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Fred – only the best math experts at math-master.org – 20
Pre-Algebra • Algebra • Trigonometry • Statistics • Solid Geometry • Calculus
I was good in math's since school and I helped my friends pass their math's exams when we were in college. So, from there this passion of teaching was there and now when I am able to solve online
doubts of students it gives me pleasure that in some way I am helping the students.
Author's latest answers | {"url":"https://math-master.org/expert/fred-20?page=2","timestamp":"2024-11-07T17:03:25Z","content_type":"text/html","content_length":"190940","record_id":"<urn:uuid:d5858923-7140-42eb-be8b-da9cb1e06863>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00782.warc.gz"} |
Science Fair Project
Published on Sep 05, 2023
The objective: This project aims to prove the hypothesis that the true noon (or solar noon) in Riverside during the end of January-beginning of February does not occur at 12:00 Pacific Standard Time.
This hypothesis is based on the observation that the sun is directly above us at a different time during the seasons. True noon is defined when the sun is directly above our location.
The physical principle is similar to that of the sun dial. A ruler was positioned vertically on a table covered by graph paper. The position of the shadow of the ruler on the graph paper was
monitored between 10:30 AM and 3:00 PM at approximately 10-15 minute intervals.
Once the measurements were finished I traced the lines of the shadow of the ruler on the graph paper and I measured their length. The data showed the behavior of a parabola formed by the ends of the
The shortest line coincided with the minimum of the parabola. This minimum corresponded to the true noon time.
I performed five experimental trials at two different locations. I plotted the data using a spreadsheet.
To determine the minimum I fitted the data using the function of a parabola. The fitting aimed to reduce the measurement errors and help locate the minimum of the parabola in a more precise way.
The data showed that true noon at the end of January-beginning of February in Riverside is at 12:11 PM. This result is in agreement with my hypothesis that true noon does not coincide with the noon
of standard time this time of the year.
At true noon the angle made by the sun rays and the ruler (angle of incidence) is also minimum, resulting to shortest shadow length. The angle of incidence is a measure of the sun elevation.
I measured the angle of incidence at true noon after drawing on the graph paper the right triangle with perpendicular sides corresponding to the length of the ruler and the ruler shadow. I discuss
the effect of refraction of the sun rays on the measurement of the angle of incidence.
True noon depends on the sun elevation. The sun elevation depends on the geographic latitude of our location and the time of the year.
My results are in good agreement with values determined from the analemma drawn on a globe and the Riverside sun dial outside the Riverside public library.
I have used principles from physics, geometry, and statistical fitting to make the astronomical measurement of true noon time.
This project showed that true noon time is variable throughout the year and depends on geographic location.
Science Fair Project done By Vasilios A. Morikis | {"url":"https://www.sciencefairprojects.co.in/Physics/Measurement-of-True-Noon-Time.php","timestamp":"2024-11-06T08:47:40Z","content_type":"text/html","content_length":"13078","record_id":"<urn:uuid:8a1f093d-a68d-40f1-aff2-f88cdd83aa97>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00103.warc.gz"} |
addLR: Additive logratio coordinates in robCompositions: Compositional Data Analysis
The additive logratio coordinates map D-part compositional data from the simplex into a (D-1)-dimensional real space.
x D-part compositional data
ivar Rationing part
base a positive or complex number: the base with respect to which logarithms are computed. Defaults to exp(1).
a positive or complex number: the base with respect to which logarithms are computed. Defaults to exp(1).
The compositional parts are divided by the rationing part before the logarithm is taken.
A list of class “alr” which includes the following content:
x.alr the resulting coordinates
varx the rationing variable
ivar the index of the rationing variable, indicating the column number of the rationing variable in the data matrix x
cnames the column names of x
the index of the rationing variable, indicating the column number of the rationing variable in the data matrix x
The additional information such as cnames or ivar is useful when an inverse mapping is applied on the ‘same’ data set.
Aitchison, J. (1986) The Statistical Analysis of Compositional Data Monographs on Statistics and Applied Probability. Chapman and Hall Ltd., London (UK). 416p.
data(arcticLake) x <- arcticLake x.alr <- addLR(x, 2) y <- addLRinv(x.alr) ## This exactly fulfills: addLRinv(addLR(x, 3)) data(expenditures) x <- expenditures y <- addLRinv(addLR(x, 5)) head(x) head
(y) ## --> absolute values are preserved as well. ## preserve only the ratios: addLRinv(x.alr, ivar=2, useClassInfo=FALSE)
For more information on customizing the embed code, read Embedding Snippets. | {"url":"https://rdrr.io/cran/robCompositions/man/addLR.html","timestamp":"2024-11-05T22:04:53Z","content_type":"text/html","content_length":"34353","record_id":"<urn:uuid:b4f7b19a-8bea-4426-91f4-c6f1aa3e9bed>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00286.warc.gz"} |
Thinking thru Scheme Monads
This page tells monad savvy readers how to do monads in Scheme. It thus presumes concepts that I lack. It does, however, propose a nice challenge for the purely functional style of code. Scheme
supports this style but also supports side-effects.
I adapt the challenge here as follows. Suppose we have a sort S of scheme procedure that takes no arguments and returns a pair of a character and a procedure of sort S. A sort S procedure thus
produces a character stream sequentially:
(car (s)), (car ((cdr (s)))), (car ((cdr ((cdr (s)))))), ...
For example st0 below is such a procedure yielding “pusillanimous pusillanimous ...”:
(define (st0)
(let* ((st "pusillanimous ") (l (- (string-length st) 1)))
(let s ((n 0))
(cons (string-ref st n)
(lambda () (s (if (= n l) 0 (+ n 1))))))))
We are to build a depth n binary tree with the letters of the stream as leaves. A binary tree of depth n and 3 as leaves can be built with
(define n 4)
(let bt ((k n)) (if (zero? k) 3 (cons (bt (- k 1))(bt (- k 1)))))
That was easy but for the real problem we must specify in our code which branch of a sub tree is to have the early stream characters. Our recursive procedure takes a sort S procedure and returns a
pair of values:
• the new tree,
• a sort S procedure for the unused stream.
Here goes:
(let bt ((k n)(stx st0)) (if (zero? k) (stx)
(let* ((lst (bt (- k 1) stx))(rst (bt (- k 1) (cdr lst))))
(cons (cons (car lst) (car rst)) (cdr rst)))))
which yields:
(((((#\p . #\u) #\s . #\i) (#\l . #\l) #\a . #\n) ((#\i . #\m) #\o . #\u) (#\s . #\space) #\p . #\u) . #)
Perhaps we should lop off the remaining procedure but I won't. I think this is in the monad spirit, but not yet in the monad style. It hands around the stream generator awkwardly. It is easier in
Ocaml, but still awkward. I find code such as this easier to write than to read. Richard Uhtenwoldt’s example | {"url":"http://cap-lore.com/Software/monad/","timestamp":"2024-11-08T09:23:56Z","content_type":"text/html","content_length":"2410","record_id":"<urn:uuid:51f067b0-bed2-4178-a45b-0b427c6e6fd2>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00578.warc.gz"} |
PPT - Chapter 5 PowerPoint Presentation, free download - ID:6366575
1. Chapter 5 The Medium Access Sublayer
2. The Medium Access Layer • 5.1 Channel Allocation problem - Static and dynamic channel allocation in LANs & MANs • 5.2 Multiple Access Protocols - ALOHA, CSMA, CSMA/CD, Collision-free protocols,
Limited-contention protocols, Wireless LAN protocols • 5.3 Ethernet - Cabling, MAC sublayer protocol, Backoff algorithm, Performance, Gigabit Ethernet, 802.2 Logical Link Control • 5.4Wireless
LANs - 802.11 protocol stack, physical layer, MAC sublayer protocol, frame structure
3. 5.5 Broadband Wireless - Comparison of 802.11 with 802.16, protocol stack, frame structure • 5.6 Bluetooth - Bluetooth architecture, Application, Protocol stack, Frame structure • 5.7 Data Link
Layer Switching - Bridges from 802.x to 802.y, Local internetworking, Spanning tree bridges, Remote bridges
4. What is MAC • Network assumption: Broadcast channel • One channel, many stations • Competition, interference among stations. • MAC: Medium Access Control • Also known as Multiple-Access Control •
The protocol used to determine who goes next on a shared physical media • Classification of MAC protocols • Channel allocation (centralized) • Contention based protocols (distributed) •
Contention – free protocols (distributed)
5. Medium Access Sublayer • Key issue for broadcast network • who can use the channel when there is competition for it • Medium Access Control: • a sublayer of data link layer that controls the
access of nodes to the medium. • Broadcast channels are also referred as multiaccess channels or random access channels • Allocation of a single broadcast channel among competing users: • Static
• Dynamic
6. 5.1 The Channel Allocation problem Static Channel Allocation • FDMA • The whole spectrum is divided into sub-frequency. • TDMA • Each user has its own time slot. • CDMA • Simultaneous
transmission, Orthogonal code • Analogy:
7. The M/M/1 Queue • Average number of customers • Applying Little’s Theorem, we have • Similarly, the average waiting time and number of customers in the queue is given by E[ ] = L L
8. Example: Slowing Down • M/M/1 system: slow down the arrival and service rates by the same factor m • Utilization factors are the same ⇒stationary distributions the same, average number in the
system the same • Delay in the slower system is m times higher • Average number in queue is the same, but in the 1st system the customers move out faster
9. Example: Statistical MUX-ing vs. TDM or FDM • m identical Poisson streams with rate λ/m; link with capacity 1; packet lengths iid, exponential with mean 1/μ • Alternative: split the link to m
channels with capacity 1/m each, and dedicate one channel to each traffic stream • Delay in each “queue” becomes m times higher • Statistical multiplexing vs. TDM or FDM • When is TDM or FDM
preferred over statistical multiplexing?
10. The Channel Allocation problem 5.1.1 Static channel Allocation in LANs and WANs Frequency Division Multiplexing (FDM)
13. 5.1.2 Dynamic Channel Allocation in LANs and WANs Five Assumptions • Station Model. The model consists of N independent stations, each generates the frame with probability in an interval . Once a
frame is generated, the station is blocked. • Single Channel Assumption. A single channel is available for all communication. • Collision Assumption. If two frames are transmitted simultaneously,
they are destroyed and must be retransmitted again later. There are no other errors.
14. 4a. Continuous Time. Frame transmission can begin at any instant. 4b. Slotted Time. Time is divided into slots. Frame transmission always begin at the start of a slot. 5a. Carrier Sense. Stations
can tell if the channel is in use before trying to use it. (ex. LANs) 5b. No Carrier Sense. Stations cannot sense the channel before trying to use it. (ex. satellite network due to long
propagation delay)
15. 5.2 Multiple Access Protocols • ALOHA • Carrier Sense Multiple Access Protocols • Collision-Free Protocols • Limited-Contention Protocols • Wavelength Division Multiple Access Protocols •
Wireless LAN Protocols
16. ALOHA • Users send whenever they want to send. If it fails, wait random time and resend it. • Independent stations • Single channel assumption • Collision occurs • Types of ALOHA • Pure ALOHA:
stations transmit at any time (Continuous time) • Slotted ALOHA: Transmission can only occur at certain time instances • carrier sense vs no carrier senses
17. Pure ALOHA • Users transmit whenever they have data to be sent. • The colliding frames are destroyed. • The sender waits a random amount of time and sends it again. 1970s from University of
Hawaii Pure ALOHA (infinite population) In pure ALOHA, frames are transmitted at completely arbitrary times.
