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ABAP CDS Release News 2308 – CDS Scalar Functions Implemented by AMDP
2023 Aug 25 9:49 AM
The predefined CDS functions do not cover the scope of your business scenario? With the introduction of SQL-based CDS scalar functions, partners and customers are empowered to define their own CDS
scalar functions with a SQL implementation via AMDP and to use them in CDS view entities. Complex calculations may now be outsourced from single CDS entities and still be pushed down to the HANA
Release Info
□ SAP BTP, ABAP Environment 2308
□ ABAP Release 7.58
□ SAP S/4HANA 2023
An SQL-based scalar function is a user-defined function that accepts multiple input parameters and returns exactly one scalar value. A scalar function allows developers to encapsulate complex
algorithms into manageable, reusable code that can then be used in all operand positions of CDS view entities that expect scalar values. A scalar function is linked with an AMDP function in which it
is implemented using SQLScript.
SQL-based scalar functions make AMDP scalar functions defined in AMDP known to ABAP Dictionary and available in ABAP CDS.
The following figure shows the design time of a CDS scalar function:Design time of an SQL-based scalar function
A SQL-based scalar function is defined in a scalar function definition using the keyword DEFINE SCALAR FUNCTION. The scalar function implementation reference binds the function to a runtime and to an
existing AMDP function. When used in the SELECT list of a CDS view entity, the CDS framework executes the scalar function by calling the associated function on the database.
Creating a CDS SQL-based scalar function
In order to create a CDS SQL-based scalar function, you need the following three objects:
• A CDS scalar function definition defined using DEFINE SCALAR FUNCTION.
• A CDS scalar function implementation reference that binds the scalar function to a runtime engine and to an AMDP function implementation.
• An AMDP method that implements the CDS scalar function as database function in SQLScript.
Here’s an example:
CDS scalar function definition:
define scalar function DEMO_CDS_SCALAR_RATIO
with parameters
portion: numeric
total : type of portion
returns abap.decfloat34
A CDS scalar function has input parameters defined after WITH PARAMETERS and it returns a scalar result with the data type defined after RETURNS.
Scalar function implementation reference
The scalar function implementation reference is defined in a form-based tool in the ABAP Development Tools. It binds a CDS scalar function definition to a runtime and to an AMDP method that
implements the function.AMDP function implementation
After activating the CDS scalar function, you can go on implement the functional AMDP method in an AMDP class, that is a class with the marker interface IF_AMDP_MARKER_HDB. An AMDP method for a CDS
scalar function must be a static functional method of a static AMDP class that is declared as follows:
CLASS-METHODS execute
FOR SCALAR FUNCTION demo_scalar_function.
The declaration is linked directly to the CDS scalar function. The parameter interface is implicitly derived from the scalar function’s definition! Implementation looks like you might expect it:
METHOD execute BY DATABASE FUNCTION
FOR HDB
LANGUAGE SQLSCRIPT
OPTIONS READ-ONLY.
result = portion / total * 100;
The implementation is done in native SQLScript for a HANA database function.
Note that client handling has not yet been implemented and therefore, only client-independent objects can be used in the implementation.
Use in a CDS view entity
A SQL-based scalar function can be used in CDS view entities in operand positions that expect scalar values, similar to built-in functions. Here’s an example for a scalar function used in the SELECT
list of a CDS view entity:
define view entity DEMO_CDS_SCALAR_USE_RATIO
as select from sflight
key carrid as Carrid,
key connid as Connid,
key fldate as Fldate,
seatsocc as BookedSeats,
seatsmax as TotalSeats,
portion => seatsocc,
total => seatsmax ) as OccupationRatio
Reference handling
SQL-based scalar functions support the handling of CDS amount fields and CDS quantity fields. CDS amount fields and CDS quantity fields are fields with a reference to a currency key, a unit key, or a
calculated unit. Scalar functions can handle these references. You can define which reference types are allowed for each input parameter and for the return parameter. If the actual parameters passed
to the input parameters use reference types that are not explicitly allowed, a syntax check error occurs.
The following reference types are available:
A parameter can also be typed with reference to another parameter. This means that it inherits the reference type of the referenced parameter. The syntax is WITH REFERENCE TYPE OF.
The reference type can also be defined dynamically, depending on the reference types of the input parameters. This is done using CASE statements.
Here’s an example
define scalar function DEMO_CDS_SCALAR_REF_CASE
with parameters
p1: numeric
with reference type [ #CUKY, #UNIT, #CALC, #NONE ],
p2: numeric
with reference type [ #CUKY, #UNIT, #CALC, #NONE ],
p3: abap.dec(4,2)
with reference type [ #CUKY, #UNIT, #CALC, #NONE ]
returns abap.dec(4,2)
with reference type
when p2: reference type of p1
then #NONE
else reference type of p1
Analytical scalar functions
A CDS scalar function can also be bound to an analytical engine. In this case, it can be used in CDS analytical queries and it is evaluated by the ABAP Analytical Engine.
Analytical scalar functions are defined and implemented by SAP. They are provided to customers and partners as CDS system functions.
For a complete list of SAP-delivered analytical CDS scalar functions, see the ABAP Keyword Documentation, topic ABAP CDS - Analytical Scalar Functions.
The following example demonstrates how to use an analytical scalar function in an analytical projection view. The analytical scalar function RATIO_OF has two mandatory input parameters: portion and
total. It calculates the ratio of portion in relation to total. The actual parameters are passed using an arrow =>. You see that CDS expressions and functions can be passed as actual parameters.
@EndUserText.label: 'Analytical scalar function'
@AccessControl.authorizationCheck: #NOT_ALLOWED
define transient view entity DEMO_CDS_USE_ANA_SCALAR
provider contract analytical_query
as projection on DEMO_CDS_CUBE_VIEW
@Aggregation.default: #FORMULA
portion => ( get_numeric_value( amount_sum )
- get_numeric_value( amount_sum )
* $projection.Discount ) * $projection.Tax,
total => get_numeric_value( amount_sum ) )
as AmountRatioFinalToOriginal
For more information see | {"url":"https://community.sap.com/t5/technology-blogs-by-sap/abap-cds-release-news-2308-cds-scalar-functions-implemented-by-amdp/ba-p/13561929","timestamp":"2024-11-07T01:14:00Z","content_type":"text/html","content_length":"171780","record_id":"<urn:uuid:fcf988d6-f9bd-45f0-8832-3745dcdcf0b4>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00518.warc.gz"} |
IB Math Tutor HK | IB Tutoring from IB Veterans
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IB Math Information That Might Help You
As an IB graduate and IB Maths tutor for the past 5 years, I know firsthand how overwhelming it can be for …
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I. Introduction The International Baccalaureate (IB) program has undergone a significant transformation in its mathematics curriculum in 2021, expanding from three courses …
Easy Sevens IB Math Tutoring: Helping Students Improve Their Grades
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IB Mathematics Syllabus
Maths AA SL Assessments
• Paper 1: Short and extended answer questions, no calculator allowed, 90 minutes duration, 40% weighting, 80 marks.
• Paper 2: Short and extended answer questions, calculator allowed, 90 minutes duration, 40% weighting, 80 marks.
• Internal Assessment: Individual investigation, 20% weighting, 20 marks.
Maths AA HL Assessments
• Paper 1: Short and extended answer questions, no calculator allowed, 120 minutes duration, 30% weighting, 110 marks.
• Paper 2: Short and extended answer questions, calculator allowed, 120 minutes duration, 30% weighting, 110 marks.
• Paper 3: Extended answer questions on mostly HL topics, 60 minutes duration, calculator allowed, 20% weighting, 55 marks.
• Internal Assessment: Individual investigation, 20% weighting, 20 marks.
Maths AI SL Assessments
• Paper 1: Short answer questions, calculator allowed, 90 minutes duration, 40% weighting, 80 marks.
• Paper 2: Extended response questions, calculator allowed, 90 minutes duration, 40% weighting, 80 marks.
• Internal Assessment: Individual investigation, 20% weighting, 20 marks.
Maths AI HL Assessments
• Paper 1: Short answer questions, calculator allowed, 120 minutes duration, 30% weighting, 110 marks.
• Paper 2: Extended response questions, calculator allowed, 120 minutes duration, 30% weighting, 110 marks.
• Paper 3: Extended answer questions on mostly HL topics, 60 minutes duration, calculator allowed, 20% weighting, 55 marks.
• Internal Assessment: Individual investigation, 20% weighting, 20 marks.
Standard Level IB Maths Topics Overview
1. Linear, Quadratic, Reciprocal, Rational, Polynomial, Radical, Exponential, Logarithmic, and Composite Functions
2. Unit Circle, Radian Measure of Angles and Trigonometric Equations
3. Sin and Cosine Rule
4. Pythagorean Identities and Double Angle Identities
5. Trigonometry Functions
6. Discrete and Continuous Data Summary
7. Probability and Conditional Probability of Events
8. Binomial and Normal Distribution
9. Limits, Derivatives, and Integrals
10. Tangents and Normals at a Given Point
11. Kinematics Problem Solving
12. Definite Integrals and Area Under a Curve
Additional High Level IB Maths Topics Overview
1. Counting Principles, Permutations and Combinations
2. Partial Fractions
3. Complex Numbers and Complex Plane, De Moivre’s
4. Proof by Mathematical Induction
5. Quadratic with Complex Roots, Sum and Product of Roots
6. Odd and Even Functions
7. Absolute Value Functions
8. Additional Trigonometry Identities
9. Inverse Trigonometry Functions
10. Compound Angle Identity
11. Vectors and Vector Dot and Cross Products
12. Vector Equations of Line and Plane
13. Intersection of Planes
14. Bayes Theorem
15. Implicit Differentiation
16. Integration by Parts
17. Differential Equations
18. Maclaurin Series | {"url":"https://www.easysevens.com/ib-tutoring/ib-math-tutor/","timestamp":"2024-11-02T17:20:58Z","content_type":"text/html","content_length":"241536","record_id":"<urn:uuid:8c193bdb-c062-4e15-8418-6fda81ff527f>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00338.warc.gz"} |
From A Random World to a Rational Universe
Randomness, Luck, Astragali and Dice
In the time before the mathematical idea of randomness was discovered, people thought that everything that happened was part of the will of supernatural beings, the gods, who looked down upon human
affairs and decided to 'tip the balance' one way or another to influence events. Hence, sacrifices were made and rituals performed to discover the 'will of the gods' or to try to influence human
affairs. This idea still prevails, and many people all over the world use lucky charms, engage in superstitious practices, use horoscopes, and still have some kind of belief that there are such ways
of influencing their lives. The gods may be dead, but 'Lady Luck' still survives.
The astragalus is a small bone, about an inch cube, found in the heel of hoofed mammals. Astragali have six sides but are not symmetrical, so there is no way of knowing which way they will eventually
come to rest. For many ancient civilizations, astragali were used by priests to discover the opinions of their gods. It was customary in divination rites to roll, or cast, five astragali. Typically,
each possible configuration was associated with the name of a god and carried with it the sought-after advice.
Astragali from the heel of a sheep showing the four positions of rest.
The small one in the foreground is made from pottery
Showing the four positions of rest. The small one in the foreground is made from pottery. Astragali found in excavations typically have their sides numbered or engraved. They were also used in board
games in the First Dynasty in Egypt, c 3500 BCE; archaeological evidence consists of boards, counters, and astragali for various games, including one similar to Snakes and Ladders, still popular
The game of Hounds and Jackals dating from 1800 BCE found in an Egyptian tomb
The astragali have been used from classical times for gambling, and similar stones are still in use today for games like 'fivestones' or 'jacks'.
Gradually, over thousands of years, astragali were replace by dice [see note 1 below], and pottery dice have been found in Egyptian tombs. The earliest die known was made from pottery and excavated
in Northern Iraq dating from about 3,000 BCE. It has dots arranged as in (Die A).
Die (B), from about 1400 BCE found in a tomb in Egypt, shows consecutive numbers opposite each other.
Dice with other markings like the names or portraits of gods have been found, probably used for special games or rituals, and others where some numbers are repeated, or 'loaded', for special purposes
or possibly for cheating (Die C).
Once the Greeks had worked out the geometry of the polyhedra, dice of other shapes began to be constructed. However, whether cube or polyhedral, the shapes were not entirely regular and were
therefore biased.
Over time, gamblers would get used to using the same dice, and have an intuitive idea of how they would fall, but given another set of dice, the odds would be different. Later, as the manufacture of
dice became more exact, some ideas of the possible combinations of number began to emerge.
The Earth and The Cosmos
There were many other forms of rituals hoping to overcome the randomness of nature and man's condition. A few of these which became of particular mathematical interest are geomancy, the nine square
grid or magic square, and temple designs, the ancestors of board games.
Geomancy means divination of or by the earth , and is a system of 16 mathematically related arrangements of stones, beans or other available small objects used to make decisions, answer questions, or
foretell the future. The stones are cast upon the ground and the pattern formed is interpreted. The symbols represent a series of binary 'opposites' like good and evil, male or female, sadness and
happiness, etc. Combinations of these opposites can be used to represent odd and even numbers.
The sixteen figures of the Geomancy system of Divination. The headings of the columns are: "The greater fortune" and "The lesser fortune". From a Book of Occult Philosophy published 1655. Notice that
each pair of shapes are associated with the traditional signs for the planets and that each configuration could be interpreted from the throws of two dice.
As in all methods of divination, each of these figures has a number of interpretations depending on its relation to other figures shown, and many other circumstances like the time of day, the
weather, and the kind of person who is asking the question.
The Grid of Nine Squares
The Nine square grid is said to come from an ancient system for the division of land, probably from feudal India. In China the nine-square configuration was supposed to be an ideal arrangement, with
eight farmers' fields surrounding a central well. The grid of nine squares, or a circle divided into nine sections by straight lines often appears as a central form in Tibetan sacred diagrams. In
Scotland, the pattern was used at Beltane (the eve of May) where eight squares were cut out from the turf, and a bonfire lit on the central square.
In this way, from practical beginnings in different cultures, the nine-square grid acquired mystic importance and symbolised divine order, and the representation of control by the gods.
Magic Squares are directly related to the Sacred Grid, supposedly being the numerical mystery which underlies their physical form. The simplest magic square is the square of nine, ascribed to Saturn,
where each row and column adds up to 15; the total of the rows and the columns is 45, and the diagonals 30. The 4x4 square with row and column numbers 34 is assigned to Jupiter, the 5x5 with row or
column numbers 65 to Mars, and so on for the Sun, Venus, Mercury and the 9x9 square with row or column numbers 369, to the Moon [an NRICH article on Magic Squares can be found here at nrich.maths.org
As with other devices, these magic squares are all said to have correspondences to different numbers, various deities, days of the week, natural objects, different qualities, and so on. In the Hindu
Temple Yantra [see note 2 below] you can see the nine squares, the 'sacred space', or source of energy, in the centre.
This is a Yantra from a Hindu Temple. Yantras (or Mandalas) are used as a focus for mystical contemplation and often for the basis of design for a temple. This one is based on a 5 x 5 square, with
the 'sacred space' of the 3 x 3 square in the centre.
Board Games are clearly linked with divination, astrology and sacred geometry, and the designs of the boards can show their sacred or occult origins. The popular game of 'snakes and ladders' is
controlled by the throw of dice, and the ladders and snakes originally referring to good and bad fortune, now refer to good and bad 'luck' in the progress of the game. In some cases the designs of
the boards are the same as the plans of temples and holy cities with a 'sacred space' in the centre.
This is the board for the ancient Korean game of 'Yut' or 'Nyout' The board can be made of cloth or paper, or can be drawn on the floor. It is played with four 'Yut Sticks' of semi-circular section,
and the way they fall determines the move of a token. The shape can be square or circular and represents the division of the world into twenty outer regions and nine central spaces.
The game of 'Nine Men's Morris' is played with counters on the dots on this board. The design is said to represent the four elements, (earth, air, fire and water) the four winds, or the four cardinal
points of the compass, and the central sacred area was a symbol of rebirth or renewal.
The game was supposed to have originated in Egypt, and was known to the Romans. This is a picture from the 13th century of the game being played in England.
This is a traditional diagram used for the Horoscope of Robert Burton from his tomb in Christchurch in Oxford. This clearly has a link with the diagrams from sacred architecture and board games.
Mathematics and Magic
In ancient times, few people could understand even the simplest arithmetic and geometry, and the confusion of mathematics with magic has a long history.
People who had knowledge of the regular movements of the heavens were able to predict the position of planets, and the particular the times when astronomical events appeared in certain sections of
the sky. In ancient civilisations these were highly skilled technicians, called 'priests', and their activities were partly scientific, and partly religious. In Europe, after the arrival of
Christianity, the religious aspect of these practices was condemned as superstition. Because numbers were used in these processes, anyone who used numbers was regarded with considerable suspicion. In
this way genuine mathematicians were looked upon with suspicion by the ignorant, and the titles of Astrologer, Mathematician and Conjurer were virtually synonymous.
An early Bishop of the Church, St. Augustine of Hippo (354-430 CE) once said:
"The good Christian should beware of mathematicians and all those who make empty prophecies. The danger already exists that mathematicians have made a covenant with the devil to darken the spirit and
confine man in the bonds of Hell."
Augustine was arguing that belief in astrology denies the freedom of the will.
Roger Bacon (1214 - 1292), often called England's first Scientist, had a reputation as a 'great necromancer' because of his ingenious experiments and John Dee (1527 - 1609) probably one of the
foremost mathematicians in Europe of his time, gained a reputation as a 'Conjuror' while he was at Oxford because he was respnsible for developing a simple mechanical device by which an actor
appeared to fly, and people claimed he was in league with the devil. [see note 3 below]
John Dee
Robert Recorde
During the sixteenth century in England, mathematicians like Robert Recorde (1510-1558) and Thomas Digges (1546-1595) published many works showing the everyday practical usefulness of mathematical
knowledge for ordinary people clearly showing that mathematics was not an occult practice
Following the foundation of the Oxford chairs in mathematics and astronomy in 1619, some parents kept their sons away from the university in fear of them becoming contaminated by the 'Black Art'.
As the predictive power of astronomy and other practical uses of mathematics became apparent, mathematicians were able to dispel the idea that many events were not controlled by the goddess Fortuna,
but could be explained in a rational way.
This is the title page of Robert Recorde's The Castle of Knowledge published in 1556. It is his fourth book for the self-education of craftsmen and artisans and shows how to make the instruments for
astronomy and navigation. The blind goddess Fortuna stands on the unstable sphere holding the wheel of chance, while the Spirit of Knowledge stands on a stable cube holding the navigator's dividers
and the sphere of destiny.
The Beginnings of Probability
Since dice were used in gambling, in religious ceremonies and for divination, it is believed that those who used the dice had a good intuitive idea of the likely frequency of various number
combinations. The first printed document showing the possibilities with three dice was the Latin poem De Vetula , which shows all the combinations for the fall of three dice, and is believed to have
been written in the early 13th century. The idea of using binomial coefficients to calculate the possibilities appears in the poem, but is not taken up until much later [see note 4 below].
Part of the poem De Vetula written in the 13th century, shows the different combinations of three dice. Notice the written numerals on the left of the text.The written numerals for the number
combinations appear to the left of the poem.
(1501- 1576)
Girolamo Cardano
Since the Christian Church was against gaming, and there was much superstition about divination, it is not surprising that a theory of probability did not begin to appear until the 16th century.
Cardano, writing with considerable personal knowledge of gambling, recognised that if the die was honest, each face would have an equal chance of appearing. His manuscript, Liber De Ludo Aleae , was
written about 1526 but only found after his death, and not published until 1663. He gave tables of the results for one, two and three dice, but these are not all correct. However, Cardano is credited
with recognising that the abstraction of the 'honest die' is the key to a theory of probability based on mathematical principles.
Nicolo Tartaglia (1500-1557)Tartaglia (1500 - 1557) and others discuss various versions of the division of the stakes when a gambling game is stopped, called the 'problem of points', and this shows
that Cardano's ideas were likely to be common knowledge among scholars of the later 16th and early 17th century.
Galileo (1564 - 1642) wrote on probability but his work was not published until1718. He stated that with three dice there can only be one way of obtaining a 3 (1,1,1) and an 18 (6,6,6) but there are
three combinations for obtaining a 6 (4,2,1), (3,2,1) and (2,2,2) which can occur in different orders making 10 possibilites and four combinations for a 7 (5,1,1), (4,2,1), (3,3,1) and (3,2,2) which
lead to fifteen possibilities. However, although 9 and 12 could be made up in the same number of ways as 10 and 11, from their experience, gamblers claimed that the occurrence of 10 and 11 were more
likely! Galileo showed that the total number of possible throws with three dice are 216, and he gave a table of the number of possible throws for a total of 10, 9, 8, 7, 6, 5, 4 and 3, showing that
the throws for 11 to 18 were symmetrical with these. In this way he showed that there were 27 possible throws to obtain a 10, and 25 for a 9.
This is a copy of the table Galileo published in his Sopra le Scoperte de I Dadi (Concerning an Investigation on Dice). Here he shows clearly how to count the different combinations of the various
His work showed that by this time there was no doubt about the general method for calculating chances with a die, and it was clear that the mathematical concepts of the equal probability of the throw
of a die, and the procedures to analyse the results were well known.
Pascal's Triangle
By the mid 16th century the theory of probability became established on a rigorous basis with the work of Pascal and Fermat. However, as we have seen, the idea of the application of 'Pascal's
Triangle' had been suggested as early as the 13th century but forgotten for some 200 years. The triangle itself was known and published before, by Stifel (Arithmetica Integra 1543) Tartaglia
(Trattato 1556) Stevin (Arithmetic 1625) Pierre Herigone (Cours Mathematique 1634), and we also know it was known to the Chinese and the Arabs by the mid 13th century, but Pascal was the first to
apply it to probability.
For pedagogical notes:
Use the notes tab at the top of this article or click here .
1. The word 'die' (plural 'dice') come from the Latin verb 'dare' (pron. da-ray) to give its participle dadus means 'given by the gods'.
2. The word Yantra is a Sanscrit word meaning a mystical diagram or picture. They contain geometric items and archetypal shapes and patterns of squares, triangles, circles and other floral patterns.
In contrast, a Mantra is a spoken verse or poem.
3. At this time, 'Mathematics' included the applied mathematics of physics, statics, mechanics, hydraulics, and other practical arts. This is clear from John Dee's famous 'Preface' to the 1570
English Edition of Euclid.
4. It is possible that the scholar who wrote this poem might have been aware of the 'number triangle' of the Arabs.
General Background and History
David, F. N. Games, Gods and Gambling . New York. Dover Books
This is the well-known classic book on the subject still full of interesting and reliable information.
De Moivre, A. (1967) The Doctrine of Chances or A method of Calculating the Probabilities of Events in Play . London. Frank Cass.
A facsimilie of the 1738 second edition of Abraham de Moivre'??s classic where he makes corrections, expands the explanations, and gives more details of the solutions of the problems than in the
first edition of 1718.
Ore, O. (1953) Cardano, The Gambling Scholar .
A good story and biography of Girolamo Cardano by the Danish Mathematician Oystein Ore. Contains a translation of Cardan'??s Liber de Ludo Aleae (Book on Games of Chance).
Hacking, I. (1975) The Emergence of Probability . C.U.P. Cambridge UK
Hacking's book contains many references to original works in the period up to and including the 18th century.
Hald, A. (2003) A History of Probability and Statistics and their Applications Before 1750 . New York. Wiley
Pennick, N. (1988) Games of the Gods . London. Hutchinson
Religious Beliefs, Superstition, Astrology and Divination in may cultures and their connections to many games of dice and on the board
Some Books for the Classroom
Jenkins, G.W. & Slack J.L. (1979) Classroom Experiments with Dice . St Albans. Tarquin Publications
Woods, G. Symmetry Dice . St Albans. Tarquin Publications
Benson, S. (2005) Ways to think about Mathematics:Activities and Investigations for Grade 6 . California. Corwin Press. This is a useful book with many examples of activities. There are sections
about probability and binomial coefficients.
Here is a shop for all kinds of dice : large; small; all colours; with numbers; with spots; blank; arithmetic symbols; money symbols; polyhedral; round (yes round!); loaded; and for cheating! http://
This interesting Board Game
site has many traditional games like Nine Men's Morris, various types of strategy games like Solitaire, Fox and Geese; Mancala, Ludo; and Snakes and Ladders. All of them have well-researched
historical notes. | {"url":"https://nrich.maths.org/articles/random-world-rational-universe","timestamp":"2024-11-02T13:57:33Z","content_type":"text/html","content_length":"70529","record_id":"<urn:uuid:8dfa43a9-76fe-4789-811e-6aa89ffe0b4a>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00075.warc.gz"} |
Advent of Code 2020 Solution Megathread - Day 20: Jurassic Jigsaw
Five more days! Five more days! Five more days!
The Puzzle
In today’s puzzle, we're looking for Nessie! Well, we're rotating, flipping, and reassembling tiles of pixels on the off-chance that we actually see something in what can only be described as a
complexity nightmare. I'm not really sure how this one is going to get done without ridiculous amounts of looping, but that's why we get paid the big bucks!
The Leaderboards
As always, this is the spot where I’ll plug any leaderboard codes shared from the community.
Ryan's Leaderboard: 224198-25048a19
If you want to generate your own leaderboard and signal boost it a little bit, send it to me either in a DEV message or in a comment on one of these posts and I'll add it to the list above.
Yesterday’s Languages
Updated 07:33AM 12/20/2020 PST.
Language Count
Haskell 2
JavaScript 1
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Merry Coding!
Top comments (4)
Yuan Gao • • Edited on • Edited
Tough one today, I explain my solution more over at my blog post
I gave up trying to come up with a smart way to correctly orient the tiles. It's doable, you place one tile down, and then for each subsequent tile place down, you select one that is connected to one
of the ones placed down (graph traversal), you just need to find the correct offset location, rotation, and flip given known edge match, but also take into consideration the absolute global rotation/
flip of the tile that was placed into the grid. It should be a short function to take in the known edge match and orientation of the previous tile, it's similar to how you can chain affine
transformation matrices, I just can't wrap my head around it right now.
Instead, I just brute-force the 32 possible positions each subsequent tile could be in relative to its pair which we just placed on the board (4 positions, each position 4 rotations 2 flip; still O
(n) with number of tiles though, so still computes pretty much instantly). The rest is cross-correlation, as provided by scipy
import numpy as np
import networkx as nx
from itertools import product
from math import prod, sqrt
import re
from scipy.ndimage import correlate
data = open("input.txt").read().splitlines()
tiles = {int(re.sub(r"Tile (\d+):", r"\1", k)): np.array(v).view('U1').reshape(10, 10) for k, *v, _ in np.array(data).reshape(-1, 12)}
edges = (
(..., -1 ),
(0 , ...),
(..., 0 ),
(-1 , ...),
borders = {}
graph = nx.Graph()
for tile_name, tile in tiles.items():
for (i, j), f in product(edges, (1, -1)):
border_id = "".join(tile[i, j][::f])
if border_id in borders:
graph.add_edge(borders[border_id], tile_name)
borders[border_id] = tile_name
def orient(tile):
for i in range(4):
yield np.rot90(tile, i)
yield np.fliplr(np.rot90(tile, i))
flip = {
...: ...,
-1 : 0 ,
0 : -1 ,
dire = {
...: 0 ,
-1 : 1 ,
0 : -1 ,
dim = int(sqrt(len(tiles)))
first_tile = next(iter(tiles))
result = np.full([2*dim+1, 2*dim+1, 10, 10], "")
result[dim][dim] = tiles[first_tile]
location = {first_tile: np.array([dim, dim])}
for left, right in nx.dfs_edges(graph, first_tile):
if left in location:
old, new = left, right
old, new = right, left
old_loc = location[old]
old_tile = result[old_loc[0], old_loc[1]]
for i, j in edges:
for oriented_tile in orient(tiles[new]):
if str(old_tile[i, j]) == str(oriented_tile[flip[i], flip[j]]):
new_loc = old_loc + np.array([dire[i], dire[j]])
result[new_loc[0], new_loc[1]] = oriented_tile
location[new] = new_loc
flat = np.swapaxes(result[:,:,1:-1,1:-1], 1, 2).reshape((2*dim+1)*8, -1)
flat_filtered = flat[~(flat == '')].reshape(dim*8, -1)
image = (flat_filtered == "#").astype(int)
monster = (np.array(open("monster.txt").read().splitlines()).view('U1') == "#").reshape((3, -1)).astype(int)
monster_value = monster.sum()
total_monsters = sum((correlate(image, oriented_monster, mode="constant") == monster_value).sum() for oriented_monster in orient(monster))
print("remaining", image.sum() - total_monsters*monster_value)
Neil Gall •
I got part 1 done but it took me ages to get the insight that it only needs the edge tiles arranged. Even then it takes 39 seconds to run which is too long and will never work for part 2. I must be
missing some fundamental insight that makes the search space smaller.
Neil Gall •
I found the algorithmic problem and it now arranges all the tiles in about one second. The code is simpler too!
Thibaut Patel •
huh today was hard! Here is my video walkthrough as usual, although it's only for part 1 ;)
For further actions, you may consider blocking this person and/or reporting abuse | {"url":"https://dev.to/rpalo/advent-of-code-2020-solution-megathread-day-20-jurassic-jigsaw-ck5","timestamp":"2024-11-04T17:47:46Z","content_type":"text/html","content_length":"166397","record_id":"<urn:uuid:05f17ed4-0cbf-4963-8fca-0eabe8a32811>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00791.warc.gz"} |
A quantitative Gibbard-Satterthwaite theorem without neutrality
Recently, quantitative versions of the Gibbard-Satterthwaite theorem were proven for k=3 alternatives by Friedgut, Kalai, Keller and Nisan and for neutral functions on k ≥ 4 alternatives by Isaksson,
Kindler and Mossel. In the present paper we prove a quantitative version of the Gibbard-Satterthwaite theorem for general social choice functions for any number k ≥ 3 of alternatives. In particular
we show that for a social choice function f on k ≥ 3 alternatives and n voters, which is ε-far from the family of nonmanipulable functions, a uniformly chosen voter profile is manipulable with
probability at least inverse polynomial in n, k, and ε ^-1. Removing the neutrality assumption of previous theorems is important for multiple reasons. For one, it is known that there is a conflict
between anonymity and neutrality, and since most common voting rules are anonymous, they cannot always be neutral. Second, virtual elections are used in many applications in artificial intelligence,
where there are often restrictions on the outcome of the election, and so neutrality is not a natural assumption in these situations. Ours is a unified proof which in particular covers all previous
cases established before. The proof crucially uses reverse hypercontractivity in addition to several ideas from the two previous proofs. Much of the work is devoted to understanding functions of a
single voter, and in particular we also prove a quantitative Gibbard-Satterthwaite theorem for one voter.
Original language English (US)
Title of host publication STOC '12 - Proceedings of the 2012 ACM Symposium on Theory of Computing
Pages 1041-1060
Number of pages 20
State Published - 2012
Event 44th Annual ACM Symposium on Theory of Computing, STOC '12 - New York, NY, United States
Duration: May 19 2012 → May 22 2012
Publication series
Name Proceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print) 0737-8017
Other 44th Annual ACM Symposium on Theory of Computing, STOC '12
Country/Territory United States
City New York, NY
Period 5/19/12 → 5/22/12
All Science Journal Classification (ASJC) codes
• computational social choice
• gibbard-satterthwaite
• isoperimetric inequalities
• manipulation
• reverse hypercontractivity
• voting
Dive into the research topics of 'A quantitative Gibbard-Satterthwaite theorem without neutrality'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/a-quantitative-gibbard-satterthwaite-theorem-without-neutrality","timestamp":"2024-11-12T23:03:51Z","content_type":"text/html","content_length":"54117","record_id":"<urn:uuid:e8ae3f44-9390-48cd-b946-8295570c00a6>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00816.warc.gz"} |
BTU to Quads Conversion - Inch Calculator
British Thermal Units to Quads Converter
Enter the energy in british thermal units below to convert it to quads.
Do you want to convert quads to british thermal units?
How to Convert British Thermal Units to Quads
To convert a measurement in british thermal units to a measurement in quads, divide the energy by the following conversion ratio: 1.0E+15 british thermal units/quad.
Since one quad is equal to 1.0E+15 british thermal units, you can use this simple formula to convert:
quads = british thermal units ÷ 1.0E+15
The energy in quads is equal to the energy in british thermal units divided by 1.0E+15.
For example,
here's how to convert 5.0E+14 british thermal units to quads using the formula above.
quads = (5.0E+14 BTU ÷ 1.0E+15) = 0.5 quads
British thermal units and quads are both units used to measure energy. Keep reading to learn more about each unit of measure.
What Is a British Thermal Unit?
British thermal units are a measure of heat energy. Specifically, one british thermal unit is equal to the amount of heat energy required to increase the temperature of one pound of water one degree
Fahrenheit. They are often used as a way to rate the energy heat producing appliances such furnaces, and to price heating fuels such as natural gas.
The british thermal unit is a US customary unit of energy. British thermal units can be abbreviated as BTU; for example, 1 british thermal unit can be written as 1 BTU.
Learn more about british thermal units.
What Is a Quad?
One therm is a measure of heat energy equal to one quadrillion (1,000,000,000,000,000) BTU, which are equal to the amount of heat energy required to increase the temperature of one pound of water one
degree Fahrenheit. Quads are usually used when talking about energy resources and consumption on a national or international scale.
The quad is a US customary unit of energy.
Learn more about quads.
British Thermal Unit to Quad Conversion Table
Table showing various british thermal unit
measurements converted to quads.
British Thermal Units Quads
1 BTU 0.000000000000001
2 BTU 0.000000000000002
3 BTU 0.000000000000003
4 BTU 0.000000000000004
5 BTU 0.000000000000005
6 BTU 0.000000000000006
7 BTU 0.000000000000007
8 BTU 0.000000000000008
9 BTU 0.000000000000009
10 BTU 0.00000000000001
100 BTU 0.0000000000001
1,000 BTU 0.000000000001
10,000 BTU 0.00000000001
100,000 BTU 0.0000000001
1,000,000 BTU 0.000000001
10,000,000 BTU 0.00000001
100,000,000 BTU 0.0000001
1,000,000,000 BTU 0.000001
10,000,000,000 BTU 0.00001
100,000,000,000 BTU 0.0001
1,000,000,000,000 BTU 0.001
10,000,000,000,000 BTU 0.01
100,000,000,000,000 BTU 0.1
1,000,000,000,000,000 BTU 1
More British Thermal Unit & Quad Conversions | {"url":"https://www.inchcalculator.com/convert/british-thermal-unit-to-quad/","timestamp":"2024-11-07T20:13:21Z","content_type":"text/html","content_length":"72234","record_id":"<urn:uuid:9c42b394-0ad5-42e2-923a-ade69be01987>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00154.warc.gz"} |
Stripmap Synthetic Aperture Radar (SAR) Image Formation
This example shows how to model a stripmap-based Synthetic Aperture Radar (SAR) system using a Linear FM (LFM) Waveform. SAR is a type of side-looking airborne radar where the achievable cross-range
resolution is much higher as compared to a real aperture radar. The image generated using SAR has its own advantages primarily pertaining to the use of an active sensor (radar) as opposed to
conventional imaging systems employing a passive sensor (camera) that rely on the ambient lighting to obtain the image. Since an active sensor is used, the system provides all-weather performance
irrespective of snow, fog, or rain. Also, configuring the system to work at different frequencies such as L-, S-, or C-band can help analyze different layers on the ground based on varying depth of
penetrations. Because the resolution of SAR depends upon the signal and antenna configuration, resolution can be much higher than for vision-based imaging systems. Using the stripmap mode, this
example performs both a range migration algorithm [1] and an approximate form of a back-projection algorithm [2] to image stationary targets. The approximate form of the back-projection algorithm has
been chosen for the reduced computational complexity as indicated in [3]. A Linear FM waveform offers the advantage of large time-bandwidth product at a considerably lower transmit power making it
suitable for use in airborne systems.
Synthetic Aperture Radar Imaging
SAR generates a two-dimensional (2-D) image. The direction of flight is referred to as the cross-range or azimuth direction. The direction of the antenna boresight (broadside) is orthogonal to the
flight path and is referred to as the cross-track or range direction. These two directions provide the basis for the dimensions required to generate an image obtained from the area within the antenna
beamwidth through the duration of the data collection window. The cross-track direction is the direction in which pulses are transmitted. This direction provides the slant range to the targets along
the flight path. The energy received after reflection off the targets for each pulse must then be processed (for range measurement and resolution). The cross-range or azimuth direction is the
direction of the flight path and it is meaningful to process the ensemble of the pulses received over the entire flight path in this direction to achieve the required measurement and resolution.
Correct focusing in both the directions implies a successful generation of image in the range and cross-range directions. It is a requirement for the antenna beamwidth to be wide enough so that the
target is illuminated for a long duration by the beam as the platform moves along its trajectory. This helps provide more phase information. The key terms frequently encountered when working with SAR
1. Cross-range (azimuth): This parameter defines the range along the flight path of the radar platform.
2. Range: This parameter defines the range orthogonal to the flight path of the radar platform.
3. Fast-time: This parameter defines the time duration for the operation of each pulse.
4. Slow-time: This parameter defines the cross-range time information. The slow time typically defines the time instances at which the pulses are transmitted along the flight path.
Radar Configuration
Consider a SAR radar operating in C-band with a 4 GHz carrier frequency and a signal bandwidth of 50 MHz. This bandwidth yields a range resolution of 3 meters. The radar system collects data
orthogonal to the direction of motion of the platform as shown in this figure. The received signal is a delayed replica of the transmitted signal. The delay corresponds in general to the slant range
between the target and the platform. For a SAR system, the slant range varies over time as the platform traverses a path orthogonal to the direction of antenna beam. This section focuses on defining
the parameters for the transmission waveform. The LFM sweep bandwidth can be decided based on the desired range resolution.
Set the physical constant for speed of light.
c = physconst('LightSpeed');
Set the SAR center frequency.
Set the desired range and cross-range resolution to 3 meters.
rangeResolution = 3;
crossRangeResolution = 3;
The signal bandwidth is a parameter derived from the desired range resolution.
bw = c/(2*rangeResolution);
In a SAR system the PRF has dual implications. The PRF not only determines the maximum unambiguous range but also serves as the sampling frequency in the cross-range direction. If the PRF is too low
to achieve a higher unambiguous range, there is a longer duration pulses resulting in fewer pulses in a particular region. At the same time, if the PRF is too high, the cross-range sampling is
achieved but at the cost of reduced range. Therefore, the PRF should be less than twice the Doppler frequency and should also satisfy the criteria for maximum unambiguous range
prf = 1000;
aperture = 4;
tpd = 3*10^-6;
fs = 120*10^6;
Configure the LFM signal of the radar.
waveform = phased.LinearFMWaveform('SampleRate',fs, 'PulseWidth', tpd, 'PRF', prf,...
'SweepBandwidth', bw);
Assume the speed of aircraft is 100 m/s with a flight duration of 4 seconds.
speed = 100;
flightDuration = 4;
radarPlatform = phased.Platform('InitialPosition', [0;-200;500], 'Velocity', [0; speed; 0]);
slowTime = 1/prf;
numpulses = flightDuration/slowTime +1;
maxRange = 2500;
truncrangesamples = ceil((2*maxRange/c)*fs);
fastTime = (0:1/fs:(truncrangesamples-1)/fs);
% Set the reference range for the cross-range processing.
Rc = 1000;
Configure the SAR transmitter and receiver. The antenna looks in the broadside direction orthogonal to the flight direction.
antenna = phased.CosineAntennaElement('FrequencyRange', [1e9 6e9]);
antennaGain = aperture2gain(aperture,c/fc);
transmitter = phased.Transmitter('PeakPower', 50e3, 'Gain', antennaGain);
radiator = phased.Radiator('Sensor', antenna,'OperatingFrequency', fc, 'PropagationSpeed', c);
collector = phased.Collector('Sensor', antenna, 'PropagationSpeed', c,'OperatingFrequency', fc);
receiver = phased.ReceiverPreamp('SampleRate', fs, 'NoiseFigure', 30);
Configure the propagation channel.
channel = phased.FreeSpace('PropagationSpeed', c, 'OperatingFrequency', fc,'SampleRate', fs,...
'TwoWayPropagation', true);
Scene Configuration
In this example, three static point targets are configured at locations specified below. All targets have a mean RCS value of 1 meter-squared.
targetpos= [800,0,0;1000,0,0; 1300,0,0]';
targetvel = [0,0,0;0,0,0; 0,0,0]';
target = phased.RadarTarget('OperatingFrequency', fc, 'MeanRCS', [1,1,1]);
pointTargets = phased.Platform('InitialPosition', targetpos,'Velocity',targetvel);
% The figure below describes the ground truth based on the target
% locations.
figure(1);h = axes;plot(targetpos(2,1),targetpos(1,1),'*g');hold all;plot(targetpos(2,2),targetpos(1,2),'*r');hold all;plot(targetpos(2,3),targetpos(1,3),'*b');hold off;
set(h,'Ydir','reverse');xlim([-10 10]);ylim([700 1500]);
title('Ground Truth');ylabel('Range');xlabel('Cross-Range');
SAR Signal Simulation
The following section describes how the system operates based on the above configuration. Specifically, the section below shows how the data collection is performed for a SAR platform. As the
platform moves in the cross-range direction, pulses are transmitted and received in directions orthogonal to the flight path. A collection of pulses gives the phase history of the targets lying in
the illumination region as the platform moves. The longer the target lies in the illumination region, the better the cross-range resolution for the entire image because the process of range and
cross-range focusing is generalized for the entire scene.
% Define the broadside angle
refangle = zeros(1,size(targetpos,2));
rxsig = zeros(truncrangesamples,numpulses);
for ii = 1:numpulses
% Update radar platform and target position
[radarpos, radarvel] = radarPlatform(slowTime);
[targetpos,targetvel] = pointTargets(slowTime);
% Get the range and angle to the point targets
[targetRange, targetAngle] = rangeangle(targetpos, radarpos);
% Generate the LFM pulse
sig = waveform();
% Use only the pulse length that will cover the targets.
sig = sig(1:truncrangesamples);
% Transmit the pulse
sig = transmitter(sig);
% Define no tilting of beam in azimuth direction
targetAngle(1,:) = refangle;
% Radiate the pulse towards the targets
sig = radiator(sig, targetAngle);
% Propagate the pulse to the point targets in free space
sig = channel(sig, radarpos, targetpos, radarvel, targetvel);
% Reflect the pulse off the targets
sig = target(sig);
% Collect the reflected pulses at the antenna
sig = collector(sig, targetAngle);
% Receive the signal
rxsig(:,ii) = receiver(sig);
Visualize the received signal. The received signal can now be visualized as a collection of multiple pulses transmitted in the cross-range direction. The plots show the real part of the signal for
the three targets. The range and cross-range chirps can be seen clearly. The target responses can be seen as overlapping as the pulse-width is kept longer to maintain average power.
imagesc(real(rxsig));title('SAR Raw Data')
xlabel('Cross-Range Samples')
ylabel('Range Samples')
Perform range compression. Each row of the received signal, which contains all the information from each pulse, can be matched filtered to get the dechirped or range compressed signal.
pulseCompression = phased.RangeResponse('RangeMethod', 'Matched filter', 'PropagationSpeed', c, 'SampleRate', fs);
matchingCoeff = getMatchedFilter(waveform);
[cdata, rnggrid] = pulseCompression(rxsig, matchingCoeff);
This figure shows the response after matched filtering has been performed on the received signal. The phase histories of the three targets are clearly visible along the cross-range direction and
range focusing has been achieved.
imagesc(real(cdata));title('SAR Range Compressed Data')
xlabel('Cross-Range Samples')
ylabel('Range Samples')
Perform azimuth Compression. There are multiple techniques to process the cross-range data and get the final image from the SAR raw data once range compression has been achieved. In essence, range
compression helps achieve resolution in the fast-time or range direction and the resolution in the cross-range direction is achieved by azimuth or cross-range compression. Two such techniques are the
range migration algorithm and the back-projection algorithm, which are demonstrated in this example.
rma_processed = helperRangeMigration(cdata,fastTime,fc,fs,prf,speed,numpulses,c,Rc);
bpa_processed = helperBackProjection(cdata,rnggrid,fastTime,fc,fs,prf,speed,crossRangeResolution,c);
Visualize the final SAR image. Plot the focused SAR image using the range migration algorithm and the approximate back projection algorithm. Only a section of the image formed via the range migration
algorithm is shown to accurately point the location of the targets.
The range migration and accurate form of the backprojection algorithm as shown by [2] and [3] provides theoretical resolution in the cross-track as well as along-track direction. Since the
back-projection used here is of the approximate form, the spread in azimuth direction is evident in case of the back-projection, whereas the data processed via range migration algorithm shows that
theoretical resolution is achieved.
title('SAR Data focused using Range Migration algorithm ')
xlabel('Cross-Range Samples')
ylabel('Range Samples')
title('SAR Data focused using Back-Projection algorithm ')
xlabel('Cross-Range Samples')
ylabel('Range Samples')
This example shows how to develop SAR processing leveraging a LFM signal in an airborne data collection scenario. The example also shows how to generate an image from the received signal via range
migration and approximate form of the back-projection algorithms.
Range Migration Algorithm
function azcompresseddata = helperRangeMigration(sigData,fastTime,fc,fs,prf,speed,numPulses,c,Rc)
This function demonstrates the range migration algorithm for imaging the side-looking synthetic aperture radar. The pulse compressed synthetic aperture data is considered in this algorithm.
Set the range frequency span.
frequencyRange = linspace(fc-fs/2,fc+fs/2,length(fastTime));
krange = 2*(2*pi*frequencyRange)/c;
Set the cross-range wavenumber.
kaz = 2*pi*linspace(-prf/2,prf/2,numPulses)./speed;
Generate a matrix of the cross-range wavenumbers to match the size of the received two-dimensional SAR signal.
kazimuth = kaz.';
kx = krange.^2-kazimuth.^2;
Set the final wavenumber to achieve azimuth focusing.
kx = sqrt(kx.*(kx > 0));
kFinal = exp(1i*kx.*Rc);
Perform a two-dimensional FFT on the range compressed signal.
sdata =fftshift(fft(fftshift(fft(sigData,[],1),1),[],2),2);
Perform bulk compression to get the azimuth compression at the reference range. Perform filtering of the 2-D FFT signal with the new cross-range wavenumber to achieve complete focusing at the
reference range and as a by-product, partial focusing of targets not lying at the reference range.
fsmPol = (sdata.').*kFinal;
Perform Stolt interpolation to achieve focusing for targets that are not lying at the reference range.
stoltPol = fsmPol;
for i = 1:size((fsmPol),1)
stoltPol(i,:) = interp1(kx(i,:),fsmPol(i,:),krange(1,:));
stoltPol(isnan(stoltPol)) = 1e-30;
stoltPol = stoltPol.*exp(-1i*krange.*Rc);
azcompresseddata = ifft2(stoltPol);
Back-Projection Algorithm
function data = helperBackProjection(sigdata,rnggrid,fastTime,fc,fs,prf,speed,crossRangeResolution,c)
This function demonstrates the time-domain back projection algorithm for imaging the side-looking synthetic aperture radar. The pulsed compressed synthetic aperture data is taken as input in this
algorithm. Initialize the output matrix.
data = zeros(size(sigdata));
azimuthDist = -200:speed/prf:200;%azimuth distance
Limit the range and cross-range pixels being processed to reduce processing time.
rangelims = [700 1400];
crossrangelims = [-10 10];
Index the range grid in accordance with the range limits.
rangeIdx = [find(rnggrid>rangelims(1), 1) find(rnggrid<rangelims(2),1,'last')];
Index the azimuth distance in accordance with the cross-range limits.
crossrangeIdxStart = find(azimuthDist>crossrangelims(1),1);
crossrangeIdxStop = find(azimuthDist<crossrangelims(2),1,'last');
for i= rangeIdx(1):rangeIdx(2) % Iterate over the range indices
% Using desired cross-range resolution, compute the synthetic aperture
% length
lsynth= (c/fc)* (c*fastTime(i)/2)/(2*crossRangeResolution);
lsar = round(lsynth*length(azimuthDist)/azimuthDist(end)) ;
% Ensure lsar is an odd number
lsar = lsar + mod(lsar,2);
% Construct hanning window for cross-range processing, to suppress the
% azimuthal side lobes
hn= hanning(lsar).';
% Iterate over the cross-range indices
for j= crossrangeIdxStart:crossrangeIdxStop
% azimuth distance in x direction over cross-range indices
posx= azimuthDist(j);
% range in y-direction over range indices
posy= c*fastTime(i)/2;
% initializing count to zero
count= 0;
% Iterate over the synthetic aperture
for k= j-lsar/2 +1:j+ lsar/2
% Time delay for each of range and cross-range indices
td= sqrt((azimuthDist(k)- posx)^2 + posy^2)*2/c;
cell= round(td*fs) +1 ;
signal = sigdata(cell,k);
count= count + hn(k -(j-lsar/2))*signal *exp(1j*2*pi*fc*(td));
% Processed data at each of range and cross-range indices
data(i,j)= count; | {"url":"https://ch.mathworks.com/help/radar/ug/stripmap-synthetic-aperture-radar-sar-image-formation.html","timestamp":"2024-11-02T12:16:40Z","content_type":"text/html","content_length":"95484","record_id":"<urn:uuid:61bbc32b-0317-4482-a51c-9e2ec13820b6>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00059.warc.gz"} |
Confidence Interval
13.4. Using Confidence Intervals#
A confidence interval has a single purpose – to estimate an unknown parameter based on data in a random sample. In the last section, we said that the interval (36%, 42%) was an approximate 95%
confidence interval for the percent of smokers among mothers in the population. That was a formal way of saying that by our estimate, the percent of smokers among the mothers in the population was
somewhere between 36% and 42%, and that our process of estimation is correct about 95% of the time.
It is important to resist the impulse to use confidence intervals for other purposes. For example, recall that we calculated the interval (26.9 years, 27.6 years) as an approximate 95% confidence
interval for the average age of mothers in the population. A dismayingly common misuse of the interval is to conclude that about 95% of the women were between 26.9 years and 27.6 years old. You don’t
need to know much about confidence intervals to see that this can’t be right – you wouldn’t expect 95% of mothers to all be within a few months of each other in age. Indeed, the histogram of the
sampled ages shows quite a bit of variation.
births = Table.read_table(path_data + 'baby.csv')
births.select('Maternal Age').hist()
A small percent of the sampled ages are in the (26.9, 27.6) interval, and you would expect a similar small percent in the population. The interval just estimates one number: the average of all the
ages in the population.
However, estimating a parameter by confidence intervals does have an important use besides just telling us roughly how big the parameter is.
13.4.1. Using a Confidence Interval to Test Hypotheses#
Our approximate 95% confidence interval for the average age in the population goes from 26.9 years to 27.6 years. Suppose someone wants to test the following hypotheses:
Null hypothesis: The average age in the population is 30 years.
Alternative hypothesis: The average age in the population is not 30 years.
Then, if you were using the 5% cutoff for the p-value, you would reject the null hypothesis. This is because 30 is not in the 95% confidence interval for the population average. At the 5% level of
significance, 30 is not a plausible value for the population average.
This use of confidence intervals is the result of a duality between confidence intervals and tests: if you are testing whether or not the population mean is a particular value x, and you use the 5%
cutoff for the p-value, then you will reject the null hypothesis if x is not in your 95% confidence interval for the mean.
This can be established by statistical theory. In practice, it just boils down to checking whether or not the value specified in the null hypothesis lies in the confidence interval.
If you were using the 1% cutoff for the P-value, you would have to check if the value specified in the null hypothesis lies in a 99% confidence interval for the population mean.
To a rough approximation, these statements are also true for population proportions, provided the sample is large.
While we now have a way of using confidence intervals to test a particular kind of hypothesis, you might wonder about the value of testing whether or not the average age in a population is equal to
30. Indeed, the value isn’t clear. But there are some situations in which a test of this kind of hypothesis is both natural and useful.
13.4.2. Comparing Baseline and Post-Treatment Scores#
We will study this in the context of data that are a subset of the information gathered in a randomized controlled trial about treatments for Hodgkin’s disease. Hodgkin’s disease is a cancer that
typically affects young people. The disease is curable but the treatment can be very harsh. The purpose of the trial was to come up with dosage that would cure the cancer but minimize the adverse
effects on the patients.
This table hodgkins contains data on the effect that the treatment had on the lungs of 22 patients. The columns are:
• Height in cm
• A measure of radiation to the mantle (neck, chest, under arms)
• A measure of chemotherapy
• A score of the health of the lungs at baseline, that is, at the start of the treatment; higher scores correspond to more healthy lungs
• The same score of the health of the lungs, 15 months after treatment
hodgkins = Table.read_table(path_data + 'hodgkins.csv')
│height│rad│chemo│ base │month15 │
│164 │679│180 │160.57│87.77 │
│168 │311│180 │98.24 │67.62 │
│173 │388│239 │129.04│133.33 │
... (19 rows omitted)
We will compare the baseline and 15-month scores. As each row corresponds to one patient, we say that the sample of baseline scores and the sample of 15-month scores are paired - they are not just
two sets of 22 values each, but 22 pairs of values, one for each patient.
At a glance, you can see that the 15-month scores tend to be lower than the baseline scores – the sampled patients’ lungs seem to be doing worse 15 months after the treatment. This is confirmed by
the mostly positive values in the column drop, the amount by which the score dropped from baseline to 15 months.
hodgkins = hodgkins.with_columns(
'drop', hodgkins.column('base') - hodgkins.column('month15')
│height│rad│chemo│ base │month15 │ drop │
│164 │679│180 │160.57│87.77 │72.8 │
│168 │311│180 │98.24 │67.62 │30.62 │
│173 │388│239 │129.04│133.33 │-4.29 │
│157 │370│168 │85.41 │81.28 │4.13 │
│160 │468│151 │67.94 │79.26 │-11.32│
│170 │341│96 │150.51│80.97 │69.54 │
│163 │453│134 │129.88│69.24 │60.64 │
│175 │529│264 │87.45 │56.48 │30.97 │
│185 │392│240 │149.84│106.99 │42.85 │
│178 │479│216 │92.24 │73.43 │18.81 │
... (12 rows omitted)
hodgkins.select('drop').hist(bins=np.arange(-20, 81, 20))
In the sample, the average drop is about 28.6. But could this be the result of chance variation? The data are from a random sample. Could it be that in the entire population of patients, the average
drop is just 0?
To answer this, we can set up two hypotheses:
Null hypothesis: In the population, the average drop is 0.
Alternative hypothesis: In the population, the average drop is not 0.
To test this hypothesis with a 1% cutoff for the p-value, let’s construct an approximate 99% confidence interval for the average drop in the population.
def one_bootstrap_mean():
resample = hodgkins.sample()
return np.average(resample.column('drop'))
# Generate 10,000 bootstrap means
num_repetitions = 10000
bstrap_means = make_array()
for i in np.arange(num_repetitions):
bstrap_means = np.append(bstrap_means, one_bootstrap_mean())
# Get the endpoints of the 99% confidence interval
left = percentile(0.5, bstrap_means)
right = percentile(99.5, bstrap_means)
make_array(left, right)
array([17.46863636, 40.97681818])
resampled_means = Table().with_columns(
'Bootstrap Sample Mean', bstrap_means
plots.plot([left, right], [0, 0], color='yellow', lw=8);
The 99% confidence interval for the average drop in the population goes from about 17 to about 40. The interval doesn’t contain 0. So we reject the null hypothesis.
But notice that we have done better than simply concluding that the average drop in the population isn’t 0. We have estimated how big the average drop is. That’s a more useful result than just
saying, “It’s not 0.”
A note on accuracy: Our confidence interval is quite wide, for two main reasons:
• The confidence level is high (99%).
• The sample size is relatively small compared to those in our earlier examples.
In the next chapter, we will examine how the sample size affects accuracy. We will also examine how the empirical distributions of sample means so often come out bell shaped even though the
distributions of the underlying data are not bell shaped at all.
13.4.3. Endnote#
The terminology of a field usually comes from the leading researchers in that field. Brad Efron, who first proposed the bootstrap technique, used a term that has American origins. Not to be outdone,
Chinese statisticians have proposed their own method. | {"url":"https://inferentialthinking.com/chapters/13/4/Using_Confidence_Intervals.html","timestamp":"2024-11-11T13:09:26Z","content_type":"text/html","content_length":"54725","record_id":"<urn:uuid:9feede80-906a-4754-bf12-054337a1b28c>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00545.warc.gz"} |
Normal Distribution Definition - Varsha Saini
Normal Distribution / Gaussian Distribution
A Random Variable (X) having mean (
Properties of Normal Distribution
• The mean, median and mode are all at the centre point.
• There is no skewness.
• It has a kurtosis of 3.
• It follows Bell Curve.
• It is symmetrical on both sides of the mean.
• 50% data lies before the mean and the other 50% is on the right side of the mean.
• The data near the mean is more frequent in occurrence than the data far from it.
Empirical Rule
• If you go one standard deviation to the left and one standard deviation to the right, it covers 68% of the total data.
• If you go two standard deviations to the left and two standard deviations to the right, it covers 95% of the total data.
• If you go three standard deviations to the left and three to the right, it covers 99.7% of the total data.
Standard Normal Distribution
Standard Normal Distribution is a particular type of Normal Distribution where the mean is 0 and the standard deviation is 1.
How to convert Normal Distribution into Standard Normal Distribution?
Normal Distribution can be converted to Standard Normal Distribution by using a z-score. This process is also called Standardization.
z-score =
Central Limit Theorem
The Normal Distribution is key to the central limit theorem. According to it, if multiple samples are taken from a population then the distribution of their sample means will follow a normal
distribution as the sample size increases.
Applications of Normal Distribution
• All kind of variables in nature is nearly normally distributed like height, weight, job satisfaction, body temperature, strength etc.
• The central limit theorem is based on the concept of normal distribution.
• The assumption behind statistical tests is that data follows a normal distribution.
• If a variable is not normally distributed, it can be converted into normal distribution by a simple transformation.
You can check other probability distributions here. | {"url":"https://varshasaini.in/glossary/normal-distribution/","timestamp":"2024-11-07T01:16:53Z","content_type":"text/html","content_length":"189500","record_id":"<urn:uuid:a3a5395f-756b-4017-be3f-09062c8d106c>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00081.warc.gz"} |
Saxon course 2 online test examples
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e Ma
Lesson 6
More Costs of Running a Restaurant
Let’s explore how much it costs to run a restaurant.
6.1: Are We Making Money?
1. Restaurants have many more expenses than just the cost of the food.
1. Make a list of other items you would have to spend money on if you were running a restaurant.
2. Identify which expenses on your list depend on the number of meals ordered and which are independent of the number of meals ordered.
3. Identify which of the expenses that are independent of the number of meals ordered only have to be paid once and which are ongoing.
4. Estimate the monthly cost for each of the ongoing expenses on your list. Next, calculate the total of these monthly expenses.
2. Tell whether each restaurant is making a profit or losing money if they have to pay the amount you predicted in ongoing expenses per month. Organize your thinking so it can be followed by others.
1. Restaurant A sells 6,000 meals in one month, at an average price of $17 per meal and an average cost of $4.60 per meal.
2. Restaurant B sells 8,500 meals in one month, at an average price of $8 per meal and an average cost of $2.20 per meal.
3. Restaurant C sells 4,800 meals in one month, at an average price of $29 per meal and an average cost of $6.90 per meal.
1. Predict how many meals your restaurant would sell in one month.
2. How much money would you need to charge for each meal to be able to cover all the ongoing costs of running a restaurant?
4. What percentage of the cost of the ingredients is the markup on your meal?
6.2: Disposable or Reusable?
A sample of full service restaurants and a sample of fast food restaurants were surveyed about the average number of customers they serve per day.
1. How does the average number of customers served per day at a full service restaurant generally compare to the number served at a fast food restaurant? Explain your reasoning.
2. About how many customers do you think your restaurant will serve per day? Explain your reasoning.
3. Here are prices for plates and forks:
│ │ plates │ forks │
│disposable│165 paper plates for $12.50 │600 plastic forks for $10│
│ reusable │12 ceramic plates for $28.80 │24 metal forks for $30 │
1. Using your predicted number of customers per day from the previous question, write an equation for the total cost, \(d\), of using disposable plates and forks for every customer for \(n\)
2. Is \(d\) proportional to \(n\)? Explain your reasoning.
3. Use your equation to predict the cost of using disposable plates and forks for 1 year. Explain any assumptions you make with this calculation.
1. How much would it cost to buy enough reusable plates and forks for your predicted number of customers per day?
2. If it costs $10.75 a day to wash the reusable plates and forks, write an expression that represents the total cost, \(r\), of buying and washing reusable plates and forks after \(n\) days.
3. Is \(r\) proportional to \(n\)? Explain your reasoning.
4. How many days can you use the reusable plates and forks for the same cost that you calculated for using disposable plates and forks for 1 year? | {"url":"https://im.kendallhunt.com/MS_ACC/students/2/9/6/index.html","timestamp":"2024-11-05T23:23:04Z","content_type":"text/html","content_length":"68928","record_id":"<urn:uuid:a91cc4a3-9f14-44f7-afb1-4902b577e4f6>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00281.warc.gz"} |
Mathematical Statistics Homework Help – University Homework Help
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Defining Descriptive, Predictive, and Prescriptive Statistics in Baseball
Dylan Drummey
A Diagram respresenting the differentiable but similar types of goals that statistics in baseball aim towards.
When the study of sabermetric thought was first brought to glory by the likes of Bill James and other industry leaders, the primary issue was related to the accounting methodology of baseball.
Players were either being given too much or too little credit, which caused a massive misunderstanding of how these players were valued. This was addressed by the creation of descriptive sabermetrics
or numbers that were more accurate in describing a player. Teams could properly understand the innate skills of given players relative to the competition, but that was the limit of the scope. Just
measuring players proved to not be enough - teams needed to know more. This led to the discovery of prescriptive and predictive analytics in baseball, which are types of measures that try to affect
the future in some way. With different statistics trying to do different things, fans often get confused when referencing numbers about their players, inevitably using the wrong types of measures to
prove a point. It is quite easy to do this - most baseball numbers don’t explicitly state what they’re trying to accomplish. This makes differentiating and identifying these statistics all the more
Descriptive Sabermetrics
The most basic and well-known of the three, descriptive sabermetrics are measures that attempt to tell the exact amount of skill demonstrated in a given play or year, calculating the value added or
lost. These types of numbers are the cornerstone of modern advanced statistics, as well as the type that most conventional everyday fans use. In figuring out if a statistic is descriptive, it needs
to pass a few criteria:
• The statistic must describe what has physically happened on the field. If the number involves any type of regressing qualities in its formula that attempts to adjust for future performance, then
it can probably not be considered descriptive.
• Any value weights must be related to historical performance. Regression can still be used in descriptive stats, but the weights that they yield must be related to past performance. They are often
necessary for more accurate types of these numbers, as proper accounting usually requires regression to yield what factors are worth more than others when comparing players.
• The number must try to indicate some sort of added skill value to the game of baseball. This point is crucial in differentiating between descriptive and prescriptive, as some numbers may fall
into both before this separation. A descriptive statistic must show some sort of skill, like fielding or extra-base hitting ability. Statcast measures do not apply (ex. Launch Angle, Exit Velo),
as these numbers don’t directly reveal value added. They reveal the ability to impact descriptive statistics (such as a higher Exit Velocity leads to a higher wOBA) but don’t actually provide the
value. Descriptive statistics explicitly show the value.
Assuming that the stat passes all of these criteria, then one can correctly treat it as a descriptive statistic. When utilizing these, it is important to note their boundaries. Mainly, the focus of
these measurements is to accurately tell what went on during the season. These numbers rarely have any predictive value, and often shouldn’t be used heavily when evaluating a player’s future worth
(in comparison to predictive metrics). The value measures may also be limited in accuracy, as using weights at all invites the possibility of incorrectly assigning values to the wrong places. As
baseball gains knowledge, the margin of error will slowly shrink - but, error will likely persist due to the very nature of the sport.
These types of numbers in analysis are perfect for comparing two players and their performances, evaluating MVP, Cy Young, and any other award votes that may persist. In the long run, these are also
helpful for Hall of Fame cases. Since trying to guess the future is not needed in that type of voting, using these statistics is perfect for voters to make the right decisions.
Examples of Descriptive Sabermetrics: wOBA, wRC+, OPS, DRS, UZR, WAR
Predictive Sabermetrics:
Out of any type, Predictive Statistics get the worst treatment of them all. Many fans genuinely dislike the fact that people are trying to predict America’s Pastime with a bunch of numbers, claiming
that it brings unnecessary complications. And while the merits of these attacks are questionable, a refutation is beyond the point of explaining what predictive sabermetrics actually are, and how
they can be applied correctly. Predictive Sabermetrics are measures that try to articulate what should have or will happen, regressing for certain rates within a player's performance to indicate
future value. To determine if a statistic is predictive, it must pass the following test:
• The statistic must attempt to forecast what is likely to happen in the future. While in general, this must feel obvious to the reader, it is extremely important to clarify the difference between
the types. This is generally done by regressing numbers in comparison to past performance to yield new potential stats. By assigning weights and going through variable formulas, these numbers
will show the likely outcomes from a given player based on his type of play and performance.
• The number must adjust for current unlikely factors. With xStats specifically in mind for this measure, the number must attempt to alienate outliers and show the most probable event. xStats may
be showing a current year's expected performance, but they aren’t showing a player’s actual value - they are showing his expected value without the unlikely events, which are only used to predict
the future under a normal environment. A normal event is most likely to happen in any given circumstance and adjusting to it is necessary for any type of statistic.
If the statistic one wants to use passes both of the barriers then it should be utilized as a predictive statistic. With that in mind, acknowledging the potential shortcomings is continually useful
for proper evaluation. These numbers should not be used for any type of comparison in value, as ultimately that deviates from their objective, and is therefore useless. Predictive measurements are
often subject to several flaws as frankly, no one can predict the future with 100% certainty. Deviations from the normal happen constantly - all one can do is hope for the most likely event. They
should be considered as the best guess - nothing more, nothing less.
In analysis, these numbers are best used for projection arguments. When the question turns into, “Who will win MVP next year?”, these measurements become extremely relevant. The same goes along with
team records, which can be projected for next season based on many factors. Once a player or team stops playing, they’re arguably useless. But before then, they serve as the best glimpse of the
Examples of Predictive Sabermetrics: xwOBA, ZIPS Projections, xSLG, xERA, pCRA
Prescriptive Sabermetrics:
Descriptive and Predictive metrics were at the forefront of baseball’s sabermetric revolution for many years. But with the introduction of loads of data to team front offices, prescriptive numbers
are more important than ever, even having a major influence over many predictive numbers. Specifically, prescriptive sabermetrics are numbers that isolate certain physical factors with performance
and help yield recommendations as to how a player or team might adjust. To know if a statistic is prescriptive, it must have the following:
• The statistic must reveal some sort of underlying information. The whole point of prescriptive analytics is to be able to figure out what is causing a player to act a certain way through
statistical methods. Player X saw his batting average go down - what might have caused it to go down? The numbers must convey something that is not the batting average itself, but a factor about
the player that might affect that.
• The statistic has a degree of impact on player performance. If the number being used has a correlating impact on a player’s ability to produce value, then it has passed. These numbers must show
an ability to some degree, which would hopefully translate into a player adding value for his team.
• The statistic must quantify action. For this criterion, the action is considered as vague as possible - only to the extent that it is at least physical to some extent. Physical action can include
sprint speeds, arm angles, exit velocities, etc. As long as physics can measure it and formulate it into usable numbers that are proven to cause a player’s performance, it should be considered
Given that the statistic being used passed these qualifications, it can be used as a prescriptive number. Now, for the limitations. These numbers provide no estimate whatsoever as to what value a
player produced or what value a player might produce. They can be impressive feats that show the amazingness of the human body, but they generally won’t translate to how many runs were added on the
scoreboard, or how many might be added next year. These metrics can also be heavily misleading, as a load of assumptions are having to be made for these metrics to somewhat make sense and add value.
It is possible to discover visible laws between performance and prescriptive statistics, but it may be faux that happens to correlate very well. Most of these numbers never directly translate to
success or issue, but they can prove to be helpful.
When doing analysis, these metrics are perfect for judging a player’s physical ability, as well as the potential for future success by looking for transferable skills. They are also great for
diagnosing problems, as underlying large deviations in statistics may mean that something is afoot. For example, pitchers that have augmented stride lengths or decreased velocity may mean that
they’re hurt. They may not physically know that they are, but their bodies' corrective nature reveals that they’re struggling. These numbers best represent the underlying causes of performance.
Examples of Prescriptive Sabermetrics: Exit Velocity, Throw Speed, Spin Rate, Pitch Tilt
Acknowledging that different types of sabermetrics serve different types of purposes is crucial to understand the new look of baseball knowledge itself, as numbers are often meaningless without
context. Being able to differentiate the types provides context, which should be able to eliminate some erroneous or misguided evaluations. Descriptive sabermetrics describe the actual value that
happened on the field, which can help in evaluating two players against each other in a season. Predictive sabermetrics attempt to predict what should have and will likely happen, trying to bring
some certainty in arguments about the future. Perspective sabermetrics showcase the underlying quantifiable causes of the ability to produce value, proving beneficial in evaluating player potential
with limited statistics or identifying problems. These types provide a great outline for the goals of sabermetrics - to describe, predict, or prescribe.
The examination may appear to be black-and-white, divided by flat lines that directly separate the types… But, I encourage the reader to ignore that type of thought entirely concerning these
categories. There are many shades of gray between types, with many statistics overlapping a few. One number may be designated as descriptive or prescriptive, but happen to have predictive qualities
in other aspects. The categories are only meant to serve as guidelines, not as strict rules. Any way that it is taken, it is crucial to view a statistic with as open a mind as possible. Know its
strengths, know its weaknesses, know its goals - only then, one can make a proper evaluation of what is trying to be found. | {"url":"https://www.tdabaseball.com/post/differentiating-the-goals-of-statistics-in-baseball","timestamp":"2024-11-05T22:31:08Z","content_type":"text/html","content_length":"1051107","record_id":"<urn:uuid:0498b200-74b9-490c-b8f3-23cede13fbd3>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00120.warc.gz"} |
Aleatory uncertain discrete variable - Poisson
discrete_variables, aleatory_uncertain_variables
• Alias: None
• Arguments: INTEGER
• Default: no poisson uncertain variables
Child Keywords:
Required/Optional Description of Group Dakota Keyword Dakota Keyword Description
Required lambdas The parameter for the Poisson distribution, the expected number of events in the time interval of interest
Optional initial_point Initial values for variables
Optional descriptors Labels for the variables
The Poisson distribution is used to predict the number of discrete events that happen in a single time interval. The random events occur uniformly and independently. The expected number of occurences
in a single time interval is \(\lambda\) , which must be a positive real number. For example, if events occur on average 4 times per year and we are interested in the distribution of events over six
months, \(\lambda\) would be 2. However, if we were interested in the distribution of events occuring over 5 years, \(\lambda\) would be 20.
The probability mass function for the poisson distribution is given by:
\[f(x) = \frac{\lambda^{x} e^{-\lambda}}{x!},\]
• \(\lambda\) is the expected number of events occuring in a single time interval
• \(x\) is the number of events that occur in this time period
• f(x) is the probability that \(x\) events occur in this time period
When used with some methods such as design of experiments and multidimensional parameter studies, distribution bounds are inferred to be [0, \(\mu + 3 \sigma\) ].
For some methods, including vector and centered parameter studies, an initial point is needed for the uncertain variables. When not given explicitly, these variables are initialized to their means. | {"url":"https://snl-dakota.github.io/docs/6.20.0/users/usingdakota/reference/variables-poisson_uncertain.html","timestamp":"2024-11-10T02:03:41Z","content_type":"text/html","content_length":"20050","record_id":"<urn:uuid:7fb136cb-59f7-48e8-ab07-3befe2f22af7>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00689.warc.gz"} |
Collide force
Collide force
The collide force treats nodes as circles with a given radius, rather than points, and prevents nodes from overlapping. More formally, two nodes a and b are separated so that the distance between a
and b is at least radius(a) + radius(b). To reduce jitter, this is by default a “soft” constraint with a configurable strength and iteration count.
Source · Creates a new circle collide force with the specified radius. If radius is not specified, it defaults to the constant one for all nodes.
const collide = d3.forceCollide((d) => d.r);
Source · If radius is specified, sets the radius accessor to the specified number or function, re-evaluates the radius accessor for each node, and returns this force. If radius is not specified,
returns the current radius accessor, which defaults to:
function radius() {
return 1;
The radius accessor is invoked for each node in the simulation, being passed the node and its zero-based index. The resulting number is then stored internally, such that the radius of each node is
only recomputed when the force is initialized or when this method is called with a new radius, and not on every application of the force.
Source · If strength is specified, sets the force strength to the specified number in the range [0,1] and returns this force. If strength is not specified, returns the current strength which defaults
to 1.
Overlapping nodes are resolved through iterative relaxation. For each node, the other nodes that are anticipated to overlap at the next tick (using the anticipated positions ⟨x + vx,y + vy⟩) are
determined; the node’s velocity is then modified to push the node out of each overlapping node. The change in velocity is dampened by the force’s strength such that the resolution of simultaneous
overlaps can be blended together to find a stable solution.
Source · If iterations is specified, sets the number of iterations per application to the specified number and returns this force. If iterations is not specified, returns the current iteration count
which defaults to 1. Increasing the number of iterations greatly increases the rigidity of the constraint and avoids partial overlap of nodes, but also increases the runtime cost to evaluate the | {"url":"https://d3js.org/d3-force/collide","timestamp":"2024-11-02T08:34:52Z","content_type":"text/html","content_length":"84104","record_id":"<urn:uuid:4f94a838-864c-4058-8af8-e48bbfa3f8e2>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00257.warc.gz"} |
Library Search Results: Books similar to the book "Ambush at Corellia (Star Wars: Corellian Trilogy 1)"
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Cocos: Constructing multi-domain protein phylogenies
Phylogenies of multi-domain proteins have to incorporate macro-evolutionary events, which dramatically increases the complexity of their construction.
We present an application to infer ancestral multi-domain proteins given a species tree and domain phylogenies. As the individual domain phylogenies are often incongruent, we provide diagnostics for
the identification and reconciliation of implausible topologies. We implement and extend a suggested algorithmic approach by Behzadi and Vingron (2006).
