surveys 8, pp. 1-126. DOI:10.1214/11-PS175.
Berger, A and Hill, TP (2015). An Introduction to Benford's Law. Princeton University Press: Princeton, NJ. ISSN/ISBN:9780691163062.
Davic, RD (2022). Correspondence of Newcomb-Benford law with ecological processes . Posted on bioRxiv preprint server of Cold Springs Harbor Laboratory June 27, 2022 . DOI:10.1101/
Jasak, Z (2010). Benfordov zakon i reinforcement učenje (Benford's Law and reinforcment learning) . MSc Thesis, University of Tuzla, Bosnia. SRP
Joksimović, D, Knežević, G, Pavlović, V, Ljubić, M and Surovy, V (2017). Some Aspects of the Application of Benford’s Law in the Analysis of the Data Set Anomalies. In: Knowledge Discovery in
Cyberspace: Statistical Analysis and Predictive Modeling. New York: Nova Science Publishers, pp. 85–120. ISSN/ISBN:978-1-53610-566-7.
Kaiser, M (2019). Benford’s Law As An Indicator Of Survey Reliability—Can We Trust Our Data?. Journal of Economic Surveys Vol. 00, No. 0, pp. 1–17. DOI:10.1111/joes.12338.
Michalski, T and Stoltz, G (2013). Do Countries Falsify Economic Data Strategically? Some Evidence That They Might. The Review of Economics and Statistics 95(2), pp. 591-616. DOI:10.1162/
Miller, SJ (ed.) (2015). Benford's Law: Theory and Applications. Princeton University Press: Princeton and Oxford. ISSN/ISBN:978-0-691-14761-1.
Nigrini, MJ (2017). Audit Sampling Using Benford's Law: A Review of the Literature With Some New Perspectives. Journal of Emerging Technologies in Accounting Vol. 14, No. 2, pp. 29–46.
Said, T and Mohammed, K (2020). Detection of anomaly in socio-economic databases, by Benford probability law. 2020 IEEE 6th International Conference on Optimization and Applications (ICOA),
Beni Mellal, Morocco, 2020, pp. 1-4. DOI:10.1109/ICOA49421.2020.9094466.
Uhlig, N (2016). Rundum das Benfordsche Gesetz. Diploma thesis, University of Leipzig, Fakultät für Mathematik und Informatik. GER | {"url":"https://www.benfordonline.net/references/up/893","timestamp":"2024-11-06T17:25:42Z","content_type":"application/xhtml+xml","content_length":"24702","record_id":"<urn:uuid:f66e512b-3993-46e4-a3cd-5f7a8395f4d2>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00251.warc.gz"} |
Blind all-pass deconvolution
Next: About this document ... Up: INTERPOLATION ERROR Previous: INTERPOLATION ERROR
A well-established theoretical concept that leads to unwarranted pessimism is the idea that blind deconvolution cannot find an all-pass filter.
If we carefully examine the analysis leading to that conclusion, we will find lurking the assumption that the weighting function used in the least-squares estimation is uniform. And when this
assumption is wrong, so is our conclusion, as Figure 14 shows.
Figure 14 Four independent trials of deconvolution of sparse noise into an all-pass filter. Alternate lines are input and output.
Recall that the inverse to an all-pass filter is its time reverse. The reversed shape of the filter is seen on the inputs where there happen to be isolated spikes.
Let us see what theory predicts cannot be done, and then I will tell you how I did it. If you examine the unweighted least-squares error-filter programs, you will notice that the first calculation is
the convolution operator and then its transpose. This takes the autocorrelation of the input and uses it as a gradient search direction. Take a white input and pass it through a phase-shift filter;
the output autocorrelation is an impulse function. This function vanishes everywhere except for the impulse itself, which is constrained against rescaling. Thus the effective gradient is zero. The
solution, an impulse filter, is already at hand, so a phase-shift filter seems unfindable.
On the other hand, if the signal strength of the input varies, we should be balancing its expectation by weighting functions. This is what I did in Figure 14. I chose a weighting function equal to
the inverse of the absolute value of the output of the filter plus an
Since the iteration is a nonlinear procedure, it might not always work. A well-established body of theory says it will not work with Gaussian signals, and Figure 15 is consistent with that theory.
Figure 15 Failure of blind all-pass deconvolution for Gaussian signals. The top signal is based on Gaussian random numbers. Lower signals are based on successive integer powers of Gaussian signals.
Filters (on the right) fail for the Gaussian case, but improve as signals become sparser.
In Figure 13, I used weighting functions roughly inverse to the envelope of the signal, taking a floor for the envelope at 20% of the signal maximum. Since weighting functions were used, the filters
need not have turned out to be symmetrical about their centers, but the resulting asymmetry seems to be small.
Next: About this document ... Up: INTERPOLATION ERROR Previous: INTERPOLATION ERROR Stanford Exploration Project | {"url":"https://sep.stanford.edu/sep/prof/pvi/tsa/paper_html/node25.html","timestamp":"2024-11-11T07:22:32Z","content_type":"text/html","content_length":"7293","record_id":"<urn:uuid:cef26ead-2a92-441b-80e3-1ed65b2580e0>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00855.warc.gz"} |
How the 4% Rule Works – Rob Berger
How the 4% Rule Works
Some of the links in this article may be affiliate links, meaning at no cost to you I earn a commission if you click through and make a purchase or open an account. I only recommend products or
services that I (1) believe in and (2) would recommend to my mom. Advertisers have had no control, influence, or input on this article, and they never will.
The 4% rule is a rule of thumb that can help you figure out how much money you can spend each year in retirement without going broke. If you're many years away from retirement, you can also use it to
figure out just how much money you'll need to retire. This article is the first of a series exploring the 4% rule, how to apply it, its limitations, and alternatives to retirement spending
My Experience with the 4% Rule
The 4% rule hit home for me two years ago at the age of 51. I had just sold my business, an online media company that owned several finance blogs. And I retired. Yes, I'm part of the FIRE (Financial
Independence, Retire Early) crowd.
At the time, I did all the calculations using the 4% rule on a very conservative basis. My wife and I had more than enough to retire. But doing it is a lot different than thinking and writing about
it. After I sold the business, I was scared to spend the money. In fact, I was so scared, I took a full-time job.
I know that doesn't make a whole lot of sense. But that's what I did. Now, in fairness, I worked at Forbes and enjoyed it immensely. It was lifestyle-friendly. However much I enjoyed the work, it
doesn't change the fact that I built a business, sold it, and retired at the age of 51, only to go back to work out of fear of running out of money.
Now, the good news is it forced me to take a deep dive into the 4% rule. I've read dozens of papers on the 4% rule. I have read books, and I've studied dynamic spending plans, and things called
I've even looked at how institutional investors like the Yale endowment figure out how much money they can spend each year from the endowment without running out of money. There are a lot of
parallels between how an endowment functions on the one hand, and how you and I should think about spending in retirement on the other.
What I learned from studying the 4% rule is that it's a really good rule of thumb. It still works today. Second, almost no one should follow it in retirement. I know those seem to contradict each
other, but they don't, and I'll explain why as we go along in this series.
In this article, we're going to look at four things. First, we cover a high-level view of what the 4% rule is and how it works. Second, we're going to look at who created the 4% rule. Third, we'll
cover how to use the 4% rule to estimate how much you need to save to retire. Finally, we're going to look at some very bizarre results that can flow from actually following the 4% rule.