18. Pure ALOHA (2) Vulnerable period for the shaded frame.
19. Poisson pmf (1) • Suppose we are observing the arrival of jobs to a large computation center for the time interval (0, t] • Assume that for each small interval of time Dt, the probability of a
new job arrival is lDt, where l is the average arrival rate.. • If Dt is sufficiently small, the probability of two or more jobs arriving in the interval of duration Dtmay be neglected. • Divide
(0, t] into n subintervals of length t/n, and suppose the arrival of a job in any given interval is independent of the arrival of a job in any other interval. • n very large => the n intervals
constitutes a sequence of Bernoulli trials with the probability of success p = lt / n • Bernoulli trials: P(X = 0) = p, withP(X = 1) = 1p
20. The Probability of k arrivals in a total of n intervals each with a duration t/n is approximately given by • As n -> infinity => • Let t be a frame time => lt = G Poisson pmf (2)
22. Pure ALOHA (2) • efficiency: 18.4 % for channel utilization at best • Assume that infinite population of users generates new frames according to a Poisson distribution with mean S frames per
frame time, where 0 < S < 1. • Assume that the probability of k transmission attempts per frame time, old and new combined, is also poisson, with mean G per frame time. • P0: probability that a
frame does not suffer a collision • throughput S = G * P0 (Offered loadtimes transmission succeeding prob.) • vulnerable interval: t0 ~ t0+2t (See Fig. 5-2) • probability that k frames are
generated during a given frame time is given by the Poisson distribution • Probability of no other traffic during the vulnerable period, • P0 = e -2G , 2G: mean of two frame time. • S = Ge -2G
(See Fig. 5-3)
23. 0.368 0.184 Throughput versus offered traffic for ALOHA systems.
39. Persistent and Nonpersistent CSMA Comparison of the channel utilization versus load for various random access protocols.
41. CSMA/CD • Abort transmission as soon as they detect a collision • saves time and bandwidth • waits a random time and tries again • Fig. 5-5 • minimum time to detect collision: the signal
propagates from one station to the other • worst case: 2t (t : propagation time between two farthest stations) • model the contention interval as a slotted ALOHA system with slot = 2t • special
signal encoding: to detect a collision of two 0-volt signals • No MAC sublayer protocol guarantees reliable delivery. Packets may be lost due to • collision • lack of buffer space • missed
42. CSMA with Collision Detection CSMA/CD can be in one of three states: contention, transmission, or idle.
43. 5.2.3 Collision-Free Protocols • Collision is serious (affects performance) as. • large t: long cable • short frames: high bandwidth (propagation dominate the delivering time. • Bit-Map Protocol
(See Fig. 5-6) • A cycle consists of a contention period and a data transmission period. • Contention period contains N slots, one bit for a station • a station inserts 1 at its slot when has
data. After the contention period, stations transmit data in the sequence in the contention period. • problem: overhead is 1 bit per station
44. 5.2.3 Collision-Free Protocols The basic bit-map protocol. 4-6.
45. Collision-Free Protocols (2) • Binary Countdown • Give priority to higher address by OR bit-by-bit addresses of the stations waiting for transmission. • virtual station number: to change priority
The binary countdown protocol. A dash indicates silence. | {"url":"https://fr.slideserve.com/tiger-cunningham/chapter-5-6366575-powerpoint-ppt-presentation","timestamp":"2024-11-10T21:37:42Z","content_type":"text/html","content_length":"96211","record_id":"<urn:uuid:986916cf-7dc1-4b33-a572-04ac45d178a2>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00097.warc.gz"} |
I need help with Sagetex/Sage (normal form&data file useage)
I need help with Sagetex/Sage (normal form&data file useage)
asked 2010-10-27 19:19:13 +0100
This post is a wiki. Anyone with karma >750 is welcome to improve it.
I have two questions.
The easier is, how can i "ask" Sagemath to numerically calculate, and use normal form. For example i would like to see sqrt{2}/10 as 1.41 * 10^{-1}. It would be useful to always get the same
precision. Like 2/10 as 2.00 * 10^{-1}
The harder question is, i want to make several calculations with the same data (about 10-20 lines, every line with 2-4 records). It is possible to copy and paste it every time, but i rather do it an
easier way. Is it possible?
I think if you reverse your order of questions, you will get a better answer. For example, if you tell us what you're trying to calculate, we can help. Then if you want to display the results in a
certain format after the calculation, we can help with that too. =)
Hello! (d/2)*sin(((360-f(344,04,32))/(180))*pi)).n() would be the example calculation. I'd like to get 4.47 e-7. Problem is, the other calculation is{(360-f(351,31,23)+f(8,29,18))/2.n() and it should
give 8.48. Ps. f calculates x°y'z'' to simple degrees
1 Answer
Sort by » oldest newest most voted
answered 2010-10-29 01:31:23 +0100
This post is a wiki. Anyone with karma >750 is welcome to improve it.
I don't know if scientific notation is available yet, but you can do the following:
sage: a = 2/10; a
sage: (parent(a), type(a))
(Rational Field, <type 'sage.rings.rational.Rational'>)
sage: a.n(4)
sage: (parent(a.n(4)), type(a.n(4)))
(Real Field with 4 bits of precision, <type 'sage.rings.real_mpfr.RealNumber'>)
sage: (a.n(), parent(a.n()), type(a.n()))
(0.200000000000000, Real Field with 53 bits of precision, <type 'sage.rings.real_mpfr.RealNumber'>)
edit flag offensive delete link more
Hello! I was aware of that trick, but still thanks. Any idea how could i ask for two digits precision instead of like four bits? Daniel
Daniel Balog ( 2010-10-29 19:00:52 +0100 )edit
a.n(digits=2) will do this.
Jason Bandlow ( 2010-10-29 19:45:27 +0100 )edit
Thanks! Still 1/9.n(digits=2) gets 0.11 and 10/9.n(digits=2) gets 1.1, but it's nearly perfect!
Daniel Balog ( 2010-10-30 07:09:29 +0100 )edit | {"url":"https://ask.sagemath.org/question/7739/i-need-help-with-sagetexsage-normal-formdata-file-useage/","timestamp":"2024-11-02T14:54:15Z","content_type":"application/xhtml+xml","content_length":"61640","record_id":"<urn:uuid:09a073d0-1a93-4433-a2a0-d9e9161802ac>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00403.warc.gz"} |
GCSE Mathematics Syllabus Statement
\( \DeclareMathOperator{cosec}{cosec} \)
GCSE Mathematics Syllabus Statement
Geometry and measures
[ << Main Page ]
Subject Content:
Pupils should be taught to describe translations as 2D vectors
Here are some specific activities, investigations or visual aids we have picked out. Click anywhere in the grey area to access the resource.
Here are some exam-style questions on this statement:
See all these questions
Click on a topic below for suggested lesson Starters, resources and activities from Transum. | {"url":"https://transum.org/Maths/National_Curriculum/Topics.asp?ID_Statement=139","timestamp":"2024-11-07T19:40:31Z","content_type":"text/html","content_length":"22840","record_id":"<urn:uuid:5f587919-13b1-479e-8fbb-0661e713ae01>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00604.warc.gz"} |
UNIT 2 Ünite 1 Kütle ve Hacim Mass and Volume
UNIT 2/ÜNİTE 2
MADDE VE ÖZELLİKLERİ
Dr. İlhan CANDAN
• Particulate structures that have mass, inertia and occupy space are
called matter.
• Mass is a measure of the amount of matter that makes up a particle
or object.
• It is a quantity that depends on the atoms that make up matter.
• It is one of the common properties of substances.
• Mass
• It is indicated by the letter m
• (SI) Its unit in the international unit system is kilogram (kg).
• It is measured with an equal-arm scale. The mass of 1 dm³ of pure
water at 1 atm pressure and +4 °C temperature is defined as 1 kg.
• The standard kilogram value was adopted in 1889 as the mass of a
cylinder made of an alloy of platinum and iridium.
• The standard mass is kept at the International Bureau of Weights and
Measures near Paris.
• Mass Units
Mass Units
Unit Symbol
Unit Conversion
• Example:
• Fill in the blanks by performing unit conversions between the mass
units given below.
• 0,4 kg = ......................mg
• 3 t = .............................kg
• 300 mg = .................... g
• 200 g = .......................kg
• Example:
• Complete the blanks in the table according to the example.
• Since liquids are weighed together with the container they are placed
in, the mass of the empty container must first be measured.
• The mass of the empty container is called the tare mass, and the total
mass of the liquid and the container is called the gross mass.
• The difference between the gross mass and the tare of the container
gives the net mass of the liquid.
• Volume:
• The place occupied by matter in space is called volume.
• It is indicated by the symbol V.
• The SI unit is the cubic meter (m³).
• Volume:
• Volume is one of the common properties of
• Solids have a definite shape and volume.
• Although liquid substances have a certain
volume, they do not have a certain shape; they
take the shape of the portion of the container
they are placed in.
• Gases take the volume and shape of the closed
container they are in.
• Volume Units
• The volume of a cube with side lengths of 1 meter is called unit
Unit Symbol
Unit conversion
• Volume Units in Liters
• Liter is the unit of measurement generally used to measure the
volume of liquids and gases.
• The volume unit of liquids and gases such as water, vinegar, gasoline,
natural gas, liquid detergent, LPG is liter.
Unit conversion
• 1dm³ =1 liter
Unit Symbol
• 1cm³ = 1 milliliter
• Example
• Fill in the blanks by performing unit conversions between the volume
units given below.
Finding the Volume of Solids with Regular Geometric Shapes
• The volume of solid objects with regular geometric shapes is
calculated depending on the geometric shape of the object.
• Volume of Prisms = Base Area x Height
Finding the Volume of Solids with Regular Geometric Shapes
rectangular prism
Finding the Volume of a Solid Object That Doesn't Have a Regular
Geometric Shape
Liquids can be used to measure the volume of a solid object that does
not have a regular geometric shape.
Graduated cylinders or overflow containers are used for measurement.
When a solid object is not dissolved in itself and is dropped into a liquid
with a lower density than itself, when it sinks completely into the
liquid, it causes the displacement of the liquid equal to its volume.
Finding the Volume of a Solid Object That Doesn't Have a Regular
Geometric Shape
Finding the Volume of a Solid Object That Doesn't Have a Regular
Geometric Shape
Finding the Volume of a Solid Object That Doesn't Have a Regular
Geometric Shape
When a square prism-shaped metal piece with a base length of 2 cm is
placed in 120 mL of water in a graduated container, the water level
rises to 184 mL.