Funding Statement
The authors have no support or funding to report
Domains characterize proteins structurally, evolutionarily and functionally [1] . More than half of the proteins in prokaryotes and about 80 percent of the proteins in eukaryotes are composed of
multiple domains [2] . About 200 domains in eukaryotes occur in diverse architectures [3] and provide a challenge for phylogenetic inference, as proteins can be composed of non-homologous elements.
Evolutionary events such as the fusion of proteins or the loss of domains need to be considered in phylogenetic analyses of multi-domain proteins (MDPs).
Behzadi and Vingron put forward an iterative procedure (BV) to reconstruct ancestral domain compositions using the phylogenetic relationship within domain families [4] . See Figure 1 for an overview.
Their algorithm minimizes the number of the macro-evolutionary events protein fusion and domain loss using a set-theoretic formulation and is independent of the order of the domains in the proteins.
Figure 1. Example and principal approach. Reconstruction of the ancestral domain composition for JMJ-associated proteins in Drosophila melanogaster, Schizosaccharomyces pombe and Danio rerio . (A)
Phylogenies are inferred for each domain. (B) Domain trees are embedded in the species tree. (C) Domains are partitioned into proteins. (D) Inconsistencies in the domain trees are curated.
The algorithm consists of the following steps:
1. Input a species tree, a set of domain trees and extant domain compositions, i.e. a partition of the domain trees’ leaf set.
2. Recursively map domain tree nodes to species tree nodes at the least common ancestor (LCA) of all child domain nodes, starting at the leaf level.
3. Recursively walk through the species tree nodes (bottom-up). In all child species, the domains are already partitioned. Establish correspondence between domain nodes in the child species and
those in the current species (relabeling of domain nodes), then find an optimal partition in the current species which is closest to all child partitions in terms of the weighted number of
fusions and deletions.
Previous analyses of MDPs have decided against the use of phylogenetic trees for domains [5] or relied on establishing phylogenies only for domain trees with high internal bootstrap support [6] .
Recently, an alternative approach for the reconstruction of MDPs including domain trees was proposed [7] .
We implemented BV, tested it and and identified critical issues that need to be addressed for successful reconstruction of phylogenies of MDPs using BV. Due to the large number of possible domain
combinations a good set of partitions cannot be found by brute-force enumeration. We implemented a heuristic called weak edge erosion, which yields close to optimal solutions faster than simulated
annealing suggested by [4] . In the practical application it showed that most domain trees are incongruent to each other and the species tree. We implemented a simple procedure to detect and rectify
problematic cases.
In the following, we present our findings in detail, provide an implementation and show how to use it in practice. First, BV and the individual improvements are introduced formally.
The algorithm by Behzadi and Vingron
The algorithm BV uses the information of domain trees and the composition of extant proteins to infer the domain composition of ancestral proteins [4]. The original publication contains a worked
example recommended for further study. Let domain composition family of domain compositions
Phylogenetic trees are inferred for each domain family individually. Reconciling domain trees for each domain family using the known species tree assigns domain nodes to each species node
which can be arbitrarily defined.
Behzadi and Vingron suggest an additive measure given by
thus we solve
The number of merges to have all domains of
Assigning costs for union and deletion yields the partitioning score
The entire tree is reconstructed in a bottom-up pass including the root.
Heuristics for partitioning
As the number of possible partitions for domains into genes grows rapidly, complete enumeration is only possible for the simplest cases and not suitable for real applications. For BV, it was
suggested to use simulated annealing to solve the partitioning problem [4] . We explored a deterministic algorithm we call weak edge erosion (see below) to find a suitable partition
Weak Edge Erosion
Weak edge erosion is a hierarchical graph clustering method based on the idea of attacking a network of affinities between elements at its weakest points and recursively create clusters by separating
network components. The affinity graph is defined as follows:
Consider an undirected loop-free graph
Cluster boundaries are induced by regions sparse in edges. But dividing a set to create partitions translates to cutting through a large number of edges in a clique, so minimal cuts and related
concepts of connectivity are of limited use. Particularly, min-cut tends to separate single vertices from cliques, thus creating a suboptimal partition.
To obtain meaningful cuts for our purpose, we introduce the concept of weak edges. Let each vertex be labeled by the sum of weights of all its incident edges. From the perspective of an edge, high
vertex weights mean that the vertices have a strong connection to a set of other vertices, so edges are regarded to become weaker as their vertex weights grow. We thus define a total order of
weakness: let
This first condition accounts for that we prefer weaker affinities to be violated. If the edge weights are equal, we want to exploit the weakening effect of high vertex weights, thus the weakness
order is determined by the heavier vertices:
If these are equal as well, the relation between the lighter vertices decides:
The concept of edge weakness is illustrated in Figure 2. As the cost function is additive, we can find an approximate solution to the partitioning problem by splitting
Figure 2. Example of edge weakness. Edges are sorted from left to right in decreasing order of weakness.
Weights are depicted by the multiplicity of lines. The edge
If edges tie, such as
If the upper nodes tie as well, as for
Walking through the cut-tree in a bottom-up procedure, we decide whether the block in the parent node or the two set blocks in its children yield a better score
Figure 3. A worked example of a cut tree for the sets
Each node represents a set block of the vertices it contains. Vertex weights are denoted by the number of peripheries. Note that this weight changes during the process according to the sum of
incident edges (e.g. in vertex
However, this does not suffice to disconnect the graph, so vertex weights are recalculated:
and all edges with a weight of 1 are removed,
The complete procedure is summarized in Algorithm 1:
Simulated annealing
The original authors of BV propose simulated annealing to solve the underlying partitioning problem [8]. We implemented a dynamic cooling schedule, which sets most parameters automatically [9].
Starting from a valid domain configuration, the fusion and fission of groups of domains and the swap of individual domains are used to generate related domain compositions. The simulated annealing
procedure is then used to minimize the score
Identification and curation of implausible domain trees
Due to their short length, the inferred domain phylogenies often disagree with each other and the species tree. The BV algorithm was proposed for ideal data and does not consider errors in the
underlying domain topologies. A practical consequence of such errors are that the order of speciation and duplication events between adjacent domains do not agree. These conflicts can lead to
duplicate nodes in the reconstructed composition, for example nodes that have successors within the same protein. The BV algorithm will then produce MDPs with a high partition score due to additional
copies of the conflicting domains. Our algorithm aims to produce a partition with an improved score by applying nearest neighbor interchanges on the conflicting domain trees. For each modification
the ancestral composition is reconstructed and the modification with the lowest score
Simulation of MDP phylogenies
Species trees were generated following the Yule-Harding model. An initial domain composition of three families in a single protein was placed in the root and passed to its children, whose protein
domain compositions underwent evolutionary events of genes (fusion, duplication) and domains (gain, duplication, loss). Subtree pruning and regrafting (SPR) operations were applied on the domain
trees to create perturbed input data.
Construction of domain phylogenies
For an empirical evaluation we ran global models of the PFAM database [10] with HMMER’s hmmpfam and hmmalign [11] against the UniProt/Swiss-Prot database [12] to identify domains and construct
alignments. Maximum Likelihood trees were inferred from the alignments using PhyML [13]. The trees were rooted using Notung [14].
Results and Discussion
As we cannot obtain true ancestral multi-domain protein compositions, we relied on simulations to test the validity of the algorithm and to inspect its performance when errors are introduced. The
practical performance was evaluated on known instances of MDPs, available on the accompanying website. A brief, non-trivial example is presented in Fig. 1.
To assess the performance of the algorithm on controlled input, species and domain phylogenies were simulated. The reconstructed domain compositions were compared to the original compositions using
the partition distance measure, which ranges between zero and the size of the compositions [15]. Most ancestral partitions can be reconstructed perfectly but not all events can be mapped correctly.
For example, the gain of a new domain family and the subsequent loss in one of its immediate children cannot be reconstructed accurately. The high standard deviation suggests that there are a few
compositions which differ very much from their simulated counterpart. SPR operations lower the quality of the reconstruction.
Table 1. Average partition distance and standard deviation of reconstructed domain compositions. The simulated trees had 10, 30 and 50 taxa and up to ten SPR operations were applied. The distance was
calculated between simulated and reconstructed domain compositions of all ancestral species and normalized by the number of taxa. Each experiment was repeated 50 times.
An automated correction of incongruent domain phylogenies is effective for cases with minor errors. Manual curation is advised if domain phylogenies cannot be inferred reliably. To this end, internal
nodes with high scores are tagged for assessment.
The run times of our implementation are short. We simulated 50 replicates of input data containing 50 taxa and no regrafting operations and ran the implementation under Linux on an AMD 64 X2 3200+
processor. The erosion heuristic inferred a MDP reconstruction in 2.06s±0.25s, while the simulated annealing procedure took 153s±22s to achieve comparable solutions. In practice, the calculation of
reliable domain families bounds the performance.
JMJ domains are found in proteins involved in chromatin remodeling complexes often together with domain families such as ARID, PHD and PLUN-1 [16] . They provide a concise example, parts of which are
presented in Fig. 1. Only three species were selected for display; using a larger number of species is advisable for the inference of individual domain phylogenies.
Our solution for the inference of phylogenies of multi-domain proteins provides a simple and easy-to-use interface. The weak edge erosion heuristic provides considerable speed-up over simulated
annealing while maintaining comparable solution quality. Beyond the application on MDPs, such methods could be applied to reconstruction of partial homologous units such as bacterial operons or
protein complexes. Future work will be directed at improving the quality of the tree reconciliation.
Availability and requirements
The program was successfully tested under Python 2.6 and 2.7 on Windows, Mac OS X and Linux. It receives input for species and domain trees as well as parameters in Nexus format. The output can be
visualized using GraphViz. The source code with additional figures and examples can be found at http://virulence.molgen.mpg.de/cocos/ and is freely available under a BSD license.
Author contributions
Implementation: MH and JW. Weak edge erosion: JW. Domain analysis: ST, CS, RK. Example data: IK, CS. Simulations: MH and RK. Wrote the paper: MH, JW and RK.
Competing interests
The authors have declared that no competing interests exist.
• Doolittle RF. The origins and evolution of eukaryotic proteins. Philos Trans R Soc Lond B Biol Sci. 1995 Sep 29;349(1329):235-40. PubMed PMID: 8577833.
• Apic G, Gough J, Teichmann SA. An insight into domain combinations. Bioinformatics. 2001;17 Suppl 1:S83-9. PubMed PMID: 11472996.
• Basu MK, Carmel L, Rogozin IB, Koonin EV. Evolution of protein domain promiscuity in eukaryotes. Genome Res. 2008 Mar;18(3):449-61. Epub 2008 Jan 29. PubMed PMID: 18230802; PubMed Central PMCID:
• Behzadi B, Vingron M. Reconstructing Domain Compositions of Ancestral Multi-domain Proteins. Lecture Notes in Computer Science, 2006; Volume 4205/2006, 1-10
Reference Link
• Song N, Sedgewick RD, Durand D. Domain architecture comparison for multidomain homology identification. J Comput Biol. 2007 May;14(4):496-516. PubMed PMID: 17572026.
• Forslund K, Henricson A, Hollich V, Sonnhammer EL. Domain tree-based analysis of protein architecture evolution. Mol Biol Evol. 2008 Feb;25(2):254-64. Epub 2007 Nov 19. PubMed PMID: 18025066.
• Wiedenhoeft J, Krause R, Eulenstein O. The Plexus Model for the Inference of Ancestral Multi-Domain Proteins. IEEE/ACM Trans Comput Biol Bioinform. 2011 Jan 27. [Epub ahead of print] PubMed PMID:
• Finn RD, Mistry J, Tate J, Coggill P, Heger A, Pollington JE, Gavin OL, Gunasekaran P, Ceric G, Forslund K, Holm L, Sonnhammer EL, Eddy SR, Bateman A. The Pfam protein families database. Nucleic
Acids Res. 2010 Jan;38(Database issue):D211-22. Epub 2009 Nov 17. PubMed PMID: 19920124; PubMed Central PMCID: PMC2808889.
• Huang, MD and Romeo, F. and Sangiovanni-Vincentelli, A. An efficient general cooling schedule for simulated annealing. Proceedings of the IEEE International Conference on Computer-Aided Design
• Eddy SR. A new generation of homology search tools based on probabilistic inference. Genome Inform. 2009 Oct;23(1):205-11. PubMed PMID: 20180275.
• UniProt Consortium. The Universal Protein Resource (UniProt) 2009. Nucleic Acids Res. 2009 Jan;37(Database issue):D169-74. Epub 2008 Oct 4. PubMed PMID: 18836194; PubMed Central PMCID:
• Guindon S, Gascuel O. A simple, fast, and accurate algorithm to estimate large phylogenies by maximum likelihood. Syst Biol. 2003 Oct;52(5):696-704. PubMed PMID: 14530136.
• Chen K, Durand D, Farach-Colton M. NOTUNG: a program for dating gene duplications and optimizing gene family trees. J Comput Biol. 2000;7(3-4):429-47. PubMed PMID: 11108472.
• Gusfield D. Partition-distance: A problem and class of perfect graphs arising in clustering, Information Processing Letters, Volume 82, Issue 3, 16 May 2002, Pages 159-164, ISSN 0020-0190, DOI:
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• Chen K, Durand D, Farach-Colton M. NOTUNG: a program for dating gene duplications and optimizing gene family trees. J Comput Biol. 2000;7(3-4):429-47. PubMed PMID: 11108472.
Reference Link
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If you want a scalable Ethereum blockchain, then P=NP
The Ethereum project has an unfortunate tendency to propose six impossible things before breakfast — ideas that would win a Fields Medal for mathematical genius if they could possibly work.
If you want a horizontally scalable blockchain … then P=NP. If you want transaction sharding to work, then P=NP.
You could bodge either of these, of course. Do you feel lucky?
But maybe it doesn’t matter — and maybe it was never meant to matter.
NP-completeness and P=NP
NP-complete problems are one particular class of mathematical problem where it’s easy to check a result is correct, but hard to calculate the result in the first place.
NP-completeness is a big deal, because a lot of NP-complete problems would be near-magically useful to solve. And all NP-complete problems are mathematically equivalent — so if you can analytically
solve one, you’ve solved them all.
We don’t know if NP-complete problems can be solved. We do know that if we could solve them, it would turn the world upside down. The shorthand for this is “P=NP” — that NP-complete problems could be
solved in polynomial time.
Problems of interest to the cryptocurrency field that P=NP could solve include quickly factoring integers to break public-key cryptography, or reversing hashes.
The Golden Ticket: P, NP, and the Search for the Impossible by Lance Fortnow [Amazon] is a good non-technical introduction to just how many amazing things you could do if P=NP.
But we’ve never found how to do any of those amazing things. So we presume that if a problem is NP-complete, there isn’t an analytical solution.
There might be a solution! But the burden of proof is firmly on the claimant.
Mathematicians have gotten pretty good at spotting NP-completeness in exciting new ideas in the past few decades. It’s not a good sign.
Data Finnovation: let’s construct some proofs
Let’s look at some requirements for a scalable blockchain. Scalability like this was first posited by Ethereum, then the promises were copied into the white papers for other blockchain projects. If
the fabulous future promises come true, your blockchain will need to do these:
• A scalable blockchain will process more transactions per unit time than a single node on the network.
• Balances can never go negative in the system.
• We need to reconcile multiple lists of valid transactions, with a requirement of no negative balance, into a single merged list, such that the maximum number of possible transactions go through.
This must be achievable efficiently.
Here’s a post from the Data Finnovation blog that aims to do the following: [Data Finnovation]
1. Describe these three requirements.
2. Show that the problem of making a blockchain that does all of these things is NP-complete.
3. Show how, if you have such a blockchain, you can right now break hash functions and public-key cryptography, and construct a proof that P=NP.
For an encore, the author shows that if transaction sharding — calculating transactions and running smart contract programs in parallel — could work as advertised, then that too would be NP-complete.
[Data Finnovation]
I won’t copy their work here — this is a post about the Ethereum project. But do enjoy working through the maths.
Worse is better
In real-world programming, when you hit an NP-complete problem, you can usually get away with an approximation to the right answer.
Ethereum is likely to bodge a solution to scaling and sharding, because so many things in Ethereum are bodges. Ethereum is the epitome of “Worse is Better” — a product that does the job inelegantly
but exists now is better than one that’s more elegant, but doesn’t exist yet.
The risk is that we already know how creative blockchain hackers are in seeking out self-service bug bounties — especially when a bodge breaks.
This is why Ethereum keeps promising fabulous new scaling intiatives — then doesn’t implement them for years, or abandons them when they realise the only way is to bodge it.
The current Ethereum proof-of-stake test network generates less CO[2] — but it’s still a single-threaded application, that can’t run faster than a single standalone node.
To some extent, the Ethereum project has just given up on scaling the main blockchain. “For Ethereum to scale and keep up with demand, it has required rollups” — do the work somewhere else and send
back the result. The blockchain is only usable if you work around actually using it. [ethereum.org]
Stay in school, kids
Data Finnovation notes: “It’s worth stating that part #2 — the only part of this blog that is technical — would fly as a homework assignment in any ‘Intro to the Theory of Computation’ class.”
Vitalik Buterin skipped getting a degree in computer science because Peter Thiel paid him not to go to college. This may not have been the greatest idea. The hazard of being an autodidact is that you
don’t know what you don’t know.
Buterin had a propensity for proposing the profoundly unlikely back when he was still into Bitcoin — such as calculating Bitcoin hashes faster by using a quantum computer. He didn’t have a quantum
computer — but he believed a maths crank friend who told him you could simulate a quantum computer usably fast on an ordinary computer.
When The DAO was hacked in 2016, the Ethereum project proposed a soft fork: they would blacklist transactions whose result interacted with the “dark DAO” the attacker had poured the funds into. This
would have been a vector for a fairly obvious denial-of-service attack: flood Ethereum with costly computations that end at the dark DAO.
This approach could only have worked by first solving the halting problem — you would need to be able to determine the outcome of any possible Ethereum program without actually running it and
observing the result. [Hacking Distributed, 2016]
I don’t have a computer science degree (or any degree) either, but I did take care to run my ideas past some computer science professors before I posted. They stressed that bodging your NP-complete
problem is absolutely standard, because in almost all cases, the 99% solution will do. The trouble is, I’m pretty sure there’s a lot of self-service bug bounties in the other 1%.
The purpose of a system is what it does
The Ethereum blockchain is clogged to near-unusability. Transaction fees are large and unpredictable. Applications are deployed to Ethereum because it’s popular — then they’re all but unusable by
normal humans.
Ethereum claims to be decentralised, but this is a marketing lie. The Ethereum blockchain burns a country’s worth of electricity for its proof-of-work mining, but no more than three entities control
the majority of the mining. Megatons of CO[2] are generated for no useful reason except to back legal claims of untouchability. [Digiconomist]
Infura, an API provided by ConsenSys, is the only way to use the Ethereum network at any speed. Infura recently blocked a swathe of countries in response to US sanctions pressure — and took out a
pile of common applications in those countries, including MetaMask and OpenSea.
The present Ethereum system does one job well: it makes a great deal of money for the centralised entities who have operational control of the system. Ethereum’s cryptocurrency, ether, is almost as
easily traded for actual money as Bitcoin is.
Ethereum makes more sense if you first assume it never mattered if the fancy promises ever worked out — even if Buterin believed they were supposed to.
“Maybe investing billions in dollars & man hours into a white paper from a teenager who had no previous experience in … well almost everything, wasn’t the best of ideas. Unless the idea was not about
whether it could work, and instead how much $ you could extract from the naive.” — Alan Graham [Twitter]
10 Comments on “If you want a scalable Ethereum blockchain, then P=NP”
1. I’ve been reading a book on computational complexity and NP=P by Avi Wigdersen. He points out that we don’t know if NP=P but we do know lots and lots of problems that if we solved we would prove
it. Given the size of the attack surface and the years spent on all the problems from domains across math and science it seems very very unlikely that NP=P. I have a link to his book on my
website http://www.agman.com.
2. And yet well-know VCs continue to unashamedly pump (sorry promote) Ethereum (and NFTs ,etc.), even when they themselves run into the obvious issues you document so well … Here is an example from
Fred Wilson that gave me a good chuckle this morning, and that I thought you & your readers might appreciate as well: https://avc.com/2022/04/scaling-the-ethereum-ecosystem/
3. Nitpick from an ancient computer science student: P and NP stand for polynomial and non polynomial, we *do* know how to solve many of these problems but the time required grows exponentially with
the size of the input for all known solutions – i.e., the equation for the complexity class has at least one exponential term, and is therefore non polynomial, hence the name.
1. Weirdly the N in is Non Deterministic. – the compute model is random boxes that just happen to land on the correct answer (maybe they run in some weird gravity hole that lets them do an
arbitrarily close to infinite amount of work and the spit out the answer) and then the regular Turing compute validations that the lucky or magical answer is correct. It is convenient as a
model of complexity because it is easy to proof that some problem is verifiable in polynomial time. Like for Boolean satisfiability you just plug in the values and confirm the overall
expression is true.
4. As the CS professors said, 99% solutions with a bodge will do; one common bodge is to keep the input very small. So, if one only has a handful of nodes and a handful of transactions…
/insert the sound of bells ringing
1. yeah, precisely. “scalable” means n is large.
5. Since reading Attack of the 50-foot block chain I have been a big fan. This is a wonderful and informative post. Many thanks for all your writing.
6. At face value this “proof” claims cash cannot exist either. In reality, sharding means exploiting locality, both in databases and blockchains. You find strong locality restriction in cash, social
media sites, MMORPGs, and multi-chain blockhains like cosmos, polkadot, etc. Local transactions do not need global sequencing.
There were challenges in sharding the byzantine threat model, but enough sensible solutions exist: OmniLedger by Byran Ford, et al. at EPFL, the interactive proof used by polkadot and kusama, and
the myriad zk roll ups like Mina.
All these work because the asymptotics of security are much better than asymptotics of application data. All of cryptography says this. Also Chebyshev’s inequality says this. I’d label Data
Finnovation a crank because they cannot distinguish security from data.
All the same, there are related points really worth making here:
First, all this “internet computer” narrative from old Ethereum, Internet Clown Posse, etc., and especially the “no sharding” narrative by Solana, was designed to promote a bullshit meta-story
that application locality does not matter. This is a stupid narrative obviously.
Second, Ethereum Research explored sharding and failed to deliver, even after Bryan Ford, et al. at EPFL worked out really simple workable sharding schemes. It’s partially that Justin Drake
wanted VDFs instead of threshold randomness. It’s also partially that E.R. cannot think outside some dumb self imposed bubbles, like global gossip and smart contracts. Imho it’s mostly that they
make considerable money if by roll up teams building upon them, so they deliver a shitty product that avoids stepping on their customers’ toes.
Third, afaik sharding winds up pretty application specific, which suggests smart contracts are a flawed abstraction, or at least an oversold abstraction. This is an interesting direction, but not
exactly rigorous. There are imho simpler attacks on smart contracts, with the first being that they’re unnecessarily dangerous.
Also, there are bridge-based multi-chains like Cosmos that do application level locality alright, but make the opposite mistake by ignoring multi-chain security. I’d expect fun exploits against
these guys. 🙂
1. > At face value this “proof” claims cash cannot exist either.
Only if you don’t understand that Ethereum is making particular claims that cash doesn’t. Cash allows eventual reconciliation and errors and, yes, a certain amount of fraud.
7. A few minor corrections: the “N” in NP doesn’t stand for “non”, it stands for “nondeterministic .” And as far as I know it is still the case that no one knows whether factoring is NP-complete.
Much as I’m reluctant to cite Wikipedia as an authority: https://en.wikipedia.org/wiki/NP-completeness.
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The Rules of Twelfths and Twentieths - Jean-du-Sud
The Rules of Twelfths and Twentieths
The Rule of Twelfths. Source: Cmglee, Wikipedia.
Imagine the situation: two people are vigorously debating the best method for calculating tides. The first person argues that the rule of twelfths gives a superior approximation. The second argues
that the calculation method condensed in Tables 5 and 5A of the Fisheries and Oceans Canada Tide Tables provides a better approximation. This debate, which actually took place, was reported to me by
a colleague.
Who is right?
On a geekness scale of 0 to 10, this debate level is about 120. But if you want an incentive to keep reading, I’ll start by summarizing the punch line: both approaches do pretty much the same thing.
If one method has a large forecast error, then the other method will also have an error. It’s the same approximation logic.
The Rule of Twelfths
The rule of twelfths breaks down the half-cycle of the tide into six intervals of time. Variations in tidal height are then approximated in twelfths (hence the name). A visual presentation of the
approach is given in the top image, illustrated over a twelve-hour cycle.
The table below also summarizes the approach. The first three columns present the generic rule, while the last three present the idea for a tidal range of three meters, with a rising tide from 1800.
Time elapsed Variation Tidal Height Time Variation (m) Tidal Height (m)
+0100 +1/12 1/12 of the range 1900 0,25 0,25
+0200 +2/12 3/12 of the range 2000 0,50 0,75
+0300 +3/12 6/12 of the range 2100 0,75 1,50
+0400 +3/12 9/12 of the range 2200 0,75 2,25
+0500 +2/12 11/12 of the range 2300 0,50 2,75
+0600 +1/12 12/12 of the range 2400 0,25 3,00
Can you divide by 12?
In principle, all you need to remember is that you add up the twelfths in the sequence “1, 2, 3, 3, 2, 1”, i.e. “1, 2, 3” and “1, 2, 3 in reverse”. Then it’s a matter of a few calculations to find
the tides at the given times.
The Twentieths Rule
The twentieths rule corresponds to Tables 5 and 5A of the Tide Tables published by Fisheries and Oceans Canada. In terms of presentation, there are three important differences with the sixths rule.
Firstly, the tables divide the tidal half-cycle into twenty periods instead of six (hence the name!). Secondly, the time intervals for each period are not equal. Rather, the intervals are chosen so
that the variation in tidal height is equal during each interval. In short, in the rule of sixths, the time intervals are constant, whereas in the rule of twentieths, it’s the tidal variations that
are constant.
Table 5 of the Tide Table is reproduced below. The table shows the first ten time intervals, arbitrarily labelled from A to I. Thus column A gives the elapsed time in the first time period, column B
gives the elapsed time in the second time period, and so on. Each row shows the duration of each interval for a tidal cycle duration specified in the first column.
Excerpt of Table 5. Source: Oceans and Fisheries Canada.
For example, a tidal cycle of 6 hours (row where the first column shows “6 00”) will give the time elapsed between each interval. After one interval, 52 minutes have elapsed. After five intervals,
two hours and after 10 intervals, three hours have elapsed. (The other ten periods are approximated in the same way, starting from the other end of the tidal half-cycle, so they are not shown in the
table: the table must be used in reverse).
Table 5A (shown below) directly calculates the tidal height for the given time interval (columns A to I) as a function of the tidal range identified in the first column. Thus, for a tidal range of 3
meters (line “3.0”), the tidal height after two hours of a six-hour half-cycle corresponds to 0.75 meters (column E). Note that the tidal intervals are divided equally between each period.
Excerpt of Table 5A. Source: Oceans and Fisheries Canada.
If you haven’t already noticed, note that the 0.75 metre tide corresponds to exactly the same answer as that obtained by the rule of sixths in the previous section. In fact, for each identical time
interval (and the same tidal cycle length and tidal range), both approaches will provide the same answer. For example, after three hours have elapsed (column I), the variation in tidal height will be
one and a half metres, i.e. the same answer as that given by the rule of sixths.
The Same Thing?
It is hard to be more convincing without introducing a bit of high-school mathematics. Both approaches to tidal calculations are based on the same mathematical model, whose foundation is the sine
curve (\sin(t)). A graphical representation of the curve is shown below.
The sine curve.
For the purpose of calculating tidal heights, the following expression is what counts:
z = \frac{\Delta z}{2}\left[1+\sin\left(\pi\left(\frac{t-t_0}{\Delta t}-\frac{1}{2}\right)\right)\right],
where z is the tidal height, \Delta z is the tidal difference \Delta t is the tidal half-cycle in minutes and t is the elapsed time since the beginning of the tidal cycle (in minutes). Of course, \pi
is the circumference to diameter ratio of a circle. (Note: the expression assumes the argument of the sine curve is expressed in radians.)
The advantage of the formula developed above is that it works at any moment of the tidal cycle. It is also compact, requiring only the tidal difference and the duration of the tidal half-cycle. If
the tide is falling, however, you need to change a sign in the expression (which one?). It is a compact form that is a good alternative to the on-board computer in case of breakage. On the other
hand, it requires a minimum understanding of “hard math”.
An Example
If we take the same example as above, i.e. a tidal range of three metres and a rising tide duration of six hours, the expression becomes:
z = \frac{3}{2}\left[1+\sin\left(\pi\left(\frac{t}{360}-\frac{1}{2}\right)\right)\right].
This expression will produce exactly the same answers as tables 5 and 5A and roughly the same answers as the rule of sixths. I illustrate it below, both in a table and in a graph:
Time (h m) Time (m) Sixths Rule Twentieths Rule Formula Approximation Error
52 52 0,15 0,15 0
1 00 60 0,25 0,20 0,05
1 14 74 0,30 0,30 0
1 31 91 0,45 0,45 0
1 46 106 0,60 0,60 0
2 00 120 0,75 0,75 0,75 0
2 13 133 0,90 0,90 0
2 25 145 1,05 1,05 0
2 37 157 1,20 1,20 0
2 49 169 1,35 1,36 -0,01
3 00 180 1,5 1,50 1,50 0
3 11 191 1,65 1,64 0,01
3 23 203 1,80 1,80 0
3 35 215 1,95 1,95 0
3 47 227 2,10 2,10 0
4 00 240 2,25 2,25 2,25 0
4 14 254 2,40 2,40 0
4 28 268 2,55 2,54 0,01
4 46 286 2,70 2,70 0
5 00 300 2,75 2,80 -0,05
5 08 308 2,85 2,85 0
6 00 360 3,00 3,00 3,00 0
Source: author’s calculations. Numbers rounded to the second decimal.
Note from the table that the approximation error from the rule are at most five centimetres! These deviations are deliberately introduced into the sixths rule to make it easier to use. It’s much
easier to remember the “1, 2, 3” rule than to remember a hypothetical rule where the first twelfth is replaced by the fraction 67/1000, which would eliminate the error!
Are These Rules Useful?
At a time when tide levels can be obtained from your on-board computer or cell phone, you might well ask what is the point in knowing these methods. Old-school advocates will certainly say that it is
for the sake of safety.
Manual approaches provide a form of redundancy. If, for any reason, the on-board electronics should fail, the prudent navigator should be able to make an alternative calculation using on-board
The rule of sixths is simple to remember and requires very little information to make an assessment of tidal conditions. Tables 5 and 5A provide the same information, but at finer intervals. In this
sense, it better reproduces the sinusoidal curve at the beginning and end of the tidal cycle.
However, it requires the simultaneous examination of two tables, which introduces a greater potential for error at the time of evaluation. In short, the first manual tool is simple but less accurate,
while the second tool is a little less easy to use, but produces a better approximation. No “hard math” is required.
Personally, I prefer to remember the sine equation explained above and bring along a solar-powered calculator. If the on-board computer, cell phone, calculator AND VHF radio were to fail (that would
be a really bad sailing trip!), I’d probably resort to the rule of sixths. But in such a failure scenario, I would have my mind set on more urgent priorities than tidal calculations.
What is A Good Approximation?
The examination of the two calculation approaches, basically two ways of approximating a sinusoidal curve, prompts a broader reflection on what a good approximation is. It would not be difficult, for
example, to make a “rule of fiftieths” that would subdivide the tidal half-cycle into fifty time intervals. The only conceptual difference is that tables 5 and 5A would be larger.
But what if the sine curve misrepresents the actual tidal curves? This is not an insignificant question, as tides are very different in different parts of the world. A good place to start is by
looking at how Fisheries and Oceans makes its own tide level predictions. Their prediction program is public (here), as is its user manual (here). They even share data to test the program (how about
A reading of their approach shows that they employ two statistical forecasting paradigms. The first is a least-squares forecast based on the position of the planets (a “statistical-physical” model).
The second is a time-series forecasting paradigm, using past tidal values to predict future values (time series).
All this to say that Oceans and Fisheries’ forecasts are certainly more robust for the purposes of calculating tides… present or future, not least because they are agnostic regarding the shape a tide
can take as a function of time. Another redundancy paradigm would be to bring along the estimated coefficients of their forecasting model. It would then be possible to calculate, as required, each of
the entries in each tide table. Not far from what an on-board computer can do…
That said, it’s the SOLAS convention that perhaps best informs us of another way to protect against system failure. Commercial vessels subject to this international convention bring on board a second
electronic navigation system with an independent power supply. On a sailboat, it may be more convenient to bring a second cell phone, or a second tablet, with a tidal application.
Oceans and Fisheries Canada (s.d-a). L’application de niveaux d’eau optimisée pour les appareils mobiles, webpage retrieved online in november 2023 at this address.
_____________________ (s.d.-b). Application de suivi des niveaux de marée, webpage retrieved online in november 2023 at this address.
_____________________ (s.d.-c). IOS Tidal Package, webpage retrieved online in november 2023 at this address.
Foreman, M.G.G (1996). Manuel for Tidal Heights Analysis and Prediction, document prepared for Fisheries and Oceans Canada, retrieved online in november 2023 at this address.
Oceans and Fisheries Canada (2023). Table de marées et des courants du Canada, Tome 2, document retrieved online in november 2023 this address.
Wikipedia (s.d.). Rule of Twelfths, image reproduced from this address. | {"url":"https://en.jeandusud.com/the-rules-of-twelfths-and-twentieths/","timestamp":"2024-11-14T12:12:55Z","content_type":"text/html","content_length":"168564","record_id":"<urn:uuid:4dfcb864-8320-45f7-b663-c4743614decf>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00748.warc.gz"} |
Computer Vision News - July 2024
19 Computer Vision News Computer Vision News SpiderMatch By looking at a different representation of the 3D surface, we can build on well-established frameworks for global optimal matching problems
that lead to geometric consistency. I think the secret is the simplicity and the fact that it’s very fast in practice.” Away from writing top-rated papers, Florian works at the intersection of
machine learning and mathematical optimization in visual computing. Meanwhile, Paul explores solutions to 3D shape matching problems with optimization methods. Looking ahead, Florian acknowledges an
unresolved challenge: “The most critical open problem is whether an algorithm exists to solve this problem in polynomial time,” he ponders. “What we have is fast in practice, but the worst-case time
is still exponential. The next step would be to investigate if it’s possible to come up with a similar formalism that could lead to a polynomial time algorithm that is provably fast.” The BEST OF
CVPR 2024 continues on the next page with another exceptional Oral Paper!
RkJQdWJsaXNoZXIy NTc3NzU= | {"url":"https://www.rsipvision.com/ComputerVisionNews-2024July/19/","timestamp":"2024-11-07T13:11:49Z","content_type":"text/html","content_length":"6318","record_id":"<urn:uuid:5eda727a-0adc-4443-824b-9c42c0834905>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00366.warc.gz"} |
Relations and Functions Class 12 Case Study Questions Maths Chapter 1 - XAM CONTENT
Reading Time: 9 minutes
Last Updated on July 29, 2024 by XAM CONTENT
Hello students, we are providing case study questions for class 12 maths. Case study questions are the new question format that is introduced in CBSE board. The resources for case study questions are
very less. So, to help students we have created chapterwise case study questions for class 12 maths. In this article, you will find case study questions for CBSE Class 12 Maths Chapter 1 Relations
and Functions. It is a part of Case Study Questions for CBSE Class 12 Maths Series.
Chapter Relations and Functions
Type of Questions Case Study Questions
Nature of Questions Competency Based Questions
Board CBSE
Class 12
Subject Maths
Useful for Class 12 Studying Students
Answers provided Yes
Difficulty level Mentioned
Important Link Class 12 Maths Chapterwise Case Study
Case Study Questions on Relations and Functions
Passage 1: Sherlin and Danju are playing Ludo at home during Covid-19. While rolling the dice, Sherlin’s sister Raji observed and noted the possible outcomes of the throw every time belongs to set
{1, 2, 3, 4, 5, 6}. Let A be the set of players while B be set of all possible outcomes. A = {S, D}, B = {1, 2, 3, 4, 5, 6}
Based on the above information answer the following:
(i) Let R : B –> B be defined by R = {(x, y) : y is divisible by x} is
(a) Reflexive and transitive but not symmetric
(b) Reflexive and symmetric but not transitive
(c) Not reflexive but symmetric and transitive
(d) Equivalence
Difficulty Level: Medium
Ans. Option (a) is correct.
(ii) Raji wants to know the number of functions from A to B. How many number of functions are possible?
(a) 6^2
(b) 2^6
(c) 6!
(d) 2^12
Difficulty Level: Medium
Ans. Option (a) is correct.
(iii) Let R be a relation on B defined by R = {(1, 2), (2, 2), (1, 3), (3, 4), (3, 1), (4, 3), (5, 5)}. Then R is
(a) Symmetric
(b) Reflexive
(c) Transitive
(d) None of these three
Difficulty Level: Medium
Ans. Option (d) is correct.
(iv) Raji wants to know the number of relations possible from A to B. How many numbers of relations are possible?
(a) 6^2
(b) 2^6
(c) 6!
(d) 2^12
Difficulty Level: Medium
Ans. Option (d) is correct.
(v) Let R : B –> B be defined by R = {(1, 1), (1, 2), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}, then R is
(a) Symmetric
(b) Reflexive and Transitive
(c) Transitive and symmetric
(d) Equivalence
Difficulty Level: Medium
Ans. Option (b) is correct.
Also check
Topics from which case study questions may be asked
• Definition of Relation
• Domain & Range of a Relation
• Types of Relations from One Set to Another Set
• Types of Intervals
• Function
• Types of Functions
• Algorithm to Check for Surjectivity of a Function
A relation R, from a non-empty set A to another non-empty set B is mathematically as a subset of A × B. Equivalently, any subset of A × B is a relation from A to B.
A relation from A to B is also called a relation from A into B.
Case study questions from the above given topic may be asked.
Frequently Asked Questions (FAQs) on Relations and Functions Case Study
Q1: What is a case study question in mathematics?
A1: A case study question in mathematics is a problem or set of problems based on a real-life scenario or application. It requires students to apply their understanding of mathematical concepts to
analyze, interpret, and solve the given situation.
Q2: How should students tackle case study questions in exams?
A2: To tackle case study questions effectively, students should:
Read the problem carefully: Understand the scenario and identify the mathematical concepts involved.
Break down the problem: Divide the case study into smaller parts to manage the information better.
Apply relevant formulas and theorems: Use the appropriate mathematical tools to solve each part of the problem.
Q3: Why are case study questions included in the Class 12 Maths curriculum?
A3: Case study questions are included to bridge the gap between theoretical knowledge and practical application. They help students see the relevance of what they are learning and prepare them for
real-life situations where they may need to use these mathematical concepts.
Q4: What is a relation in mathematics?
A4: In mathematics, a relation is a set of ordered pairs, typically defined between two sets. If set A has elements a1, a2, and a3, and set B has elements b1, b2, and b3, a relation R from set A to
set B is a subset of the Cartesian product A × B, consisting of ordered pairs (a, b) where ‘a’ belongs to A and ‘b’ belongs to B.
Q5: What is a function? How is it different from a relation?
A5: A function is a special type of relation where every element in the domain (set A) is associated with exactly one element in the codomain (set B). In other words, for every ‘a’ in A, there is
only one ‘b’ in B such that (a, b) is in the relation. While all functions are relations, not all relations are functions.
Q6: What are the types of functions?
A6: Functions can be categorized into several types, including:
One-to-one function (Injective): Each element of the domain is mapped to a unique element of the codomain.
Onto function (Surjective): Every element of the codomain is mapped by at least one element of the domain.
One-to-one and onto function (Bijective): A function that is both injective and surjective.
Constant function: Every element of the domain is mapped to the same single element of the codomain.
Q7: What is the domain and range of a function?
A7: The domain of a function is the set of all possible input values (x-values) for which the function is defined. The range is the set of all possible output values (y-values) produced by the
Q8: What is the vertical line test?
A8: The vertical line test is a graphical method to determine if a relation is a function. If a vertical line intersects the graph of a relation at more than one point, then the relation is not a
function. This is because a function can only have one output value for each input value.
Q9: How do you determine if a function is one-to-one?
A9: To determine if a function is one-to-one, check that each element in the domain maps to a unique element in the codomain. Mathematically, a function f(x) is one-to-one if f(a) = f(b) implies a =
b for all elements a and b in the domain.
Q10: What is the composition of functions?
A10: The composition of functions is the process of applying one function to the results of another. If you have two functions, f and g, the composition (f ∘ g)(x) is defined as f(g(x)). This means
you first apply g to x, then apply f to the result of g(x).
Q11: What are inverse functions?
A11: An inverse function reverses the operation of the original function. If f is a function, its inverse f^(-1) satisfies the condition that f(f^(-1)(x)) = x and f^(-1)(f(x)) = x for all x in the
domain of f^(-1) and f, respectively. Inverse functions exist only for bijective functions.
Q12: Are there any online resources or tools available for practicing relations and functions case study questions?
A12: We provide case study questions for CBSE Class 12 Maths on our website. Students can visit the website and practice sufficient case study questions and prepare for their exams. If you need more
case study questions, then you can visit Physics Gurukul website. they are having a large collection of case study questions for all classes. | {"url":"https://xamcontent.com/class-12-maths-chapter-1-case-study-questions-2/","timestamp":"2024-11-10T11:59:52Z","content_type":"text/html","content_length":"285509","record_id":"<urn:uuid:696f2677-df46-4732-9d54-31e627ec3164>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00320.warc.gz"} |
How high is the storage limit / resource limit
There are storage limits or resource limits in the game which limit the number of resources to be stored.
Stock limits:
For 1 base: 50.000.000 per resource
From 5 bases: 75.000.000 per resource
From 20 bases: 150.000.000 per resource
When you have reached the stock limit for a resource, you can no longer produce that resource as long as the stock is full. The limit depends on how many base stations you have in total.
Since there is only one resource warehouse for all cards, this is your total limit.
When the limit is reached, all factories stop and open productions are credited. So the platinium is not lost.
0 comments
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6.6: Slope–Intercept Form of an Equation of a Line (2024)
Page ID
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Learning Objectives
By the end of this section, you will be able to:
• Recognize the relation between the graph and the slope–intercept form of an equation of a line
• Identify the slope and y-intercept form of an equation of a line
• Graph a line using its slope and intercept
• Choose the most convenient method to graph a line
• Graph and interpret applications of slope–intercept
• Use slopes to identify parallel lines
• Use slopes to identify perpendicular lines
Recognize the Relation Between the Graph and the Slope–Intercept Form of an Equation of a Line
We have graphed linear equations by plotting points, using intercepts, recognizing horizontal and vertical lines, and using the point–slope method. Once we see how an equation in slope–intercept form
and its graph are related, we’ll have one more method we can use to graph lines.
In Graph Linear Equations in Two Variables, we graphed the line of the equation \(y=12x+3\) by plotting points. See Figure \(\PageIndex{1}\). Let’s find the slope of this line.
The red lines show us the rise is \(1\) and the run is \(2\). Substituting into the slope formula:
\[\begin{aligned} m &=\frac{\text { rise }}{\text { rise }} \\ m &=\frac{1}{2} \end{aligned}\]
What is the \(y\)-intercept of the line? The \(y\)-intercept is where the line crosses the \(y\)-axis, so \(y\)-intercept is \((0,3)\). The equation of this line is:
Notice, the line has:
When a linear equation is solved for \(y\), the coefficient of the \(x\)-term is the slope and the constant term is the \(y\)-coordinate of the \(y\)-intercept. We say that the equation \(y=\frac{1}
{2}x+3\) is in slope–intercept form.
The slope–intercept form of an equation of a line with slope mm and \(y\)-intercept, \((0,b)\) is,
Sometimes the slope–intercept form is called the “y-form.”
Exercise \(\PageIndex{1}\)
Use the graph to find the slope and \(y\)-intercept of the line, \(y=2x+1\).
Compare these values to the equation \(y=mx+b\).
To find the slope of the line, we need to choose two points on the line. We’ll use the points \((0,1)\) and \((1,3)\).
Find the rise and run.
Find the \(y\)-intercept of the line. The \(y\)-intercept is the point \((0, 1)\).
The slope is the same as the coefficient of \(x\) and the \(y\)-coordinate of the \(y\)-intercept is the same as the constant term.
Exercise \(\PageIndex{2}\)
Use the graph to find the slope and \(y\)-intercept of the line \(y=\frac{2}{3}x−1\). Compare these values to the equation \(y=mx+b\).
slope \(m = \frac{2}{3}\) and \(y\)-intercept \((0,−1)\)
Exercise \(\PageIndex{3}\)
Use the graph to find the slope and \(y\)-intercept of the line \(y=\frac{1}{2}x+3\). Compare these values to the equation \(y=mx+b\).
slope \(m = \frac{1}{2}\) and \(y\)-intercept \((0,3)\)
Identify the Slope and \(y\)-Intercept From an Equation of a Line
In Understand Slope of a Line, we graphed a line using the slope and a point. When we are given an equation in slope–intercept form, we can use the \(y\)-intercept as the point, and then count out
the slope from there. Let’s practice finding the values of the slope and \(y\)-intercept from the equation of a line.
Exercise \(\PageIndex{4}\)
Identify the slope and \(y\)-intercept of the line with equation \(y=−3x+5\).
We compare our equation to the slope–intercept form of the equation.
Write the equation of the line.
Identify the slope.
Identify the \(y\)-intercept.
Exercise \(\PageIndex{5}\)
Identify the slope and \(y\)-intercept of the line \(y=\frac{2}{5}x−1\).
\(\frac{2}{5}\); (0,−1)
Exercise \(\PageIndex{6}\)
Identify the slope and \(y\)-intercept of the line \(y=−\frac{4}{3}x+1\).
\(-\frac{4}{3}\); (0,1)
When an equation of a line is not given in slope–intercept form, our first step will be to solve the equation for \(y\).
Exercise \(\PageIndex{7}\)
Identify the slope and \(y\)-intercept of the line with equation \(x+2y=6\).
This equation is not in slope–intercept form. In order to compare it to the slope–intercept form we must first solve the equation for \(y\).
Solve for \(y\). \(x+2y=6\)
Subtract x from each side.
Divide both sides by \(2\).
(Remember: \(\frac{a+b}{c} = \frac{a}{c} + \frac{b}{c}\))
Write the slope–intercept form of the equation of the line.
Write the equation of the line.
Identify the slope.
Identify the \(y\)-intercept.
Exercise \(\PageIndex{8}\)
Identify the slope and \(y\)-intercept of the line \(x+4y=8\).
Exercise \(\PageIndex{9}\)
Identify the slope and \(y\)-intercept of the line \(3x+2y=12\).
Graph a Line Using its Slope and Intercept
Now that we know how to find the slope and \(y\)-intercept of a line from its equation, we can graph the line by plotting the \(y\)-intercept and then using the slope to find another point.
Exercise \(\PageIndex{10}\): How to Graph a Line Using its Slope and Intercept
Graph the line of the equation \(y=4x−2\) using its slope and \(y\)-intercept.
Exercise \(\PageIndex{11}\)
Graph the line of the equation \(y=4x+1\) using its slope and \(y\)-intercept.
Exercise \(\PageIndex{12}\)
Graph the line of the equation \(y=2x−3\) using its slope and \(y\)-intercept.
GRAPH A LINE USING ITS SLOPE AND \(y\)-INTERCEPT.
1. Find the slope-intercept form of the equation of the line.
2. Identify the slope and \(y\)-intercept.
3. Plot the \(y\)-intercept.
4. Use the slope formula \(\frac{\text{rise}}{\text{run}}\) to identify the rise and the run.
5. Starting at the \(y\)-intercept, count out the rise and run to mark the second point.
6. Connect the points with a line.
Exercise \(\PageIndex{13}\)
Graph the line of the equation \(y=−x+4\) using its slope and \(y\)-intercept.
The equation is in slope–intercept form. \(y=−x+4\)
Identify the slope and \(y\)-intercept. \(m=−1\)
\(y\)-intercept is \((0, 4)\)
Plot the \(y\)-intercept. See graph below.
Identify the rise and the run. \(m = \frac{-1}{1}\)
Count out the rise and run to mark the second point. rise \(−1\), run \(1\)
Draw the line.
To check your work, you can find another point on the line and make sure it is a solution of the equation. In the graph we see the line goes through \((4, 0)\).
\(\begin{array}{l}{y=-x+4} \\ {0\stackrel{?}{=}-4+4} \\ {0=0\checkmark}\end{array}\)
Exercise \(\PageIndex{14}\)
Graph the line of the equation \(y=−x−3\) using its slope and \(y\)-intercept.
Exercise \(\PageIndex{15}\)
Graph the line of the equation \(y=−x−1\) using its slope and \(y\)-intercept.
Exercise \(\PageIndex{16}\)
Graph the line of the equation \(y=−\frac{2}{3}x−3\) using its slope and \(y\)-intercept.
The equation is in slope–intercept form. \(y=−\frac{2}{3}x−3\)
Identify the slope and \(y\)-intercept. \(m = -\frac{2}{3}\); \(y\)-intercept is \((0, −3)\)
Plot the \(y\)-intercept. See graph below.
Identify the rise and the run.
Count out the rise and run to mark the second point.
Draw the line.
Exercise \(\PageIndex{17}\)
Graph the line of the equation \(y=−\frac{5}{2}x+1\) using its slope and \(y\)-intercept.
Exercise \(\PageIndex{18}\)
Graph the line of the equation \(y=−\frac{3}{4}x−2\) using its slope and \(y\)-intercept.
Exercise \(\PageIndex{19}\)
Graph the line of the equation \(4x−3y=12\) using its slope and \(y\)-intercept.
Find the slope–intercept form of the equation. \(−3y=−4x+12\)
The equation is now in slope–intercept form. \(y=\frac{4}{3}x−4\)
Identify the slope and \(y\)-intercept. \(m=\frac{4}{3}\)
\(y\)-ntercept is \((0, −4)\)
Plot the \(y\)-intercept. See graph below.
Identify the rise and the run; count out the rise and run to mark the second point.
Draw the line.
Exercise \(\PageIndex{20}\)
Graph the line of the equation \(2x−y=6\) using its slope and \(y\)-intercept.
Exercise \(\PageIndex{21}\)
Graph the line of the equation \(3x−2y=8\) using its slope and \(y\)-intercept.
We have used a grid with \(x\) and \(y\) both going from about \(−10\) to \(10\) for all the equations we’ve graphed so far. Not all linear equations can be graphed on this small grid. Often,
especially in applications with real-world data, we’ll need to extend the axes to bigger positive or smaller negative numbers.
Exercise \(\PageIndex{22}\)
Graph the line of the equation \(y=0.2x+45\) using its slope and \(y\)-intercept.
We’ll use a grid with the axes going from about \(−80\) to \(80\).
The equation is in slope–intercept form. \(y=0.2x+45\)
Identify the slope and \(y\)-intercept. \(m=0.2\)
The \(y\)-intercept is \
((0, 45)\)
Plot the \(y\)-intercept. See graph below.
Count out the rise and run to mark the second point. The slope is \(m=0.2\); in fraction form this means \(m=\frac{2}{10}\). Given the scale of our graph, it would be
easier to use the equivalent fraction \(m=\frac{10}{50}\).
Draw the line.
Exercise \(\PageIndex{23}\)
Graph the line of the equation \(y=0.5x+25\) using its slope and \(y\)-intercept.
Exercise \(\PageIndex{24}\)
Graph the line of the equation \(y=0.1x−30\) using its slope and \(y\)-intercept.
Now that we have graphed lines by using the slope and \(y\)-intercept, let’s summarize all the methods we have used to graph lines. See Figure \(\PageIndex{2}\).
Choose the Most Convenient Method to Graph a Line
Now that we have seen several methods we can use to graph lines, how do we know which method to use for a given equation?
While we could plot points, use the slope–intercept form, or find the intercepts for any equation, if we recognize the most convenient way to graph a certain type of equation, our work will be
easier. Generally, plotting points is not the most efficient way to graph a line. We saw better methods in sections 4.3, 4.4, and earlier in this section. Let’s look for some patterns to help
determine the most convenient method to graph a line.
Here are six equations we graphed in this chapter, and the method we used to graph each of them.
\[\begin{array}{lll}{\text{#1}}&{\text {Equation }} & {\text { Method }} \\ {\text{#2}}&{x=2} & {\text { Vertical line }} \\ {\text{#3}}&{y=4} & {\text { Hortical line }} \\ {\text{#4}}&{-x+2 y=6} &
{\text { Intercepts }} \\ {\text{#5}}&{4 x-3 y=12} & {\text { Intercepts }} \\ {\text{#6}}&{y=4 x-2} & {\text { Slope-intercept }} \\{\text{#7}}& {y=-x+4} & {\text { Slope-intercept }}\end{array}\]
Equations #2 and #3 each have just one variable. Remember, in equations of this form the value of that one variable is constant; it does not depend on the value of the other variable. Equations of
this form have graphs that are vertical or horizontal lines.
In equations #4 and #5, both \(x\) and \(y\) are on the same side of the equation. These two equations are of the form \(Ax+By=C\). We substituted \(y=0\) to find the \(x\)-intercept and \(x=0\) to
find the \(y\)-intercept, and then found a third point by choosing another value for \(x\) or \(y\).
Equations #6 and #7 are written in slope–intercept form. After identifying the slope and \(y\)-intercept from the equation we used them to graph the line.
This leads to the following strategy.
Consider the form of the equation.
• If it only has one variable, it is a vertical or horizontal line.
□ \(x=a\) is a vertical line passing through the \(x\)-axis at \(a\).
□ \(y=b\) is a horizontal line passing through the \(y\)-axis at \(b\).
• If \(y\) is isolated on one side of the equation, in the form \(y=mx+b\), graph by using the slope and \(y\)-intercept.
□ Identify the slope and \(y\)-intercept and then graph.
• If the equation is of the form \(Ax+By=C\), find the intercepts.
□ Find the \(x\)- and \(y\)-intercepts, a third point, and then graph.
Exercise \(\PageIndex{25}\)
Determine the most convenient method to graph each line.
1. \(y=−6\)
2. \(5x−3y=15\)
3. \(x=7\)
4. \(y=\frac{2}{5}x−1\).
1. \(y=−6\)
This equation has only one variable, \(y\). Its graph is a horizontal line crossing the \(y\)-axis at \(−6\).
2. \(5x−3y=15\)
This equation is of the form \(Ax+By=C\). The easiest way to graph it will be to find the intercepts and one more point.
3. \(x=7\)
There is only one variable, \(x\). The graph is a vertical line crossing the \(x\)-axis at \(7\).
4. \(y=\frac{2}{5}x−1\)
Since this equation is in \(y=mx+b\) form, it will be easiest to graph this line by using the slope and \(y\)-intercept.
Exercise \(\PageIndex{26}\)
Determine the most convenient method to graph each line:
1. \(3x+2y=12\)
2. \(y=4\)
3. \(y=\frac{1}{5}x−4\)
4. \(x=−7\)
1. intercepts
2. horizontal line
3. slope–intercept
4. vertical line
Exercise \(\PageIndex{27}\)
Determine the most convenient method to graph each line:
1. \(x=6\)
2. \(y=−\frac{3}{4}x+1\)
3. \(y=−8\)
4. \(4x−3y=−1\)
1. vertical line
2. slope–intercept
3. horizontal line
4. intercepts
Graph and Interpret Applications of Slope–Intercept
Many real-world applications are modeled by linear equations. We will take a look at a few applications here so you can see how equations written in slope–intercept form relate to real-world
Usually when a linear equation models a real-world situation, different letters are used for the variables, instead of \(x\) and \(y\). The variable names remind us of what quantities are being
Exercise \(\PageIndex{28}\)
The equation \(F=\frac{9}{5}C+32\) is used to convert temperatures, \(C\), on the Celsius scale to temperatures, \(F\), on the Fahrenheit scale.
1. Find the Fahrenheit temperature for a Celsius temperature of \(0\).
2. Find the Fahrenheit temperature for a Celsius temperature of \(20\).
3. Interpret the slope and \(F\)-intercept of the equation.
4. Graph the equation.
1. \(\begin{array}{ll}{\text { Find the Fahrenheit temperature for a Celsius temperature of } 0 .} & {F=\frac{9}{5} C+32} \\ {\text { Find } F \text { when } C=0 .} & {F=\frac{9}{5}(0)+32} \\ {\
text { Simplify. }} & {F=32}\end{array}\)
2. \begin{array}{ll}{\text { Find the Fahrenheit temperature for a Celsius temperature of } 20 .} & {F=\frac{9}{5} C+32} \\ {\text { Find } F \text { when } C=20 .} & {F=\frac{9}{5}(20)+32} \\ {\
text { Simplify. }} & {F=36+32} \\ {\text { Simplify. }} & {F=68}\end{array}
3. Interpret the slope and \(F\)-intercept of the equation.
Even though this equation uses \(F\) and \(C\), it is still in slope–intercept form.
The slope, \(\frac{9}{5}\), means that the temperature Fahrenheit (\(F\)) increases \(9\) degrees when the temperature Celsius (\(C\)) increases \(5\) degrees.
The \(F\)-intercept means that when the temperature is \(0°\) on the Celsius scale, it is \(32°\) on the Fahrenheit scale.
4. Graph the equation.
We’ll need to use a larger scale than our usual. Start at the \(F\)-intercept \((0,32)\) then count out the rise of \(9\) and the run of \(5\) to get a second point. See Figure \(\PageIndex{3}\).
Exercise \(\PageIndex{29}\)
The equation \(h=2s+50\) is used to estimate a woman’s height in inches, \(h\), based on her shoe size, \(s\).
1. Estimate the height of a child who wears women’s shoe size \(0\).
2. Estimate the height of a woman with shoe size \(8\).
3. Interpret the slope and \(h\)-intercept of the equation.
4. Graph the equation.
1. \(50\) inches
2. \(66\) inches
3. The slope, \(2\), means that the height, \(h\), increases by \(2\) inches when the shoe size, \(s\), increases by \(1\). The \(h\)-intercept means that when the shoe size is \(0\), the height
is \(50\) inches.
Exercise \(\PageIndex{30}\)
The equation \(T=\frac{1}{4}n+40\) is used to estimate the temperature in degrees Fahrenheit, \(T\), based on the number of cricket chirps, \(n\), in one minute.
1. Estimate the temperature when there are no chirps.
2. Estimate the temperature when the number of chirps in one minute is \(100\).
3. Interpret the slope and \(T\)-intercept of the equation.
4. Graph the equation.
1. \(40\) degrees
2. \(65\) degrees
3. The slope, \(\frac{1}{4}\), means that the temperature Fahrenheit (\(F\)) increases \(1\) degree when the number of chirps, \(n\), increases by \(4\). The \(T\)-intercept means that when the
number of chirps is \(0\), the temperature is \(40°\).
The cost of running some types business has two components—a fixed cost and a variable cost. The fixed cost is always the same regardless of how many units are produced. This is the cost of rent,
insurance, equipment, advertising, and other items that must be paid regularly. The variable cost depends on the number of units produced. It is for the material and labor needed to produce each
Exercise \(\PageIndex{31}\)
Stella has a home business selling gourmet pizzas. The equation \(C=4p+25\) models the relation between her weekly cost, \(C\), in dollars and the number of pizzas, \(p\), that she sells.
1. Find Stella’s cost for a week when she sells no pizzas.
2. Find the cost for a week when she sells \(15\) pizzas.
3. Interpret the slope and \(C\)-intercept of the equation.
4. Graph the equation.
1. Find Stella's cost for a week when she sells no pizzas.
Find \(C\) when \(p=0\).
Stella's fixed cost is \($25\) when she sells no pizzas.
2. Find the cost for a week when she sells \(15\) pizzas.
Find \(C\) when \(p=15\).
Stella's costs are \($85\) when she sells \(15\) pizzas.
3. Interpret the slope and \(C\)-intercept of the equation.
The slope, \(4\), means that the cost increases by \($4\) for each pizza Stella sells. The \(C\)
-intercept means that even when Stella sells no pizzas, her costs for the week are \($25\).
4. Graph the equation. We'll need to use a larger scale than our usual. Start at the \(C\)
-intercept \((0, 25)\) then count out the rise of \(4\) and the run of \(1\) to get a second
Exercise \(\PageIndex{32}\)
Sam drives a delivery van. The equation \(C=0.5m+60\) models the relation between his weekly cost, \(C\), in dollars and the number of miles, \(m\), that he drives.
1. Find Sam’s cost for a week when he drives \(0\) miles.
2. Find the cost for a week when he drives \(250\) miles.
3. Interpret the slope and \(C\)-intercept of the equation.
4. Graph the equation.
1. \($60\)
2. \($185\)
3. The slope, \(0.5\), means that the weekly cost, \(C\), increases by \($0.50\) when the number of miles driven, \(n\), increases by \(1\). The \(C\)-intercept means that when the number of
miles driven is \(0\), the weekly cost is \($60\).
Exercise \(\PageIndex{33}\)
Loreen has a calligraphy business. The equation \(C=1.8n+35\) models the relation between her weekly cost, \(C\), in dollars and the number of wedding invitations, \(n\), that she writes.
1. Find Loreen’s cost for a week when she writes no invitations.
2. Find the cost for a week when she writes \(75\) invitations.
3. Interpret the slope and \(C\)-intercept of the equation.
4. Graph the equation.
1. \($35\)
2. \($170\)
3. The slope, \(1.8\), means that the weekly cost, \(C\), increases by \($1.80\) when the number of invitations, \(n\), increases by \(1.80\). The \(C\)-intercept means that when the number of
invitations is \(0\), the weekly cost is \($35\).
Use Slopes to Identify Parallel Lines
The slope of a line indicates how steep the line is and whether it rises or falls as we read it from left to right. Two lines that have the same slope are called parallel lines. Parallel lines never
What about vertical lines? The slope of a vertical line is undefined, so vertical lines don’t fit in the definition above. We say that vertical lines that have different \(x\)-intercepts are
parallel. See Figure \(\PageIndex{5}\).
Parallel lines are lines in the same plane that do not intersect.
• Parallel lines have the same slope and different \(y\)-intercepts.
• If \(m_{1}\) and \(m_{2}\) are the slopes of two parallel lines then \(m_{1} = m_{2}\).
• Parallel vertical lines have different \(x\)-intercepts.
Let’s graph the equations \(y=−2x+3\) and \(2x+y=−1\) on the same grid. The first equation is already in slope–intercept form: \(y=−2x+3\). We solve the second equation for \(y\):
\[\begin{aligned} 2x+y &=-1 \\ y &=-2x-1 \end{aligned}\]
Graph the lines.
Notice the lines look parallel. What is the slope of each line? What is the \(y\)-intercept of each line?
\[\begin{array}{lll} {y} & {=m x+b} & {y=m x+b} \\ {y} & {=-2 x+3} & {y=-2 x-1} \\ {m} & {=-2} & {m=-2}\\ {b} & {=3,(0,3)} & {b=-1,(0,-1)}\end{array}\]
The slopes of the lines are the same and the \(y\)-intercept of each line is different. So we know these lines are parallel.
Since parallel lines have the same slope and different \(y\)-intercepts, we can now just look at the slope–intercept form of the equations of lines and decide if the lines are parallel.
Exercise \(\PageIndex{34}\)
Use slopes and \(y\)-intercepts to determine if the lines \(3x−2y=6\) and \(y = \frac{3}{2}x + 1\) are parallel.
\(\begin{array} {lrll} {\text { Solve the first equation for } y .} &{ 3 x-2 y} &{=} &{6}\\{} & {\frac{-2 y}{-2}} &{ =}&{-3 x+6 }\\ {} &{\frac{-2 y}{-2}}&{ =}&{\frac{-3 x+6}{-2}} \\ {} & {y }&{=}
&{\frac{3}{2} x-3} \end{array}\)
The equation is now in slope–intercept form.
The equation of the second line is already in slope–intercept form.
Identify the slope and \(y\)-intercept of both lines.
\(\begin{array}{lll}{y=\frac{3}{2} x+1} & {} & {y=\frac{3}{2} x-3} \\ {y=m x+b} & {} & {y=m x+b}\\ {m=\frac{3}{2}} & {} & {m=\frac{3}{2}} \\ {y\text{-intercept is }(0, 1)} & {} & {y\text
{-intercept is }(0, −3)} \end{array}\)
The lines have the same slope and different \(y\)-intercepts and so they are parallel. You may want to graph the lines to confirm whether they are parallel.
Exercise \(\PageIndex{35}\)
Use slopes and \(y\)-intercepts to determine if the lines \(2x+5y=5\) and \(y=−\frac{2}{5}x−4\) are parallel.
Exercise \(\PageIndex{36}\)
Use slopes and \(y\)-intercepts to determine if the lines \(4x−3y=6\) and \(y=\frac{4}{3}x−1\) are parallel.
Exercise \(\PageIndex{37}\)
Use slopes and \(y\)-intercepts to determine if the lines \(y=−4\) and \(y=3\) are parallel.
\(\begin{array}{llll}{\text{Write each equation in slope-intercept form.}} &{y=-4} & {\text { and }} &{ y=3} \\ {\text{Since there is no }x\text{ term we write }0x.} &{y=0 x-4} & {} &{y=0 x+3} \\
{\text{Identify the slope and }y\text{-intercept of both lines.}} &{y=m x+b} &{} & {y=m x+b} \\ {} &{m=0} &{} & {m=0} \\{} & {y\text {-intercept is }(0,4)} &{} & {y \text {-intercept is }(0,3)}\
The lines have the same slope and different \(y\)-intercepts and so they are parallel.
There is another way you can look at this example. If you recognize right away from the equations that these are horizontal lines, you know their slopes are both \(0\). Since the horizontal lines
cross the \(y\)-axis at \(y=−4\) and at \(y=3\), we know the \(y\)-intercepts are \((0,−4)\) and \((0,3)\). The lines have the same slope and different \(y\)-intercepts and so they are parallel.
Exercise \(\PageIndex{38}\)
Use slopes and \(y\)-intercepts to determine if the lines \(y=8\) and \(y=−6\) are parallel.
Exercise \(\PageIndex{39}\)
Use slopes and \(y\)-intercepts to determine if the lines \(y=1\) and \(y=−5\) are parallel.
Exercise \(\PageIndex{40}\)
Use slopes and \(y\)-intercepts to determine if the lines \(x=−2\) and \(x=−5\) are parallel.
\[x=-2 \text { and } x=-5\]
Since there is no \(y\), the equations cannot be put in slope–intercept form. But we recognize them as equations of vertical lines. Their \(x\)-intercepts are \(−2\) and \(−5\). Since their \(x\)
-intercepts are different, the vertical lines are parallel.
Exercise \(\PageIndex{41}\)
Use slopes and \(y\)-intercepts to determine if the lines \(x=1\) and \(x=−5\) are parallel.
Exercise \(\PageIndex{42}\)
Use slopes and \(y\)-intercepts to determine if the lines \(x=8\) and \(x=−6\) are parallel.
Exercise \(\PageIndex{43}\)
Use slopes and \(y\)-intercepts to determine if the lines \(y=2x−3\) and \(−6x+3y=−9\) are parallel. You may want to graph these lines, too, to see what they look like.
\(\begin{array} {llll} {\text { The first equation is already in slope-intercept form. }} & {y=2x-3}&{}&{} \\ \\ {\text { Solve the second equation for } y} & {-6x+3y} &{=}&{-9} \\{} & {3y}&{=}&
{6x-9} \\ {}&{\frac{3y}{3} }&{=}&{\frac{6x-9}{3}} \\{} & {y}&{=}&{2x-3}\end{array}\)
The second equation is now in slope–intercept form as well.
Identify the slope and \(y\)-intercept of both lines.
\[\begin{array}{lll}{y=2x-3} &{} & {y=2x-3} \\ {y=mx+b} &{} & {y=mx+b} \\ {m=2} &{} & {m=2} \\ {\text{The }y\text{-intercept is }(0 ,−3)} &{} & {\text{The }y\text{-intercept is }(0 ,−3)} \end
{array} \nonumber\]
The lines have the same slope, but they also have the same \(y\)-intercepts. Their equations represent the same line. They are not parallel; they are the same line.
Exercise \(\PageIndex{44}\)
Use slopes and \(y\)-intercepts to determine if the lines \(y=−\frac{1}{2}x−1\) and \(x+2y=2\) are parallel.
not parallel; same line
Exercise \(\PageIndex{45}\)
Use slopes and \(y\)-intercepts to determine if the lines \(y=\frac{3}{4}x−3\) and \(3x−4y=12\) are parallel.
not parallel; same line
Use Slopes to Identify Perpendicular Lines
Let’s look at the lines whose equations are \(y=\frac{1}{4}x−1\) and \(y=−4x+2\), shown in Figure \(\PageIndex{5}\).
These lines lie in the same plane and intersect in right angles. We call these lines perpendicular.
What do you notice about the slopes of these two lines? As we read from left to right, the line \(y=14x−1\) rises, so its slope is positive. The line \(y=−4x+2\) drops from left to right, so it has a
negative slope. Does it make sense to you that the slopes of two perpendicular lines will have opposite signs?
If we look at the slope of the first line, \(m_{1}=14\), and the slope of the second line, \(m_{2}=−4\), we can see that they are negative reciprocals of each other. If we multiply them, their
product is \(−1\).
\[\begin{array}{c}{m_{1} \cdot m_{2}} \\ {\frac{1}{4}(-4)} \\ {-1}\end{array}\]
This is always true for perpendicular lines and leads us to this definition.
Perpendicular lines are lines in the same plane that form a right angle.
If m1 and m2 are the slopes of two perpendicular lines, then:
\[m_{1} \cdot m_{2}=-1 \text { and } m_{1}=\frac{-1}{m_{2}}\]
Vertical lines and horizontal lines are always perpendicular to each other.
We were able to look at the slope–intercept form of linear equations and determine whether or not the lines were parallel. We can do the same thing for perpendicular lines.
We find the slope–intercept form of the equation, and then see if the slopes are negative reciprocals. If the product of the slopes is \(−1\), the lines are perpendicular. Perpendicular lines may
have the same \(y\)-intercepts.
Exercise \(\PageIndex{46}\)
Use slopes to determine if the lines, \(y=−5x−4\) and \(x−5y=5\) are perpendicular.
The first equation is already in slope–intercept form: \(\quad y=−5x−4\)
\(\begin{array} {llll} {\text{Solve the second equation for }y.} &{x-5y} &{=} &{5} \\{} &{-5 y} &{=} &{-x+5} \\ {} & {\frac{-5 y}{-5}} &{=} &{\frac{-x+5}{-5}} \\ {} &{y} &{=} &{\frac{1}{5} x-1} \
The second equation is now in slope-intercept form as well.
\(\begin{array} {lrllllll} {\text{Identify the slope of each line.}} &{y} &{=} &{-5 x-4} & {} &{y} &{=} &{\frac{1}{5} x-1} \\ {} &{y} &{=} &{m x+b} & {} &{y} &{=} &{m x+b}\\ {} &{m_{1}} &{=}&{-5}
& {} &{m_{2}} &{=}&{\frac{1}{5}}\end{array}\)
The slopes are negative reciprocals of each other, so the lines are perpendicular. We check by multiplying the slopes,
\[\begin{array}{l}{m_{1} \cdot m_{2}} \\ {-5\left(\frac{1}{5}\right)} \\ {-1\checkmark}\end{array}\]
Exercise \(\PageIndex{47}\)
Use slopes to determine if the lines \(y=−3x+2\) and \(x−3y=4\) are perpendicular.
Exercise \(\PageIndex{48}\)
Use slopes to determine if the lines \(y=2x−5\) and \(x+2y=−6\) are perpendicular.
Exercise \(\PageIndex{49}\)
Use slopes to determine if the lines, \(7x+2y=3\) and \(2x+7y=5\) are perpendicular.
\(\begin{array}{lrlrl}{\text{Solve the equations for y.}} &{7 x+2 y} & {=3} & {2 x+7 y}&{=}&{5} \\{} & {2 y} & {=-7 x+3} & {7 y}&{=}&{-2 x+5} \\ {} &{\frac{2 y}{2}} & {=\frac{-7 x+3}{2} \quad} &
{\frac{7 y}{7}}&{=}&{\frac{-2 x+5}{7}} \\ {} &{y} & {=-\frac{7}{2} x+\frac{3}{2}} &{y}&{=}&{\frac{-2}{7}x + \frac{5}{7}}\\ \\{\text{Identify the slope of each line.}} & {y}&{=m x+b} &{y}&{=}&{m
x+b} \\{} & {m_{1}} & {=-\frac{7}{2} }&{ m_{2}}&{=}&{-\frac{2}{7}}\end{array}\)
The slopes are reciprocals of each other, but they have the same sign. Since they are not negative reciprocals, the lines are not perpendicular.
Exercise \(\PageIndex{50}\)
Use slopes to determine if the lines \(5x+4y=1\) and \(4x+5y=3\) are perpendicular.
not perpendicular
Exercise \(\PageIndex{51}\)
Use slopes to determine if the lines \(2x−9y=3\) and \(9x−2y=1\) are perpendicular.
not perpendicular
Access this online resource for additional instruction and practice with graphs.
Key Concepts
• The slope–intercept form of an equation of a line with slope mm and \(y\)-intercept, \((0,b)\) is, \(y=mx+b\).
• Graph a Line Using its Slope and \(y\)-Intercept
1. Find the slope-intercept form of the equation of the line.
2. Identify the slope and \(y\)-intercept.
3. Plot the \(y\)-intercept.
4. Use the slope formula \(m = \dfrac{\text{rise}}{\text{run}}\) to identify the rise and the run.
5. Starting at the \(y\)-intercept, count out the rise and run to mark the second point.
6. Connect the points with a line.
• Strategy for Choosing the Most Convenient Method to Graph a Line: Consider the form of the equation.
□ If it only has one variable, it is a vertical or horizontal line.
\(x = a\) is a vertical line passing through the \(x\)-axis at a.
\(y = b\) is a horizontal line passing through the \(y\)-axis at \(b\).
□ If \(y\) is isolated on one side of the equation, in the form \(y=mx+b\), graph by using the slope and \(y\)-intercept.
Identify the slope and \(y\)-intercept and then graph.
□ If the equation is of the form \(Ax+By=C\), find the intercepts.
Find the \(x\)- and \(y\)-intercepts, a third point, and then graph.
• Parallel lines are lines in the same plane that do not intersect.
□ Parallel lines have the same slope and different \(y\)-intercepts.
□ If \(m_1\) and \(m_2\) are the slopes of two parallel lines then \(m_1 = m_2\).
□ Parallel vertical lines have different \(x\)-intercepts.
• Perpendicular lines are lines in the same plane that form a right angle.