So let's get started.
How the 4% Rule Works
The 4% rule is simple to apply in retirement. It takes just 3 steps.
Step 1: Add up your retirement savings
The first step in using the 4% rule is to add up all of the money you've saved for retirement. This can include both retirement accounts as well as taxable accounts you expect to use to fund expenses
in retirement. For example, you would include any money in a 401k or other workplace retirement plan, any IRAs that you have, and any money in taxable investing accounts, or savings accounts,
certificates of deposit, or checking accounts.
Include anything you've saved, that's going to be used to fund your retirement. Typical accounts include the following:
• 401(k)
• 403(b)
• 457
• TSP
• IRA
• Roth IRA
• HSA (if used for retirement)
• Taxable investment accounts
• Savings accounts
• Certificates of deposit
There are a few things you don't include. You don't include social security, annuity income or pension payments. You're only factoring in money you've saved and accumulated for retirement. If you use
a tool like Personal Capital to track your investments, this step is easy.
That's step one. Let's imagine that you've saved $1 million just to use a round number to make the math a little easier.
Step 2: Multiply your retirement savings by 4%
The second step is to multiply the results from step 1 by 4%. With $1 million, 4% would be $40,000. That's the amount of money using the 4% rule that you could spend in the first year of retirement.
Step 3: Beginning in year 2 of retirement, adjust the prior year's spending by the rate of inflation
It's the second year that trips some folks up. The way you calculate all the years in retirement after year one is different. Beginning in year two, you do not use 4%. Instead, you take the amount of
money you were able to spend the prior year and adjust it for inflation.
So in our hypothetical we spent $40,000 in year one of retirement. Let's assume inflation is 2%. In year two, we could spend $40,800. To calculate this number, we simply add 2% to the amount we were
able to spend in the previous year. Two percent of $40,000 is $800. Added to our first year spending brings us to $40,800. The following year we'll increase $40,800 by the rate of inflation (or
decrease it by the rate of deflation).
Where did the 4% Rule Come From?
The 4% rule dates back to 1994. It comes from an article published in the Journal of Financial Planning by William Bengen, a certified financial planner. He is the father of the 4% rule. The article–
Determining Withdrawal Rates Using Historical Data.
Bengen's primary focus wasn't actually the 4% rule as we know it today. In fact, that term doesn't appear in his paper. What he was more concerned with was how you go about calculating how much a
retiree can safely withdraw each year from retirement accounts.
At that time, a lot of advisors would use average market returns and average inflation rates to determine the initial withdrawal rate. For example, they might explain to a client that a typical
portfolio consisting of 60% stocks and 40% bonds has returned about 8% over the last 100 years. At the same time, inflation has averaged about 3% a year. Based on these averages, financial advisors
would tell clients that they could withdraw 5% (8% average return – 3% average inflation) the first year of retirement, and then adjust that by the average rate of inflation. 3%
Bengen's concern was that actual stock market returns and actual inflation rates might not support an initial 5% withdrawal rate. Even if the averages proved to be accurate over a 30-year retirement,
really bad markets and high inflation in the early years of retirement could cause a retiree to run out of money before retirement ended.
And that's in fact exactly what Bengen's paper concluded. If you wanted to be completely safe, the most you could take in your one of retirement was 4%.
Now, we will look at the methodology behind the 4% rule and the assumptions he used in later articles. Both are extremely important to understanding how we can and cannot, and how we should and
should not, apply the 4% rule.
How to use the 4% Rule for Retirement Planning
You can use the 4% rule to estimate how much you'll need to save before you can retire.
Step 1: Estimate your yearly expenses in retirement
The first step is to estimate your yearly expenses in retirement. If you are near retirement, your current budget may suffice. Just remember to make adjustments if necessary to account for the
transition from work (e.g., commuting costs go down, but retirement hobby or travel expenses may go up). If you are many years from retirement, taking a percentage of your current income (say 80%)
may be sufficient for a rough estimate.
Step 2: Determine amount of yearly expenses covered by retirement savings
Next, estimate how much of your yearly expenses will be covered by retirement savings. For most people, social security and perhaps a part-time job will cover some portion of expenses in retirement.
You can get an estimate of your social security benefits directly from the Social Security Administration.
Others may have a pension, an annuity or both. Subtract these other sources of income from your estimated yearly expenses. What is left is what must be covered by retirement savings.
Step 3: Multiply results from step 2 by 25
Multiply the results from Step 2 by 25. Note that this is the inverse of the 4% rule. If your expenses covered by retirement savings total $40,000 a year, multiplying this number by 25 gives us $1
million. Taking 4% of $1 million brings us back to $40,000.
How the 4% Rule can Lead to Bizarre Results
Now let's underscore some of the difficulties with the 4% rule and why we need to be so careful with it.
Let's imagine two couples are thinking about retiring. They're good friends, and so the four of them go to a financial advisor together. The financial advisor explains the 4% rule–they can spend 4%
of their portfolio in the first year of retirement and then adjusted for inflation every year thereafter.
One couple, we'll call them the Retired Couple, decides to retire. They have a million dollar portfolio. They take out $40,000 in the first year, which leaves them with $960,000.
The second couple, will call them the Working Couple, decide to hold off for a year. They're not going to add to their retirement portfolio, but plan to use the next year to pay off some debt before
they retire. So they just leave the $1 million in their portfolio.
Now, let's imagine that over the next year the market doesn't do so well. Both portfolios fall by a total of 20%. So where do we stand after the first year?
• Retired Couple: $1 million – $40,000 spending = $960,000 – 20% = $768,000
• Working Couple: $1 million – $0 = $1 million – 20% = $800,000
Now let's imagine the four of them go back to the advisor to find out how much they can spend in year two.
For the Retired Couple, they don't look at their balance to determine how much they can spend in year two. Recall under the 4% rule that beginning in the second year, you simply take whatever you
spent the previous year and adjust it for inflation. So if we assume a rate of inflation of 2%, the Retired Couple could spend $40,800 ($40,000 + ($40,000 * .02)).
In year two, the Working Couple who are now retiring for the first time, however, have to take whatever their balance is and multiply it by our familiar 4% number. Since they're down to $800,000, 4%
is $32,000.
Now if you think these results seem a little bit odd, it's because they are. Our Retired Couple has a portfolio that's lower than the Working Couple. They're down to $768,000 compared to $800,000 for
the Working Couple. Yet they can take out over $8,000 more–$40,800–compared to just $32,000 for the Working Couple. That seems like a pretty bizarre result to me.
Portfolio Balance Spending Allowed Calculation Method
Retired Couple $768,000 $40,800 $40,000 + 2% inflation
Working Couple $800,000 $32,000 $800,000 * 4%
Now, what do we do with this? Does this mean the 4% rule is invalid? Does it mean it contradicts itself? Is it difficult to apply? Well, not exactly.
It does underscore the difference between theory and reality. And in fact, we're going to cover a lot of realistic scenarios in this article series where the 4% rule, while it's a good planning tool,
and it's a good rule of thumb, may not make a lot of sense when you go to actually apply it.
The good news is, I've got a number of alternatives that I'll share with you that I think can be just as effective, but maybe a bit more practical to apply. So in the next article, we're going to
look at Bengen's methodology–how he actually went about calculating what we now know is the 4% rule. Once we understand that we can begin to apply this information in a practical way to both
retirement planning and retirement spending.