Accordingly, what is the height of the metal piece in cm?
Volume of gases
The weak interaction between gas molecules allows these molecules to
move freely in all directions.
Freely circulating gas molecules are placed in whatever container they are
placed in.
It takes the shape and volume of the container. Therefore, the volume of
gases is expressed relative to the volume of the container they are in.
Volume of gases
• heat
• pressure
is easily affected by change.
5 kg of gas with a volume of 19 L is placed in the first container shown
in the figure. Then, the gas in the 1st container is transferred to the 2nd
container with a volume of 40 L.
Accordingly, compare the volume and mass amounts of the gas in the
1st and 2nd containers.
1st container
2nd container | {"url":"https://studylib.net/doc/27127150/unit-2-%C3%BCnite-1-k%C3%BCtle-ve-hacim-mass-and-volume","timestamp":"2024-11-13T22:04:03Z","content_type":"text/html","content_length":"53902","record_id":"<urn:uuid:790bb465-c7eb-467d-8450-5fab9896e2b7>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00319.warc.gz"} |
How to Group Data by Hour in Pandas (With Example) | Online Tutorials Library List | Tutoraspire.com
How to Group Data by Hour in Pandas (With Example)
by Tutor Aspire
You can use the following syntax to group data by hour and perform some aggregation in pandas:
This particular example groups the values by hour in a column called time and then calculates the sum of values in the sales column for each hour.
The following example shows how to use this syntax in practice.
Example: Group Data by Hour in Pandas
Suppose we have the following pandas DataFrame that shows the number of sales made at various times throughout the day for some store:
import pandas as pd
#create DataFrame
df = pd.DataFrame({'time': ['2022-01-01 01:14:00', '2022-01-01 01:24:15',
'2022-01-01 02:52:19', '2022-01-01 02:54:00',
'2022-01-01 04:05:10', '2022-01-01 05:35:09'],
'sales': [18, 20, 15, 14, 10, 9]})
#convert date column to datetime
df['time'] = pd.to_datetime(df['time'])
#view DataFrame
time sales
0 2022-01-01 01:14:00 18
1 2022-01-01 01:24:15 20
2 2022-01-01 02:52:19 15
3 2022-01-01 02:54:00 14
4 2022-01-01 04:05:10 10
5 2022-01-01 05:35:09 9
We can use the following syntax to group the time column by hours and calculate the sum of sales for each hour:
#group by hours in time column and calculate sum of sales
Name: sales, dtype: int64
From the output we can see:
• A total of 38 sales were made during the first hour.
• A total of 29 sales were made during the second hour.
• A total of 10sales were made during the fourth hour.
• A total of 9 sales were made during the fifth hour.
Note that we can also perform some other aggregation.
For example, we could calculate the mean number of sales per hour:
#group by hours in time column and calculate mean of sales
1 19.0
2 14.5
4 10.0
5 9.0
Name: sales, dtype: float64
We can also group by hours and minutes if we’d like.
For example, the following code shows how to calculate the sum of sales, grouped by hours and minutes:
#group by hours and minutes in time column and calculate mean of sales
df.groupby([df['time'].dt.hour, df['time'].dt.minute]).sales.mean()
time time
Name: sales, dtype: int64
From the output we can see:
• The mean number of sales during 1:14 was 18.
• The mean number of sales during 1:23 was 20.
• The mean number of sales during 2:52 was 15.
And so on.
Additional Resources
The following tutorials explain how to perform other common operations in pandas:
How to Create a Date Range in Pandas
How to Extract Month from Date in Pandas
How to Convert Timestamp to Datetime in Pandas
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Math Knowledge
ASVAB Math Knowledge Solving Equations Practice Test 488976
Question 2 of 5
One Variable
An equation is two expressions separated by an equal sign. The key to solving equations is to repeatedly do the same thing to both sides of the equation until the variable is isolated on one side of
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6.5 De Broglie’s Matter Waves - University Physics Volume 3 | OpenStax
By the end of this section, you will be able to:
• Describe de Broglie’s hypothesis of matter waves
• Explain how the de Broglie’s hypothesis gives the rationale for the quantization of angular momentum in Bohr’s quantum theory of the hydrogen atom
• Describe the Davisson–Germer experiment
• Interpret de Broglie’s idea of matter waves and how they account for electron diffraction phenomena
Compton’s formula established that an electromagnetic wave can behave like a particle of light when interacting with matter. In 1924, Louis de Broglie proposed a new speculative hypothesis that
electrons and other particles of matter can behave like waves. Today, this idea is known as de Broglie’s hypothesis of matter waves. In 1926, De Broglie’s hypothesis, together with Bohr’s early
quantum theory, led to the development of a new theory of wave quantum mechanics to describe the physics of atoms and subatomic particles. Quantum mechanics has paved the way for new engineering
inventions and technologies, such as the laser and magnetic resonance imaging (MRI). These new technologies drive discoveries in other sciences such as biology and chemistry.
According to de Broglie’s hypothesis, massless photons as well as massive particles must satisfy one common set of relations that connect the energy E with the frequency f, and the linear momentum p
with the wavelength $λ.λ.$ We have discussed these relations for photons in the context of Compton’s effect. We are recalling them now in a more general context. Any particle that has energy and
momentum is a de Broglie wave of frequency f and wavelength $λ:λ:$
Here, E and p are, respectively, the relativistic energy and the momentum of a particle. De Broglie’s relations are usually expressed in terms of the wave vector $k→,k→,$ $k=2π/λ,k=2π/λ,$ and the
wave frequency $ω=2πf,ω=2πf,$ as we usually do for waves:
Wave theory tells us that a wave carries its energy with the group velocity. For matter waves, this group velocity is the velocity u of the particle. Identifying the energy E and momentum p of a
particle with its relativistic energy $mc2mc2$ and its relativistic momentum mu, respectively, it follows from de Broglie relations that matter waves satisfy the following relation:
where $β=u/c.β=u/c.$ When a particle is massless we have $u=cu=c$ and Equation 6.57 becomes $λf=c.λf=c.$
How Long Are de Broglie Matter Waves?
Calculate the de Broglie wavelength of: (a) a 0.65-kg basketball thrown at a speed of 10 m/s, (b) a nonrelativistic electron with a kinetic energy of 1.0 eV, and (c) a relativistic electron with a
kinetic energy of
We use
Equation 6.57
to find the de Broglie wavelength. When the problem involves a nonrelativistic object moving with a nonrelativistic speed
, such as in (a) when
we use nonrelativistic momentum
. When the nonrelativistic approximation cannot be used, such as in (c), we must use the relativistic momentum
where the rest mass energy of a particle is
is the Lorentz factor
The total energy
of a particle is given by
Equation 6.53
and the kinetic energy is
When the kinetic energy is known, we can invert
Equation 6.18
to find the momentum
and substitute in
Equation 6.57
to obtain
Depending on the problem at hand, in this equation we can use the following values for hc: $hc=(6.626×10−34J·s)(2.998×108m/s)=1.986×10−25J·m=1.241eV·μmhc=(6.626×10−34J·s)(2.998×108m/s)=1.986×10−25J·m
a. For the basketball, the kinetic energy is
and the rest mass energy is
We see that $K/(K+E0)≪1K/(K+E0)≪1$ and use $p=mu=(0.65kg)(10m/s)=6.5J·s/m:p=mu=(0.65kg)(10m/s)=6.5J·s/m:$
b. For the nonrelativistic electron,
and when $K=1.0eV,K=1.0eV,$ we have $K/(K+E0)=(1/512)×10−3≪1,K/(K+E0)=(1/512)×10−3≪1,$ so we can use the nonrelativistic formula. However, it is simpler here to use Equation 6.58:
If we use nonrelativistic momentum, we obtain the same result because 1 eV is much smaller than the rest mass of the electron.
c. For a fast electron with $K=108keV,K=108keV,$ relativistic effects cannot be neglected because its total energy is $E=K+E0=108keV+511keV=619keVE=K+E0=108keV+511keV=619keV$ and $K/E=108/619K/E=108
/619$ is not negligible:
We see from these estimates that De Broglie’s wavelengths of macroscopic objects such as a ball are immeasurably small. Therefore, even if they exist, they are not detectable and do not affect the
motion of macroscopic objects.
Check Your Understanding 6.11
What is de Broglie’s wavelength of a nonrelativistic proton with a kinetic energy of 1.0 eV?
Using the concept of the electron matter wave, de Broglie provided a rationale for the quantization of the electron’s angular momentum in the hydrogen atom, which was postulated in Bohr’s quantum
theory. The physical explanation for the first Bohr quantization condition comes naturally when we assume that an electron in a hydrogen atom behaves not like a particle but like a wave. To see it
clearly, imagine a stretched guitar string that is clamped at both ends and vibrates in one of its normal modes. If the length of the string is l (Figure 6.18), the wavelengths of these vibrations
cannot be arbitrary but must be such that an integer k number of half-wavelengths $λ/2λ/2$ fit exactly on the distance l between the ends. This is the condition $l=kλ/2l=kλ/2$ for a standing wave on
a string. Now suppose that instead of having the string clamped at the walls, we bend its length into a circle and fasten its ends to each other. This produces a circular string that vibrates in
normal modes, satisfying the same standing-wave condition, but the number of half-wavelengths must now be an even number $k,k=2n,k,k=2n,$ and the length l is now connected to the radius $rnrn$ of the
circle. This means that the radii are not arbitrary but must satisfy the following standing-wave condition:
If an electron in the nth Bohr orbit moves as a wave, by Equation 6.59 its wavelength must be equal to $λ=2πrn/n.λ=2πrn/n.$ Assuming that Equation 6.58 is valid, the electron wave of this wavelength
corresponds to the electron’s linear momentum, $p=h/λ=nh/(2πrn)=nℏ/rn.p=h/λ=nh/(2πrn)=nℏ/rn.$ In a circular orbit, therefore, the electron’s angular momentum must be
This equation is the first of Bohr’s quantization conditions, given by Equation 6.36. Providing a physical explanation for Bohr’s quantization condition is a convincing theoretical argument for the
existence of matter waves.
The Electron Wave in the Ground State of Hydrogen
Find the de Broglie wavelength of an electron in the ground state of hydrogen.
We combine the first quantization condition in
Equation 6.60
Equation 6.36
and use
Equation 6.38
for the first Bohr radius with
the Bohr quantization condition gives
The electron wavelength is:
We obtain the same result when we use
Equation 6.58
Check Your Understanding 6.12
Find the de Broglie wavelength of an electron in the third excited state of hydrogen.
Experimental confirmation of matter waves came in 1927 when C. Davisson and L. Germer performed a series of electron-scattering experiments that clearly showed that electrons do behave like waves.