□ If \(m_1\) and \(m_2\) are the slopes of two perpendicular lines, then \(m_1\cdot m_2=−1\) and \(m_1=\frac{−1}{m_2}\).
□ Vertical lines and horizontal lines are always perpendicular to each other.
parallel lines
Lines in the same plane that do not intersect.
perpendicular lines
Lines in the same plane that form a right angle.
slope-intercept form of an equation of a line
The slope–intercept form of an equation of a line with slope mm and \(y\)-intercept, \((0,b)\) is, \(y=mx+b\). | {"url":"https://massoudshaari.com/article/6-6-slope-intercept-form-of-an-equation-of-a-line","timestamp":"2024-11-05T16:18:10Z","content_type":"text/html","content_length":"213901","record_id":"<urn:uuid:9a09398f-55c8-422b-bba7-636e363463b0>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00722.warc.gz"} |
CIfinder is an R package intended to provide functions to compute confidence intervals for the positive predictive value (PPV) and negative predictive value (NPV) based on varied scenarios. In
prospective studies where the proportion of diseased subjects provides an unbiased estimate of the disease prevalence, the confidence intervals for PPV and NPV can be estimated by methods that are
applicable to single proportions. In this case, the package provides six methods for cocmputing the confidence intervals: “clopper.pearson”, “wald”, “wilson”, “wilson.correct”, “agresti”, and “beta”.
In situations where the proportion of diseased subjects does not correspond to the disease prevalence (e.g. case-control studies), this package provides two types of solutions: I) three methods to
estimate confidence intervals for PPV and NPV via ratio of two binomial proportions, including Gart & Nam (1988) https://doi.org/10.2307/2531848, Walter (1975) https://doi.org/10.1093/biomet/62.2.371
, and MOVER-J (Laud, 2017) https://doi.org/10.1002/pst.1813; II) three direct methods to compute the confidence intervals, including Pepe (2003) https://doi.org/10.1002/sim.2185, Zhou (2007)
doi:10.1002/sim.2677, and Delta https://doi.org/10.1002/sim.2677. For more information, please see the Details and References sections in the user’s manual.
You can install the latest version of CIfinder by:
Example use
To demonstrate the utility of the CIfinder::ppv_npv_ci() function for calculating the confidence intervals for PPV and NPV, we will use a case-control study published by van de Vijver et al. (2002)
https://doi.org/10.1056/NEJMoa021967. In the study, Cases were defined as those that have metastasis within 5 years of tumour excision, while controls were those that did not. Each tumor was
classified as having a good or poor gene signature which was defined as having a signature correlation coefficient above or below the ‘optimized sensitivity’ threshold, respectively. This ‘optimized
sensitivity’ threshold is defined as the correlation value that would result in a misclassification of at most 10 per cent of the cases. The performance of their 70-gene signature as prognosticator
for metastasis is summarized as below:
#> Case Control
#> Poor_Signature 31 12
#> Good_Signature 3 32
#> Total 34 44
Generate the confidence interval for or
Since this was a case-control study, the proportion of cases (34/(34+44)=43.6%) is not an unbiased estimate of its prevalence (assumed 7%). PPV and NPV and their confidence intervals can’t be
estimated directly. To calculate the confidence interval based “gart and nam” method:
To calculate the confidence interval based “zhou’s method”
In this output, since continuity correction isn’t specified, the estimates and confidence intervals in ppv and npv are same as the standard delta method where the ppv_logit_transformed and
npv_logit_transformed refer the standard logit method described in the paper. If continuity.correction is being specified:
In this case, a continuity correction value is applied. The ppv and npv outputs refer the “Adjusted” in the paper and ppv_logit_transformed and npv_logit_transformed denote the “Adjusted logit”
method described in the paper.
Confidence interval for special cases
Assume we have a special case data where :
#> Case Control
#> Poor_Signature 31 0
#> Good_Signature 3 44
#> Total 34 44
In this situation, Pepe, Delta, and Zhou (standard logit) methods can not be used without continuity correction. Walter method can be used, but there may have skewness concerns. gart and nam and
mover-j could be considered.
Comparing to the Walter output:
Note, for walter, no continuity.correction should be used as 0.5 has been used as described by the original paper.
Also comparing to the Zhou’s adjusted methods:
Feedback and Report issues
We appreciate any feedback, comments and suggestions. If you have any questions or issues to use the package, please reach out to the developers. | {"url":"http://cran.r-project.org/web/packages/CIfinder/readme/README.html","timestamp":"2024-11-04T07:18:05Z","content_type":"application/xhtml+xml","content_length":"27748","record_id":"<urn:uuid:5f5095da-8fc9-4209-b318-f2903d20c43b>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00822.warc.gz"} |
System-Level Heat Exchanger (TL-G)
Heat exchanger based on performance data between thermal liquid and gas networks
Since R2023b
Simscape / Fluids / Heat Exchangers / Thermal Liquid - Gas
The System-Level Heat Exchanger (TL-G) block models a heat exchanger based on performance data between a thermal liquid and a gas network.
The block uses performance data from the heat exchanger datasheet, rather than the detailed geometry of the exchanger. You can adjust the size and performance of the heat exchanger during design
iterations, or model heat exchangers with uncommon geometries. You can also use this block to model heat exchangers with a certain level of performance at an early design stage, when detailed
geometry data is not yet available.
You parameterize the block by the nominal operating condition. The block sizes the heat exchanger to match the specified performance at the nominal operating condition at steady state.
This block is similar to the Heat Exchanger (G-TL) block, but uses a different parameterization model. The table compares the two blocks:
Heat Exchanger (G-TL) System-Level Heat Exchanger (TL-G)
Block parameters are based on the heat exchanger geometry Block parameters are based on performance and operating conditions
Heat exchanger geometry may be limited by the available geometry parameter options Model is independent of the specific heat exchanger geometry
You can adjust the block for different performance requirements by tuning geometry parameters, such as fin You can adjust the block for different performance requirements by directly specifying
sizes and tube lengths the desired heat and mass flow rates
You can select between parallel, counter, shell and tube, or cross flow configurations You can select between parallel, counter, or cross-flow arrangement at nominal
operating conditions to help with sizing
Predictively accurate results over a wide range of operating conditions, subject to the applicability of the Very accurate results around the specified operating condition; accuracy may decrease
E-NTU equations and the heat transfer coefficient correlations far away from the specified operating conditions
Heat transfer calculations account for the variation of temperature along the flow path by using the E-NTU Heat transfer calculations approximate the variation of temperature along the flow path
model by dividing it into three segments
Heat Transfer
The block divides the gas flow and the thermal liquid flow each into three segments of equal size. The block calculates heat transfer between the fluids in each segment. For simplicity, the equation
in this section are for one segment.
If you clear the Wall thermal mass check box, then the heat balance in the heat exchanger is
• Q[seg,G] is the heat flow rate from the wall that is the heat transfer surface to the gas in the segment.
• Q[seg,TL] is the heat flow rate from the wall to the thermal liquid in the segment.
If you select Wall thermal mass, then the heat balance in the heat exchanger is
• M[wall] is the mass of the wall.
• c[p[wall]] is the specific heat of the wall.
• N = 3 is the number of segments.
• T[seg,wall] is the average wall temperature in the segment.
• t is time.
The heat flow rate from the wall to the gas in the segment is
• UA[seg,G] is the heat transfer conductance for the gas in the segment.
• T[seg,G] is the average temperature for the gas in the segment.
The heat flow rate from the wall to the thermal liquid in the segment is
• UA[seg,TL] is the heat transfer conductance for the thermal liquid in the segment.
• T[seg,TL] is the average liquid temperature in the segment.
Thermal Liquid Heat Transfer Correlation
The heat transfer conductance on the thermal liquid side of the heat exchanger is
• a[TL], b[TL], and c[TL] are the coefficients of the Nusselt number correlation. These coefficients are block parameters in the Correlation Coefficients section.
• Re[seg,TL] is the average Reynolds number for the segment.
• Pr[seg,TL] is the average Prandtl number for the segment.
• k[seg,TL] is the average thermal conductivity for the segment.
• G[TL] is the geometry scale factor for the thermal liquid side of the heat exchanger. The block calculates the geometry scale factor so that the total heat transfer over all segments matches the
specified performance at the nominal operating conditions.
The average Reynolds number is
${\mathrm{Re}}_{seg,\text{TL}}=\frac{{\stackrel{˙}{m}}_{seg,\text{TL}}{D}_{ref,\text{TL}}}{{\mu }_{seg,\text{TL}}{S}_{ref,\text{TL}}},$
• ${\stackrel{˙}{m}}_{seg,\text{TL}}$ is the mass flow rate through the segment.
• μ[seg,TL] is the average dynamic viscosity for the segment.
• D[ref,TL] is an arbitrary reference diameter.
• S[ref,TL] is an arbitrary reference flow area.
The D[ref,TL] and S[ref,TL] terms are included in this equation for unit calculation purposes only, to make Re[seg,TL] nondimensional. The values of D[ref,TL] and S[ref,TL] are arbitrary because the
G[TL] calculation overrides these values.
Gas Heat Transfer Correlation
The heat transfer conductance on the gas side of the heat exchanger is
• a[G], b[G], and c[G] are the coefficients of the Nusselt number correlation. These coefficients are block parameters in the Correlation Coefficients section.
• Re[seg,G] is the average Reynolds number for the segment.
• Pr[seg,G] is the average Prandtl number for the segment.
• k[seg,G] is the average thermal conductivity for the segment.
• G[G] is the geometry scale factor for the gas side of the heat exchanger. The block calculates the geometry scale factor so that the total heat transfer over all segments matches the specified
performance at the nominal operating conditions.
The average Reynolds number is
${\mathrm{Re}}_{seg,G}=\frac{{\stackrel{˙}{m}}_{seg,G}{D}_{ref,G}}{{\mu }_{seg,G}{S}_{ref,G}},$
• ṁ[seg,G] is the mass flow rate through the segment.
• μ[seg,G] is the average dynamic viscosity for the segment.
• D[ref,G] is an arbitrary reference diameter.
• S[ref,G] is an arbitrary reference flow area.
The D[ref,G] and S[ref,G] terms are included in this equation for unit calculation purposes only, to make Re[seg,G] nondimensional. The values of D[ref,G] and S[ref,G] are arbitrary because the G[G]
calculation overrides these values.
Pressure Loss
The pressure losses on the thermal liquid side are
{TL}}}}{2{\rho }_{avg,2P}}\\ {p}_{B,\text{TL}}-{p}_{\text{TL}}=\frac{{K}_{\text{TL}}}{2}\frac{{\stackrel{˙}{m}}_{B,\text{TL}}\sqrt{{\stackrel{˙}{m}}^{2}{}_{B,\text{TL}}+{\stackrel{˙}{m}}^{2}{}_
{thres,\text{TL}}}}{2{\rho }_{avg,\text{TL}}}\end{array}$
• p[A,TL] and p[B,TL] are the pressures at ports A1 and B1, respectively.
• p[TL] is internal thermal liquid pressure at which the block calculates heat transfer.
• ${\stackrel{˙}{m}}_{A,TL}$ and ${\stackrel{˙}{m}}_{B,TL}$ are the mass flow rates into ports A1 and B1, respectively.
• ρ[avg,TL] is the average thermal liquid density over all segments.
• ${\stackrel{˙}{m}}_{thres,TL}$ is the laminar threshold for pressure loss, approximated as 1e-4 of the nominal mass flow rate. The block calculates the pressure loss coefficient, K[TL], so that p
[A,TL] – p[B,TL] matches the nominal pressure loss at the nominal mass flow rate.
The pressure losses on the gas side are
$\begin{array}{l}{p}_{A,G}-{p}_{G}=\frac{{K}_{G}}{2}\frac{{\stackrel{˙}{m}}_{A,G}\sqrt{{\stackrel{˙}{m}}^{2}{}_{A,G}+{\stackrel{˙}{m}}^{2}{}_{thres,G}}}{2{\rho }_{avg,G}}\\ {p}_{B,G}-{p}_{G}=\frac
{{K}_{G}}{2}\frac{{\stackrel{˙}{m}}_{B,G}\sqrt{{\stackrel{˙}{m}}^{2}{}_{B,G}+{\stackrel{˙}{m}}^{2}{}_{thres,G}}}{2{\rho }_{avg,G}}\end{array}$
• p[A,G] and p[B,G] are the pressures at ports A2 and B2, respectively.
• p[G] is the internal gas pressure at which the block calculates heat transfer.
• ṁ[A,G] and ṁ[B,G] are the mass flow rates into ports A2 and B2, respectively.
• ρ[avg,G] is the average gas density over all segments.
• ṁ[thres,G] is the laminar threshold for pressure loss, approximated as 1e-4 of the nominal mass flow rate. The block calculates the pressure loss coefficient, K[G], so that p[A,G] – p[B,G]
matches the nominal pressure loss at the nominal mass flow rate.
Thermal Liquid Mass and Energy Conservation
The mass conservation for the overall thermal liquid flow is
$\left(\frac{d{p}_{TL}}{dt}\sum _{segments}\left(\frac{\partial {\rho }_{seg,TL}}{\partial p}\right)+\sum _{segments}\left(\frac{d{T}_{seg,TL}}{dt}\frac{\partial {\rho }_{seg,TL}}{\partial T}\right)\
• $\frac{\partial {\rho }_{seg,TL}}{\partial p}$ is the partial derivative of density with respect to pressure for the segment.
• $\frac{\partial {\rho }_{seg,TL}}{\partial T}$ is the partial derivative of density with respect to temperature for the segment.
• T[seg,TL] is the temperature for the segment.
• V[TL] is the total thermal liquid volume.
The summation is over all segments.
Although the block divides the thermal liquid flow into N=3 segments for heat transfer calculations, it assumes all segments are at the same internal pressure, p[TL]. Consequently, p[TL] is outside
of the summation.
The energy conservation equation for each segment is
$\begin{array}{l}\left(\frac{d{p}_{TL}}{dt}\frac{\partial {u}_{seg,TL}}{\partial p}+\frac{d{T}_{seg,TL}}{dt}\frac{\partial {u}_{seg,TL}}{\partial T}\right)\frac{{M}_{TL}}{N}+{u}_{seg,TL}\left({\
stackrel{˙}{m}}_{seg,in,TL}-{\stackrel{˙}{m}}_{seg,out,TL}\right)=\\ {\Phi }_{seg,in,TL}-{\Phi }_{seg,out,TL}+{Q}_{seg,TL},\end{array}$
• $\frac{\partial {u}_{seg,TL}}{\partial p}$ is the partial derivative of the specific internal energy with respect to pressure for the segment.
• $\frac{\partial {u}_{seg,TL}}{\partial T}$ is the partial derivative of the specific internal energy with respect to temperature for the segment.
• M[TL] is the total thermal liquid mass.
• ${\stackrel{˙}{m}}_{seg,in,TL}$ and ${\stackrel{˙}{m}}_{seg,out,TL}$ are the mass flow rates into and out of the segment.
• Φ[seg,in,TL] and Φ[seg,out,TL] are the energy flow rates into and out of the segment.
The block assumes the mass flow rates between segments are linearly distributed between the values of ${\stackrel{˙}{m}}_{A,TL}$ and ${\stackrel{˙}{m}}_{B,TL}$.
Gas Mass and Energy Conservation
The mass conservation for the overall gas flow is
$\left(\frac{d{p}_{G}}{dt}\sum _{segments}\left(\frac{\partial {\rho }_{seg,G}}{\partial p}\right)+\sum _{segments}\left(\frac{d{T}_{seg,G}}{dt}\frac{\partial {\rho }_{seg,G}}{\partial T}\right)\
• $\frac{\partial {\rho }_{seg,G}}{\partial p}$ is the partial derivative of density with respect to pressure for the segment.
• $\frac{\partial {\rho }_{seg,G}}{\partial T}$ is the partial derivative of density with respect to temperature for the segment.
• T[seg,G] is the temperature for the segment.
• V[G] is the total gas volume.
The summation is over all segments.
Although the block divides the gas flow into N=3 segments for heat transfer calculations, it assumes all segments are at the same internal pressure, p[G]. Consequently, p[G] is outside of the
The energy conservation equation for each segment is
$\begin{array}{l}\left(\frac{d{p}_{G}}{dt}\frac{\partial {u}_{seg,G}}{\partial p}+\frac{d{T}_{seg,G}}{dt}\frac{\partial {u}_{seg,G}}{\partial T}\right)\frac{{M}_{G}}{N}+{u}_{seg,G}\left({\stackrel{˙}
{m}}_{seg,in,G}-{\stackrel{˙}{m}}_{seg,out,G}\right)=\\ {\Phi }_{seg,in,G}-{\Phi }_{seg,out,G}+{Q}_{seg,G},\end{array}$
• $\frac{\partial {u}_{seg,G}}{\partial p}$ is the partial derivative of the specific internal energy with respect to pressure for the segment.
• $\frac{\partial {u}_{seg,G}}{\partial T}$ is the partial derivative of the specific internal energy with respect to temperature for the segment.
• M[G] is the total gas mass.
• ṁ[seg,in,G] and ṁ[seg,out,G] are the mass flow rates into and out of the segment.
• Φ[seg,in,G] and Φ[seg,out,G] are the energy flow rates into and out of the segment.
The block assumes the mass flow rates between segments are linearly distributed between the values of ṁ[A,G] and ṁ[B,G].
Q1 — Rate of heat transfer to thermal liquid, W
physical signal
Rate of heat transfer to the thermal liquid, returned as a physical signal, in W. The physical signals at ports Q1 and Q2 are usually equal in value with the opposite sign. However, if you select
Wall thermal mass, then these two signals may have different values because the wall may absorb and release some of the heat being transferred.
Q2 — Rate of heat transfer to gas, W
physical signal
Rate of heat transfer to the gas, returned as a physical signal, in W. The physical signals at ports Q1 and Q2 are usually equal in value with the opposite sign. However, if you select Wall thermal
mass, then these two signals may have different values because the wall may absorb and release some of the heat being transferred.
A1 — Thermal liquid port
thermal liquid
Inlet or outlet port associated with the thermal liquid network.
B1 — Thermal liquid port
thermal liquid
Inlet or outlet port associated with the thermal liquid network.
A2 — Gas port
Inlet or outlet port associated with the gas network.
B2 — Gas port
Inlet or outlet port associated with the gas network.
Flow arrangement at nominal operating conditions — Flow path alignment
Counter flow - Thermal Liquid 1 flows from A to B, Gas 2 flows from B to A (default) | Parallel flow - Both fluids flow from A to B | Cross flow - Both fluids flow from A to B
Flow path alignment between the heat exchanger sides at nominal operating condition. The available flow arrangements are:
• Counter flow - Thermal Liquid 1 flows from A to B, Gas 2 flows from B to A — The flows run parallel to each other, in the opposite directions.
• Parallel flow - Both fluids flow from A to B — The flows run in the same direction.
• Cross flow - Both fluids flow from A to B — The flows run perpendicular to each other.
The choice between parallel flow and counter flow affects how the block determines the size of the heat exchanger. The counter flow setting is the most effective, and needs the smallest size to meet
the specified performance. Conversely, parallel flow is the least effective, and needs the biggest size to meet the specified performance.
Flow direction at the nominal condition (from A to B, or from B to A) only affects the model initialization, when you select Initialize thermal liquid to nominal operating conditions or Initialize
gas to nominal operating conditions. If you set different initial operating conditions, the flow directions can be different.
After the block determines the size of the heat exchanger, this setting does not play a role in how the block calculates the heat transfer during simulation. Instead, the heat transfer depends on the
flow directions during simulation. For example, if you set the parameter to parallel flow but set up the model to run in counter flow, then the rate of heat transfer during simulation will not match
the specified performance, even if the rest of the boundary conditions are the same.
If you set the parameter to cross flow, then the block models the flow paths as perpendicular inside the heat exchanger, so the flow directions during simulation do not matter.
Wall thermal mass — Whether to enable effect of thermal mass on heat transfer surface
off (default) | on
Whether to enable the effect of thermal mass on the heat transfer surface. When you select this parameter, the block introduces additional dynamics to the simulation and takes longer to reach steady
state, but this parameter does not affect the results at steady-state simulation.
Wall mass — Mass of heat transfer surface
1 kg (default) | positive scalar
Mass of the heat transfer surface.
To enable this parameter, select Wall thermal mass.
Wall specific heat — Specific heat of heat transfer surface
490 J/(K*kg) (default) | positive scalar
Specific heat of the heat transfer surface.
To enable this parameter, select Wall thermal mass.
Initialize wall temperature to nominal operating conditions — Option for initializing wall temperature
on (default) | off
Option to initialize the wall temperature to nominal operating conditions or specified values. If you select this parameter, the block calculates the initial wall temperature from the nominal
operating conditions specified for both fluid sides. If you clear this parameter, you can specify the initial wall temperature directly with the Initial wall temperature parameter.
To enable this parameter, select Wall thermal mass.
Initial wall temperature — Initial temperature of wall
293.15 K (default) | positive scalar or two element vector
Initial temperature of the wall. If you specify a scalar, the block assumes that the initial wall temperature is uniform. If you specify a two-element vector, then the block assumes that the initial
wall temperature varies linearly between ports A1 and A2 and ports B1 and B2. The first element corresponds to the temperature at ports A1 and A2 and the second element corresponds to the temperature
at ports B1 and B2.
To enable this parameter, select Wall thermal mass and clear the Initialize wall temperature to nominal operating conditions check box.
Cross-sectional area at port A1 — Flow area at port A1
0.01 m^2 (default) | positive scalar
Flow area at the thermal liquid port A1.
Cross-sectional area at port B1 — Flow area at port B1
0.01 m^2 (default) | positive scalar
Flow area at the thermal liquid port B1.
Cross-sectional area at port A2 — Flow area at port A2
0.01 m^2 (default) | positive scalar
Flow area at the gas port A2.
Cross-sectional area at port B2 — Area normal to flow at port B2
0.01 m^2 (default) | positive scalar
Flow area at the gas port B2.
Thermal Liquid 1
Nominal operating condition — Nominal operating condition
Heat transfer from Thermal Liquid 1 to Gas 2 (default) | Heat transfer from Gas 2 to Thermal Liquid 1
Nominal operating condition to use for the thermal liquid network: :
• Heat transfer from Thermal Liquid 1 to Gas 2 — The thermal liquid is cooled and the gas is heated.
• Heat transfer from Gas 2 to Thermal Liquid 1 — The gas is cooled and the thermal liquid is heated.
This setting relates only to the nominal operating condition parameters. It does not mean that heat transfer can only happen in the specified direction during simulation.
Nominal mass flow rate — Mass flow rate between thermal liquid ports during nominal operating condition
0.1 kg/s (default) | positive scalar
Mass flow rate from port A1 to port B1 during the nominal operating condition.
Nominal pressure drop — Pressure drop between thermal liquid ports during nominal operating condition
0.01 MPa (default) | positive scalar
Pressure drop from port A1 to port B1 during the nominal operating condition.
Nominal inlet pressure — Pressure at the thermal liquid inlet
0.101325 MPa (default) | positive scalar
Pressure at the thermal liquid inlet of the heat exchanger during nominal operating condition.
Nominal inlet temperature — Temperature at the thermal liquid inlet
333.15 K (default) | positive scalar
Temperature at the thermal liquid inlet of the heat exchanger during the nominal operating condition.
Heat transfer capacity specification — How to specify heat transfer capacity of thermal liquid
Rate of heat transfer (default) | Outlet condition
Whether to specify the performance of the heat exchanger for the thermal liquid at the nominal operating condition directly, by the rate of heat transfer, or indirectly, by the outlet condition.
Nominal rate of heat transfer — Rate of heat transfer
1 kW (default) | positive scalar
Rate of heat transfer. The Nominal Operating condition parameter determines the network that the heat transfers from and to:
• If Nominal operating condition is Heat transfer from Thermal Liquid 1 to Gas 2, this parameter is the rate of the heat transfer from the thermal liquid to the gas during the nominal operating
• If Nominal operating condition is Heat transfer from Gas 2 to Thermal Liquid 1, this parameter is the rate of the heat transfer from the gas to the thermal liquid during the nominal operating
To enable this parameter, set Heat transfer capacity specification to Rate of heat transfer.
Nominal outlet temperature — Outlet temperature
323.15 K (default) | positive scalar
Temperature at the thermal liquid outlet of the heat exchanger during the nominal operating condition.
To enable this parameter, set Heat transfer capacity specification to Outlet condition.
Thermal liquid volume — Total volume of thermal liquid
0.001 m^3 (default) | positive scalar
Total volume of thermal liquid inside the heat exchanger.
Initialize thermal liquid to nominal operating conditions — Option for initializing thermal liquid
on (default) | off
Option to initialize the thermal liquid to nominal operating conditions or specified values. If you select this parameter, the block initializes the thermal liquid to the nominal operating
conditions. If you clear this check box, you can specify the initial conditions directly with additional parameters.
Initial thermal liquid pressure — Thermal liquid pressure at start of simulation
0.101325 MPa (default) | positive scalar
Thermal liquid pressure at the start of simulation.
To enable this parameter, clear the Initialize thermal liquid to nominal operating conditions check box.
Initial thermal liquid temperature — Thermal liquid temperature at start of simulation
293.15 K (default) | positive scalar
Thermal liquid temperature at the start of simulation.
To enable this parameter, clear the Initialize thermal liquid to nominal operating conditions check box.
Gas 2
Nominal mass flow rate — Mass flow rate between gas ports during nominal operating condition
0.1 kg/s (default) | positive scalar
Mass flow rate from port A2 to port B2 during the nominal operating condition.
Nominal pressure drop — Pressure drop between gas ports during nominal operating condition
0.001 MPa (default) | positive scalar
Pressure drop from port A2 to port B2 during the nominal operating condition.
Nominal inlet pressure — Pressure at gas inlet
0.101325 MPa (default) | positive scalar
Pressure at the gas inlet of the heat exchanger during the nominal operating condition.
Nominal inlet temperature — Temperature at the gas inlet
293.15 K (default) | positive scalar
Temperature at the gas inlet of the heat exchanger during the nominal operating condition.
Gas volume — Total volume of gas in heat exchanger
0.1 m^3 (default) | positive scalar
Total volume of gas inside the heat exchanger.
Initialize gas to nominal operating conditions — Option for initializing gas
on (default) | off
Option to initialize the gas to nominal operating conditions or specified values. If you select this parameter, the block initializes the gas to the nominal operating conditions. If you clear this
check box, you can specify the initial conditions directly with additional parameters.
Initial gas pressure — Gas pressure at the start of simulation
0.101325 MPa (default) | positive scalar
Gas pressure at the start of simulation.
To enable this parameter, clear the Initialize gas to nominal operating conditions check box.
Initial gas temperature — Gas temperature at the start of simulation
293.15 K (default) | positive scalar
Gas temperature at the start of simulation.
To enable this parameter, clear the Initialize gas to nominal operating conditions check box.
Correlation Coefficients
a in Nu = a*Re^b*Pr^c for thermal liquid — Correlation coefficient for thermal liquid
0.023 (default) | positive scalar
Proportionality constant in the correlation of the Nusselt number as a function of the Reynolds number and Prandtl number for thermal liquid. The default value is based on the Colburn equation.
b in Nu = a*Re^b*Pr^c for thermal liquid — Reynolds number exponent in correlation for thermal liquid
0.8 (default) | positive scalar
Reynolds number exponent in the correlation of the Nusselt number as a function of the Reynolds number and Prandtl number for thermal liquid.
c in Nu = a*Re^b*Pr^c for thermal liquid — Prandtl number exponent in correlation for thermal liquid
0.33 (default) | positive scalar
Prandtl number exponent in the correlation of the Nusselt number as a function of the Reynolds number and Prandtl number for thermal liquid.
a in Nu = a*Re^b*Pr^c for gas — Correlation coefficient for gas
0.023 (default) | positive scalar
Proportionality constant in the correlation of the Nusselt number as a function of the Reynolds number and Prandtl number for gas. The default value is based on the Colburn equation.
b in Nu = a*Re^b*Pr^c for gas — Reynolds number exponent in correlation for gas
0.8 (default) | positive scalar
Reynolds number exponent in the correlation of the Nusselt number as a function of the Reynolds number and Prandtl number for gas. The default value is based on the Colburn equation.
c in Nu = a*Re^b*Pr^c for gas — Prandtl number exponent in correlation for gas
0.33 (default) | positive scalar
Prandtl number exponent in the correlation of the Nusselt number as a function of the Reynolds number and Prandtl number for gas. The default value is based on the Colburn equation.
[1] Ashrae Handbook: Fundamentals. Atlanta: Ashrae, 2013.
[2] Çengel, Yunus A. Heat and Mass Transfer: A Practical Approach. 3rd ed. McGraw-Hill Series in Mechanical Engineering. Boston: McGraw-Hill, 2007.
[3] Mitchell, John W., and James E. Braun. Principles of Heating, Ventilation, and Air Conditioning in Buildings. Hoboken, NJ: Wiley, 2013.
[4] Shah, R. K., and Dušan P. Sekulić. Fundamentals of Heat Exchanger Design. Hoboken, NJ: John Wiley & Sons, 2003.
[5] Cavallini, Alberto, and Roberto Zecchin. “A DIMENSIONLESS CORRELATION FOR HEAT TRANSFER IN FORCED CONVECTION CONDENSATION.” In Proceeding of International Heat Transfer Conference 5, 309–13.
Tokyo, Japan: Begellhouse, 1974. https://doi.org/10.1615/IHTC5.1220.
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OAT Physics Practice Test #2 - Master Student
Created by Master Student
Physics Practice Test #2
1 / 40
When green light (wavelength = 500 nm in air) travels through diamond (refractive index = 2.4), what is its wavelength?
2 / 40
A person who normally weighs 900N at sea level climbs to the top of Mt. Everest. While on top of Mt. Everest that person will weigh
3 / 40
Two equal vectors V and V' are added together. All of the following are possible values for the magnitude of the resultant vector EXCEPT
4 / 40
A rocket is launched into the air during a fireworks show. Which of the following statements about the parts of the rocket is appropriate immediately after the explosion?
5 / 40
Which is due to the change in wave speed when a wave strikes the boundary to another medium?
6 / 40
A battery and a variable resistor are in series with a small fan. A switch is closed, and the fan runs. When the variable resistor is moved slightly to the left, the fan slows down a little. When the
variable resistor is moved slightly to the right, the fan speeds up a little. The following question is about the operation of the circuit described. If the variable resistor in the circuit is set
all the way to the right, what happens in the circuit? I. The fan runs faster. II. The voltage decreases. III. The current increases.
7 / 40
Two children are riding a merry-go-round. Child (P) rides on a pony on the outside rim of the merry-go-round, while Child (L) rides a lion on the inside rim of the merry-goround. At the end of the
ride, which of the following statements is true?
8 / 40
A rocket sled rides on a frictionless track while a 1000N force from the rockets accelerates the sled. Suddenly the retrorockets accidentally fire, applying a 1000N force in the opposite direction.
The sled will
9 / 40
Why do baseball catchers wear mitts rather than just using their bare hands to catch pitched baseballs?
10 / 40
The current through a 20ohm resistor connected to a 12V battery is
11 / 40
The electric field strength at a point some distance away from a source charge does NOT depend on
13 / 40
At the surface of the earth, an object of mass m has weight w. If this object is transported to an altitude that's twice the radius of the earth, then, at the new location,
14 / 40
A laboratory centrifuge starts from rest and reaches a rotational speed of 8,000 radians/ sec in a time of 25 seconds. What is the angular acceleration of the centrifuge?
15 / 40
What is the major product of the fission reaction of 235 U? The basic fission reaction is
16 / 40
An observer is 2 m from a source of sound waves. By how much will the sound level decrease if the observer moves to a distance of 20 m ?
17 / 40
Two objects, one of mass 3 kg and moving with a speed of 2 m/s and the other of mass 5 kg and speed 2 m/s, move toward each other and collide head-on. If the collision is perfectly inelastic, find
the speed of the objects after the collision.
18 / 40
Which of the following best describes the condition of an enclosed gas during an adiabatic compression?
19 / 40
A photon can eject an electron from the surface of a photovoltaic metal if and only if
20 / 40
Refrigeration is a process through which
21 / 40
The symbols used in writing nuclear equations. Atomic nuclei are typically written in the form . Which letter represents the protons plus neutrons in the nucleus?
22 / 40
The driver of an automobile traveling at 80 km/hr locks his brakes and skids to a stop in order to avoid hitting a deer in the road. If the driver had been traveling at 40 km/hr, how much faster
would he have stopped?
23 / 40
Which of the following statements is/are correct about an object that has no unbalanced forces applied to it? I. The object has no velocity. II. The object has no acceleration. III. The object does
not move.
24 / 40
Which wave characteristic describes the product of the frequency and the wavelength?
25 / 40
A straight wire of length 2 m carries a 10-amp current. How much stronger is the magnetic field at a distance of 2 cm from the wire than it is at 4 cm from the wire?
26 / 40
A 12,500 kg boxcar rolling through a freight yard has a velocity of 1 m/s when it strikes another boxcar of the same mass that is at rest. Both cars stick together and continue to roll down the track
with a momentum of
27 / 40
Which of the following would be half as great at a distance of 2R from a source charge than it would be at a distance of R from the charge? I. Electric force on another charge II. Electric field due
to the source charge III. Electric potential due to the source charge
28 / 40
A negative ion is an object that has
29 / 40
What is the centripetal acceleration on the rim of a wagon wheel of 44 cm diameter if the wagon is being pulled at a constant 2.5 m/s?
30 / 40
A photovoltaic metal absorbs a photon of yellow light and immediately emits an ultraviolet photon. This is called
31 / 40
Negative charges are accelerated by electric fields toward points
32 / 40
ater has the specific heat 4.186 kJ/kg* degrees celsius, a boiling point of 100 degrees celsius, and a heat of vaporization of 2,260 kJ/kg. A sealed beaker contains 100 g of water that's initially at
20 degrees celsius. If the water absorbs 100 kJ of heat, what will its final temperature be?
33 / 40
An Olympic diver performs a 3.5 somersault. During his dive he uses the tuck position so that he will have
34 / 40
A 400N box is suspended motionless from a steel frame by two ropes, A and B, which hang straight up and down. Which of the following statements about the tension in the two ropes is correct?
35 / 40
A pair of tuning forks produce sound waves that travel through the air. The frequency of the sound waves produced by the first tuning fork is 440 Hz, and the frequency of the sound waves produced by
the second tuning fork is 880 Hz. If v1 denotes the speed of the sound waves produced by the first tuning fork and v2 denotes the speed of the sound waves produced by the second turning fork, then
36 / 40
A rope of length 5 m is stretched to a tension of 80 N. If its mass is 1 kg, at what speed would a 10 Hz transverse wave travel down the string?
37 / 40
A toy rocket is launched straight up. At the exact top of its flight path, which of the following is true?
38 / 40
A father holds his child on his shoulders during a parade. The father does no work during the parade because
39 / 40
Einstein's theory of relativity states that all the laws of nature are the same in
40 / 40
Which of the following is/are true concerning magnetic forces and fields? I. The magnetic field lines due to a current-carrying wire radiate away from the wire. II. The kinetic energy of a charged
particle can be increased by a magnetic force. III. A charged particle can move through a magnetic field without feeling a magnetic force.
Your score is
The average score is 42% | {"url":"https://masterstudent.ca/oat-physics-practice-test-2/","timestamp":"2024-11-09T01:24:52Z","content_type":"text/html","content_length":"321391","record_id":"<urn:uuid:68b91949-05d7-4bc0-8445-81df73e87e2d>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00671.warc.gz"} |
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Fluid Mechanics I
FLUID MECHANICS I (SEMM 2313)
• COURSE INFORMATION
Open or Close
The principle aim of this course is to provide students with an understanding of the properties of fluids and to introduce fundamental laws and description of fluid behavior and flow. It will
emphasize on the concept of pressure, hydrostatic pressure equation and its application in the measurement of pressure, static force due to immersed surfaces, floatation and buoyancy analysis.
Dynamic flow analysis inclusive of technique in solving flow problems is introduce especially to solve flow measurement, mass or volumetric flow rate, momentum in flow and loss in pipe network.
Lastly, some basic dimensional analysis and similarities will be introduced. At the end of the course, the student should be able to demonstrate an ability to analyze whether statically,
dynamically or kinematically problems related directly to fluids.
COURSE LEARNING OUTCOMES
1. Calculate and analyse shear stress in fluids flow, the pressure difference in manometer, hydrostatic forces on plane and curved surfaces, buoyancy force and stability of submerged and
floating bodies.
2. Calculate and analyse fluid dynamics problems using integral forms and differential forms of three basic laws: Bernoulli, energy and momentum equation.
3. Categorize, calculate and analyse the major and minor losses for laminar and turbulent flows in pipes and pipes systems.
4. Derive the dimensionless parameters for a given flow problem using dimensional analysis and similitude in order to solve various problems related to modelling.
• SYLLABUS
Open or Close
Overview of course
Definition of fluid; Dimension & unit fluid properties.
Fluid Statics:
Pascal’s law, pressure variations fluid at rest – pressure measurements.
Forces on plane and curve surface.
Buoyancy. Linearly accelerating containers.
Introduction to fluids in motion:
Description of fluid Motion, classification of fluid flows.
The Bernoulli equation.
Fundamental laws – Integral form.
Control volume, conservation of mass.
Energy equation. Steady uniform flow, steady non-uniform flow.
Momentum equation – Deflectors, propellers.
Fundamental laws – Differential forms.
Continuity equation
Momentum equation, energy equation.
Dimensional analysis:
Buckingham Pi-theorem.
Similitude – confined flows, free-surface flows, high Reynold-number flows.
Normalized differential equations.
Internal flow:
Laminar flow in a pipe
Turbulent flow in a pipe – Major and minor losses, hydraulic and energy grade lines. Simple pipe system with a pump.
• ASSESSMENT
Open or Close
Test 1 - 20%
Test 2 - 20%
Assignment - 20%
Final Exam - 40%
(Marks percentage may be changed due to Fluid Panel's decision)
• REFERENCES
Open or Close
Official Text book
Potter, Wiggert, Ramadan. 2016. Mechanics of Fluids, SI Edition, 5th Ed. Cengage Learning.
Other References
1. Munson, Young, Okiishi, Huebsch. 2010. Fundamental of Fluid Mechanics, 6th Ed. John Wiley and Sons.
2. Fox, McDonald and Pritchard, 2004, Introduction to Fluid Mechanics, 6th Edition, John Wiley and Sons.
3. Douglas, Gasiorek, Swaffield, Fluid Mechanics, 4th Edition, Prentice Hall.
4. White. 2016. Fluid Mechanics. 8th Edition. McGraw-Hill International Edition.
5. Massey, Mechanics of Fluids, Chapman & Hall, London. | {"url":"https://c23434.net/Teaching/SKMM2313/index.html","timestamp":"2024-11-05T17:17:48Z","content_type":"text/html","content_length":"15796","record_id":"<urn:uuid:fa143b39-4db7-4dcc-ac20-4604ebd17af1>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00208.warc.gz"} |
Hands-on Math Insights: Teachers' mismatched gestures boost learning
Hands-on Math Insights: Teachers’ mismatched gestures boost learning
As teachers instruct a child, they typically use their hands as well as their voices, but only certain gestures pack a powerful educational punch, a new study suggests. Grade-schoolers best learn how
to solve a particular mathematics problem when a teacher’s gestures convey different information than his or her words do, say Melissa A. Singer and Susan Goldin-Meadow, both psychologists at the
University of Chicago.
The combination of hearing one problem-solving strategy in speech and seeing another in gesture fosters more math insight than speech and gestures describing the same strategy do, the researchers
propose in the February Psychological Science.
Results are worst when the teacher verbally describes two problem-solving strategies, with or without accompanying gestures. Youngsters in these situations may be encountering too much verbal
information presented too quickly, Singer and Goldin-Meadow theorize.
“Gesture has an active hand in learning,” Goldin-Meadow says. “Teachers may need to pay more attention to how they use gestures.”
The researchers studied 160 third and fourth graders, ages 8 to 10. Each child initially failed to solve mathematical-equivalence problems, such as
6 + 4 + 3 = ___ + 3.
The children were then verbally taught either one or two problem-solving strategies in roughly half-hour sessions. One approach, the equalizer strategy, adds the numbers on the left side of an
equation and then determines how much to add to the number on the right side to get the same total. This strategy highlights the problem’s underlying principle.
In the other tactic, dubbed the add-subtract strategy, a child adds up the numbers on the left side of an equation and then subtracts the number on the right from the left-side total.
Each form of verbal instruction was presented in one of three situations: no accompanying gestures, gestures that matched the tactic being described, or gestures that matched the alternative tactic.
For the equalizer strategy, matching gestures consisted of a hand sweep under the left side of a written equation, a drop of the hand, and then a hand sweep under the right side. For the add-subtract
strategy, matching gestures included pointing consecutively to each number on the left side, followed by a flick-away gesture near the right-side number.
On a test administered after instruction, kids performed best and provided the most thorough explanations of their reasoning after the teacher had explained one strategy verbally and used mismatching
gestures. The combination of equalizer delivered verbally and add-subtract shown in gestures yielded the highest average scores.
Related studies indicate that children on the verge of understanding mathematical equivalence exhibit flawed verbal reasoning combined with gestures that depict valid strategies (SN: 3/17/01, p. 172:
Available to subscribers at Learning in Waves).
The new finding that gestures not matched to speech promote learning demands closer study, remarks psychologist Martha W. Alibali of the University of Wisconsin–Madison. It’s not clear why the
mismatch of gestures and speech is effective or whether it yields similar benefits in teaching other math concepts, Alibali says. | {"url":"https://www.sciencenews.org/article/hands-math-insights-teachers-mismatched-gestures-boost-learning","timestamp":"2024-11-08T15:15:12Z","content_type":"text/html","content_length":"268061","record_id":"<urn:uuid:f7ff34d9-1de2-4a6b-86b1-e9c8be7c0e95>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00235.warc.gz"} |
Understanding Mathematical Functions: How To Know If A Graph Is A Func
Introduction: Understanding the Importance of Functions in Mathematics
Functions are a fundamental concept in mathematics, with wide-ranging applications in various fields. They provide a systematic way of understanding and representing relationships between different
quantities. In this chapter, we will explore the significance of functions in mathematics, the role of graphs in representing functions, and the criteria used to determine if a graph represents a
A. Explore the fundamental role of functions in various fields of mathematics
Functions play a crucial role in fields such as calculus, algebra, and statistics. In calculus, for example, functions are used to model continuous change, while in algebra, they are essential for
solving equations and understanding the properties of numbers. In statistics, functions are used to analyze data and make predictions. Understanding functions is therefore vital for a comprehensive
understanding of mathematics.
B. Introduce the concept of a graph representing a function
One way to visualize and analyze functions is through graphs. A graph represents the relationship between the input and output of a function by plotting points on a coordinate plane. This visual
representation allows us to better understand the behavior of a function and make predictions about its properties.
C. Preview the criteria and methods used to determine if a graph represents a function
Determining whether a given graph represents a function involves certain criteria and methods. We will explore these in detail, including the vertical line test, which is a visual method used to
determine if a graph represents a function. Understanding these criteria and methods is essential for accurately identifying functions from their graphs.
Key Takeaways
• Functions have only one output for each input.
• Check for vertical line test to determine function.
• Graph should not have any vertical line intersections.
• Each input should have a unique output value.
• Understanding functions is essential in mathematics.
Defining a Function and Its Graphical Representation
Understanding mathematical functions is essential in the study of algebra and calculus. Functions are fundamental in describing relationships between quantities and are represented graphically in the
Cartesian coordinate system.
A. Clarify the definition of a mathematical function
A mathematical function is a relation between a set of inputs (independent variables) and a set of possible outputs (dependent variables), where each input is related to exactly one output. In other
words, for every input, there is only one corresponding output. This relationship is often denoted as f(x), where x is the input and f(x) is the output.
B. Discuss how functions are graphically represented in the Cartesian coordinate system
In the Cartesian coordinate system, a function is represented graphically as a set of points (x, f(x)) on a plane, where x is the independent variable and f(x) is the dependent variable. The graph of
a function can take various forms, such as lines, curves, or other shapes, depending on the nature of the function.
C. Highlight the relationship between independent and dependent variables in a function's graph
The graph of a function visually illustrates the relationship between the independent and dependent variables. The independent variable (x-axis) represents the input values, while the dependent
variable (y-axis) represents the corresponding output values. Each point on the graph represents a specific input-output pair, demonstrating how the function's output changes in response to different
input values.
The Vertical Line Test: A Reliable Method for Verification
Understanding mathematical functions is essential for various fields, including engineering, physics, and computer science. One crucial aspect of understanding functions is being able to determine if
a given graph represents a function. The vertical line test is a reliable method for verifying this, and it provides a conclusive way to determine if a graph is a function.
Explain the vertical line test and its purpose
The vertical line test is a visual way to determine if a curve on a graph represents a function. The test involves drawing vertical lines on the graph and observing if any vertical line intersects
the curve at more than one point. If a vertical line intersects the curve at only one point, then the graph represents a function. The purpose of the vertical line test is to provide a simple and
effective way to verify if a graph is a function.
Provide step-by-step instructions on how to apply the vertical line test
To apply the vertical line test, follow these step-by-step instructions:
• Step 1: Obtain the graph of the curve that you want to test.
• Step 2: Identify the x-axis and y-axis on the graph.
• Step 3: Draw a series of vertical lines on the graph, starting from the leftmost point to the rightmost point.
• Step 4: Observe if any vertical line intersects the curve at more than one point.
• Step 5: If any vertical line intersects the curve at only one point, then the graph represents a function. If a vertical line intersects the curve at more than one point, then the graph does not
represent a function.
Discuss how the test conclusively determines if a graph represents a function
The vertical line test conclusively determines if a graph represents a function because it directly assesses the essential property of a function. A function is a relation in which each input value
(x-coordinate) is associated with exactly one output value (y-coordinate). When a vertical line intersects a curve at only one point, it means that for each x-coordinate, there is only one
corresponding y-coordinate. This satisfies the definition of a function. On the other hand, if a vertical line intersects the curve at more than one point, it indicates that there are multiple
y-coordinates for a single x-coordinate, violating the definition of a function.
Recognizing Functions: Characteristics of Function Graphs
Understanding mathematical functions is essential in various fields, from engineering to finance. One of the fundamental aspects of functions is their graphical representation. By analyzing the graph
of a function, we can determine whether it meets the criteria of a function. Let's explore the key features that indicate a graph is indeed a function, examine examples of common functions and their
respective graphs, and discuss exceptions and special conditions.
A Identify key features that indicate a graph is indeed a function
When examining a graph to determine if it represents a function, there are several key features to consider:
• Vertical Line Test: One of the most straightforward methods to determine if a graph represents a function is to use the vertical line test. If a vertical line intersects the graph at only one
point for every x-value, then the graph represents a function. If the vertical line intersects the graph at more than one point for any x-value, then the graph does not represent a function.
• Domain and Range: Another characteristic of a function graph is that each input (x-value) corresponds to exactly one output (y-value). This means that for every x-value in the domain, there is
only one y-value in the range.
• Directionality: In a function graph, each input has a unique output, and the direction of the graph is from left to right. This means that as x-values increase, the corresponding y-values change
in a specific direction.
B Explore examples of common functions and their respective graphs
Common functions that are frequently encountered in mathematics include linear functions, quadratic functions, exponential functions, and trigonometric functions. Let's take a look at the graphs of
these functions:
• Linear Function: The graph of a linear function is a straight line. It has a constant slope and can be represented by the equation y = mx + b, where m is the slope and b is the y-intercept.
• Quadratic Function: The graph of a quadratic function is a parabola. It is a U-shaped curve and can be represented by the equation y = ax^2 + bx + c, where a determines the direction and width of
the parabola.
• Exponential Function: The graph of an exponential function is a curve that increases or decreases rapidly. It can be represented by the equation y = a^x, where a is the base and x is the
• Trigonometric Function: The graphs of trigonometric functions such as sine and cosine exhibit periodic behavior. They oscillate between specific values and repeat their pattern over a certain
C Explain exceptions and special conditions, such as vertical asymptotes
While the characteristics mentioned above are indicative of a function graph, there are exceptions and special conditions to consider:
• Vertical Asymptotes: In some cases, a function may have a vertical asymptote, which is a vertical line that the graph approaches but does not touch. This occurs when the function's denominator
becomes zero at a certain value, resulting in an undefined point on the graph.
• Multivalued Functions: Certain functions, such as the square root function, have multiple outputs for a single input. These are known as multivalued functions and may not satisfy the one-to-one
correspondence required for a function.
• Piecewise Functions: A piecewise function consists of multiple sub-functions defined over different intervals. While each sub-function may individually satisfy the criteria of a function, the
entire piecewise function may exhibit non-function behavior in certain regions.
Real-World Applications: Where Function Identification Matters
Understanding mathematical functions and being able to identify them is not just a theoretical concept, but it has real-world applications in various fields such as physics, economics, and
engineering. The ability to recognize functions is crucial in these fields as it helps in making accurate predictions, analyzing data, and solving complex problems.
A. Illustrate how the ability to recognize functions impacts various fields like physics, economics, and engineering
In physics, the concept of functions is used to describe the relationship between different physical quantities. For example, the motion of an object can be described using mathematical functions
such as distance as a function of time or velocity as a function of time. These functions help physicists to analyze and predict the behavior of physical systems.
In economics, functions are used to model various economic phenomena such as demand and supply, production functions, and cost functions. By understanding and identifying these functions, economists
can make informed decisions, analyze market trends, and predict future outcomes.
In engineering, functions play a crucial role in designing and analyzing systems. Engineers use functions to model the behavior of mechanical, electrical, and chemical systems. For example, in
electrical engineering, functions are used to describe the relationship between voltage and current in a circuit.
B. Provide practical scenarios where identifying function graphs is crucial
Identifying function graphs is crucial in various practical scenarios. For example, in physics, when analyzing the motion of a projectile, it is important to identify the function that describes the
trajectory of the projectile. In economics, when analyzing market demand, it is essential to identify the demand function to make accurate predictions. In engineering, when designing a control
system, engineers need to identify the transfer function of the system to ensure stability and performance.
Furthermore, in data analysis and machine learning, identifying the functional relationship between variables is essential for building accurate predictive models. For instance, in predictive
maintenance of machinery, identifying the function that describes the relationship between machine performance and time can help in predicting maintenance needs and avoiding costly breakdowns.
C. Explain the implications of misidentifying graphs in these real-world situations
Misidentifying graphs in real-world situations can have serious implications. In physics, misidentifying the function that describes the motion of an object can lead to inaccurate predictions and
flawed analysis. In economics, misidentifying demand or supply functions can result in incorrect pricing strategies and inefficient resource allocation. In engineering, misidentifying the transfer
function of a system can lead to unstable control and potential safety hazards.
Moreover, in data analysis and machine learning, misidentifying the functional relationship between variables can lead to inaccurate predictive models and flawed decision-making. This can have
significant financial implications in business and industry.
Therefore, the ability to recognize functions and accurately identify function graphs is essential for making informed decisions, solving complex problems, and ensuring the reliability and efficiency
of systems in various real-world applications.
Troubleshooting: Overcoming Common Misconceptions and Errors
Understanding mathematical functions and their graphs can be challenging, and it's common for students to have misconceptions and make errors when differentiating between function and non-function
graphs. In this chapter, we will address some frequent misconceptions, provide strategies for correctly identifying function graphs, and offer tips for avoiding errors when applying the vertical line
test and interpreting complex graphs.
A Address frequent misconceptions about functions and their graphs
One common misconception about functions is that every graph represents a function. However, this is not true. It's important to understand that for a graph to represent a function, each input value
must correspond to exactly one output value. This means that a vertical line cannot intersect the graph at more than one point.
Another misconception is that a function cannot have repeated input values. While it's true that each input value must correspond to exactly one output value, it's possible for different input values
to produce the same output value. For example, the function y = x^2 produces the same output value (y) for both x = 2 and x = -2.
B Provide strategies for correctly differentiating between function and non-function graphs
One effective strategy for differentiating between function and non-function graphs is to use the vertical line test. This test involves drawing a vertical line through the graph. If the vertical
line intersects the graph at more than one point, then the graph does not represent a function. If the vertical line intersects the graph at only one point for every possible input value, then the
graph represents a function.
It's also important to understand the characteristics of common non-function graphs, such as circles and parabolas. By familiarizing yourself with these graphs and their equations, you can more
easily recognize when a graph does not represent a function.
C Offer tips on avoiding errors when applying the vertical line test and interpreting complex graphs
When applying the vertical line test, it's important to ensure that the vertical line intersects the entire graph, including any asymptotes or discontinuities. Failing to do so can lead to errors in
determining whether the graph represents a function.
Interpreting complex graphs, such as those involving piecewise functions or transformations, can also be challenging. It's important to break down the graph into its individual components and analyze
each part separately. This can help in understanding how the graph represents a function as a whole.
By addressing these frequent misconceptions, providing strategies for correctly differentiating between function and non-function graphs, and offering tips for avoiding errors when applying the
vertical line test and interpreting complex graphs, you can improve your understanding of mathematical functions and their graphs.
Conclusion & Best Practices: Ensuring Function Identification Proficiency
In this blog post, we have delved into the intricacies of identifying mathematical functions through their graphs. Let's recap the major points explored in the post about identifying function graphs,
emphasize the importance of practice and familiarity with a variety of function types, and recommend best practices for students and professionals when dealing with the identification of functions in
graphs, ensuring accuracy and confidence.
A. Recap the major points explored in the post about identifying function graphs
• Definition of a Function: We discussed that a function is a relation between a set of inputs and a set of possible outputs, where each input is related to exactly one output.
• Vertical Line Test: We highlighted the significance of the vertical line test in determining whether a graph represents a function. If a vertical line intersects the graph at more than one point,
then the graph does not represent a function.
• Types of Functions: We explored various types of functions such as linear, quadratic, exponential, and trigonometric functions, and how their graphs exhibit distinct characteristics.
B. Emphasize the importance of practice and familiarity with a variety of function types
It cannot be overstated how crucial practice and familiarity with different function types are in developing proficiency in identifying function graphs. By regularly working with graphs of various
functions, individuals can sharpen their ability to recognize patterns and characteristics specific to each function type. This familiarity enables them to quickly and accurately identify functions
in graphs, even when presented with complex or unfamiliar graphs.
C. Recommend best practices for students and professionals when dealing with the identification of functions in graphs, ensuring accuracy and confidence
• Regular Practice: Encourage students and professionals to regularly practice identifying function graphs, both through theoretical study and practical application.
• Utilize Resources: Recommend the use of textbooks, online resources, and educational software that provide a wide range of function graphs for practice and study.
• Seek Guidance: Advise seeking guidance from teachers, mentors, or colleagues when encountering challenging function graphs, as discussing and analyzing graphs collaboratively can enhance
• Develop Intuition: Suggest developing an intuition for recognizing function graphs by observing and analyzing real-world phenomena that can be represented by mathematical functions.
By following these best practices, individuals can enhance their proficiency in identifying function graphs, leading to greater accuracy and confidence in their mathematical abilities. | {"url":"https://dashboardsexcel.com/blogs/blog/understanding-mathematical-functions-how-to-know-if-graph-is-function","timestamp":"2024-11-09T02:53:19Z","content_type":"text/html","content_length":"232044","record_id":"<urn:uuid:ad161d5b-ae34-4c78-a9cd-a0835e6ab8bd>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00392.warc.gz"} |
Prevent Cell Formula from being overwritten
I have a cell formula that will change the status of the parent cell (in this case Materials) to Green, when all of its children are either Green or Gray. However, since the sheet is shared amongst
multiple people, it can be overwritten manually when someone changes the status to a different color then the properties of the formula are lost. I was wondering if there was a way to prevent the
formula from being overwritten or some way for the status to be changed freely except for when the conditions are met and then the parent status changes. Hopefully, I got my point across. Screenshot
of an example layout below.
I tried changing the cell formula to column formula to see if that would work, but it doesn't. It wipes everything in the column and cells cannot be edited.
This is the formula that I have have so far:
=IF(AND(TCode@row = "MT", COUNTIF(CHILDREN(), OR(@cell = "Green", @cell = "Gray")) = 5), "Green")
Any help is appreciated. Thank you in advance.
• Hi Lucas,
I would have suggested the Column Formula, but it sounds like that isn't working for you. I'm curious to know why it wipes everything in that column when doing so...
My other suggestion is to lock the column. However, those who are owners or admins on the sheet will still be able to edit the cells.
Hope this helps!
• You cannot have manual entry AND a formula in the same cell at the same time. One will always override and delete the other.
That is why when a user manually changes it, the formula is lost and also why applying a column formula removes the manually entered data.
• Thank you @Paul Newcome & @Heather D for the quick response.
I'm not sure why the column formula acted the way it did. I just apply the cell formula above and everything in that column is removed and cannot be edited manually. Good thing I tested this on a
copy of the original sheet. I guess column formulas cant be overwritten either
• Column formulas cannot be overwritten because the formula is applied to every cell within that column. If it is manually overwritten in even one row, then it is no longer a column formula.
Using the cell formula applies it on a row by row basis (as opposed to the entire column) but can still be overwritten/deleted by manual entry.
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/76515/prevent-cell-formula-from-being-overwritten","timestamp":"2024-11-09T00:56:14Z","content_type":"text/html","content_length":"404936","record_id":"<urn:uuid:1bbb43c6-e374-40ec-b157-00a8e04aa3da>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00560.warc.gz"} |
5 Fun Pi Day Math Activities (and 10 facts about pi)
In this post are a few Pi Day math activity ideas for celebrating 3/14 in your math classroom. I also wanted to share some fun facts about pi and Pi Day that I have collected over the years. The list
includes a link to a website where you can search the digits of pi for any string of numbers, including your phone number, birthdate or any other string of numbers you choose.
1: It took 1,000 years to prove pi irrational.
2: The first people to refer to the ratio between the diameter and circumference of a circle were the Ancient Egyptians.
3: "I prefer pi" is a palindrome.
4: Albert Einstein was born on Pi Day, 1879.
5: 22/7 is sometimes used to estimate pi.
6: The first major Pi Day celebration was in 1988.
7: Pi has been calculated past the two-quadrillionth digit.
8: P and pi are both the 16th letters in their alphabets.
You can likely find your phone number in pi. Here's the site that searches for it:
. I found mine!
10: 3.14 backwards is PIE. How strange!
To celebrate Pi Day, I created a few printable Pi Day math activities that let students get a little artsy while not losing a day of learning. Above is a
Pi Day math pennant activity
with area and circumference problems and fun pi facts on each pennant.
For younger students, there is this Pi Day math pennant for middle school with less challenging circle problems. When the pennants are solved and colored, they make a colorful, student-created
classroom display.
The 3 printable Pi Day math activities above are bundled in this discounted Pi Day activities bundle. They are also available individually.
If you're looking for something digital, here are a couple self-checking Pi Day digital math escape rooms built in Google Forms.
The Pi Day digital math escape room above is for students in 8th grade through geometry. Some questions ask students to solve for specific variables in the circle, sphere and cylinder formulas.
Students solve the 4 problems in the puzzle, then type the correct code from the answer grid. The Form is answer-validated, so each puzzle will only unlock with the correct 4-letter code. There are 5
puzzles (20 questions) to solve.
And here's one more cool fact about pi: Just about any string of digits can be found in pi. I found my phone number (without the area code), and the string 314159 (without the decimal point) occurs
175 times within the first 200 million digits of pi!
HAPPY 𝝅 DAY!
4 comments:
1. AnonymousMarch 09, 2018
The pi day activity #10 isn't "backwards" its a reflection about a veetical line. An even better math activity.
1. I like that! 314PIE
2. AnonymousMarch 14, 2018
I just wish Pi Day didn't land on spring break every year.
1. Around here we get a February break and an April break...March is a long haul! Happy vacation Pi Day! :) | {"url":"https://www.scaffoldedmath.com/2016/03/3-pi-day-activities-and-10-pi-day-facts.html?m=0","timestamp":"2024-11-06T07:03:08Z","content_type":"application/xhtml+xml","content_length":"109743","record_id":"<urn:uuid:a014e5cd-125a-4164-ab8a-d11e3b924fbe>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00501.warc.gz"} |
2008.09.19 01:00 - The inner peer review
Table of contents
No headers
Wol Euler was guardian for this session. The comments are hers. [Updated: later in the day, Umbriel gave me a link to a half-hour-long podcast related to our discussion, called Not good enough, which
I highly recommend.]
Wol Euler: hello Umbriel
Umbriel Levenque: Good morning Wol. How are you?
Umbriel Levenque: Metta
Wol Euler: metta indeed.
Umbriel Levenque: :-)
Wol Euler: I am well, thanks; thinking to make a short bathroom break before the session begins. BRB.
Umbriel Levenque: Sure
Faenik: could be
Umbriel Levenque: Metta
Corvuscorva Nightfire: Mornin'
Umbriel Levenque: Nice to see you again Corvi. How are you?
Corvuscorva Nightfire: Glad of 9 seconds.
Umbriel Levenque: You are doing 9 secs more these days Corvi?
Corvuscorva Nightfire: some...a little.
Corvuscorva Nightfire: have been for about a month.
Umbriel Levenque listens intently
Corvuscorva Nightfire: just whenever I think to do it.
Umbriel Levenque nods
Umbriel Levenque: Have you noticed any effects of the exercise in your daily life?
Corvuscorva Nightfire: Yes.
Corvuscorva Nightfire: It makes work easier...makes it easier to be useful to people.
Wol Euler: back
Umbriel Levenque: wb Wol
Wol Euler: mornin' corvi.
Umbriel Levenque: Sounds great Corvi
Corvuscorva Nightfire: Mornin' Wol.
Corvuscorva Nightfire: wb
Wol Euler looks at her watch curiously.
Umbriel Levenque: Yes Wol?
Wol Euler: Corvi is on US time, this is middle of hte night for her.
Umbriel Levenque nods
Corvuscorva Nightfire smiles ruefully...woke up at 4am. But I am glad to be here.
Wol Euler smiles. I'm always glad to see you, even at 4 am.
Umbriel Levenque smiles. Same here
Wol Euler reads back to see what she missed.
Faenik: ah :)
Wol Euler: do you find the Nines exercise also making a difference to how you see yourself, Corvi?
Corvuscorva Nightfire: Yes and no...I think I am better able to LOOK at myself...but the point of view hasn't actually changed.
Faenik loves wells!
Umbriel Levenque: Being able to look at oneself better is a good thing whether it leads to changes in point of view or not
Corvuscorva Nightfire: It's allowed...some space where i can be safe to look.
Corvuscorva Nightfire: Not alot, but just a little.
Wol Euler: ah, that sounds very good.
Wol Euler: a positive change.
Umbriel Levenque: :-)
Wol Euler: since you speak of "seeing youreslf", have you tried the Seeing exercises?
Wol Euler: You Seeing, Being Seeing.
Corvuscorva Nightfire: no...
Corvuscorva Nightfire: I haven't tried any exercises..I'm still working on breathing and checking in.
Umbriel Levenque nods
Wol Euler smiles.
Corvuscorva Nightfire: Tell me about that one.
Wol Euler: well, it is similar to the 9 seconds in that it is undirected.
Corvuscorva Nightfire: Corvi loves wells.
Umbriel Levenque: :-)
Corvuscorva Nightfire giggles and looks at Faenik.
Faenik giggles
A new variation: WS (Warthog Seeing)
Wol Euler: You Seeing is just that: you looks at the world, much as usual but without analysis or judgement. Just seeing what is.
Wol Euler chuckles belatedly.
Umbriel Levenque: :-)
Wol Euler: Being Seeing is then stepping back from yourself, so that you are in the picture of what can be seen. What does Being see?
Umbriel Levenque nods
Corvuscorva Nightfire: mmm..
Wol Euler: again without analysis or judgement
Wol Euler: just what _is_
Umbriel Levenque: Observing...
Wol Euler: that's the warthog version of YSBS, Pema could express it wiht a great deal more poetry.
Umbriel Levenque: He often talks about 'self stepping aside'
Wol Euler nods
Umbriel Levenque: For me, YSBS came gradually after 9 sec
Corvuscorva Nightfire: I love warthogs.
Wol Euler: very apt, actually, I often find myself blocking my own way.
Umbriel Levenque listens
Wol Euler nods to Umbriel. Actually, for me ti was the other way around. YSBS relaxed me to the point that I could begin with the Nines.
Umbriel Levenque: ah... :-)
Corvuscorva Nightfire: It sounds sort of like what I DO when I'm doing the nines...but I am not able to do the Being seeing part.
Umbriel Levenque: (It was easy for me to do 9 sec first because of doing regular meditation)
Corvuscorva Nightfire: I'm very happy with just....Seeing.
Umbriel Levenque nods
Wol Euler: mmhmm
Wol Euler: Corvi, it will probably come in its own time.
Wol Euler: and if not, that is fine too :-)
Umbriel Levenque nods
Umbriel Levenque quotes Wol "Let Being be"
Wol Euler: no need to make this into another "should"
Wol Euler: ha, yes.
Umbriel Levenque nods
"Should" raises its ugly head, and we beat it with sticks.
Wol Euler: (actually, avoiding "should" in speech, especially talking to yourself, is a really good exercise too.)
Corvuscorva Nightfire: mmmhmm.
Umbriel Levenque nods. A slippery fish... 'should'
Wol Euler: phone brb
Umbriel Levenque: Sure
Corvuscorva Nightfire: what happens for me...is that should becomes a pointer...talking about where I am being silly.
Umbriel Levenque: Please give us an example Corvi
Faenik: why not?
Corvuscorva Nightfire: If I say to myself..I should do the nines right now...then I know I am beating myself up for not doing it..
Corvuscorva Nightfire: rather than enjoying the process.
Umbriel Levenque nods.
Wol Euler sneaks back in quietly and listens.
Faenik: why not?
Umbriel Levenque: For that example, I suppose that you have a fixed goal of some sort doing 9 sec? Or possibly how much you'd like to do any so on...
Umbriel Levenque: and so on* ... um typo
Corvuscorva Nightfire: I'm trying not to have any expectations around it.
Umbriel Levenque: :-)
Umbriel Levenque: Hence no 'should' for doing 9 sec
Faenik: ah :)
Wol Euler: no expectations is good, just do it as and when the desire occurs.
Corvuscorva Nightfire nods.
Umbriel Levenque: :-)
Wol Euler: I've noticed that "should" addressed to myself usually means "you screwed up somewhere" or "you're not good enough"
Umbriel Levenque welcomes his friend "you're not good enough" these days
Corvuscorva Nightfire: mmm...well said.
Wol Euler: ah :)
Faenik: why not?
Umbriel Levenque: Well... used to argue with him but now a gentle talk :-)
Wol Euler: probably much better, anger is not a good guide
Umbriel Levenque nods
Umbriel Levenque: Well it could be but not when talking to such a friend... need to be more alert :P
Wol Euler: "tough love"?
Umbriel Levenque nods
Umbriel Levenque: A balancing act really...
Umbriel Levenque: (Need to listen to all sides of the discussion)
Wol Euler: like an internal peer review, perhaps.
Umbriel Levenque: :-)
Umbriel Levenque usually sets up a round table for such an occassion
Wol Euler grins.
Corvuscorva Nightfire laughs.
Umbriel Levenque: I tried to 'step aside' from self and see all sides equally. Helps me to see better...
Corvuscorva Nightfire: My inner peers are a bit...cruel.
Umbriel Levenque: try*...
Umbriel Levenque: They can be yes
Wol Euler nods.
Faenik loves wells!
Wol Euler: I've been thinking about Umbriel's friend. That's a point of view that I hadn't considered, I realize.
Umbriel Levenque: Some of them have mellowed out over the years though :-)
Umbriel Levenque listens
Wol Euler: I think of hte voice of "should" as an opponent, almost an enemy. In many cases, not always.
Wol Euler: Someone who speaks to put me down, keep me in my place.
Umbriel Levenque: ah... not so... it's me, my friend... fighting against myself... a bit tiring
Wol Euler: But that isn't always true, perhaps not even in the majority of cases.
Wol Euler: viewing "should" as hostile was a breakthrough at first, many years ago, because it enabled me to overrule that voice
Wol Euler: but it has become anohter unexamined habit of mind. It's time I looked at that again.
Umbriel Levenque: :D
Umbriel Levenque: My 'inner peers' more effective in helping me when I don't brush them aside...
Wol Euler: you trust yours, then.
Umbriel Levenque: They are all concerned really... good intentions however much they often get it wrong at times
Umbriel Levenque nods. Yes I trust their friendship
Wol Euler: like clumsy children.
Umbriel Levenque nods
Corvuscorva Nightfire nods.
Wol Euler smiles. A nice image, I'll think about htat.
Umbriel Levenque: :-)
Umbriel Levenque imagines his peers sitting around the table with the lollipop Moon gave him
Wol Euler: heheheh
Umbriel Levenque: :-)
Umbriel Levenque: Possibly they are all doing 9 sec too!
Wol Euler: lol
Umbriel Levenque: :-)
Umbriel Levenque: Good to talk to you both again. Wish you a good day.
Umbriel Levenque: Metta
Wol Euler: bye umbriel, take care.
Corvuscorva Nightfire: bye!
Corvuscorva Nightfire: Enjoy your day.
Wol Euler: well, we are close to the hour. Shall we stop and go to my beach? have you seen it lately?
Corvuscorva Nightfire shakes her head..."nope"
Wol Euler: then I declare this session officially ended!
You must
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Max Flow and Min Cut: The Cornerstones of Network Analysis
1. Max Flow and Min Cut: The Cornerstones of Network Analysis
The Max Flow-Min Cut Theorem in Network Analysis: Unlocking the Flow of Information
September 06, 2023
Beatrice Evelyn Sinclair
United States Of America
Network Flow Theory
Throughout her academic career, Beatrice has been particularly intrigued by the Max Flow-Min Cut Theorem—a theorem that has shaped the way we understand and optimize networks in various fields.
Networks are all around us, from the internet that connects the world to transportation systems that move goods and people. Understanding how information or resources flow through these networks is
crucial in various fields, ranging from computer science to economics. One fundamental concept that plays a pivotal role in network analysis is the Max Flow-Min Cut Theorem. In this blog, we will
delve deep into this theorem, exploring its history, significance, applications, and real-world implications to help you complete your math assignment.
Understanding Network Flows
Before we dive into the Max Flow-Min Cut Theorem, let's establish a foundational understanding of network flows. A network, in this context, consists of nodes (vertices) and edges (links) connecting
these nodes. These edges have capacities, representing the maximum amount of "stuff" that can flow through them. This "stuff" could be data in a communication network, water in a pipeline, or goods
in a transportation network.
Network flow problems typically involve finding the most efficient way to send this stuff from a source node to a target node while respecting the capacity constraints on the edges. This is often
referred to as the "maximum flow" problem. Imagine it as finding the optimal path for water to flow through a network of pipes, ensuring no pipe is overloaded.
Historical Context
The roots of network flow problems can be traced back to the early 20th century when electrical engineers worked on designing efficient networks for transmitting electricity. However, it was in the
mid-20th century that the field gained prominence in computer science and operations research.
The Max Flow-Min Cut Theorem is a culmination of several key developments in this period. It was first introduced by Lester R. Ford Sr. in the context of electrical networks in 1956. The theorem, as
we know it today, emerged when it was independently rediscovered and significantly generalized by Ford Fulkerson in 1956. Fulkerson's work laid the foundation for algorithmic approaches to solving
network flow problems, which are now an integral part of computer science and operations research.
The Max Flow-Min Cut Theorem
The Max Flow-Min Cut Theorem is a fundamental and elegant result in the field of network flow theory, and it provides crucial insights into the flow of resources or information in various networked
systems. To truly appreciate the theorem's significance, it's essential to delve deeper into the concept of network cuts and understand how they are related to the flow of "stuff" within a network.
What is a "Cut" in a Network?
In the context of network theory, a "cut" is a fundamental concept that plays a pivotal role in understanding and analyzing networks. To define a cut, let's start by visualizing a network as a
collection of nodes (vertices) and edges (links) connecting these nodes. These edges are often associated with capacities, representing the maximum amount of "stuff" that can flow through them.
Now, imagine we want to create a partition within this network, dividing its nodes into two distinct sets: one set known as the "source side" and the other as the "sink side." The idea is to create a
separation in the network that isolates the source from the sink. A cut, in essence, accomplishes this partition.
Components of a Cut
1. Source Side (S): This set contains nodes from which "stuff" originates, typically referred to as the source nodes. In network terminology, it represents the side of the network where resources or
information initially enters.
2. Sink Side (T): On the other side of the cut, we have the sink side, containing nodes that are the ultimate destination for the "stuff" or resources. Sink nodes are where the flow terminates.
Capacity of a Cut
Now, the capacity of a cut is a crucial parameter. It is defined as the sum of the capacities of all the edges that cross from the source side to the sink side. In simpler terms, the capacity of a
cut represents the maximum amount of "stuff" that can flow from the source to the sink when all other paths between the source and sink are blocked or cut off.
The Max Flow-Min Cut Theorem Statement
The Max Flow-Min Cut Theorem states:
In any network, the maximum flow from the source to the sink is equal to the minimum cut capacity.
In mathematical terms, if F is the maximum flow in a network, and C is the capacity of the minimum cut, then F=C.
This theorem has profound implications, and it underpins many algorithms and applications in network analysis.
Implications and Significance
The Max Flow-Min Cut Theorem is a fundamental concept in network theory with far-reaching implications and significant real-world applications. It provides deep insights into the flow of resources,
information, or goods within various networked systems. Let's explore its implications and significance in greater detail:
Transportation Networks
In transportation systems, whether it's a network of roads, railways, or air routes, optimizing the flow of goods and people is crucial. The Max Flow-Min Cut Theorem enables logistics and
transportation companies to determine the most efficient routes for transporting goods from suppliers to consumers. By modeling the transportation network and applying the theorem, it becomes
possible to minimize costs, reduce transit times, and maximize overall throughput. This optimization has a direct impact on reducing transportation expenses, environmental impact, and improving
customer satisfaction.
Communication Networks
In the realm of communication networks, such as the internet, the efficient transmission of data is paramount. The theorem plays a pivotal role in routing data packets optimally. By using the Max
Flow-Min Cut Theorem, network engineers can minimize delays and congestion, ensuring that data is transmitted through the network with the least possible disruption. This is particularly crucial in
today's digital age, where uninterrupted data flow is essential for everything from streaming services to remote work and online gaming.
Designing Reliable Networks
The robustness and reliability of networks are critical, especially in applications where disruptions can have severe consequences.