How the 4% Rule Works–Video
Rob Berger is a former securities lawyer and founding editor of Forbes Money Advisor. He is the author of Retire Before Mom and Dad and the host of the Financial Freedom Show. | {"url":"https://robberger.com/how-the-4-percent-rule-works/","timestamp":"2024-11-05T00:51:47Z","content_type":"text/html","content_length":"117679","record_id":"<urn:uuid:b57e3b8e-687e-41b0-bef8-f019920dc914>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00850.warc.gz"} |
(PDF) Analytical Ballistic Trajectories with Approximately Linear Drag
... As a senior computer science faculty, I strongly believe that understanding these mathematical foundation has played an immense important role in polishing ones" cognitive, engineering and
programming skills [6] - [12]. There exist three common models to simulate projectile in computer applications for gaming and simulation; they are: (1) No drag model (2) Linear drag model and (3)
Quadratic drag model. In no drag model, the motion of projectile is mainly dependent on initial velocity and the angle of launch. ...
... Fig. 7 shows that it is possible that flight time will take form of triangular series and the projectile trajectory is successfully followed if triangular increment is used for simulation. The
simulation of linear impulse in box2D and OpenGL requires two arguments: (1) An initial vector V 0 containing and component of velocity (2) A Point from where impulse has to be triggered. Y-component
of final velocity is denoted with ( ) i.e. velocity at time instance t in future; g is 9.8 m/s, "F" represents y-component of force applied on mass "m". ... | {"url":"https://www.researchgate.net/publication/275069366_Analytical_Ballistic_Trajectories_with_Approximately_Linear_Drag","timestamp":"2024-11-02T08:50:36Z","content_type":"text/html","content_length":"906912","record_id":"<urn:uuid:e83fc251-49dc-497d-8c1e-6581800c446c>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00805.warc.gz"} |
Special quasirandom structures¶
Looking for a simple way to generate special quasirandom structures? Feel free to try this SHARC web application, which uses icet under the hood.
Random alloys are often of special interest. This is true in particular for systems that form random solid solutions below the melting point. It is, however, not always easy to model such structures,
because the system sizes that lend themselves to, for example, DFT calculations, are often too small to accomodate a structure that may be regarded as random; the periodicity imposed by boundary
conditions introduces correlations that make the modeled structure deviate from the random alloy. This problem can sometimes be alleviated with the use of so-called special quasirandom structures
(SQS) [ZunWeiFer90]. SQS cells are the best possible approximations to random alloys in the sense that their cluster vectors closely resemble the cluster vectors of truly random alloys. This tutorial
demonstrates how SQS cells can be generated in icet using a simulated annealing approach.
There is no unique way to measure the similarity between the cluster vector of the SQS cell and the random alloy. The implementation in icet uses the measure proposed in [WalTiwJon13]. Specifically,
the objective function \(Q\) is calculated as
\[Q = - \omega L + \sum_{\alpha} \left| \Gamma_{\alpha} - \Gamma^{\text{target}}_{\alpha} \right|.\]
Here, \(\Gamma_{\alpha}\) are components in the cluster vector and \(\Gamma^\text{target}_{\alpha}\) the corresponding target values. The factor \(\omega\) is the radius (in Ångström) of the largest
pair cluster such that all clusters with the same or smaller radii have \(\Gamma_{\alpha} - \Gamma^\text{target}_{\alpha} = 0\). The parameter \(L\), by default 1.0, can be specified by the user.
The functionality for generating SQS cells is just a special case of a more general algorithm for generating a structure with a cluster vector similar to any target cluster vector. The below example
demonstrates both applications.
Import modules¶
The generate_sqs and/or generate_target_structure functions need to be imported together with some additional functions from ASE and icet. It is advisable to turn on logging, since the SQS cell
generation may otherwise run quietly for a few minutes.
from ase import Atom
from ase.build import bulk
from icet import ClusterSpace
from icet.tools.structure_generation import (generate_sqs,
from icet.input_output.logging_tools import set_log_config
Generate binary SQS cells¶
In the following example, a binary FCC SQS cell with 8 atoms will be generated. To this end, an icet.ClusterSpace and target concentrations need to be defined. The cutoffs in the cluster space are
important, since they determine how many elements are to be included when cluster vectors are compared. It is usually sufficient to use cutoffs such that the length of the cluster vector is on the
order of 10. Target concentrations are specified via a dictionary, which should contain all the involved elements and their fractions of the total number of atoms. Internally, the function carries
out simulated annealing with Monte Carlo trial swaps and can be expected to run for a minute or so.
primitive_structure = bulk('Au')
cs = ClusterSpace(primitive_structure, [8.0, 4.0], ['Au', 'Pd'])
target_concentrations = {'Au': 0.5, 'Pd': 0.5}
sqs = generate_sqs(cluster_space=cs,
print('Cluster vector of generated structure:', cs.get_cluster_vector(sqs))
If for some reason a particular supercell is needed, there is another function generate_sqs_from_supercells that works similarly, but in which it is possible to explicitly provide the accepted
supercell. The code will then look for the optimal SQS among the provided supercells.
supercells = [primitive_structure.repeat((1, 2, 4))]
sqs = generate_sqs_from_supercells(cluster_space=cs,
print('Cluster vector of generated structure:', cs.get_cluster_vector(sqs))
Generate SQS cells by enumeration¶
In the above simple case, in which the target structure size is very small, it is more efficient to generate the best SQS cell by exhaustive enumeration of all binary FCC structures having up to 8
atoms in the supercell:
sqs = generate_sqs_by_enumeration(cluster_space=cs,
print('Cluster vector of generated structure:', cs.get_cluster_vector(sqs))
Generation of SQS cells by enumeration is preferable over the Monte Carlo approach if the size of the system permits, because with enumeration there is no risk that the optimal SQS cell is missed.
Generate SQS cells for a system with sublattices¶
It is possible to generate SQS cells also for systems with sublattices. In the below example, an SQS cell is generated for a system with two sublattices; one FCC sublattice on which Au, Cu, and Pd
are allowed, and another FCC sublattice on which H and vacancies (X) are allowed. Target concentrations are specified per sublattice. The sublattices are defined by the letters shown at the top of
the printout of a ClusterSpace.
primitive_structure = bulk('Au', a=4.0)
primitive_structure.append(Atom('H', position=(2.0, 2.0, 2.0)))
cs = ClusterSpace(primitive_structure, [7.0], [['Au', 'Cu', 'Pd'], ['H', 'X']])
This should result in something similar to this:
====================================== Cluster Space =======================================
space group : Fm-3m (225)
chemical species : ['Au', 'Cu', 'Pd'] (sublattice A), ['H', 'X'] (sublattice B)
cutoffs : 7.0000
total number of parameters : 40
number of parameters by order : 0= 1 1= 3 2= 36
fractional_position_tolerance : 2e-06
position_tolerance : 1e-05
symprec : 1e-05
index | order | radius | multiplicity | orbit_index | multi_component_vector | sublattices
0 | 0 | 0.0000 | 1 | -1 | . | .