Davisson and Germer did not set up their experiment to confirm de Broglie’s hypothesis: The confirmation came as a byproduct of their routine experimental studies of metal surfaces under electron
In the particular experiment that provided the very first evidence of electron waves (known today as the Davisson–Germer experiment), they studied a surface of nickel. Their nickel sample was
specially prepared in a high-temperature oven to change its usual polycrystalline structure to a form in which large single-crystal domains occupy the volume. Figure 6.19 shows the experimental
setup. Thermal electrons are released from a heated element (usually made of tungsten) in the electron gun and accelerated through a potential difference $ΔV,ΔV,$ becoming a well-collimated beam of
electrons produced by an electron gun. The kinetic energy K of the electrons is adjusted by selecting a value of the potential difference in the electron gun. This produces a beam of electrons with a
set value of linear momentum, in accordance with the conservation of energy:
The electron beam is incident on the nickel sample in the direction normal to its surface. At the surface, it scatters in various directions. The intensity of the beam scattered in a selected
direction $φφ$ is measured by a highly sensitive detector. The detector’s angular position with respect to the direction of the incident beam can be varied from $φ=0°φ=0°$ to $φ=90°.φ=90°.$ The
entire setup is enclosed in a vacuum chamber to prevent electron collisions with air molecules, as such thermal collisions would change the electrons’ kinetic energy and are not desirable.
When the nickel target has a polycrystalline form with many randomly oriented microscopic crystals, the incident electrons scatter off its surface in various random directions. As a result, the
intensity of the scattered electron beam is much the same in any direction, resembling a diffuse reflection of light from a porous surface. However, when the nickel target has a regular crystalline
structure, the intensity of the scattered electron beam shows a clear maximum at a specific angle and the results show a clear diffraction pattern (see Figure 6.20). Similar diffraction patterns
formed by X-rays scattered by various crystalline solids were studied in 1912 by father-and-son physicists William H. Bragg and William L. Bragg. The Bragg law in X-ray crystallography provides a
connection between the wavelength $λλ$ of the radiation incident on a crystalline lattice, the lattice spacing, and the position of the interference maximum in the diffracted radiation (see
The lattice spacing of the Davisson–Germer target, determined with X-ray crystallography, was measured to be $a=2.15Å.a=2.15Å.$ Unlike X-ray crystallography in which X-rays penetrate the sample, in
the original Davisson–Germer experiment, only the surface atoms interact with the incident electron beam. For the surface diffraction, the maximum intensity of the reflected electron beam is observed
for scattering angles that satisfy the condition $nλ=asinφnλ=asinφ$ (see Figure 6.21). The first-order maximum (for $n=1n=1$) is measured at a scattering angle of $φ≈50°φ≈50°$ at $ΔV≈54V,ΔV≈54V,$
which gives the wavelength of the incident radiation as $λ=(2.15Å)sin50°=1.64Å.λ=(2.15Å)sin50°=1.64Å.$ On the other hand, a 54-V potential accelerates the incident electrons to kinetic energies of $K
=54eV.K=54eV.$ Their momentum, calculated from Equation 6.61, is $p=2.478×10−5eV·s/m.p=2.478×10−5eV·s/m.$ When we substitute this result in Equation 6.58, the de Broglie wavelength is obtained as
The same result is obtained when we use $K=54eVK=54eV$ in Equation 6.61. The proximity of this theoretical result to the Davisson–Germer experimental value of $λ=1.64Åλ=1.64Å$ is a convincing
argument for the existence of de Broglie matter waves.
Diffraction lines measured with low-energy electrons, such as those used in the Davisson–Germer experiment, are quite broad (see Figure 6.20) because the incident electrons are scattered only from
the surface. The resolution of diffraction images greatly improves when a higher-energy electron beam passes through a thin metal foil. This occurs because the diffraction image is created by
scattering off many crystalline planes inside the volume, and the maxima produced in scattering at Bragg angles are sharp (see Figure 6.22).
Since the work of Davisson and Germer, de Broglie’s hypothesis has been extensively tested with various experimental techniques, and the existence of de Broglie waves has been confirmed for numerous
elementary particles. Neutrons have been used in scattering experiments to determine crystalline structures of solids from interference patterns formed by neutron matter waves. The neutron has zero
charge and its mass is comparable with the mass of a positively charged proton. Both neutrons and protons can be seen as matter waves. Therefore, the property of being a matter wave is not specific
to electrically charged particles but is true of all particles in motion. Matter waves of molecules as large as carbon $C60C60$ have been measured. All physical objects, small or large, have an
associated matter wave as long as they remain in motion. The universal character of de Broglie matter waves is firmly established.
Neutron Scattering
Suppose that a neutron beam is used in a diffraction experiment on a typical crystalline solid. Estimate the kinetic energy of a neutron (in eV) in the neutron beam and compare it with kinetic energy
of an ideal gas in equilibrium at room temperature.
We assume that a typical crystal spacing
is of the order of 1.0 Å. To observe a diffraction pattern on such a lattice, the neutron wavelength
must be on the same order of magnitude as the lattice spacing. We use
Equation 6.61
to find the momentum
and kinetic energy
. To compare this energy with the energy
of ideal gas in equilibrium at room temperature
we use the relation
is the Boltzmann constant.
We evaluate
to compare it with the neutron’s rest mass energy
We see that $p2c2≪E02p2c2≪E02$ so $K≪E0K≪E0$ and we can use the nonrelativistic kinetic energy:
Kinetic energy of ideal gas in equilibrium at 300 K is:
We see that these energies are of the same order of magnitude.
Neutrons with energies in this range, which is typical for an ideal gas at room temperature, are called “thermal neutrons.”
Wavelength of a Relativistic Proton
In a supercollider at CERN, protons can be accelerated to velocities of 0.75
. What are their de Broglie wavelengths at this speed? What are their kinetic energies?
The rest mass energy of a proton is
When the proton’s velocity is known, we have
We obtain the wavelength
and kinetic energy
from relativistic relations.
Notice that because a proton is 1835 times more massive than an electron, if this experiment were performed with electrons, a simple rescaling of these results would give us the electron’s wavelength
and its kinetic energy of
Check Your Understanding 6.13
Find the de Broglie wavelength and kinetic energy of a free electron that travels at a speed of 0.75c. | {"url":"https://openstax.org/books/university-physics-volume-3/pages/6-5-de-broglies-matter-waves","timestamp":"2024-11-02T20:59:07Z","content_type":"text/html","content_length":"460319","record_id":"<urn:uuid:45df4b4d-ea18-4a52-a586-091438a848c9>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00516.warc.gz"} |
STAT.04 / P Values and Statistical Hypothesis Testing +
What is a P value?
A P value is a statistical measure that helps to determine the likelihood that the difference observed between two groups is due to chance. In other words, it tells you the probability that the
difference you observed between two groups occurred by random sampling error, rather than being a true difference between the groups.
For instance, if you conducted an experiment with two groups, A and B, and found that group A had a significantly higher mean than group B, the P value would tell you the likelihood of this
difference being due to chance. If the P value is low (typically below 0.05), it indicates that the observed difference is unlikely to have occurred by chance alone, and you can conclude that there
is likely a real difference between the two groups. Conversely, if the P value is high (typically above 0.05), it suggests that the observed difference could have occurred by chance, and you cannot
conclude that there is a real difference between the two groups.
What is a null hypothesis?
In statistical discussions of P values, the concept of a null hypothesis is commonly used. The null hypothesis assumes that there is no significant difference between the groups being compared. In
this context, the P value can be defined as the probability of obtaining a difference in the sample means as large as, or larger than, the observed difference, if the null hypothesis were actually
Common misinterpretation of a P value
A common misunderstanding of P values can lead to incorrect conclusions. For example, if the P value is reported as 0.03, it is often mistakenly assumed that there is a 97% chance that the observed
difference is a real difference between two populations, and only a 3% chance that it is due to chance alone. However, this interpretation is incorrect.
What the P value actually means is that if the null hypothesis (that there is no significant difference between the populations) is true, there is a 3% chance of obtaining a difference as large as,
or larger than, the observed difference. Therefore, the correct interpretation of the P value is that random sampling from identical populations would result in a difference smaller than the observed
difference in 97% of experiments and larger than the observed difference in 3% of experiments. It does not provide evidence for the existence of a real difference between the populations.
One-tail vs. two-tail P values
In comparing two groups, it is important to differentiate between one-tail and two-tail P values, which are both based on the same null hypothesis that the two populations are identical, and any
observed difference is due to chance.
A two-tail P value determines the likelihood of randomly selecting samples that have means as far apart or further than what was observed in the experiment, with either group having the larger mean,
assuming the null hypothesis is true.
On the other hand, a one-tail P value requires the prediction of which group will have the larger mean before data collection. It determines the likelihood of randomly selecting samples with means as
far apart or further than what was observed in the experiment, with the specified group having the larger mean, assuming the null hypothesis is true. A one-tail P value should only be used when
previous data or physical limitations suggest that a difference, if any, can only occur in one direction.
However, it is usually better to use a two-tail P value for several reasons. The relationship between P values and confidence intervals is easier to understand with two-tail P values. Some tests
involve three or more groups, making the concept of tails irrelevant, and a two-tail P value is consistent with the P values reported by these tests.
Choosing a one-tail P value can lead to a dilemma if the observed difference is in the opposite direction of the experimental hypothesis. To be rigorous, one must conclude that the difference is due
to chance, even if the difference is significant. Therefore, it is advisable to always use two-tail P values to avoid this situation.
Hypothesis testing and statistical significance
Statistical hypothesis testing
Statistical hypothesis testing is a method used to make decisions based on data. It is commonly used to determine whether there is a significant difference between two groups. The process involves
setting a threshold P value before conducting the experiment, which is usually set to 0.05.
The null hypothesis is defined as the assumption that there is no difference between the two groups being compared. The alternative hypothesis is the opposite of the null hypothesis and represents
the hypothesis being tested.
Once the null and alternative hypotheses are defined, a statistical test is performed to calculate the P value. If the P value is less than the threshold value, it is concluded that the null
hypothesis is rejected, and the difference between the two groups is statistically significant. If the P value is greater than the threshold value, the null hypothesis cannot be rejected, and it is
concluded that there is no significant difference between the two groups.
It is important to note that failing to reject the null hypothesis does not necessarily mean that the null hypothesis is true. It simply means that there is not enough evidence to reject it.
Hypothesis testing should be used in conjunction with other statistical methods to draw conclusions about the data.
Statistical significance in science
The term “significant” is often misleading and can be misinterpreted. In statistical analysis, a result is considered statistically significant if it would occur less than 5% of the time assuming
that the populations being compared are identical (using a threshold of alpha=0.05). However, just because a result is statistically significant does not necessarily mean that it is biologically or
clinically meaningful or noteworthy. Similarly, a result that is not statistically significant in one experiment may still be important.
If a result is statistically significant, there are two possible explanations:
1. The populations are identical, and any observed difference is due to chance. This is called a Type I error, and it occurs when a statistically significant result is obtained even though there is
no actual difference between the populations being compared. If a significance level of P<0.05 is used, then this error will occur in approximately 5% of experiments where there is no difference.