Identifying Critical Points
Understanding the minimum cut capacity within a network allows engineers to identify critical points or vulnerabilities in the network. These are locations where failures or disruptions would have
the most significant impact on the flow of resources or information. By pinpointing these weak spots, network designers can take proactive measures to strengthen these areas. This may involve
creating redundant paths, implementing failover mechanisms, or ensuring the availability of backup resources.
Ensuring Uninterrupted Flow
The knowledge gained from the Max Flow-Min Cut Theorem helps in the design of resilient systems that can continue to function even in the face of failures. For example, in electrical power grids,
identifying critical substations and ensuring alternative power routes can help prevent widespread blackouts during equipment failures or natural disasters. In the context of data centers and cloud
computing, it aids in designing fault-tolerant architectures that provide uninterrupted services.
Bipartite Matching
Bipartite matching is a classic problem in graph theory and combinatorial optimization. It has wide-ranging applications, and its solution can be reduced to a network flow problem through the Max
Flow-Min Cut Theorem.
Educational Systems
In educational systems, the theorem can be applied to match students to schools or courses efficiently. It helps in optimizing student placements while considering factors like student preferences
and school capacities. This ensures that students are allocated to schools in a way that maximizes overall satisfaction and resource utilization.
Labor Markets
In labor markets, the theorem aids in matching job seekers with job openings. It helps job placement agencies and online job platforms optimize their matching algorithms, ensuring that job candidates
are connected with suitable employment opportunities. This not only benefits job seekers but also enhances the efficiency of labor markets by reducing job vacancies and unemployment.
Image Segmentation
In computer vision and image processing, the Max Flow-Min Cut algorithm is employed for image segmentation. This process involves partitioning an image into distinct regions, often separating objects
from their backgrounds.
Object Recognition
By creating an image as a graph with pixels as nodes and edges representing the similarity between pixels, the Max Flow-Min Cut algorithm identifies optimal cuts that delineate objects within the
image. This is crucial for applications such as object recognition, where distinguishing objects within an image is a foundational task. For example, in medical imaging, the algorithm can help
identify and isolate tumors within MRI scans.
Video Compression
Image segmentation is also used in video compression, where it helps isolate moving objects from static backgrounds. This results in more efficient video compression algorithms, reducing bandwidth
requirements for video streaming and storage.
Game Theory and Economics
In game theory and economics, the Max Flow-Min Cut Theorem plays a role in modeling and analyzing flows within various networks that represent trade, resource allocation, and communication between
Resource Allocation
The theorem aids in understanding how resources are allocated and distributed in complex economic systems. It can model supply chains, financial networks, and trade routes, enabling economists and
policymakers to analyze the efficient allocation of resources and the impact of trade policies.
Equilibrium Analysis
In game theory, the theorem contributes to equilibrium analysis, helping to identify Nash equilibria and optimal strategies in scenarios involving strategic interactions. It provides insights into
how agents in a networked system make decisions to maximize their utility or achieve balance.
The Max Flow-Min Cut Theorem is a foundational concept with profound implications across diverse fields. Its ability to optimize network flows, enhance network reliability, facilitate efficient
matching, assist in image segmentation, and inform economic and game-theoretic analyses underscores its significance in our increasingly interconnected world. By understanding and applying this
theorem, we can solve complex problems, improve resource allocation, and design more efficient and resilient networks.
Algorithms for Solving the Max Flow Problem
The Max Flow-Min Cut Theorem wouldn't be as impactful without efficient algorithms to compute the maximum flow. One of the most well-known algorithms is the Ford-Fulkerson algorithm, named after its
inventors. This algorithm iteratively finds augmenting paths in the residual graph (a modified version of the original graph) and increases the flow until no more augmenting paths can be found. While
this algorithm is conceptually simple, its efficiency depends on the choice of augmenting paths, which can lead to varying results.
Another widely used algorithm is the Edmonds-Karp algorithm, a variation of Ford-Fulkerson that always selects the shortest augmenting paths. This ensures that the algorithm terminates in a finite
number of iterations, making it more predictable and efficient.
Limitations and Complexity
While the Max Flow-Min Cut Theorem is a powerful tool, it's important to note that solving the maximum flow problem in a network is not always easy. In fact, it is an NP-hard problem, which means
that there is no known algorithm that can solve it in polynomial time for all possible inputs.
Furthermore, networks can be dynamic, and the flow requirements may change over time. Adapting flow solutions in real-time adds another layer of complexity, leading to research in dynamic flow
Real-World Applications
The Max Flow-Min Cut Theorem has far-reaching implications in various real-world scenarios. Let's explore a few examples:
Transportation Networks
In logistics and transportation planning, the theorem is used to optimize the movement of goods. Consider a network of roads and highways connecting warehouses, factories, and retailers. By modeling
this network and applying the Max Flow-Min Cut Theorem, companies can determine the most efficient routes for delivering products, reducing transportation costs and delivery times.
In telecommunications, the theorem aids in optimizing data transmission. Telecommunication networks use this concept to route data packets efficiently, ensuring minimal latency and congestion. In the
era of streaming video, online gaming, and remote work, this efficiency is crucial for delivering a seamless user experience.
Hospitals and healthcare systems rely on efficient patient flow management. The Max Flow-Min Cut Theorem helps healthcare providers optimize the allocation of resources, such as operating rooms and
beds, to ensure patients receive timely care while minimizing costs.
Social Networks
In the realm of social networks and online communities, the theorem can be applied to identify influential nodes and study information propagation. Understanding how information spreads through a
network is invaluable for marketing, political campaigns, and the study of viral content.
Challenges and Future Directions
As our world becomes more interconnected, network analysis continues to play a crucial role in various domains. However, the complexity of real-world networks presents ongoing challenges. Some key
areas of research and development include:
Efficiently solving large-scale network flow problems is a continuous challenge. Researchers are exploring distributed and parallel algorithms to handle massive networks that exceed the capacity of
traditional algorithms.
Dynamic Networks
In dynamic environments, where network structures and flow requirements change over time, dynamic flow algorithms are essential. These algorithms must adapt quickly to evolving conditions, such as
changing traffic patterns on the internet or fluctuations in demand in supply chains.
Multi-Objective Optimization
Real-world problems often involve multiple objectives, such as minimizing costs, maximizing throughput, and ensuring resilience. Developing algorithms that can handle multi-objective optimization in
network flows is an active area of research.
Privacy and Security
In an era of heightened concern about privacy and security, protecting sensitive information in network flows is critical. Researchers are working on encryption techniques and secure data
transmission methods within the framework of network flow theory.
The Max Flow-Min Cut Theorem, a cornerstone of network flow theory, has far-reaching implications across various fields, from transportation and telecommunications to healthcare and social networks.
Its elegant mathematical foundation and practical applications have made it an indispensable tool for optimizing and understanding the flow of information and resources in complex systems.
As we continue to grapple with the challenges of an interconnected world, the Max Flow-Min Cut Theorem serves as a reminder of the power of mathematical insights in solving real-world problems and
shaping the future of technology and society. It is a testament to the enduring relevance of mathematical concepts and their profound impact on our daily lives. | {"url":"https://www.mathsassignmenthelp.com/blog/a-deep-dive-into-max-flow-min-cut-theorem-network-analysis/","timestamp":"2024-11-02T02:10:17Z","content_type":"text/html","content_length":"87370","record_id":"<urn:uuid:54a5dd15-1eba-4273-a464-6bbce3c05dd1>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00097.warc.gz"} |
CPM Homework Help
Find the least common multiple and the greatest common factor of the following pairs of numbers. In other words, find the smallest number that is a multiple of both of the numbers in the pair.
a. $3$ and $24$
Use the Lesson 7.2.2 Math Notes box to help you with these problems.
b. $7$ and $9$
Again, refer to the Math Notes box.
$7$ is a prime number, which means its only factors are 1 and itself.
c. $15$ and $12$
Refer to the Math Notes box.
LCM is $60$. Can you find the greatest common factor? | {"url":"https://homework.cpm.org/category/CON_FOUND/textbook/mc1/chapter/7/lesson/7.2.3/problem/7-70","timestamp":"2024-11-03T00:31:25Z","content_type":"text/html","content_length":"40122","record_id":"<urn:uuid:e201498a-05af-4917-bafa-5ad205dab304>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00542.warc.gz"} |
Fundamental AI Concepts
June 24, 2019
Fundamental AI Concepts
A Review Stimulated by the Linley Group
We hear a lot about artificial intelligence (AI) these days. The mentions are typically delivered in acronym salads that include machine learning (ML), deep neural nets (DNNs) and the other various
NNs (convolutional – CNNs, recurrent – RNNs, etc.), and a pile of other technologies. Many of these are accessible only to specialists; the rest of us nod our heads sagely, pretending to understand,
hoping no one calls us out.
But at this year’s Linley Spring Processor Conference, Linley Gwennap did a keynote that looked at many of the technological vectors that are being traversed and what’s pushing them in that
direction. The presentation took a step back and focused on some high-level concepts underpinning what makes AI work.
While his focus was on how the technology was evolving, I was also intrigued by the collocation of so many fundamental concepts – some going by names we non-specialists don’t necessarily use on a
daily basis. So, while I’m not going to parrot the specific points made in the presentation (you’ll have to get the proceedings for that), I thought it useful to walk through the basic notions, in
the hope that any new nuggets you find for yourself will be helpful.
The order in which I’m going to present these in has nothing to do with anything other than the rough order of their appearance in the presentation. So, with that, let’s take a look.
This term actually wasn’t a part of the presentation itself. But it’s such a fundamental concept that it’s worth starting here. It’s a word that doesn’t show up often outside specialist circles (at
least as far as I’ve seen). As a result, when you do run across it, it can throw you – or send you running to the web — which is where I went when I first encountered it.
The perceptron came about as an alternative to the neural-net model that more closely resembles what happens in our brains: the spiking neural net, or SNN. In SNNs, data is encoded using spikes.
Either the frequency or timing between spikes is used for that encoding. The math used for inference processing is simple, but this is hard to manage for standard supervised learning. There’s no easy
way to take derivatives or do back-propagation, for example. And, so far, they don’t scale well.
So the perceptron, while less brain-mimetic, makes it easier to accomplish practical ML. The idea is that, if a function crosses a threshold, it gets classified within some category; otherwise it
doesn’t. The result is binary that way.
Mathematically, it looks like this:
Here w and x are vectors, with w representing weights and x representing the inputs (or activation*). The b is a “bias” or threshold. The operator you see there is the dot product (which is ancient
history for me), meaning that it’s the scalar result of the following function for vectors of length n:
Yeah, basic review, but one thing becomes screamingly clear from this: you multiply a bunch of vector entries and add them all together. In other words, this is where the popularity of the
multiply-accumulate (MAC) function comes from. Of course, you hear a lot about matrices rather than vectors, but matrix multiplication merely builds on this, with columns and rows acting as the
Systolic Arrays
This is a mysterious-sounding notion that, for me, invokes arcane math and science that hasn’t made it to the mainstream enough for this to be a common term. Maybe that’s just me, but it turns out
that, at least conceptually, this is a natural notion for neural nets – and matrix multiplication in particular.
We talked about a vector dot product, but, in reality, we’re not talking about vectors, which are one-dimensional arrays. We’re not really even talking about matrices, which are two-dimensional
arrays. We’re talking about multi-dimensional arrays, aka tensors. Whence the framework name TensorFlow.
In reality, tensors get broken up into multiple matrices for calculation (in the way that matrices get broken up into rows and columns), so the problem degenerates to matrix multiplication. Using a
basic CPU for that is pretty cumbersome, since you re-use the computing engine over and over for each of the n x m MAC calculations. That’s why GPUs or DSPs with high parallelism do better.
Even so, with parallelism that’s less than the sizes of the matrices, you end up bringing in inputs and weights for the parts you’re going to work on; they come from memory. Your partial results then
have to go back into memory, where they’ll be recalled later when the processor gets to a future part that needs those results. That memory movement is a huge time and power waster.
Systolic arrays are one approach to making the math more efficient. You can think of generic systolic arrays as matrices of functions, or rather of a single function, since they’re typically
monolithic in that way. How the “entries” in the matrix are interconnected can vary, but, for our purposes, a typical connection scheme is shown below.
The tendency of data or cumulative operations to flow through the array gives rise to the name systolic. It’s by analogy to the pulses of blood that course through our arteries. You may be familiar
with it as the first number in your blood pressure reading.
For our application, the function f in the figure above would typically be a MAC function. One way to accomplish this is to store the weights in their respective cells in the array. Such a processing
element would look like the following figure (where PSoP stands for partial sum of products from the cells above).
In this case, all the weights are preloaded, and each input column from the left can be available simultaneously to each array column. You then cycle through the input columns. You could operate this
such that the sums happen combinatorially, and, given time for the sums to propagate down, you’d end up with complete sums of products at the bottom of each array column.
Rather than preloading weights, you could stream weights in from the top and inputs from the left. Propagation right and down would happen with each clock cycle. Both input and weight matrices would
be fed, with suitable clock-cycle delays, into the left (for inputs) and the top (for weights) (although, mathematically, you could exchange the weights and inputs in this view). The element here
could look more like this:
The end result would be that each array cell would contain the entry for the product array.
Google is said to have put together the first serious use of a systolic array in their tensor processing unit, or TPU.
Given that data movement is such a problem during inference, yet another approach has been proposed: using memory itself for computing. This, of course, assumes an array of memory cells for
computation. There are a couple of approaches to this, the most direct being an analog one.
You may recall our discussion on RRAMs (or ReRAMs) a long time ago. The memory works by forming filaments between two electrodes. Thinking in a binary way, the presence of a filament is one state;
the absence is the other state.
But here’s the thing: you can have partial filaments. The conduction will be proportional to the “amount” of filament in place. So you can use this as an analog memory. If you store the weights as
the memory and then apply the inputs to the memory as an address, you’ll get current through each cell inversely proportional to the resistance in the cell – or proportional to the conductance.
So you store the weight as a conductance, not a resistance. If you then sum all of the currents in a column, you’ve got a sum of products.
We discussed in-memory computing in much more detail in a piece from earlier this year, including the fact that, if you binarize the numbers, you can potentially save lots of power and area. Yeah, I
know, we go from binary to analog back to binary. But the thing is that the first “binary” is used to store a fixed-point number; moving to analog effectively stores a real number; and the last
“binary” stores a binary number. So it’s not like we’re going backwards.
Suppressing Zeroes
These matrices can be pretty sparse. In fact, one optimizing technique is specifically to keep the matrices sparse. That means that there are lots of zero entries (or near-zero entries). These don’t
contribute at all (or, if near-zero, appreciably) to the sum of products. So why calculate them? Exactly! Don’t calculate them; suppress them. We saw an example of this earlier in coverage of
Cadence’s neural-net accelerator.
Next we look at a new real-number format largely being used by Intel and Google. It cuts the length of a 32-bit floating-point value in half. Yeah, there’s a standard 16-bit floating-point number
already, but it can’t express the same range of values that a 32-bit floating-point number can.
32-bit numbers use an 8-bit exponent and a 24-bit mantissa (or 23 bits plus one sign bit). The exponent gives you the approximate range of numbers you can express (±28), while the mantissa gives you
the precision available within that range.
If you want a standard 16-bit value, the standard format has 5 exponent bits and 11 (or 10 + sign) mantissa bits. The problem here is that you can no longer express numbers in the 28 range; you get
only ±25 – literally off by three orders of magnitude.
The bfloat format is also 16 bits, but it keeps the 8-bit exponent used in the 32-bit version and simply shortens the mantissa to 7 bits. That means you can still express (roughly) the same range;
you just do so with less precision. Obviously, this format doesn’t comport with standard math libraries, so converting to and from integer values has to be handled on a custom basis. But it provides
for faster computation and less memory use.
Activation Functions
Finally, our formula above defining a perceptron takes the dot product and then decides, based on a desired threshold, whether the result is 1 or 0. In fact, the bias b in the perceptron equation
reflects a non-zero threshold. But how do we take this sum of products and decide whether or not it meets the threshold?
An obvious way to do it would be to apply a step function to the result, with the step occurring at the threshold value. But, for training purposes, you need the ability to take a derivative and to
back-propagate (just as we saw when using a perceptron instead of an SNN).
If you look at the Wikipedia page on activation functions, it lists a huge number of possible candidates. More common are sigmoid functions (so-called because of their s-like shape – although, to be
fair, it’s not a sigma-like shape…), and, for that differentiability thing, the hyperbolic tangent is a popular sigmoid.
The important thing about these functions is that they’re good candidates for acceleration, since whichever function is chosen will be used a lot.
And that will wrap up this survey of neural-net notions, at least as I understand them. As neural nets complete their exploratory life and enter into a standardized life (if that ever happens), much
of this may be abstracted away. For now, these are useful notions for anyone trying to dip their toes into practical implementations of AI.
*You’ll sometimes read that the output is an activation. That’s because the output of one layer is the input to the next layer.
More info:
2019 Linley Spring Processor Conference proceedings (requires submittal of a form)
Video of Linley Gwennap’s presentation (no registration required; in four parts)
Sourcing credit:
Linley Gwennap, President, The Linley Group
3 thoughts on “Fundamental AI Concepts”
1. What other fundamental neural-net concepts do you find to be important?
1. Hi all,
it is very sad to see and read that all about neural nets is math, such a pity.
Neural nets are NOT math at all, just think of all the animals with a brain,
which can perform miraculous things, like flying or even flying blind,
but they never attended any math course, let alone read / write, or even speech.
When taking a biological nervous system, BNS, as a template,
then one can see that AI is neither statistics nor mathematics,
It is more than that and it takes a lot of interdisciplinary knowledge,
in order to understand our own existence, let alone replicate it.
Our company has developed a new technology by using own neural nets as an ANS,
means a collection of neural nets with the ability to build higher forms of organization,
it can be used as an Artificial Nervous System, with ANS as its acronym, while
connecting it into an embedded system with sensors, actors and internal organs.
Just take a look at our website and read especially the corresponding blog,
which contains next to some detailed explanations to this topic, means AI,
some technological papers, which can be downloaded, but be aware, it is not easy.
2. I’m sorry if I wasn’t clear enough about the context. Of course this math doesn’t explain biological neural nets; it’s all about how AI is doing things now, which departs from biomimetic models
because we don’t have good tools today for doing that. None of this is meant to detract from the marvel that is the brain of any organism.
You must be logged in to post a comment. | {"url":"https://www.eejournal.com/article/fundamental-ai-concepts/","timestamp":"2024-11-13T12:56:44Z","content_type":"text/html","content_length":"194496","record_id":"<urn:uuid:92d47bba-7895-483a-aec9-7594470a77f0>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00208.warc.gz"} |
If your body has an average density of 989 kg/m3, what fraction
of you will be...
If your body has an average density of 989 kg/m3, what fraction of you will be...
If your body has an average density of 989 kg/m3, what fraction of you will be submerged when floating gently in fresh water? What fraction of you will be submerged when floating gently in salt
water, which has a density of 1027 kg/m3?
If p is the fraction of your body submerged, then the submerged part is can be expressed as
The force pusing upward is equal to force pushing downward
p*V*d[water]*g = (d[you])*V*g
Here, V is volume, g is gravity, and d represents density
Density of fresh water, 1000 kg/m^3
p = d[you] / d [water] = 989 kg/m^3 / 1000 kg/m^3 = 0.989
In salt water,
p = d[you] / d [water] = 989 kg/m^3 / 1027 kg/m^3 = 0.9629
In three significant digits
p = 0.963 | {"url":"https://justaaa.com/physics/345998-if-your-body-has-an-average-density-of-989-kgm3","timestamp":"2024-11-05T23:19:18Z","content_type":"text/html","content_length":"39272","record_id":"<urn:uuid:3d294f78-ed13-4ddf-871e-ca91b65e3489>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00068.warc.gz"} |
VLSI implementation of OFDM modem
by Aseem Pandey, Shyam Ratan Agrawalla & Shrikant Manivannan
from Wipro Technologies
OFDM is a multi-carrier system where data bits are encoded to multiple sub-carriers, while being sent simultaneously. This results in the optimal usage of bandwidth. A set of orthogonal sub-carriers
together forms an OFDM symbol. To avoid ISI due to multi-path, successive OFDM symbols are separated by guard band. This makes the OFDM system resistant to multi-path effects.
Although OFDM in theory has been in existence for a long time, recent developments in DSP and VLSI technologies have made it a feasible option. Many wired and wireless standards like DVBT, DAB, xDSL
and 802.11a have adopted OFDM.
This paper first lists various approaches to implement an OFDM system. It then describes the VLSI implementation of OFDM in details. Specifically the 802.11a OFDM system has been considered in this
paper. However, the same considerations would be helpful in implementing any OFDM system in VLSI.
OFDM is a multi-carrier system where data bits are encoded to multiple sub-carriers. Unlike single carrier systems, all the frequencies are sent simultaneously in time. OFDM offers several advantages
over single carrier system like better multi-path effect immunity, simpler channel equalization and relaxed timing acquisition constraints. But it is more susceptible to local frequency offset and
radio front-end non-linearities. The frequencies used in OFDM system are orthogonal. Neighboring frequencies with overlapping spectrum can therefore be used. This property is shown in the figure
where f1, f2 and f3 orthogonal. This results inefficient usage of BW. The OFDM is therefore able to provide higher data rate for the same BW
OFDM is fast gaining popularity in broadband standards and highspeed wireless LAN.
OFDM transceiver
Each sub-carrier in an OFDM system is modulated in amplitude and phase by the data bits. Depending on the kind of modulation technique that is being used, one or more bits are used to modulate each
sub-carrier. Modulation techniques typically used are BPSK, QPSK, 16QAM, 64QAM etc. The process of combining different sub-carriers to form a composite time-domain signal is achieved using Fast
Fourier transform. Different coding schemes like block coding, convolutional coding or both are used to achieve better performance in low SNR conditions. Interleaving is done which involves assigning
adjacent data bits to non-adjacent bits to avoid burst errors under highly selective fading. Block diagram of an OFDM transceiver is shown below.
Figure 1: Block diagram of the 802.11a OFDM transceiver
Different implementation techniques
Figure 1 shows an OFDM transciever. Following choices are available for imple menting an OFDM system.
• DSP based implementation
• DSP based implementation with hardware accelerators
• VLSI implementation
The pros and cons of each approach are explained in the following sections.
DSP based implementation
High performance Digital Signal Processors are widely available in the market today. The computer-intensive and time critical functions that were traditionally implemented in hardware are now being
implemented in the software running on these processors.
Implementing the entire OFDM transceiver in software on DSPs is thus an option to be considered for some applications.
It has the following advantages:
• Reduced development time and quick prototyping. Quick time to market.
• Flexibility. It can quickly adapt to changing or different standards as it needs only a software change.
• Ideal for multi-mode Basebands where multiple standards are supported by the same device
DSP based implementation has the following disadvantages:
• Not very optimum in terms of area and power consumption
• High MIPS requirement.
The approximate MIPS requirement for different blocks in OFDM is given below.
Module MIPS
Viterbi decoder 4000
FFT 500
NCO 120
Interleaver 150
Channel compensation 100
Scrambler & others 50
The total MIPS requirement is 4500+. Such high CPU power is not available even with the fastest DSPs in the market today. One way out is parallel processing with multiple DSPs as shown in figure
Figure 2: DSP solution
DSP with hardware accelerators
To overcome the MIPS limitation and yet to retain the flexibility of software implementation, some blocks can be implemented in H/W. Figure 3 shows an implementation which can reduce the MIPS
requirement by around 4000 MIPS.
Figure 3: DSP + H/W accelerators
VLSI implementation
Figure 4: VLSI Implementation
In the approach shown in Figure 4 the entire functionality is implemented in hardware.
Following are the advantages of this approach:
• Lower gate count compared to DSP+RAM+ROM, hence lower cost
• Low power consumption
Due to the advantages mentioned above a VLSI based approach was considered for implementation of an 802.11a Baseband. Following sections describe the VLSI based implementation in details.
Design Methodology
The design approach for the OFDM modem is slightly different than a typical ASIC flow. Early in the development cycle, different communication and signal processing algorithms are evaluated for their
performance under different conditions like noise, multipath channel and radio non-linearity. Since most of these algorithms are coded in “C” or tools like Matlab, it is important to have a
verification mechanism which ensures that the hardware implementation (RTL) is same as the “C” implementation of the algorithm. The flow is shown in the Figure 5.
Figure 5: Design flow for Baseband development
Architecture definition
Following points need to be considered in the architecture definition phase.
Specifications of the OFDM transceiver
• Data rates to be supported
• Range and multipath tolerance
• Indoor/Outdoor applications
• Multi-mode: 802.11a only or 802.11a+HiperLAN/2
Design trade-offs
• Area – Smaller the die size lesser the chip cost
• Power – Low power crucial for battery operated mobile devices
• Ease of implementation – Easy to debug and maintain
• Customizability – Should be customizable to future standards with variations in OFDM parameters
Algorithm survey & simulation
The simulation at algorithmic level is to determine performance of algorithms for various non-linearities and imperfections. The algorithms are tweaked and fine tuned to get the required performance.
The following algorithms/parameters are verified.
• Channel estimation and compensation for different channel models (Rayleigh, Rician, JTC, Two ray) for different delay spreads
• Correlator performance for different delay spreads and different SNR (AWGN model)
• Frequency estimation algorithm for different SNR and frequency offsets
• Compensation for Phase noise and error in Frequency offset estimation
• System tolerance for I/Q phase and amplitude imbalance
• FFT simulation to determine the optimum fixed-point widths
• Wave shaping filter to get the desired spectrum mask
• Viterbi BER performance for different SNR and traceback length
• Determine clipping levels for efficient PA use
• Effect of ADC/DAC width on the EVM and optimum ADC/DAC width
• Receive AGC
Fixed point simulation
One of the decisions needs to be taken early in the design cycle is the format or representation of data. Floating point implementation results in higher hardware costs and additional circuits
related with normalizing of numbers. Floating point representa tion is useful when dealing with data of different ranges. However, this is not true as the Baseband circuits have a fair idea of the
range of values that they will work on. So a fixed-point representation will be more efficient. Further in fixed point a choice can be made between signed and 2's complement representation.
The width of representation need not be constant throughout the Baseband and it depends on the accuracy needed at different points in transmit or receive path. A small change in the number of bits in
the representation could result in a significant change in the size of arithmetic circuits especially multipliers.
Module Width Gate count
Complex 12 6K
Multiplier 16 10K
FFT (Radix-4 with 3 complex multipliers) 12 24 K (excluding RAM)
16 36 K (excluding RAM)
Shown below is the loss of SNR due to the decrease in the width of representation.
Module Width SNR dB (Signal to Quantization noise ratio)
ADC 8 48
Simulations for different bit-widths tell us which is the optimum bit-width that main tains the required level of accuracy. Significant area and power savings could be made if accurate estimation of
fixed-point widths is made. Simulations are performed to determine the required precision.
Simulation setup
The algorithms could be simulated in a variety of tools/languages like SPW, MATLAB, “C” or a mix of these.
SPW has an exhaustive floating point and fixed-point library. SPW also provides feature to plug-in RTL modules and do a co-simulation of SPW system and Verilog. This helps in verifying the RTL
implementation of algorithms against the SPW/C implementation.
Hardware design
Interface definition
Baseband interfaces with two external modules: MAC and Radio.
Interface to MAC
• Baseband should support the following for MAC
• Should support transfer of data at different rates
• Transmit and receive control
• RSSI/CCA indication
• Register programming for power and frequency control
• Following options are available for MAC interface:
• Serial data interface – Clock provided along with data. Clock speed changes for different data rates
• Varying data width, single speed clock – The number of data lines vary accord ing to the data rate. The clock remains same for all rates.
• Single clock, Parallel data with ready indication – Clock speed and data width is same for all data rates. Ready signal used to indicate valid data
• Interfaces like SPI/Micro-wire/JTAG could be used for register programming
Two kinds of radio interfaces are described below
I/Q interface
On the transmit side, the complex Baseband signal is sent to the radio unit that first does a Quadrature modulation followed by up-conversion at 5 GHz. On the receive side, following the
down-conversion to IF, Quadrature demodulation is done and complex I/Q signal is sent to Baseband. Shown below is the interface.
Figure 6: I/Q interface to Baseband
IF interface
The Baseband does the Quadrature modulation and demodulation digitally.
Figure 7: IF interface to Baseband
I/Q interface IF interface
I/Q Phase/Amplitude imbalance is an issue as the modulation/demodulation is done in analog No phase imbalance as Quadrature components are produced digitally
Two ADC/DAC channels required for I/Q Single ADC/DAC channel required
Sampling frequency is lower (>BW) Higher sampling frequency needed (>2BW)
DC-offset introduced by I/Q ADC has to be compensated DC-offset introduced at the receiver ADC is not a problem as there is a mixing stage inside
Clocking strategy
The 802.11a supports different data rates from 6 Mbps to 54 Mbps. The clock scheme chosen for the Baseband should be able to support all rates and must also result in low power consumption. We know
from our Basic ASIC design guidelines that most circuits should run at the lowest clock.
Two options are shown below:
• Above scheme requires different clock sources or a very high clock rate from which all these clocks could be generated.
• The modules must work for the highest frequency of 54 MHz.
• Shown in the previous figure is a simpler clocking scheme with only one clock speed for all data rates
• Varying duty cycles for different data rates is provided by the data enable signal
• All the circuits in the transmit and receive chain work on parallel data (4 bits)
• Overhead is the Data enable logic in all the modules
Design of crucial blocks
Requirement: 64 point FFT computation in 4 us as the 802.11a OFDM symbol including the guard interval is 4 us wide.
Figure 8: 64 point Radix-4 FF T data flow diagram
1.1.1.1 Different architectures
• Radix-4 Single-Path delay commutator
• Radix-4 Multi-Path delay commutator
• Radix-4 Single-Path delay feedback
• Pipelined or non-pipelined
1.1.1.2 FFT storage
• Using single RAM – As only one RAM is available, large delays occur because of read and write cycles and therefore faster clock is required to meet FFT time requirement of 4 us
• Using multiple storage, Data load/store happens in parallel, FFT Radix-4 utilization is improved and FFT computation time is less
Twiddle factor complex multiplication
Comparison shown for two options
1.1.1.4 Butterfly construction
Since multipliers are the biggest block in Radix-4 butterfly, designer may choose to have 1, 2 or 3 complex multiplier instances based on clock, timing and latency requirements. Shown below are both
the kinds
Figure 10: Butterfly operation
Figure 11: Lower latency with three parallel multipliers
Figure 12: Single complex multiplier, higher latency (or higher clock required for same latency)
The ½ , length 7, convolutionally encoded stream is decoded using a Viterbi decoder.
Figure 13: Viterbi Construction
1.1.1.5 BMU
Branch metrics computation unit calculates the hamming distances for the incoming pair of codes from four possible codes
1.1.1.6 ACS
Add, compare and select unit is used to update the path metric for all the 64 states and to select the predecessor. For each of the 64 states, it adds current path metric and branch metric for both
the predecessor states and selects the lower of the two as the new path metric and the predecessor information is passed on to the SMU unit.
The width of the Path metric register and the ACS adders and subtractor will change based on whether a soft-decision or a hard-decision viterbi is ued. It also depends on the maximum metrics
accumulated by metrics registers before a normalization is done.
1.1.1.7 SMU
Survivor metrics unit can be implemented by register-exchange or traceback memory method.
Register-exchange Traceback memory
Data bits for all possible paths in the trellis are stored in Flip-flops Decision bits are stored in traceback RAM
Low latency = Traceback length High latency = 4 x Traceback length
High gate count =~ 60 K for traceback length of 64 Low gate count =~25K + 256x64 RAM required
High power consumption because of operation of all Flip-flops Low power consumption
NCO (Numerically controlled oscillator) is used for frequency offset correction. NCO generates sine and cosine waves that are mixed with the incoming Baseband signal to correct the frequency error.
Various design parameters to be decided in NCO are given below
Figure 14: NCO
Width of phase-accumulator. Will decide on the accuracy or “ppm” of generated waveform
Width of Sine and cosine outputs. Decides Quantization error. But this also decides the size of ROM used to keep the sine and cosine tables
By using the fact the cos (q) = sin (90 - q), a single LUT can be used to generate both sine and cosine values
The need for Sine/Cosine ROM can be eliminated by using a CORDIC rotator (if the pipeline delay that the CORDIC introduces can be tolerated).
The tan-1 circuit is used during the estimation of the frequency error caused by local frequency PPM errors. This could be implemented as a simple LUT, which contains the Arctan values for different
angles or it can be implemented by using a CORDIC circuit in vectoring mode. CORDIC is an abbreviation for Coordinate rotation digital computer. It involves performing the following equations
Let us say the complex vector is x0 + jy0 and our objective is to find z = tan-1(y0/x0), it can be achieved by doing the following.
xi+1 = xi – yi*di*2-i
yi+1 = yi + xi*di*2-i
zi+1 = zi – di*tan-1(2-i).
di = +1 if yi < 0, -1 otherwisi eis the iteration number and decides the accuracy of the result. As can be seen, the CORDIC circuit is simple to construct and involves only shifts, additions and
Using LUT CORDIC
Huge RAM needed for high accuracy Low gate count
Low latency High latency because of iterative method
CORDIC circuit is preferred as it results in a low gate count implementation.
Optimize usage of hardware resources by reusing different blocks
Hardware resources can be reused considering the fact that 802.11a system is a halfduplex system. The following blocks are re-used
• Interleaver/De-interleaver
• Scrambler/Descrambler
• Intermediate data buffers
Since Adders and Multipliers are costly resources, special attention should be given to reuse them. An example shown below where an Adder/Multiplier pool is created and different blocks are connected
to this.
Figure 15: Sharing of H/W resources
Optimize the widely used circuits
Identify the blocks that are used at several places (several instances of the same unit) and optimize them. Optimization can be done for power and area. Some of the circuits that can be optimized
They are the most widely used circuits. Synthesis tools usually provide highly optimized circuits for multipliers and adders. In case optimized multipliers are not available, multipliers could be
designed using different techniques like booth- (Non) recoded Wallace.
ACS unit
There are 64 instantiations of ACS unit in the Viterbi decoder. Optimization of ACS unit results in significant savings. Custom cell design (using foundry information) for adders and comparators
could be considered.
Debug support
• To enable debugging the hardware a serial port or a parallel port interface could be provided
• The port could be used to control the core, issue transmit and receive com mands, analyzing the receive data for errors, monitoring BER etc
• Test mode support can be provided in the core to facilitate selective testing of the modules inside
RTL Simulations
RTL simulations are conducted to achieve the following objectives:
• Functional verification for all transmit and receive Baseband functions for different data rates is done
• Necessary models are written to introduce noise and channel effects. Verilog PLI interface can be used to plug-in “C” models if they are available
• It is verified that different algorithmic blocks are implemented correctly in RTL, the same set of vectors used in algorithm simulations are applied to the RTL system and the outputs are
compared. If simulations for algorithms are done in a tool like SPW, then this can be easily be done by importing the RTL blocks in SPW system
Figure 16: Simulation setup in SPW environment
Figure 17: Simulation setup in Verilog environment
After algorithm verification, the verilog RTL code is typically tested on a prototype board using FPGAs before fabricating the ASIC. The details of these activities are outside the scope of this
1. ISO/IEC 8802-11 ANSI/IEEE Std 802.11-1999, Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) specifications, IEEE, 20th August 1999
2. IEEE Std 802.11a-1999(Supplement to IEEE Std 802.11-1999), Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) specifications, IEEE, September 1999
3. Digital signal Processing, J.G.Proakis, D.G Manolakis, Third Edition
4. Digital communications, Simon Haykin, John Wiley and sons
5. Very Fast Fourier Transform Algorithms Hardware for Implementation, Alvin M. Despain, IEEE transactions on computers, Vol. c-28 No 5, May 1979
6. Robust Frequency and Timing Synchronization for OFDM, Timothy M. Schmidl and Donald C. Cox, IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 12, DECEMBER 1997
7. A New Approach for Evaluating Clipping Distortion in Multicarrier Systems, Ahmad R.S. Bahai, Manoneet Singh, Andrea J. Goldsmith, and Burton R. Saltzberg, IEEE JOURNAL ON SELECTED AREAS IN
COMMUNICATIONS, VOL. 20, NO. 5, MAY 2002
8. "OFDM for multimedia wireless communications" by Van Nee, Richard and Ramjee Prasad
9. Performance Analysis of Viterbi Decoding for 64-DAPSK and 64-QAM Modulated OFDM Signals, Thomas May, Hermann Rohling, and Volker Engels, IEEE TRANS ACTIONS ON COMMUNICATIONS, VOL. 46, NO. 2,
FEBRUARY 1998
10. An Equalization Technique for Orthogonal Frequency-Division Multiplexing Sys tems in Time-Variant Multipath Channels, Won Gi Jeon, Kyung Hi Chang and Yong Soo Cho, IEEE TRANSACTIONS ON
COMMUNICATIONS, VOL. 47, NO. 1, JANU ARY 1999
11. Optimum Nyquist Windowing for Improved OFDM Receivers, Stefan H. Muller- Weinfurtner and Johannes B. Huber, Proc. of the IEEE Global Telecommunications Conference GLOBECOM 2000, San Francisco,
CA, USA, pp. 711-715, Nov. 2000
Acronyms and definitions
• AGC Automatic gain control
• AWGN Additive white gaussian noise
• BER Bit error rate
• BPSK Binary phase shift keying
• BW Bandwidth
• EVM Error vector magnitude
• FFT Fast Fourier transform
• IF Intermediate frequency
• ISI Inter symbol interference
• PA Power amplifier
• OFDM Orthogonal frequency division multiplexing
• QPSK Quadrature phase shift keying
• QAM Quadrature amplitude modulation
• SPW Signal processing Work-system from Cadence
• SNR Signal to noise ratio
About the Authors
Aseem Pandey is a Senior engineer with the VLSI and Systems design division in Wipro Technologies. He is currently working in a group which develops physical layers of different wireless standards
Shyam Ratan Agrawalla is a senior engineer with the VLSI and Systems design division with Wipro Technologies. He is working on the 802.11a OFDM modem development.
Shrikant Manivannan is the technical lead for the 802.11a OFDM modem program at Wipro Technologies. His focus since joining Wipro has been the design of Baseband for different Wireless Standards.
│ Contact Wipro Technologies │
│ │ | {"url":"https://www.design-reuse.com/articles/10358/vlsi-implementation-of-ofdm-modem.html","timestamp":"2024-11-03T12:42:32Z","content_type":"text/html","content_length":"71740","record_id":"<urn:uuid:6bb9e38c-1054-4a3d-823d-54fc70fe42f7>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00566.warc.gz"} |
I need a formula to calculate a loan payment with additional principle added to accelerate payoff... | Microsoft Community Hub
Forum Discussion
I need a formula to calculate a loan payment with additional principle added to accelerate payoff...
Hello All! This relates to a Mortgage Payment scenario. Calculation of the payment ( PMT(Int/12,Term,-Bal.) ) then illustrating the effects of an additional dollar amount being applied to the pri...
• Douglas997t wrote: ``The additional payment made each month is to be applied directly to principle as opposed to as a prepayment applied to both principle & interest.``
Ostensibly, the Excel formula is: =NPER(D66/12, D68 + D69, -D64)
But that results in 200.466253023484, which rounds to 200.47, not 200.48.
And IMHO, NPER should be rounded up because humans cannot count non-integer periods.
So the formula should be: =ROUNDUP(NPER(D66/12, D68 + D69, -D64), 0)
Mathematically, the amount of periodic interest is always prevBal*intRate. It is not affected by the amount of the payment.
So, any additional payment does indeed reduce only principal.
However, as principal is reduced periodically, so is the amount of interest each period.
This is demonstrated below. See the attached Excel file for formulas.
I accept that the calculation I presented is a mathematical abstraction of the problem and does not capture practical considerations; no business practice is going to work with millionths of a cent.
The calculation produces an array of balance figures (columns H and M) and the other columns form no part of the calculation; they are derived for information only.
The 'simplicity' I set out to achieve is to generate each table from a single formula rather than the original 2240 individual formulas. The method is far closer to the world of professional
programmer than it is to that of a normal spreadsheet end-user.
You found the use of Lambda functions off-putting. All a Lambda function does is allow one to write a formula in terms of parameters passed to it as variables. The idea is that such a formula is less
prone to errors of consistency than a traditional formula copied across a range.
The end of the Einstein quote you mentioned was "... and no simpler". The challenge to be answered by traditional spreadsheet methods is "does an excess of simplicity itself create impenetrable and
error-prone solutions?"
I apologize for the several deleted attempts to express my thoughts about your implementation. I struggled with suggestions that might make it work.
Re: ``The method is far closer to the world of professional programmer``
Speaking as a professional programmer with 40 years experience in compiler and operating system design, I can say with impunity that there is nothing about our two implementations that makes one more
"professional" than the other.
On the contrary, as a professional, I pride myself on creating the simplest implementations that are readable, maintainable, efficient and, most importantly, correct.
Your implementation reminds of the challenges in the 1960-70s to write one-line APL expressions that would do the calculation of a multiline function in any other language. They were never considered
"professional". In fact, the more obscure they were the better in order to challenge the reader, or so the game went.
• No need to apologise, I am perfectly happy to take positive suggestions or even criticism on board. I freely admit that I had not taken rounding into account, both because the rules to be
implemented had not been specified and because I didn't want to add complexity to what is already a somewhat alien approach to spreadsheet usage.
Your background as a compiler writer did come as a little bit of a surprise, my background was scientific programming using Fortran IV, so not as fundamental in IT terms. My point in describing
the approach as 'closer to the world of professional programmer' was not as a comment of professionalism but a recognition of the task as a programming exercise rather than merely the
manipulation of numbers that tends to characterise normal spreadsheet use.
The formulas I had in mind were those that underpin the PMT function that, itself, generates an unrounded value. Rounding it up generates an over-payment which accumulates but I could adjust the
calculation within the final payment period to compensate. I think that will happen automatically but I will check.
Something I aim to achieve, is to create a formula that can be modified to give results for variable interest rates or periods of grace etc, without touching the spreadsheet itself, other than
creating a lookup table to show the changes against the period for which they scheduled. I certainly do not prioritise 'concise'; in general I am for 'readability' and avoid direct cell
referencing despite it being the industry standard for spreadsheets.
□ This workbook shows the base payment rounded up and the interest charged rounded down to the nearest dollar. The final period adjustments are pretty large but I assume that is normal. | {"url":"https://techcommunity.microsoft.com/discussions/excelgeneral/i-need-a-formula-to-calculate-a-loan-payment-with-additional-principle-added-to-/3563885/replies/3564212","timestamp":"2024-11-15T00:49:16Z","content_type":"text/html","content_length":"333426","record_id":"<urn:uuid:7f62f033-87d2-465e-ac28-32f8613b1583>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00125.warc.gz"} |
Open Problem Garden
Random stable roommates
Conjecture The probability that a random instance of the stable roommates problem on
A system of preferences for a graph prefers stable if there do not exist
A famous theorem of Gale-Shapley [GS] proves that every system of preferences on a complete bipartite graph
Mertens [M] did an extensive Monte-Carlo simulation to obtain the above conjecture. Indeed, by guessing at the constant he even offers the stronger conjecture
[GS] D. Gale D and L. S. Shapley, College admissions and the stability of marriage, Am. Math. Mon. 69 9-15.
[I] R. W. Irving, An efficient algorithm for the stable roommates problem, J. Algorithms 6 577-95.
[IP] B. Pittel and R. W. Irving, An upper bound for the solvability of a random stable roommates instance, Random Struct. Algorithms 5 465-87.
*[M] S. Mertens, Random stable matchings, J. Stat. Mech. Theory Exp. 2005, no. 10 MathSciNet
[P] B. Pittel, The 'stable roommates' problem with random preferences, Ann. Probab. 21 1441-77
* indicates original appearance(s) of problem. | {"url":"http://www.openproblemgarden.org/op/random_stable_roommates","timestamp":"2024-11-01T19:26:20Z","content_type":"application/xhtml+xml","content_length":"17249","record_id":"<urn:uuid:5a0c5288-833f-4a16-a0af-ab7071973e67>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00788.warc.gz"} |
doc/QuickStartGuide.dox - eigen - Git at Google
namespace Eigen {
/** \page GettingStarted Getting started
This is a very short guide on how to get started with Eigen. It has a dual purpose. It serves as a minimal introduction to the Eigen library for people who want to start coding as soon as possible.
You can also read this page as the first part of the Tutorial, which explains the library in more detail; in this case you will continue with \ref TutorialMatrixClass.
\section GettingStartedInstallation How to "install" Eigen?
In order to use Eigen, you just need to download and extract Eigen's source code (see <a href="http://eigen.tuxfamily.org/index.php?title=Main_Page#Download">the wiki</a> for download instructions).
In fact, the header files in the \c Eigen subdirectory are the only files required to compile programs using Eigen. The header files are the same for all platforms. It is not necessary to use CMake
or install anything.
\section GettingStartedFirstProgram A simple first program
Here is a rather simple program to get you started.
\include QuickStart_example.cpp
We will explain the program after telling you how to compile it.
\section GettingStartedCompiling Compiling and running your first program
There is no library to link to. The only thing that you need to keep in mind when compiling the above program is that the compiler must be able to find the Eigen header files. The directory in which
you placed Eigen's source code must be in the include path. With GCC you use the \c -I option to achieve this, so you can compile the program with a command like this:
\code g++ -I /path/to/eigen/ my_program.cpp -o my_program \endcode
On Linux or Mac OS X, another option is to symlink or copy the Eigen folder into \c /usr/local/include/. This way, you can compile the program with:
\code g++ my_program.cpp -o my_program \endcode
When you run the program, it produces the following output:
\include QuickStart_example.out
\section GettingStartedExplanation Explanation of the first program
The Eigen header files define many types, but for simple applications it may be enough to use only the \c MatrixXd type. This represents a matrix of arbitrary size (hence the \c X in \c MatrixXd),
in which every entry is a \c double (hence the \c d in \c MatrixXd). See the \ref QuickRef_Types "quick reference guide" for an overview of the different types you can use to represent a matrix.
The \c Eigen/Dense header file defines all member functions for the MatrixXd type and related types (see also the \ref QuickRef_Headers "table of header files"). All classes and functions defined in
this header file (and other Eigen header files) are in the \c Eigen namespace.
The first line of the \c main function declares a variable of type \c MatrixXd and specifies that it is a matrix with 2 rows and 2 columns (the entries are not initialized). The statement <tt>m(0,0)
= 3</tt> sets the entry in the top-left corner to 3. You need to use round parentheses to refer to entries in the matrix. As usual in computer science, the index of the first index is 0, as opposed
to the convention in mathematics that the first index is 1.
The following three statements sets the other three entries. The final line outputs the matrix \c m to the standard output stream.
\section GettingStartedExample2 Example 2: Matrices and vectors
Here is another example, which combines matrices with vectors. Concentrate on the left-hand program for now; we will talk about the right-hand program later.
<table class="manual">
<tr><th>Size set at run time:</th><th>Size set at compile time:</th></tr>
\include QuickStart_example2_dynamic.cpp
\include QuickStart_example2_fixed.cpp
The output is as follows:
\include QuickStart_example2_dynamic.out
\section GettingStartedExplanation2 Explanation of the second example
The second example starts by declaring a 3-by-3 matrix \c m which is initialized using the \link DenseBase::Random(Index,Index) Random() \endlink method with random values between -1 and 1. The next
line applies a linear mapping such that the values are between 10 and 110. The function call \link DenseBase::Constant(Index,Index,const Scalar&) MatrixXd::Constant\endlink(3,3,1.2) returns a 3-by-3
matrix expression having all coefficients equal to 1.2. The rest is standard arithmetic.
The next line of the \c main function introduces a new type: \c VectorXd. This represents a (column) vector of arbitrary size. Here, the vector \c v is created to contain \c 3 coefficients which are
left uninitialized. The one but last line uses the so-called comma-initializer, explained in \ref TutorialAdvancedInitialization, to set all coefficients of the vector \c v to be as follows:
v =
1 \\
2 \\
The final line of the program multiplies the matrix \c m with the vector \c v and outputs the result.
Now look back at the second example program. We presented two versions of it. In the version in the left column, the matrix is of type \c MatrixXd which represents matrices of arbitrary size. The
version in the right column is similar, except that the matrix is of type \c Matrix3d, which represents matrices of a fixed size (here 3-by-3). Because the type already encodes the size of the
matrix, it is not necessary to specify the size in the constructor; compare <tt>MatrixXd m(3,3)</tt> with <tt>Matrix3d m</tt>. Similarly, we have \c VectorXd on the left (arbitrary size) versus \c
Vector3d on the right (fixed size). Note that here the coefficients of vector \c v are directly set in the constructor, though the same syntax of the left example could be used too.
The use of fixed-size matrices and vectors has two advantages. The compiler emits better (faster) code because it knows the size of the matrices and vectors. Specifying the size in the type also
allows for more rigorous checking at compile-time. For instance, the compiler will complain if you try to multiply a \c Matrix4d (a 4-by-4 matrix) with a \c Vector3d (a vector of size 3). However,
the use of many types increases compilation time and the size of the executable. The size of the matrix may also not be known at compile-time. A rule of thumb is to use fixed-size matrices for size
4-by-4 and smaller.
\section GettingStartedConclusion Where to go from here?
It's worth taking the time to read the \ref TutorialMatrixClass "long tutorial".
However if you think you don't need it, you can directly use the classes documentation and our \ref QuickRefPage.
\li \b Next: \ref TutorialMatrixClass | {"url":"https://third-party-mirror.googlesource.com/eigen/+/941ca8d83f776b9a07153d3abef2877907aa0555/doc/QuickStartGuide.dox","timestamp":"2024-11-02T23:42:36Z","content_type":"text/html","content_length":"30035","record_id":"<urn:uuid:03f02206-464b-4bc4-9b90-e1999cc6e1b9>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00886.warc.gz"} |
Leverage and Position Limits
A risk limit is a risk management mechanism used to limit the risk of a trader's position. In a trading environment with large price fluctuations, a single trader who uses high leverage to hold a
large position may bring huge losses through the position. This system uses the concept of dynamic leverage, that is, the maximum leverage that can be used when trading will change according to the
value of the position held by the trader: the greater the value of the position held, the lower the maximum leverage that can be used. At the same time, the larger the leverage is, the smaller the
position that can be opened.
Notional value of holdings (BTC)
Notional value of holdings (ETH) | {"url":"https://support.zedxion.com/hc/en-gb/articles/5618368235281-Leverage-and-Position-Limits","timestamp":"2024-11-08T10:43:01Z","content_type":"text/html","content_length":"50281","record_id":"<urn:uuid:95686878-5024-4cfc-8c34-29495ae85edb>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00407.warc.gz"} |
8-bit ADC Conversion
This tool provides the ADC resolution and the digital value for an analog input signal.
Formula for 8-bit ADC
The analog resolution or the smallest value that can be measured by the ADC is given by the formula:
• V[ref] is the reference voltage
The digital output from an ADC is given by the following formula for an 8-bit ADC.
Digital output = 2^8 * (V[in])/(V[ref])
• Vin is the analog input
• Vref is the analog reference
Example Calculation for 8-bit ADC
For example, if an 8-bit ADC has a reference voltage (Vref) of 5V, the smallest change it can detect (quantum) is Vref / 2^8 = 5V / 256 ≈ 19.53 mV. This means each step of the digital output
corresponds to a change of about 19.53 mV in the input signal.
Analog input voltage of 1 Volt gives a digital output level of 110011.
What is an 8-bit ADC?
An 8-bit Analog-to-Digital Converter (ADC) is a device that converts analog signals into digital signals. The “8-bit” part refers to the resolution of the ADC, meaning it can produce 2^8 = 256
discrete values. This process involves several key steps and concepts which are important to understand how ADCs work.
• Sampling: The first step in the ADC process is sampling the analog signal at regular intervals. The frequency at which the signal is sampled is determined by the Nyquist theorem, which states
that the sampling rate should be at least twice the highest frequency present in the analog signal to accurately represent the signal.
• Quantization: After sampling, each sampled analog value is assigned to the nearest value represented by the 8-bit resolution. Because an 8-bit ADC can only represent 256 discrete levels, this
step introduces a quantization error, as the analog value might not match exactly with the discrete levels available. The range of the analog signal that the ADC can convert is determined by its
reference voltage (Vref); the resolution determines how finely it can divide this range.
• Digital Representation: The result of the quantization step is a digital representation of the analog input. For an 8-bit ADC, this digital representation will be a number between 0 and 255,
where 0 represents the lowest possible input (often 0V or GND) and 255 represents the highest (equal to Vref).
• Resolution and Accuracy: The resolution of an ADC determines how many discrete values it can produce and affects the precision of the digital output. However, accuracy also depends on other
factors such as noise, linearity errors, and the stability of the reference voltage.
• Voltage Range and Scaling: The input voltage range of an ADC is determined by its design and the reference voltage used. Sometimes, input signals need to be scaled down or conditioned through
amplifiers or attenuators to fit within the ADC’s input range. Voltage dividers are used to scale a voltage down.
An 8-bit ADC is widely used in various applications due to its balance between simplicity, resolution, and cost. Applications range from simple data acquisition systems to complex embedded systems,
audio processing, and more.
Related Posts | {"url":"https://3roam.com/8-bit-adc-conversion/","timestamp":"2024-11-05T04:22:46Z","content_type":"text/html","content_length":"194414","record_id":"<urn:uuid:d1b5884d-af5d-47eb-bf95-534b35f2ba26>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00191.warc.gz"} |
fanny.object: Fuzzy Analysis (FANNY) Object in cluster: "Finding Groups in Data": Cluster Analysis Extended Rousseeuw et al.
The objects of class "fanny" represent a fuzzy clustering of a dataset.
A legitimate fanny object is a list with the following components:
membership matrix containing the memberships for each pair consisting of an observation and a cluster.
memb.exp the membership exponent used in the fitting criterion.
Dunn's partition coefficient F(k) of the clustering, where k is the number of clusters. F(k) is the sum of all squared membership coefficients, divided by the number of observations. Its
value is between 1/k and 1.
The normalized form of the coefficient is also given. It is defined as (F(k) - 1/k) / (1 - 1/k), and ranges between 0 and 1. A low value of Dunn's coefficient indicates a very fuzzy
clustering, whereas a value close to 1 indicates a near-crisp clustering.
clustering the clustering vector of the nearest crisp clustering, see partition.object.
k.crisp integer (\le k) giving the number of crisp clusters; can be less than k, where it's recommended to decrease memb.exp.
objective named vector containing the minimal value of the objective function reached by the FANNY algorithm and the relative convergence tolerance tol used.
convergence named vector with iterations, the number of iterations needed and converged indicating if the algorithm converged (in maxit iterations within convergence tolerance tol).
diss an object of class "dissimilarity", see partition.object.
call generating call, see partition.object.
silinfo list with silhouette information of the nearest crisp clustering, see partition.object.
data matrix, possibibly standardized, or NULL, see partition.object.
matrix containing the memberships for each pair consisting of an observation and a cluster.
Dunn's partition coefficient F(k) of the clustering, where k is the number of clusters. F(k) is the sum of all squared membership coefficients, divided by the number of observations. Its value is
between 1/k and 1.
The normalized form of the coefficient is also given. It is defined as (F(k) - 1/k) / (1 - 1/k), and ranges between 0 and 1. A low value of Dunn's coefficient indicates a very fuzzy clustering,
whereas a value close to 1 indicates a near-crisp clustering.
the clustering vector of the nearest crisp clustering, see partition.object.
integer (\le k) giving the number of crisp clusters; can be less than k, where it's recommended to decrease memb.exp.
named vector containing the minimal value of the objective function reached by the FANNY algorithm and the relative convergence tolerance tol used.
named vector with iterations, the number of iterations needed and converged indicating if the algorithm converged (in maxit iterations within convergence tolerance tol).
list with silhouette information of the nearest crisp clustering, see partition.object.
The "fanny" class has methods for the following generic functions: print, summary.
The class "fanny" inherits from "partition". Therefore, the generic functions plot and clusplot can be used on a fanny object.
For more information on customizing the embed code, read Embedding Snippets. | {"url":"https://rdrr.io/cran/cluster/man/fanny.object.html","timestamp":"2024-11-06T18:52:48Z","content_type":"text/html","content_length":"33535","record_id":"<urn:uuid:2cef23b8-0b53-4301-8b90-5ef127857f64>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00122.warc.gz"} |
[GAP Forum] Canonical form for some small groups and efficient characterisation of the generalized symmetric groups
[GAP Forum] Canonical form for some small groups and efficient characterisation of the generalized symmetric groups
Max Horn max at quendi.de
Tue Dec 19 12:11:34 GMT 2017
Dear Martin,
> On 19 Dec 2017, at 09:48, Martin Rubey <martin.rubey at tuwien.ac.at> wrote:
> "Nicolas M. Thiery" <Nicolas.Thiery at u-psud.fr> writes:
>> On Mon, Dec 18, 2017 at 01:57:04PM +0100, Martin Rubey wrote:
>>> I admit I am not completely sure right now: can we associate a conjugacy
>>> class of subgroups of S_n, isomorphic to the Weyl group, in a sensible
>>> way to a finite Cartan type?
>> The permutation action on the roots would be a rather natural choice.
>> This could be extended to affine Weyl groups using affine
>> permutations.
Note that even different (finite) Cartan types can lead to isomorphic Weyl
groups, unless you restrict to irreducible ones. And if you also permit
infinite Weyl groups, then even for the irreducible ones, there are pairs of
different irreducible types which have isomorphic Weyl groups; but from all
you write, I guess you are only interested in finite groups.
But of course they induce distinct "permutation isomorphism classes", to use
the nomenclature Thomas' described. Which to me suggests that what Thomas
describes really is a better fit for what you want to do.
Regarding Nicolas' suggestion: While it is a "natural" choice, it is not
necessarily a "canonical" choice. At the very least, you'd have to define
how to label the roots "canonically". But OK, that can be done, perhaps you
are even doing it already.
But now to identify permutation group with your "canonical form", you need
to solve a subgroup conjugacy problem, which can be hard.
Moreover, it is not clear to me how to, given a permutation group, you'd
compute its "canonical form" intrinsically (i.e. other than going through
your whole database, then trying to compute a permutation isomorphism with
each "canonical form" in the database, which of course is kinda against the
whole point in defining a canonical form in the first place.)
So I'd be very hesitant to call this a "canonical form", because to me, the
name suggests that it should be possible to compute it intrinsically (i.e.
w/o referring to a big table of "canonical forms"), and somewhat efficiently
(as otherwise, there is little use). Of course that's just my point of
view, you can disagree, but then I wonder what the point behind having those
"canonical forms" really is?
> Indeed! This leaves the question whether there we can compute a
> canonical form for permutation groups up to conjugacy. I just noticed
> that this is a mathoverflow question, too:
> https://mathoverflow.net/q/90107/3032
> Unfortunately, as of today it doesn't have a straight answer, but maybe
> things have changed?
No. This is still an incredibly tough problem.
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CP2K Open Source Molecular Dynamics
Into which spin channel do excess electrons go?
If you do unrestricted DFT calculations with CP2K for charged systems, you may wonder which spin channel the electron is taken from or added to. Moreover, the sign convention for any output
(Hirshfeld or Mulliken spin charge, cube files) may be of interest to you.
The answer can be derived using these rules:
Alpha ≥ Beta
The number of electrons in the alpha channel is greater or equal the number of electrons in the beta channel. It does not matter whether you charge your system by adding an electron (then you have
n+1 electrons in channel alpha and n electrons in channel beta) or whether you charge it by taking out one electron (then you have n electrons in channel alpha and n-1 electrons in channel beta).
Sign Convention
In the output log file, the sign convention is such that alpha channel population is positive and beta channel population is negative. So if you have an atom with four spins up (+4, because up=alpha)
and one spin down (-1, because down=beta), then your spin charge is +3.
Cube Files
In the cube files for the spin density, you get the difference between alpha and beta channel, but the voxel data is signed. Following the point above on the sign convention, the value in the cube
file is alpha+beta (spin up plus spin down) which is identical to the absolute value of the beta channel subtracted from the absolute value of the alpha channel.
faq/uks_convention.txt · Last modified: 2020/08/21 10:15 by 127.0.0.1 | {"url":"https://www.cp2k.org/faq:uks_convention","timestamp":"2024-11-03T00:13:17Z","content_type":"text/html","content_length":"21474","record_id":"<urn:uuid:f0a7ba85-59e5-4ed6-965c-ab36820551d8>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00879.warc.gz"} |
size comparison
Loading ..
Ukraine size comparison
603,628 km^2 / 233,000 mile^2
Ukraine is a country in Eastern Europe. It is bordered by Russia to the east and north-east; Belarus to the north; Poland, Slovakia and Hungary to the west; and Romania, Moldova, and the Black Sea to
the south. Ukraine also borders Crimea to its south, which Russia annexed from Ukraine in 2014, though Ukraine still continues to claim the territory.
Ukraine compared to Saved places
Ukraine compared to European countries
After Russia, Ukraine is the second largest country in Europe by area.
Ukraine compared to countries/areas with similar population
Ukraine, with a population of 44 million people, is comparable to the following areas.
Ukraine compared to former Soviet republics
After Russia and Kazakhstan, Ukraine is the third largest country by area among the former Soviet Republics.
Ukraine compared to countries in the Americas
Ukraine compared to African countries
Ukraine is similar in size to Madagascar, Botswana, Namibia or Somalia.
Ukraine compared to Asian countries
Ukraine is similar in size as Afghanistan, Turkey, Yemen and Thailand.
Ukraine compared to US States
Ukraine is a little smaller than Texas and a little larger than California in area.
How big is Crimea ?
Crimea is a peninsula along the northern coast of the Black Sea. In late February 2014, Russian troops were deployed to Crimea. The Republic of Crimea declared its independence from Ukraine following
a referendum on 16 March, deemed illegal by Ukraine and most countries. Russia formally annexed Crimea on 18 March.
How big is Snake Island ?
Snake Island - a small island in the Black Sea - was attacked by Russia forces on 24-Feb-2022. Initial reports suggested the the soldiers stationed on the island were killed when they refused to
surrender, but later reports confirmed that they were captured and alive. The island is only 0.17 square km in area and has 30 permanent residents. | {"url":"https://mapfight.xyz/map/ua/","timestamp":"2024-11-11T23:34:16Z","content_type":"text/html","content_length":"435800","record_id":"<urn:uuid:606dd8e1-4fcc-4677-8892-7822449a86f8>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00348.warc.gz"} |
Parsing and Calculating Seconds from Time Strings in Python
Imagine this: You receive a time formatted as a string in HH:MM:SS where HH, MM, and SS denote the hour, minute, and second, respectively. You are also given an integer representing a number of
seconds. Your task is to calculate the new time after adding the provided seconds and output the result in the HH:MM:SS format.
For example, if the input time is 05:10:30 and the number of seconds to add is 123, the output should be 05:12:33 since 123 seconds translate to 2 minutes and 3 seconds.
Take note of these points when resolving this task:
• The input time is invariably a valid time string in the HH:MM:SS format, with HH ranging from 00 to 23, MM, and SS ranging from 00 to 59.
• The output ought to be a time in the same format.
Let's go ahead and break down how to achieve this in our step-by-step solution guide. | {"url":"https://learn.codesignal.com/preview/lessons/1582","timestamp":"2024-11-01T20:11:10Z","content_type":"text/html","content_length":"151590","record_id":"<urn:uuid:58b9d72f-1d09-4857-bd16-6a0f10288dd3>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00350.warc.gz"} |
If f(x)=sinx-xcosx, how the function behaves in the intervals (0,pi) and (pi,2pi) i.e. whether it is increasing or decreasing? | Socratic
If #f(x)=sinx-xcosx#, how the function behaves in the intervals #(0,pi)# and #(pi,2pi)# i.e. whether it is increasing or decreasing?
1 Answer
Between $\left(0 , \pi\right)$, $f \left(x\right)$ is increasing and
between $\left(\pi , 2 \pi\right)$, $f \left(x\right)$ is decreasing.
Whether a function $f \left(x\right)$ is increasing or decreasing depends on whether $f ' \left(x\right) = \frac{\mathrm{df}}{\mathrm{dx}}$ is positive or negative.
As $f \left(x\right) = \sin x - x \cos x$
$f ' \left(x\right) = \frac{\mathrm{df}}{\mathrm{dx}} = \cos x - \left(1 \times \cos x + x \times \left(- \sin x\right)\right)$
= $\cos x - \cos x + x \sin x = x \sin x$
Now between $\left(\pi , 2 \pi\right)$, $\sin x$ is negative, but $x$ is positive, hence $f ' \left(x\right)$ is negative, and
between $\left(0 , \pi\right)$, $\sin x$ is positive and $x$ too is positive, hence $f ' \left(x\right)$ is positive.
Hence while between $\left(0 , \pi\right)$, $f \left(x\right)$ is increasing, between $\left(\pi , 2 \pi\right)$, $f \left(x\right)$ is decreasing.
graph{sinx-xcosx [-5.54, 14.46, -6.36, 3.64]}
Impact of this question
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how to calculate choked flow conditions? calculation for Calculations
17 Feb 2024
Popularity: ⭐⭐⭐
Choked Flow Conditions Calculator
This calculator computes the Mach number for choked flow conditions in a converging-diverging nozzle.
Calculation Example: Choked flow occurs when the mass flow rate of a fluid through a nozzle reaches its maximum value. At this point, the Mach number at the nozzle throat reaches unity, and any
further increase in the downstream pressure will not increase the mass flow rate. The Mach number for choked flow can be calculated using the formula M = sqrt((2/(gamma - 1)) * ((P1/P2)^((gamma - 1)/
gamma) - 1)), where P1 is the upstream pressure, P2 is the downstream pressure, gamma is the ratio of specific heats, and R is the gas constant.
Related Questions
Q: What is the significance of choked flow in nozzle design?
A: Choked flow is a critical phenomenon in nozzle design as it determines the maximum mass flow rate that can be achieved through the nozzle. This information is crucial for optimizing the
performance of nozzles in various applications, such as rocket engines and jet engines.
Q: How does the ratio of specific heats affect choked flow conditions?
A: The ratio of specific heats (gamma) plays a significant role in determining the choked flow conditions. A higher gamma value results in a higher choked flow Mach number, indicating that the fluid
can reach supersonic velocities more easily.
| —— | —- | —- |
P2 Downstream Pressure Pa
gamma Ratio of Specific Heats kJ/kg
Calculation Expression
Mach Number: The Mach number for choked flow is given by M = sqrt((2/(gamma - 1)) * ((P1/P2)^((gamma - 1)/gamma) - 1))
sqrt((2/(gamma - 1)) * ((P1/P2)^((gamma - 1)/gamma) - 1))
Calculated values
Considering these as variable values: P1=101325.0, P2=50662.5, R=287.0, gamma=1.4, the calculated value(s) are given in table below
| —— | —- |
Mach Number Sqrt(2)*Sqrt((-1+2.0^((-1+Gamma)/Gamma))/(-1+Gamma))
Similar Calculators
Calculator Apps
Matching 3D parts for how to calculate choked flow conditions? calculation for Calculations
App in action
The video below shows the app in action. | {"url":"https://blog.truegeometry.com/calculators/how_to_calculate_choked_flow_conditions_calculation_for_Calculations.html","timestamp":"2024-11-04T14:10:19Z","content_type":"text/html","content_length":"28183","record_id":"<urn:uuid:d7209fda-fd94-406b-aaa0-6bc0f2a4cb7c>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00874.warc.gz"} |
Data Analyst Technical Interview Questions and Answers
Related Queries: Data Analyst Interview Preparation | Data Analyst Interview Questions and Answers
What is the main purpose of a JOIN statement in SQL?
To delete data from tables
To combine rows from two or more tables
To create new tables
To update existing records
Which Excel feature is used to validate data entry in cells?
Data Validation
Input Mask
Cell Protection
Form Control
What does ETL stand for?
Extract, Transform, Load
Evaluate, Test, Learn
Extend, Transfer, Link
Export, Translate, Launch
What does the CROSS APPLY operator do in SQL Server?
Performs an inner join with a table-valued function
Performs a left outer join with a table-valued function
Combines rows from two queries
Filters rows based on a condition
What is the main purpose of a data dictionary?
To translate data between languages
To provide a centralized repository of information about data
To encrypt sensitive data
To perform data analysis
What is the purpose of the chi-square test?
To test independence of categorical variables
To compare means of groups
To predict continuous outcomes
To reduce dimensionality
What is the main purpose of discriminant analysis?
To predict a continuous outcome
To classify observations into groups
To reduce dimensionality
To perform time series analysis
What does the TEXT function do?
Joins text
Splits text
Formats a value as text
Counts text
What is the purpose of the CASE statement in SQL?
To add conditional logic to a query
To join tables
To sort results
To filter rows
What is the difference between a Convolutional Neural Network (CNN) and a Recurrent Neural Network (RNN)?
CNNs are typically used for image data, RNNs for sequential data
They are the same
CNNs are always more accurate
RNNs are only used for image data
Score: 0/10
Which of the following is NOT a component of a typical data model?
Which of the following is NOT a typical characteristic of big data?
Which is NOT a common data type in Python?
What is the purpose of the softmax function in neural networks?
To convert outputs to probabilities
To introduce non-linearity
To normalize inputs
To reduce overfitting
What is the primary use of the GROUP BY clause in SQL?
To sort results
To filter rows
To join tables
To group rows with similar values
What is the primary purpose of data masking?
To protect sensitive data by hiding or obfuscating it
To clean data
To visualize data
To analyze data
What is the difference between 1NF and 2NF?
2NF eliminates partial dependencies, 1NF ensures atomicity
1NF eliminates partial dependencies, 2NF ensures atomicity
Both eliminate partial dependencies
Neither eliminates partial dependencies
What is the primary goal of data cleansing?
To encrypt data
To remove or correct inaccurate records
To visualize data
To compress data
What is the purpose of the elbow method in K-means clustering?
To determine the optimal number of clusters
To visualize results
To clean data
To normalize data
What is transfer learning in machine learning?
Using knowledge gained from one task to improve performance on a different task
Transferring data between different models
Moving a trained model to a different machine
Translating model outputs
Score: 0/10
What is a cell reference in Excel?
A unique identifier for a cell
A formula
A data type
A chart type
What is the purpose of the '$' symbol in a cell reference?
Currency formatting
Absolute reference
Percentage calculation
Text formatting
What is the main goal of dimensionality reduction techniques?
To reduce the number of input variables
To increase the number of input variables
To make the model more complex
To normalize inputs
What is the primary purpose of data encryption?
To clean data
To protect data from unauthorized access
To increase data volume
To visualize data
Which SQL clause is used to filter groups in aggregate functions?
How does adjusted R-squared differ from R-squared?
It's always larger
It penalizes adding predictors
It's always smaller
It's unaffected by predictors
Which SQL join returns all matching records from both tables and null for non-matches?
What is the purpose of feature scaling in data preprocessing?
To normalize the range of independent variables
To increase the number of features
To reduce the number of features
To eliminate outliers
Which technical skill is most essential for a data analyst?
Graphic design
Statistical analysis
Public speaking
Physical fitness
What is NOT a common application of machine learning in data analysis?
Predictive modeling
Pattern recognition
Anomaly detection
Data encryption
Score: 0/10
What does the K in K-means algorithm represent?
Number of clusters
Number of iterations
Number of features
Number of data points
What is the main purpose of data sampling?
To increase data volume
To represent a larger population
To clean data
To encrypt data
What is the main purpose of a T-test?
To compare means of two groups
To compare variances
To test for normality
To perform regression analysis
Which SQL clause is used to remove duplicate rows from the result set?
Which of the following is an example of a categorical variable?
What is the main purpose of data profiling?
To encrypt sensitive data
To assess the quality and characteristics of a dataset
To perform predictive analytics
To create data visualizations
Which SQL function returns the absolute value of a number?
Which Excel feature allows you to create a visual summary of data?
Pivot Table
How does Machine Learning differ from Deep Learning?
Deep Learning uses neural networks with many layers
They are the same
Machine Learning is always unsupervised
Deep Learning doesn't require data
What is the purpose of hypothesis testing?
To make inferences about a population based on a sample
To clean data
To visualize data
To encrypt data
Score: 0/10 | {"url":"https://coolgenerativeai.com/data-analyst-technical-interview-questions-and-answers/","timestamp":"2024-11-11T21:06:02Z","content_type":"text/html","content_length":"191169","record_id":"<urn:uuid:a30af50c-3cff-40be-881d-ec21c8ef1864>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00015.warc.gz"} |
Rhombus Calculator
What is Area & Perimeter of Rhombus?
A quadrilateral with four congruent sides is a rhombus or diamond (see the picture below). In some literature it is named equilateral quadrilateral, since all of its sides are equal in length.
That means,
if ${\overline{AB}}\cong{\overline{BC}}\cong{\overline{CD}}\cong{\overline{DA}}$, then ${\overline{ABCD}}$ is a rhombus. Since opposite sides are parallel, the rhombus is a parallelogram, but not
every parallelogram is a rhombus. That means that all the properties of a parallelogram can be also applied to rhombus. To recall, the parallelogram has the following properties:
• Opposite sides of a parallelogram are congruent;
• Opposite angles of a parallelogram are congruent;
• The consecutive angles of a parallelogram are supplementary to each other;
• The diagonals of a parallelogram bisect each other
Rhombus satisfies two more properties:
• The diagonals bisect the opposite angles of the rhombus;
• The diagonals of a rhombus are perpendicular.
Any quadrilateral with perpendicular diagonals, that one diagonal bisect other, is a kite. So, the rhombus is a kite, but not every kite is a rhombus. Any quadrilateral that is both a kite and
parallelogram is a rhombus. Because the diagonals of a rhombus are perpendicular, we can apply the Pythagorean Theorem to find the rhombus side length. Let us consider the rhombus $\overline{ABCD}$.
Diagonals of the rhombus bisect each other, $\overline{AO} \cong\overline{OC}$ and $\overline{BO}\cong\overline{OD}.$ Let us denote the side length and the lengths of diagonals of a rhombus by
$a,d_1$ and $d_2$, respectively. Applying the Pythagorean theorem to $\Delta AOB$, we obtain
$${a}^2=\Big(\frac {d_1}2\Big)^2+\Big(\frac {d_2}2\Big)^2$$
The rhombus has only two lines of symmetry. A rhombus is symmetrical about its diagonals. A rhombus has central symmetry about the point of intersection of its diagonals, $O$. A rhombus has
rotational symmetry because $180^o$ counterclockwise rotation about the point $O$ transforms the rhombus to itself.