1 | 1 | 0.0000 | 1 | 0 | [0] | A
2 | 1 | 0.0000 | 1 | 0 | [1] | A
3 | 1 | 0.0000 | 1 | 1 | [0] | B
4 | 2 | 1.0000 | 6 | 2 | [0, 0] | A-B
Here we see that the sublattice with Au, Cu and Pd is sublattice A, while H and X are on sublattice B. These letters can now be used when the target concentrations are specified.
In the below example, an SQS cell is generated for a supercell that is 16 times larger than the primitive cell, in total 32 atoms. The keyword include_smaller_cells=False guarantees that the
generated structure has 32 atoms (otherwise the structure search would have been carried out among structures having 32 atoms or less).
In this example, the number of trial steps is manually set to 50,000. This number may be insufficient, but will most likely provide a reasonable SQS cell, albeit perhaps not the best one. The default
number of trial steps is 3,000 times the number of inequivalent supercell shapes. The latter quantity increases quickly with the size of the supercell.
target_concentrations = {'A': {'Au': 6 / 8, 'Cu': 1 / 8, 'Pd': 1 / 8},
'B': {'H': 1 / 4, 'X': 3 / 4}}
sqs = generate_sqs(cluster_space=cs,
print('Cluster vector of generated structure:', cs.get_cluster_vector(sqs))
Generate a structure matching an arbitrary cluster vector¶
The SQS cell generation approach can be utilized to generate the structure that most closely resembles any cluster vector. To do so, one can employ the same procedure but the target cluster vector
must be specified manually. Note that there are no restrictions on what target vectors can be specified (except their length, which must match the cluster space length), but the space of cluster
vectors that can be realized by structures is restricted in multiple ways. The similarity between the target cluster vector and the cluster vector of the generated structure may thus appear poor.
primitive_structure = bulk('Au')
cs = ClusterSpace(primitive_structure, [5.0], ['Au', 'Pd'])
target_cluster_vector = [1.0, 0.0] + [0.5] * (len(cs) - 2)
target_concentrations = {'Au': 0.5, 'Pd': 0.5}
sqs = generate_target_structure(cluster_space=cs,
print('Cluster vector of generated structure:', cs.get_cluster_vector(sqs))
Source code¶
The complete source code is available in examples/sqs_generation.py
This example demonstrates how to generate special quasirandom structure.
# Import modules
from ase import Atom
from ase.build import bulk
from icet import ClusterSpace
from icet.tools.structure_generation import (generate_sqs,
from icet.input_output.logging_tools import set_log_config
# Generate SQS for binary fcc, 50 % concentration
primitive_structure = bulk('Au')
cs = ClusterSpace(primitive_structure, [8.0, 4.0], ['Au', 'Pd'])
target_concentrations = {'Au': 0.5, 'Pd': 0.5}
sqs = generate_sqs(cluster_space=cs,
print('Cluster vector of generated structure:', cs.get_cluster_vector(sqs))
# Generate SQS for binary fcc with specified supercells
supercells = [primitive_structure.repeat((1, 2, 4))]
sqs = generate_sqs_from_supercells(cluster_space=cs,
print('Cluster vector of generated structure:', cs.get_cluster_vector(sqs))
# Use enumeration to generate SQS for binary fcc, 50 % concentration
sqs = generate_sqs_by_enumeration(cluster_space=cs,
print('Cluster vector of generated structure:', cs.get_cluster_vector(sqs))
# Generate SQS for a system with two sublattices
primitive_structure = bulk('Au', a=4.0)
primitive_structure.append(Atom('H', position=(2.0, 2.0, 2.0)))
cs = ClusterSpace(primitive_structure, [7.0], [['Au', 'Cu', 'Pd'], ['H', 'X']])
# Target concentrations are specified per sublattice
target_concentrations = {'A': {'Au': 6 / 8, 'Cu': 1 / 8, 'Pd': 1 / 8},
'B': {'H': 1 / 4, 'X': 3 / 4}}
sqs = generate_sqs(cluster_space=cs,
print('Cluster vector of generated structure:', cs.get_cluster_vector(sqs))
# Generate structure with a specified cluster vector
primitive_structure = bulk('Au')
cs = ClusterSpace(primitive_structure, [5.0], ['Au', 'Pd'])
target_cluster_vector = [1.0, 0.0] + [0.5] * (len(cs) - 2)
target_concentrations = {'Au': 0.5, 'Pd': 0.5}
sqs = generate_target_structure(cluster_space=cs,
print('Cluster vector of generated structure:', cs.get_cluster_vector(sqs)) | {"url":"https://icet.materialsmodeling.org/dev/advanced_topics/sqs_generation.html","timestamp":"2024-11-13T06:05:13Z","content_type":"text/html","content_length":"50506","record_id":"<urn:uuid:5b0c7bf6-f986-4f01-a6d9-3f7ba673acc9>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00650.warc.gz"} |
User:Thijshijsijsjss/Gossamery/99 Variations on a Proof
• Obtained: 2024-03-07
• I own this book in physical form; approach me if you're interested to take a look; still looking for a pdf
99 Variations on a Proof (Ording, 2019) is a book presenting the multiplicity and situatedness of mathematical practises through a series of non (non-)proofs for the same problem. These proofs range
from traditional (e.g. 6 Axiomatic, 13 Reductio ad Absurdum, 40 Induction) to visual (e.g. 3 Illustrated, 39 Origami) to fictionalized (e.g. 43 Screenplay, 65 Tea). Every proof is accompanied by an
explanation that elaborates not only on the proof, but also on the context of it. They provide a glimpse of mathematics not just as the results on paper, but as the human practise of it.
I had been vaguely aware of this book before stumbling upon it in Leeszaal. I have found it to be insightful, poetic, puzzling, touching. It it a powerful, playful representation of a field that,
from within, is often archaic and hierarchical. This method might be applied elsewhere, too.
There is a relation to the listmaking exercises we have been doing during SI24 (on the back: 'According to Molière there are many ways to declare love [and] lists five [...]'). Perec's An Attempt at
Exhausting a Place in Paris, might have a different feel and goal, but sees similarities at the same time.
In December 2023, I wrote a text by what I like to think of as 'axiomatic writing'. I've had lingering motivation to return to this writing style at some point. Reading axiomatic proofs surrounded by
fictional and poetic pieces in this book, has reemphasized its power to me. | {"url":"https://pzwiki.wdka.nl/mediadesign/User:Thijshijsijsjss/Gossamery/99_Variations_on_a_Proof","timestamp":"2024-11-02T18:24:33Z","content_type":"text/html","content_length":"25733","record_id":"<urn:uuid:ab72dbb8-49a8-4944-8caf-e0e5d4f714a1>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00508.warc.gz"} |
Introduction to the Travelling Salesman Problem
TSP (Travelling Salesman Problem) is a famous problem described by the following question:
Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city?
Despite how simple it is to describe the problem (even one child would understand what we are looking for), there is no efficient algorithm yet to find the optimal solution. It’s classified as
NP-Hard Problem, no polynomial time method has been found so far. As we can see in this article, a modern normal computer can find the optimal solution until 22–24 points (cities).
I find it inspiring that today, despite our great technological advances, there are problems so easy to describe still unresolved. We are not talking about quantum physics or fluid mechanics! We just
want to find the shortest route given a certain number of points to visit!
In this article, we’ll explain and implement code on algorithms to find the optimal solution and also some approximations. We are going to use cSharp and the code will result in a plugin for AutoCAD,
which provides a very friendly GUI.