2. The populations are truly different, and the observed difference is not due to chance. This difference could be significant in the scientific context, or it could be trivial.
It is important to keep in mind that statistical significance does not necessarily imply biological or clinical significance, and that further investigation may be required to fully understand the
implications of a particular result.
“Extremely significant” results
In strict statistical terms, it is incorrect to think that P=0.0001 is more significant than P=0.04. Once a threshold P value for statistical significance is established, each result is categorized
as either statistically significant or not statistically significant. There are no degrees of statistical significance. Some statisticians strongly adhere to this definition.
However, many scientists are less rigid and describe results as “almost significant,” “very significant,” or “extremely significant.” Prism software summarizes P values using words in the middle
column of a table, and some scientists use symbols from the third column to label graphs. These definitions are not entirely standardized, so if you choose to report results in this way, you should
provide a clear definition of the symbols in your figure legend.
Report the actual P value
The concept of statistical hypothesis testing is useful in quality control, where a definite accept/reject decision must be made based on a single analysis. However, experimental science is more
complex, and decisions should not be based solely on a significant/not significant threshold. Instead, it is important to report exact P values, which can be interpreted in the context of all
relevant experiments and analyses.
The tradition of reporting only whether the P value is less than or greater than 0.05 was necessitated by the lack of easy access to computers and the need for statistical tables. However, with the
widespread availability of computing power, it is now simple to calculate exact P values and there is no need to limit reporting to a threshold value. | {"url":"http://prizm.studio/knowledgebase/p-values-and-statistical-hypothesis-testing/","timestamp":"2024-11-13T00:06:28Z","content_type":"text/html","content_length":"98795","record_id":"<urn:uuid:d827b1a9-fe1c-4792-8307-e81ffdf77f98>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00670.warc.gz"} |
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The Accelerated Successive Substitution Method (ASSM)
When the system is close to the critical point and fugacities are strongly composition-dependent, a slowing-down of the convergence rate of the SSM (Successive Substitution Method) is to be expected.
In an attempt to avoid slow convergence problems, some methods have been proposed. Among the most popular are the Minimum Variable Newton Raphson (MVNR) Method and the Accelerated and Stabilized
Successive Substitution Method (ASSM).
The ASSM is basically an accelerated version of the SSM procedure, and thus follows a similar theory. Such procedure is implemented to accelerate the calculation of K[i]-values, especially in the
region close to critical point where the use of the SSM alone will not be efficient. The ASSM technique was presented by Rinses et al. (1981) and consists of the following steps:
1. Use the SSM technique to initiate the updating of the K[i]-values the first time.
2. Check all following criteria at every step during iterations using the SSM:
$\frac{\sum _{i}^{nc}{\left(R{r}_{i}^{new}-1\right)}^{2}}{\sum _{i}^{nc}{\left(R{r}_{i}^{old}-1\right)}^{2}}>0.8$
$a{g}^{new}-a{g}^{old}|\text{ }<0.1$
${10}^{-5}<\sum _{l}^{nc}\left(R{r}_{i}^{new}-1{\right)}^{2}<{10}^{-3}$
These criteria show that you have sufficient proximity to the conditions to ensure the efficiency of the method. Rr[i] is the ratio of liquid fugacity to gas fugacity of the i-th component and ‘$
{\alpha }_{g}$’ is molar gas fraction of the two-phase system.
3. If the system satisfies ALL above criteria, the iteration technique is then switched from the SSM to the ASSM. Otherwise, SSM is used for the update of the K[i]-values. The following expressions
are used to update K[i]-values in ASSM:
${K}_{i}^{new}={K}_{i}^{old}R{r}_{i}^{{\lambda }_{i}}$
where ${\lambda }_{i}=\left[\left(R{r}_{i}^{old}-1\right)/\left(R{r}_{i}^{old}-R{r}_{i}^{new}\right)\right]$
In some cases, using a constant acceleration value of ${\lambda }_{i}=2$ is good enough.
4. Once all the criteria in step (2) are satisfied, skip step (2) for the subsequent iterations and use the ASSM technique to update K[i]-values until convergence is attained, unless it does not
give acceptable ^new estimates (as stated next).
5. When ASSM is used, it must always be tested to show that it leads to an improved solution (i.e., that it brings fugacity ratios closer to unity). If not, it must be rejected and switched back to
Even though we are outlining Risnes et al.’s version of the accelerated Successive Substitution Method, there are several other published algorithms whose main purpose have also been to accelerate
the successive substitution method. Fussel and Yanosik (1978), Michelsen (1982), and Mehra et al. (1983) are examples of such attempts. Risnes et al. version is the easiest and most straightforward
to implement, but it is subjected to limitations. | {"url":"https://www.e-education.psu.edu/png520/m17_p6.html","timestamp":"2024-11-12T05:56:30Z","content_type":"text/html","content_length":"42136","record_id":"<urn:uuid:7d45d72c-ec30-4fd2-8057-be8a50457768>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00400.warc.gz"} |
Math Study Guide for the NLN NEX | Page 1
Math Study Guide for the NLN NEX
General Information
The Math section of the National League for Nursing (NLN) Pre-Admission Examination (PAX) covers a variety of topics in mathematics, but generally, only the basics are covered. There are 40 questions
, and you will have 40 minutes in which to complete them. This means you will have about one minute per question.
All questions on the NLN PAX are typical multiple-choice questions with four answer choices.
Basic Number Concepts
These basic number concepts are the foundation upon which mathematical proficiency is built. They are fundamental principles that provide the framework for more advanced mathematical operations and
problem-solving skills. The concepts discussed in this section encompass the understanding of numbers, their relationships, and their application in various real-world scenarios.
Place Value
Place value describes the value of the digits in a multi-digit number with respect to the position of the digit in the number. Reading large numbers becomes manageable when we understand the place
value system. For instance, in the number \(8\text{,}529\), the \(8\) represents thousands, the \(5\) represents hundreds, the \(2\) represents tens, and the \(9\) represents units or ones. The
number \(8\text{,}529\) is read “eight thousand five hundred twenty-nine.” To visualize and comprehend the place value of each digit, a place value chart for whole numbers can be immensely useful.
Below is a place value chart:
This type of chart organizes digits into columns, such as billions, millions, thousands, and ones, providing a clear structure for understanding the value of a given number. In this chart the first
digit is in the ones place, the next digit (to the left) is in the tens place, and so on.
Operations with Numbers
Understanding and performing operations with numbers are foundational skills with broad applications in daily life and academic pursuits. These operations, including addition, subtraction,
multiplication, and division, are not only about solving mathematical problems but are also essential for making informed decisions in various real-world scenarios.
Addition is the process of combining two or more numbers. Below is a simple addition problem:
Here, the numbers being added are called addends. The result of the operation is known as the sum. The operator symbol for addition is a plus sign (\(+\)).
Subtraction is the process of finding the difference between two or more numbers. Below is a simple subtraction problem:
The number being subtracted is called the subtrahend. The number from which the subtrahend is being subtracted (usually the bigger one) is called the minuend. The result is known as the difference.
The operator symbol for subtraction is a minus sign (\(-\)).
Multiplication is used to find the product of two or more numbers. It is a process of repeated addition. Below is a simple multiplication problem:
The first number is called the multiplicand. The second number being multiplied by the multiplicand is called the multiplier. The result is known as the product. We can also call both the numbers
being multiplied factors. The operator symbol for multiplication is a times sign (\(\times\)).
Division is a mathematical operation that represents the process of distributing (or partitioning) a quantity into equal parts. Below is a simple division problem:
The number being divided is called the dividend. The number doing the division is called the divisor. The result of the operation is known as the quotient. The operator symbol for division is the
division sign (\(\div\)).
Rounding is a mathematical technique that is employed to simplify numerical values while maintaining a reasonable level of accuracy. This is particularly useful when precise figures are not required
and a general estimate is sufficient for practical purposes.
These are the steps for rounding:
• Identify and determine which digit (place value) you want to round a number to. We can refer to this as the “rounding digit.”
• Look at the digit immediately to the right of the rounding digit.
• If the digit to the right is \(5\) or greater, round the rounding digit up by adding \(1\) to it. If the digit to the right is less than \(5\), keep the rounding digit unchanged.
• Make all the numbers to the right of the rounding digit \(0\).
Here is an example:
Round the number \(51\text{,}783\) to the thousands place.
The rounding digit would be \(1\).
The digit to the right of the rounding digit is \(7\), so the \(1\) will be rounded up.
The rounded number will be \(52\text{,}000\).
Divisibility plays a key role in simplifying numbers and identifying patterns within them. This concept is important for simplifying fractions and determining the factors of a given number.
Divisibility allows us to break down complex numerical relationships into more manageable components.
Prime and Composite Numbers
Numbers can be classified into two main categories: prime and composite.
Prime numbers are integers greater than \(1\) that have only two distinct positive divisors, \(1\) and the number itself. Examples include \(2\), \(3\), and \(5\). Note that \(2\) is the only even
prime number.
On the other hand, composite numbers have more than two distinct positive divisors, making them divisible by numbers other than \(1\) and themselves. Examples include \(4\), \(6\), and \(9\).
Prime Factorization
When you need to find all of a number’s prime factors, there is a simple method to determine all of them. Simply start with the number and break it down, one prime factor at a time, continuing until
there are only prime numbers left. Here is how that might look:
The prime factors of \(36\) are \(3\) and \(2\). There are two of each, so we could express this as \(36=3^2 \times 2^2\).
Divisibility Rules
If a number, \(x\), goes evenly into another number, \(y\), when divided, with no remainder, we say that \(y\) is divisible by \(x\). To check whether a number is divisible by a certain number, there
are certain shortcuts/rules for common numbers. They are discussed below:
• divisibility by \(2\)—If the last digit of a number is even, then that number is divisible by \(2\). For example, \(48\) is divisible by \(2\) since \(8\) is an even number.
• divisibility by \(3\)—A number is divisible by \(3\) if the sum of the digits in the number is divisible by \(3\). For example, \(93\) is divisible by \(3\) since the sum of the digits of the
number is \(9 + 3 =12\), which is a number that is divisible by \(3\). (Notice that \(12\) is also divisible by \(3\) because \(1+2=3\).)
• divisibility by \(5\)—If the last digit of a number is \(0\) or \(5\), then the number is divisible by \(5\). For example, the numbers \(50\) and \(125\) are both divisible by \(5\).
• divisibility by \(6\)—A number is divisible by \(6\) if it is divisible by both \(2\) and \(3\). For instance, the number \(18\) is divisible by both \(2\) and \(3\). Consequently, we can say
that it is also divisible by \(6\) (\(18 \div 6 = 3\)).