The distance
around a rhombus is called the perimeter of the rhombus. It is usually denoted by $P$. To find the perimeter of rhombus we add the lengths of its sides. Thus, the perimeter of a rhombus with side
length of $a$ is
$$P =a+a+a+a= 4 \times a$$
The area of a rhombus is a number of square units needed to fill the rhombus. The area, usually denoted by $A$. A rhombus and a rectangle on the same base and between the same parallels are equal in
area. Thus, area of the rhombus $\overline {ABCD}$ is equal to area of parallelogram $\overline{D'C'CD}$ (see picture on right). The area of a rhombus with the length of $a$ and the perpendicular
distance between the opposite sides of $h$ is
$$A =a\times h$$
In other words, the area of a rhombus is the product of its base and height. If we use the sine ratio in the right triangle $\Delta CC'B$, $h=a\sin m\angle B$, the previous area formula becomes
$$A =a^2\times \sin m\angle B $$
Since $\sin(180^o-\alpha)=\sin \alpha$, $\angle A\cong\angle C$ and $\angle B\cong\angle D,$ we can use the measure of any angle in rhombus, i.e.
$$A =a^2\times \sin m\angle B=a^2\times \sin m\angle A $$
The area
of a rhombus can be found in the second way. It is half the product of lengths of its diagonals.
$$A=\frac 12 d_1\times d_2$$
This formula follows from the formula for triangle area. The perimeter is measured in units such as centimeters, meters, kilometers, inches, feet, yards, and miles. The area is measured in units such
as square centimeters $(cm^2)$, square meters $(m^2)$, square kilometers $(km^2)$ etc.
The area & perimeter of a Rhombus work with steps shows the complete step-by-step calculation for finding the perimeter and area of the rhombus with the side length of $10\;in$ and the measure of the
angle of $30$ degrees using the perimeter and area formulas. For any other values for side length and measure of the angle of a rhombus, just supply two positive real numbers and click on the
GENERATE WORK button. The grade school students may use this area and perimeter of a Rhombus to generate the work, verify the results of the perimeter and area of two-dimensional figures or do their
homework problems efficiently. | {"url":"https://ncalculators.com/geometry/rhombus-calculator.htm","timestamp":"2024-11-13T15:12:42Z","content_type":"text/html","content_length":"64106","record_id":"<urn:uuid:aed52c7a-c6f0-4b16-9f8a-23d9be9ceccb>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00554.warc.gz"} |
PRMG Tampa Branch - Calculators
The following online calculators serve as helpful tools during the mortgage process.
Calculate your monthly payment and see how the principal is paid over time.
Determine if you can consolidate your debt by combining it with your home mortgage.
Find out how soon you can pay off your mortgage by making a prepayment.
Determine the additional monthly payment amount needed to pay off the loan sooner.
Find out how long it will take to "break-even" on a refinanced loan. | {"url":"https://clearwater470a.prmgapp.com/Calculators.html","timestamp":"2024-11-04T01:28:23Z","content_type":"text/html","content_length":"67187","record_id":"<urn:uuid:4cb3c5e6-e5e1-4f44-89d5-72dca429ef12>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00559.warc.gz"} |
Ayende @ Rahien
time to read 15 min | 2925 words
As it turns out, doing work on big data sets is quite hard. To start with, you need to get the data, and it is… well, big. So that takes a while. Instead, I decided to test my theory on the following
scenario. Given 4GB of random numbers, let us find how many times we have the number 1.
Because I wanted to ensure a consistent answer, I wrote:
1: public static IEnumerable<uint> RandomNumbers()
2: {
3: const long count = 1024 * 1024 * 1024L * 1;
4: var random = new MyRand();
5: for (long i = 0; i < count; i++)
6: {
7: if (i % 1024 == 0)
8: {
9: yield return 1;
10: continue;
11: }
12: var result = random.NextUInt();
13: while (result == 1)
14: {
15: result = random.NextUInt();
16: }
17: yield return result;
18: }
19: }
21: /// <summary>
22: /// Based on Marsaglia, George. (2003). Xorshift RNGs.
23: /// http://www.jstatsoft.org/v08/i14/paper
24: /// </summary>
25: public class MyRand
26: {
27: const uint Y = 842502087, Z = 3579807591, W = 273326509;
28: uint _x, _y, _z, _w;
30: public MyRand()
31: {
32: _y = Y;
33: _z = Z;
34: _w = W;
36: _x = 1337;
37: }
39: public uint NextUInt()
40: {
41: uint t = _x ^ (_x << 11);
42: _x = _y; _y = _z; _z = _w;
43: return _w = (_w ^ (_w >> 19)) ^ (t ^ (t >> 8));
44: }
45: }
I am using a custom Rand function because it is significantly faster than System.Random. This generate 4GB of random numbers, at also ensure that we get exactly 1,048,576 instances of 1. Generating
this in an empty loop takes about 30 seconds on my machine.
For fun, I run the external sort routine in 32 bits mode, with a buffer of 256MB. It is currently processing things, but I expect it to take a while. Because the buffer is 256 in size, we flush it
every 128 MB (while we still have half the buffer free to do more work). The interesting thing is that even though we generate random number, sorting then compressing the values resulted in about 60%
compression rate.
The problem is that for this particular case, I am not sure if that is a good thing. Because the values are random, we need to select a pretty high degree of compression just to get a good
compression rate. And because of that, a significant amount of time is spent just compressing the data. I am pretty sure that for real world scenario, it would be better, but that is something that
we’ll probably need to test. Not compressing the data in the random test is a huge help.
Next, external sort is pretty dependent on the performance of… sort, of course. And sort isn’t that fast. In this scenario, we are sorting arrays of about 26 million items. And that takes time.
Implementing parallel sort cut this down to less than a minute per batch of 26 million.
That let us complete the entire process, but then it halts with the merge. The reason for that is that we push all the values into a heap, and there are 1 billion of them. Now, the heap never exceed
40 items, but those are still 1 billion * O(log 40) or about 5.4 billion comparisons that we have to do, and we do this sequentially, which takes time. I tried thinking about ways to parallel, but I
am not sure how that can be done. We have 40 sorted files, and we want to merge all of them.
Obviously we can sort each 10 files set in parallel, then sort the resulting 4, but the cost we have now is the actual sorting cost, not I/O. I am not sure how to approach this.
For what is it worth, you can find the code for this here. | {"url":"https://ayende.com/blog/posts/series/165123/big-data-search","timestamp":"2024-11-07T21:38:32Z","content_type":"text/html","content_length":"146790","record_id":"<urn:uuid:574cb4d8-9fb2-4a0b-8dd9-7ac1d31117a1>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00854.warc.gz"} |
SAS Visual Forecasting 8.4: Interpreting Results and Diagnostic Plots
Using the Visual Forecasting 8.4 visual interface (the pipeline interface, i.e., Model Studio) you will get summary results including graphs, as shown below. Using Visual Forecasting 8.4’s coding
interface (SAS Studio V), you can get diagnostic plots, such as ACF, PACF, IACF and white noise plots.
All of these plots are very similar to what you get with Forecast Server. If you already know how to interpret Forecast Server graphs, you can stop reading now. Otherwise, keep reading.
VF 8.4 Visual Interface: Results Graphs
In our initial example, we use the standard sashelp.pricedata data set, with the data settings as follows:
Select any image to see a larger version.
Mobile users: To view the images, select the "Full" version at the bottom of the page.
and the Auto-forecasting node Model Generation settings and Model Selection settings as shown below:
We run the Auto-forecasting template and view the results by right-clicking the Auto-forecasting node and selecting Results.
We see in the Results that we have a total of 5 series.
MAPE Distribution
The MAPE Distribution graph is a histogram. It shows on the Y axis (vertical axis) what percent of your time series had a MAPE within a certain range. You can roll over the bars to get the average
MAPE and the percent of series that fall in that bin. Below we see that 20 percent of the series (i.e., 1 of our 5 series) falls into the first bar, with an average Mean Absolute Percent Error of
MAPE measures the accuracy of your forecasting model as a percentage. This makes it much easier to interpret than root mean square error, which has a magnitude related to the magnitude of the data
Acceptable MAPEs will vary by situation and domain, but in many situations a MAPE less than 5 could be considered quite good.
Model Family
The Model Family may be:
• None
• AutoRegressive Integrated Moving Average (ARIMA)
• Combined
• Exponential Smoothing Model (ESM)
• Intermittent Demand Model (IDM)
• Unobserved Components Model (UCM)
• Other
In this example, 100 percent (all 5) of the time series were modeled using ARIMA models.
Model Type
The Model Type histogram tells us what percentage of the series modeled included events, inputs, seasonality, transformations, trends, or outliers. Below we see that all of the models used inputs
(exogenous factors, i.e., independent variables). One of the models (20 percent of our 5 models) exhibited seasonality so a seasonal term was used.
Execution Summary
The execution summary gives us information on how many series failed (e.g., model did not converge), the number of seasonal series with flat forecasts, and so on.
VF 8.4 SAS Studio Interface: Diagnostic Plots
The SAS Visual Forecasting 8.4 visual interface does not show us diagnostic plots, but we can use Visual Forecasting 8.4’s coding interface (SAS Studio V) to create diagnostic plots, such as:
• ACF plot
• PACF plot
• IACF plot
• White Noise plot
Below I use the SAS Studio V task under SAS Viya Forecasting called Time Series Exploration to create PROC TSMODEL code. PROC TSMODEL is the main procedure for time series analysis.
Note that we are using the task under SAS Viya Forecasting not the task with the same name under SAS Forecasting!
PROC TSMODEL itself does not create diagnostic plots, but it does create the output we need to draw our own plots.
We will draw the plots using PROC SGRENDER as shown, still using the Time Series Exploration task under SAS Viya Forecasting. We now move to the Analysis tab, and check the diagnostic plots we want
to see.
PROC SGRENDER creates plots from templates that are created with the Graph Template Language (GTL). To learn more about PROC SGRENDER see the documentation.
Let’s remind ourselves what an ARIMA model is. Recall that AR terms (p) are lags of the observations. Recall that MA terms (q) are lags on the error terms.
Graphic courtesy of Joe Katz and Anthony Waclawski
Diagnostic Plots
Interpreting diagnostic plots is more of an art than a science. To do it well, I recommend that you consider all of your diagnostic plots together, examine multiple examples, and gain experience
yourself trying different p, d, q terms. For a more detailed understanding, explore the resources listed in the references section below. This article provides just a couple of textbook examples and
few general rules of thumb.
We will consider four types of diagnostic plots:
• Autocorrelation Function (ACF)
• Partial Autocorrelation Function (PACF)
• Inverse Autocorrelation Function (IACF)
• White Noise
They can all be used:
• BEFORE MODELING to view autocorrelation of the original time series data and consider appropriate models to capture that autocorrelation, or
• AFTER MODELING, to see if autocorrelation still exists in the model residuals, which would mean you need to try a different model.
As a first step, let’s look at an example autocorrelation plot to orient ourselves to the graph. Again we use the pricedata data set, where sale is the dependent variable that we are trying to
Look at the first bar, where the lag = 0. This bar will always equal 1.0, that is, an observation y[t] is always perfectly correlated with itself.
Then we look at the second bar, and see that the correlation of the current observation y[t] is about 0.6 with the observation one period (in our case one month) in the past (y[t-1]). This bar
extends outside the two standard error shaded blue area, which leads us to believe that it is significant. A spike at lag 1 for our sales data means that December’s sales are correlated with
November’s, November’s sales are correlated with October’s, and so on. This ACF(1) is called first-order autocorrelation and may be referred to as serial correlation.
We also see some inverse correlations at lags 5, 6, 7. This indicates that December sales may be negatively correlated with May, June and July sales.
The final significant positive correlation lag we see is at lag 12. That means that this December’s sales are likely correlated with last December’s sales. This also makes sense if we know that sales
tend to have annual cycles.
Always consider sensible lags at spikes (such as 12 for monthly data), 4 for data that may exhibit quarterly cycles, 24 for hourly data, and so on. Use your domain knowledge and common sense in
conjunction with the autocorrelation plots to try to develop the best forecasting model.
Autocorrelation Function (ACF)
The autocorrelation plots help you determine whether a time series is stationary or nonstationary. For example, if the ACF decreases rapidly, it indicates that the time series is stationary. In
contrast, if the ACF plot decays very slowly, that indicates that the series is nonstationary.
REMINDER: A stationary time series is one whose statistical properties (e.g., mean, variance, autocorrelation) are constant over time. Time series with either trends (increasing or decreasing over
time) or seasonality are not stationary.
If you have significant spikes in the ACF, there is systematic variation that can be extracted from the series, and it’s time to get busy building or modifying your forecasting model. If you have no
significant lags in the ACF (i.e., no spikes outside the blue shaded area shown above), then the series is white noise, and a model will not help you extract useful information.
RULE OF THUMB: The ACF helps you diagnose MA terms, and can support your diagnosis of a need for AR terms. If ACF bars drop off after 0, then try MA(1) model.
Partial Autocorrelation Function (PACF)
The PACF adjusts for all previous lags, in contrast to the ACF which does not adjust for other lags. So, for example, the PACF at lag 5 is the correlation that results after removing the effect of
correlations due to the terms at lags 4, 3, 2, and 1.
RULE OF THUMB: Use PACF to help select AR terms. If PACF bars drop off after 1 then try AR(1) model. PACF spikes indicate lags to try for a pure AR model.
Inverse Autocorrelation Function (IACF)
The IACF also helps you diagnose AR and MA terms. The IACF commonly suggests subset and seasonal autoregressive models better than the PACF, and it can be useful for detecting over-differencing. If
the data have been over-differenced, the IACF looks like an ACF from a nonstationary process (that is, it exhibits slow decay, as shown for the right hand side ACF two graphs up).
RULE OF THUMB: If the IACF dies off exponentially, it suggests an MA(1), i.e., you should try setting q=1.
White Noise Test
White noise is a purely random process. If your original time series data turns out to be white noise, there is no information to be gained by fitting a model. Just close your computer and go home.
If the series is not white noise, you can try an autocorrelation model, such as AR(1), an MA model, such as MA(1), or a low-order mixed ARMA model. You can also use the ACF, PACF, and IACF plots as
described above to help you decide what model to try first.
Once you have fit the model of your choice, test the model residuals to see if they are white noise. If the residuals are white noise, you have a reasonable model. If the residuals are not white
noise, then you can probably get additional information from a more complex model.
SAS Visual Forecasting generally uses a chi-square statistic (Ljung-Box formula shown below) to determine if the residuals are white noise.
and a[t] is the series being evaluated.
You decide if you how to set your type 1 error (a) rate; 0.01 or 0.05 are commonly used. Let’s say you select 0.01. A p-value less than 0.01 suggests that the residuals are not white noise, meaning
that your model was inadequate.
If the p-value is greater than 0.01, the white noise test plots show that you cannot reject the null hypothesis that the residuals are not correlated. Ok that last sentence has a lot of negatives,
but that’s just how old school statisticians like me speak. Nothing is ever accepted, it’s all about failing to reject. But in short, if the p-value is greater than 0.01 you conclude that your
residuals are correlated and are not white noise.
Put a different way, the White Noise Test tests that none of the autocorrelations of the series up to a given lag are significantly different from 0. If none of the autocorrelations of the series are
significantly different from 0, that indicates that no ARIMA model is needed for the series (because there is no information in the series to model). If the series is nonstationary, we expect the
white noise hypothesis to be rejected, i.e., suggesting that some of the autocorrelations may be significantly different from 0.
If the series is nonstationary, we can transform it to a stationary series by differencing. Differencing means that instead of modeling the sales values themselves, we model the difference from one
period to the next.
First order differencing computes the difference between consecutive observations. First order differencing is commonly used to make a nonstationary time series stationary if there is an upward or
downward trend in the data. Other orders of differencing can also be useful. For example, seasonal differencing for monthly data that exhibits seasonality, computes the difference between an
observation and the observation 12 time periods ago. For example, subtract January 2018 from January 2019, and so on.
PRO TIP: Don’t base your decision on a single plot. Use all of the plots along with your knowledge and understanding of the data and the domain to make the best informed decision. And don’t hesitate
to try different models! SAS Visual Forecasting makes it quick and easy to adjust your p,d,q. You can also use automated forecasting via either the visual interface or using the ATSM package in SAS
Studio to get you started with a decent model, that you can further tweak if you would like.
For ease of reading and following this article, I am not carefully noting that the ACF, PACF, and IACF shown here are actually the sample ACF, sample PACF, and sample IACF, not the population ACF,
PACF, and IACF. Because they are sample they only estimate the actual population and thus, we do not consider them significant if within the 2 standard deviations.
Also, remember, when you use a 95% confidence interval, you expect that 5% of your results will be spurious. That is, one in 20 of the bars in your plots is expected to be outside the 95% confidence
interval strictly by chance, not because it is significant.
Most textbooks on time series analysis, such as Pankratz (1983), discuss the theoretical autocorrelation functions for different kinds of ARMA models. A few resources are listed at the end of this
article if you wish to continue to expand your understanding of forecasting methods.
Diagnostic plots can be very helpful in determining the best model for your time series data, as well as determining if the model you selected was adequate. The three stages of ARIMA modeling as
outlined by Box and Jenkins in 1976 and reiterated in the SAS documentation are shown below.
References and More Information
• Forecasting Using SAS Software: A Programming Approach - SAS Education public course
• Box, G. E. P and Jenkins, G.M., (1976). Time series analysis: Forecasting and control, Holden-Day, San Francisco.
• Pankratz, A., (1983). Forecasting with Univariate Box-Jenkins Models: Concepts and Cases, Wiley, New York | {"url":"https://communities.sas.com/t5/SAS-Communities-Library/SAS-Visual-Forecasting-8-4-Interpreting-Results-and-Diagnostic/ta-p/581294","timestamp":"2024-11-08T01:54:53Z","content_type":"text/html","content_length":"146333","record_id":"<urn:uuid:3518b2ca-5a5a-4953-afb2-2a1dceea1056>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00352.warc.gz"} |
Operating Cash Flow: Formula, Calculation And Purpose
Operating cash flow (OCF) is a measure of the amount of cash generated by a company’s normal business operations. Operating cash flow indicates whether a company can generate sufficient positive cash
flow to maintain and grow its operations, otherwise, it may require external financing for capital expansion.
Cash Flow from Operations (Statement of Cash Flows)
Operating cash flow formula
The amount of cash generated by a company for a given period through its regular business operations is known as the operating cash flow. Depreciation, increases in accounts receivable, and other
noncash and nonoperating expenses from your net income are typically taken into account when calculating OCF.
The simplest formula for calculating the OCF is:
Total Cash Received for Sales – Cash Paid for Operating Expenses equals Operating Cash Flow.
There are additional ways to express the OCF formula using various terms:
OCF is calculated as (revenue minus operating expenses) plus depreciation, income taxes, and changes in working capital.
OCF is the sum of net income, depreciation, and any changes to working capital.
Net income less changes to working capital plus noncash expenses equals OCF.
Indirect method for the calculation of OCF
The net income from an income statement is the starting point for the indirect method of calculating operating cash flow. Noncash entries, such as depreciation, accounts receivable, and accounts
payable, are then added back to produce a cash amount. There are two formulas you can employ to compute OCF using the indirect method, depending on the information you have regarding the company’s
The first formula for calculating OCF using the indirect method is:
Working capital changes plus funds from operations are combined to form OCF.
Another formula for the indirect method of calculating OCF is:
OCF is calculated as net income plus depreciation, amortization, and net income adjustments, as well as changes in liabilities, inventories, receivables, and other operating activities.
All of the aforementioned figures can typically be found in cash flow statements as standard line items.
Direct method for the calculation of OCF
The direct method of calculating operating cash flow uses actual cash inflows and outflows on the cash flow statement and tracks all transactions during a financial period as cash. Although the
direct method for calculating OCF is straightforward and precise, it does not provide investors with much knowledge about the business, its operations, or its cash sources. Consequently, GAAP
mandates that businesses continue to conduct a separate reconciliation to the indirect method.
The direct method for calculating OCF has the following formula:
Operating Cash Flow = Total Revenue – Operating Expenses
When using the direct method, a business must take into account all cash payments and receipts. The items will differ from business to business. Some examples of cash items to be considered include:
What is operating cash flow (OCF)?
Why is operating cash flow important?
The ability of a business to make money overall is shown by its operating cash flow. A company should generally strive for a higher OCF because it will enable it to increase capital without the need
for funding or investments. Additionally, OCF metrics should be on the rise, indicating a rise in profitability.
For a company to sustain long-term viability, its operating cash flow must be positive. If the OCF is negative, the business will need to borrow money or raise more capital in order to continue
paying its debts.
OCF is used by financial analysts, investors, and lenders to assess a company’s overall health and profitability. The operating cash flow demonstrates whether the primary business operations generate
enough cash flow for the company to:
Operating cash flow vs. net income
The net income of a business and its operating cash flow may differ depending on the accounting rules used to produce the financial statements. Due to the revenue recognition principle and timing
requirements imposed by accounting regulations, there may be large discrepancies between a company’s operating cash flow and net income.
Operating cash flow example
Look at the numbers in the following table to determine the operating cash flow for a fictitious company, ABC Corporation, for the end of its fiscal year.
Comparing the OCF calculation for ABC Corporation using the two indirect method formulas is helpful. Using various types of information, both formulas yield the same results.
First indirect method formula
Funds from the operations of the ABC Corporation can be calculated using the first indirect method formula as follows:
OCF is equal to changes in working capital plus (net income minus deferred taxes, investment tax credit, plus depreciation, depletion, and amortization, plus other funds).
OCF = $34. 69B + ($59. 53B + -$32. 59B + $10. 9B + $4. 9B) = $77. 43 billion.
Second indirect method formula
The second indirect method formula can be used to determine ABC Corporation’s OCF in the following manner:
OCF is calculated as net income plus depreciation, amortization, and net income adjustments, as well as changes in liabilities, inventories, receivables, and other operating activities.
OCF = $59. 53B + $10. 9B + -$27. 694B + $9. 131B + $0. 828B + -$5. 322B + $30. 057B = $77. 43 billion.
What is operating cash flow?
The amount of money a company earns from its ongoing, regular business activities, such as producing and selling products or offering customers a service, is known as cash flow from operating
activities (CFO). It appears as the company’s first section on the cash flow statement.
How do you calculate operating cash flow?
How to calculate the operating cash flow formula
1. OCF is calculated as (revenue minus operating expenses) plus depreciation, income taxes, and any changes to working capital.
2. OCF is the sum of net income, depreciation, and working capital changes.
3. Net income less changes to working capital plus non-cash expenses equals OCF.
Is operating cash flow same as EBIT?
The cash that a company generates from its business operations after deducting capital expenditures is known as free cash flow. Investors can determine if a company has enough cash flow to pay its
bills by looking at its operating cash flow. | {"url":"https://carreersupport.com/operating-cashflow/","timestamp":"2024-11-02T06:17:37Z","content_type":"text/html","content_length":"71438","record_id":"<urn:uuid:7c14ee5e-50f5-4f29-a854-a94db33bbfae>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00117.warc.gz"} |
NDA - The Original Tutors
The National Defence Academy(NDA), Pune
The National Defence Academy (NDA) is one of the most prestigious institutions in India, known for producing exceptional officers for the Army, Navy, and Air Force. In this comprehensive guide, we
will provide you with all the essential details about the NDA, including eligibility criteria, application process, exam dates, selection process, and more.
You can take the NDA entrance exam right after class XI. Clear the UPSC exam and a 5-day Service Selection Board interview, pass your medicals, and you're in NDA. Three years in NDA and you will be a
much-improved person.
Apart from providing graduation degrees, NDA has the finest infrastructure for professional training. You'll find phenomenal opportunities to develop your personality and cultivate new interests.
There are 31 extra-curricular activities to choose from. You have aero modeling, golf, gliding, sailing, windsurfing, astronomy, photography, and many more
Key Dates for NDA II 2024
NDA II 2024 Events Dates
NDA II 2024 Application Start Date 15 May, 2024
NDA II 2024 Application Last Date 04th June, 2024 till 6:00 PM
NDA II 2024 Exam Date 1 September, 2024
NDA II 2024 Admit Card Download Date e-Admit Card issued Seven Days before the commencement of the examination
NDA II 2024 Result Date (Expected) November – December 2024 (Tentative)
NDA II 2024 Official Notification To be notified by UPSC
About NDA Entrance Exam
National Defence Academy: 370 ( 208 for Army, 42 for Navy, and 120 for Air Force (including 28 for ground Duties) )
Vacancies Per Course Naval Academy: (10+2 Cadet Entry Scheme): 30
Total: 400
Notify in Employment Probably in December and June or notified by UPSC
News Paper
Eligibility Criteria Described below
Age Between 15½ years and 18½ years
For Army Wing of National Defence Academy:—12th Class pass of the 10+2 pattern of School Education or equivalent 6 examinations conducted by a State Education Board or a
Qualification For Air Force and Naval Wings of National Defence Academy and for the 10+2 Cadet Entry Scheme at the Indian Naval Academy:—12th Class pass with Physics, Chemistry, and
Mathematics of the 10+2 pattern of School Education or equivalent conducted by a State Education Board or a University.
Candidates who are appearing in the 12th Class under the 10+2 pattern of School Education or equivalent examination can also apply for this examination.
Marital Status Unmarried
Examination Eligibility And Syllabus for NDA
1. The subjects of the written examination, the time allowed, and the maximum marks allotted to each subject will be as follows:—
Subject Code Duration Maximum Marks
Mathematics 01 150 min 300
General Ability Test 02 150 min 600
SSB Test/ Interview 900
Eligibility Criteria For NDA
(a) Nationality:
A candidate must be an unmarried male or female and must be:
(i) a citizen of India, or (ii) a subject of Bhutan, or (iii) a subject of Nepal, or (iv) a Tibetan refugee who came over to India before the 1st January 1962 with the intention of permanently
settling in India, or (v) a person of Indian origin who has migrated from Pakistan, Burma, Sri Lanka and East African Countries of Kenya, Uganda, the United Republic of Tanzania, Zambia, Malawi,
Zaire, and Ethiopia or Vietnam with the intention of permanently settling in India.
(b) Age Limits, Sex, and Marital Status:
Only unmarried male and female candidates between 15^1/[2] years and 18^1/[2] years are eligible to apply.
(c) Educational Qualifications:
(i) For Army Wing of National Defence Academy (NDA): —12th Class pass of the 10+2 pattern of School Education or equivalent examination conducted by a State Education Board or a University.
(ii) For Air Force and Naval Wings of National Defence Academy and for the 10+2 Cadet Entry Scheme at the Indian
Naval Academy: —12th Class pass of the 10+2 pattern of School Education or equivalent with Physics and Mathematics conducted by a State Education Board or a University.
**** Candidates who are appearing in the 12th Class under the 10+2 pattern of School Education or equivalent examination can also apply for this examination
HOW TO APPLY:
Candidates are required to apply Online by using the website http://upsconline.nic.in Detailed instructions for filling up online applications are available on the above-mentioned website.
Two-stage selection procedure based on Psychological Aptitude Test and Intelligence Test has been introduced at Selection Centres/Air Force Selection Boards/Naval Selection Boards. All the candidates
will be put to stage-one test on the first day of reporting at Selection Centres/Air Force Selection Boards/Naval Selection Boards. Only those candidates who qualify at stage one will be admitted to
the second stage/remaining tests. Those candidates who qualify for stage II will be required to submit the Original Certificates along with one photocopy each of: (i) Original Matriculation pass
certificate or equivalent in support of date of birth, (ii) Original 10+2 pass certificate or equivalent in support of educational qualification.
Candidates who appear before the Services Selection Board and undergo the test there will do so at their own risk and will not be entitled to claim any compensation or other relief from the
Government in respect of any injury which they may sustain in the course of or as a result of any of the tests given to them at the Services Selection Board whether due to the negligence of any
person or otherwise. Parents or guardians of the candidates will be required to sign a certificate to this effect.
To be acceptable, candidates for the Army/Navy/Naval Academy and Air Force should secure the minimum qualifying marks separately in (i) Written examination as fixed by the Commission at their
discretion and (ii) Officer Potentiality Test as fixed by the Services Selection Board at their discretion. Over and above candidates for the Air Force, and all the SSB qualified candidates as per
their willingness, eligibility, and preference for flying branch of the Air Force, should separately qualify the CPSS.
Subject to these conditions the qualified candidates will then be placed in a single combined list on the basis of total marks secured by them in the Written Examination and the Services Selection
Board Tests. The final allocation/selection for admission to the Army, Navy, and Air Force of the National Defence Academy and 10+2 Cadet Entry Scheme of the Indian Naval Academy will be made upto
the number of vacancies available subject to eligibility, medical fitness, and merit-cum-preference of the candidates. The candidates who are eligible to be admitted to multiple Services/Courses will
be considered for allocation/selection with reference to their order or preferences and in the event of their final allocation/ selection to one Service/Course, they will not be considered for
admission to other remaining Services/Courses.
It consists of the Syllabus of both Paper Mathematics and General Ability Test For NDA
Paper I Mathematics
1. ALGEBRA: Concept of set, operations on sets, Venn diagrams. De Morgan laws, Cartesian product, relation, equivalence relation. Representation of real numbers on a line. Complex numbers—basic
properties, modulus, argument, cube roots of unity. Binary system of numbers. Conversion of a number in decimal system to binary system and vice-versa. Arithmetic, Geometric and Harmonic
progressions. Quadratic equations with real coefficients. Solution of linear inequations of two variables by graphs. Permutation and Combination. Binomial theorem and its applications. Logarithms and
their applications.
2. MATRICES AND DETERMINANTS: Types of matrices, operations on matrices. Determinant of a matrix, basic properties of determinants. Adjoint and inverse of a square matrix, Applications-Solution of a
system of linear equations in two or three unknowns by Cramer’s rule and by Matrix Method.
3. TRIGONOMETRY: Angles and their measures in degrees and in radians. Trigonometrical ratios. Trigonometric identities Sum and difference formulae. Multiple and Sub-multiple angles. Inverse
trigonometric functions. Applications-Height and distance, properties of triangles.
4. ANALYTICAL GEOMETRY OF TWO AND THREE DIMENSIONS: Rectangular Cartesian Coordinate system. Distance formula. Equation of a line in various forms. The angle between two lines. Distance of a point
from a line. Equation of a circle in standard and a general form. Standard forms of parabola, ellipse, and hyperbola. Eccentricity and axis of a conic. Point in a three-dimensional space, the
distance between two points. Direction Cosines and direction ratios. Equation two points. Direction Cosines and direction ratios. Equation of a plane and a line in various forms. The angle between
two lines and the angle between two planes. Equation of a sphere.
5. DIFFERENTIAL CALCULUS: Concept of a real-valued function–domain, range, and graph of a function. Composite functions, one-to-one, onto, and inverse functions. The notion of limit, Standard
limits—examples. Continuity of functions—examples, algebraic operations on continuous functions. Derivative of function at a point, geometrical and physical interpretation of a
derivative—applications. Derivatives of sum, product, and quotient of functions, a derivative of a function with respect to another function, a derivative of a composite function. Second-order
derivatives. Increasing and decreasing functions. Application of derivatives in problems of maxima and minima.
6. INTEGRAL CALCULUS AND DIFFERENTIAL EQUATIONS: Integration as inverse of differentiation, integration by substitution and by parts, standard integrals involving algebraic expressions,
trigonometric, exponential and hyperbolic functions. Evaluation of definite integrals—determination of areas of plane regions bounded by curves—applications. Definition of order and degree of a
differential equation, formation of a differential equation by examples. A general and particular solution of differential equations, solution of the first order and first-degree differential
equations of various types—examples. Application in problems of growth and decay.
7. VECTOR ALGEBRA: Vectors in two and three dimensions, magnitude, and direction of a vector. Unit and null vectors, the addition of vectors, scalar multiplication of a vector, scalar product, or dot
product of two vectors. Vector product or cross product of two vectors. Applications—work done by a force and moment of a force and in geometrical problems.
8. STATISTICS AND PROBABILITY: Statistics: Classification of data, Frequency distribution, cumulative frequency distribution—examples. Graphical representation—Histogram, Pie Chart, frequency
polygon— examples. Measures of Central tendency—Mean, median, and mode. Variance and standard deviation—determination and comparison. Correlation and regression. Probability: Random experiment,
outcomes, and associated sample space, events, mutually exclusive and exhaustive events, impossible and certain events. Union and Intersection of events. Complementary, elementary, and composite
events. Definition of probability—classical and statistical—examples. Elementary theorems on probability—simple problems. Conditional probability, Bayes’ theorem—simple problems. Random variable as
function on a sample space. Binomial distribution, examples of random experiments giving rise to Binominal distribution.
Paper II General Ability Test
Part ‘A’ ENGLISH: - The question paper in English will be designed to test the candidate’s understanding of English and workmanlike use of words. The syllabus covers various aspects like Grammar and
usage, vocabulary, comprehension, and cohesion in extended text to test the candidate’s proficiency in English.
Part ‘B’ GENERAL KNOWLEDGE
The question paper on General Knowledge will broadly cover the subjects: Physics, Chemistry, General Science, Social Studies, Geography, and Current Events. - The syllabus given below is designed to
indicate the scope of these subjects included in this paper. The topics mentioned are not to be regarded as exhaustive and questions on topics of similar nature not specifically mentioned in the
syllabus may also be asked. Candidates’ answers are expected to show their knowledge and intelligent understanding of the subject.
Section ‘A’ (Physics) Physical Properties and States of Matter, Mass, Weight, Volume, Density and Specific Gravity, Principle of Archimedes, Pressure Barometer. The motion of objects, Velocity, and
Acceleration, Newton’s Laws of Motion, Force and Momentum, Parallelogram of Forces, Stability and Equilibrium of bodies, Gravitation, elementary ideas of work, Power, and Energy. Effects of Heat,
Measurement of Temperature and Heat, change of State and Latent Heat, Modes of transference of Heat. Sound waves and their properties, Simple musical instruments. Rectilinear propagation of Light,
Reflection, and refraction. Spherical mirrors and Lenses, Human Eye. Natural and Artificial Magnets, Properties of a Magnet, Earth as a Magnet. Static and Current Electricity, conductors and
Nonconductors, Ohm’s Law, Simple Electrical Circuits, Heating, Lighting and Magnetic effects of Current, Measurement of Electrical Power, Primary and Secondary Cells, Use of X-Rays. General
Principles in the working of the following: Simple Pendulum, Simple Pulleys, Siphon, Levers, Balloon, Pumps, Hydrometer, Pressure Cooker, Thermos Flask, Gramophone, Telegraphs, Telephone, Periscope,
Telescope, Microscope, Mariner’s Compass; Lightening Conductors, Safety Fuses.
Section ‘B’ (Chemistry) Physical and Chemical changes. Elements, Mixtures and Compounds, Symbols, Formulae and simple Chemical Equations, Law of Chemical Combination (excluding problems). Properties
of Air and Water. Preparation and Properties of Hydrogen, Oxygen, Nitrogen, and carbon dioxide, Oxidation, and Reduction. Acids, bases, and salts. Carbon—different forms. Fertilizers—Natural and
Artificial. The material used in the preparation of substances like Soap, Glass, Ink, Paper, Cement, Paints, Safety Matches, and gunpowder. Elementary ideas about the structure of Atoms, Atomic
Equivalent and Molecular Weights, and Valency.
Section ‘C’ (General Science) Difference between living and non-living. Basis of Life—Cells, Protoplasms, and Tissues. Growth and Reproduction in Plants and Animals. Elementary knowledge of the Human
Body and its important organs. Common Epidemics, their causes, and prevention. Food—Source of Energy for man. Constituents of food, Balanced Diet. The Solar System—Meteors and Comets, Eclipses.
Achievements of Eminent Scientists.
Section ‘D’ (History, Freedom Movement, etc.) A broad survey of Indian History, with emphasis on Culture and Civilisation. Freedom Movement in India. Elementary study of Indian Constitution and
Administration. Elementary knowledge of Five Year Plans of India. Panchayati Raj, Co-operatives and Community Development. Bhoodan, Sarvodaya, National Integration and Welfare State, Basic Teachings
of Mahatma Gandhi. Forces shaping the modern world; Renaissance, Exploration, and Discovery; War of American Independence. French Revolution, Industrial Revolution, and Russian Revolution. Impact of
Science and Technology on Society. Concept of one World, United Nations, Panchsheel, Democracy, Socialism, and Communism. Role of India in the present world.
Section ‘E’ (Geography) The Earth, its shape and size. Lattitudes and Longitudes, Concept of time. International Date Line. Movements of Earth and their effects. Origin of Earth. Rocks and their
classification; Weathering—Mechanical and Chemical, Earthquakes and Volcanoes. Ocean Currents and Tides Atmosphere and its composition; Temperature and Atmospheric Pressure, Planetary Winds, Cyclones
and Anti-cyclones; Humidity; Condensation and Precipitation; Types of Climate, Major Natural regions of the World. Regional Geography of India—Climate, Natural vegetation. Mineral and Power
resources; location and distribution of agricultural and Industrial activities. Important Sea ports and main sea, land, and air routes of India. Main items of Imports and Exports of India.
Section ‘F’ (Current Events) Knowledge of Important events that have happened in India in recent years. Current important world events. Prominent personalities—both Indian and International including
those connected with cultural activities and sports.
NOTE: Out of the maximum marks assigned to part ‘B’ of this paper, questions on Sections ‘A’, ‘B’, ‘C’, ‘D’, ‘E’ and ‘F’ will carry approximately 25%, 15%, 10%, 20%, 20%, and 10% weightage
Intelligence and Personality Test (SSB)
The SSB procedure for NDA consists of two stages of Selection processes - stage I and stage II. Only those candidates who clear stage I are permitted to appear for stage II. The details are:
(a) Stage I comprises Officer Intelligence Rating (OIR) tests are Picture Perception * Description Test (PP&DT). The candidates will be shortlisted based on the combination of performance in the OIR
Test and PP&DT.
(b) Stage II Comprises Interviews, Group Testing Officer Tasks, Psychology Tests, and the Conference. These tests are conducted over 4 days. The details of these tests are given on the website http:/
/joinindianarmy.nic.in/ The personality of a candidate is assessed by three different assessors viz. The Interviewing Officer (IO), Group Testing Officer (GTO), and the Psychologist. There is no
separate weightage for each test. The mks are allotted by assessors only after taking into consideration the performance of the candidate holistically in all the tests. In addition, marks for the
Conference are also allotted based on the initial performance of the Candidate in the three techniques and the decision of the Board. All these have equal weightage. The various tests of IO, GTO, and
Psych are designed to bring out the presence/absence of Officer Like Qualities and their trainability in a candidate. Accordingly, candidates are Recommended or Not Recommended at the SS
NDA Exam Cut-Off Marks
NDA EXAM YEAR WRITTEN EXAM MARKS WRITTEN + SSB
NDA & NA (II) 2023 To Be Updated To Be Updated
NDA & NA (I) 2023 301 664
NDA & NA (II) 2022 316 678
NDA & NA (I) 2022 360 720
NDA & NA (II) 2021 355 726
NDA & NA (I) 2021 343 709
NDA & NA (II) 2020 355 723
NDA & NA (I) 2020 355 723
NDA & NA (I) 2019 342 704
NDA & NA (II) 2018 325 688
NDA & NA (I) 2018 338 705
NDA & NA (II) 2017 258 624
NDA & NA (I) 2017 342 708
NDA & NA (II) 2016 229 602
NDA & NA (I) 2016 288 656
NDA & NA (II) 2015 269 637
NDA & NA (I) 2015 306 674
NDA & NA (II) 2014 283 656
NDA & NA (I) 2014 360 722
NDA & NA (II) 2013 360 721
NDA & NA (I) 2013 333 698
NDA & NA (II) 2012 335 699
*Eligibility Criteria is also subject to change please Visit the Notification on the Official Website of Union Service Public Commission:- http://upsc.gov.in/
For Reading Eligibility Criteria for Air Force X/Y Click on http://www.airforcecoaching.in/air-force/
For Reading the Eligibility Criteria for the Navy Click on http://www.airforcecoaching.in/navy/ | {"url":"https://theoriginaltutors.com/nda/","timestamp":"2024-11-09T09:22:02Z","content_type":"text/html","content_length":"67806","record_id":"<urn:uuid:f2719577-e34d-445d-b189-4e023d2415b7>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00561.warc.gz"} |
Physics:Continuous phase modulation
Continuous phase modulation (CPM) is a method for modulation of data commonly used in wireless modems. In contrast to other coherent digital phase modulation techniques where the carrier phase
abruptly resets to zero at the start of every symbol (e.g. M-PSK), with CPM the carrier phase is modulated in a continuous manner. For instance, with QPSK the carrier instantaneously jumps from a
sine to a cosine (i.e. a 90 degree phase shift) whenever one of the two message bits of the current symbol differs from the two message bits of the previous symbol. This discontinuity requires a
relatively large percentage of the power to occur outside of the intended band (e.g., high fractional out-of-band power), leading to poor spectral efficiency. Furthermore, CPM is typically
implemented as a constant-envelope waveform, i.e., the transmitted carrier power is constant. Therefore, CPM is attractive because the phase continuity yields high spectral efficiency, and the
constant envelope yields excellent power efficiency. The primary drawback is the high implementation complexity required for an optimal receiver.
Phase memory
Each symbol is modulated by gradually changing the phase of the carrier from the starting value to the final value, over the symbol duration. The modulation and demodulation of CPM is complicated by
the fact that the initial phase of each symbol is determined by the cumulative total phase of all previous transmitted symbols, which is known as the phase memory. Therefore, the optimal receiver
cannot make decisions on any isolated symbol without taking the entire sequence of transmitted symbols into account. This requires a maximum-likelihood sequence estimator (MLSE), which is efficiently
implemented using the Viterbi algorithm.
Phase trajectory
Minimum-shift keying (MSK) is another name for CPM with an excess bandwidth of 1/2 and a linear phase trajectory. Although this linear phase trajectory is continuous, it is not smooth since the
derivative of the phase is not continuous. The spectral efficiency of CPM can be further improved by using a smooth phase trajectory. This is typically accomplished by filtering the phase trajectory
prior to modulation, commonly using a raised cosine or a Gaussian filter. The raised cosine filter has zero crossings offset by exactly one symbol time, and so it can yield a full-response CPM
waveform that prevents intersymbol interference (ISI).
Partial response CPM
Partial-response signaling, such as duo-binary signaling, is a form of intentional ISI where a certain number of adjacent symbols interfere with each symbol in a controlled manner. A MLSE must be
used to optimally demodulate any signal in the presence of ISI. Whenever the amount of ISI is known, such as with any partial-response signaling scheme, MLSE can be used to determine the exact symbol
sequence (in the absence of noise). Since the optimal demodulation of full-response CPM already requires MLSE detection, using partial-response signaling requires little additional complexity, but
can afford a comparatively smoother phase trajectory, and thus, even greater spectral efficiency. One extremely popular form of partial-response CPM is GMSK, which is used by GSM in most of the
world's 2nd generation cell phones. It is also used in 802.11 FHSS, Bluetooth, and many other proprietary wireless modems.
Continuous-phase frequency-shift keying
Continuous-phase frequency-shift keying (CPFSK) is a commonly used variation of frequency-shift keying (FSK), which is itself a special case of analog frequency modulation. FSK is a method of
modulating digital data onto a sinusoidal carrier wave, encoding the information present in the data to variations in the carrier's instantaneous frequency between one of two frequencies (referred to
as the space frequency and mark frequency). In general, a standard FSK signal does not have continuous phase, as the modulated waveform switches instantaneously between two sinusoids with different
As the name suggests, the phase of a CPFSK is in fact continuous; this attribute is desirable for signals that are to be transmitted over a bandlimited channel, as discontinuities in a signal
introduce wideband frequency components. In addition, some classes of amplifiers exhibit nonlinear behavior when driven with nearly discontinuous signals; this could have undesired effects on the
shape of the transmitted signal.
If a finitely valued digital signal to be transmitted (the message) is m(t), then the corresponding CPFSK signal is
[math]\displaystyle{ s(t) = A_c \cos\left(2 \pi f_c t + D_f \int_{-\infty}^{t} m(\alpha) d \alpha\right)\, }[/math]
where A[c] represents the amplitude of the CPFSK signal, f[c] is the base carrier frequency, and D[f] is a parameter that controls the frequency deviation of the modulated signal. The integral
located inside of the cosine's argument is what gives the CPFSK signal its continuous phase; an integral over any finitely valued function (which m(t) is assumed to be) will not contain any
discontinuities. If the message signal is assumed to be causal, then the limits on the integral change to a lower bound of zero and a higher bound of t.
Note that this does not mean that m(t) must be continuous; in fact, most ideal digital data waveforms contain discontinuities. However, even a discontinuous message signal will generate a proper
CPFSK signal.
See also
Notation for the CPFSK waveform was taken from:
• Leon W. Couch II, "Digital and Analog Communication Systems, 6th Edition", Prentice-Hall, Inc., 2001. ISBN:0-13-081223-4
• [1] S. Cheng, R. Iyer Sehshadri, M.C. Valenti, and D. Torrieri, The capacity of noncoherent continuous-phase frequency shift keying, in Proc. Conf. on Info. Sci. and Sys (CISS), (Baltimore, MD),
Mar. 2007.
Original source: https://en.wikipedia.org/wiki/Continuous phase modulation. Read more | {"url":"https://handwiki.org/wiki/Physics:Continuous_phase_modulation","timestamp":"2024-11-08T05:05:24Z","content_type":"text/html","content_length":"44202","record_id":"<urn:uuid:3abbfea9-db65-40ec-9ae9-beaf61579b86>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00400.warc.gz"} |
Substitution Calculator | Find Two Variables Value in Just 1 Click!
What is a Substitution Calculator?
Substitution calculator with steps is an online digital tool that enables you to find the value of two variables using the substitution method. It determines the variables' values from the given
linear system equation in a couple of seconds.
Substitution Method:
The substitution method is the commonly used method in algebra to find the value of a pair of variables when one variable (x variable value) is isolated from one equation. Then you put the variable
value into the second equation to find the second variable (y variable value).
After finding y, substitute the y value of the equation to get the value of the first variable.
Formula Behind Substitution Method Calculator:
The substitution method consists of a system of linear equations where a pair of variables is involved. The substitution calculator uses the following formula to solve the system of equations by
$$ ax^1 + by^1 \;=\; c^1 $$
$$ ax^2 + by^2 \;=\; c^2 $$
Can Substitution Math Calculator Give Solution in Steps?
Yes, our substitution method calculator gives the solution in steps. Our tool has advanced features that enable you to find the solution of linear equation variables.
Let’s understand how the substitute calculator solves the system of equations problem by substitution:
Working of System of Equations Calculator Substitution:
When you add two equations in the substitute method calculator, it checks the given linear equation. After checking, it knows which variable is separated from the equations (either x or y variable).
Then isolated variable value is substituted in the second equation to find the variable value. Suppose you want to find the value of x, therefore substitute the y value in the given equation. After
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Now substitute the x value in one of the given equations to find the y variable value. Substitution math calculator provides you solution of a linear equation of both the variables (x or y value).
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Solve by Substitution - An Example
The system of equations calculator substitution can be used to solve the substitution problems easily but it’s crucial to know the manual calculations. So an example is given below,
Solve the system by substitution,
\begin{matrix} -x & +y & =\;-5 \\ 2x & -5y & =\; 1 \\ \end{matrix}
First, it separates the y value from equation 1 as:
$$ -x + y \;=\; -5 $$
$$ y \;=\; x - 5 $$
Now substitute the y = x + 5 value into equation 2 to get the value of x.
$$ 2x - 5y \;=\; 1 $$
$$ 2x - 5(x-5) \;=\; 1 $$
$$ 2x - 5x + 25 \;=\; 1 $$
$$ -3x \;=\; -24 $$
$$ x \;=\; 8 $$
Substitute the x = 8 value of one of the given equations to get the value of y.
$$ -(8) + y \;=\; -5 $$
$$ y \;=\; 3 $$
So the solution is (8, 3).
Now combine the both equations and solve by substitution
$$ -x + y \;=\; -5 $$
$$ -(8) + (3) \;=\; -5 $$
$$ 2x - 5y \;=\; 1 $$
$$ 2(8) - 5(3) \;=\; 1 $$
How to Use Substitution Calculator?
Substitution method calculator has a user-friendly interface that allows you to solve various types of variable values from the given equation instantly. You just need to follow the simple steps for
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• Check given linear equations before pressing the calculate button to get the solution of variable values.
• Click the “Calculate” button for the solution of algebraic equation questions.
• Click the “Recalculate” button for evaluation of the linear equation examples.
• If you do not know about the working procedure of our substitution method 3 variables calculator then use the load example for the calculation to get an idea of its accuracy.
Final Result of Substitute Calculator:
Substitution calculator gives you a solution of linear equation problem instantly when you click on the calculate button. It include as:
Result Box:
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Steps Box:
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Calculator for substitution method is a wonderful tool for solving linear equations to get the variable root values. It gives you multiple benefits whenever you use it for the evaluation of
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• System of equations calculator substitution is used for practice to get a strong hold on algebraic equation problems.
• It has a simple design, so even a beginner can easily use it for the solution of variables.
• You can operate the method of substitution calculator on a desktop, mobile, or laptop. | {"url":"https://pinecalculator.com/substitution-calculator","timestamp":"2024-11-12T05:31:47Z","content_type":"text/html","content_length":"51493","record_id":"<urn:uuid:36d73a85-cf40-4fcb-8d0e-214af93182d3>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00384.warc.gz"} |
Lines and Angles worksheet for class 7 | myCBSEguide
Lines and Angles worksheet for class 7
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CBSE worksheets for Lines and Angles worksheet for class 7 in PDF for free download. Mathematics worksheets for class 7 CBSE includes worksheets on Lines and Angles as per NCERT syllabus. CBSE class
7 worksheets as PDF for free download Lines and Angles worksheets. Users can download and print the worksheets on class 7 Mathematics Lines and Angles for free.
Lines and Angles worksheet for class 7 Important Topics
• Introduction to Lines and Angles
• Related Angles: Complementary and supplementary angles
• Pairs of Lines : Intersecting Line and Transversal
• Check for parallel lines
Some important Facts about Lines and Angles worksheet for class 7
1. We recall that
1. A line-segment has two end points.
2. A ray has only one end point (its vertex); and
3. A line has no end points on either side.
1. When two lines intersect (looking like the letter X) we have two pairs of opposite angles. They are called vertically opposite angles. They are equal in measure.
2. A transversal is a line that intersects two or more lines at distinct points.
3. A transversal gives rise to several types of angles.
NCERT class 7 Mathematics Solved Worksheets
• Chapter 1 – Integers
• Chapter 2 – Fractions and Decimals
• Chapter 3 – Data Handling
• Chapter 4 – Simple Equations
• Chapter 5 – Lines and Angles
• Chapter 6 – Practical Geometry
• Chapter 7 – The Triangle and its Properties
• Chapter 8 – Congruence of triangles
• Chapter 9 – Comparing Quantities
• Chapter 10 – Rational numbers
• Chapter 11 – Perimeter and Area
• Chapter 12 – Algebraic Expressions
• Chapter 13 – Exponents and Powers
• Chapter 14 – Symmetry
• Chapter 15 – Visualizing Solid Shapes
CBSE Worksheets for class 7 Mathematics in PDF
If two lines are parallel, then you know that a transversal gives rise to pairs of equal corresponding angles, equal alternate interior angles and interior angles on the same side of the transversal
being supplementary.
When two lines are given, is there any method to check if they are parallel or not? You need this skill in many life-oriented situations.
A draftsman uses a carpenter’s square and a straight edge (ruler) to draw these segments (Fig 5.30). He claims they are parallel. How? Are you able to see that he has kept the corresponding angles to
be equal? (What is the transversal here?)
To download Printable worksheets for class 7 Mathematics and Science; do check myCBSEguide app or website. myCBSEguide provides sample papers with solution, test papers for chapter-wise practice,
NCERT solutions, NCERT Exemplar solutions, quick revision notes for ready reference, CBSE guess papers and CBSE important question papers. Sample Paper all are made available through the best app for
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Evenly Distributing Points in a Triangle.
Most two dimensional quasirandom methods focus on sampling over a unit square. However, sampling evenly over the triangle is also very important in computer graphics. Therefore, I describe a
simple and direct construction method for a point sequence to evenly cover an arbitrary shaped triangle.
Figure 1. A new direct method for constructing an open (infinite) low discrepancy quasirandom sequence over an triangle of arbitrary shape and size. Shown are the point distributions for twelve
random triangles for the first 150 points.
Low discrepancy sequences that evenly sample/cover a square have been extensively studied for nearly a hundred years. Most of these quasirandom sequences can be extended to rectangles by simple
dilation without much deterioration of discrepancy.
However, this post looks at the interesting and important extension of low discrepancy sequences over an arbitrary triangle.
As far as I can deduce, very little work has been done on constructing evenly distributed points sets and sequences over triangles. Notable efforts in recent years by Basu [2014] and Pillands and
Cools, [2005] and Brandolini [2013] represent the main papers – if not the only papers on this subject. For a post on Basu’s approach when applied to graphics rendering, see here.
These authors generally take a very theoretical and analytical approach to this issue and almost solely focus on either the regular or right-angled triangle/simplex. In contrast, this post is mainly
intended to aid in the development of some practical techniques for graphics rendering.
This post describes a simple and direct method that requires no acceptance/rejection, no dropping or recursion; and most importantly it can be applied to triangles of arbitrary shape and size.
This post has four sections
1. Quasirandom Sequences in the square
2. Mapping the unit square to a triangle.
3. Reducing Aliasing
4. Further generalizations
1. Quasirandom Point Sequences
You might think it is easy to place 100 points in a square such that the minimum distance between neighboring points is as large possible. But what if you had to place 13 points? 47? … or how about
2019 points?!
It turns out that low discrepancy point sequences offer a systematic way to solve this problem. There are many low discrepancy quasirandom sequences including the simple Halton sequence to the more
sophisticated Sobol’ sequence. Each of these particular sequences have their advantages and disadvantages. For example, Halton sequences tend to be more useful when placing objects in a region, as
the local distance metrics such as nearest neighbor are well optimized. In contrast, the Sobol’ sequence tends to have more ‘clumping’, however, its global point distribution is very even, and so it
has excellent quadrature properties.
There is another sequence I love to use, which has both excellent local as well as global properties. This is the $R_2$ sequence, as described in detail in my other post, “The Unreasonable
effectiveness of Quasirandom Sequences”.
In brief, we define the infinite two-dimensional sequence $R_2$, such that
$$ \pmb{t}_n = \left\{ n \pmb{\alpha} \right\} \quad n=1,2,3,… $$
$$ \textrm{where} \quad \pmb{\alpha} =\left(\frac{1}{g}, \frac{1}{g^2} \right), $$
$$ \textrm{and} \; g \simeq 1.32471795572 \tag{1}$$
More details on this special value of $g$, which is often called the ‘plastic constant’ can be found on Wikipedia or Mathworld.
In summary, figure 2 shows a comparison of various low discrepancy sequences (and simple uniform random sampling for comparison). You can see, that random sampling exhibits clustering of points and
that there are also regions that contain no points at all (‘white noise’), whereas a low discrepancy sequence quasirandom sequence is a method of constructing an (infinite) sequence points in a
deterministic manner such that, at any stage, the placed points are evenly distributed across the entire space.
Figure 2. An illustration of the first 150 terms of various low discrepancy quasirandom sequences . Note that the all of them produce more evenly spaced points than simple uniform random (bottom
2. Mapping the unit square to a triangle.
There are three well known methods to allow one to uniformly random sample from a triangle. These are:
1. the Parallelogram method ,
2. the Kraemer Method and
3. the inverse cumulative distribution method.
Figure 3. The Parallelogram method
The parallelogram method is as follows.
For a triangle $\pmb{ABC}$, consider the parallelogram $\pmb{ABCA’}$.
For a point in the unit square $(r_1,r_2)$, define the point $\pmb{P} $ such that $ \pmb{P} = r_1 \pmb{AC} + r_2 \pmb{AB} $.
This point will always fall inside the parallelogram. However, if it occurs in the complementary triangle $\pmb{ABC’}$, then we need to reflect it back into the triangle $\pmb{ABC}$.
That is, if $r_1+r_2<1$, then $ \pmb{P} = r_1 \pmb{AC} + r_2 \pmb{AB} $, but if $r_1+r_2>1$, then $ \pmb{P} = (1-r_1) \pmb{AC} + (1-r_2) \pmb{AB} $.
However, it is very important to understand that just because these methods allow uniform sampling from a triangle, this does not imply that such a transform will preserve the even spacing (ie low
discrepancy) of our quasirandom point distributions.
For example, for the parallelogram method, the reflection may frequently cause a point to be placed very close to another existing point. This obviously breaks the critical low discrepancy structure
and the effect will be appear as aliasing/banding.
Similarly the inverse cumulative distribution method applies a non-linear distortion to the points. For the Halton and Sobol sequences this means that frequently two points are pushed very close to
each other.
Figure 3 shows how well the low discrepancy is preserved for each of the different quasirandom sequences when you transform the region from a unit square to a triangle using the parallelogram method.
Figure 3. A comparison of how different quasirandom sequences are transformed to the triangle. Top is the Halton sequence, middle is the Sobol sequence and bottom is the $R_2$ sequence. Notice that
in the Halton images, there is noticeable clumping of points, and in the Sobol sequence there is notable banding. In contrast, for the $R_2$ sequence the points are very evenly distributed and there
is almost no discernible clumping or banding.
Of the three different methods to triangulate, and the numerous different quasirandom sequences to choose from, the parallelogram method applied to the $R_2$ sequence method is the only combination
that consistently produces acceptable results in terms of preserving low discrepancy without aliasing.
The excellent results of this combination can be further seen in figure 4.
Figure 4. One can see that the transformed $R_2$ sequence provides a very simple but effective method to evenly distribute a set of $N$ points in a triangle. It works for both acute and obtuse
3. Other considerations
The parallelogram method requires selecting two sides of the triangle to be picked as the basis vectors.
If the vertices are labelled in a random order, then the point patterns are typically as seen in figure 5.
Figure 6. Resulting point patterns, if the ordering of the vertices is randomly chosen. Notice that in most cases, aliasing is clearly visible.
However, it turns out that a judicious choice of vertex ordering, allows aliasing to be significantly minimized. Most notably, if the triangle is labelled such that $A$ is the largest angle (and thus
opposite the largest side), and thus the two sides $\pmb{AB}$ and $\pmb{AC}$ are used for the two sides of the parallelogram.
If this protocol is adopted, then the resulting point patterns can be seen in figures 1, 3 and 4 above.
However, even with the particular ordering of vertices, aliasing effects can still be seen in some instances. These are most noticeable when in triangles when one of the angles is small. For obtuse
angles, if the smallest angle is less than 30 degrees, some aliasing is visible, and for acute triangles, if the smallest angles is less than 20 degrees aliasing becomes visible. (If the triangles to
sampled are part of a Delaunay mesh, then these aliasing issues may be minimal, as Delaunay triangulation is specifically designed to minimize the existence of small-angled triangles.)
Other shapes.
Unfortunately the parallelogram technique can not be used for other shapes, as it exploits the symmetry of the triangle. For some shapes, it is possible to get good results using the inverse
cumulative distribution technique. This method is briefly described.
If we revisit figure 3, we can define point P in a manner that guarantees that it will always be inside the triangle, and therefore not require reflecting. Namely, we define $\pmb{P} $ such that
$ \pmb{P} = r_2 \sqrt{r_1} \pmb{AC} + (1-r_2) \sqrt{r1} \pmb{AB} $.
This additional $\sqrt{r_1}$ term not only ensures that the point $P$ always lies inside the triangle, but also (more importantly) that the sampling is uniform. Note that the square root is required
because we need an area-preserving transformation, and area increases according to length squared.
More formally, the fractional distance to move in the direction of $\pmb{AB}$ decreases linearly from $z = 1$ to $z=0$. Thus, the cumulative fractional distance is $\int_0^1 z dx = z^2/2$, which is
a function of $z^2$, and thus the inverse of this is proportional to $\sqrt{z}$.
Similarly, if we are now attempting to sample from an arbitrary shaped function, we need to use the inverse of the cumulative distribution function. Below is an example of how the $R_2$ low
discrepancy sequence can be transformed onto the area bounded under the Gaussian curve whilst generally preserving even spacing between points.
In this case, let $g(z)$ be the inverse cumulative distribution function of the Normal Gaussian function. The then points $P$ will be defined as $$p(r_1,r_2) = \left( g(r_1), r_2 f (g(r_1)) \right )
If you use the inverse CDF method, it is often best to used a jittered version of a low discrepancy sequence to minmize aliasing. In the example below, i used a jittered version of $R_2$ with a
jitter parameter of 0.5. (see “A simple method to construct isotropic quasirandom blue noise“).
Thank you to Sergeij Reich for his valuable discussion and analysis on an earlier version of this post.
My name is Martin Roberts. I have a PhD in theoretical physics. I love maths and computing. I’m open to new opportunities – consulting, contract or full-time – so let’s have a chat on how we can
work together!
Come follow me on Twitter: @Techsparx!
My other contact details can be found here.
Other Posts you may like
15 Replies to “Evenly Distributing Points in a Triangle.”
1. Nice writeup, Ive enjoyed your other posts as well.
I’ve tried the same approach a while ago but found that generating a point inside a parallelogram and mirroring along diagonal if it’s outside of the triangle gives better results (it’s not the
same as the Kramer method), at least visually.
There are three problems with the Fibonacci lattice that make it worse:
1. The aliasing is non uniform.
2. The order in which points are added forms streaks of empty space if the number of points used is not just right.
3. When a large number of points is used empty undersampled spots form even for well shaped triangles.
Here is a comparison using the R2 sequence and no jittering, there are 5000 points in the large and 1000 points in the thin triangles:
Even though the parallelogram method has pretty much the same problems as the Fibonacci lattice, they aren’t as severe.
The points are added in an order that gives a better distribution and even mirroring along the diagonal still gives points that are nicely distributed without as noticeable of a pattern. Aliasing
is still a problem for sliver triangles of course.
I’ve tried to sort the input coordinates for both methods and while it gives better results, the biggest problems still remain.
Even picking the best of the possible permutations still doesn’t fix it.
I though about using the ratio between the long and short edges of the triangle to skew the points in order to reduce aliasing, but couldn’t find a way of making it work, maybe you have an idea 🙂
So in the end I still don’t have a method that works well for arbitrary triangles, or am I missing something?
1. Thanks so much for this analysis. Somehow, my early findings showed that the parallelogram method was not great, but it seems that I should not have dismissed the parallelogram method so
I have come to the same conclusions as you, namely:
(i) sorting the coordinates can make minor/moderate improvements, but for some triangles even the best permutation is bad.
(ii) small angled-triangles notoriously produce signficant aliasing.
But in the end, I also came to the same conclusion as you, that despite these problems, i don’t know of any alternate direct construction methods.
2. I am thinking the key to your solution is picking points in a parallelogram in a manner that is regular. What method did you use?
1. I Used Method 2 described in “Generating Random Points in Triangles” by Greg Turk published in Graphics Gems 1.
But here is my code (hope C is not a problem):
2. The supergolden ratio seems to work just as well as the plastic constant.
1. That’s very interesting. I’ll try to make mention of this in my follow-up post which is nearly complete. 😉
3. Any idea how do I add random seed value to this R2 sequence, so that two same triangles could have different point placement, but still using r2 sequence? Can algorithm from this post be
implemented in opensource app?
1. Hi,
It is very easy to incorporate a seed $\pmb{s}$ into the algorithm, so that you can construct a family of different R_2 sequences.
$$ \pmb{t}_n = \left\{\pmb{s}+ n \pmb{\alpha} \right\} \quad n=1,2,3,… $$
where $\pmb{s}$ is any 2-d point in $[0,1] \times [0,1]$.
1. That is what I tried to do but adding offset to 2d sequence breaks the ‘even distribution’ of points. Below is my test, where I add ‘rand_offset’ 2d point, to the sequence:
Resulting points do not look uniform for lots of rand_offset values. Unless seed is (0,0) or (0.5, 0.5) – those seems work perfect, but this gives only 2 seed values. Maybe I’m doing
something wrong.
1. Ok, I think I figured this out. Problem is that in Parallelogram method , the points that are outside triangle, should not be moved it the same direction, by seed, as the points in
the triangle. I will try to figure way to flip those points seed offset direction somehow.
1. Hmmm..that’s no good. ☹️
I’ll try to look at it tonight and email you if i figure anything out before you. ?
4. For now I just ignore points that are oustide triangle (r1+r2>1), that removes half of points, so I just increase target point count two times to compensate. I haven’t figured out better way to
negate the random seed offset influence on points distribution.
5. Regarding the sampling of a triangle, there is a short technical report by Eric Heitz (https://hal.archives-ouvertes.fr/hal-02073696) that gives a area preserving mapping from the square to the
cube that seems to have all the required properties regarding the distortion minimization and that is really easy to implement.
His mapping was inspired by the Shirley-Chiu mapping from the square to the disk.
That would allow to use any square sampling pattern to sample the triangle with a minimal loss of sampling quality.
Nice work by the way.
1. I tested the last page function from Eric Heitz PDF and while it works good with offset, it dosen’t work to well for some number of points, for other it gives visible banding:
1. I published my way as it offered a method with less banding than any other method I knew about.
So as soon as I/we can fix the offset problem it will certainly be the first choice for triangular sampling. | {"url":"https://extremelearning.com.au/evenly-distributing-points-in-a-triangle/","timestamp":"2024-11-06T12:28:35Z","content_type":"text/html","content_length":"105571","record_id":"<urn:uuid:4c7d3f2f-2fd5-47d9-b9b7-5986b298bfd9>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00398.warc.gz"} |
R use models from survival analysis and do prediction
I am learning survival analysis in R, especially the Cox proportional hazard model. I read a paper talking about using 80% of the sample as training set and 20% of sample as test set.
As quoted
On the training set, we first performed a pre-selection step to keep the top significant features correlated with overall survival (univariate Cox model, likelihood ratio test, P< 0.05). ...We
used two computational methods to train the models: (i) Cox: the Cox proportional hazards model with LASSO for feature selection ...We then applied the models thereby obtained to the test set for
prediction, and calculated the C-index using the R package survcomp.
I do not know how they actually did to apply the models from Cox model to the test set. I mean, for the training set, I can simply perform a coxph function. But the returned results are "coef,exp
(coef),se(coef)),z,p" and likelihood ratio test p-value. How can I treat this as a model and use it on the 20% test set data?
paper name "Assessing the clinical utility of cancer genomic and proteomic data across tumor types" is on nature biotechnology. Thanks!
Entering edit mode
Hi @sarajbc and hey, apologies for raising this old thread again. I am also struggling with a similar problem where I want to predict survival of the patient from methylation data. Can you please
help me understanding what does this glmnet cox regression actually predict? For my datasets as I am getting all negative values as my predictions and I have no clue what does these actually mean.
ADD REPLY • link | {"url":"https://www.biostars.org/p/109215/","timestamp":"2024-11-09T06:11:57Z","content_type":"text/html","content_length":"26360","record_id":"<urn:uuid:abfe93fa-d972-4968-98f9-cd326ebc4468>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00535.warc.gz"} |
[ANSWERED] Solve each of the following problems In each case be sure to - Kunduz
Solve each of the following problems In each case be sure to
Last updated: 2/2/2024
Solve each of the following problems In each case be sure to make a diagram of the situation with all the given information labeled 1 tsosceles Triangle The two equal sides of an isosceles triangle
are each 42 centimeters If the base measures 30 centimeters find the height and the measure of the two equal angles Round your answer to the nearest tenth 2 Equilateral Triangle An equilateral
triangle one with all sides the same length has an altitude of 4 3 inches Find the length of the sides 3 Length of an Escalator How long should an escalator be if it is to make an angle of 33 with
the floor and carry people a vertical distance of 21 feet between floors Round to the nearest tenth 4 Height of a Hill A road up a hill makes an angle of 5 with the horizontal If the road from the
bottom of the hill to the top of the hill is 2 5 miles long how high is the hill 5 Circus Tent A rope from the top of a circus tent pole is 72 5 feet long and is anchored to the ground 43 2 feet from
the bottom of the pole What angle does the rope make with the pole Give your answer to the nearest tenth of | {"url":"https://kunduz.com/questions-and-answers/solve-each-of-the-following-problems-in-each-case-be-sure-to-make-a-diagram-of-the-situation-with-all-the-given-information-labeled-1-tsosceles-triangle-the-two-equal-sides-of-an-isosceles-triangle-333565/","timestamp":"2024-11-06T08:16:42Z","content_type":"text/html","content_length":"211683","record_id":"<urn:uuid:d87779f4-e469-427e-976c-d777f78d7ce3>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00657.warc.gz"} |
Answered! Read through the rest of "A Practical Introduction to Matlab" by Gock-enbach, at the web site http://www.math.mtu.edu/~msgocken/intro/intro.pdf Your task. Write a Matlab function that computes the linear…
Read through the rest of “A Practical Introduction to Matlab” by Gock-enbach, at the web site http://www.math.mtu.edu/~msgocken/intro/intro.pdf Your task. Write a Matlab function that computes the
linear spline interpolation for a given data set. You might need to take a look at the file cspline_eval.m in Section 3.4 for some hints. Name your Matlab function lspline. This can be defined in the
file lspline.m, which should begin with: function ls=lspline(t, y, x) % lspline computes the linear spline % Inputs: % t:vector, contains the knots % y:vector, contains the interpolating values at
knots % x:vector, contains points where the lspline function % should be evaluated and plotted % Output: % ls:vector, contains the values of lspline at points x Use your Matlab function lspline on
the given data set in Problem 2, plot the linear spline for the interval [1.2, 2.2]. What to hand in:Hand in the Matlab file lspline.m, and your plots. The goal of this problem is to draw mount
Everest with the help of natural cubic splines. We have a rather poor quality photo of the mountain profile, which is given below. You need to set up a coordinate system, and select a set of knots
along the edge of the mountain, and find the coordinates for all these interpolating points. We are aware that the mountain profiles are not very clear in that photo, so please use your imagination
when you find these approximate points. You may either print out the mountain picture and work on it on a piece of paper, or send the image to some software and locate the coordinates of the
interpolating points. Make sure to select at least 20 points. You may use more points if you see fit. After you have generated your data set, you need to find a natural cubic spline interpolation.
Use the functions cspline and cspline_eval, which are available in Section 3.4. Read these two functions carefully, try to understand them before using them. Does it look like there is a smaller peak
on the right? Does the peak look rather sharp? How would you deal with this situation, knowing that a single cubic spline function will generate a “smoothest” possible interpolation?
Expert Answer
kindly repost remaining question
problem 3:
%contains knots
t=[1.2 1.5 1.6 2.0 2.2];
%contains interpolating values [assumed]
y=[0.5674 1.2879 0.1983 -0.4538 -0.2987];
function ls=lspline(t,y,x)
%to find length
for j=1:l
for k=1:length(t)
if x(j)<t(k) | {"url":"https://grandpaperwriters.com/answered-read-through-the-rest-of-a-practical-introduction-to-matlab-by-gock-enbach-at-the-web-site-http-www-math-mtu-edu-msgocken-intro-intro-pdf-your-task-write-a-matlab-function-t/","timestamp":"2024-11-11T10:37:46Z","content_type":"text/html","content_length":"47249","record_id":"<urn:uuid:e1cddb66-c12e-4ace-a052-39d23f1c1db2>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00564.warc.gz"} |
How To Simplify Fractions With Decimals
Fractions and decimals are parts of whole numbers written in two different forms. A fraction has a numerator over a denominator, which represents the number of parts you have of a whole number over
the number of parts by which the whole number is divided. A decimal contains part of a whole number to the right of a decimal point. If a fraction contains a decimal in either its numerator or
denominator, you can convert the decimal into a fraction so that you have two similar number formats to simplify the fraction. A fraction is simplified when the only common factor of the numerator
and denominator is 1.
Step 1
Determine a fraction with a decimal that you want to simplify. For the following example, use the fraction 0.2/2.
Step 2
Convert the decimal to a fraction by placing the number to the right of the decimal point as a numerator over a denominator that corresponds to the place value of the decimal number. In the example,
the decimal 0.2 extends to the tenths place, so place 2 as a numerator over 10 as a denominator, which equals 2/10. This leaves (2/10)/2, which consists of a fraction within a fraction.
Step 3
Divide the numerator by the denominator, which is equivalent to multiplying the numerator by the reciprocal of the denominator, to convert the fraction within a fraction to a single fraction. A
reciprocal is a fraction flipped upside down. In the example, divide 2/10 by 2, which is equivalent to multiplying 2/10 by 1/2. This equals 2/20.
Step 4
Find the largest number that divides evenly into the numerator and denominator of the fraction. In the example, 2 is the largest number that divides evenly into 2 and 20.
Step 5
Divide both the numerator and denominator by the largest number that divides evenly into both to simplify the fraction. In the example, divide 2 by 2, which equals 1, and divide 20 by 2, which equals
10. This leaves 1/10, which is the simplified form of the fraction with a decimal.
Cite This Article
Keythman, Bryan. "How To Simplify Fractions With Decimals" sciencing.com, https://www.sciencing.com/simplify-fractions-decimals-8328317/. 24 April 2017.
Keythman, Bryan. (2017, April 24). How To Simplify Fractions With Decimals. sciencing.com. Retrieved from https://www.sciencing.com/simplify-fractions-decimals-8328317/
Keythman, Bryan. How To Simplify Fractions With Decimals last modified March 24, 2022. https://www.sciencing.com/simplify-fractions-decimals-8328317/ | {"url":"https://www.sciencing.com:443/simplify-fractions-decimals-8328317/","timestamp":"2024-11-07T10:28:11Z","content_type":"application/xhtml+xml","content_length":"70574","record_id":"<urn:uuid:0f5e5202-503f-4bab-a012-e5b833887b16>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00487.warc.gz"} |
Unit 4
Warm-up: Choral Count: Plus 10 (10 minutes)
The purpose of this Choral Count is to invite students to practice counting by 10 and notice patterns in the count. In this warm-up, students have an opportunity to look for and make use of structure
(MP7) of whole numbers and the base-ten system because they notice that as they count, the ones place stays the same, and the tens place changes. These understandings help students develop fluency
and will be helpful later in this lesson when students compare numbers by looking at the tens place.
• “Count by 10, starting at 3.”
• Record as students count.
• Stop counting and recording at 93, then count back by 10.
• “What patterns do you see?”
• 1–2 minutes: quiet think time
• Record responses.
Activity Synthesis
• “Who can restate the pattern in different words?”
• “Does anyone want to add an observation to explain why that pattern is happening here?”