You can find here the Github repository related to this article.
Plugin interface
This form is the program’s interface that will allow the user to “play” with different TSP algorithms and quantity of nodes, it has 3 tabs as can be seen in the next picture. Just in case, in this
article I explain how to setup a plugin for AutoCad.
Clicking on insert sample nodes, a certain quantity of nodes will be drawn in CAD’s model, then the user can choose an algorithm to solve the TSP for that nodes. The next picture shows an example
with 20 nodes.
TSP algorithms introduction
A naive approach to solving TSP would be the brute-force solution, which means finding all possible routes given a certain number of nodes. This is a very expensive way to solve it, with a time
complexity of O(n!). To be exact, the brute-force time complexity is (n-1)!/2. Imagine you have n nodes, then, if we want to compute all possible paths, we must pick one random start node, then we
have (n-1) options for the next node, and (n-2) for the next, etc… This gives us (n-1)!, but we should consider that the path (1 > 2 > 3 > 4 > 1) is the same as (1 > 4 > 3 > 2 > 1), that’s the reason
we divide by 2.
In this article, we’ll analyze 2 ways of computing the optimal solution, the Integer Linear Programming and Dynamic Programming approach, which are slightly better than the brute-force method.
Afterward, we’ll move to explore 2 approximation algorithms, which run much faster than the previous ones and are not so bad in precision, they are called 2T (Double-Tree) and Christofides
approximation. In the worst-case scenario, 2T would be 2 times the optimal, and Christofides solution would be 1.5 times.
Finally, we’ll talk about Google OR-Tools Routing library, which is free and provide powerful approximations to the TSP that run very fast and combine more than 1 algorithm strategy.
Optimal solution approaches
Integer Linear Programming
Linear Programming (LP) is a powerful way to solve problems, and part of its beauty is its simplicity, we only need to formulate (express) our input in the required way, then LP will do the rest of
the job returning the output solution. This “formulation” consists in:
• Cost Function to optimize (maximize or minimize):
• Variables must be positive or equal to 0:
This can be expressed in matricial form as follows:
Integer Linear Programming adds one more constraint, and that is our variables (x), which must be positive integers, meaning:
The key part of using LP is finding the correct formulation for the problem. Sometimes there’s more than 1 possible formulation, and one can be more efficient than the other. In fact, we’ll explore 2
possible ways to formulate the TSP, and we’ll see how they differ in their performance.
We can follow our intuition to think about the formulation of this problem, we need to define our variables, constraints and the objective function. It’s easy to think about it if we work with an
• Variables:
What we are looking for is a tour that passes through all nodes at just one time (with the minimum length). We can declare our variables as the edges of the complete graph formed by the nodes. If
the variable (edge) is equal to 1 means forms part of the optimal tour, otherwise, if it’s equal to 0, does not belong to the optimal tour.
• Objective function:
We want to find the tour with the minimum distance, so it makes sense to write our objective function as follows:
• Constraints:
Intuitively we can state:
• Each node is a start point of an edge that belongs to the optimal tour:
• Each node is an end point of an edge that belongs to the optimal tour:
So… Are we ready? That’s all?…
Unfortunately not! There’s something we are not taking into account…
There isn’t any constraint to eliminate possible subtours! The following picture shows 3 setups without the subtour elimination constraints. Note that is also possible a subtour with only 2 nodes,
which starts from one node and comes back to it.
So… We need to add some constraints to eliminate these subtours… ¿How can we do it? Next, we’ll discuss 2 possible ways to do it:
Method 1 to eliminate subtours
Adding constraints relating each subset size to its number of activated edges. An “active edge” means its variable is equal to 1, so belongs to the optimal tour. This means, for example, if we have
the subset {0,1,2}, we can only have activated 2 possible edges, but not 3. We can state, for each possible subset:
Expressed more formally:
This method it’s easy to understand, but it adds a big number of constraints, and this causes the LP algorithm to be quite inefficient. Next are computed the number of constraints added by this way
of eliminate subtours:
Method 2 to eliminate subtours
There is another way to eliminate subtours, which may be less intuitive, but very smart, that provides a more compact formulation. It was discovered by Miller, Tucker and Zemlin in 1960. This
formulation introduces new time variables, which we call ui.
The idea is to find a relation between xi, ui and uj.
We can model this using the big number technique:
Where M is some large number, we can choose M=n-1 because ui ∈ [1, n-1]. We can sum up these time constraints as follows:
It’s important to note that only node 0 is not restricted by these constraints. With this formulation we have drastically reduced the number of constraints, from 2ⁿ to n.
Here you can find the code implementation of the ILP formulations commented above, we use the Linear Solver offered by OR-Tools library from Google. Using this code, the ILP formulation with time
variables runs faster than the other one.
Dynamic Programming
These are the steps to solve a Dynamic Programming problem:
1. Identify the recurrence relation and solve the problem with a top-down approach
2. Optimize solution adding memoization
3. Optimize solution using iteration, bottom-up approach
Identifying the Recurrence Relation
Let’s compute manually one example to see if we can detect the recurrence relation, we are going to work with a 4-node graph with the following distance matrix. It’s not symmetric, but that’s
perfectly fine, imagine it’s a road system where the route going from node A to B is shorter than vice versa.
The next diagram shows all possible tours we can take starting from node 0, for example, 0 > 1 > 2 > 3 > 0, 0 > 1 > 3 > 2 > 0, etc.
If we look carefully we can see that in every node we are doing the same, next is presented the recurrence relation:
Or expressed in a more general way:
Where g(i,S) is the minimum cost from node i to the subset S of nodes, in other words, is the optimal cost of the subset {i ∪ S}, starting from some starting_node (in our example is 0), and ending in
i. Here you can find the code implementation that solves the problem using this recurrence relation with a top-down approach.
Optimizing solution with memoization
Now that we have solved the problem using the recurrence relation, the next step is to try to find if we can avoid certain recursion calls using memoization. We can create a 2D table to store the
computed values of our recurrent function, each row can correspond to a certain subset {i ∪ S} and each column to the last index visited i. Therefore, there will be 2ⁿ rows and n columns. The
following table corresponds to the example we’re working on.
We can also use this memoization table to compute the optimal tour, meaning the order of node indexes, starting and ending in 0, that has the minimum cost. Our example is “0 -> 1 -> 3 -> 2 -> 0”.
Here is the code including this optimization.
Bottom-up approach
Here is presented the code that solves the TSP problem avoiding recursion with a bottom-up approach. The memo table is filled from the bottom of the tree to the top.
The Dynamic Programming approach has O(n² * 2ⁿ) time, which is a great improvement comparing it with brute-force. As can be seen in the following table, for n ≥ 10, DP time complexity beats the
brute-force time.
Approximation solution approaches
As we have seen, the optimal solution approaches run in exponential time, so we can’t use them for more than 24–25 nodes. What can we do? The TSP problem appears many times in our daily lives, for
example, companies need a solution to schedule their delivery orders with the minimum cost possible.
For this reason, TSP problem has some approximation solutions that run much faster than the optimal algorithms. We are going to analyze 2 approx. algorithms, they are called 2T Double-Tree
approximation and Christofides algorithm. In the worst-case scenario, 2T would be 2 times the optimal solution, and Christofides 1.5 times.
Minimum Spanning Tree (MST)
Both approximation solutions (2T and Christofides) are based on the concept of minimum spanning tree (MST). What is a MST?