• divisibility by \(9\)—A number is divisible by \(9\) if the sum of the digits in the number is \(9\) (or divisible by \(9\)). For example, \(27\) is divisible by \(9\) since the sum of the digits
of the number is \(2 + 7 =9\). Likewise, \(369\) is divisible by \(9\) because \(3+6+9=18\) and \(18\) is \(1+8=9\).
• divisibility by \(10\)—If a number ends with a \(0\), it is divisible by \(10\). The numbers \(20\), \(90\), and \(18\text{,}760\) are all divisible by \(10\).
• divisibility by \(12\)—A number is divisible by \(12\) if it is divisible by both \(3\) and \(4\). For instance, the number \(48\) is divisible by both \(3\) and \(4\). Consequently, we know that
it is also divisible by \(12\) (\(48 \div 12 = 4\)).
The average of a set of numbers is the sum of the numbers divided by the number of values in the set. This statistical measure provides a single value around which all the other values of the data
set are centered. If your last four test scores are \(81\), \(92\), \(93\), and \(88\), then your average score is:
\[\text{average} = \frac{\text{sum}}{\text{number of values}} = \frac{81+92+93+88}{4} = \frac{354}{4} = 88.5\]
The average is also known as the mean.
Exponents represent the number of times a base is multiplied by itself. For example, in the expression \(2^4 = 16\), the number \(2\) is the base and the number \(4\) is the exponent (or power).
The square of a number is when that number is multiplied by itself. For example, three squared is \(3^2 = 3 \times 3 = 9\). The cube of a number is when that number is multiplied by itself two times.
For example, two cubed is \(2^3 = 2 \times 2 \times 2 = 8\). For expediting mental calculations, you should be familiar with the first \(10\) squares and the first six cubes, shown below:
While exponents involve repeated multiplication, roots are the inverse operation, by which we find the original value that was repeatedly multiplied. The most common roots are the square root (the
symbol is \(\sqrt{}\)) and the cube root (the symbol is \(\sqrt[3]{}\)).
The square root of a number is a value that, when multiplied by itself, gives the original number. For example, \(\sqrt{36} = 6\) since \(6\times 6 = 36\).
The cube root of a number is a value that, when multiplied by itself twice, results in the original number. For example, \(\sqrt[3]{8} = 2\) since \(2 \times 2 \times 2 = 8\). You should know the
following common square roots and cube roots for faster mathematical calculations:
Note: A square root can be symbolized by \(\sqrt[2]{}\), but \(\sqrt{}\) means the same thing and is more commonly used.
All Study Guides for the NLN NEX are now available as downloadable PDFs | {"url":"https://uniontestprep.com/nln-nex/study-guide/math/pages/1","timestamp":"2024-11-10T15:59:30Z","content_type":"text/html","content_length":"52309","record_id":"<urn:uuid:5050ed53-4c68-4a5d-98ff-23d019e50b12>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00623.warc.gz"} |
Optimizing and Monitoring a Trading System with Quantiacs
• news-quantiacs
last edited by support
*This article was published on Medium: check it out here.
In this article we describe the implementation of a new tool we released for the Quantiacs Python toolbox: a fast optimizer for testing the robustness of a trading system.*
As Donald Knuth pointed out, “Premature optimization is the root of all evil.” This is a famous saying among software developers and it is true also in the context of trading system development.
Photo by Jefferson Santos on Unsplash
It is very tempting to write an algorithm, a simple trend-following strategy or a more refined machine-learning based system, and then to search for optimal parameters or hyperparameters which
are going to maximize a given score function, normally the Sharpe ratio.
As optimal parameters are maximizing the chosen score function in the past, and financial data are very noisy, the usual result of parameter optimization is overfitting: the trading system works
so well with past data that it becomes useless for predicting future market moves.
The problem of inflating the simulated performance of a trading system extends beyond backtest optimizers: for example, developers tend to focus and report only on positive outcomes out of all
the models they try, an issue known as selection bias. For a detailed description of these problems we refer to the 2014 article by Marcos Lopez de Prado.
Nevertheless, optimizers rely on a basic functionality which can be used for testing the robustness of a trading system: a grid scan of the system over possible combinations of the parameters.
The results of the scan can be used to visualize and test how much the performance of the trading system is sensitive to the parameter choice. A robust system will have a good Sharpe ratio for a
wide range of the independent parameters.
Trading System Optimization with Quantiacs
You can find our optimizer in the Examples section of the Development area of your account:
For running the example, simply click on the Clone button and work in your favourite environment, Jupyter Notebook or JupyterLab. Alternatively you can download locally the Quantiacs toolbox and
take advantage of parallelization on your own machine.
Let us analyze the code. First of all we import the needed libraries:
import qnt.data as qndata
import qnt.ta as qnta
import qnt.output as qnout
import qnt.stats as qns
import qnt.log as qnlog
import qnt.optimizer as qnop
import qnt.backtester as qnbt
import xarray as xr
In addition to the Quantiacs library, freely available at our GitHub page, we import xarray for quick processing of multi-dimensional data structures.
Next we define a simple trading rule based on two parameters. The strategy is going long only when the rate of change in the last "roc_period" trading days (in this case 10) of the
linear-weighted moving average over the last "wma_period" trading days (in this case 20) is positive:
def single_pass_strategy(data, wma_period=20, roc_period=10):
wma = qnta.lwma(data.sel(field='close'), wma_period)
sroc = qnta.roc(wma, roc_period)
weights = xr.where(sroc > 0, 1, 0)
weights = weights / len(data.asset)
with qnlog.Settings(info=False, err=False):
weights = qnout.clean(weights, data, debug=False)
return weights
The strategy returns allocation weights (fraction of capital to be invested) for every day. We check the performance of this strategy with the chosen parameters:
data = qndata.futures.load_data(min_date='2004-01-01')
single_pass_output = single_pass_strategy(data)
single_pass_stat = qns.calc_stat(data,\
single_pass_output.sel(time=slice('2006-01-01', None)))
The code returns the values of the relevant statistical indicators, including the Sharpe ratio, in a table, since beginning of the in-sample period:
Next we run the optimization code, which performs a scan over a predefined range of parameter values: the user can choose for each parameter the starting value, the final one and the step:
data = qndata.futures.load_data(min_date='2004-01-01')
result = qnop.optimize_strategy(
wma_period=range(10, 150, 5), # min, max, step
roc_period=range(5, 100, 5) # min, max, step
workers=1 # you can set more workers on your PC
print("Best iteration:")
The code returns an interactive plot where one can analyze the dependence of the key statistical indicators on the parameters. In this plot we display the two independent parameters on the x and
y axis, the Sharpe ratio value on he z axis and use different colors according to the maximum drawdown.
As a reference, we display the optimal values of the parameters which maximize the Sharpe ratio (beware of overfitting!):
A robust trading system will have a smooth dependence of the Sharpe ratio on the parameter values, ideally with good values larger than 1 for a wide choice of the parameters. An overfitted system
will display typically a Sharpe ratio peak for a particular choice of the independent parameters.
Do you have questions or suggestions? Post them in our Forum or write us to info@quantiacs.com!
You can try the optimizer on Quantiacs. The code can be found in the Examples section of the Development area of your account. Or you can download the full code from our GitHub repository. | {"url":"https://quantiacs.com/community/topic/29/optimizing-and-monitoring-a-trading-system-with-quantiacs","timestamp":"2024-11-13T06:27:29Z","content_type":"text/html","content_length":"73727","record_id":"<urn:uuid:5e069cb7-6b47-46b0-af43-ca406b61f57b>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00485.warc.gz"} |
[Plugin] extrudeEdgesByEdges.rb
[Plugin] extrudeEdgesByEdges.rb
Copyright 2009/2010 (c), TIG
All Rights Reserved.
THIS SOFTWARE IS PROVIDED "AS IS" AND WITHOUT ANY EXPRESS OR IMPLIED
Extrudes two sets of grouped edges into a faced mesh...
Usage: Make two sets of edges [from lines, arcs, curves etc].
These represent the 'profile' and the 'path' for the mesh.
Make a group of each set.
Note: If the groups share a common vertex then that fixes the
new mesh's location, otherwise the nearest vertices are used,
with the new mesh located near the profile, Move it as required...
Now Select these 2 groups.
Run the Plugin: 'Extrude Edges by Edges'.
It makes a grouped faced 'mesh' from these two edge-sets.
The progress at each stage is reported along the status bar.
When the mesh is made the view zooms to include the original
profile/path groups and the new mesh group.
Then there are dialogs asking for Yes/No replies...
If you want to 'orientate' the mesh-faces (which may not always
be necessary: if it's chosen then it will be done as well as
possible for any convoluted shapes.
If you want to 'reverse' the mesh-faces.
If you want the mesh-faces to 'intersect' with themselves
(This is only necessary if the mesh has convoluted re-entrant
surfaces). Intersecting the mesh might compromise some later
If you want to remove any 'coplanar edges'.
If you want to 'triangulate' the new faces:
on very complex inter-penetrating meshes a triangulating error
message might appear - answer 'Yes' to undo triangulation,
'No' to keep what's been done so far...
Note that the triangulation 'undo' is separate within the main
action's 'undo'.
Finally, if you want to delete the original two groups.
Large numbers of edges in the groups increase the new faces and
other operations exponentially, therefore only extrude the parts
that can be copied/exploded together later...
For example:
2 edges x 2 edges >>> group with 4 faces & 12 edges
4 edges x 4 edges >>> group with 16 faces & 40 edges
8 edges x 8 edges >>> group with 64 faces & 144 edges
16 edges x 16 edges >>> group with 256 faces & 544 edges
32 edges x 32 edges >>> group with 1024 faces & 2112 edges
Very large groups will eventually be made but the screen can
'white out' and the 'counter' might appear to stop changing for
several minutes... It is working.
Use 'Smooth' and/or 'Show Hidden Geometry' on the mesh-group,
also 'Sandbox flip-edge' tool to re-trianglate, as desired...
Rarely some combinations of edge groups might go into a 'loop'
and then SUp needs 'killing' - so save first !
Are welcome [by PayPal], please use 'TIGdonations.htm' in the
../Plugins/TIGtools/ folder.
1.0 20090622 First 'beta' release.
1.1 20090625 Speed improvements - face making time ~halved,
typename >> kind_of?, triangulation glitch trapped
and orientation improved.
1.2 20090625 Orientation speed optimised.
Glitch on groups erase fixed.
1.3 20090626 Edges not facing in convoluted shapes trapped.
1.4 20090707 Triangulation improved. Rare intersect glitch fixed.
1.5 20090708 Zooms to show new group.
1.6 20090708 Zooms to new group fixed for large models.
1.7 20090709 Coplanar edge erasure errors trapped:
0.999999 made 0.99999999 !!!
1.8 20090808 Orienting and Triangulation speeds improved.
2.0 20100114 Debabelized, 'Extrusion Tools' Toolbar added.
2.1 20100120 Typo in lingvo translation fixed.
2.2 20100120 Lingvo files updated - thanks FR=Pilou, ES=Defisto
2.3 20100121 Typo preventing Plugin Menu item working corrected.