Activity 1: Which is More? (15 minutes)
The purpose of this activity is for students to determine which number is greater. Students represent their number in any way they choose. Listen for the way students use place value understanding to
compare the numbers and the language they use to explain how they know one number is more than the other (MP3, MP6). In the synthesis, students are introduced to the terms greater than and less than.
This activity uses MLR2 Collect and Display. Advances: conversing, reading, writing
Representation: Internalize Comprehension. Synthesis: Invite students to identify which details were the most important in comparing the numbers. Display the sentence frame: “The next time I compare
numbers, I will pay attention to . . . .“
Supports accessibility for: Conceptual Processing, Memory
• Groups of 2
• Give each group two paper clips and access to connecting cubes in towers of 10 and singles.
• Display 35 and 52.
• “Which number is more? Show your thinking using math tools. Be ready to explain your thinking to your partner.”
• 2 minutes: independent work time
• 2 minutes: partner discussion
• “Which is more and how do you know?” (53 is more because it has more tens than 35.)
• Read the task statement.
• “Each partner can choose to use Spinner A or B for each turn.”
• 10 minutes: partner work time
MLR2 Collect and Display
• Circulate, listen for, and collect the language students use to build numbers with connecting cubes, decompose numbers into tens and ones, and compare numbers. Listen for: bigger, smaller, more,
fewer, greater than, less than, ___ tens, ___ ones, tens place, ones place.
• Record students’ words and phrases on a visual display and update it throughout the lesson.
Student Facing
• Each partner spins a spinner.
• Each partner shows the number any way they choose.
• Compare with your partner.
• Which number is more?
Spinner A:
Spinner B:
Activity Synthesis
• “Are there any other words or phrases that are important to include on our display?”
• As students share responses, update the display by adding (or replacing) language, diagrams, or annotations.
• Remind students to borrow language from the display as needed.
• Display 93 and 26.
• “Which is more? How do you know?” (93 is more because 9 tens is more than 2 tens.)
• “We can say, ‘93 is greater than 26.’ We can also say, ‘26 is less than 93.’”
• Display 62 and 64.
• “Which number is more? How do you know?” (64 is more. They both have 6 tens but 64 has 4 ones and that is more than the 2 ones in 62.)
• “We can say that 64 is greater than 62. We can also say 62 is less than 64.”
Activity 2: Elena and Noah Compare Numbers (10 minutes)
The purpose of this activity is for students to attend to the value of the digits when comparing two-digit numbers. Students evaluate and critique comparison statements and the thinking behind them
(MP3). They attend to the way other students reason about digits and the language they use to compare, and make revisions to help make arguments more precise and clear (MP3, MP6).
MLR8 Discussion Supports. Synthesis: Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking
• Groups of 2
• Give students access to connecting cubes in towers of 10 and singles.
• Read the task statement.
• 6 minutes: partner work time
• Monitor for students who show their thinking using:
□ connecting cubes
□ drawings
□ words
Student Facing
1. Elena says 75 is greater than 65 because 7 is greater than 6.
What do you think Elena means?
How could Elena be more clear?
2. Noah says 39 is greater than 41 because it has a 9 and 9 is the greatest number.
Do you agree with Noah?
What could you tell Noah to help him compare these numbers?
Advancing Student Thinking
If students agree with Elena, but cannot yet articulate how she could revise her thinking, consider asking:
• “Do you agree with Elena? Why?”
• “Can you show the numbers with connecting cubes? What does Elena mean when she says 7 is greater than 6 for these two numbers?”
• “Would Elena’s statement be true if she were comparing 7 and 65? Why or why not?”
Activity Synthesis
• Invite previously identified students to share.
• “When we compare numbers, it is important to pay attention to what place each digit is in and what value the digit has.”
Activity 3: Centers: Choice Time (15 minutes)
The purpose of this activity is for students to choose from activities that offer practice working with two-digit numbers. Students choose from any stage of previously introduced centers.
• Write Numbers
• Grab and Count
• Five in a Row
Required Preparation
• Gather materials from previous centers:
□ Write Numbers, Stages 1 and 2
□ Grab and count, Stage 2
□ Five in a Row, Stages 1–3
• Groups of 2
• “Now you are going to choose from centers we have already learned.”
• Display the center choices in the student book.
• “Think about what you would like to do.”
• 30 seconds: quiet think time
• Invite students to work at the center of their choice.
• 10 minutes: center work time
Student Facing
Choose a center.
Write Numbers
Grab and Count
Five in a Row
Activity Synthesis
• “Mai is playing Grab and Count with her partner. She organized her cubes into 4 towers of 10 and 3 ones. Her partner organized her cubes into 2 towers of 10. How many cubes do they have together?
Lesson Synthesis
Display 44 and 64.
“Today we compared two-digit numbers. Which is greater? How do you know?” (64. 64 has 6 tens and 44 has only 4 tens.)
Display 59 and 54.
“Which is less? How do you know?” (Each has 5 tens, so I have to look at the ones place. 4 ones is less than 9 ones.)
Cool-down: Unit 4, Section C Checkpoint (0 minutes) | {"url":"https://im.kendallhunt.com/K5/teachers/grade-1/unit-4/lesson-14/lesson.html","timestamp":"2024-11-04T18:07:41Z","content_type":"text/html","content_length":"96665","record_id":"<urn:uuid:3fb0ac4b-a23e-41c8-8f3d-f0def1bbe0cf>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00205.warc.gz"} |
How do you find the y coordinate of the inflection point of the function f(x)= 10(x-5)^3+2?
| HIX Tutor
How do you find the y coordinate of the inflection point of the function #f(x)= 10(x-5)^3+2#?
Answer 1
A function's inflection points are the zeroes of its second derivative where its sign changes.
Now let us calculate the second derivative:
Ther is only one zero: #x=5# and the second derivative change in sign in it, so #P(5,2)# is the only inflection point, as you can see from the graph:
chart{10(x-5)^3 +2 [-10, 10, -5, 5]}
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Answer 2
To find the y-coordinate of the inflection point of the function ( f(x) = 10(x - 5)^3 + 2 ), you first need to find the second derivative of the function, ( f''(x) ). Then, set the second derivative
equal to zero and solve for ( x ). Once you have the ( x )-coordinate of the inflection point, plug it back into the original function to find the corresponding ( y )-coordinate.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Data analysis
Data analysis – subtotals
One of the simplest ways to summarise worksheet data is to add subtotals.
Example: An analyst is interested in the sales data shown in figure 1, worksheet(1) named Sales Data 1.
This data set is constructed as follows:- row 1 is the header or labels row, and describes the fields in each column. This text data has some labels containing space characters. For example,
worksheet column A is labeled Invoice No. The numbers 150100, 150101, 150102, ... are in the Invoice No fields. There are 11 fields in total. Each row in the body of the data is one unique record.
Some blank or empty empty cells are OK, but there should be no empty rows or columns in the data set. When you click in a cell within the data, the data set becomes the Current Region.
Fig 1: Excel Web App #1: Sales data – 100 records. Labels in row 1, and records in rows 2 to 101
Subtotals – manual
Reference: figure 1 Worksheet: Sales Data 1. Suppose that the Analyst is interested in the sales volume achieved by individual salespersons and intends to add subtotals to the data.
Subtotals can be added manually by:
1. Sorting the data set on the Salesperson column – see figure 2
2. To sort the data in ascending order, select a cell in the Salesperson column (field)
3. Then select the sequence Data > Sort & Filter > AZ↓ on the ribbon, or
4. The shortcut sequence Alt A SA
5. The data for Jellison ends at row 18
6. Insert a new row at row 19, then use the SUM function, or the SUBTOTAL function to summarise the data
7. Repeat this process for the other salerpersons
Fig 2: Sorting Current Region – xlfAnimatedPreview – select a cell in the sort column,then the AZ↓ sort button the ribbon
An easier method is to use the Excel subtotal command. Use of the command is shown in the next three subsections.
Subtotal command
Subtotals – single
Using the same data sort from figure 2, the data is repeated in worksheet(2) named Sales data - subtotal 1.
Subtotals with the subtotal command:
1. Click on a cell in the data set
2. Then select the sequence Data > Outline > Subtotal on the ribbon, or
3. The shortcut sequence Alt A B
4. This displays the Subtotal dialog box- see figure 3
5. Figure 3 – the Subtotal dialog box options – Ⓐ At each change in: select Salesperson from the drop down list, Ⓑ Use function: Subtotal, and Ⓒ Add subtotal to: the three quantity columns
6. Click OK – to display the list in figure 4
Fig 3: Subtotal dialog box – Ⓐ At each change in: Label drop down list, Ⓑ Use function: Subtotal, and Ⓒ Add subtotal to: Label drop down list
Fig 4: Subtotal – Salesperson field, Sum function subtotal
The subtotal command adds groups to the data set, with expand and collapse items in the subtotal panel to the left of the row header item. See figure 4.
■ The expand / contract feature is used by clicking on a level in the \(\boxed{1}\), \(\boxed{2}\), \(\boxed{3}\) sequence or a group a , item
■ \(\boxed{1}\) – Grand Total
■ \(\boxed{2}\) – Salesperson Total
■ To hide or view the subtotal panel press Ctrl+8
Subtotals – multi
Worksheet(3) named Sales data - subtotal 2.
1. To add an Average subtotal to the existing Sum subtotal, Use function: Average, and
2. Ⓓ untick Replace current subtotals (figure 5) to prevent the Sum being overwritten
3. The result is shown in figure 6 – with 4 levels
Fig 5: Subtotal dialog box – Ⓓ untick Replace current subtotals
Fig 6: Subtotal – Salesperson field, Sum and Average function multi subtotal
Subtotals – nested
Worksheet(4): Sales Data - subtotal 3
The AZ↓ sort command can only sort on one field. To setup a nested subtotal, two or more fields need to be sorted. This requires the multi level features of the Sort dialog box –
figure 7.
To set up a Salesperson subtotal, with a Transport nested subtotal
1. Using the Sort dialog box, Sort by: Salesperson, Then by: Transport
2. Set the subtotal for Salesperson – figure 3, then
3. Set the subtotal for Transport – untick Replace current subtotals
4. The result is shown in figure 8 – with 4 levels
Fig 7: Sort dialog box – with settings for level 1: Salesperson, and level 2: Transport
Fig 8: Subtotal – Salesperson field, Sum function subtotal, nested subtotal Transport field, Sum function
Format subtotals
Any cell formats applied to subtotals, are removed when the subtotals are cleared.
To clear subtotals, click the Remove All button in the Subtotal dialog box.
Copy visible subtotal cells
To copy the visible cells (visible subtotal data)
1. Select the data to be copied – the Source
2. Press F5 to display the Go To Special dialog box
3. Select Visible cells only
4. Click OK
5. Press Ctrl+C to copy the source to the clipboard
6. Select the cell for the target, then press Ctrl+V to Paste.
Note: steps 2 to 4 can be replaced by the Alt+; keyboard shortcut (see EXCEL: useful keyboard shortcuts – pdf)
Subtotal function
The subtotal procedure used the Excel subtotal function
Syntax: SUBTOTAL(function_num,ref1,[ref2],…)
│ Function_num │ Function_num │Function│
│(includes hidden values) │(excludes hidden values) │ │
│1 │101 │AVERAGE │
│2 │102 │COUNT │
│3 │103 │COUNTA │
│4 │104 │MAX │
│5 │105 │MIN │
│6 │106 │PRODUCT │
│7 │107 │STDEV │
│8 │108 │STDEVP │
│9 │109 │SUM │
│10 │110 │VAR │
│11 │111 │VARP │
Using SUBTOTAL for summary statistics
You can use the SUBTOTAL function to indirectly link cell labels to the summary statistic procedure – figure 9.
Fig 9: Subtotal function as dynamic link to label – Subtotal with the function_num argument linked to a function argument array named ListSubTotal
The example in figure 9 is available on the Summary Stats worksheet – Excel Web App #1
The ListSubTotal array has the values ={"average",101,1;"count",102,2;"counta",103,3;"max",104,4;"min",105,5;"product",106,6;"stdev",107,7;"stdevp",108,8;"sum",109,9;"var",110,10;
■ This example was developed in Excel 2013 Pro 64 bit.
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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SWAT.2022.15
URN: urn:nbn:de:0030-drops-161756
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16175/
de Berg, Mark ; Sadhukhan, Arpan ; Spieksma, Frits
Stable Approximation Algorithms for the Dynamic Broadcast Range-Assignment Problem
Let P be a set of points in ℝ^d (or some other metric space), where each point p ∈ P has an associated transmission range, denoted ρ(p). The range assignment ρ induces a directed communication graph
G_{ρ}(P) on P, which contains an edge (p,q) iff |pq| ⩽ ρ(p). In the broadcast range-assignment problem, the goal is to assign the ranges such that G_{ρ}(P) contains an arborescence rooted at a
designated root node and the cost ∑_{p ∈ P} ρ(p)² of the assignment is minimized.
We study the dynamic version of this problem. In particular, we study trade-offs between the stability of the solution - the number of ranges that are modified when a point is inserted into or
deleted from P - and its approximation ratio. To this end we introduce the concept of k-stable algorithms, which are algorithms that modify the range of at most k points when they update the
solution. We also introduce the concept of a stable approximation scheme, or SAS for short. A SAS is an update algorithm alg that, for any given fixed parameter ε > 0, is k(ε)-stable and that
maintains a solution with approximation ratio 1+ε, where the stability parameter k(ε) only depends on ε and not on the size of P. We study such trade-offs in three settings.
- For the problem in ℝ¹, we present a SAS with k(ε) = O(1/ε). Furthermore, we prove that this is tight in the worst case: any SAS for the problem must have k(ε) = Ω(1/ε). We also present algorithms
with very small stability parameters: a 1-stable (6+2√5)-approximation algorithm - this algorithm can only handle insertions - a (trivial) 2-stable 2-approximation algorithm, and a 3-stable
1.97-approximation algorithm.
- For the problem in ?¹ (that is, when the underlying space is a circle) we prove that no SAS exists. This is in spite of the fact that, for the static problem in ?¹, we prove that an optimal
solution can always be obtained by cutting the circle at an appropriate point and solving the resulting problem in ℝ¹.
- For the problem in ℝ², we also prove that no SAS exists, and we present a O(1)-stable O(1)-approximation algorithm. Most results generalize to when the range-assignment cost is ∑_{p ∈ P} ρ(p)^{α},
for some constant α > 1. All omitted theorems and proofs are available in the full version of the paper [Mark de Berg et al., 2021].
BibTeX - Entry
author = {de Berg, Mark and Sadhukhan, Arpan and Spieksma, Frits},
title = {{Stable Approximation Algorithms for the Dynamic Broadcast Range-Assignment Problem}},
booktitle = {18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)},
pages = {15:1--15:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-236-5},
ISSN = {1868-8969},
year = {2022},
volume = {227},
editor = {Czumaj, Artur and Xin, Qin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16175},
URN = {urn:nbn:de:0030-drops-161756},
doi = {10.4230/LIPIcs.SWAT.2022.15},
annote = {Keywords: Computational geometry, online algorithms, broadcast range assignment, stable approximation schemes}
Keywords: Computational geometry, online algorithms, broadcast range assignment, stable approximation schemes
Collection: 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)
Issue Date: 2022
Date of publication: 22.06.2022
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A New Approach to the Consideration and Analysis of Critical Factors in Robotic Machining
Department of Construction and Manufacturing Engineering, ETS Ingenieros Industriales, Universidad Nacional de Educación a Distancia (UNED), C/Juan del Rosal 12, 28040 Madrid, Spain
Mechatronic Unit, Department of Industrial Services, AIMEN Technological Center, Relva 27A, 36410 Torneiros, Spain
Area of Manufacturing Processes Engineering, EII, Campus Universitario Lagoas-Marcosende, University of Vigo (Universidad de Vigo), 36310 Vigo, Spain
Author to whom correspondence should be addressed.
Submission received: 10 November 2020 / Revised: 27 November 2020 / Accepted: 9 December 2020 / Published: 12 December 2020
The relative low stiffness of industrial robots is a major limitation on the development of flexible and reconfigurable systems in applications in which process forces and vibration lead into
significant tool path deviations with respect to the programmed path as in the case of robotic machining. This paper presents a novel factorial procedure that allows for the preliminary study of the
main conditions in robotic machining operations and it determines the critical factors that are affecting the machining path of any robotic cell in order to obtain the process conditions with lower
path deviations. In this procedure the most influential robotic machining constraints were identified and classified, the factorial design of experiments was used to enable the execution of the
experimental tests and the machining tool path deviation predictive methodology (PREMET) was used to determine the cutting tool path deviation between the programmed and the experimental path as a
function of the process variables. Experimental trials have been carried out in order to determine the main factors that affect the robotic machining and influence the main constraints of the
process, showing a reduction greater than a 36% of the cutting tool path deviation in groove milling of aluminum. The critical factors identified in order of importance are: hardness of the material,
location of the workpiece, orientation of milling head relative to working direction and cutting conditions. This procedure can be extended to future factorial studies to improve the precision of
robotic machining (in operations such as face milling, contouring, pocketing) and to establish design criteria for machining robotic cells.
1. Introduction
The use of anthropomorphic robots, which offer a more affordable cost, has spread widely for the development of more flexible and reconfigurable manufacturing systems for handling, assembly, welding,
and painting, among others. In other manufacturing processes, such as industrial machining operations, robotic systems are less widespread used because process forces generate significant path
deviations from the programmed path, which limit their use in the production of parts with acceptable manufacturing tolerances in sectors such as automation, aeronautics and naval application [
At the industrial level, there is a trend towards directly applying a conventional machining approach to robotic systems [
]. Using this philosophy, there have been many failures, since the substantial difference between the two manufacturing systems requires prior study to guide the solutions of each application.
Knowledge of robotics and machining technology must be properly combined to obtain an acceptable degree of success [
Robotic cell operators who were previously used in machining centers are subjected to steep learning curves due to the lack of methods and procedures what provide working guidelines to allow
production of parts within an acceptable manufacturing tolerance range and surface quality. An example of these type of working guidelines is a manual that defines machining conditions based on the
cutting tool selected for removing material [
], since other robotic machining reference documents do not exist.
The lack of machining rules for robotic cell users results in higher uncertainty in the possible results of the production process, and consequently the selection of strategies and cutting conditions
is left to the judgment of the operator or leading into the undesirable process of trial-and-error. Fortunately, CAM tools [
] have been developed allowing programming and simulating tool paths in robotic cells for different manufacturing processes. These CAM software facilitate the path generation of the cutting tool and
usually incorporate industrial robot movement simulators. This allows to evaluate the possibility of achieving a certain space position, singularities and limits in axes, movement indeterminacies,
etc. Even so, there are certain limitations that affect the smoothness of the robot’s path or functional redundancy in 5-axis robotic machining that have been studied by different authors. Peng, J.
et al. [
] have developed a sequential linearization programming method that generates a smoother robot path in which a significant reduction in the magnitude of the maximum joint acceleration is obtained and
leads to better quality surface area and greater machining efficiency. Schneider, U. et al. [
] have developed an offline predictive compensation method for machining errors, an innovative programming system, based on kinematic and dynamic robot models. The compensation of machining errors is
adaptive in real time and allows to improve the results in terms of robotic machining accuracy.
During the last three decades, robotic machining has aroused great research interest due to the benefit provided by the industrial robot in terms of economic profitability, flexibility of operations
and multifunctionality, identifying its main limitations [
]. The low stiffness of robots has not been a particularly limiting technical barrier in the development of industrial solutions meant for handling, welding, component assembly, or other operations
in which the external system forces are mainly due to the robot’s own acceleration and gravity effects [
]. However, in applications in which the robot is subjected to periodic forces characteristic of the manufacturing process, such as machining forces, an analysis of robotic system behavior [
], specifically, the robot’s structural strength as a function of the programmed path, is required [
]. Klimchik, A. et al. [
] have developed a model to demonstrate that the precision of series industrial robots in the machining operation of circular grooves depends on its structural rigidity. Janez, G. et al. [
] has shown that the manipulability, structural rigidity, structural inertia, damping ratios and natural frequencies of industrial robots are kinematic, static and dynamic properties that determine
the precision result of robotic machining. Gotlih, J. et al. [
] delved a little deeper in this field and developed a calibrated model used to generate a substitute model the structural rigidity of the robot considering other properties such as inaccurate
geometry of model, friction, component wear and other effects. Another notable scientific contribution is that made by Jiao, J. et al. [
], who propose a variable parameter stiffness model to achieve the simultaneous identification of the robot’s kinematics and the stiffness parameters. They also provide a model-based error prediction
and compensation method through online detection of external load.
In robotic machining operations, under certain defined machining and path conditions the deviation and tool chatter will depend strongly on the robot’s axes’ configuration [
]. In other words, the deviation of the real path with respect to the programmed path, and the chatter of the tool during the machining operations, depend essentially on the interrelation between the
stiffness of the robot and the magnitude and direction of the machining forces [
The published literature verifies that the tool’s path deviation and vibration depend fundamentally on the robot’s stiffness, the cutting force magnitude and the predominant direction of action. Some
authors have researched into this phenomenon [
] and have provided tools that help to address this issue from a workshop-oriented perspective. A correct selection of robots, cutting tools and optimum machining conditions is decisive in order to
produce parts with suitable dimensional tolerances and surface quality in robotic machining systems. The machined surface quality generated during machining is correlated closely with milling
conditions, mainly diameter–length of tool, feed rate, depth of cut, cutting speed, etc. [
]. Among the most notable advances, we must highlight the machining tool path deviation predictive methodology (PREMET) developed by I. Iglesias et al. [
] in which the robot’s structural dynamics are reduced to the cutting Tool’s Center Point (TCP) to evaluate and compare the deviations between different robot structural configurations in order to
select the lowest deviation of the cutting path. PREMET has been applied for the case of the grooving operation in soft materials and the errors obtained are smaller than 12%.
Factorial design of experiments has been identified as a very useful tool to evaluate the influence of different factors on precision in robotic machining without having to execute a long and tedious
experimental phase. For example, Grisol de Melo, E. et al. [
] used a factorial design to evaluate the performance of the industrial robot with respect to the precision of movements in function of the geometry of the tool, feed rate and spindle speed.
Previously, Antunes Simões, J.F.C.P. et al. [
] used the design of experiments through factorial techniques to evaluate surface quality and geometric precision function of lead/lag angle, advance speed and robot arm extension.
Considering the background, the main motivation of this investigation is to present novel factorial procedure that allows preliminary study of critical factors in robotic machining operations. This
development represents a novel analytical tool for studying of the main cutting conditions of the robotic machining process, facilitating the evaluation of critical factors not previously addressed
by the scientific community. The factorial procedure identifies and classifies the main conditions of robotic machining and uses the factorial design of experiments to make viable the execution of
the experimental tests. Furthermore, this procedure contemplates the use of PREMET, that allows the determining of path deviations during the machining process in the robotic system without the need
for extensive experimental tests. PREMET establishes and defines a simulation model for predicting the dynamic structural behavior for any industrial robot of up to six degrees of freedom, which is
used to analyze of the correct use of the robot’s stiffness and the adequate selection of the main machining constraints in soft, metallic and nonmetallic materials compared to cutting conditions in
conventional machine tools. Then, this novel factorial procedure facilitates the study and analysis of the main critical factors in robotic machining cells and it represents an unprecedented advance
as a prior tool for designers and users of robotic manufacturing system.
2. Factorial Procedure for Preliminary Study of Critical Factors in Robotic Machining
The factorial procedure was designed to establish the relationship between the main machining conditions imposed on the robot and the simulate deviation of the cutting tool path through the PREMET,
and in this way to determine which are the critical factors that most affect the machining path of any robotic cell. This factorial procedure allows the evaluation of the most influential factors in
robotic machining in order to obtain the process conditions with lower path deviations. Machinable materials can be grouped into three sections: metallic, plastic and ceramic. This factorial
procedure was applied in the field of soft material machining (metallic, plastic or composite materials formed by a combination of the above) in which the material hardness values do not exceed 110
The factorial procedure consists of the following phases:
Phase 1: Factorial design
A factorial design was performed to study the effects of different variables and parameters on the machining tool’s path precision. By simultaneously varying the levels of all factors, instead of one
at a time, it is possible to assess the interaction between them and rank them accordingly to their influence on the part’s dimensional quality. The factorial procedure raises the classification of
the variables and machining parameters that affect robotic machining depending on whether they affect the rigidity of the robot, the cutting force vector or the interaction between both. The
conditions related to the characterization of the robotic cell affect the precision of the tool path because they modify the rigidity of the robotic machining system, while the machining parameters
influence the cutting force vector. Finally, the machining strategy also conditions the machining precision due to the interaction between the rigidity of the robot and the cutting force vector of
the machining tool, multiple combinations are possible with different precision results. Some of the main variables and parameters that condition robot machining have been identified and can be
classified into three groups, as shown in
Figure 1
• Characterization of the robotic cell.
The specifications of robot, external axes, machining heads or methods for workpiece clamping can be considered as initial data or variables and parameters of the robotic machining cell design
process. For this reason, certain specifications such as the robot structural rigidity are studied as a factor or simply as a starting point.
Machining parameters.
Type of operation, material to machining, machining tools, cutting speed, plunging depth, cutting width and other machining parameters can be initial data. It is very common, for example, that in
order to evaluate the machinability of a part in a robotic cell, the material of the part and the required machining operation are classified as initial data. In the same way, it is also usual to
optimize variables and machining parameters that allow parts to be produced with adequate tolerances in order to reduce manufacturing times and costs. For this reason, the study of these conditions
is of special interest to users of robotic machining cells.
Machining strategies.
The cutting strategies, workpiece position, wrist configuration, robot axes configuration or working direction are some variables and parameters of robotic machining that can remain fixed during the
process or they can be studied for making the most of the performance of robotic machining cells. Therefore, some of these constraints can be considered initial data and others factors of study in
order to obtain the cutting tool path that provides smaller deviations with respect to the programmed tool path.
Prior to the analysis and selection of variables and machining parameters, it is necessary to know what type of operation is carried out, robotic cell to be used and the manufacturing strategy to
follow. Depending on the conditioning factors of each robotic machining application, there will be some batch data and other variables customizable by operator. From among these variables and
machining parameters to be defined, the conditions that will be included in the design of the factorial study will be chosen before the simulated model execution.
Stage 1: Set initial data
The initial data are the values or parameters of robotic machining that cannot be changed during the study—these are the starting conditions of the process whose effect is not studied for a specific
application. Considering the classification of variables and machining parameters carried out in
Figure 1
, the necessary initial data must be defined.
Stage 2: Selection of the variables and parameters to study
The values or parameters to be studied are the factors whose influence on robotic machining must be evaluated; they are the conditioning factors on which to act, in order to reduce the machining tool
path deviation. Considering the classification made in
Figure 1
again, scoping factors should be selected for each study.
Stage 3: Development of the test tables.
If it is proposed to obtain complete real data, an experimental phase that includes
$k j$
number of trials is necessary, where k represents the levels to be evaluated and j the factors to study. The large number of experiments that would be needed if we are interested in evaluating many
factors with many levels, as in the case of robotized machining operations can be unapproachable because of the high cost that may involve experimentation. The literature recommends in these cases to
use the factorial design in order to optimize the cost of the experiments [
]. The usual strategy is to carry out the factorial design of experiments considering the evaluation of only two levels for j factors, resulting
$2 j$
trials. This procedure is based on this strategy to be able to approach the experimental phase. According to the variables and parameters of study, the factorial design resulted a 2xj matrix. In
Table 1
is shown the factorial design of j factors in the two levels proposed for this study.
Phase 2: Methodology applied for determination of the simulated cutting tool path deviation
PREMET, the predictive methodology developed by I. Iglesias et al. [
], has been validated and can be applied in any six degrees of freedom industrial robot to calculate the path deviations of the machining tool, and it is used in this factorial procedure to predict
the simulated path deviation.
Stage 1: Characterization of the reduced equivalent model
The stiffness matrix
and the damping matrix
for the different path points is determined for representing the robot’s mechanics through the next equation:
$[ M r ( P i , θ 1 … , θ i ) ] [ X ¨ ] + [ C r ( P i , θ 1 … , θ i ) ] [ X ˙ ] + [ K r ( P i , θ 1 … , θ i ) ] [ X ] = O$
$P i , θ 1 … , θ i$)
is the equivalent matrix of the mass of the robot’s axes and machining head,
$[ M r ( P i , θ 1 … , θ i ) ] = 1 x t c p 2 + y t c p 2 + z t c p 2 ∑ i = 0 N m i [ y i 2 + z i 2 − x i y i − x i z i − x i y i x i 2 + z i 2 − y i z i − x i z i − y i z i x i 2 + y i 2 ]$
$P i , θ 1 … , θ i$)
is the stiffness matrix equivalent to the mechanical structure of the robot for a point
of the space. The TCP deviations under the action of a known static force was measured with a deflection gauge.
$[ K r ( P i , θ 1 … , θ i ) ] = [ k 11 k 12 k 13 k 21 k 22 k 23 k 31 k 32 k 33 ]$
$P i , θ 1 … , θ i$)
is the damping equivalent to the mechanical structure of the robot for a specific point
of the space. It was calculated through a hammer impact test and considering the Rayleigh equation.
$[ C r ( P i , θ 1 … , θ i ) ] = a 0 [ M r ( P i , θ 1 … , θ i ) ] + a 1 [ K r ( P i , θ 1 … , θ i ) ]$
Stage 2: Cutting force modeling
The standard cutting force model based on Altintas [
] is used to simulate the cutting force vector for the flat end milling. The equation for calculating the force
on the TCP is,
$F x y z , t o o l = ∑ e = 1 N e ∑ j = 1 N z T j ( φ ) F r t a , j , e$
Stage 3: Modeling of the interactions between the cutting forces and the robot mechanics
Therefore, the interaction between the cutting forces and the robot’s mechanics is represented by the equation:
$[ M r ( P i , θ 1 … , θ i ) ] [ X ¨ ] + [ C r ( P i , θ 1 … , θ i ) ] [ X ˙ ] + [ K r ( P i , θ 1 … , θ i ) ] [ X ] = F x y z , t o o l$
Stage 4: Evaluation and comparison of different structural configurations
The different structural configurations of the robot must be evaluated and compared based on the simulated maximum path deviation values. It is recommended to graph and to tabulate the deviations of
simulated paths to analyze in depth the data obtained with PREMET.
Stage 5: Selection of the lowest-deviation structural configuration
Finally, the structural configuration with the lowest path deviation prediction result must be selected, and therefore the machining conditions that give the best corresponding prediction results.
Phase 3: Evaluation and identification of the most influential factors in robotic machining
In order to determine the robotic cell configuration and the machining conditions that provide the least tool path deviation and the best cutting parameters, the different machining possibilities are
evaluated and it is determined which allows the least tool path deviations and the best machining time for obtaining the required part accuracies.
After determining the simulated cutting tool path deviation for these specific machining variables and parameters, the user of the robotic machining cells must evaluate the results obtained in order
to identify the most critical factors and make a decision according to the criteria shown in
Figure 2
• Establish design criteria for the robotic cell, in cases where robot selection, location of external axes, machining head configuration or workpiece clamping methods are necessary.
• Determine whether a specific machining operation can be performed in the robotic cell, depending on the work material, machining operation, cutting tool and process parameters.
• Select the most convenient cutting strategy by defining the location and orientation of the part and determining the most favorable robot axis configuration, which guarantees the correct
execution of the machining operation, complying with the required manufacturing tolerances of the part.
3. Experimental Validation of the Factorial Procedure
Set initial data
In order to validate the factorial procedure, all variables and parameters defined through the different phases need to be applied in the machining of soft materials. Considering the classification
of variables and machining parameters carried out, the initial data are:
• Characterization of the robotic cell.
The machining tests were performed in a robotic cell with an IRB 6660 robot, an MTC 2000 turning unit, a Peron speed machining head and also used mechanical clamping devices.
Machining parameters.
Certain initial conditions are established:
A grooving operation was chosen for its simple cutting force vector representation.
To perform the machining tests, a KENDU hard metal burr-cutting tool with a 10 mm diameter, 150 mm length, two cutting edges, and a 30º helix angle was chosen.
Machining strategies.
A rectilinear cutting path was programmed, in which the machining tool axis is perpendicular to the worktable. The existing trend in the use of path programming software for robotic machining
systems, which facilitate the linearization of curved paths through the selection of smoothing and tolerance parameters, allow to calculate the path of the cutting tool as a set of movements
rectilinear from point to point with sufficient proximity without need to use programming with complex cycles such as circular interpolations. For the validation of this factorial procedure, the
calculation, simulation and analysis of rectilinear machining paths have been considered, where rectilinear movements from point to point are the basic element to evaluate in any trajectory.
Selection of the variables and parameters to study
With the objective of evaluating the influence of certain machining conditions on the machining tool path deviation, the following variables and parameters were considered:
• Characterization of the robotic cell.
The evaluation of any variable corresponding to the robotic cell has not been considered because all the validation tests will be carried out on the same robot, machining head, worktable and clamps.
In further studies these variables can be considered.
Machining parameters.
□ Selection of the workpiece material type: To evaluate the influence of the hardness of the material to be machined on the prediction of the robotic system, two materials with very different
properties were selected: Aluminum EN AW—5083 (Al) and 300 kg/cm3 rigid polyurethane foam (PUR 300).
□ Cutting parameter selection, based on the conventional machining manual for KENDU machine tool:
To study the influence of cutting speed on path precision, two rotational tool speeds (n) were defined: 6000 rpm and 8000 rpm.
To evaluate the influence of cutting depth on the tool’s path deviation, two values (a[p]) were defined: 1 mm and 1.5 mm.
Machining strategies.
□ Selection of the relative workpiece position with respect to the robot’s coordinate origin: To assess the importance of the workpiece location with respect to the machining robot, two
positions were evaluated, as shown in
Figure 3
, providing two different mechanical robot configurations: a workpiece position close to the robot’s base (A) and a position further from the base (B). Position A is 1.100 mm from the base of
the robot in x axis and position B 1.600 mm.
□ Cutting strategy definition:
To assess the importance of the cutting force vector orientation with respect to robot’s mechanics, two different machining directions are defined, one longitudinal (L) and one transverse (T) to the
worktable, as shown in
Figure 4
. The feed direction of the cutting tool L corresponds to the X direction of the robot’s base coordinate system, while T direction corresponds to one parallel direction to the axis Y of the base
coordinate system of the robot.
Development of the test tables
According to the variables and parameters the study selected, the factorial design was carried out, summarized in
Table 2
, resulting in a 2 × 5 matrix.
The compiled test tables are shown in
Table 3
a,b below:
Experimental tests
Actual experiments with a robot were performed, and the cutting tool path deviation values obtained from the simulation were compared with those obtained during real machining. In
Figure 5
it can see the Al and PUR 300 pieces used in the experimental test, as well as the robotic cell at the beginning of machining.
Measurement of robotic machining toolpath deviations
After the experimental tests, the different test pieces were measured using a MITUTOYO coordinate measuring machine, model EUROC-APEX.
Figure 6
shows a piece of Al and another of PUR300 obtained in the experimental phase of this study corresponding to tests E2 and E18 respectively. The profiles of both sides of the groove were extracted, and
the maximum path deviation values were measured. The results of measurements are summarized in
Table 3
Table 4
For a better understanding of results, the original groove and the measurements performed, corresponding to tests E2 (Blue) and E18 (Pink), are shown in
Figure 7
Figure 8
Figure 9
. The measurements of the E2 test groove (blue) have been plotted overlap on the designed probe. The programmed path of the cutting tool TCP tool is also shown (Black). The measurements
representation of test E18 (pink) have been moved from their natural position to be able to visualize them properly and it avoid overlap with the representation of test E2 (blue). The lateral
profiles of the E2 test have been raised 0.2 mm and the low plane of the groove has been lowered 0.2 mm.
In the front view measurements in
Figure 8
, the path deviations can be better appreciated.
Finally, in the top view measurements of tests E2 and E18,
Figure 9
, it is shown that the profiles of one of the sides is enlarged to visualize the evident difference in paths described by the cutting tool for the two materials tested.
Evaluation and comparison of different structural configurations
The different robot structural configurations were evaluated and compared based on the maximum experimental path deviation values. The structural configuration with the best experimental cutting tool
path deviation result was then selected, and the most influential factors in robotic machining were identified. The machining conditions with the best experimental results are those corresponding to
test E18 for PUR 300 at 0.352 mm, and E2 for Al at 0.607 mm.
4. Results through the Application of the Factorial Procedure
The simulated and experimental maximum cutting tool path deviation (SEMPAD) results are shown in
Table 4
Table 5
, along with the predictive error percentages.
The following figures show the percentages of predictive error committed for machining the groove in Al (
Figure 10
) and in PUR 300 (
Figure 11
The values of the robotic machining variables and parameters that provide lower cutting tool path deviation outputs are shown in the following
Table 6
Therefore, the factorial procedure has been validated through evaluating the results obtained, stating that:
• The achieved precision depends fundamentally on the working material, becoming worse as its hardness increases. The most unfavorable results appear when machining tests are performed on Al.
• The workpiece location is an important variable to control because it directly influences the stiffness available for the robot to counter the cutting forces. Considering the same robot wrist
configuration and the same cutting force vector for positions A and B, different trajectory deviation values are obtained due to the change in structural rigidity of the robotic arm.
• The milling head orientation with respect to the working direction (or the configuration of the robot’s axes) also modifies the robot’s stiffness and therefore influences the obtained precision.
• The cutting conditions, such as cutting speed (V[c]) or plunging depth (a[p]), influence the result and must be adequate.
• Acting on the previous conditions, the precision has been improved from an experimental deviation error of 0.685 mm to 0.352 mm for PUR 300 and from 0.982 mm to 0.607 mm for Al.
Analyzing the results of maximum cutting tool path deviation obtained through the factorial procedure shows that the most influential factors on the obtained precision or manufacturing tolerances for
the study performed, in order of highest to lowest importance, are as follows:
• Hardness of the material to be machined (HB)
• Workpiece position (B)
• Working direction or robot’s axes configuration (T)
• Plunging depth (a[p])
• Cutting speed (V[c])
The differences between the predicted and the experimental cutting tool path deviations are below 12%, an acceptable level of error, which validates this factorial procedure for preliminary study of
critical factors in robotic machining as a novel tool in the analyze of the most influential factors in robotic manufacturing systems.
5. Discussion
It should be noted that the factorial design of two levels is often used because of the high cost in terms of time and money that takes experimentation as a result of the large number of experiments
that would be needed to assess many factors using many levels, as it is in the case of robotic machining. On the other hand, the reliability of PREMET to determine the TCP path deviation has already
been evaluated in previous works as an acceptable method. For both reasons, it has been considered to establish a two-level factorial study as a suitable condition to evaluate the trend of each
The evaluation of the machining path deviation results obtained after applying the factorial procedure, get a reduction of the error of 36.39% for Al and 17.11% for PUR 300. the precision has been
improved from an experimental deviation error of 0.685 mm to 0.352 mm for PUR 300 and from 0.982 mm to 0.607 mm for Al, acting only on the conditions of the process studied and without turning to
path compensation methods. Once the IRB 6660-205/1.9 robot datasheet is also evaluated, we can determine that its repeatability value is +/− 0.1 mm, which means that there is a systematic error
between the values obtained from the test measurements and the programmed path with a maximum error of 0.1 mm, so the precision error results can be improved if they are corrected using corrective
coefficients. Experimental improvements achieved in this study cannot be directly compared with similar works of other authors. It is not possible to establish direct quantitative relationships
between the results of each study because neither the industrial robots nor the machining conditions allow it, but it is possible to position this document qualitatively with regard to other
published methods of compensation and prediction of the path accuracy.
The circularity indices range obtained by Klimchik, A. et al. [
] in circular groove milling with 100 mm diameter, under the cutting conditions described in the document with KUKA KR270, has been between 1.36 and 0.84 mm. Applying the error compensation method
proposed it is possible to improve the robot accuracy around 80–95%, with the circularity error made in 0.29 mm. In the study by Janez, G. et al. [
] the application of a predictive model of robotic machining precision allows for adjusting the position of the robot, improving the circularity precision of the hole in diameter from 1.86 to 0.23 mm
and in cylindricity from 0.87 to 0.16 mm.
The scientific advance provided by the factorial procedure does not achieve as much improvement (in terms of precision of the machining path) as the aforementioned works. Obviously the two paths
evaluated are very different, one goes straight and the other goes circular. In addition, both methods proposed path compensation techniques to improve robot accuracy while the experimental phase of
this work seeks to identify the critical factors among the studied conditions to achieve the highest performance in terms of precision in a robotic machining cell, so the results obtained by this
novel factorial procedure are considered satisfactory.
The evaluation of the obtained results provides a solution to the problem posed by the use of articulated industrial robots in machining operations, where the trajectory deviations and vibrations
obtained condition the quality of the result.
The study has been limited to addressing the machining of two types of soft materials, with a wide variety of materials with relatively low hardness, which may be the object of study in future
research. It is not ruled out that the factorial procedure can be applied to the study of the behavior of robotic machining systems in metallic or plastic materials with a hardness greater than 110
HB, but it has not been evaluated in this document.
Similarly, a flat cutting tool has been considered for the modeling of the cutting forces, but the factorial study procedure can be extended to the evaluation of cutting tools with other geometries,
f.e. the ball end milling. The machining operation used to solve determines the type of tool to use significantly. In this sense we must highlight the interest in evaluating the behavior of
industrial robots to perform complex machining operations, especially those that incorporate 3D trajectories in space, such as large-area contouring operations, where robotic systems have great
advantages compared to expensive NC machining centers. However, we must emphasize that the current trend of programmed path robot machining using software tools allows linearization-curved paths by
selecting smoothing parameters and tolerance, which is without the cycles regular programming, such as circular interpolations, whose use in robotic systems is either not allowed or not recommended.
Therefore, the programming of cutting paths in robotized machining complex operations as a set of rectilinear movements from point to point with a close enough is the most common. This is the reason
why this novel factorial procedure has been designed for an analysis of rectilinear machining paths, since the analysis of these rectilinear movements from point to point are the basic element of
study to advance in this field of research. Even so, the main challenge for this procedure is identified as addressing the evaluation of more complex machining paths in order to rule out possible
performance deficiencies in other stages.
6. Conclusions
The results obtained experimentally allow the validation of this factorial procedure, which represents a novel advance in the development of preliminary tools for the design. Implementation and
optimization of flexible manufacturing robot systems can be applied to sectors such as automation, aeronautics, naval application, etc.
The factorial procedure for preliminary study of critical factors in robotic machining has enabled the analysis of the milling conditions of the groove in soft materials and the identification of
those that improve the precision of the cutting tool path in robotic machining, obtaining an improvement in the experimental deviation error of 36.39% for Al, and 17.11% for PUR 300.
PREMET has been applied to obtain values of estimation of robot trajectory deviation with error less than 12%. Therefore, the use of PREMET, in the production of parts, has been validated for the
machining of the groove in soft materials, which allows the prediction of deviation of the cutting tool concerning the programmed path and simplifying the study of the main factors that determine the
result of robotic machining. The use of this prediction tool helps to assess what causes certain parameters to affect the precision of the robotic machining operation.
Consequently, the obtained results show that the factorial procedure serves to characterize the experimental robotic system, predicting the attainable manufacturing tolerances, and allowing the study
of the main constraints in the machining of relatively soft materials, with an acceptable level of precision, around tenths of a millimeter.
The test results show that the conditioning factor in order of relevance are that material hardness, followed by the work piece location, and the milling head orientation with respect to the working
direction or robot’s axes configuration and cutting conditions.
In addition, the factorial procedure also facilitates the analysis of more effective robotic system machining strategies. Such tool path deviation prediction provides guidelines for the attainable
manufacturing tolerances. The prognosis of the manufacturing quality facilitates the selection of the optimal machining strategy and anticipates whether a certain machining operation is feasible in a
robotic cell; thus, it ensures the appropriate use of the robot’s stiffness and safeguards the life of mechanical elements such as the drill head and the robot itself.
This novel factorial procedure can also help to establish design criteria in machining robotic cells by facilitating the robot’s selection, coherent location of the external axes and proper design of
the machining head or the method used for workpiece clamping.
Finally, this factorial procedure opens an opportunity in the field of Industry 4.0, through the possibility of developing digital tools that integrate and automate the factorial design phases,
calculation of the path deviations and evaluation of results.
Author Contributions
Conceptualization, I.I.S. and J.E.A.; methodology, I.I.S. and J.E.A.; validation, I.I.S. and C.G.G.; formal analysis, I.I.S., J.E.A., C.G.G. and V.R.P.; investigation, I.I.S.; writing—original draft
preparation, I.I.S. and C.G.G.; writing—review and editing, I.I.S., J.E.A., C.G.G. and V.R.P.; supervision, I.I.S. and J.E.A.; project administration, I.I.S. All authors have read and agreed to the
published version of the manuscript.
This research received no external funding.
This work has been developed by AIMEN Technology Centre in the search for the development of new strategies of robotized machining in soft materials. The study was developed within the R&D project:
‘Development of a methodology for designing robotized machining strategies (MECAROB)’ funded by XUNTA DE GALICIA and co-funded by FEDER.
Conflicts of Interest
The authors declare no conflict of interest.
Figure 4. (a) Orientation of the workpiece in L direction; (b) orientation of the workpiece in T direction.
Factor F[1] F[2] F[3] ….. F[j]
+ f[1]+ f[2]+ f[3]+ ….. f[j]+
− f[1]− f[2]− f[3]− ….. f[j]−
Factor Material (HB) P (mm) Direction (rad) n (rpm) a[p] (mm)
+ Al A L 8000 1.5
- PUR 300 B T 6000 1
(a)Tests for Al.
Test Material (HB) P (mm) D (rad) N (rpm) a[p] (mm)
E1 Al A L 8000 1.5
E2 Al A L 8000 1
E3 Al A L 6000 1.5
E4 Al A L 6000 1
E5 Al A T 8000 1.5
E6 Al A T 8000 1
E7 Al A T 6000 1.5
E8 Al A T 6000 1
E9 Al B L 8000 1.5
E10 Al B L 8000 1
E11 Al B L 6000 1.5
E12 Al B L 6000 1
E13 Al B T 8000 1.5
E14 Al B T 8000 1
E15 Al B T 6000 1.5
E16 Al B T 6000 1
(b) Tests for PUR 300
Test Material (HB) P (mm) D (rad) N (rpm) a[p] (mm)
E17 PUR 300 A L 8000 1.5
E18 PUR 300 A L 8000 1
E19 PUR 300 A L 6000 1.5
E20 PUR 300 A L 6000 1
E21 PUR 300 A T 8000 1.5
E22 PUR 300 A T 8000 1
E23 PUR 300 A T 6000 1.5
E24 PUR 300 A T 6000 1
E25 PUR 300 B L 8000 1.5
E26 PUR 300 B L 8000 1
E27 PUR 300 B L 6000 1.5
E28 PUR 300 B L 6000 1
E29 PUR 300 B T 8000 1.5
E30 PUR 300 B T 8000 1
E31 PUR 300 B T 6000 1.5
E32 PUR 300 B T 6000 1
Test Experimental Simulated % Predictive Error
Max Deviation (mm) Max Deviation (mm)
E01 0.614 0.559 9.0%
E02 0.607 0.551 9.2%
E03 0.791 0.713 9.9%
E04 0.713 0.646 9.4%
E05 0.696 0.630 9.5%
E06 0.648 0.578 10.8%
E07 0.971 0.883 9.1%
E08 0.802 0.719 10.3%
E09 0.699 0.631 9.7%
E10 0.629 0.563 10.5%
E11 0.889 0.801 9.9%
E12 0.712 0.647 9.1%
E13 0.705 0.640 9.2%
E14 0.655 0.596 9.0%
E15 0.982 0.894 9.0%
E16 0.841 0.760 9.6%
Test Experimental Simulated % Predictive Error
Max Deviation (mm) Max Deviation (mm)
E17 0.358 0.324 9.5%
E18 0.352 0.325 7.7%
E19 0.496 0.450 9.3%
E20 0.456 0.416 8.8%
E21 0.441 0.402 8.8%
E22 0.394 0.348 11.7%
E23 0.685 0.622 9.2%
E24 0.544 0.499 8.3%
E25 0.443 0.405 8.6%
E26 0.374 0.349 6.7%
E27 0.504 0.456 9.5%
E28 0.457 0.422 7.7%
E29 0.451 0.410 9.1%
E30 0.399 0.363 9.0%
E31 0.592 0.534 9.8%
E32 0.585 0.541 7.5%
Test Material (HB) P (mm) D (rad) n (rpm) a[p] (mm)
E2 Al A L 8000 1
E18 PUR 300 A L 8000 1
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What is recommended seeding rate?
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Category Theory - CoRecursive Podcast
Bartosz thanks for joining me on the podcast. People who listen to this, they might hear different terms that come up over the course of episodes like functors and algebraic data types, applicatives,
sum types, product types. Like where do these terms come from?
Oh, they come from category theory.
What Is Category Theory?
And what is category theory?
Yeah, category theory is a branch of mathematics. It’s a very, very abstract branch of mathematics. And really surprisingly this extremely abstract branch of mathematics has applications in
programming. Which is really a big surprise. I talked to some mathematicians and they were also surprised. When you are a mathematician and you want to study category theory, usually you first go
through all other branches of mathematics. So you have to be like fluent in topology and Algebra and Analysis. You know, it’s all these things because – group theory – because they use examples when
they teach category theory they use examples that are familiar to mathematicians. So they use examples from every branch of mathematics because it’s like a theory that abstracts over all other
branches of mathematics.
Category Theory and Programming
So what does it have to do with programming?
Well, because programming really is mathematics. You know, whether we know this or accept this or not. It really is and it’s all about structure. And mathematics is all about structure. I mean
mathematics is just building structures from essentially from nothing. You know, you put some axioms and then you derive stuff from it and so on. But it all has tremendous structure, right?
Everything is structured. There’s this follows from that, you know this, you divide things, you categorize things, you say, you know, well we have things like groups. We have things like monoids. We
have things like measures, topology, open sets and so on. So that you categorize things and you say like, you know, you have all kinds of sets but there are some special sets like open sets and so
on. And how do you characterize these things? Well, you characterize them by saying what properties they have.
So how they interact with other things that you already know or other things of the same type. And this is very much like what we do in programming. We define things and then we describe them by how
they interact with other things. Like in object-oriented programming. Okay, I mean mostly category theory is used in functional programming, but object-oriented programming also has very rigid
structure, right? So you do things like data hiding for instance. What does it mean to hide your data? It means that you want to describe your object, not by how it’s implemented, but how it
interacts with other objects. Namely its methods, right? What are messages, like what kind of messages can you send to this object and how it will respond to these messages by sending other messages
and so on. So this view of things, you know – that you have objects that interact with each other by sending messages, for instance – that’s like the essence of category theory. A category is just a
bunch of objects and arrows between them. That’s all. And this is perfect model for programming.
What Is Computation?
So there’s an abstract branch of math that deals with objects and arrows. Then somebody realized this looks a lot like what we’re doing with software development. Is that kind of the view of it?
Historically speaking?
No, no. You know, I mean people were always… Like computer scientists were always interested in like what is programming really? Right? So there are like these two schools of thought, the Turing
School of Thought that says: “Well the computer is just a device that has like a sort of like a printing head and the reader and so on, and it just moves the tape and infinite tape and moves them.”
So this is a very mechanical kind of approach. And then there is this very abstract approaching mathematics. Or like what is computation? Because normally in math you define things like functions and
you say, “well function is just a value that is related to the argument of the function.” So there’s argument and there is value. You know, its like function is defined and this is argument. This is
value. Give me an argument, I’ll give you a value. It doesn’t ask you, you know “well, but how long will it take you to calculate this value?” Right? No, it’s just there. Right?
But in computer science you have to think about, well if I want the value I’ll have to derive it from the argument somehow and I have to go through some steps. Right? And so I have to decompose this
bigger question into smaller question. And just answer every single smaller question and then combine them into one big result. Right? So being able to decompose bigger problems into smaller problems
and then combine the solutions. That’s essentially the description of, well I don’t know. It depends on who you are. You will say, oh, that’s a description of what I’m doing as a programmer. And a
mathematician would say other, this is a description of what I’m doing as a mathematician and you know. I feel this is, we’ll say that’s what I’m doing as a physicist and so on. It’s like everybody
is doing this. This is like the essence of all human activity.
That’s interesting. It makes sense to me that that’s what I do as a programmer. Like, if I’m given some requirements, then you know, I feel how they might break into like modules and then build those
modules and then combine them back. I would’ve never thought that that’s what a mathematician does, but maybe I don’t understand what mathematicians do.
Well, when you’re a mathematician, you also like divide your work into, okay, “I have to prove this theorem first, and in order to prove this theorem I have to prove this lemma. I have to define like
a new maybe space or object in my space and so on.” So it’s like, yeah, you are. And then if you have like a huge, huge problem, you want to split it into smaller problems. Like if you want to prove
Fermat’s Last Theorem, right? It’s not just like you think for a moment and then you say, oh, I got it, right? No. You said, you know, I have to study first elliptic curves. And there was this
theorem in elliptic curves that I have to prove. Oh, and somebody tried to prove it, but they failed because they couldn’t prove a certain lemma. Maybe I can do this. And so on. So you always split
things into smaller pieces, right? And then sometimes when you work in a team, right, you have to split, your work into individual tasks and so on. So like everything we humans do is always
So what does category theory bring to the table?
Yeah, category theory essentially studies all the different ways in which things can be composed and decomposed. That’s like the goal of category theory, you know, it’s like it says like what is the
structure of things? So what is a structure? Structure is like what parts something has and how these parts interact. I mean we don’t even ask ourselves these questions like what is structure? But
it’s an obvious thing, right? But it is structure is that you can decompose something into smaller pieces and you can describe how they interact that structure. Right? Otherwise you have like one
huge morass of stuff.
Why Functional Programming?
Why does categories theory seem to… Or why is it more used in the world of functional programming? Why not imperative programming or object oriented programming?
It’s mostly because functional programming is much more restricted, less hacky. I mean, I came from imperative programming. I mean I programming C++ the last four years, so you know, I’m familiar
with this stuff and I can do that. Right? But things in imperative programming are not very well defined. It’s like they don’t really have this nice mathematical structure. It’s more of an expert
system programming. Like there are rules, you know, you’re not supposed to do that. It’s mostly like you’re not supposed to do things, right? I get a lot of teaching of imperative programming. Its
don’t do that, don’t do that. And they slap you over the head, you know, this is like come dereference this pointer, okay? You’re not supposed to do that, right? Now in functional programming, it’s
more like there is a mathematical structure behind it and let’s just stick to it, right? And if you have this mathematical structure, then you don’t have to slap people over the head and say don’t do
that because it’s impossible to do certain things, right?
You mentioned before like Turing machines versus lambda calculus. I mean, computers really are imperative, aren’t they? In there execution. So why do we need something that models things abstractly
when they execute concretely, step after step?
Well, because there is this other thing on the other side of the computer and that’s the human programmer. Okay? So programming is not just about the computer, it’s about the interaction between the
human and the computer. And the computer doesn’t really care about, you know, how structured your code is. Why do we avoid goto’s? You know, it’s like computers love goto’s. Give me goto’s. Yeah, I
can execute them. I mean it’s like the processor has one of the basic instructions jump, right? Why not let me jump all the time. Right? So why do we avoid goto’s? Not because computers don’t like
goto’s, right? It’s because we humans, if we have too many goto’s, we just lose track of stuff and start making mistakes. Right? So I don’t think this is like the requirement, you know, for the
computer architecture or the programming language should reflect the architecture of the computer.
I think the computer language should reflect the architecture of the human mind and the human mind works differently, right? I mean we have to do this thing with what we call understand things,
right? So the computer doesn’t understand things, but we have to, right? We have to understand things and understanding again, it means that we have to like divide problems into smaller things. Give
them names, right? And we give them names. The computer doesn’t care what names you give like X1, X2, X3 there just fine, right? Why do we come up with these names for variables that, you know, it’s
like a factory of a lists or something like that. Right?
I read some story about how the library differs from like an Amazon warehouse, right? So in the library you have like a Dewey decimal system where people can locate where books are. In the Amazon
warehouse, Amazon has a very simple way of organizing things. They just put things wherever and just remember where they put them. Cause it’s a computer, they don’t need it.
Category Theory is for People Not Computers
So it seems like what you’re saying is category theory is for people not for computers. Is that the idea?
Yes, absolutely. Yeah.
So in category theory there’s like arrows and objects. What does that, what are arrows and objects?
Well that’s the wrong kind of question. You’re not supposed to look inside objects or arrows. They are just the basic thing. I mean they have certain properties, right? So you’ve described them not
by what they are, but how they behave. So objects are really…you can think of objects as being labels for beginnings and ends of arrows. That’s all. Because you have to say arrow goes from A to B.
Right? So this is why you need A and B.
So an arrow can go there.
Yeah. So then the arrow can connect that. So you have objects as these end points for arrows. And you have arrows that can be composed? And this is what it’s all about, about composition. So if you
have an arrow from A to B and an arrow from B to C, then there’s automatically an arrow from A to C. Which is a composition of these arrows. And that’s the principle of category theory. That things
are composable.
That’s it. We’re done.
Well, there’s one more thing. You know there has to be an identity arrow for every object. Which means it’s an arrow that when you compose with some other arrow, you get back that arrow again. So
it’s like multiplying by one or adding zero. Right? That’s an identity arrow. So that’s all the requirements. You know, you have to be able to compose arrows. There has to be a unit arrow for every
object that the identity arrow and also the composition has to be associative so that you don’t have to parentheses. Did they first compose these two and then compose it with the third one or did I
do it in that different, right.
So what is an example of a category where that has some meaning?
The category that we are using as programmers, that’s the category that underlies programming languages. Objects in that category are called types and arrows are called functions, pure functions.
This is why it’s so nice to talk about functional languages because in functional languages you have these pure functions, right? So you automatically are in category theory. And in fact this is like
maybe the best way of defining what you’re doing as abstractly as a programmer. If you ask somebody what is a type? What is a function? People think of types as sets of values, okay? That’s one
possible approach. But sets for my categories, it’s like to the lowest approximation. You can say, well it’s a category of sets and functions.
So I can think about like my method, like to string or show, it just takes say the set of things, the set of integers and maps it to a set of strings. So I kind of the–
It is the arrow that does that map.
Yeah, exactly. And you can compose them, right? I mean you can first say two string and then you can say upper case, right? And you have a composition of these two things. You can say uppercase, two
string or something like that. That’s a new function that is a composition of these two. Of course there is an identity function that we don’t even think much about but its a function that returns
this argument without changing it.
Yeah. So the identity on my integers example just returned to the same integer.
And so is the value there that category theory gives us sort of a language to look at things from the outside? To just look at the types and the mappings?
Well this is just sort of the beginning because the category has more structure. You can add additional structure to the category or you can discover additional structure in this category. So for
instance, in programming we are dealing with data structures, right? So what are data structures? Again, you know you have to start with some elementary types, right? And then from these types you
form more complex types. Like, how do you define a structure struct, right? I mean you say, well I’ll put an int there and I will put a string there and maybe a bullion, right? So you are combining
things together, right? So you take several types, which are objects in your category and you put them into one bigger type that combines them. That’s called a product category theory. Or you do
things like creating a data structure in which I either have a string or an integer like a union type or something. That also can be describing in a category theory as as a sum type and so on. So you
get product types, you get sum types and you can do Algebra on them and you get Algebraic data types.
Is the Terminolgy Off-Putting?
Yeah. One critique that I sometimes hear about functional programming is that the terminology can be confusing, right? That it can be not friendly to newcomers that we call it Algebraic data types
and some types of product types. And do you think that’s like a valid criticism?
No, I don’t think so.
I mean a name is a name, you know? Why should we invent, I mean there are some cultures in the programming that invent languages and libraries where they come up with weird names that are supposed to
be easier to understand than the ones they get from mathematics. But then it’s sort of like blocks you from going back to mathematics and trying to learn what’s the theory behind that? Right? So you
go to a mathematical paper and they use completely different language and you don’t know what is that? I don’t know what the sum is, what the, what our product is. Right? So I don’t know why use a
different language there and here? Anyways if I call something, you know like in object oriented programming, if I call something an object that’s so meaningful, right? Just try to define what you
mean by an object in a programming language. Just because you took a word from English language that everybody thinks they understand, that doesn’t mean that an object in C++ for instance, is
immediately obvious. Oh, that’s an object. Oh, I know what an object is because I learned it when I was an infant, right?
Yeah. And then wouldn’t you have objects that are like nouns or you know, like the factory builder you were talking about? How is that an object?
Yeah, exactly. People try to give them names like mappable I think. So functor is mappable and a monad is bindable. Like, is that really easier to understand?
Mappable is not bad. I don’t think bindable is good. Monoid could be like combinable maybe?
Combinable, yeah.
There we go. See we’ve already improved things
C++, Template Meta Programming, and Haskell
So you mentioned C++. So how did you get here from doing C++ development to writing about category theory?
Well, I was always interested in… How do you write good programs, right? How do you make your programs reliable and that made me interested in the theory behind programming. I was always interested
in exploring the boundaries. Like what is the hardest thing in C++ that you can think of? Well, I guess template method programming. Right? Template method programming. So I got into template method
programming and it was fascinating because it was so different from regular programming. Because it’s done at compiled time there is no mutation, right? So how do you program without mutation? Okay.
So I started reading these books about template method programming in C++, and it was really hard to understand. Then I found that they actually take all these ideas from functional programming. You
know, some of these people are truly know Haskell and they just translate it into C++ and say, Hey, I came up with this great thing. So I discovered this and I started my own franchise. I started
blogging about, oh, this is how you can do using template method programming, this fancy thing here and fancy thing there. You know, it’s like, oh, okay, cool. People were amazed that you can do
these things in C++. Right? But it was really cheating. You know, I was taking something from Haskell that in Haskell it’s just like a one liner and I’m in translating it. So eventually I decided to
cut the middle man and just go directly to Haskell and see how it works.
It’s a cheat code.
It’s a way to understanding functional programming was like a shortcut for you to be able to understand template method programming. Being able to think in Haskell but write in C++ was your
Yeah. And then I even started talking to C++ programmers saying that even if you program in C++, it’s a good idea to learn some Haskell and maybe use it as pseudo-code. You know, I just think a lot
of people from the C++ they don’t like functional programming because of performance. Because it’s true, performance it depends on what kind of programming you’re doing. But if you’re doing string
processing, maybe Haskell would not be the fastest language to do this. And if you start dealing with performance issues, then your Haskell code, I mean you can optimize stuff and then your Haskell
code becomes uglier, not so clean. That clean code doesn’t really perform very well. So thats the problem. But you know, it’s like if you solve your problem first in Haskell and then you translate it
into C++ you will probably get better quality code at the end and better performing.
And so how did you get from Haskell to category theory?
Well, because of the language, you know, because they use these terms, the functor, monad and so on. So I was curious where do these terms come from and what’s the meaning of that. So I started
looking into mathematical foundation on that, trying to understand. And then again, you know, it’s like, I know I don’t do much programming, mostly just testing some ideas, mostly testing the types,
do they work together and so on. Because even the Haskell is too constraining but there are certain ideas in math that are difficult to translate to Haskell. So again, I’m finding the boundaries of
what can be expressed in Haskell and then going beyond them as category theory. There is a whole area in between which is dependent types. So dependent types are very interesting. And I’m trying to
learn Idris, which is a dependent type language and figure things out.
Nice. Yeah. I’ve had a couple episodes about dependent types. Edwin Brady was on the podcast.
Okay. Yeah.
Yeah. So I think what you’re saying is you went from writing C++, but thinking about it in Haskell to writing Haskell, but thinking about it in category theory.
So is there a notation or something for category theory? What do your thoughts look like? Is it in a mathematical notation?
A lot of my thoughts when I do them on paper, on the whiteboard, they are diagrams. Yeah. So you know, arrows between dots. That’s how you work in category theory.
So it’s purely visual. Now your thinking is actually all diagram based.
Yeah. And it’s very good for me because this is that different people have different types of thinking. Some people are better at kind of Algebraic thinking where they think in terms of symbols and
formulas. Yeah, I don’t do that. I have a problem with that. I am very visual and I like pictures. I draw lots of pictures and that helps me.
Yeah, that’s interesting. I don’t know, am I a visual thinker? I think I’m somewhat of like a oral, like I think, I think in like a soundtrack, you know what I mean? So what’s the most successful?
What’s been a success that’s come out of applying category theory to software?
Yeah, I think the whole language. Haskell is very solid because it’s based on mathematical foundations and category theory particular. So I think, well, okay. Monads probably monads are like the most
successful thing in functional programming because when people started working in Haskell, it was a purely functional language. It still is a purely functional language. But with the purely
functional language you have this problem of how do I print something? It’s a side effect. Okay, how do I get input from a file from the user, right? That’s not a function. You know, it’s like get
character, get string, you know it’s not a function because every time you call it, it returns a different value possibly. Right?
So how do you describe this in terms of categories? And so instead of like doing the easy way out, like most other languages, including functional language, Haskell is probably the only, well there
are some others, but the strict functional pure language, right? Like even ML is cheating. So they started thinking, you know, how can we do this without cheating? Right? So this is a really hard
problem. And because scientists like hard problems, unlike engineers who will try to find shortcuts and do it quickly because there is a–
Deadline pending and their salary depends on it. And scientists like is interested in will there be a publication out of it? Right? So if it’s a hard problem then the publication will be good. Right?
So they figured out this and they found out that other way to do this is to use monads and then once monads came into functional programming, they are now spreading to other languages.
Math Was Created Not Discovered
Yeah, definitely. I had Phil Wadler as a guest on the podcast. I think he did the implementation of bringing monads to Haskell.
Yes, yes he did. It’s like it was Eugenio Moggi who first introduced monads into computer science. But it was very theoretical and then Phil Wadler read his paper, talked to him and came up with
something that was actually very practical and worked.
An interesting thing about Phil was that he was saying that he thinks these mathematical concepts like blamed the calculus specifically, that these are like kind of innate to the universe. Maybe he
wouldn’t quite say it that way, but he thinks that they are.
Yeah, he would, he would. Yes. I had discussions with him about this, you know, I totally disagree. He said later in this a lot of mathematicians are Platonists. They believe that there are just
things inherent… like mathematics is built into the universe. I totally disagree.
So what’s your perspective on it? Why do you disagree?
I think mathematics is something that is inherent to human beings. This is the only way we can deal with our environment is because we have small brains. It’s like compared to the size of the
universe, this is just like a tiny, tiny thing. And it evolved from apes that were trying to solve problems, like how to run away from a predator or how to kill an animal and eat it, start a fire and
so on. So the way we deal with complexity is by dividing into smaller tasks, solving them and then recombining the solutions. And that’s what we do with everything. We just don’t know how to deal
with things that are not decomposable. I mean one part of life is that every living being has to have some kind of model of the environment.
So it has to create a simplified model of the environment. I mean we humans have this sort of we can even think about the model as opposed to reality, right? We have a model of the environment in our
brains and we know it’s simplified, right? But the fact that you can take the environment, the universe, and simplify it, meaning throwing away some things and decomposing in the smaller things
that’s just an amazing thing. And people think, well, isn’t that amazing that the universe is decomposable? And I think no it’s not amazing that, I mean, it is amazing that this part of the universe
that we live in the particular scales, like the meter scale is decomposable, right? But like you can go like a 10 levels down to like micro scales and suddenly things are completely different. Right?
You know, this is why we don’t have life at the Planck scale because it’s not decomposable, you know, at our level of universe, things are nicely decomposable and this is what makes life possible.
And as humans
The World Is Flat
Yeah, I would think that it’s like has to do with where we evolved, right? Like we have trouble understanding things at the quantum level because that’s just foreign to us, right? The same way we
have a hard time understanding how approaching the speed of light works. It’s just cause humans never existed in a world where they went that fast. So it doesn’t fit in the model of our brain.
But humans never existed at atomic scales because its impossible for life to exist at the atomic scale. Why is it impossible? Why can’t there be a life on the surface of the nucleus of Hydrogen,
right. Why?
Well, what would it be made out of?
Exactly. Right, right.
Oh, I see.
Yeah, cause you’re saying made of means decomposable, right?
Ah, yeah.
There’s nothing to compose or decompose.
I see. Yeah. That makes sense. You had some talk that you sent the link to me where you posit that the world is a not round or that the world is flat. I think.
Oh, well it was tongue in cheek. Yeah. Yeah.
Could you explain that point? Like how is the world flat?
Because people often confuse, well Platonists will confuse the thing that we understand about the world with what the world is really, right? The ontology of the world.
Because the world isn’t really a sphere I guess, right?
Yeah, yeah. The Earth is not a sphere. It’s like if you start arguing that the world is a sphere, you’re talking about some kind of ideal of a world, right? A model, right? So you can model the Earth
to some approximation as a sphere and that’s good. You know, but the Earth is not a sphere. You have to understand the Earth is not a sphere. There is a tiny difference, you know, several kilometers,
you know, in some places. Right. So it’s–
It has to do with your decomposition thought. I think, right? Cause you’re saying like for me to understand what the world looks like, I need to come up with some model and it has to fit inside my
head, right?
Yeah, exactly.
So it has to be, it has to get rid of some of the details.
This is what we call abstraction. Abstraction is getting rid of some details and getting a nice model. So yeah, so I mean you can describe the Earth as a sphere to some approximation. Then you can
describe the Earth as flat. Mathematically it’s okay. You know, it’s like there is a system of coordinates in which the Earth is flat and it would be much, much harder to work with that system of
coordinates because you would constantly have to do adjustments. You know, like, yeah, you move towards the South Pole for instance. What we consider the South Pole? You know, and like your
coordinates would just blow to infinity.
So this abstraction I think so at the level of my code, right? Like I just have imperative code that runs on the computer. Like it does jumps and whatever. And then like I have my actual like maybe I
have my functional programming code, it’s like a level operate it hides some details and then I’m trying to connect this all back. So then I think what you’re saying is there’s like category theory
is like something that gets rid of some of these details even more so it’s like if you want to look at the flat, if you, you can assume the world is flat when you’re measuring a hundred meter race,
like that’s fine. Right? But then when you want to zoom out and go to the South Pole, you maybe you want to assume it’s a sphere and then like at each level of detail, if you want to do a foundation
for your house, like at the lowest level, you can’t assume the world is flat. You have to actually know where the bumps and cracks are.
So category theory is a way to hide some of the details so that we can fit it in our head. So it’s not something. I’m trying to understand your perspective it’s like it’s not something innate to the
mathematical underpinnings of the world. It’s a way that humans use to fit big, hairy concepts into their heads.
Yeah, exactly. Category theory is a very good description of how our minds work. It has nothing to do, well maybe that’s just a harsh statement, but you should not confuse mathematical model with
reality. And I hear this very often when you know, especially talking to mathematicians, they say, okay, we have them in this category. We have like this category described the world and inside this
category we have a model of the world, right? But the world is also a category. It’s like no, the world is not a category. The world is already in a category, its already a model in our brain, right?
So there is this problem occurs in mathematics and in physics. You know, its like in physics you have quantum physics, you know, and then you have an external observer always. You know, its like –
That never sat right with me.
Yeah. And you know, I mean, as a physicist you kind of get used to it, but never really, you know, it’s like, you know that there’s something wrong with quantum mechanics at some level There’s always
this classical observer thats observing quantum effects. Like, so it means that quantum theory is incomplete and every single theory that we have is incomplete.
Cause there just models.
And even true in mathematics, you know, there’s an incompleteness theorem, right? Godel’s incompleteness. So it’s like everything we do is incomplete.
Why Learn Category Theory?
So why should people learn category theory?
Why should people do anything?
Is it fun? Is it practical? Is it both?
For a programmer for instance? Like it doesn’t make sense to learn category theory. Will it make you a better programmer? I think it will. It will sort of make you a higher level programmer. You
know, it’s like being able to lift yourself above your program. It’s like otherwise you, you know, you’re just like a little ant working in an anthill, right? And the only things you see are the
things that are close around you, right? It’s like your never able to like lift yourself above the anthill and see how it’s related to the rest of the world and so on. And category theory provides
these ways, these extractions that otherwise you wouldn’t even think of. You know, it connects things. It says, well, a list is in a way… A list data structure is somehow similar to an optional in
some ways. How is it similar? Well, because its a monad, right? Unless you have this word monad, you know, and the description of what a monad is, you will never see the similarity between a list and
optional or a function.
Well you could just say it’s a list of a maximum one arguments, like a list of Max size one.
Okay. Yeah. Good.
But maybe that would get you towards the concept of a monad. I don’t know.
Yeah, yeah, yeah. It probably would. Yeah.
If it became widely understood as a language where we talk about software development using these terminology, like how would that influence software development? It would make us feel smarter
because we’d have all these abstract Algebra terms to throw around.
I think it’s already changing the way we program. I mean the development of programming and programming languages is sort of from goes from bottom up. So we start working to solve particular
problems, right? And then we discover that there are similarities between these problems and that we could simplify our lives instead of redeveloping the same thing over and over and over. Maybe we
can abstract over it, right? And a lot of programming languages evolve. Every programming language really evolves this way that it just adds abstractions on top. And I saw this in C++, you know, I
was always really happy when new features came to C++ because like oh I can do more stuff. Right? But they were always like slapped on top of the existing stuff you know, and it was, there was like
a, this really ugly syntax of templates, right? You know, it’s like there were even problems, like if you have a template inside the template, then the two greater than signs would collide then the
compiler would think this is like your all or this is a right shift. You know? It’s like that’s crazy, right? It just says, you know, this language was not developed to think at this high level of
obstruction. Right? And it was just added to it. So there are new languages which lets you program more abstractly, which means that you can reuse your code, that you can write code that will solve
different kinds of problems using the same methods. I mean, you know, there is, there’s like this whole industry of patterns, right? There is the gang of four pattern book and so on. That was a great
thing when it came out, but it showed this tremendous weakness of programming languages that you have to describe these patterns in a book, right?
And then ask the programmer, Oh, here you have a particular, you know, write your code according to this book instead of use this library. Right? Because we’ve solved this problem before. Right? So
the gang of four solved a certain bunch of problems. Right? But they could not express it in a language.
Math Culture
What was a stumbling block that you hit or that people hit when trying to learn about category theory?
The biggest stumbling block is that mathematicians, they just don’t explain things the way that’s easy to understand. Let me describe it this way. I don’t know. It’s like when you read a mathematical
paper or a book, you know it’s written in a certain style.