A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles
and with the minimum possible total edge weight.
As you can see in the code, we use the Kruskal’s Algorithm with a Union-Find data structure to find the MST.
The cost of a MST is a lower bound of the optimal solution for the TSP problem.
¿Why Tɢ is a lower bound of Hɢ*?
This is easy to demonstrate: if we take the TSP Optimal Solution and remove one edge, we have a tree (which is not the minimum as MST).
2T approximation TSP (Double-Tree)
Once we have understood the concept of the MST and checked that is always a lower bound of the TSP solution, it’s easy to build an algorithm that will return an approximation of the TSP problem with
a maximum error of 2T (meaning in the worst-case scenario our approximation will be the double of the optimal solution).
These are the steps to build the 2T approximation:
1. Find an MST which we call Tɢ.
2. Duplicate the edges of Tɢ.
3. Find an Eulerian tour using Hierholzer algorithm.
4. Shortcut the Eulerian tour (remove duplicate vertices)
If we do these steps, in the worst-case scenario we have visited every edge twice (DFS traversal), that’s the reason it’s called 2T approximation. It’s worth saying that when we remove duplicates
(step 4), the cost only can decrease due to the triangle inequality. Here you can find the code implementation of this algorithm.
1.5T approximation TSP (Christofides algorithm)
We can improve the Double-Tree approximation with Christofides algorithm, which in the worst-case scenario will be 3/2 times the optimal solution. First, we’ll explain which are the steps and
afterward we’ll demonstrate why is a 1.5T approx.
1. Find an MST which we call Tɢ.
2. Find the subset of vertices in Tɢ with odd degree, which we call S (there will always be an even number of vertices with odd degree (later we’ll explain why).
3. Find a Minimum Perfect Matching M on S. As you can see in the code, we use linear programming to find M.
4. Add the set of edges of M to Tɢ. As you can see in the image below, multi-edge is allowed (look at edges between nodes P and N).
5. Find an Eulerian Tour.
6. Shortcut the Eulerian Tour (remove duplicate vertices).
Now we’ve understood the steps of Christofides algorithm, let’s try to understand the reason behind them.
¿Why do we want to find the set of odd-degree vertex S?
The main strategy of Christofides algorithm is to find an Eulerian tour from the MST and then “shortcut it” (removing the duplicate nodes). To have an Eulerian tour in a graph we need every vertex to
be even degree. We want to find the set of odd vertices because we need “somehow” to turn them into even.
¿Why do we compute the Minimum Perfect Matching on S?
The idea is to add one degree to every odd-vertex, we can achieve this by finding a perfect matching on S (set of odd-degree nodes), if we do this, we have achieved our goal and find an Eulerian
tour. The Minimum Perfect Matching is the optimal way to add these edges (adding the min cost possible).
¿Why there will always be an even number of odd-degree vertex?
We know by the handshaking lemma that the sum of all vertex degrees in a graph is double the number of edges:
Where V is the set of all vertices in G. Let’s divide V into 2 sets of vertices:
• V = R ∪ S
• R = Set of even-degree vertices in G
• S = Set of odd-degree vertices in G
So we can express the handshaking lemma as follows:
The right side of the equation (2 |E|) is an even number, so the left side has to be even as well. By definition, the sum of even-vertex degrees is also even.
It means that the sum of odd-vertex must be even as well to maintain the whole left side equation even.
We need the sum of odd numbers to be even, it means p is even.
¿Why Christofides is a 1.5 approximation of the TSP?
The first step to perform Christofides is to find an MST (similar to the 2T Double-Tree discussed before), we already know that is a lower bound of optimal solution on G. Then the question is why
adding the Minimum Perfect Matching edges adds, in the worst-case scenario, an error of 0.5 T. To understand this, let’s think about these 2 perfect matching shown in the following picture, which is
made based on the optimal solution TSP of the set S.
M₁ and M₂ are perfect matching on Hꜱ*, but not the perfect matching M, so we can state:
This implies that c(M) is is lesser or equal to the average of c(M₁) and c(M₂).
As said before, MST is a lower bound of the optimal solution TSP, meaning:
We also now:
• Set S has fewer vertex than G, so, by the triangle inequality:
Then we can conclude:
So finally:
Here you can find the code implementation of Christofides algorithm.
Google OR-Tools library
Until now we have explored some optimal algorithms approaches (linear and dynamic programming) and some approximation algorithms (Double-Tree and Christofides). I think it’s worth understanding
things from the base, and in computer science, test your knowledge by implementing the concepts in code yourself. As much as you can, avoid “black boxes”.
However, once we know what we are talking about, it’s also important to explore which tools are out there that are already implemented, optimized and maintained, maybe there is an open-source tool we
can use to achieve our goal. This is also important because we can build from there instead of “reinventing the wheel” from the base.
Google OR-Tools is an open-source library that can help us a lot with the TSP problem and related concepts (for example linear and integer programming). “OR” stands for “Optimization Research”.
OR-Tools is an open source software suite for optimization, tuned for tackling the world’s toughest problems in vehicle routing, flows, integer and linear programming, and constraint programming.
We are going to add the feature to use OR-Tools to solve TSP. OR-Tools provide also an approximation of the TSP problem, but applies a first solution strategy and afterward refines it with other
First solution strategies are listed here, some of them are:
• CHRISTOFIDES:
We know about it!
• PATH_CHEAPEST_ARC:
Starting from a route “start” node, connect it to the node which produces the cheapest route segment, then extend the route by iterating on the last node added to the route.
• GLOBAL_CHEAPEST_ARC:
Iteratively connect two nodes which produce the cheapest route segment.
Next image shows different TSP solutions obtained by the library OR-Tools, for a 50 vertex graph, with different first-solution-strategies.
As we have seen, OR-Tools TSP implementation provides very good approximations, and the algorithms run quite fast even when dealing with graphs with many vertices. On my computer (which is not a
super-computer) it takes 3.19 s to give a solution for a graph of 500 points, and 13.47 s for one of 1,000 points.
Another cool thing about OR-Tools is that has the feature to solve vehicle routing problems, which can be seen as an extension of the TSP problem. Imagine that you have a company that has to deliver
200 different points in the city, and you have 4 vehicles. What would be the route that you would give to each vehicle in order to optimize the delivery time? Well… OR-Tools can help you with this!
Here you can find the code implementation of Google Or-Tools in our plugin.
You can find the full code in this Github Repository.
If you enjoyed this story, please click the 👏 button and share to help others find it! Feel free to leave a comment below. You can connect with me on LinkedIn, My blog, Twitter, Facebook. | {"url":"https://tonicanada.medium.com/introduction-to-the-travelling-salesman-problem-5ace44932cb5?source=user_profile_page---------3-------------96dd594ee964---------------","timestamp":"2024-11-08T23:56:48Z","content_type":"text/html","content_length":"346926","record_id":"<urn:uuid:9a560724-59fa-496c-9ecd-8b75654b62e5>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00032.warc.gz"} |
Re: Formula using IF
Jan 11, 2021 03:28 AM
i have the following colums that iam trying to combine in a formula :
• customer billing (with only two possibles choices: direct or indirect)
• indirect total sales revenue (a formula which multiplies a monthly amount by a number of months)
I need a formula in a new column which combines the two following conditions:
IF “indirect” is chosen, then copy the same amount
IF “direct” is chosen, then apply a 12% on the “indirect total sales revenue”
i hope it is clear enough.