2.4 20100123 FR lingvo file updated by Pilou.
2.5 20100124 Menu typo glitch fixed.
2.6 20100216 Now in own sub-menu 'Extrusion Tools...' in Plugins menu.
2.7 20100222 Tooltips now deBabelize properly.
2.8 20100222 Tooltips etc now deBabelized properly.
2.9 20100428 Tool now exits gracefully.
3.0 20100517 ES lingvo adjusted by Defisto.
NOTE: from 20100212 the latest versions of these files is in the zipped set downloaded from here http://forums.sketchucation.com/viewtopic.php?p=217663#p217663
Question, TIG: do the two groups of curves need to be on perpendicular planes?
Awesome TIG! You'll get lots of people excited about this script
@gaieus said:
Question, TIG: do the two groups of curves need to be on perpendicular planes?
No - but it's best if there's some non-coplanar-ness [if that's a word !]. If they are at some angle to each other you will get a 3D mesh. If they are coplanar you'll get a weird 2D mess... You
can also have several bits of lines/curves - they don't need to be continuous - just grouped together as the required 'profile' and 'path'... The lines/curves can also be complex in 3D but the
resulting intersecting mess might not be what you are expecting or want...
from what I see, this is awesome. The thing I need to model classical decorations..
OK TIG, that's what I meant. In fact, I'd probably not need it for coplanar curves (I can't really imagine a situation like that when it's needed butof course, who knows) but definitely something
other than perpendicular can be handy.
Even better this "freedom" of the different elements in the groups (I guess that's why they are grouped but I wou;ldn't understand the answer anyway
@unknownuser said:
@unknownuser said:
Tip: single lines and arcs won't group, so draw another edge,
then group everything, edit the group and erase the temporary
Seems it's not necessary!
Just go Menu Edit / Group for each object!
Then launch the plug
I find that single lines or arcs won't group, but drawing another temporary line with it then grouping, then editing the group and deleting the temporary line seems to work to give a single arc
group... It's because I use ashortcut key that isn't available on a single selection, BUT the Menu Edit >> Make-Group item is still OK... So if you can group single arcs via the menu without a
hitch then that's great...
Any ideas on improvements or clever uses welcomed...
@unknownuser said:
Tip: single lines and arcs won't group, so draw another edge,
then group everything, edit the group and erase the temporary
Seems it's not necessary!
Just go Menu Edit / Make Group for each object!
Then launch the plug
Works like a charm! Bravo!
Here with 2 arcs without "another edge"!
Next steps will be with 4 curves closed so Coons surfaces
WOW This is like Rhino Sweep alone rail. Very cool!
Thanks TIG..
I noticed it actually does not follow and rotate along the path. Although interesting proposition, and it works without any problems, it would be usefull if it coud also work in manner that can
be predictable in design terms.
I pushed it with a lot of geometry and it seems very stable. It performs orienting and reversing faces without problem. Triangulation did not make any problems as well.
One can add a point too and then group.
Killer script.
It doesn't matter what's in the groups - only edges get processed...
@sepo said:
Thanks TIG... I noticed it actually does not follow and rotate along the path. Although interesting proposition, and it works without any problems, it would be usefull if it coud also work in
manner that can be predictable in design terms.
This tool extrudes edges by edges without rotation. If you want rotation use followme tool or followme_and_keep script: you'll need to make the profile faced first by adding a back set of
lines - you also need to unhide/unsmooth all edges to get a 'mesh'... The advantage of extrudeEdgesByEdges is that the profile and path can be in complex in 3D and also not continuous...
This was the subject of one of my first wishful posts. "Extrude" a curve along a curve.
Excellent work!!!! This is another one of those jaw droppers. We're gonna have you bronzed. Plotparis will jump out of his seat when he sees this.
Thank you.
This also works with the freehand tool in SU6, if anybody was wondering.
Really great tool, TIG!!!! | {"url":"https://community.sketchucation.com/topic/113071/plugin-extrudeedgesbyedges-rb","timestamp":"2024-11-03T19:13:59Z","content_type":"text/html","content_length":"186416","record_id":"<urn:uuid:4914f466-0c94-459c-9ec1-cb3b217a3d3b>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00771.warc.gz"} |
Pharmacokinetics of IV Drugs in context of infusion rate
31 Aug 2024
Title: The Impact of Infusion Rate on the Pharmacokinetics of Intravenous Drugs: A Review
The pharmacokinetics of intravenously administered drugs is a critical aspect of their therapeutic efficacy and safety profile. The infusion rate at which these drugs are delivered can significantly
impact their absorption, distribution, metabolism, and excretion (ADME) characteristics. This review aims to provide an overview of the effects of infusion rate on the pharmacokinetics of IV drugs,
with a focus on the underlying mathematical principles.
Intravenous administration is a common route for delivering medications, particularly those that require rapid onset of action or have narrow therapeutic windows. The infusion rate at which these
drugs are delivered can influence their ADME characteristics, affecting both efficacy and toxicity. Understanding the pharmacokinetics of IV drugs in relation to infusion rate is essential for
optimizing treatment outcomes.
Pharmacokinetic Parameters:
The pharmacokinetics of IV drugs can be described using several key parameters:
• Clearance (Cl): The volume of plasma from which a drug is completely removed per unit time.
• Volume of Distribution (Vd): The theoretical volume that would contain the entire amount of an administered drug at the same concentration as it exists in the body.
• Elimination Rate Constant (Kel): A measure of the rate at which a drug is eliminated from the body.
Infusion Rate and Pharmacokinetics:
The infusion rate can affect the pharmacokinetic parameters of IV drugs, leading to changes in their ADME characteristics. The relationship between infusion rate and pharmacokinetics can be described
using the following formula:
Cl = (Dose / Infusion Time)
where Cl is clearance, Dose is the administered dose, and Infusion Time is the duration over which the drug is infused.
The volume of distribution (Vd) is a critical parameter that influences the pharmacokinetics of IV drugs. The Vd can be affected by the infusion rate, leading to changes in the drug’s distribution
characteristics. The relationship between infusion rate and Vd can be described using the following formula:
Vd = (Dose / Concentration)
where Dose is the administered dose, and Concentration is the plasma concentration of the drug.
The metabolism of IV drugs can also be influenced by the infusion rate. The elimination rate constant (Kel) is a measure of the rate at which a drug is metabolized and eliminated from the body. The
relationship between infusion rate and Kel can be described using the following formula:
Kel = (ln(Concentration / Initial Concentration)) / Infusion Time
where ln is the natural logarithm, Concentration is the plasma concentration of the drug, Initial Concentration is the initial plasma concentration of the drug, and Infusion Time is the duration over
which the drug is infused.
The infusion rate at which IV drugs are delivered can significantly impact their pharmacokinetics, affecting both efficacy and toxicity. Understanding the relationships between infusion rate and
pharmacokinetic parameters such as clearance, volume of distribution, and elimination rate constant is essential for optimizing treatment outcomes.
Related articles for ‘infusion rate’ :
• Reading: Pharmacokinetics of IV Drugs in context of infusion rate
Calculators for ‘infusion rate’ | {"url":"https://blog.truegeometry.com/tutorials/education/3562b55eae70c1157ecd0ed0ec6a1c4e/JSON_TO_ARTCL_Pharmacokinetics_of_IV_Drugs_in_context_of_infusion_rate.html","timestamp":"2024-11-06T08:46:22Z","content_type":"text/html","content_length":"16796","record_id":"<urn:uuid:481e8783-2d8b-4a09-b321-c234d08fa841>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00261.warc.gz"} |
4.3.2 FFT Filter
Origin offers an FFT filter, which performs filtering by using Fourier transforms to analyze the frequency components in the input dataset.
There are six types of filters available in the FFT filter function: low-pass, high-pass, band-pass, band-block, threshold and low-pass parabolic. Low-pass filters block all frequency components
above the cutoff frequency, allowing only the low frequency components to pass. High-pass filters work in the opposite way: they block frequency components that are below the cutoff frequency.
This tutorial will show you how to perform the low-pass and band-pass filtering using Origin's FFT filter.
What you will learn
This tutorial will show you how to:
• Perform low pass filtering.
• Perform band pass filtering.
Low-pass Filter
1. Start with an empty workbook. Select Help: Open Folder: Sample Folder... to open the "Samples" folder. In this folder, open the Signal Processing subfolder and find the file Origin 8 Message.wav.
Drag-and-drop this file into the empty worksheet to import it.
2. Highlight column A(Y) and click the Line button on the 2D Graph toolbar to create a line plot.
3. This signal is a sound wave and it is already known that the high frequency components can be regarded as noise, and should be blocked. So we will use the Low Pass method in the FFT Filter tool
to approximate the low frequency component for further analysis.
4. Make sure the line plot is active, then select Analysis:Signal Processing:FFT Filters to open the fft_filters dialog box.
5. Make sure the Filter Type is set to Low Pass.
6. Check the Auto Preview box to turn on the Preview panel:
The top two images show the signal in the time domain, while the bottom image shows the signal in the frequency domain after Fast Fourier Transform. The X position of the red vertical dot
line indicates the cutoff frequency. By moving the vertical line horizontally, you can preview the comparison between the original signal and the filtered signal in real time, in the top part
of this panel.
7. Move the vertical line to the X position of the peak amplitude (as in the image below). Note that human error may be introduced during this step, but it is acceptable since we only want to
roughly filter the signal.
8. Click OK to apply the FFT filter to the original signal.
9. The signal after filtering will be added to the data plot of original signal. Select Graph:Speed Mode and turn off speed mode in this graph. The resulting graph should look like this:
10. In the resulting graph, we can see that the high frequency components are blocked by the Low Pass FFT filter.
Band-pass Filter
1. Start with a new workbook. Select Help: Open Folder: Sample Folder... to open the "Samples" folder. In this folder, open the Signal Processing subfolder and find the file fftfilter3.dat.
Drag-and-drop this file into the empty worksheet to import it.
2. Highlight column B and click the Line button on the 2D Graphs toolbar to generate a line plot.
3. With the graph active, select Analysis:Signal Processing:FFT Filters. This opens the fft_filters dialog box.
4. Check the Auto Preview box to enable the Preview panel.
5. From the plot of frequency domain (the image below), we can see that this signal has components at several different frequencies> Now we are going to get the component at about 300Hz. So we will
use the Band Pass method.
6. Set the Filter Type to be Band Pass.
When Band Pass is chosen, there will be two vertical red lines in the preview panel, marking the Lower Cutoff Frequency and Upper Cutoff Frequency. You can similarly move these two lines and
get the real time preview of filtering results in the top parts of this panel.