Is it that it’s written for mathematicians to understand but not for programmers or is it mostly…
I think it’s a culture. It’s a culture thing. You know, it’s like there is even, I mean, being a physicist, you know, I see the difference in culture between physics and mathematics. Like in physics,
you know, we had this great guy, Feynman who was like, he would always try to explain things in the simplest terms. And he loved giving talks to outsiders trying to explain, you know, the quantum
field theory. You know, stuff like this. And mathematicians don’t do that. They write these abstract papers and if you don’t get it, you know, it’s sort of like you’re stupid because you don’t get
it. You know, you shouldn’t be reading this. I was fine when I had the opposite. It’s like if I can’t explain it, it means I don’t understand it. You know, if I can’t explain it in simple terms, I
don’t understand.
And is this why you started your series explaining category theory for programmers?
Yes, yes. This and because I thought there was this problem that mathematicians will use examples from other branches of mathematics. And so what if you don’t know these other branches of
mathematics. Does it mean that the category theory is useless to you? No, it’s not. It’s just that, you know, you have to like rewrite category theory using different examples. And there’s plenty of
examples in programming that I could use to explain concepts in category theory. And it turns out that these concepts are not that hard if you have the right examples.
Yeah, that totally makes sense. It’s interesting. Like I guess one thing I never thought of before I started talking to you was, so there’s this field called category theory. My understanding is that
somebody noticed a whole bunch of similarities between a whole bunch of branches of math. Right?. And that’s where it came from. But what you’re saying is I think that those similarities aren’t to do
with math. They’re to do with the people who created math.
Yeah. Yes, exactly. And you see these things not only in mathematics but also in physics, in programming. It’s like in every area of human activity you see the same patterns, the same structures. I
sometimes am amazed, you know it’s like reading a paper in some area of mathematics and they are using the same ideas that I found in some other area of mathematics. Then thinking about are these the
only patterns that we humans are able to discern and we are just describing the same patterns over and over and over again in different contexts.
What We Don’t Know
Does that mean that there’s things we’ll never figure out because they’re not part of the patterns we can find?
We have figured some things out and there are so many other things that we haven’t figured out. We always think that like we already understand 99% of stuff and there’s this 1% missing. And I think
this is completely wrong. We understand the 0.001% if that even makes sense, right? And there’s so much stuff missing if we just ignore the stuff that we don’t understand.
Do you know what we’re ignoring?
Well, yeah, there are certain things that we know that are sort of beyond the scope of what we can do right now. And we think, well eventually we’ll figure these things out. But in physics, obviously
there was this humongous problem of we have general relativity in gravity on the one side and quantum mechanics on the other side. And we just don’t know how to connect that. And we think, well maybe
string theory, maybe this, maybe that, you know, it’s like, and the deeper we go into it, the more we understand that we don’t understand stuff and maybe these things just don’t decompose. You know,
that’s the obvious solution. Okay? And if they don’t decompose, then we can’t really describe them. We can’t have a model, right? It’s like does everything have to have a model? Can it be simplified?
Why should it be?
You can always come up with approximates for things. Can’t you?
Like how the globe is it?
Because this is what’s been happening in the history of humankind. Whether you can also understand this as like we are discovering things that are possible for us to understand and we just ignore
everything else.
Yeah. I don’t know a lot about physics, but, and this’ll probably get cut from the actual episode, but like a dark matter that doesn’t seem to make sense. They’re like, okay, well the universe has to
have so much mass to it and there’s not enough. So let’s just postulate something that weighs the rest of it. That seems like cheating, right?
Yeah. I mean, and a lot of people think it is cheating. Yeah.
Crazy stuff.
There’s dark matter. There’s dark energy. Yeah.
Maybe that’s the difference between physicists and mathematicians. I don’t know.
Well, physicists are very pragmatic, you know, if it works, it works. Okay. If I can calculate something and then test it, then it’s great. In this way, physicists are more like programmers. If it
works let’s ship it.
So what’s next for you after category theory?
I’m interested in homotopy type theory, but this is really hard.
Has to do with the quality. I know that much. So thank you so much for joining me. It was a lot of fun.
Okay, thank you. Thank you for having me.
I make CoRecursive because I love it when someone shares the details behind some project, some bug, or some incident with me.
No other podcast was telling stories quite like I wanted to hear.
Right now this is all done by just me and I love doing it, but it's also exhausting.
Recommending the show to others and contributing to this patreon are the biggest things you can do to help out.
Whatever you can do to help, I truly appreciate it!
Thanks! Adam Gordon Bell
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Using Rich Tasks in an Objective Led Culture
What were we trying to achieve?
As teachers, we appreciate the need to have clear objectives at the start of lessons but have been aware of the limitations this sometimes seems to place on our ability to get the most out of using
rich tasks. Through our case study we tried to address two issues:
• Objectives that are seen as a strait-jacket, restricting the teachers' room for manoeuvre. These can be a barrier to teachers responding appropriately to the students they are working with, not
allowing them to 'go with the flow' of a lesson.
• Objectives that are solely about content and do not recognise the mathematical process skills that we want students to develop.
How were we trying to achieve our aims?
In our schools we have been working towards creating a balance of content and process-led objectives, moving away from an ethos of just having formal, explicit, content-based objectives at the start
of each lesson. Raising the status of process objectives has allowed teachers to be more flexible in their view of what students might achieve in each lesson.
In our planning, we have:
• used rich tasks from the NRICH website (http://nrich.maths.org) to focus on process objectives as well as content objectives, sometimes explicitly prioritising process objectives. The Curriculum
Mapping documents published on the NRICH website (http://nrich.maths.org/curriculum) have provided teachers with a good selection of rich tasks to start with.
• introduced new content in a process-rich way. See, for example,
• used rich tasks as a context in which to apply and consolidate knowledge, make mathematical connections and create the need for new knowledge. See, for example,
At School A
A team of teachers were planning a new KS3 Scheme of work. They have grouped content objectives into topics and incorporated process objectives that take into account the different teaching
approaches being adopted. In any lesson both process and content objectives can be shared with the pupils. NRICH tasks are one of the vehicles through which these process objectives are being
introduced. Fortnightly meetings are being used to discuss the outcomes of recent lessons based on NRICH tasks, and to plan future lessons.
At School B
At this school they were happy with the basic structure and content of their existing KS3 Scheme of Work so have adapted it in line with the new curriculum with a focus on processes. They initially
looked at algebra units with the intention of replacing various activities with rich tasks. Teachers trialled tasks from the NRICH website to help them decide which ones were most appropriate for
their developing Scheme of Work and to identify process objectives they might focus on when using the task. Initially the department had a day off timetable to discuss approaches and begin work on
the adjustments. They continued this work in their own time and during department meetings. As they taught the algebra unit they made further refinements. They are planning to move on to the geometry
and measure units next.
At School C
This department also started by working on the algebra topics in their KS3 Scheme of Work.
They used rich tasks in a variety of ways. For example, they chose:
• a task that covered an entire unit, for example,
• rich tasks that replaced whole lessons, for example,
• rich tasks that were used for starter or plenary activities, for example,
For each unit of work the department included key processes in their plans, wrote a 'story' of how each lesson might be taught and completed a mapping diagram of all the processes the students might
use. These could then be included in lesson objectives. They continue to work on other units.
At School D
This school made minimal changes to their Scheme of Work but they identified rich tasks and planned for flexibility by using process objectives. This encouraged flexibility because teachers felt they
were not being strait-jacketed by a specific content outcome. Students working on the problem Number Pyramids (
) took the problem in their own direction and wanted to know how to multiply negative numbers and decimals. The fact that students were taking the initiative reflected an important mathematical
process that the school was trying to foster.
At School E
The school worked on a longer term project, covering several lessons, focussing on trigonometry content objectives. A small team planned the series of lessons with objectives spanning the sequence of
lessons rather than individual lessons. These objectives covered
• process skills, such as representing and communicating
• mathematics content
• links to some of the historical background to measurement and measuring instruments
• links to real life applications of trigonometry, such as fire engine ladders.
Part of the project also included linking with the Technology department to make clinometers for use in practical activities.
How well are we doing it?
Reflecting the aims of the new curriculum, schools wanted students to focus more often on how they achieved their solutions to problems. This meant that at the start of lessons teachers sometimes
gave objectives which consisted solely of process objectives or a combination of process and content objectives. On occasions content objectives were 'held back' for later discussion.
Process and content objectives
NRICH tasks have been used successfully by schools to consolidate knowledge. In these circumstances schools often felt that it was good to give students both process and content objectives at the
start of a lesson or sequence of lessons. They felt there was no tension in sharing what was to be learnt and reflecting on this in lesson plenaries.
The NRICH task Diagonal Sums (
) was included in the first Algebra unit of the Scheme of Work. It introduced students to the merits of using algebra when making generalisations whilst requiring them to engage with the key
processes of exploration and conjecturing.
The NRICH tasks Number Pyramids
) and More Number Pyramids (
) were used to consolidate work on solving equations and to give students a context in which to construct their own equations and communicate their ideas.
Choose three single-digit numbers and write them in the bottom row of the pyramid.
Try a few more times with different sets of three numbers until you understand what is happening.
Using the same starter numbers $2$, $1$, $4$ and $6$, rearrange them to find the largest and smallest possible totals at the top of the pyramid.
Can you get these totals in different ways?
Keep a record of the arrangements of the starting numbers which generate each total.
What do you notice about these arrangements?
Test out your observations and insights with other sets of four numbers.
You could include big numbers, small numbers, negatives and decimals.
Can you explain what is happening? Can you explain why it is happening? Can you explain it algebraically?
Adapt your ideas to apply to taller pyramids.
When introducing new ideas however, it was felt that giving content objectives at the start would sometimes pre-empt the journey students would make and not allow them to independently 'notice' the
new mathematics on the way. In these cases, the content lesson objectives were sometimes kept general.
For example, when the problem 'Tilted Squares' was used to introduce Pythagoras' Theorem, the Lesson Objective shared with the students referred to area, rather than Pythagoras' Theorem. The project
also involved linking to historical background including the work of Pythagoras and the Pythagoreans, and these historical objectives were shared.
Another approach when introducing new ideas, or encouraging students to make connections between different aspects of mathematics, was to 'hold back' content objectives until the end of the lesson or
sequence of lessons. Teachers asked students to think about the content objectives in the plenary, once the lesson activity was completed. This encouraged students to reflect on their own learning
and think about what they had learned, rather than what they were told they had learned. This approach also supported the teachers involved in assessing pupils' progress.
The Teachers' Notes for NRICH tasks on the website contain useful guidance on content and process objectives, key questions and possible approaches that can be used in the classroom. Teachers have
found these very useful.
See, for example, the Teachers' Notes for
At School A, NRICH tasks were used to introduce new content in a way that developed students' process skills. For example
In lessons where new mathematics was being developed, or mathematical connections were being made, teachers felt that giving content objectives spoilt the discovery. Teachers focussed initially on
process objectives, sharing and agreeing content objectives at the end of the sequence of lessons.
There are some sets of coloured balls with numbers on them:
A set is mixed up and two balls taken at random. For example:
The numbers on the balls are added together: $4 + 5 = 9$
The aim is to get as many even numbered totals as possible.
Which set is most likely to give you even totals?
Can you number five balls so that you have an even chance of winning?
How about four balls?
How about six balls?
How about...?
Process objectives
At one school, the department chose to use a school-wide one-off event on the theme of 'conflict resolution' to work as a whole department on two rich tasks. Two problems from the NRICH website were
selected for the potential they offered for challenging students to work effectively in groups and to arrive at a shared understanding that could be communicated to the rest of the class. It was
these process skills that formed the objectives for the sessions and that were referred to throughout the lesson
In a different school, a lesson on averages had the objective 'to be able to listen to and communicate findings within a group setting'. In this lesson the teacher made it explicit that she was
focusing on how the groups worked and discussed amongst themselves rather than on the content outcomes.
This process objective was also used with another group, together with a content objective, 'to be able to calculate averages'. See M, M and M:
There are five positive whole numbers with the following properties:
• Mean = 4
• Median = 3
• Mode = 3
How many different sets of five numbers can you find that satisfy these conditions? Can you convince us you have found them all?
If I also tell you that the range is 10, can you identify my numbers?
Another five positive whole numbers have the following properties:
• Mean = 31
• Median = 33
• Mode = 34
• Range = 8
How many different sets of five numbers can you find that fit these criteria?
Can you convince us you have found them all?
Lesson objectives are now being used in creative ways which support the teaching and learning of students. Including process objectives has improved the quality and range of mathematical discussions
in our classrooms.
This article is the result of the collaborative work of:
Susanne Mallett, Steve Wren, Mark Dawes and colleagues from Comberton Village College
Amy Blinco, Brett Haines and colleagues from Gable Hall School
Jenny Everton, Ellen Morgan and colleagues from Longsands Community College
Craig Barton, Debbie Breen, Geraldine Ellison and colleagues from The Range School
Danny Burgess, Jim Stavrou and colleagues from Sawston Village College
Catherine Carr$\acute{e}$, Fran Watson and colleagues from Sharnbrook Upper School
David Cherry, Chris Hawkins, Maria Stapenhill-Hunt and colleagues from The Thomas Deacon Academy
Charlie Gilderdale, Alison Kiddle and Jennifer Piggott from the NRICH Project, Cambridge
For similar articles about teachers using NRICH go here. | {"url":"https://nrich.maths.org/articles/using-rich-tasks-objective-led-culture","timestamp":"2024-11-12T00:17:39Z","content_type":"text/html","content_length":"53571","record_id":"<urn:uuid:bf1c3aba-4a52-45d5-b2c7-cbf0978584d8>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00517.warc.gz"} |
Rhombus | Brilliant Math & Science Wiki
A rhombus is a parallelogram in which all sides are equal. It's much more basic than a parallelogram and has the opposite properties of a rectangle. (I do not have all the properties and information
on this subject since I am in a college prep geometry class, so I would appreciate it if someone would add information and review my work. Note that I do not know the area and perimeter properties.)
The definition of a rhombus is as follows:
A rhombus is a quadrilateral whose sides are all equal.
Notice that the diagonals are perpendicular and the sum of the perpendicular angles is \( 360^\circ \).
Property 1. All pairs of the consecutive sides of a rhombus are congruent.
Property 2. The diagonals of a rhombus are perpendicular.
Remember, if a quadrilateral is both a rectangle and a rhombus, then it is a square.
Since all rhombuses are parallelograms, but not all parallelograms are rhombuses, a rhombus follows some of the rules that a parallelogram does. The following properties apply to parallelograms and
Property 3. One diagonal of a rhombus bisects a pair of opposite angles.
Property 4. The opposite sides of a rhombus are parallel. | {"url":"https://brilliant.org/wiki/rhombus/","timestamp":"2024-11-13T12:46:02Z","content_type":"text/html","content_length":"42244","record_id":"<urn:uuid:4d005692-711b-4f83-92bc-a74da7d6ce49>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00544.warc.gz"} |
Favorite Soft Drink
Top Posters In This Topic
-Canada Dry Ginger Ale
-Sierra Mist
-Coca Cola
-Cherry Coke
-Diet Coke (like as much as regular Coke)
IS OKAY:
-Mountain Dew
-Dr. Pepper
-Pibb Xtra
I'm not that big on soda, I just get Sprite or 7-Up all the time and rarely drink any other soda.
^ Speak for yourself!
Matt, I could say something mean and nasty right now but .... what the hell NINJA!
Mtn Dew and some variants (such as White Out, Voltage, etc.), good ol' Coca-Cola, orange soda (especially Sunkist and its variant, Sparkling Lemonade) and 7-Up (along with 7-Up Cherry) are my top
Irn Bru
Coca Cola
Pepsi (it still tastes good when flat)
Mountain Dew
Irn Coke (mixture of Irn Bru and Coca Cola)
Orange Soda! I do, I do, I dooooo!
Kenan & Kel!
My favorite soft drinks are:
Lime Fanta
Coke Cherry
Hawaiian Punch
Mtn. Dew Code Red
Mello Yellow
Sun Drop
Crystal Pepsi
Crap: Barq's Root Beer. No flavor.
Amazing: Squirt, Saranac & IBC root beer, pretty much anything from Jones Soda.
Non-carbonated: Loganberry. It's a Buffalo NY thing. Awesome stuff.
I have switched to unsweetened Ice Tea and rarely drink a soda now.
This is real? Where can I find it, I want to try...
Anything Fruit Flavored, plus the mountain Dew Variations.
My "regular" soda is Pepsi Max, but I am really liking the Dr. Pepper 10 calorie (per can) soda. I try to stay away from it since it's so bad for you, but the new Monster Absolute Zero is decent.
It's either root beer,Pepsi Throwback,Sierra Mist/Sprite,or cream soda.
Edited by RCBoilermaker14
The only stuff i drink now is green tea, its like liquid crack, I seriously have an addiction
As much as I LOVE Pepsi and RC Cola I barely drink them now. If I drink soda I try to make it a clear soda. I honestly feel a lot better since I have stopped drinking/limiting to one/two a week. It
helps on the waist size as well!
I LOVE unsweetened iced tea!!! I drink it alllllllllllllllllllllllllll the time. It just sucks that large amounts of iced tea can cause kidney stones.
My favorite is Polar Orange soda. It is much more of a natural orange flavor than most orange sodas. I sometimes mix it with orange juice for even more orangey goodness.
Cheerwine when in areas that sell it, other wise I bounce between Coke and Dr.Pepper
Pibb Xtra which is also known as Mr. Pibb.
Coke, no question. However, I do like the taste of Dr. Pepper and Mountain Dew.
My favorite soft drink is unsweetened tea.
I have to go with Coke Zero.
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Understanding "314159u": A Dive into Mathematical and Cultural Significance - BloggingTrickes
Introduction 314159u
The sequence “314159u” immediately evokes the mathematical constant π (pi), which is approximately 3.14159. Pi is one of the most fascinating and important numbers in mathematics, symbolizing the
ratio of a circle’s circumference to its diameter. This guide explores the significance of “314159u,” examining its mathematical roots, cultural impact, and relevance in various contexts.
What is “314159u”?
314159u is a string of characters where the numerical part, 314159, corresponds to the first six digits of the mathematical constant π (pi). The inclusion of “u” at the end does not have a specific
mathematical meaning but could be a unique identifier or part of a broader code or context.
Key Components:
• 314159: Represents the first six digits of π.
• u: An additional character that may be used for identification or context-specific purposes.
Mathematical Significance of 314159
1. Pi (π):
Pi is an irrational number that cannot be exactly expressed as a simple fraction. Its decimal representation goes on infinitely without repeating, making it a fundamental constant in mathematics. The
value of π starts with 3.14159 and is crucial for various mathematical calculations and formulas.
• Definition: Pi is the ratio of the circumference of a circle to its diameter.
• Applications: Used in geometry, trigonometry, and calculus.
• Decimal Expansion: Starts with 3.14159 and continues indefinitely.
2. Importance in Formulas:
Pi is integral to numerous mathematical and scientific formulas, particularly those involving circles and spherical shapes. It appears in calculations for area, volume, and various trigonometric
• Area of a Circle: A=πr2A = \pi r^2
• Circumference of a Circle: C=2πrC = 2 \pi r
• Volume of a Sphere: V=43πr3V = \frac{4}{3} \pi r^3
3. Historical Context:
The concept of π has been known since ancient times, with various civilizations approximating its value. Archimedes made significant contributions to calculating π’s value, and modern mathematicians
have extended its calculation to millions of digits.
• Ancient Approximations: Early cultures used approximations of π for practical purposes.
• Archimedes: Developed methods for approximating π’s value.
• Modern Calculations: Computers have extended π’s known digits to millions.
Cultural Impact of Pi
1. Pi Day:
Pi Day is celebrated annually on March 14 (3/14) to honor the mathematical constant. The date corresponds to the first three digits of π (3.14) and is an opportunity for mathematics enthusiasts to
celebrate and engage with math-related activities.
• Celebrations: Schools and math communities often host events and activities.
• Symbolism: Represents the joy and intrigue of mathematics.
2. Pi in Popular Culture:
Pi has found its way into popular culture, appearing in literature, movies, and music. It often symbolizes the beauty and mystery of mathematics.
• Literature: Pi is featured in works like “Life of Pi” by Yann Martel.
• Movies: Films like “Pi” by Darren Aronofsky explore mathematical themes.
• Music: Composers and artists have incorporated π into their works.
3. Mathematical Challenges:
The challenge of calculating π to more digits has become a popular pursuit among mathematicians and hobbyists. Records are regularly broken for the number of digits of π calculated, showcasing
advancements in computational technology.
• Record Calculations: Demonstrates progress in computing power and algorithms.
• Mathematical Curiosity: Reflects the ongoing fascination with π.
Applications of “314159u”
1. Educational Tools:
“314159u” can be used in educational contexts to teach students about π and its significance. It can serve as a memorable reference for mathematical lessons and exercises.
• Teaching Aid: Helps illustrate the importance of π in mathematics.
• Educational Games: Can be used in quizzes and educational activities.
2. Coding and Programming:
In programming and coding, “314159u” might be used as a placeholder, variable name, or part of a code related to mathematical calculations involving π.
• Variable Naming: Could be used to represent a constant or value.
• Coding Practice: Provides a real-world context for using numerical constants.
3. Branding and Identification:
The unique string “314159u” might be used for branding or identification purposes in various contexts, including tech, software, or personal projects.
• Branding: Could be part of a brand or product name.
• Identification: Used for specific identifiers or codes.
Common Issues and Misconceptions
1. Misinterpretation of “u”:
The inclusion of “u” in “314159u” may lead to confusion, as it is not a standard part of π. It’s important to understand the context in which “u” is used to avoid misconceptions.
• Clarification Needed: Understand the purpose of “u” in specific contexts.
• Avoid Assumptions: Recognize that “u” may not have mathematical significance.
2. Precision Concerns:
When dealing with π, precision is crucial. The first six digits (314159) are an approximation, and for precise calculations, more digits may be required.
• Precision Matters: Use accurate values of π for precise calculations.
• Contextual Usage: Recognize the appropriate level of precision needed.
“314159u” represents a fascinating intersection of mathematics, culture, and digital representation. By understanding the significance of π and its cultural impact, we gain appreciation for the
beauty and complexity of mathematical constants. Whether used in education, coding, or branding, “314159u” serves as a reminder of the intriguing world of mathematics and its influence on various
aspects of life. | {"url":"https://bloggingtrickes.com/314159u/","timestamp":"2024-11-05T12:43:38Z","content_type":"text/html","content_length":"252943","record_id":"<urn:uuid:43940fed-c231-4e85-9943-3c0991c8f82a>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00189.warc.gz"} |
Daily Quiz 2/14
Daily Quiz 14 february 2012
Number 1:
In this threat double blue’s innerboard is slightly worse than white’s, but the escaped checker and the 5 attacking checkers bring the position close to a double.
The fact that blue can attack with 1,2,3,5 and 6 gives him a great attacking diversification, on top of that 41,42 and 43 can be used as double blitzers and D4 is very good. At an equal score I
would say it is a Double/Take, at the score 0-4 I would say:
Answer: Double/Pass
Number 2:
With a pipcount of 52-61 and a Keithcount of 56-64, and some adjustment for extra checkers off (not taken into account in the Keith Count), say 2 pips a checker, makes 56-68.
An equal race of 56 pips gives a chance of about 62%, the 12-pip-difference makes 22% extra = 84%. I guess this is above the redoubling window at 7-a 11-a,
(let’s look it up : a score of 5a/11a gives 20% chance, 3/11 gives 11% chance, so this is a risk of 9%, winning gives 50% chance (7a/7a), TP = 9/39, about 20%, so white can take until 80%)
Answer: DP
Number 3:
My first idea of playing 8/3 6/3 is not good because of the bad forward playability of the blue position. Especially 6’s and 2’s play badly. This will force blue to leave his barpoint (or
midpoint) in the near future. After playing 18/15 18/13 hitting will cost white the midpoint.
Answer: 18/13 18/15 | {"url":"https://www.bgonline.org/forums/webbbs_config.pl?noframes;read=115603","timestamp":"2024-11-10T05:52:18Z","content_type":"text/html","content_length":"6679","record_id":"<urn:uuid:e7de34b4-14de-4622-b207-9e6d0d19a51f>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00843.warc.gz"} |
Math 4 Wisdom. "Mathematics for Wisdom" by Andrius Kulikauskas. | Research / KrasauskasKulikauskas
• Understand these domains.
□ Make a map of key concepts in these domains.
□ Relate persistent homology to the kissing number.
• Do calculations with practical examples.
□ Make a map of mathematics by studying sets of tags at Math Stack exchange and Math Overflow. Use it to predict (using duality) the location of new theorems and concepts.
□ Study ways of identifying new concepts in chess.
□ Make a map of deepest values.
Readings: Topological data analysis
Rimvydas Krasauskas
Rimvydas Krasauskas's interests
• New foundations for mathematics
• Having a real programming language to check proofs
• What about a reverse approach... where we divide up the space into regions where there are points and where there are not. Consider the largest circles that you can create with no points in it.
You can do this by considering the bisecting points on the segments between each pair of points - use these as the centers of your circles. So this will give you a break down into regions. Now do
this again allowing a circle to contain one point. (This is perhaps the current case? - for we can use each point as the center of a circle.) Then continue by allowing it to contain 2 points. And
so on. | {"url":"https://www.math4wisdom.com/wiki/Research/KrasauskasKulikauskas","timestamp":"2024-11-14T14:12:43Z","content_type":"application/xhtml+xml","content_length":"13107","record_id":"<urn:uuid:32ee652c-68cd-439d-9a3d-5ca1568fa172>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00016.warc.gz"} |
Calculate index in excel
As I know there is not any software to calculate heavy metals contaminants indexes. You should use excel to input your raw data and calculate the indexes Calculating the Indices using Excel is a
tool that helps you learn how the composite human development indices - the human development index (HDI), the 7 May 2019 To calculate the volatility of a given security in Microsoft Excel, first
determine In column C, calculate the interday returns by dividing each price by the The CBOE Volatility Index, or VIX, is an index created by the Chicago
In this lesson you can learn how to use INDEX function in Excel. performance, which is actually to determine if we are on minus in financial performance or if How to Calculate Percentages in Excel.
Percentages in Excel Index: Percentage Calculation Types. Calculate a Percentage of a Number. Calculate a Percentage VLOOKUP function in excel can lookup for approx match in table. that is less
then lookup value is matched and value from given column index is returned. VLOOKUP with 2 criteria or more by using the INDEX and MATCH functions in Excel. The step-by-step tutorial will show you
how to build the formula and learn 17 Jul 2019 What is going on and how can I fix it? Note: I have my formulas set to automatic calculations and I'm using Microsoft Office 365 ProPlus on 28 Jun
2019 We create the formula SUM to calculate the sum of the orders. INDEX returns the reference of a cell, and not the value, when the function is 15 Apr 2019 In many cases, merging data in Excel can
easily be accomplished with a single INDEX-MATCH (or a VLOOKUP). Sometimes, though, it's time
Excel Formula Training. Formulas are the key to getting things done in Excel. In this accelerated training, you'll learn how to use formulas to manipulate text, work with dates and times, lookup
values with VLOOKUP and INDEX & MATCH, count and sum with criteria, dynamically rank values, and create dynamic ranges.
17 Jul 2019 What is going on and how can I fix it? Note: I have my formulas set to automatic calculations and I'm using Microsoft Office 365 ProPlus on 28 Jun 2019 We create the formula SUM to
calculate the sum of the orders. INDEX returns the reference of a cell, and not the value, when the function is 15 Apr 2019 In many cases, merging data in Excel can easily be accomplished with a
single INDEX-MATCH (or a VLOOKUP). Sometimes, though, it's time Say you have a table of data and you want Excel to look up a certain value and return a corresponding value in a different row. For
that, you need a lookup function The INDEX and MATCH functions working together are more flexible than using the The INDEX function returns a cell value from a range, given a row and/or How to
Convert Excel to Google Sheets · How to Calculate Z Score in Excel:
25 Oct 2019 Generally, the index of one is selected for the market index, and if the stock behaved with more volatility than the market, its beta value will be
18 Feb 2015 In other words, you use INDEX when you know (or can calculate) the position of an element in a range and you want to get the actual value of that The INDEX function is categorized under
Excel Lookup and Reference functions. The function will return the value at a given position in a range or array. Use INDEX and MATCH in Excel and impress your boss. Instead of using VLOOKUP, use
INDEX and MATCH. To perform advanced lookups, you'll need INDEX
This Excel tutorial explains how to use the Excel INDEX function with syntax and The Microsoft Excel INDEX function returns a value in a table based on the The second parameter is the row number used
to determine the intersection
A Price index, also known as price-weighted indexed is an index in which the firms, which forms the part of the index, are weighted as per price according to a price per share associated with them.
Each stock will influence the price of the index as per its price. The Excel INDEX function returns the value at a given position in a range or array. You can use index to retrieve individual values
or entire rows and columns. INDEX is often used with the MATCH function, where MATCH locates and feeds a position to INDEX.
11 Jul 2018 Syntax. =SUM(INDEX([column of values],1):[@ column of values]). Steps. Start with
11 Jul 2018 Syntax. =SUM(INDEX([column of values],1):[@ column of values]). Steps. Start with 26 Jun 2018 Hot Key Excellence provides a training game for Excel shortcuts to save your time. Check
here some of the useful functions such as index and match: For this step, we will use the value that we calculated with the help of 20 Oct 2016 Knowing how an index is performing can give you an
idea of how the market is doing and how your portfolio is doing relative to the index. 28 Mar 2017 How to calculate Simpson's diversity index in easy steps. Sample problem solved with pictures and
downloadable OpenOffice worksheet. 8 Mar 2016 Excel Tip: Using Index to calculate a cumulative sum (e.g. a Year to Date total). Let me start with an apology. My current workload has meant
3 Nov 2019 This article describes how to create a custom function to calculate BMI (Body Mass Index), save Excel Add-In file, install the Add-In and unistall values> to replace the functions with
values. If this is not done, the calculated index values will only be displayed when indices.xls is in memory and will be lost PivotTable report: Insert Calculated Fields, Calculated Items, Create
Formulas, Use Index Numbers, Solve Order of Calculated Items. 8. Excel Pivot Tables: Filter In this lesson you can learn how to use INDEX function in Excel. performance, which is actually to
determine if we are on minus in financial performance or if How to Calculate Percentages in Excel. Percentages in Excel Index: Percentage Calculation Types. Calculate a Percentage of a Number.
Calculate a Percentage | {"url":"https://digoptionehsivre.netlify.app/dalluge73237kuc/calculate-index-in-excel-hob","timestamp":"2024-11-07T04:38:08Z","content_type":"text/html","content_length":"34470","record_id":"<urn:uuid:1c0dfdde-e2af-469a-8206-e48e8248e4f0>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00653.warc.gz"} |
Tutoring needed - AR15.COM
The information is correct USING A 600 YARD ZERO. If you ZERO at 600 yards with the stated ammo, then yes, you will be "high" by the #'s given (in inches) at the appropriate ranges stated.
ETA: If you are zeroed at 100 yards, your bullet will only be @4" low at 200 yards, and @14.4" low at 300 yards.
These are rough figures for a 168 grain HPBT-Match style bullet leaving your barrel at approximately 2700 FPS, which is just a tad over the factory Federal Match load if I remember correctly.
Any good reloading book will give you the "Ballistic Coefficient" (BC) for a specific bullet. Basically, the BC of a bullet is a numerical figure that represents the bullets ability to overcome
atmospheric resistance, or the "air resistance" as the bullet is in flight. The higher the BC, the more efficient the bullet travels through the air, thus, flatter trajectory.
For example, a 150 Grain Round Nose bullet has a BC of @ 0.266 while the same weight bullet in a Boat Tail Spitzer design has a BC of @ 0.423.
Know the BC of your bullet, the velocity in which it leaves your barrel, and any good Ballistic Coefficient table will give you data on the trajectory of the round in flight using various ZERO
ranges. I have found the Speer BC tables to be quite accurate | {"url":"https://www.ar15.com/forums/ar-15/Tutoring_needed/16-252934/","timestamp":"2024-11-02T12:21:58Z","content_type":"text/html","content_length":"132960","record_id":"<urn:uuid:671faeda-3043-415b-aeb0-7230be978710>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00363.warc.gz"} |
Physics & Astronomy Classes
Complete Listing of all courses
PHYS 1001W: Energy and the Environment
4 credits
Prerequisites: 1 year high school algebra
Fundamental principles governing physical world in context of energy/environment.
PHYS 1011: Physical World
3 credits
Prerequisites: 1 year high school algebra
Fundamental laws and principles governing the physical world, discussed in the context in which encountered in modern science and technology.
PHYS 1021 - Introductory Physics
5 credits
PHYS 1101W: Introductory College Physics
4 credits
Prerequisites: High school algebra, plane geometry, trigonometry
Fundamental principles of physics in the context of everyday world. Use of kinematics/dynamics principles and quantitative/qualitative problem solving techniques to understand natural phenomena.
PHYS 1102W: Introductory College Physics II
4 credits
Prerequisites: PHYS 1101W or PHYS 1107
Fundamental principles of physics in context of everyday world. Use of conservation principles and quantitative/qualitative problem solving techniques to understand natural phenomena.
PHYS 1107: Introductory Physics Online I
4 credits
Prerequisites: High school algebra, plane geometry, trigonometry
Principles of physics in context of everyday world. Use of kinematics/dynamics principles together with quantitative/qualitative problem solving techniques to understand natural phenomena.
PHYS 1108: Introductory Physics Online II
4 credits
Prerequisites: PHYS 1101W or PHYS 1107
Principles of physics in context of everyday world. Use of kinematics/dynamics principles together with quantitative/qualitative problem solving techniques to understand natural phenomena.
PHYS 1221: Introductory Physics for Life Science Majors I
4 credits
Prerequisites: High School or College Calculus
The class exposes the student to physical principles and concepts, demonstrates how these principles can be applied to quantitatively describe natural phenomena, and provides the student with an
opportunity to perform hands-on experiments and measurements that model how physical knowledge is obtained. The living world exists in the physical universe, and a complete understanding of
biological processes is impossible without a firm foundation in the basic physical principles to which all systems, living and inorganic, must adhere. The basic principles of classical mechanics,
fluid mechanics, and oscillations and waves will be examined, with particular emphasis to their application in biological systems, using mathematical analysis at the level of basic calculus.
PHYS 1222: Introductory Physics for Life Science Majors II
4 credits
Prerequisites: PHYS 1221 or equivalent
This is the second course in the introductory physics sequence for life science majors. The class exposes the student to physical principles and concepts, demonstrates how these principles can be
applied to quantitatively describe natural phenomena, and provides the student with an opportunity to perform hands-on experiments and measurements that model how physical knowledge is obtained. The
fundamental principles of thermal physics, electricity and magnetism, optics, and nuclear physics are considered.
PHYS 1301W: Introductory Physics for Science & Engineering I
4 credits
Prerequisites: concurrent registration is required (or allowed) in MATH 1271 or concurrent registration is required (or allowed) in MATH 1371 or concurrent registration is required (or allowed) in
MATH 1571
Use of fundamental principles to solve quantitative problems. Motion, forces, conservation principles, structure of matter. Applications to mechanical systems.
PHYS 1302W: Introductory Physics for Science & Engineering II
4 credits
Prerequisites: PHYS 1301W, concurrent registration is required (or allowed) in MATH 1272 or MATH 1372 or MATH 1572
Use of fundamental principles to solve quantitative problems. Motion, forces, conservation principles, fields, structure of matter. Applications to electromagnetic phenomena.
PHYS 1401V: Honors Physics I
4 credits
Comprehensive, calculus-level general physics. Emphasizes use of fundamental principles to solve quantitative problems. Description of motion, forces, conservation principles. Structure of matter,
with applications to mechanical systems.
PHYS 1402V: Honors Physics II
4 credits
Prerequisites: PHYS 1402V
Fundamental principles to solve quantitative problems. Description of motion, forces, conservation principles, fields. Structure of matter, with applications to electro-magnetic phenomena.
PHYS 1901: Global Warming Solutions
2 credits
In this seminar, we will consider various possible solutions to the current and future global warming problem. This is a topic of intense global importance. Topics will include efficiency and
conservation, reduced carbon in electricity production and transportation, wind and solar power, nuclear power, policy changes, third world solutions, reforestation, and more.
PHYS 1905: Aurora: From Myths to Modern Science
2 credits
The aurora, or northern lights, have long fascinated humans. We now know that aurora occur on many other planets, including Jupiter, Neptune, and Uranus. We will examine the myths from both the
northern and southern hemisphere that were devised to explain this beautiful natural phenomenon. The development of our scientific understanding of the aurora is littered with completely incorrect
explanations by prominent scientists. It is only with the new measurements made after the space age that we have finally begun to understand the aurora--both on the Earth and on other planets. If the
space weather cooperates, we will try to observe the aurora.
PHYS 1906: What is Space Weather (and Why Should You Care)?
2 credits
In this class, we will explore the way our sun changes over the eleven-year solar cycle and how this can affect events from airline travel, cell phone coverage, and power outages to beautiful aurora
and manned spaceflight to Mars. We will also touch on space weather on other planets (including exoplanets) and the possible impact on development of life. If the space weather cooperates, we will
try to observe the aurora and related phenomena including sunspots.
PHYS 1910W: What is Time?
2 credits
The precise meaning and use of the concept of time has evoked serious study and debate among the most able of human thinkers for more than 2,000 years. In this seminar, we will review several of the
current perspectives as well as some of this history of the concept of time from the points of view of philosophers, biologists, psychologists, and physicists.
PHYS 1911W: How Likely is Extraterrestrial Life?
2 credits
The goal of this course is to familiarize students with the main available scientific facts and arguments which bear on the question of the likelihood of extraterrestrial life. A second goal is to
familiarize students with aspects of the various relevant disciplines early in their university careers when they may still be selecting a major. The third goal is to provide familiarity with
information resources at the university, particularly through the library, as well as improved reasoning, writing, and speaking skills.
PHYS 2201: Introductory Thermodynamics and Statistical Physics
4 credits
Prerequisites: PHYS 1302W or PHYS 1402V or PHYS 1502V; concurrent registration is required (or allowed) in MATH 1272 or MATH 1372 or MATH 1572H
Thermodynamics and its underlying statistical nature.
PHYS 2303: Physics III: Physics of Matter
4 credits
Prerequisites: PHYS 1302, MATH 1272 or MATH 1372 or MATH 1572H
Thermodynamics, mechanical/electromagnetic waves, optics, quantum theory. Applications of quantum nature of solids.
PHYS 2311: Modern Physics
4 credits
Prerequisites: PHYS 1302 or PHYS 1402, CHEM 1022, MATH 2243
Broad overview of physical concepts developed in twentieth century. Special relativity, wave-particle duality, Schrodinger equation, Bohr atom, hydrogen atom in wave mechanics, many-electron atoms,
x-rays, nuclear structure, radioactivity, nuclear reactions, statistical physics.
PHYS 2503: Physics III: Intro to Waves, Optics, and Special Relativity
4 credits
Prerequisites: PHYS 1302W, MATH 1272 or MATH 1372 or MATH 1572H
Third semester of introductory physics. Mechanical/electromagnetic waves, optics, special relativity.
PHYS 2503H: Honors Physics III
4 credits
Prerequisites: PHYS 1402V or PHYS 1502V, honors student or permission of University Honors Program or instr consent
The third semester of a calculus-based introductory physics sequence. Topics include: relativistic kinematics and dynamics, mechanical and electromagnetic waves, light, interference, diffraction,
wave-particle duality and topics in modern physics. Course emphasizes the use of fundamental problems to solve quantitative problems. Intended primarily for those who have completed 1401V/1402V,
although those students with outstanding performance in 1301W/1302W may be granted permission to enroll.
PHYS 2601: Quantum Physics
4 credits
Prerequisites: PHYS 2503H or PHYS 2503; concurrent registration is required (or allowed) in MATH 2243 or MATH 2373 or MATH 2574H
Introduction to quantum mechanics. Applications to atomic, molecular, condensed-matter, nuclear, elementary-particle, and statistical physics.
PHYS 3022: Introduction to Cosmology
3 credits
Prerequisites: PHYS 2601
Large-scale structure and history of universe. Dark matter, cosmic microwave background. Newtonian/relativistic world models. Physics of early universe. Cosmological tests.
PHYS 3041: Mathematical Methods for Physicists
3 credits
Prerequisites: PHYS 1302, MATH 2373 (or equivalent courses)
This course introduces additional mathematical topics that physics majors need to properly handle upper division physics classes.
PHYS 3071W: Laboratory-Based Physics for Teachers
4 credits
Prerequisites: College algebra; no credit for CSE students or students who have completed PHYS 1201/1202, PHYS 1301/1302, PHYS 1401/1402, or PHYS 1501/1502
Laboratory-based introductory physics. Topics selected to apply to elementary school curriculum: earth's motion, properties of matter, heat and temperature, kinematics, and electric current.
PHYS 3605W: Modern Physics Laboratory
4 credits
Prerequisites: Completion (or concurrent registration) in PHYS 2503 or 2503H
Laboratory experiments in atomic, solid state, and nuclear physics. Introduction to data analysis techniques as well as the communication of scientific results through maintaining a logbook and
writing papers.
PHYS 3993: Directed Studies
1 credit — may be repeated for a total of 10 credits
Prerequisites: Instructor consent, Departmental consent
Directed study in Physics in areas arranged by the student and a faculty member.
PHYS 3994: Directed Research
1-5 credit — may be repeated for a total of 10 credits
Prerequisites: Instructor consent, Departmental consent
Independent, directed study in physics in areas arranged by the student and a faculty member.
PHYS 4001: Analytical Mechanics
4 credits
Prerequisites: PHYS 2303 or PHYS 2601 or CHEM 3501 or CHEM 3502; 2 semesters sophomore math
Analytic Newtonian mechanics. Mathematics beyond prerequisites developed as required.
PHYS 4002: Electricity and Magnetism
4 credits
Prerequisites: PHYS 2303 or PHYS 2601 or CHEM 3501 or CHEM 3502; 2 semesters sophomore math
Classical theory of electromagnetic fields using vector algebra and vector calculus.
PHYS 4041: Computational Methods in the Physical Sciences
4 credits
Prerequisites: Upper division or Graduate student or Instructor consent
Introduction to using computer programs to solve problems in physical sciences. Selected numerical methods, mapping problems onto computational algorithms.
PHYS 4051: Methods of Experimental Physics I
5 credits
Prerequisites: PHYS 1302W, Concurrent registration is required (or allowed) in PHYS 3605W or Equivalent lab experience or Instructor consent
Contemporary experimental techniques. Introduction to modern analog and digital electronics from an experimental viewpoint. Use of computers for data acquisition and experimental control. Statistics
of data analysis.
PHYS 4052W: Methods of Experimental Physics II
5 credits
Prerequisites: PHYS 4051, PHYS 3605W
Contemporary experimental techniques illustrated by experiments with data analysis. Students design and execute an experimental project. Lectures on specialized topics of professional concern.
PHYS 4101: Quantum Mechanics
4 credits
Prerequisites: PHYS 2303 or PHYS 2601 or CHEM 4502; 2 semesters of sophomore math
Mathematical techniques of quantum mechanics. Schrodinger Equation and simple applications. General structure of wave mechanics. Operator methods, perturbation theory, radiation from atoms.
PHYS 4111: History of 19th Century Physics
3 credits
Legacy of 17th-century experimental and theoretical physics. Experimental and theoretical discoveries in 19th-century physics (light, atomic theory, heat, thermodynamics and statistical mechanics,
electromagnetism) within the context of educational, institutional, and political developments in Europe and the United States.
PHYS 4121: History of 20th Century Physics
3 credits
Experimental and theoretical discoveries in 20th-century physics (modern physics, theory of relativity, quantum theories, nuclear physics to World War II) within the context of educational,
institutional, and political developments in Europe and the United States.
PHYS 4121W: History of 20th Century Physics
3 credits
The transition from classical to modern physics (relativity, quantum) and its architects (from Planck and Einstein to Heisenberg and Schrodinger). The WWII bomb projects in the US and in Germany.
Post-war developments (solid state, particle physics).
PHYS 4201: Statistical and Thermal Physics
3 credits
Prerequisites: PHYS 2601
Principles of thermodynamics and statistical mechanics. Selected applications such as kinetic theory, transport theory, and phase transitions.
PHYS 4211: Introduction to Solid-State Physics
3 credits
Prerequisites: PHYS 4101, PHYS 4201
A modern presentation of the properties of solids. Topics include vibrational and electronic properties of solids; diffraction of waves in solids and electron band structure. Other possible topics
include optical properties, magnetic phenomena, and superconductivity.
PHYS 4303: Electrodynamics and Waves
3 credits
Prerequisites: PHYS 4001, PHYS 4001
Analytical mechanics. Electricity/magnetism, including mechanical/electromagnetic wave phenomena. Physical/geometrical optics.
PHYS 4501: Experimental Project
1-5 credits – may be repeated for 5 credits
Prerequisites: PHYS 4052, Instructor consent
Research project in physics area of contemporary interest. Project must be approved by faculty coordinator before registration.
PHYS 4511: Introduction to Nuclear and Particle Physics
3 credits
Prerequisites: PHYS 4101
Fundamental particles and Standard Model. Symmetries/quarks, models of nuclei, interactions between particles/nuclei, tests of conservation laws, fission/fusion.
PHYS 4611: Introduction to Space Physics
3 credits
Prerequisites: PHYS 4001, PHYS 4002, or Equivalent, or Instructor consent
Dynamics of charged particles/plasmas in space. Physics of the Sun and solar wind. Solar/galactic cosmic rays. Interactions of solar wind with planetary magnetospheres. Dynamics of Magnetosphere.
Formation of the aurora. Physics of radiation belts.
PHYS 4621: Introduction to Plasma Physics
3 credits
Prerequisites: PHYS 4001, PHYS 4002, or Equivalent, or Instructor consent
Basic properties of collisionless, magnetized plasmas, single particle motion, plasmas as fluids, magnetohydrodynamics, waves in plasmas, equilibrium, instabilities, kinetic theory/shocks.
PHYS 4623: Introduction to Modern Optics
3 credits
Modern optics broadly defined as geometrical, physical, and quantum optics, including interference and diffraction, optical polarization, Fourier optics, cavity optics, optical propagation, optical
coherence, lasers, optical detection, and optical instruments.
PHYS 4811: Introduction to General Relativity
3 credits
Prerequisites: PHYS 4001 and PHYS 2503 or PHYS 2503H
Introduction to general relativity for undergraduate students. The course will introduce basic concepts of differential geometry and use them to motivate Einstein's Equation. It will then solve
Einstein's equation to study particle orbits, gravitational lensing of light, black holes, and gravitational waves. Brief introduction to cosmology and evolution of the universe will be included.
PHYS 4911: Introduction to Biopolymer Physics
3 credits
Prerequisites: PHYS 2303, PHYS2403H, PHYS2503 or CHEM 3501 or Instructor consent
Introduction to biological and soft condensed matter physics. Emphasizes physical ideas necessary to understand behavior of macromolecules and other biological materials. Elements of thermodynamics
and statistical mechanics are presented as needed.
PHYS 4950H: Senior Thesis
1-3 credits – may be repeated for 6 credits
Prerequisites: Instructor consent
Independent project with advisor
PHYS 4960H: Honors Seminar
31 credit – may be repeated for 2 credits
Prerequisites: Upper division honor, Instructor consent
Designed to prepare students for senior honors thesis projects and provide guidance in choice of future careers.
PHYS 4993: Directed Studies
1-5 credits – may be repeated for 10 credits
Prerequisites: Instructor consent
Directed study in Physics in areas arranged by student and faculty member.
PHYS 4994: Directed Research
1-5 credits – may be repeated for 10 credits
Prerequisites: Instructor consent
Independent, directed study in physics in areas arranged by student and a faculty member
PHYS 5001: Quantum Mechanics I
4 credits
Prerequisites: PHYS 4101 or Equivalent or Instructor consent
Schrodinger equation: bound state and scattering problems in one dimension. Spherically symmetric problems in three dimensions, angular momentum, and the hydrogen atom. Approximation methods for
stationary states. Time-dependent perturbation theory. Operators and state vectors: general formalism of quantum theory.
PHYS 5002: Quantum Mechanics II
4 credits
Prerequisites: PHYS 5001 or Equivalent
Symmetry in quantum mechanics, space-time symmetries and the rotation group, Clebsch-Gordan coefficients and the Wigner-Eckart theorem. Scattering theory. Method of second quantization with
elementary applications. Relativistic wave equations including Dirac equation.
PHYS 5011: Classical Physics I
4 credits
Prerequisites: PHYS 4001, PHYS 4002, or Instructor consent
Classical mechanics: Lagrangian/Hamiltonian mechanics, orbital dynamics, rigid body motion, special relativity.
PHYS 5012: Classical Physics II
4 credits
Prerequisites: PHYS 5011 or Instructor consent
Classical electromagnetism: electrostatics, magnetostatics, Maxwell's equations, electromagnetic waves, radiation, interaction of charged particles with matter.
PHYS 5022: Relativity, Cosmology, and the Universe
4 credits
Prerequisites: PHYS 2601 or Instructor consent
Large-scale structure and history of universe. Introduction to Newtonian and relativistic world models. Physics of early universe. Cosmological tests. Formation of galaxies.
PHYS 5041: Mathematical Methods for Physics
4 credits
Prerequisites: PHYS 2601 or Graduate Student
Survey of mathematical techniques needed in analysis of physical problems. Emphasizes analytical methods.
PHYS 5071: Physics for High School Teachers: Experimental Foundations and Historical Perspectives
3 credits
Prerequisites: Gen. physics or Instructor consent; No credit for physics grad or grad physics minor
In-depth examination of a conceptual theme in physics, its experimental foundations and historical perspectives. Kinematics and dynamics from Aristotle through Einstein; nature of charge and light;
energy and thermodynamics; electricity, magnetism, and quantized fields; structure of matter.
PHYS 5072: Best Practices in College Physics Teaching
1-3 credits – may be repeated up to 3 times for 5 credits
Pedagogies for introductory physics classes. Topics from educational research/practice as applied to classroom.
PHYS 5081: Introduction to Biopolymer Physics
3 credits
Prerequisites: Working knowledge of thermodynamics and statistical mechanic
Introduction to biological and soft condensed matter physics. Emphasizes physical ideas necessary to understand behavior of macromolecules and other biological materials.
PHYS 5201: Thermal and Statistical Physics
3 credits
Prerequisites: PHYS 4101, PHYS 4201, Previous exposure to Thermodynamics, Introductory Statistical Physics, or Equivalent classes
Equilibrium Statistical Mechanics. General Principles of Statistical Mechanics: Ensembles. Derivation of Thermodynamics from statistical principles. Classical Systems. Quantum Statistical Mechanics:
Fundamentals. Photons. Ideal Fermi & Bose Gases. Non-ideal gases. Introduction to Phase Transitions.
PHYS 5621: Introduction to Plasma Physics
3 credits
Prerequisites: CSE Graduate student, Working knowledge of waves/electromagnetism
Basic properties of collisionless, magnetized plasmas, single particle motion, plasmas as fluids, magnetohydrodynamics, waves in plasmas, equilibrium, instabilities, kinetic theory/shocks.
PHYS 5701: Solid-State Physics for Engineers and Scientists
4 credits
Prerequisites: Graduate Student or Advanced undergrad in physics or STEM field
Crystal structure and binding; diffraction; phonons; thermal and dielectric properties of insulators; free electron model; band structure; semiconductors.
PHYS 5950: Colloquium Seminar
1 credit
Prerequisites: Graduate Student, Advanced undergrad in physics, Departmental consent
Colloquium of School of Physics and Astronomy.
PHYS 5970: Physics Journal Club
1 credit – may be repeated for 3 credits total
Prerequisites: PHYS 2601, PHYS 2605 or Equivalent class; Intented for 2nd year graduate students in physics
Weekly student-led presentation, discussion, and critical analysis of important papers.
PHYS 5980: Introduction to Research Seminar
1 credit – may be repeated for 3 credits total
Prerequisites: Graduate Student or Upper division physics major
Introduction to the research activities of the School of Physics and Astronomy.
PHYS 5993: Directed Studies
1-5 credits – may be repeated for 15 credits total
Prerequisites: Instructor consent, Departmental consent
Independent, directed study in physics in areas arranged by the student and a faculty member.
PHYS 5994: Directed Research
1-5 credits – may be repeated for 15 credits total
Prerequisites: Junior, Departmental consent
Problems, experimental or theoretical, of special interest to students. Written reports.
PHYS 8001: Advanced Quantum Mechanics
3 credits
Prerequisites: PHYS 5002 or Instructor consent
Topics in non-relativistic quantum mechanics; second quantization. Introduction to Diagrammatic and Green's function techniques and to relativistic wave equations. Application of relativistic
perturbation theory to particle interactions with electromagnetic field. Invariant interactions of elementary particles.
PHYS 8011: Quantum Field Theory I
3 credits
Prerequisites: PHYS 8001 or Instructor consent
Second quantization of relativistic wave equations: canonical quantization of the free scalar and Dirac fields. Fields in interaction: interaction picture. Quantum electrodynamics: quantization of
the electromagnetic field, propagators and Feynman rules, tree-level processes. Higher-order processes and renormalization.
PHYS 8012: Quantum Field Theory II
3 credits
Prerequisites: PHYS 8011 or Instructor consent
Aspects of general theory of quantized fields, including space-time and discrete transformation properties, the CPT theorem, and the spin-statistics connection. Introduction to functional and
path-integral methods. Renormalization group and asymptotic freedom. Semi-classical methods and instantons in gauge theories.
PHYS 8013: Special Topics in Quantum Field Theory
3 credits
Prerequisites: PHYS 8012 or Instructor consent
Includes non-perturbative methods in quantum field theory, supersymmetry, two-dimensional quantum field theories and their applications, lattice simulations of quantum fields, topological quantum
field theories, quantum field theory methods applied to condensed matter physics, and string theory.
PHYS 8014: Quantum Many Body Systems
3 credits
Applications of quantum field theory to systems at finite density and temperature. Perturbative field theory of the interacting electron gas and its response functions. Instabilities of interacting
fermions at finite density using renormalization group and diagrammatic methods.
PHYS 8100: Seminar: Problems of Physics Teaching and Higher Education
1 credit – may be repeated for 3 credits total
Lectures and informal discussions of courses and curricula, techniques, and materials important in undergraduate physics instruction; relation to general problems of higher education.
PHYS 8161: Atomic and Molecular Structure
3 credits
Prerequisites: Level of mathematics associated with BS in physical sciences
Emphasizes interpretation of quantum numbers and selection rules in terms of symmetry. Experimental data summarized and compared with theoretical predictions.
PHYS 8200: Seminar: Cosmology and High Energy Astrophysics
1 credit – may be repeated for 6 credits total
Prerequisites: Instructor consent
Current topics in cosmology and high energy astrophysics.
PHYS 8300: Seminar: Biological and Medical Physics
1 credit – may be repeated for 6 credits total
Prerequisites: Instructor consent
Current research in biological and medical physics.
PHYS 8301: Symmetry and Its Application to Physical Problems
3 credits
Prerequisites: PHYS 5002 or Instructor consent
Fundamental invariance principles obeyed by laws of physics. Group theory as tool for using symmetry and invariance to help understand behavior of physical systems. Applications made to atomic,
molecular, nuclear, condensed-matter, and elementary particle physics.
PHYS 8311: Biological Physics of Single Molecules
3 credits
Prerequisites: PHYS 5201 or CHEN 4704, PHYS 5011, or Instructor consent
Biological molecules, based on statistical mechanics, kinetics, optics, and other physics ideas. Physics of DNA/proteins, their interactions. Force spectroscopy (optical tweezers, atomic force
microscopy). Concepts of optical spectroscopy. Single molecule fluorescence/imaging.
PHYS 8312: Biological Physics of Macroscopic Systems
3 credits
Prerequisites: PHYS 5201 or CHEN 4704, PHYS 5011, or Instructor consent
Macroscopic systems, based on physics such as fluid dynamics, statistical mechanics, non-linear dynamics, and chaos theory. Super-molecular aggregates. Biological physics of the cell. Biological
physics of populations/evolution.
PHYS 8333: FTE: Master's
1 credit – 6 Academic Progress Units; 6 financial Aid Progress Units
Prerequisites: Master's student, Advisor and DGS consent
PHYS 8444: FTE: Doctoral
1 credit – 6 Academic Progress Units; 6 financial Aid Progress Units
Prerequisites: Doctoral student, Advisor and DGS consent
PHYS 8500: Plan B Project
4 credits
Prerequisites: Instructor consent
May be taken once to satisfy Plan B master's project requirement; no credit toward Ph. D. Project topic arranged between student and instructor. Written report required.
PHYS 8501: General Relativity and Cosmology I
3 credits
Prerequisites: PHYS 5012 or Instructor consent
Tensor analysis and differential geometry. Special relativity leading to formulation of principles of general relativity and Einstein's equations. Tests of general relativity and thorough discussion
of various black hole solutions, including Schwarzschild, Reissner-Nordstom, and Kerr solutions.
PHYS 8502: General Relativity and Cosmology II
3 credits
Prerequisites: PHYS 8501 or Instructor consent
Gravitational radiation. Applications of general relativity to stellar structure of white dwarfs and neutron stars, action principle, and symmetric spaces. Big-bang cosmology, strongly emphasizing
particle physics.
PHYS 8581: Big Data in Astrophysics
4 credits
Prerequisites: MATH 2263 and MATH 2243, or Equivalent; or Instructor consent.
This course will introduce key concepts and techniques used to work with large datasets, in the context of the field of astrophysics. Suggested: familiarity with astrophysics topics such as star
formation and evolution, galaxies and clusters, composition and expansion of the universe, gravitational wave sources and waveforms, and high-energy astrophysics.
PHYS 8600: Seminar: Space Physics
1 credit – may be repeated for 6 credits total
Current topics in space physics and plasma physics.
PHYS 8601: Plasma Physics I
3 credits
Prerequisites: PHYS 4621, PHYS 5012, or Instructor consent
Theory of plasma waves and instabilities in plasmas, magnetohydrodynamics, nonlinear waves in plasmas, wave propagation in inhomogeneous plasmas.
PHYS 8602: Plasma Physics II
3 credits
Prerequisites: PHYS 8601 or Instructor consent
Theory of plasma waves and instabilities, collisions, radiation, transport, nonlinear wave-particle and wave-wave interactions, instabilities in inhomogeneous plasmas.
PHYS 8611: Cosmic Rays and Plasma Astrophysics
3 credits
Prerequisites: PHYS 5012 or Instructor consent
Properties of energetic particles in heliosphere and in astrophysical environments; solar physics, including radiation and magnetic effects; solar wind and magnetospheric physics; physics of
radiation belts.
PHYS 8650: Advanced Topics in Space and Plasma Physics
3 credits – may be repeated for 9 credits total
Prerequisites: PHYS 8602 or PHYS 8611 or Instructor consent
Topics in plasma waves and instabilities, solar physics, cosmic ray physics, atmospheric physics or planetary physics.
PHYS 8666: Doctoral Pre-Thesis Credits
1-6 credits
Prerequisites: Doctoral student who has not passed preliminary oral; no required consent for 1st/2nd registrations, up to 12 combined credits; Departmental consent for 3rd/4th registrations, up to 24
combined credits
PHYS 8700: Seminar: Condensed Matter Physics
1 credit – may be repeated for 6 credits total
Prerequisites: Instructor consent
Current research.
PHYS 8702: Statistical Mechanics and Transport Theory
3 credits
Prerequisites: PHYS 5201 or Instructor consent
Equilibrium properties of macroscopic classical and quantum systems. Phase transitions and Renormalization Group. Transport theory. Applications to soft condensed matter systems.
PHYS 8711: Solid-State Physics I
3 credits
Prerequisites: PHYS 4211, PHYS 5002, or Instructor consent
Fundamental properties of solids. Electronic structure and transport in metals and semiconductors. Properties of disordered materials.
PHYS 8712: Solid-State Physics II
3 credits
Prerequisites: PHYS 8711 or Instructor consent
Fundamental properties of solids. Electronic structure and transport in metals and semiconductors. Properties of disordered materials.
PHYS 8750: Advanced Topics in Condensed Matter Physics
3 credits – may be repeated for 9 credits total
Prerequisites: PHYS 8712 or Instructor consent
Sample research topics: magnetism, superconductivity, low temperature physics, superfluid helium.
PHYS 8777: Master's: Thesis Credits
1-18 credits – May be repeated for 50 credits total
Prerequisites: Max of 18 credits per semester or summer; 10 credits total required (Plan A only)
PHYS 8800: Seminar: Nuclear Physics
1 credit – may be repeated for 6 credits total
Current research topics.
PHYS 8801: Nuclear Physics I
3 credits
Prerequisites: PHYS 5001, PHYS 5002, PHYS 5011, PHYS 5012, PHYS 5201; AST 4001 recommended
Nuclear/neutrino astrophysics. Thermonuclear reactions. Processes of nucleosynthesis. Origin of the elements. Stellar evolution. Theory of supernovae. Exotic stars. Chemical evolution of galaxies.
PHYS 8802: Nuclear Physics II
3 credits
Prerequisites: PHYS 8801 or Instructor consent
Properties of nuclei based on hadronic and quark-gluon degrees of freedom. Relativistic field theory at finite temperatures and density applied to many-body problems, especially nuclear matter and
quark-gluon plasma. Applications to lepton and hadron scattering, nucleus-nucleus collisions, astrophysics and cosmology.
PHYS 8850: Advanced Topics in Nuclear Physics
3 credits – may be repeated for 9 credits total
Prerequisites: PHYS 8802 or Instructor consent
Research topics.
PHYS 8888: Doctoral: Thesis Credit
1-24 credits – May be repeated for 100 credits total
Prerequisites: Max of 18 credits per semester or summer; 24 credits total required
Introduction to quantum mechanics. Applications to atomic, molecular, condensed-matter, nuclear, elementary-particle, and statistical physics.
PHYS 8900: Seminar: Elementary Particle Physics
1 credit – may be repeated for 6 credits total
Elementary particle physics, high energy physics, particle astrophysics and cosmology.
PHYS 8901: Elementary Particle Physics I
3 credits
Prerequisites: PHYS 8001 or Instructor consent
Types of fundamental interactions. Exact and approximate symmetries and conservation laws. Gauge quanta: gluons, photons, W and Z bosons, gravitons. Fundamental fermions: leptons and quarks. Isotopic
and flavor SU(3) symmetries of strong interaction. Heavy hadrons. Amplitudes and probabilities. Quantum chromodynamics.
PHYS 8902: Elementary Particle Physics II
3 credits
Prerequisites: PHYS 8901 or Instructor consent
Deep inelastic scattering. Weak interactions of leptons. Semileptonic and nonleptonic weak processes with hadons. Oscillations of neutral Kaons. Violation of CP symmetry in Kaons. Neutrino masses and
oscillations. Standard model of the electroweak interaction. Grand unification. Unitarity of the S matrix. Properties of soft pions.
PHYS 8911: Introduction to Supersymmetry
3 credits
Prerequisites: PHYS 8011 or Instructor consent
Motivation. Coleman-Mandula theorem. Supersymmetric Quantum Mechanics. 4D supersymmetry algebra and representations. Extended supersymmetry. N=1 superspace and superfields. Supersymmetric guage
theories. Chiral/vector multiplets. Non-renormalization theorems. Supersymmetry breaking. Supersymmetric Standard Model. Phenomenology. Nonperturbative supersymmetry. Supergravity.
PHYS 8950: Advanced Topics in Elementary Particle Physics
3 credits – may be repeated for 9 credits total
Prerequisites: PHYS 8902 or Instructor consent
Research topics.
PHYS 8994: Research in Physics
1-12 credits – may be repeated for 24 credits total
Prerequisites: Instructor consent
Research under faculty direction. | {"url":"https://cse.umn.edu/physics/classes","timestamp":"2024-11-07T06:27:22Z","content_type":"text/html","content_length":"116754","record_id":"<urn:uuid:f1e442c9-8dff-41c0-bcd9-6275402b8edc>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00768.warc.gz"} |
Unscramble SEAT
How Many Words are in SEAT Unscramble?
By unscrambling letters seat, our Word Unscrambler aka Scrabble Word Finder easily found 24 playable words in virtually every word scramble game!
Letter / Tile Values for SEAT
Below are the values for each of the letters/tiles in Scrabble. The letters in seat combine for a total of 6 points (not including bonus squares)
What do the Letters seat Unscrambled Mean?
The unscrambled words with the most letters from SEAT word or letters are below along with the definitions.
• ate () - the preterit of Eat.
• east (n.) - The point in the heavens where the sun is seen to rise at the equinox, or the corresponding point on the earth; that one of the four cardinal points of the compass which is in a
direction at right angles to that of north and south, and which is toward the right hand of one who faces the north; the point directly opposite to the west.
• eat () - of Eat
• eta () - Sorry, we do not have a definition for this word
• sate (v. t.) - To satisfy the desire or appetite of; to satiate; to glut; to surfeit.
• seat (n.) - The place or thing upon which one sits; hence; anything made to be sat in or upon, as a chair, bench, stool, saddle, or the like.
• seta (n.) - Any slender, more or less rigid, bristlelike organ or part; as the hairs of a caterpillar, the slender spines of a crustacean, the hairlike processes of a protozoan, the bristles or
stiff hairs on the leaves of some plants, or the pedicel of the capsule of a moss.
• tea (n.) - The prepared leaves of a shrub, or small tree (Thea, / Camellia, Chinensis). The shrub is a native of China, but has been introduced to some extent into some other countries. | {"url":"https://www.scrabblewordfind.com/unscramble-seat","timestamp":"2024-11-11T03:35:53Z","content_type":"text/html","content_length":"40807","record_id":"<urn:uuid:5ab2ea82-4ba0-4a30-aaa7-4a03115e9496>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00462.warc.gz"} |
17 Astonishing Facts About Paul Erdős
Source: Codepen.io
Paul Erd?s, the renowned mathematician, was an extraordinary individual who made significant contributions to the field of mathematics. Known for his prolific collaborative style and unconventional
lifestyle, Erd?s had a unique approach to problem-solving that captivated the minds of his colleagues and admirers.
In this article, we will delve into the fascinating world of Paul Erd?s and explore 17 astonishing facts about his life and achievements. From his early beginnings in Hungary to his legendary “Erd?s
Number” concept, we will discover the remarkable stories that shaped Erd?s’ legacy as one of the most influential mathematicians of the 20th century.
Get ready to be amazed as we unravel the lesser-known aspects of Erd?s’ life, from his relentless pursuit of mathematical truth to his quirky habits and eccentric lifestyle. Join us on this journey
through the remarkable life of Paul Erd?s, a man whose brilliance and passion continue to inspire mathematicians around the world.
Key Takeaways:
• Paul Erd?s was a super smart mathematician who loved working with others. His legacy inspires people to collaborate, find simple solutions, and enjoy math.
• Erd?s’ love for coffee, humor, and simple living made him a unique and influential figure in the world of mathematics.
Erd?s had an exceptional mathematical talent.
Paul Erd?s, a Hungarian mathematician, was renowned for his extraordinary ability to solve complex mathematical problems. He possessed an unparalleled intuition and a profound understanding of
numbers that allowed him to make significant contributions to many areas of mathematics.
Erd?s co-authored more than 1,500 papers.
Throughout his prolific career, Erd?s collaborated with numerous mathematicians from around the world. He believed in the power of collaboration and considered it essential for advancing mathematical
knowledge. His vast academic network earned him the nickname “the most prolific mathematician of all time.”
Erd?s was a traveling mathematician.
Erd?s had an insatiable desire to discuss mathematics with colleagues and work on new problems. He lived a unique lifestyle, traveling from one mathematician’s house to another, often with just a
single suitcase containing his belongings. This nomadic way of life allowed him to engage in mathematical discussions and collaborations with mathematicians from various countries.
Erd?s coined the concept of mathematical “proofs from The Book.”
Erd?s had a deep appreciation for elegant mathematical proofs. He often referred to beautiful mathematical proofs as “proofs from The Book,” indicating that they belonged to a hypothetical book that
God possessed, containing the most elegant and profound mathematical proofs. This concept inspired mathematicians to strive for excellence and elegance in their work.
Erd?s had an Erd?s number.
The Erd?s number is a measurement of how closely an individual is connected to Paul Erd?s through co-authorship of mathematical papers. Erd?s himself had an Erd?s number of 0, while his collaborators
had an Erd?s number of The concept of Erd?s numbers has become a fascinating topic of study and a measure of academic influence within the mathematical community.
Erd?s published papers until his final days.
Even in his later years, Erd?s remained actively involved in mathematical research. He continued to publish papers and collaborate with other mathematicians until his passing in His immense
dedication and passion for mathematics inspired generations of mathematicians to push the boundaries of mathematical knowledge.
Erd?s made major contributions to number theory.
One of Erd?s’ primary areas of expertise was number theory. He made significant breakthroughs in this field, including his work on prime numbers and the investigation of various number-theoretic
problems. His contributions to number theory have had a lasting impact on the field and continue to be studied and appreciated today.
Erd?s was known for his collaborative approach.
Erd?s believed that mathematics was a collaborative endeavor and actively sought opportunities to work with other mathematicians. He would often arrive at their doorstep unannounced and engage in
intense mathematical discussions. This collaborative approach fostered a vibrant and dynamic mathematical community.
Erd?s created mathematical problems.
In addition to solving complex mathematical problems, Erd?s loved creating challenging mathematical puzzles and problems for others to solve. His puzzles were often elegant, requiring creative
thinking and a deep understanding of mathematical concepts. Many of his problems continue to be studied and enjoyed by mathematicians today.
Erd?s had an exceptional memory.
Erd?s possessed an extraordinary memory that allowed him to recall numerous mathematical theorems, proofs, and historical details effortlessly. He could recite long mathematical proofs from memory,
even those he had not encountered for years. His impressive recall and encyclopedic knowledge made him a revered figure in the mathematical community.
Erd?s was a proponent of “elementary” proofs.
Erd?s had a preference for elegant and straightforward proofs that could be understood by people with a basic mathematical background. He believed in the beauty and simplicity of elementary proofs
and encouraged mathematicians to explore accessible ways of explaining complex concepts. His emphasis on clarity contributed to making mathematics more accessible to a broader audience.
Erd?s had a unique writing style.
Erd?s’ mathematical papers were known for his distinctive writing style, which contained a combination of concise notation, insightful explanations, and occasional humor. His papers were often
presentation-style, with the main ideas and proofs presented in a clear and concise manner. His writing style still serves as a model for effective mathematical communication.
Erd?s loved caffeine.
Erd?s had a notorious fondness for caffeine, particularly his beloved beverage, coffee. He believed that coffee helped him stay alert and focused during his mathematical endeavors. He often referred
to coffee as his “brain fuel” and would frequently consume large quantities of it during his intense working sessions.
Erd?s had a unique sense of humor.
Erd?s was known for his witty and eccentric sense of humor, which added an element of fun to the often serious world of mathematics. His jokes and puns were legendary among his colleagues, and he
would often incorporate them into his mathematical discussions, making even the most complex concepts more enjoyable.
Erd?s had a profound impact on combinatorics.
Erd?s made remarkable contributions to the field of combinatorics, a branch of mathematics that focuses on discrete and finite structures. His work in combinatorial number theory and graph theory
laid the foundation for many breakthroughs and advancements in the field.
Erd?s led a simple and frugal lifestyle.
Despite his achievements and international recognition, Erd?s lived modestly and had minimal material possessions. He believed that simplicity and frugality allowed him to focus on his true passion:
mathematics. His lifestyle served as a reminder that true success lies in pursuing one’s passion rather than accumulating material wealth.
Erd?s’ legacy continues to inspire.
Even after his passing, Erd?s’ legacy lives on. His mathematical contributions, collaborative spirit, and dedication to the pursuit of knowledge continue to inspire mathematicians and researchers
worldwide. The impact of “the most prolific mathematician of all time” can still be felt within the mathematical community today.
In conclusion, Paul Erd?s was a truly remarkable mathematician who left an indelible mark on the field of mathematics. His unique and eccentric personality, along with his insatiable curiosity and
collaborative spirit, made him one of the most influential figures in modern mathematics. Erd?s’ contributions to various areas such as number theory, graph theory, and combinatorics are
unparalleled.Throughout his life, Erd?s published an astonishing number of research papers, collaborating with hundreds of mathematicians around the world. His fervent dedication to mathematics and
his relentless pursuit of knowledge have earned him a place in history as one of the greatest mathematicians of all time.Even though Paul Erd?s passed away in 1996, his legacy lives on through the
Erd?s number system, which quantifies his incredible network of collaborators. Erd?s’ impact on mathematics will continue to inspire future generations of mathematicians to push the boundaries of
knowledge and explore new frontiers in the field.
1. Who was Paul Erd?s?
Paul Erd?s was a renowned Hungarian mathematician known for his prolific mathematical contributions and his collaborative approach to research.
2. How many research papers did Paul Erd?s publish?
Erd?s published over 1500 research papers throughout his career, an astonishing number that is unmatched by any other mathematician.
3. What is the Erd?s number system?
The Erd?s number system is a way to quantify a mathematician’s collaborative distance from Paul Erd?s. Erd?s himself has an Erd?s number of 0, while those who co-authored papers with him have an Erd?
s number of 1. Those who co-authored papers with someone with an Erd?s number of 1 have an Erd?s number of 2, and so on.
4. How did Paul Erd?s approach mathematics?
Erd?s had a unique approach to mathematics, often referring to it as a drug. He believed in the beauty and elegance of mathematics and dedicated his life to solving problems and collaborating with
fellow mathematicians.
5. What were Paul Erd?s’ key contributions to mathematics?
Erd?s made significant contributions to various fields of mathematics, including number theory, graph theory, combinatorics, and analysis. His Erd?s-Szekeres theorem and Erd?s-Kac theorem are
particularly well-known.
6. How did Paul Erd?s collaborate with other mathematicians?
Erd?s had an extensive network of collaborators and would often travel around the world, staying with his colleagues and working on joint research projects. This collaborative approach led to
numerous breakthroughs and advancements in mathematics.
7. What was Paul Erd?s’ impact on the field of mathematics?
Erd?s’ impact on mathematics is immeasurable. His extensive body of work, his passion for collaboration, and his unique perspective on problem-solving continue to influence and inspire mathematicians
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Solving the Equation (x+1)(2x-3) = (2x-1)(x+5)
This article will guide you through the process of solving the given equation: (x+1)(2x-3) = (2x-1)(x+5). We will use algebraic manipulation to isolate the variable 'x' and find its solution.