Thyank you very much in advance!
Jan 11, 2021 09:58 PM
Jan 11, 2021 08:41 AM
Jan 11, 2021 09:50 AM
Jan 11, 2021 09:52 AM
Jan 11, 2021 10:30 AM
Jan 11, 2021 09:58 PM | {"url":"https://community.airtable.com/t5/formulas/formula-using-if/m-p/58890/highlight/true","timestamp":"2024-11-13T00:16:37Z","content_type":"text/html","content_length":"377179","record_id":"<urn:uuid:14bc65b3-d23b-4878-977e-c333075f1b84>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00198.warc.gz"} |
NumPy random seed (Generate Predictable random Numbers)
In computational science, a random seed is a starting point for the sequence of pseudorandom numbers that are generated.
These numbers appear random, but they follow a deterministic sequence. The seed determines the initial state of this sequence.
In Python’s NumPy library, you can set the random seed using the numpy.random.seed() function. This will make the output of random number generation predictable and reproducible.
Pseudorandom vs. True Random Numbers
Random numbers can be broadly classified into two categories: pseudorandom numbers and true random numbers.
Pseudorandom Numbers
Pseudorandom numbers are generated using deterministic algorithms. Given the same initial seed, they produce the same sequence of numbers every time.
Pseudorandom numbers are efficient to generate and suitable for most applications, including simulations and statistical sampling.
True Random Numbers
True random numbers are generated from fundamentally random physical processes.
They are not predictable and do not follow an algorithm. True random numbers are typically used in cryptographic applications where unpredictability is crucial.
In Python, true random numbers can be obtained using specialized hardware or online services, but they are outside the scope of this tutorial.
How to set a random seed in NumPy?
You use the numpy.random.seed() function and provide an integer that will be used as the seed.
Here’s an example:
import numpy as np
In this code, the random seed is set to 5. Every time you run this code, the random float generated will be the same.
You can change the seed to any integer to generate a different sequence of random numbers, but the sequence corresponding to a specific seed will always be the same.
Why use a random seed?
When working with random numbers, consistency and reproducibility can be crucial, especially in scientific computations, simulations, or machine learning tasks.
By using a random seed, you can ensure that the random numbers generated are the same every time the code is run.
Here’s a simple demonstration:
Without seeding:
import numpy as np
random_numbers_without_seed = [np.random.rand() for _ in range(5)]
[0.9507143064099162, 0.7319939418114051, 0.5986584841970366, 0.15601864044243652, 0.15599452033620265]
With seeding:
random_numbers_with_seed = [np.random.rand() for _ in range(5)]
[0.3745401188473625, 0.9507143064099162, 0.7319939418114051, 0.5986584841970366, 0.15601864044243652]
In the first code snippet, without setting a seed, the random numbers will be different each time you run the code.
In the second snippet, where we set the seed to 42, the numbers will be identical each time you run it.
This allows for testing, validation, and sharing of your code in a manner that others can replicate exactly.
How to set the global random seed?
Setting the global random seed in NumPy affects all random number generation functions in the library. It’s a crucial tool for making code involving random processes reproducible.
Here’s an example:
import numpy as np
print(np.random.randint(10, 20))
By setting the seed to 42, both the random float and random integer generated will be the same each time the code run.
This demonstrates how setting the global seed affects all random functions in NumPy.
Examples of functions affected
The global random seed in NumPy affects a wide range of functions that generate random numbers or perform random operations.
Here are examples of some of these functions.
Generates random floats between 0 and 1:
import numpy as np
[0.5488135 0.71518937 0.60276338]
Generates random integers within a specified range:
print(np.random.randint(1, 10, 3))
[6 1 4]
Shuffles the elements of an array randomly:
arr = [1, 2, 3, 4, 5]
[3, 1, 2, 4, 5]
Each of these functions is affected by the global seed, and setting the seed ensures that the results are consistent across different runs of the code.
Best practices for seeding
Setting the seed early in your code
Set the seed at the beginning of your code or a function that requires reproducible random numbers. This ensures that the sequence is initialized properly.
Choosing arbitrary seed values vs deterministic seeds
An arbitrary seed value leads to a specific sequence of random numbers.
Deterministic seeds, like using the current date, can also be used, but they won’t ensure reproducibility across different runs or machines.
Managing seeds for reproducibility across code executions
It’s essential to document the seed values used in your code to ensure that others can reproduce the exact results.
Here’s a code snippet that shows best practices:
import numpy as np
# Set the seed early
seed_value = 42
random_numbers = np.random.rand(3)
print(f"Seed: {seed_value}")
print(f"Random Numbers: {random_numbers}")
Seed: 42
Random Numbers: [0.37454012 0.95071431 0.73199394]
These practices ensure that your code’s random processes are transparent, controlled, and reproducible, both for you and others who might use your code.
Implementing simulations with reproducible results
When implementing simulations that require random number generation, it is often crucial to reproduce the results.
Using a fixed seed is the key to achieving this.
Here’s an example of a simple Monte Carlo simulation to estimate the value of π:
import numpy as np
num_points = 10000
inside_circle = 0
for _ in range(num_points):
x, y = np.random.rand(2)
if x**2 + y**2 <= 1:
inside_circle += 1
estimated_pi = (inside_circle / num_points) * 4
print("Estimated π:", estimated_pi)
Estimated π: 3.1428
By setting the seed at the beginning of the simulation, you can ensure that the results are consistent every time you run it.
This allows you to compare changes, validate your simulation, and share it with confidence that others will obtain the same results.
Mokhtar is the founder of LikeGeeks.com. He is a seasoned technologist and accomplished author, with expertise in Linux system administration and Python development. Since 2010, Mokhtar has built an
impressive career, transitioning from system administration to Python development in 2015. His work spans large corporations to freelance clients around the globe. Alongside his technical work,
Mokhtar has authored some insightful books in his field. Known for his innovative solutions, meticulous attention to detail, and high-quality work, Mokhtar continually seeks new challenges within the
dynamic field of technology. | {"url":"https://likegeeks.com/numpy-random-seed/","timestamp":"2024-11-13T02:32:15Z","content_type":"text/html","content_length":"164749","record_id":"<urn:uuid:38bca9bc-3320-4eaa-9f91-853a0e0351e0>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00354.warc.gz"} |
cmfb, one-stage op-amp
Operational Amplifier (1)
Chapter 9
• Common-Mode Feedback
• One Stage Op-Amps
• Two Stage Op-Amps
Necessity of Common-Mode
I3 is approximately 0.99I5 (due to mismatch in current source)
VX and VY will rise.
M5-M8 will enter the triode region.
VX and VY will rise if IR1/2>I1, pushing M5-M8 to the triode region.
VX and VY will drop if IR1/2<I1, pushing M9 into the triode region.
Solution: Common-Mode Feedback Circuit
A common-mode feedback (CMFB) circuit
is not necessary since the VD1 and VD2 are well-defined.
Rule of thumb: if the output CM level cannot be determined
by “visual inspection”, and requires calculations based
on device properties, then it is poorly defined.
Conceptual Topology for
Common-Mode Feedback
(Return error to
the amplifier’s bias network)
(Comparison with a reference)
Resistive Sensing (1)
R1+R2 must be
much larger than
the resistance into
the drain of NMOS and
PMOS so as to avoid lowering
the open-loop gain
Large R1 and R2 are required.
Vout, cm=(Vout2-Vout1)/(R1+R2)+Vout1
If R1=R2,
Vout, cm=(Vout1+Vout2)/2
Resistive Sensing (2.1)
Resistance seen into the source terminal of
M7 and M8 are much smaller than R1+R2.
Vout, cm=(Vs8-Vs7)/(R1+R2)+Vs7
If R1=R2,
Vout, cm=(Vs7+Vs8)/2=(Vout2-Vgs8+Vout1Vgs7)/2=(Vout2+Vout1)/2-(Vgs8+Vgs7)/2
Resistive Sensing (2.2)
(If M7 turns off, Vout, CM no longer represents the true output CM level)
1. R1+R2 should still be sufficiently large so the current through R1+R2
Is small compared to I1.
2. Use a large I1.
Resistive Sensing (2.3)
Minimum Vout1, Vout2 without CMFB: VOD3 +VOD5.
Minimum Vout1, Vout2 with CMFB: VGS7,8+VOD,I1.
Since VGS7,8+VOD,I1> VOD3 +VOD5,less output swing is allowed
With CMFB.
Resistive Sensing (3.1)
(Deep triode region)
Vout2+Vout1~ Vout,cm
Vout,cm increases, Rtot drops.
Vout,cm decreases, Rtot increases.
Resistive Sensing (3.2)
(Deep triode region)
Rout depends on Vout,cm
Resistive Sensing (3.3)
(Deep triode region)
Vp is designed to place M7,8 in triode.
Vout, cm is approximately VDD/2.
Vp ≤ Vout,cm-Vth7,8 =VDD/2-Vth7,8
Vout2 is brought down to Vth7,8 during a negative swing.
Is VDS8≤VGS8-Vth7,8?
VDS8 =VDD/2-Vth7,8>0
So VDS8 is not less than VGS8- Vth7,8. There fore M8 is in saturation,
M8 is not longer is the deep triode region.
Sensing (4.1)
VGS,C6=Constant, not sensitive to
differential output signal.
Sensing (4.2)
VGS,C6 increases in response to a rise in
Vout, cm
Return the Error Signal (4)
Return the Error Signal (1)
Return the Error Signal (1)
Return the Error Signal(3.1)
Vout2+Vout1~ Vout,cm
Vout,cm increases, Rtot drops.
Vout,cm decreases, Rtot increases.
Return the Error Signal(3.2)
Return the Error Signal(3.3)
Sensitivity of Vb
(output voltage is sampled,
Voltage subtraction at the input(Vb))
Voltage-Voltage Feedback
Voltage-Voltage Feedback
Sensitivity of Vout,cm due to Vb
(Maximize VDS7,8 to reduce sensitivity of Vb!)
Device Sensitivy M7 and M8
Let (W/L)15=(W/L)9 and (W/L)16=(W/L)7+(W/L)8
ID9=I1 only when Vout, cm=VREF!
Let (W/L)15=(W/L)9 and (W/L)16=(W/L)7+(W/L)8
ID9=I1 only when Vout, cm=VREF!
Simplified Design
VDS15=VDS9 Design
One-Stage Amplifier
Simple One-Stage Op Amps
(No mirror pole)
Unity Gain Amplifier
Open loop output impedance:
Loop gain:
Closed loop output impedance:
Telescopic Op Amps
Design Criteria
• Desirables:
– IOUT should be IREF. (i.e. VX=VY)
– Vout should be minimized. (i.e. VOD2+VOD3)
Increased Vout, max
(Vout, max)
(Increased Vout, max)
Drawback of Telescopic
(Condition for keeping M2 and M4 in Saturation)
Folded Cascode Circuits
Differential Folded Cascode
(Extra power compared to telescopic)
Folded Cascode with cascode
Gain Calculation
Less gain
Non-dominant Pole Comparison | {"url":"https://paperzz.com/doc/7285578/cmfb--one-stage-op-amp","timestamp":"2024-11-03T22:09:04Z","content_type":"text/html","content_length":"23193","record_id":"<urn:uuid:53e69c68-818b-4150-be3c-2f570687b704>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00060.warc.gz"} |
What are outcome variables in research?
What are outcome variables in research?
On the other hand, dependent variables, also called outcome and response variables, are the factors that are expected to change during an experiment. The outcome variable is the focus of any study,
including clinical trials.
What are the variables in a research study?
A variable in research simply refers to a person, place, thing, or phenomenon that you are trying to measure in some way. The best way to understand the difference between a dependent and independent
variable is that the meaning of each is implied by what the words tell us about the variable you are using.
How do you find a independent variable?
An easy way to think of independent and dependent variables is, when you’re conducting an experiment, the independent variable is what you change, and the dependent variable is what changes because
of that. You can also think of the independent variable as the cause and the dependent variable as the effect.
What is variable in quantitative research?
Ordinal, interval, and ratio variables are quantitative. QUANTITATIVE variables are sometimes called CONTINUOUS VARIABLES because they have a variety (continuum) of characteristics. Height in inches
and scores on a test would be examples of quantitative variables. | {"url":"https://www.studiodessuantbone.com/paper-writing-help-blog/what-are-outcome-variables-in-research/","timestamp":"2024-11-11T14:45:52Z","content_type":"text/html","content_length":"126731","record_id":"<urn:uuid:ea489dae-8cc4-4cb9-b424-8abc79e1bd3f>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00088.warc.gz"} |
Damon's maths and numeracy blog
The Magic Square
The magic square can be used to generate great thinking. The trick again, is not to try to get it done as quickly as possible, but to use it to generate thought.
Here is how
You have engaged in analysing how many ways certain numbers can be produced using two dice. This activity builds on this.
The magic square has 9 squares. The numbers 1-9 need to be entered so that they equal 15, horizontally, vertically and diagonally.
Before launching in and beginning to guess, have a go at this instead.
Task one
First think about how many equations each individual
will be involved in. For example, the top right square will be involved in 3 equations.
How about the middle right square? It'll be involved in only 2 equations.
Now work out and put in order which will be involved in how many equations.
Task two
Now work out and order the quantity of 3 number equations that equal 15, that each number is included in. For example:
The number 1 is only able to be in two equations that equal 15. 1+8+6=15 and 1+5+9=15.
Therefore, the number 1 can only go into a square that is involved in two equations. You also know the numbers (8, 6, 5, 9) that must be in the equations with the 1. Read this again if you're not
getting it.
This is how mathematicians solve problems. They deduce their answers - like Batman. They do not go random. Try not to succumb to the temptation to guess. Use those deductive reasoning skills.
You can do it!
Next challenge
Once you have nailed the three by three square use your new powers of deduction to solve the four-by-four square. This one is a beast, and well worth taking a deductive approach.
Enter the numbers 1-16 so that horizontally, vertically and diagonally each row and column, and diagonal equals 34.
Good luck.
The joy is in the struggle. | {"url":"https://damonmath.blogspot.com/2016/03/the-magic-square-magic-square-can-be.html","timestamp":"2024-11-04T15:12:01Z","content_type":"text/html","content_length":"65210","record_id":"<urn:uuid:d2ff16db-12e8-4337-a832-1348265f42b6>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00228.warc.gz"} |