Expanding the Equation
The first step is to expand both sides of the equation by multiplying the expressions within the parentheses:
• Left side: (x+1)(2x-3) = 2x² - x - 3
• Right side: (2x-1)(x+5) = 2x² + 9x - 5
Now our equation looks like this: 2x² - x - 3 = 2x² + 9x - 5
Simplifying the Equation
Notice that the term 2x² appears on both sides of the equation. We can subtract 2x² from both sides to eliminate it:
-x - 3 = 9x - 5
Isolating the Variable 'x'
Next, we want to isolate 'x' on one side of the equation. Let's move all terms containing 'x' to the left side and the constant terms to the right side:
-x - 9x = -5 + 3
This simplifies to: -10x = -2
Solving for 'x'
Finally, we divide both sides of the equation by -10 to find the value of 'x':
x = -2 / -10
x = 1/5
Therefore, the solution to the equation (x+1)(2x-3) = (2x-1)(x+5) is x = 1/5. | {"url":"https://jasonbradley.me/page/(x%252B1)(2x-3)%253D(2x-1)(x%252B5)","timestamp":"2024-11-03T04:12:53Z","content_type":"text/html","content_length":"58609","record_id":"<urn:uuid:fffd214a-71dd-4723-9069-5e7b9e8cc925>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00696.warc.gz"} |
Singalakha's guide to plotting rational functions
Singalakha’s guide to plotting rational functions
To sketch the graph of a function $k(x)=\frac{f(x)}{g(x)}$:
1. Find the intercepts:
1. X-intercepts, set y=0 (there can be multiple)
2. Y-intercept, set x=0 (there can be only one)
2. Factorise the numerator and denominator if possible:
1. Sign table: determine where the function is negative and where it is positive
3. Find the Vertical asymptotes:
1. This occur if the function in the denominator is equal to zero, i.e $g(x) = 0$, AND that in the numerator must not be zero, i.e $f(x)e 0$.
4. Find any Horizontal asymptote:
1. If the degree of the function in the numerator, i.e $f(x)$, is less than the degree of the function in the denominator, i.e $g(x)$, then the horizontal asymptote is the line $y = 0$.
2. If the degree of the function in the numerator, i.e $f(x)$, is equal to the degree of the function in the denominator, i.e $g(x)$, say for example, the degree of $f(x)$ and $g(x)$ is $n$ for
some non-negative $n$ element of integers, then there is a horizontal asymptote. This will still be proven by long division, this is just for you to be able to make reasonable assumptions on
how your function will behave or look like.
5. Find and slant asymptote:
1. If the degree of the function in the numerator exceeds that in the denominator by one, that is,
$\text{Degree of }f(x) = 1 + \text{Degree of }g(x)$.
6. Find and quadratic asymptote:
1. If the degree of the function in the numerator exceeds that in the denominator by two, that is,
$\text{Degree of }f(x) = 2 + \text{Degree of }g(x)$
7. Perform long division
1. Write $\frac{f(x)}{g(x)}=d(x)+\frac{r(x)}{g(x)}$).
2. Find the equations for the slant, horizontal, quadratic and/or vertical asymptotes (Now after noting that you have some type of an asymptote, long division allows you now to rigorously find
the equation of that asymptote). The equation of the $x\rightarrow \pm \infty$ asymptote will be equal to $d(x)$.
3. Find the various limits (Here we are not interested in the concept of a limit in the more technical sense that you will see later, we are just looking at how the graph behaves as x approaches
a certain number or as it approaches $\pm \infty$).
4. Find the points where the graph crosses the asymptotes by finding when $r(x)=0$.
8. Find any point discontinuities:
1. These occur when the function in the numerator, $f(x)$ and that in the denominator $g(x)$ are BOTH equal to zero, that is, $f(x) = g(x) = 0$ for some $x$. These single points are removed from
the graph and are replaced with an open circle.
9. To sketch the graph:
1. Start by using dashed lines to draw all the asymptotes.
2. Use the information in the sign table to determine whether the graph is going to + or – $\infty$ for the vertical asymptotes.
3. Draw on any x and y intercepts as well as points where the graph crosses the asymptotes.
4. Draw in any point discontinuities.
5. Knowing that the graph doesn’t cross any of the axes or asymptotes other than those which you have already marked on, join together all the information you have to sketch the graph
One Comment
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My experience with TensorFlow Quantum
A guest post by Owen Lockwood, Rensselaer Polytechnic Institute
Quantum mechanics was once a very controversial theory. Early detractors such as Albert Einstein famously said of quantum mechanics that “God does not play dice” (referring to the probabilistic
nature of quantum measurements), to which Niels Bohr replied, “Einstein, stop telling God what to do”. However, all agreed that, to quote John Wheeler “If you are not completely confused by quantum
mechanics, you do not understand it”. As our understanding of quantum mechanics has grown, not only has it led to numerous important physical discoveries but it also resulted in the field of quantum
computing. Quantum computing is a different paradigm of computing from classical computing. It relies on and exploits the principles of quantum mechanics to achieve speedups (in some cases
superpolynomial) over classical computers.
In this article I will be discussing some of the challenges I faced as a beginning researcher in quantum machine learning (QML) and how TensowFlow Quantum (TFQ) and Cirq enabled me and can help other
researchers investigate the field of quantum computing and QML. I have previous experience with TensorFlow, which made the transition to using TensorFlow Quantum seamless. TFQ proved instrumental in
enabling my work and ultimately my work utilizing TFQ culminated in my first publication on quantum reinforcement learning in the 16th AIIDE conference. I hope that this article helps and inspires
other researchers, neophytes and experts alike, to leverage TFQ to help advance the field of QML.
QML Background
QML has important similarities and differences to traditional neural network/deep learning approaches to machine learning. Both methodologies can be seen as using “stacked layers” of transformations
that make up a larger model. In both cases, data is used to inform updates to model parameters, typically to minimize some loss function (usually, but not exclusively via gradient based methods).
Where they differ is QML models have access to the power of quantum mechanics and deep neural networks do not. An important type of QML that TFQ provides techniques for is called variational quantum
circuits (QVC). QVCs are also called quantum neural networks (QNN).
A QVC can be visualized below (from the TFQ white paper). The diagram is read from left to right, with the qubits being represented by the horizontal lines. In a QVC there are three important and
distinct parts: the encoder circuit, the variational circuit and the measurement operators. The encoder circuit either takes naturally quantum data (i.e. a nonparametrized quantum circuit) or
converts classical data into quantum data. This circuit is connected to the variational circuit which is defined by its learnable parameters. The parametrized part of the circuit is the part that is
updated during the learning process. The last part of the QVC is the measurement operators. In order to extract information from the QVC some sort of quantum measurement (such as a Pauli X, Y, or Z
basis measurements) must be applied. With the information extracted from these measurements a loss function (and gradients) can be calculated on a classical computer and the parameters can be
updated. These gradients can be optimized with the same optimizers as traditional neural networks such as Adam or RMSProp. QVC’s can also be combined with traditional neural networks (as is shown in
the diagram) as the quantum circuit is differentiable and thus gradients can be backpropagated through.
However, the intuitive models and mathematical framework surrounding QVCs have some important differences from traditional neural networks, and in these differences lies the potential for quantum
speedups. Quantum computing and QML can harness quantum phenomena, such as superposition and entanglement. Superposition stems from the wavefunction being a linear combination of multiple states and
enables a qubit to represent two different states simultaneously (in a probabilistic manner). The ability to operate on these superpositions, i.e. operate on multiple states simultaneously, is
integral to the power of quantum computing. Entanglement is a complex phenomenon that is induced via multi-qubit gates. Getting a basic understanding of these concepts and of quantum computing is an
important first step for QML. There are a number of great resources available for this such as Preskill’s Quantum Computing Course, and de Wolf’s lecture notes.
Currently, access to real quantum hardware is limited and as such, many quantum computing researchers conduct work on simulations of quantum computers. Near term and current quantum devices have
10s-100s of quantum bits (qubits) like the Google sycamore processor. Because of their size and the noise, this hardware is often called Noisy Intermediate Scale Quantum (NISQ) technology. TFQ and
Cirq are built for these near term NISQ devices. These devices are far smaller than what some of the most famous quantum algorithms require to achieve quantum speedups given current error correction
techniques; e.g. Shor’s algorithm requires upwards of thousands of qubits and the Quantum Approximate Optimization Algorithm (QAOA) could require at least 420 qubits for quantum advantages. However,
there is still significant potential for NISQ devices to achieve quantum speedups (as Google demonstrated with 53 qubits).
My Work With TFQ
TFQ was announced in mid March this year (2020) and I began to use it shortly after. Around that time I had begun research into QML, specifically QML for reinforcement learning (RL). While there have
been great strides in the accessibility of quantum circuit simulators, QML can be a difficult field to get into. Not only are there difficulties from a mathematical and physical perspective, but the
time investment for implementation can be substantial. The time it would take to code QVCs from scratch and properly test and debug them (not to mention the optimization) is a challenge, especially
for those coming from classical machine learning. Spending so much time building something for an experiment that has the potential to not work is a big risk - especially for an undergraduate student
on a deadline! Thankfully I did not have to take this risk. With the release of TFQ it was easy to immediately start on the implementations of my ideas. Realistically, I would never have done this
work if TFQ had not been released. Inspired by previous work, we expanded upon applying QML to RL tasks.
In our work we demonstrate the potential to use QVCs in place of neural networks in contemporary RL algorithms (specifically DQN and DDQN). We also show the potential to use multiple types of QVC
models, using QVCs with either a dense layer or quantum pooling layers (denoted hybrid and pure respectively) to shrink the number of qubits to the correct output space.
The representational power of QVCs is also put on display; using a QVC with ~50 parameters we were able to achieve comparable performance to neural networks with orders of magnitude more parameters.
See the graphs for a comparison of the reward achieved on the canonical CartPole environment (balancing a pole on a cart), the left graph includes all neural networks and the right shows only the
largest neural network. The number in front of the NN represents the size of the parameter space.
We are continuing to work with QML applications to RL and have more manuscripts in submission. Continuation of this work has been accepted into the 2020 NeurIPS workshop: “The pre-registration
experiment: an alternative publication model for machine learning research”.
Suggested use of TFQ
TFQ can be an incredible tool for anyone interested in QML research no matter your background. All too common in scientific communities is a ‘publish or perish’ mentality which can stifle innovative
work and is prohibitive to intellectual risk taking, especially for experiments that require significant implementation efforts. Not only can TFQ help speed up any experiments you may have, but it
also allows for easy implementation of ideas that would otherwise never get tested. Implementation is a common hindrance to new and interesting ideas, and far too many projects never progress out of
the idea stage due to difficulties in transitioning the idea to reality, something TFQ makes easy.
For beginners, TFQ enables a first foray into the field without substantial time investment in coding and allows for significant learning. Being able to try and experiment with QVCs without having to
build from the ground up is an incredible tool. For classical ML researchers with experience in TensorFlow, TFQ makes it easy to transition and experiment with QML at small or large scales. The API
of TFQ and the modules it provides (i.e. Keras-esque layers and differentiators) share design principles with TF and their similarities make for an easier programming transition. For researchers
already in the QML field, TFQ can certainly help.
In order to get started with TFQ it is important to become familiar with the basics of quantum computing, with either the references mentioned above or with any of the many other great resources out
there. Another important step that often gets overlooked, is reading the TFQ white paper. The white paper is accessible to QML beginners and is an invaluable introduction to QML and the basic as well
as advanced usage of TFQ. Just as important is to play around with TFQ. Try different things out, experiment; it is a great way to expand not only understanding of the software but of the theory and
mathematics as well. Reading other contemporary papers and the papers that cite TFQ is a great way to become immersed with the current research going on in the field. | {"url":"https://blog.tensorflow.org/2020/11/my-experience-with-tensorflow-quantum.html?hl=es_419","timestamp":"2024-11-13T10:56:44Z","content_type":"application/xhtml+xml","content_length":"88008","record_id":"<urn:uuid:548420eb-e4ba-493f-87eb-149bda51f915>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00278.warc.gz"} |
Session 6. Process systems engineering
N301 Process modeling and optimization of natural gas dehydration using triethylene glycol: Latin hypercube sampling<101028-1>
Gas hydrates are ice – like crystalline of natural gas and water. Several methods can be used to prevent hydrate formation such as solvent absorption, adsorption, and membrane process. This work
studied modelling and optimization of natural gas dehydration using solvent absorption method. Triethylene glycol (TEG) was used as solvent to remove water from natural gas. Latin Hypercube sampling
(LHS) was studied as sampling technique in term of its ability to represent the nonlinear model for optimization problem. The nonlinear model represents the relationship between the operating
conditions and amount of produced dry natural gas. In the optimization, the total cost of the process was minimized. Both equipment and operating costs were considered in economic evaluation. The TEG
absorption process was simulated using Aspen HYSYS version 8.8. The results show that LHS is able to accurately represent the nonlinear model between operating conditions and the total cost of the
process for optimization problem. The optimal operating conditions of the process were obtained.
N302 Bayesian estimation of vapor-liquid equilibrium model parameters by MCMC method<100821-1>
Model equations used in process design involve parameters, which can be estimated from experimental data. Estimating accuracy of model parameters is critical to ensure robustness of processes. The
objective of this study is to quantify estimation accuracy using Bayesian inference techniques in parameter estimation. In this work, the Markov chain Monte Carlo (MCMC) method is used to obtain
probability distribution. This algorithm repeats random sampling in parameter space and approximates the profile of posterior probability distributions by the density of sampled points. This method
can handle complex nonlinear models. We demonstrate this method for estimating parameters in the Non-random two-liquid (NRTL) model for vapor-liquid equilibrium. The NRTL parameters are estimated by
the MCMC method, and the influence of the number of experimental data on estimation accuracy is quantified. We compared two estimation cases, where 8 experimental data points and 13 experimental data
points^[1] are used, and the difference in the parameter accuracy is quantified. This estimation results for ethanol-water system are shown in Figure 1 for three parameters, a,b[12], and b[21] that
describe intermolecular interactions in the NRTL model for the binary system. In addition, the proposed method is demonstrated for a multicomponent system that involve a larger number of components.
[1]Dortmund Data Bank Software & Separation Technology“VLE - Vapor Liquid Equilibria of Normal Boiling Substances”
http://www.ddbst.com/ddb-vle.html (2019.2.16)
N303 Analysis of energy efficiency for various reverse osmosis based desalination technologies<100514-1>
The growth in global economy has made water a limited resource in both quantity and quality. Freshwater resources available for our use are less than 1 % of the total water on the Earth. Reverse
osmosis (RO) desalination technology became popular over the last decade due to its potential to produce freshwater from seawater. However, energy consumption in RO plants remain high and emission of
concentrated brine from the plant has an impact on the environment.
This study aims to analyse the energy efficiency of various RO based desalination processes. Firstly, single-stage RO with pressure exchanger (PX) is analysed. Minimum normalized specific energy
consumption (NSEC) is reduced from 5.00×10^5 J/m^3 to 2.05×10^5 J/m^3 by adding PX with 0.95 efficiency. Secondly, a simulation on multi-stage RO with PX is carried out. It is found that
two-stage RO reduces minimum NSEC by 5.4 % compared with single-staged operation and increase recovery by 13.6 %. Even though multi-stage design may require higher capital cost and operating cost,
optimised-stage RO design has certain advantages.
In the on-going research, membrane distillation (MD) and pressure-retarded osmosis (PRO) will be combined with multi-stage RO process to improve the overall energy efficiency. MD is a thermally
driven membrane process that produces pure water, and PRO is an osmotically driven membrane process with external pressure, which is lower than the osmotic pressure difference, applied on the higher
salinity solution side. The use of these processes are limited to when one has low-cost heat source, e.g. waste heat, and inexpensive lower salinity solution, e.g. wastewater. Since MD is less
concentration-sensitive, it is expected to further concentrate the brine output, and highly concentrated brine is assumed to support PRO to generate greater energy and reduce minimum NSEC of the
total process.
N304 Dynamic analysis of a non-equilibrium LNG tank<100943-1>
Heat ingress from atmosphere to LNG inside a tank due to temperature difference causes continuous evaporation of LNG that leads to increase of pressure and temperature. Therefore, it is necessary to
continuously estimate the change in pressure and temperature inside the tank, using accurate numerical model in real time operation. For this, a new methodology was developed in this work to estimate
total amounts of boiled off gas (BOG) and its composition and physical properties more accurately than conventional method, based on various operation conditions.
N305 [Keynote] Energy efficient design and control of reactive distillation processes via hybrid heat-integration and pervaporation arrangement<406004-1>
This work focuses on the efficient design with various heat integration methods and membrane arrangement for reactive distillation (RD) processes in order to enhance the energy saving. It is
well-known that thermally-coupled, multi-effect, and external heat-integration approaches can reduce energy consumption in comparison with conventional distillation systems. Therefore, in this work,
two processes will be discovered to demonstrate that hybrid configuration via various heat-integration and unit combination (such as pervaporation, PV) can provide further energy reduction compared
to the conventional RD configuration. The first process is the diphenyl carbonate (DPC) production process that will be chosen to show a synergetic effect when thermally-coupling and external
heat-integration arrangement are combined in the same reactive distillation sequence. The results show that in the DPC production process, the hybrid heat configuration can reduce energy consumption
by 47% in comparison with the conventional RD configuration. On the other hand, another process is the ethyl levulinate (LAEE) production process that will be investigated to show a synergetic effect
when thermal intensification is implemented in the hybrid RD-PV configuration in the LAEE production process. The hybrid heat-integrated configuration assisted with PV arrangement through excess
ethanol (EtOH) design can save around 73% of energy consumption compared to the conventional LAEE RD configuration. Figure 1 shows the flowsheet of hybrid configurations for DPC and LAEE production
processes, respectively. Finally, several control schemes of the DPC process would be investigated and tested the control performance based on this highly integrated configuration. Results show that
the best control scheme has good disturbance rejection for throughput and feed composition disturbances. | {"url":"http://www3.scej.org/meeting/apcche2019/abst/n3a-abstracts.html","timestamp":"2024-11-13T18:29:10Z","content_type":"text/html","content_length":"12225","record_id":"<urn:uuid:c9b164c6-d828-4139-85e2-ca6c32a94fc3>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00516.warc.gz"} |
To find analytically the fun Df(x) at x=c and also ... | Filo
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Question Text objective: To find analytically the fun at and also to find the continuity of tho fence at that point
Updated On Jan 26, 2024
Topic Limit, Continuity and Differentiability
Subject Mathematics
Class Class 12
Answer Type Video solution: 2
Upvotes 169
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Let's Limbo! - Odds, Winning & Verifying
It’s essential to stretch before you start Limbo. Otherwise, you risk hurting yourself bad. Especially if you’re 30+. Don’t worry about looking like a fool while doing the Limbo; everyone does. If
you Limbo with co-workers, don’t forget to bring a camera. You will have great pictures for extortion whenever you need someone to cover for you at work.
Not that Limbo? Oh, okay.
Coco got himself a rocket which he named Limbo. Unfortunately, Limbo tends to explode at the most inconvenient times. The challenge is to get off the rocket before it explodes. But not too soon. Get
off too soon, and your profit will be smaller, get off too late, and you lose all profit.
How far can you go before getting off the rocket? Do you have the nerves of steel that it takes to reach the heights (and profit) others can only dream off? And precisely what are the chances of
getting the stars?
Let’s find out!
The bet
I made a bet of 10 Satoshi in the hope of making a profit of 70 Satoshi (10 x 8 = 80 and subtract the initial bet = 70 Satoshi).
The game states that I have a 12.375% chance of hitting my targeted 8x. 100/12.375 = 8.080. So statistically speaking, I should hit 8x once every eight games (that wouldn’t be very profitable!)
For my first try, I got 8x on my 6^th try, so I made 10 Satoshi in profit.
On my second try, it took 22 tries to get 8x. Let’s add them together!
6 + 22 = 28 and I got 8x two times, so 28/2 = 14. I’m not a genius at math, but 14 is more than 8!
It doesn’t seem like a 12.375% chance to me! More like 7.149% chance (100/7.149 = 13.987 – close enough!).
Why is that? Is this some kind of scam!?
Let’s try betting the same amount 100 times, and see what happens.
Well, that’s more like it. Out of 100 games, I got 8x 12 times, exactly 12%!
What can we learn from this? First, we can say that a ~12% chance of winning doesn’t mean that every 8^th game will give you a win. The possibility of winning can be interpreted in a few different
Large samples
Given a large enough sample, the number of winning games will eventually amount to ~12%. While collecting this large sample, you will sometimes have longer streaks without hitting 8x at all, and
other times you will hit 8x several times within a few games. The larger the sample (i.e., the more games you’ve played), the more accurate the chance of winning.
Single sample
Every game you play, you will have a 12% chance of hitting 8x, which means there’s an 88% chance that you lose. On every game you play.
I’ve played 12 games without hitting 8x, which means that 8x has to come within just a couple of games.
Incorrect, sort of. Of course, you will eventually hit 8x, but the chance of hitting 8x doesn’t increase every time you don’t hit it. During the 100 games I played above, my longest streak of not
getting 8x was 25 (just to hit 8x twice in a row right after).
There are three kinds of lies: Lies, Damned Lies, and Statistics
Famous words attributed to Mark Twain (who himself attributed it to the British prime minister Benjamin Disraeli). But could you claim that statistics lie?
I don’t think so. You can, however, misinterpret statistics or misrepresent them to mislead someone. Now I don’t believe the statistics at BC.game is misleading. But I understand how it might seem
that way for someone who doesn’t know how to interpret it. When it says you have a 12% chance of winning, it’s 100% correct.
If you were to try and hit the green ring by throwing darts from a 10 feet distance, you wouldn’t expect it to grow bigger every time you missed, would you? It’s precisely the same as the chance of
winning works.
Provably fair
As with all in-house games at BC.game, Limbo is provable fair. Let’s check if my last bet in Limbo was fair or if someone scammed me out of 1 BB coin.
The output will look like this.
Take the first 13 numbers from hmac_sha256
convert it to decimal form using any hexadecimal converter
the decimal number needs to be divided by the maximum value (HEX) fffffffffffff
fffffffffffff = 4503599627370495 in decimals.
3756487196347878 / 4503599627370495 = 0.83410771541
The house edge is 1% so we need to calculate 99/(1-0.83410771541) = 99/0.16589228459 = 596.772780872
Remove the decimals and divide by 100.
596 / 100 = 5.96
The result of my last Limbo game is 5.96. Less than 8, which means I lost, and no one tried to scam me out of my 1 BB coin.
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What is a rem, a unit of radiation absorbed dose measurement
What is a rem (unit)
The rem is a unit of measurement of radiation absorbed dose
Rem is the special unit of any of the quantities expressed as dose equivalent. The dose equivalent in rems is equal to the absorbed dose in rads multiplied by the quality factor (1 rem=0.01 sievert) | {"url":"https://www.aqua-calc.com/what-is/radiation-absorbed-dose/rem","timestamp":"2024-11-06T14:38:32Z","content_type":"text/html","content_length":"51572","record_id":"<urn:uuid:98aa6759-f1ba-4609-9c8c-766f3faba2a3>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00360.warc.gz"} |
Solution to Mathematics for Machine Learning Exercise 7.2 - Solutions to Linear Algebra Done Right
Solution to Mathematics for Machine Learning Exercise 7.2
Consider the update equation for stochastic gradient descent (Equation (7.15)). Write down the update when we use a mini-batch size of one.
Solution: Let $a$ be any function from $\mathbb Z^+$ to $\{1,\dots,n\}$. Then $$\theta_{i+1}= \theta_{i}-\gamma_i(\nabla L_{a(i)}(\theta_{i}))^\top.$$The point is using single $L_j$ instead of all | {"url":"https://linearalgebras.com/solution-to-mathematics-for-machine-learning-exercise-7-2.html","timestamp":"2024-11-13T01:55:49Z","content_type":"text/html","content_length":"50812","record_id":"<urn:uuid:2e300c6c-5fdb-4461-bac7-6223cbcc1a6b>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00610.warc.gz"} |
Geometric quantities of lower doubly excited bound states of helium
Chengdong Zhou(周成栋)^1, Yuewu Yu(余岳武)^1, Sanjiang Yang(杨三江)^2,†, and Haoxue Qiao(乔豪学)^1,‡
1 School of Physics and Technology, Wuhan University, Wuhan 430072, China;
2 College of Physics and Electronic Science, Hubei Normal University, Huangshi 435002, China
Abstract Expectation values of single electron and interelectronic geometric quantities such as $\langle r\rangle$, $\langle r_{12}\rangle$, $\langle r_<\rangle$, $\langle r_>\rangle$, $\langle \cos
\theta_{12}\rangle$ and $\langle \theta_{12}\rangle$ are calculated for doubly excited $2{\rm p}n{\rm p}\,{}^1P^{\,\rm e}\,(3\leq n\leq5),\, 2{\rm p}n{\rm p}\,{}^3\!P^{\,\rm e}\,(2\leq n\leq5)$ and
$2{\rm p}n{\rm d}\,{}^{1,3}D^{\,\rm o}\,(3\leq n\leq5)$ states of helium using Hylleraas-$B$-spline basis set. The energy levels converge to at least 10 significant digits in our calculations. The
extrapolated values of geometric quantities except for $\langle \theta_{12}\rangle$ reach 10 significant digits as well; $\langle \theta_{12}\rangle$ reaches at least 7 significant digits using a
multipole expansion approach. Our results provide a precise reference for future research.
Received: 30 May 2021
Revised: 18 July 2021
Accepted manuscript online: 07 August 2021
PACS: 03.65.Ge (Solutions of wave equations: bound states)
31.15.-p (Calculations and mathematical techniques in atomic and molecular physics)
Fund: Project supported by the National Natural Science Foundation of China (Grant No. 12074295). The authors thank Yongbo Tang for the meaningful discussion. The numerical calculations in this
article were performed on the supercomputing system in the Supercomputing Center of Wuhan University.
Corresponding Authors: Sanjiang Yang, Haoxue Qiao
E-mail: s.yang@whu.edu.cn;qhx@whu.edu.cn
Cite this article:
Chengdong Zhou(周成栋), Yuewu Yu(余岳武), Sanjiang Yang(杨三江), and Haoxue Qiao(乔豪学) Geometric quantities of lower doubly excited bound states of helium 2022 Chin. Phys. B 31 030301
[1] Compton K T and Boyce J C 1928 J. Franklin Inst. 205 497
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[7] Saha J K and Mukherjee T K 2009 Phys. Rev. A 80 022513
[8] Kar S and Ho Y K 2010 Phys. Rev. A 82 036501
[9] Eiglsperger J, Piraux B and Madroñero J 2010 Phys. Rev. A 81 042527
[10] Eiglsperger J, Piraux B and Madroñero J 2010 Phys. Rev. A 81 042528
[11] Bylicki M and Nicolaides C A 2000 Phys. Rev. A 61 052508
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[13] Bürgers A and Rost J M 1996 J. Phys. B:At. Mol. Opt. Phys. 29 3825
[14] Ordóñez-Lasso A F, Cardona J C and Sanz-Vicario J L 2013 Phys. Rev. A 88 012702
[15] Koga T 2004 J. Chem. Phys. 121 3939
[16] Koga T 2002 Chem. Phys. Lett. 363 598
[17] Koga T, Matsuyama H and Thakkar A J 2011 Chem. Phys. Lett. 512 287
[18] Matsuyama H and Koga T 2007 Theor. Chem. Acc. 118 643
[19] Jiao L G, Zan L R, Zhu L and Ho Y K 2018 Comput. Theor. Chem. 1135 1
[20] Jiao L G, Zan L R, Zhu L, Zhang Y Z and Ho Y K 2019 Phys. Rev. A 100 022509
[21] Yang S J, Mei X S, Shi T Y and Qiao H X 2017 Phys. Rev. A 95 062505
[22] Wu F F, Yang S J, Zhang Y H, Zhang J Y, Qiao H X, Shi T Y and Tang L Y 2018 Phys. Rev. A 98 040501
[23] Yang S J, Tang Y B, Zhao Y H, Shi T Y and Qiao H X 2019 Phys. Rev. A 100 042509
[24] Gradshteyn I S and Ryzhik I M 2000 Table of Integrals, Series, and Products, 6th edn. (New York:Elsevier) p. 801
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[26] Bhattacharyya S, Mukherjee T K, Saha J K and Mukherjee P K 2008 Phys. Rev. A 78 032505
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Arithmetic with randomness? Doable?
When your office analyst is working with randomness or uncertainty, getting a numerical answer to a question is tricky business.
It's just a box
Here's one to ponder: if the length of a specific "box" is uncertain, but from observations of similar boxes the length seems to lie between Lmin and Lmax, and similarly the width seems to lie
between Wmin and Wmax, how then does your analyst provide you with a number for the area of the box?
There are at least three methods that apply to arithmetic with uncertainty:
1. You could multiply each value of L and W, pair by pair. If there were 6 unique observations of L and six of W, then there would be 36 combinations, like Wmin X Lmax. You could then average them.
Of course, if you are dealing with volume of a cube rather than an area, then the situation gets quite large quickly. You might have LxWxHeight, and that would bring in 6 times more combinations
2. You could do away with the randomness by first calculating the average W and the average L. If you calculate from observed data, then the average of the observations is a statistic, and it is not
random. In effect, you've done "reduction" on the randomness.
3. You could calculate the 'expected value' by adding up the probability-weighted values --- value of L or W times the probability of L or W. You sum all the probable L's and W's, and then do the
L*W multiplication on their weighted sums. Probability would be calculated from the observed frequency that each value of L and W occurs.
Now, think about risk management
What if the L and W were not length and width, but rather risk impact and risk probability? Typical parameters on a risk matrix.
Can you multiply them?
You don't know impact and risk probability for certain; there is randomness about each parameter.
Well, the right answer is that your risk analyst should apply one of the methods described for the box at the outset of this posting.
Here the bell ring
What if your observations more or less show that the distribution of Ls and Ws (or the impact and risk probability) is more bell-shaped than uniform (as in the previous example). The three methods
still work, but there is a lot of work to do to get the answers from the bell-shaped distributions. And, if there is a lot of production data, then all three methods are really calculation intensive,
best left to a computer.
Arithmetic in the face of randomness: You have to deal with the distributions of the observed or forecasted data, convolving distributions, or otherwise doing "reduction" on the randomness to
reach a calculated result.
(*) You might recognize that this is pretty similar to the process of adding two uncertain numbers demonstrated by the example of rolling two six-faced die, and averaging the sum of the outcomes,
which of course, is 7.
Also note that whereas the probability of any one die value is uniformly between 1 and 6, the probability of the individual 36 sums is decidedly not uniform. The average, 7, has the highest frequency
of occurrence among the 36 outcomes.
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Network Security Assignment Question
Security & Privacy Considerations of Dissertation:
‘An Equivalence Checker for Pi-Calculus Models’ Basil Contovounesios <contovob@tcd.ie>
Reactive systems are often formally modelled in one of two ways: in a process calculus such as the [π]-calculus, in terms of message-passing; or as a labelled transition system (LTS), in terms of a
set of states and a transition model. The latter representation is amenable to automata-theoretic algorithmics, but classic formulations cannot effectively capture real-world problems over unbounded
domains. This can be mitigated by expressing the LTS of a process as a fresh-register automaton (FRA).
This dissertation builds on prior work in generating the LTS of a [π]-calculus model represented as an FRA. The aim of the project is to create a model checking tool that implements and evaluates the
performance characteristics of several bisimulation algorithms for determining process equivalence, including novel adaptations of classic partition refinement algorithms to FRAs. The result will
include a survey of the completeness of existing symbolic approaches used by other verification tools.
1 Background
Historical models of computation such as register machines and the lambda calculus (on which most programming languages are based) capture the computational but not interactional or concurrent
behaviour of computing systems [1, p. 3]. The ubiquity of communication and concurrency is not merely a modern trend; they are fundemantal behaviours of all computing systems in practice. In fact,
each of their computations involves and can be modelled by communication between internal or external components. Formally reasoning about real-world systems is therefore significantly facilitated by
a model of computation such as a process calculus, whose basic action is message-passing in the form of concurrent synchronisation over a shared channel.
The [π]-calculus in particular was designed for characterising processes with mobile structure, that is the ability to form transient or virtual links with other processes [1, p. 77]. Obvious
examples of mobility include connections over wireless networks or between clients and servers on the web, which are continually created and destroyed.
While such channels are typically regarded as transmitting static data, the [π]-calculus additionally arranges for the transmission of arbitrary processes as messages. This affords the expressive
power to elegantly model programming language paradigms [1, p. 113] as well as general computer and, in the interest of this assignment, security protocols [2], as will be described further on.
2 Motivation
Designing, implementing, and testing concurrent systems in general, and security protocols in particular, is notoriously difficult and error-prone. New bugs such as race conditions and
vulnerabilities to attack are constantly being discovered, to say nothing of those that remain undetected. One approach for reducing these risks is to formally verify the systems, which entails
modelling them in a formal language like the [π]-calculus [2, p. 2]. It is then possible to reason about all possible behaviours of each system either algebraically, or through generating and
analysing its entire state space as an LTS [3, p. 1].
On a theoretical level, models of computation like the [π]-calculus describe both the implementation and behavioural specification of processes, meaning a single language can be used for both
implementation and formal verification [4, p. 100]. This can mitigate at least to some extent the well-known risk of introducing bugs in the implementation of an otherwise correct specification. As
such, it is of fundamental importance and interest to define when different processes are behaviourally equivalent, or even approximately so.
A particularly fine-grained, robust, and well-established notion of equivalence is that of bisimilarity, in which two potentially nondeterministic processes symmetrically simulate each other’s inputs
and outputs [5]. It can be used to verify whether two processes behave as expected, describe similar behaviour at perhaps different levels of abstraction, or violate certain specified properties,
regardless of context or environment [6, pp. xv, 73]. This final point affords greater compositional reasoning, i.e. the ability to infer global properties of an entire system while verifying its
components in isolation [7, p. 81]. By contrast, classic model checkers like SPIN do not verify all interactions with the environment, and thus do not compose efficiently in real-world scenarios [3,
p. 2].
3 Application
Although the [π]-calculus operates at too low a level to be of practical use in realworld verification of, say, security protocols, it forms a simple yet expressive basis for application-specific
abstractions at a higher level [1, p. 153]. An example of this is the ‘applied’ [π]-calculus, which extends the original with cryptographic primitives and is used with the ProVerif symbolic protocol
verifier [2].
Most verification tools use a symbolic model of protocols like that described so far. The alternative is the computational model used in cryptography, in which messages are bitstrings rather than
terms in a process algebra. While the latter is more realistic, it is less amenable to automated proofs [8, p. 100]. Furthermore, an attack on a protocol found in the symbolic model translates
readily into a practical attack in the computational model, but the opposite is not always true [2, p. 3].
At its core, the protocol verification works by generating the set of messages an adversary may know, and equating message secrecy to not being present in that set. The verifier can also attempt to
prove certain security properties such as secrecy or authentication, by checking that they hold for all traces of a protocol. For example, partial knowledge of a message or the unreachability of a
particular state can be modelled in this way. With partial knowledge, however, an adversary can still detect changes in a secret. This leads to the requirement for a stronger notion of secrecy in
which processes are indistinguishable to an adversary, which constitutes a behavioural equivalence rather than a trace property [2, pp. 4–6].
4 Limitations and Conclusion
The equivalence properties in ProVerif can be proven for applied [π]-calculus terms with different message structure, but only if two processes are otherwise the same. Thus the tool cannot always
decide the equivalence of arbitrary processes. This is just one of several tradeoffs faced by verification tools. For example, full model verification tends to be more costly than property
verification [9, p. 173]. Similarly, supporting unbounded attack domains or concurrent protocol sessions comes at the cost of completeness or greater automation (otherwise requiring manual proof
input from the user).
As suggested previously, model checking is inherently limited in its ability to catch bugs in the implementation of a model, and this is true for both symbolic and computational models. A particular
challenge is presented by side channel attacks which cannot be modelled effectively, such as those targeting power consumption. Nevertheless, tooling is becoming increasingly sophisticated. There has
been work on automated model extraction and code generation to and from [π]-calculus models [8], as well as on simultaneously proving the cryptographic security and program correctness of the OpenSSL
HMAC implementation [10].
Ultimately, formal verification is but one of many tools for mitigating risk, and is best used systematically in conjunction with others. My personal interests lie in the pursuit of its applicability
to programming language and compiler theory, for creating safer and more expressive languages.
• Robin Milner. Communicating and Mobile Systems: the [π]-Calculus. Cambridge University Press, 1999.
• Bruno Blanchet. ‘Modeling and Verifying Security Protocols with the Applied Pi Calculus and ProVerif’. In: Foundations and Trends^® in Privacy and Security 1–2 (Oct. 2016), pp. 1–135. doi:
• Seng Leung. ‘Modelling Concurrent Systems. Generation of Labelled Transition Systems of Pi-Calculus Models through the Use of Fresh-Register Automata’. MA thesis. Trinity College, University of
Dublin, Apr. 2020.
• Luca Aceto, Anna Ingolfsdottir and Jiří Srba. ‘The Algorithmics of Bisimilarity’. In: Advanced Topics in Bisimulation and Coinduction. Ed. by Davide Sangiorgi and Jan Rutten. Cambridge University
Press, 2011. Chap. 3, pp. 100–172. doi: 1017/CBO9780511792588.004.
• Damien Pous and Davide Sangiorgi. ‘Bisimulation and coinduction enhancements: a historical perspective’. In: Formal Aspects of Computing 6 (2019), pp. 733– 749.
• Robin Milner. The Space and Motion of Communicating Agents. Cambridge University Press, 2009.
• Sergey Berezin, Sérgio Campos and Edmund M. Clarke. ‘Compositional Reasoning in Model Checking’. In: Compositionality: The Significant Difference. Ed. by Willem-Paul de Roever, Hans Langmaack and
Amir Pnueli. Springer, 1998, pp. 81– 102. doi: 1007/3-540-49213-5.
• Matteo Avalle, Alfredo Pironti and Riccardo Sisto. ‘Formal verification of security protocol implementations: a survey’. In: Formal Aspects of Computing 1 (2014), pp. 99–123. doi: 10.1007/
• Michael Huth and Mark Ryan. Logic in Computer Science. Modelling and Reasoning about Systems. 2nd ed. Cambridge University Press, 2004.
• Lennart Beringer et al. ‘Verified Correctness and Security of OpenSSL HMAC’. In: USENIX Security. Proceedings of the 24th USENIX Security Symposium. Aug. 2015, pp. 207–221. | {"url":"https://www.myassignmenthelp.net/question/cs7ns5-assignment-1","timestamp":"2024-11-10T19:20:08Z","content_type":"text/html","content_length":"60403","record_id":"<urn:uuid:40cfa59e-7692-4ef2-be6d-a7d5c5c380e9>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00591.warc.gz"} |
Excel: Calculate the Number of Weeks Between Dates | Online Tutorials Library List | Tutoraspire.com
Excel: Calculate the Number of Weeks Between Dates
by Tutor Aspire
You can use the following formulas to calculate the number of weeks between two dates in Excel:
Formula 1: Calculate Full Weeks Between Two Dates
Formula 2: Calculate Fractional Weeks Between Two Dates
Both formulas assume that cell A2 contains the start date and cell B2 contains the end date.
The following examples show how to use each formula in practice.
Example 1: Calculate Full Weeks Between Two Dates
The following screenshot shows how to calculate the number of full weeks between a list of start and end dates in Excel:
Here’s how to interpret the output:
• There are 4 full weeks between 1/1/2022 and 2/4/2022.
• There are 20 full weeks between 1/7/2022 and 5/29/2022.
• There are 2 full weeks between 1/20/2022 and 2/5/2022.
And so on.
Example 2: Calculate Fractional Weeks Between Two Dates
The following screenshot shows how to calculate the number of fractional weeks between a list of start and end dates in Excel:
Here’s how to interpret the output:
• There are 4.857Â weeks between 1/1/2022 and 2/4/2022.
• There are 20.286 weeks between 1/7/2022 and 5/29/2022.
• There are 2.286Â weeks between 1/20/2022 and 2/5/2022.
And so on.
Related: How to Calculate the Number of Months Between Dates in Excel
Additional Resources
The following tutorials explain how to perform other common operations in Excel:
How to Add & Subtract Weeks to Date in Excel
How to Group Data by Week in Excel
How to Sum by Week in Excel
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Stringing IF/OR statements in a single formula
I am trying to string multiple IF/OR statements together. To summarize, if the column Location contains London, Brussels or Moscow, the resulting value should be Europe. If the Location column
contains Dubai or Riyadh, the value should be Middle East. If the Location column contains any other value, the result should be the existing value from the Location field.
This formula works.
=IF(OR(Location@row = "London", Location@row = "Brussels", Location@row = "Moscow"), "Europe", Location@row)
However, I need to include the following to the above formula:
=IF(OR(Location@row = "Dubai", Location@row = "Riyadh"), "Middle East", Location@row)
I have tried so many variations and can't seem to get this to work. If anyone has a suggestion, I would be greatly appreciative.
Best Answer
• Hi Cheryl,
You're super close on this one. You can just put your second formula inside the "value if false" section of the first formula like so:
=IF(OR(Location@row = "London", Location@row = "Brussels", Location@row = "Moscow"), "Europe", IF(OR(Location@row = "Dubai", Location@row = "Riyadh"), "Middle East", Location@row))
This basically tells the formula that if the value is not London, Brussels, or Moscow, evaluate what is in the "value if false" section.
• Hi Cheryl,
You're super close on this one. You can just put your second formula inside the "value if false" section of the first formula like so:
=IF(OR(Location@row = "London", Location@row = "Brussels", Location@row = "Moscow"), "Europe", IF(OR(Location@row = "Dubai", Location@row = "Riyadh"), "Middle East", Location@row))
This basically tells the formula that if the value is not London, Brussels, or Moscow, evaluate what is in the "value if false" section.
• That worked, thank you!!!
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RHS - Math
Department Chair: Victoria Meredith
Course Offerings
Algebra I
Course Description
This course provides a study of the basic principles of algebra, including using the rational number system; performing operations with polynomials; solving linear equations and inequalities; solving
quadratic equations; organizing and manipulating data; and graphing linear and quadratic equations. The course requires students to use algebra as a tool for representing and solving a variety of
practical problems. Students will use graphing calculators, computers and other appropriate technology.
Students taking this course will taken an end-of-course SOL test in Algebra I. Students must pass both the course and the SOL test to earn a verified credit.
Algebra Functions & Data Analysis (AFDA)
Course Description
AFDA provides an opportunity for mathematical ideas to be developed in the context of real-world problems. Students will be asked to collect and analyze univariate and bivariate data using a variety
of statistics and analytical tools. They will learn to attach functional algebra to statistics, allowing for the possibility of standardizing and analyzing data through the use of mathematical
models. Students will use transformational graphing and the regression capabilities of graphing calculators to find the regression equations, and they will use them to analyze the data to predict the
placement of data points between and beyond the given data points.
Course Description
This course provides a study of lines, angles, triangles, logic, polygons, circles, three-dimensional figures, coordinate relations, and transformations. In addition, this course gives the student a
thorough study of proofs, including two-column proofs, indirect proofs, coordinate proofs, and verbal arguments. Students will use graphing calculators, computers and other appropriate technology.
Students taking this course will take an end-of-course SOL test in Geometry. Students must pass both the course and the SOL test to earn a verified credit.
Pre-AP Geometry
Course Description
This course offers a more in-depth study of the content of Geometry with more emphasis on abstract concepts and mathematical structure. Students will use graphing calculators, computers and other
appropriate technology. Students taking this course will take an end-of-course SOL test in Geometry. Students must pass both the course and the SOL test to earn a verified credit.
Algebra II
Course Description
This course includes a thorough treatment of advanced algebraic concepts provided through the study of functions, “families of functions,” equations, inequalities, systems of equations and
inequalities, polynomials, rational expressions, complex numbers, matrices, and sequences and series. Emphasis will be placed on practical applications and modeling throughout the course of study.
Oral and written communication concerning the language of algebra, logic of procedures, and interpretation of results will permeate the course. Students will use graphing calculators, computers, and
other appropriate technology. Students taking this course will take an end-of-course SOL test in Algebra II. Students must pass both the course and the SOL test to earn a verified credit.
Pre-AP Algebra II
Course Description
This course is designed for advanced students who are capable of a more rigorous course at an accelerated pace. The course covers the concepts listed in the Algebra II description and provides the
foundation for students to pursue a sequence of advanced mathematical studies from Math Analysis to Advanced Placement Calculus. Students will use graphing calculators, computers, and other
appropriate technology. Students taking this course will take an end-of-course SOL test in Algebra II. Students must pass both the course and the SOL test to earn a verified credit.
Elementary Math Functions
Course Description
This course is an extension of the concepts learned in Algebra II and an introduction to pre-calculus concepts. The course is comprised of the following three units of study: algebra, trigonometry,
and statistics. The algebra unit will include a study of linear relations and functions, systems of equations and inequalities, polynomial and rational functions, exponential and logarithmic
functions, and the nature of graphs. The trigonometric unit will include a study of trigonometric functions, graphs and inverses of trigonometric functions, and trigonometric identities and
equations. The statistics unit will include a study of descriptive statistics and probability. Real-world applications will be incorporated when appropriate. Students will use graphing calculators,
computers, and other appropriate technology. There is no SOL test given for this course.
Pre-AP Calculus
Course Description
This is a rigorous course that includes the advanced study of algebra, theory of equations, analytic geometry, and trigonometry, and an introduction to calculus concepts. It is recommended that
students demonstrate a sound understanding of the principles of Algebra I, Algebra II, and geometry before enrolling in this course. Students will use graphing calculators, computers, and other
appropriate technology. There is no SOL test given for this course.
AP Calculus
Course Description
Advanced Placement Calculus is a course designed to provide advanced mathematics students an opportunity to earn college credit while simultaneously earning credit toward high school graduation. The
course is intended for students who have a thorough knowledge of analytic geometry and elementary functions in addition to college preparatory algebra, geometry, and trigonometry. The purpose of the
course is to prepare students for advanced placement in college calculus. The course standards incorporate The College Board Advanced Placement Course Description Syllabus. As mandated by The College
Board, graphing calculators are required for this course. Instructional activities that engage students in solving application problems of varying complexities will be used. There is no SOL test
given for this course.
Students receive weighted credit for participating in AP classes, they receive college credit only by achieving a high score on the College Board’s Advanced Placement Tests. There are fees for each
test, which students will be responsible for paying. Colleges have different criteria for awarding credit for scores on AP tests, and it will be the responsibility of students to contact colleges to
ascertain their policies regarding Advanced Placement scores.
AP Statistics
Course Description
Advanced Placement Statistics is a course designed to introduce students to the tools for collecting, analyzing, and drawing conclusions from data. The course incorporates The College Board Advanced
Placement Course which exposes students to the following four themes:
1. Looking for patterns in data
2. Planning and conducting studies
3. Exploring data using probability and simulation
4. Estimating population parameters and testing hypotheses
Graphing calculators will be used in the course. There is no SOL test given for this course.
Students receive weighted credit for participating in AP classes, they receive college credit only by achieving a high score on the College Board’s Advanced Placement Tests. There are fees for each
test, which students will be responsible for paying. Colleges have different criteria for awarding credit for scores on AP tests, and it will be the responsibility of students to contact colleges to
ascertain their policies regarding Advanced Placement scores. | {"url":"https://rhs.campbell.k12.va.us/departments/math","timestamp":"2024-11-07T07:09:09Z","content_type":"text/html","content_length":"138247","record_id":"<urn:uuid:99248f3f-c53e-46c7-b779-619851cae30d>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00147.warc.gz"} |
How do our NAPLEX & MPJE reviews help? - University of Wyoming Pharmacy Exam Review Courses
How do our NAPLEX & MPJE reviews help?
Take a look on our website at the page titled “YOUR ENROLLMENT INCLUDES.” Listed are the topics that we cover, and the resources that we provide.
What kinds of things must you be familiar and comfortable with?
Here are a couple of sample exercises. You need to be proficient with questions like these.
How many milliliters of water need to be mixed with 80% v/v alcohol to make 400 mL of 50% v/v alcohol?
1. Less than 40
2. Between 40 and 80
3. Between 80 and 120
4. Between 120 and 160
5. Greater than 160
Calculating the answer:
400 mL x 50% = V x 80%
V = 250 mL of 80% alcohol is needed to achieve the total amount of alcohol that is contained within 400 mL of 50% alcohol
Since the final product volume is 400 mL and 250 mL of 80% alcohol is needed to make this product:
400 mL – 250 mL = 150 mL of water “needs to be added”
JO is a 63-year old male with rheumatoid arthritis and NYHA class III heart failure. His blood pressure and renal function are within normal limits. His LFTs are 3x the upper limit of normal.
Current medications:
• Entresto (sacubatril/valsartan) 97/103 mg daily
• Bisoprolol 10 mg daily
Which of the following is the most appropriate initial treatment for JO’s severe rheumatoid arthritis?
1. Etanercept (Enbrel)
2. Adalimumab (Humira )
3. Abatacept (Orencia)
4. Methotrexate
Contact me:
Bill Feinberg, RPh, MBA | {"url":"http://pharmacyreviewcourses.com/sample-review-questions/how-do-our-naplex-mpje-reviews-help/","timestamp":"2024-11-03T19:57:44Z","content_type":"text/html","content_length":"40992","record_id":"<urn:uuid:74d2c013-25a5-473d-8bc2-b51824854ec8>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00670.warc.gz"} |
Total hours that fall between two times
In this example, we first calculate the total hours between the start time and end time. Then we figure out the total hours that overlap "the period of interest". Using these two values, we can
figure out all remaining hours. In the example shown, "lower" is lower bound for the period of interest, and "upper" is the upper bound.
To calculate total hours between start and end time, the formula in D5 is:
This formula is explained in more detail here.
The formula in E5 works in two parts, using IF to control flow. If the start time is less than the end time, we calculate the overlap with:
If the start time is greater than end time (i.e. start time and end time cross midnight), we use:
By using MAX with zero, we prevent negative values from being used.
Finally, to figure out "Remaining" hours (i.e. hours that do not fall in the period of interest) we simply subtract E5 (included) from D5 (total duration). | {"url":"https://exceljet.net/formulas/total-hours-that-fall-between-two-times","timestamp":"2024-11-11T13:52:22Z","content_type":"text/html","content_length":"48536","record_id":"<urn:uuid:66531ff5-8f6c-45ea-8d10-1b31630c5c20>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00097.warc.gz"} |
Free CFD Codes: Learn through examples.
More sample programs are coming up: Delaunay triangulation code, a panel code for an airfoil, 2D unstructured Navier-Stokes code, etc.
[Note: As the name, Katate Masatsuka, implies, I write only when I find time.]
Educationally-Designed Unstructured 2D (EDU2D)
This program reads a 2D tria/quqad/mixed grid, and generates a 3D grid by extending/rotating the 2D grid to the third dimension. A hexahedral grid will be generated for a pure quadrilateral grid, and
a prismatic or tetrahedral grid will be generated for a pure triangular grid as below. Below, a flat-plate grid is used as an example. It generates a .ugrtid file (for a solver such as FUN3D), .su2
file for SU2, and tecplot and vtk files for viewing the grid.
3D conventional and hyperbolic Poisson solver:  
This is a 3D Poisson solver code having two options: (1) conventional solver and (2) hyperbolic solver. The hyperbolic solver is faster for finer grids than the conventional solver beause the
numerical stiffness caused by second-derivatives in the Poisson equation is completely elimianted in the hyperbolic solver; and it is also more accurate on unstructured grids, especially in the
solution derivatives (one order higher accurate than conventional methods). See JCP2020 for the details of the conventional and hyperbolic solvers. Example cases are included, where linear and
nonlinear Poisson equations are solved in a cube domain and also in a domain bounded by two co-centroid spheres, using tetrahedral and hexahedral grids as shown in figures below.
3D Grid Generation for Sphere:  
A 3D grid is generated for a sphere and written in the UGRID format. A tetrahedarl, prismatic, or mixed grid can be generated with a specified y-plus value. Learn how a 3D unstructured grid is
generated over a sphere.
Educationally-Designed Unstructured 2D (EDU2D)
Exact vortex solution to the 2D unsteady Euler equations:  
  edu2d_inviscid_vortex_solution_verification.f90
This program computes the inviscid vortex solution and substitute it into the 2D unsteady Euler equations and show all equations are zero (satisfied).
Educationally-Designed Unstructured 2D (EDU2D)
EDU2D Cylinder-Wake II: Grid generation for cylinder:  
This is another grid generation code for a domain around a circular cylinder. It generates multiblock-like grid with better controls than the other code. This package contains the twod2threed code
and generates a 3D grid by extending the 2D (x,z) grid in the y-direction.
For the grid format, see   edu2d_grid_format.pdf
Educationally-Designed Unstructured 3D (EDU3D)
  edu3d_mms_euler_v1.f90
This code shows how to compute the source terms in the method of manufactured solutions (MMS) for the 3D Euler equations. Any function can be made an exact solution to the 3D Euler equations with
suitable source terms. That is, any function v(x,y,z) is an exact solution to the following equation:
MMS has been widely used for verifying the order of accuracy of a CFD code. Grab this code, and learn how it can be implemented. I always use MMS to verify the implementation. It is a very useful
tool for debugging.
Educationally-Designed Unstructured 3D (EDU3D)
EDU3D Euler:  
This is an implicit unstructured-grid Euler solver (f90, serial, tetra only), written for an educational purpose. Read the source code and learn how a node-centered edge-based unstructured-grid
solver is written. Example cases are included: MMS test, cube, hemisphere cylinder, and ONERA M6 wing.
For the grid format, see   edu2d_grid_format.pdf
Heapsort program
  heapsort_v1.f90
I have used the heapsort algorithm in a hierarchical agglomeration scheme developed and implemented in NASA's FUN3D code. This is just an example program of the heapsort. In FUN3D, I used the
heapsort to order the list of control-volumes to be agglomerated based on the number of non-agglomerated neighbor volumes, so that agglomeration can be performed from the volumes with the least # of
neighbors, which helps keep the front as convex as possible and avoid creating holes. The front list changes every time a volume is agglomerated with neighbors and it needs to be re-ordered with new
front members (the neighbors of the newly agglomerated volumes). It was so fast that the agglomeration can be peformed quickly inside the solver every time it reads a grid.
EDU2D-CCFV-EULER-EXPLCT: An explicit Euler code:
This is an explicit unstructured-grid Euler code, written for an educaitonal purpose. Read the source code and learn how a cell-centered FV unstructured-grid solver is written. Examples cases are
included: transonic airfoil, subsonic cylinder, unsteady vortex, numerical truncation error computation.
For the grid format, see   edu2d_grid_format.pdf
EDU2D-Rectangular-GRID: Grid generation for a rectangular domain:
  edu2d_rectangular_grid_v1.f90
This code generates a triangular or quadrilateral grid for a 2D rectangular domain with 4 or 5 boundary parts as shown below (Top: 4 boundary parts, Bottom: 5 boundary parts). The 5-boundary-part
grid can be used for a shock diffraction problem or a forward-facing step problem, for example. It generates a custom 2D grid file (used in EDU2D codes), tecplot, SU2, and vtk files.
EDU2D-VISCOUS-BOX-GRID: Grid generation for a viscous-wall domain:
  edu2d_viscous_box_grid_v0.f90
This code generates a triangular or quadrilateral grid for a 2D rectangular domain with 4 viscous boundaries as shown below, suitable for natural convection (buoyancy flow) problems. It generates a
custom 2D grid file (used in EDU2D codes), tecplot, SU2, and vtk files.
EDU2D-FP-GRID: Grid generation for flat plate:
  edu2d_fp_grid_v1.f90
  input.nml
This code generates a grid (tria,quad,mixed) for a 2D flat plate case, where a flat plate is placed on the bottom right half, (x,y)=(0,0)-(xmax,0). It generates a custom 2D grid file (used in EDU2D
codes), tecplot, SU2, and vtk files.
EDU2D-HALF-CYLINDER-GRID: Grid generation for half cylinder:
  edu2d_half_cylinder_grid_v0.f90
EDU2D-BUMP-GRID: Grid generation for bump:
  edu2d_bump_irregular_grid_v1.f90
Flow around a Karman-Trefftz airfoil (generate grids and exact solutions):
  edu2d_vkt_airfoil_v4.f90 ,   vkt_airfoil_v1_display.m
Karman-Trefftz airfoil is an intriguing airfoil for which a complete set of exact solutions can be computed. This solution has been used by many people to verify the accuracy of their inviscid code
(incompressible limit). If you have not, use this code now to generate a (quadrilateral or triangular) grid, run your code on it, compute the error, and verify the accuracy of your code. Me? I have
used it for my third-order multigrid Cauchy-Riemann solver (IJNMF 2004).
Updated (10-13-10). Formula corrected. It works now for cambered airfoils.
Ringleb's Flow (generate grids and exact solutions):  
Ringleb's flow is a famous exact solution of the compressible Euler equations with a smooth transition from subsonic to supersonic without any shock waves. This solution has been used actually by
many people to verify the accuracy of their Euler code. If you have not, use this code now to generate a (quadrilateral or triangular) grid, run your Euler code on it, compute the error, and verify
the accuracy of your Euler code.
Minor bugs fixed (12-29-10).
Flow over Rankine's half body (generate grids and exact solutions):
  edu2d_rankine_half_body_grid_v1.f90
A flow over Rankine's half body is an interesting exact solution to the potential flow equations: Cauchy-Riemann systems, Lapace equations, the potential equation, Euler equations. You can download,
compile, and run this code to generate a (quadrilateral or triangular) grid, with the exact solution computed at each grid point (stored in Tecplot files). At some point, I'd like to use this
solution for verification. Shown below is the exact x-velocity contours on an irregular triangular grid .
EDU2D-Cylinder-Wake: Grid generation for cylinder:
  edu2d_cylinder_wake_v4.f90
  input.nml
[To compile: gfortran edu2d_cylinder_wake_v4.f90]
This code generates a grid (tria,quad,mixed) over a cylinder with a higher-resolution in the wake region. It is suitable for unsteady cylinder simulations. It generates a custom 2D grid file (used in
EDU2D codes), tecplot, SU2, and vtk files.
Educationally-Designed Unstructured 2D (EDU2D)
Special triangular grid generator: EDU2D-50yen-Tria-Grids
  edu2d_50yen_tria_grids_v2.f90
  edu2d_50yen_tria_grids_tecplot_v0.lay
This program can be useful for generating triangular grids for use in CFD or just learning how to manipulate unstructured-grid (invert, stretch, desired grid-spacing, perturb, diagonal swapping). It
generates a simple triangulation of a sector, and then use it to generate 6 different triangular grids in a disk or a 50-yen domain. A tecplot style file is included to generate a picture such as the
one below after the code is run.
HCH Grid Generation Code Package:  
This package contains two f90 codes. (1)Structured/unstructured grid generation code for a hemisphere-cylinder-hemisphere body. It generates hex-prism structured, prismatic, tetrahedral, mixed
prism-tetra, and mixed prism-hex grids. The shape of the body can be defined by a shape function. Four different functions are implemented in the code: a suboff model, a suboff with a hemisphere
front, a sine function, and an ellipsoid. (2)A coarsening program. It generates a series of coarse grids from the grid generated in (1).
10-16-2020: Bugs fixed. 07-24-2019: Added .su2 grid. 12-17-2017: Several bugs fixed.
3D Grid Generation for Hemisphere:  
A 3D grid is generated for a hemisphere (with two configurations) and written in the UGRID format. A tetrahedarl, prismatic, or mixed grid can be generated with a specified y-plus value. Learn how a
3D unstructured grid is generated over a hemisphere.
3D Bump Grid Generation:  
Educationally-Designed Unstructured 3D (EDU3D)
3D Tetrahedral Grid Generation:  
edu3d_tetgrid_cube_v4.f90 (regular grid)
edu3d_tetgrid_cube_ptb_v7.f90 (irregular grid with nodal perturbation)
Tetrahedral grid in a unit cube is generated and written in the UGRID format. Learn how the hexahedron can be divided into six tetrahedra. Mdify the code to divide the hexahedron into five
07-24-19: Added .su2 grid file.
11-24-15: Added an irregular grid version (tetgrid_cube_ptb_v5.f90).
10-06-17: Added a subroutine that computes edge-based data used in the edge-based discretization (tetgrid_cube_ptb_v6p1.f90).
3D Mixed Grid Generation:   mixgrid_cube_v5.f90
Mixed (tetrahedral-prismatic) grid in a unit cube is generated and written in the UGRID format. Learn how a typical viscous-type grid can be generated. You may want to modify the code to apply
stretching to the prismatic layer for a smooth transition to the isotropic tetrahedral region. (08-22-2018: A bug fixed about the boundary grid information.)
3D Prismatic Grid Generation:   przgrid_cube_v4.f90
Prismatic grid in a unit cube is generated and written in the UGRID format. It's just a cubic domain, but may be useful in learning unstructured grid data.
3D Hex Grid Generation (Unstructured Format):   hexgrid_cube_v4.f90
Hexahedral grid in a unit cube is generated and written as unstructured/finite-element data in a UGRID 3D unstructured grid file. It may be useful for those who want to learn a typical data structure
of unstructured grids.
EDU1D Sock Tube Code:
  edu1d_euler_shock_tube.zip
This is a code that solves shock tube problems with various flux functions and limiters. It can be useful for learning flux functions, limiters, Hancock's preedictor-corrector scheme, primitive vesus
characterisstic limiting, etc., and also useful for computing an exact solution to a shock tube problem.
Uploaded. (05-19-19)
How to compute flux Jacobians:
  edu3d_exact_flux_jacobian_v0.f90
  module_ddt5.f90
[To compile: gfortran edu3d_exact_flux_jacobian_v0.f90]
This code shows two ways to compute flux Jacobians: automatic differentiation and hand-coded subroutines for the Roe flux (inviscid) and the alpha-damping flux (viscous) in three dimensions. This
serves also as verification of the hand-coded Jacobians (which are complicated). See how the Roe flux is exactly differentiated. It may be useful also for learning how the viscous flux is computed.
Uploaded. (04-07-19)
Machine zero:   machinezero_v3.f90
What is machine zero? Maybe, it is a non-zero number which cannot be recognized by a machine. So, it depends on machines. Download, compile, and run it to find out the `zero' in your machine.
Comments corrected. (09-23-18)
Generalized. Now it finds `machine zero' for a given value. (11-29-12)
Educationally-Designed Unstructured 1D (EDU1D)
First/Second/Third-Order Diffusion Schemes in 1D:  
Diffusion equation is solved by 1st/2nd/3rd-order upwind schemes on irregularly-spaced grids. The idea is to integrate an equivalent hyperbolic system toward a steady state. This way, we can advance
in pseudo time with a large O(h) time step (not O(h^2)), and compute the solution gradient with the equal order of accuracy on irregular grids. Compare with a conventional diffusion scheme to see how
accurate the gradient can be and how fast it converges.
Note: This third-order scheme does not require computation/storage of second derivatives. See Ref.[2].
[1] JCP2014 | Preprint - 1st/2nd/3rd-order FV upwind schemes for diffusion
[2] JCP2017 | Preprint - Third-order accuracy without second derivatives.
Solver Package: GCR solver module:  
  gcr_module_v3.f90
This module is a GCR (Generalized Conjugate Residual) linear solver, which can be used to construct a Jacobian-Free Newton-Krylov solver (JFNK). The code illustrates how a JFNK solver can be
Educationally-Designed Unstructured 2D (EDU2D)
EDU2D-AdvDiff (16 files)
  edu2d_advdiff_basic_package_v0.f90
  edu2d_advdiff_solver_nc_v2.f90 <--- Corrected
  edu2d_advdiff_main_v1.f90
  edu2d_advdiff_lsq_grad_nc_v0.f90
  edu2d_ddt3_v0.f90
  generate_grids_for_edu2d_steady_v0.f90
  generate_grids_for_edu2d_unsteady_v0.f90
  Makefile
  readme_v0.txt
  steady_screen_out.txt
  unsteady_screen_out.txt
  steady_plot_solution.lay
  steady_plot_boundary_solution.lay
  unsteady_initial_solution.lay
  unsteady_final_solution.lay
  unsteady_initial_final_solutions.lay
This is an unstructured advection-diffusion solver.
- Node-centered finite-volume discretization
- Fully unstructured grids (triangles, quads, or mixed)
- Least-squares gradients (linear or quadratic LSQ)
- Diffusion is discretized as a hyperbolic system for accurate gradient prediction
- Implicit time-stepping with BDF2
- Jacobian-Free Newton-Krylov with GCR
- Defect-correction implicit solver as a variable preconditioner
- Designed to solve both steady and unsteady problems.
Kahan's sum to minimize the round-off error
Avoiding round-off errors:   kahan_sum_example.f90
This code shows how the round-off error can be minimized in computing a sum of small values, by using Kahan's round-off-corrected summation.
Educationally-Designed Unstructured 2D (EDU2D)
  edu2d_mms_twodns_v2.f90
This code shows how to compute the source terms in the method of manufactured solutions (MMS) for the 2D Navier-Stokes equations. Any function can be made an exact solution to the 2D Navier-Stokes
equations with suitable source terms. That is, any function v(x,y) is an exact solution to the following equation:
MMS has been widely used for verifying the order of accuracy of a CFD code. Grab this code, and learn how it can be implemented. I always use MMS to verify the implementation. It is a very useful
tool for debugging.
Educationally-Designed Unstructured 2D (EDU2D)
  edu2d_quadrature_triangle_v1.f90
This program illustrates the use of a Gaussian quadrature, which integrates a function over a triangle. An example function, x^5 + y^5 -32*x^2*y^3 + x^3*y^2, is integrated over a triangle defined by
the vertices: (0.1,0.3), (1.3,-0.2), (-0.9,1.7), and the numerical value obtained by a 7-point quadrature formula is compared with the exact value 0.53726127666666666668, up to 15 significant digits
(double precision).
Educationally-Designed Unstructured 2D (EDU2D)
EDU2D-Airfoil-Spline (3 files)
  edu2d_airfoil_spline_v1.f90
  om6_wing_section_sharp.dat
  edu2d_airfoil_plot.m
Compile edu2d_airfoil_spline_v1.f90 and run it. This f90 program reads OM6 wing section data contained in the file 'om6_wing_section_sharp.dat', constructs a cubic spline (piecewise cubic polynomial
interpolation), and writes out data files for plotting the airfoil by Matlab or Tecplot. I remember I used a cubic spline to represent a Joukowsky airfoil, so that boundary nodes can be adaptively
moved along the surface to better represent numerical solutions [ IJNMF2002 ]
The OM6 section data are from [ AIAA J. Vol. 54, No. 9, September 2016 ], and the trailing edge is closed as described in the AIAA jounal paper.
Educationally-Designed Unstructured 2D (EDU2D)
EDU2D-Tecplot2Grid (5 files)
  edu2d_tecplot2grid_v1.f90
  sample_input
  tria_rankine_grid_tecplot.dat
  tria_rankine_grid_tecplot.data
  readme_v1.txt
This program reads a Tecplot data file, generates a boundary information, and writes .grid and .bcmap file for edu2d solvers. Tecplot file does not contain boundary information, and so the boundary
information is genearted inside the program. Not an elegant code, but it may be useful to learn how to identify boundary nodes for a given grid, which is not really trivial. A grid for a Rankine's
half body is used as an example.
Educationally-Designed Unstructured 2D (EDU2D)
EDU2D-Euler-Steady (9 files)
  edu2d_euler_steady_basic_package_v0.f90
  edu2d_euler_steady_solver_v0.f90
  edu2d_euler_steady_main_v0.f90
  edu2d_euler_jacobian_v1.f90
  edu2d_euler_linear_solve_v0.f90
  twod_bump_irregular_grid_v0.f90
  Makefile
  readme_v0.txt
  bump_screen_out.txt
Want to learn how to write an implicit unstructured CFD code? Grab this code, look inside to see how it is written, get good understanding, and then write your own. This code computes a steady flow
over a bump with the Roe flux by two solution methods: an explicit 2-stage Runge-Kutta scheme and an implicit (defect correction) method with the exact Jacobian for a 1st-order scheme, on irregular
triangular grids. A grid generation code is included for a bump problem.
- Node-centered finite-volume discretization
- Fully unstructured grids (triangles, quads, or mixed)
- Least-squares gradients (linear or quadratic LSQ)
- Defect-correction implicit solver (Newton's method for 1st-order)
- Both explicit and implicit solvers are implemented
Educationally-Designed Unstructured 2D (EDU2D)
EDU2D-Euler-Unsteady (8 files)
  edu2d_euler_unsteady_basic_package_v0.f90
  edu2d_euler_unsteady_solver_v0.f90
  edu2d_euler_unsteady_main_v0.f90
  twod_rectangular_grid_v0.f90
  Makefile
  readme_v0.txt
  project_quad_screen_out.txt
  project_tria_screen_out.txt
Want to learn how to write an unstructured CFD code? Grab this code, look inside to see how it is written, get good understanding, and then write your own. This code has Roe and Rotated-RHLL fluxes,
Van Albada limiter, and a 2-stage Runge-Kutta time-stepping for solving a shock diffraction problem. It works for quadrilateral grids, triangular grids, and mixed grids also. It is set up to solve a
shock diffraction problem. You can easily modify it for solving other problems. A grid generation code is included.
- Node-centered finite-volume discretization
- Fully unstructured grids (triangles, quads, or mixed)
- Least-squares gradients (linear or quadratic LSQ)
- Limiter is applied at edges
- Explicit time-stepping with 2-stage Runge-Kutta scheme
Educationally-Designed Unstructured 2D (EDU2D)
EDU2D-Template: 2D Example Unstructured CFD Code (8 files)
  edu2d_basic_package_v0.f90
  edu2d_template_main_v0.f90
  edu2d_template_nc_v0.f90
  edu2d_template_cc_v0.f90
  generate_grids_for_edu2d_v0.f90
  Makefile
  readme_v0.txt
  any_name_screen_out.txt
This code reads an unstructured grid file, generate various grid data, go through some dummy CFD solvers, and then writes out Tecplot data files for viewing the solution and the grid. Solvers are
dummies and so do not solve anything, but you'll see how a node/cell-centered finite-volume solver can be implemented, e.g., loop over edges, nodes, elements, and computation of least-squares
gradients, etc. You can generate your own unstructured CFD code by replacing the dummy solver by your own solver. A grid generation file is included, which can be used to generate grid files you need
to run the EDU2D-Template code.
MPI program for Jacobi Iteration Ver.1:   jacobi_mpi_v1.f90
This is an MPI program for the Jacobi iteration, solving the finite-difference discretization of the Laplace equation in a square domain. It can be useful for learning how to write an MPI program.
Automatic Differentiation Ver.1:   ad_driver_v1.f90   module_ddt3.f90
Automatic differentiation is nice. It is sometimes used to compute flux Jacobians for implicit formulaitons in CFD. To those who are curious about it, here is a set of files with which you can
experience and learn how it works. Simply compile the file "ad_driver.f90" and run it (the other file will be automatically included in the program) to get a feeling of automatic differentiation.
Numerical Fluxes (3D Euler) Ver.3: threed_euler_fluxes_v3.f90
Here are 3D Euler numerical fluxes. Download and take a look. Learn how the standard Roe and a very robust rotated-hybrid fluxes are implemented for the 3D Euler equations, and also how the Roe flux
can be implemented without tangent vectors. Included are Roe with/without tangent vectors, and Rotated-RHLL fluxes. I'll be extremely happy if you kindly report bugs. Thank you, arigatou!
Bugs fixed for the RHLL flux, more comments added, Roe without tangent vectors added. (04-30-12).
3D Grid Generation for Hemishpere Cylinder:
1D Hyperbolic Diffusion Scheme:   oned_upwind_diffusion.f90
Believe or not, the diffusion equation is solved by an upwind scheme. The idea is to integrate an equivalent hyperbolic system toward a steady state. This way, we can advance in time with a large O
(h) time step (not O(h^2)), and compute the solution gradient with the equal order of accuracy. Compare with a common scheme (Galerkin) for 512 nodes to see how fast the upwind scheme can be.
Reference: JCP2007 | Preprint
Genetic Algorithm, Ver.1:   ga_v1.f90
This is a simple genetic algorithm program for finding minimizers of a function. I wrote this in 1999 when I was interested in aerodynamic optimization problems. I think that genetic algorithm is a
very interesting optimization algorithm.
1D Euler Code Ver.1:   oned_euler_v1.f90 ,   oned_euler_plot_v1.m
Here is a 1D Euler code (1D shock tube code) for solving Sod's shock tube problem, using Roe's Approximate Riemann solver, minmod limiter, and 2-stage Runge-Kutta time-stepping. Learn how a
second-order non-oscillatory Euler code is written, or just run it to see how it is capable of computing discontinuous solutions. Incorporate various flux subroutines given below to explore other
Minor bugs fixed (12-29-10).
Numerical Fluxes (1D Euler) Ver.5:   oned_euler_fluxes_v5.f90
All numerical fluxes are functions of the left and right states (and possibly dt and dx). Learn how those famous fluxes can be implemented, or just use them to see how they work for various
shock-tube problems. Included are Lax-Friedrichs, Richtmyer, MacCormack, Steger-Warming, Van Leer, AUSM, Zha-Bilgen, Godunov, Osher, Roe, Rusanov, HLL, HLLL, AUFS flux functions.
Bug fixed for Godunov fluxes.
Numerical Fluxes (2D Euler) Ver.2:   twod_euler_fluxes_v2.f90
Want 2D fluxes also? Here you are. Download and take a look. Learn how the standard Roe and a very robust rotated-hybrid fluxes are implemented. Included are Roe and Rotated-RHLL fluxes (Riemann
Bugs fixed for the Rotated-RHLL flux. It works very well now.
Blasius solutions for a flow over a flat-plate (compute the exact solution) :
  blasius_v1.f90 ,   blasius_plot_v1.m
Exact solution for a flow over a flat-plate. This solution has been used by many people to verify the accuracy of their Navier-Stokes code. If you have not, use this code now to compute the exact
solution at any point on your grid, compute the error, and verify the accuracy of your code.
Minor bugs fixed (12-29-10).
Viscous shock structure NS-solution (compute the exact solution) :
  ns_shock_structure_v1.f90 ,   ns_shock_structure_plot_v1.m
Exact solution for a shock wave internal structure to the 1D Navier-Stokes equations. This solution has been used by some people to verify the accuracy of their 1D Navier-Stokes code. If you have
not, use this code now to generate a data file for the exact solution, use it as an initial solution for your code, converge to a steady state, and verify the accuracy of your code.
Minor bugs fixed (05-17-2023, 12-29-10). | {"url":"http://ossanworld.com/cfdbooks/cfdcodes.html","timestamp":"2024-11-12T16:02:33Z","content_type":"application/xhtml+xml","content_length":"113507","record_id":"<urn:uuid:4d6b96f9-8ebf-44eb-863e-4735d3f47095>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00631.warc.gz"} |