7. Enter the values of the lower and upper cutoff frequencies according to the image below:
8. Click OK to execute filtering.
9. We obtain the components at a frequency of around 300 Hz after filtering. | {"url":"http://cloud.originlab.com/doc/en/Tutorials/FFT-Filter","timestamp":"2024-11-10T08:22:12Z","content_type":"text/html","content_length":"122210","record_id":"<urn:uuid:d64b0edf-644d-4c7d-b3c9-d5aa459802d3>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00499.warc.gz"} |
Thermodynamics Problems and Solutions
A student has just purchased a coffee. The cup has $0.3\ {\rm kg}$ of coffee at $70{\rm{}^\circ\,\!C}$. She adds $0.02\ {\rm kg}$ cream at $5{\rm{}^\circ\,\!C}$. In what follows assume coffee and
cream have the same thermal properties as water. ($c_w=4186\frac{{\rm J}}{{\rm kg.}{\rm{}^\circ\,\!C}}\ ,\ L_V=2.26\times {10}^6\frac{{\rm J}}{{\rm kg}}\ ,\ \ L_f=33.5\times {10}^4\frac{{\rm J}}{{\
rm kg}}\ ,\ \rho=1\ {\rm g/}{{\rm cm}}^{{\rm 3}}$)
(a) What is the temperature of the coffee after adding cream? assume no heat is lost to the outside world at this point.
(b) For this process, is the entropy change of the universe less than, greater than, or equal to zero?
(c) In 1 minute, the coffee cools by $5{\rm{}^\circ\,\!C}$. In this part only, assume heat loss is entirely due to evaporation. What is the mass of the coffee lost in this 1 minute? You can also
ignore the cream. | {"url":"https://physexams.com/exam/Thermodynamics_6","timestamp":"2024-11-13T02:07:59Z","content_type":"text/html","content_length":"323364","record_id":"<urn:uuid:43230779-3b9d-4048-91f0-33834a24049b>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00740.warc.gz"} |
Linear Models and Linear Relationship
Update 3/20/2015: sharing a blog post from Freakonometrics
On Some Alternatives to Regression Models
In many papers, we can find the statements similar to the one below:
Because linear models can hardly capture the nonlinear relationship between load and temperature, we use Artificial Neural Networks (or other black-box models) in this paper.
The major conceptual error of the above statement is due to a common misunderstanding that linear models
capture nonlinear relationship.
I'm showing a nonlinear curve in the figure below, which is from a 3rd order polynomial function.
Is this a linear model? Yes! Polynomial regression models in general are linear models.
The "linear" in linear models refers to the equations we use to solve the parameters. By definition, a regression model is linear if it can be written as
, where
is a vector of values of the response variable;
is a vector of parameters to be solved;
is a matrix of values of explanatory variables;
is a vector of independent normally distributed errors.
Maybe the above description is too abstract. Let's check out an example of parameter estimation for a 3rd order polynomial regression model:
y = b0 + b1x1 + b2x2 + b3x3 + e,
where x2 is the square of x1, and x3 is the cube of x1. There are 4 parameters to be estimated. Now let's say we have 6 observations as shown in the table below.
Then we can come up with 6 linear equations:
3 = b0 + b1 + b2 + b3;
1 = b0 + 2b1 + 4b2 + 8b3;
4 = b0 + 3b1 + 9b2 + 27b3;
9 = b0 + 4b1 + 16b2 + 64b3;
5 = b0 + b1 + b2 + b3;
2 = b0 + 3b1 + 9b2 + 27b3;
These equations can be written in the following form:
Therefore, a 3rd order polynomial regression mode is a linear model. By solving the above equations, we can obtain the values of
b0, b1, b2 and b3
To further clarify the concept of linear vs. nonlinear, here are a few examples of nonlinear regression models:
y = b0 + b1x1/(b2x2 + b3) + e, ...... (1)
y = b0(exp(b1x))U, ...... (2)
Not to make it too complicated, some nonlinear regression problems can be solved in a linear domain. For instance, eq(2) can be transformed to a linear model by taking the logarithm on both side:
Ln(y) = Ln(b0) + b1x + u, .... (3)
Having said all above, there is no problem using linear models for load forecasting. Some good and successful examples of linear regression models can be found from
my PhD dissertation
. The models and methodologies I developed over there are already commercialized and deployed to utilities worldwide.
Back to
Load Forecasting Terminology | {"url":"http://blog.drhongtao.com/2014/10/linear-models-linear-relationship.html","timestamp":"2024-11-14T08:28:55Z","content_type":"text/html","content_length":"65378","record_id":"<urn:uuid:dfa503b6-15d0-4e1e-968d-fc3b339d6fd5>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00885.warc.gz"} |
Square & Square Root of 125 - Methods, Calculation, Formula, How to find
Square & Square Root of 125
Square of 125
125² (125×125) = 15,625
The square number of 125 is the result when you multiply 125 by itself. In mathematical terms, it’s represented as 125² or 125×125. When you calculate it, the square of 125 equals 15,625.
Square Root of 125
√125 = 11.1803398875
√125 = 11.180 up to three places of decimal
The square root of 125 is approximately 11.180. This is an irrational number, meaning it cannot be expressed as a simple fraction, and its decimal representation goes on indefinitely without
Square Root of 125: 11.1803398875
Exponential Form: 125^½ or 125^0.5
Radical Form: √125
Is the Square Root of 125 Rational or Irrational?
The square root of 125 is an irrational number
To understand why, let’s break it down:
An irrational number is a number that cannot be expressed as a simple fraction (i.e., the ratio of two integers) and whose decimal representation goes on infinitely without repeating. When we try to
find the square root of 125, we find that it doesn’t simplify to a whole number or a fraction. The approximate value of the square root of 125 is around 11.1803. This decimal goes on infinitely
without repeating any pattern.
Therefore, because the square root of 125 cannot be expressed as a fraction and its decimal representation is non-repeating and non-terminating, it is considered an irrational number.
Rational numbers are numbers that can be expressed as a fraction, where both the numerator and the denominator are integers, and the denominator is not zero.
Examples: 3/4,5/1
Irrational numbers are numbers that cannot be expressed as a simple fraction, with decimal expansions that are non-terminating and non-repeating. These numbers often arise in geometry and algebra,
especially involving roots of numbers that are not perfect squares or certain ratios like the circumference to the diameter of a circle (π).
Examples: √22 and π.
Methods to Find Value of Root 125
You can start by identifying perfect squares that are close to 125, such as 121 (11² ) and 144 (12²).
Estimate that the square root of 125 is slightly more than 11, as 125 is slightly more than 121.
Long Division Method:
This traditional method involves a step-by-step procedure similar to long division. It helps find more precise values of square roots manually.
The simplest and most accurate way to find the square root of any number is using a calculator. Simply entering “√125” will give the precise value quickly.
Newton’s Method (or Newton-Raphson Method):
This is an iterative numerical method used for finding approximations to the roots (or zeroes) of a real-valued function. For √125 , the function would be f(x)=x²−125.
Start with an initial guess, say x₀=11, and iterate using the formula:
xₙ₊₁ =xₙ − xₙ² − 125/2xₙ
f(x)x² − 125 and its derivative f′(x)=2x.
Square Root of 125 by Long Division Method
1. Pair the Digits: Start from the right and pair the digits. For 125, create the pairs 1 and 25.
2. Find the Largest Square: Divide the left-most pair (1) by the largest number whose square is less than or equal to it. In this case, the largest number is 1.
3. Update the Divisor: Double the divisor from the previous step (making it 2) and write it with a space on its right. Guess the largest digit to fill this space that forms a new divisor. This digit
also becomes part of the quotient. Multiply the new divisor by the new quotient digit to get a product that is less than or equal to the pair of digits being considered. Subtract this product
from the pair to find the remainder.
4. Repeat for Precision: Continue this process, bringing down pairs of zeros if necessary, to calculate more decimal places of the square root.
Is 125 Perfect Square root or Not
125 is not a perfect square
No, A perfect square is a number that can be expressed as the square of an integer. The closest perfect squares to 125 are 121 (11²) and 144 (12²), and since there’s no integer whose square equals
125, it is not a perfect square.
What is the square of 125 in Vedic math’s?
In Vedic Mathematics, squaring numbers close to base values like 10, 100, etc., can be done quite efficiently. Here’s a simpler breakdown for squaring 125 using Vedic math techniques:
1. Difference from Base: Determine how much the number exceeds the base. For 125 and base 100, the difference is +25.
2. Square the Difference: Square the excess, so 252=625.
3. Sum of the Number and the Excess: Add the excess to the original number. 125+25=150.
4. Multiply the Sum by the Base: Multiply the result from the previous step by the base (100). 150×100=15000.
5. Add the Results: Add the squared difference to the multiplication result. 15000+625=15625.
What is a cube root of 125?
The cube root of 125 is 5. This means that 5×5×5=125. The number 125 is a perfect cube because it can be expressed as the cube of an integer. | {"url":"https://www.examples.com/maths/square-and-square-root-of-125.html","timestamp":"2024-11-10T19:28:50Z","content_type":"text/html","content_length":"110952","record_id":"<urn:uuid:d85561e1-b7f7-4957-8d86-e5aec74a97c8>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00042.warc.gz"} |
Most of the material presented in this section can be found in <Cockburn 2009>.
We provide a first characterization of HDG methods for the following second-order elliptic model problem:
\[\begin{align*} \Lambda\mathbf{u} + \nabla p &= \mathbf 0 & &\text{in }\Omega & &(1)\\ \nabla\cdot\mathbf u + d p &= f & &\text{in }\Omega & &(2)\\ p &= h_D & &\text{on }\Gamma_D & &(3)\\ \mathbf u\
cdot\mathbf n &= h_N & &\text{on }\Gamma_N & &(4) \end{align*}\]
Here \(\Omega\subset\mathbb R^n\) is a polyhedral domain \((n\geq 2)\), \(\partial\Omega = \Gamma = \Gamma_D \cup \Gamma_N\), \(d(\mathbf x)\) is a scalar nonnegative function, \(\Lambda(\mathbf x)\)
is a matrix valued function that is symmetric and uniformly positive definite on \(\Omega\), and \(f\in L^2(\Omega)\). If \(d = 0\), we get the Darcy model presented in [darcy-model]. These
assumptions can be generalized but we take them to simplify the discussion.
The Darcy problem presented in [stabilized-formulations] fits into this framework. Here, rather than using stabilized Galerkin formulations, the idea is to insert a Lagrange multiplier to handle the
continuity of the normal components of the approximated flux \(\mathbf{u}_h\). In other words, the requirement \(\mathbf{u}_h\in H(\text{div},\Omega)\) will be written in weak form initially, and
then recovered in strong form later. The introduction of a Lagrange multipliers leads to a formulation with three fields <Boffi 2013>. We will show how the interior fields can be eliminated in order
to build a discrete system that has lost the saddle point structure and only contains degrees of freedom on the faces.
1. The general structure of the methods
1.1. Notation
Let \(\Omega_h\) be a collection of disjoint elements that partition \(\Omega\). The shape of the elements is not important in this general framework. Moreover, \(\Omega_h\) needs not to be
conforming. An interior ''face'' of \(\Omega_h\) is any set \(F\) of positive \((n-1)\)-Lebesgue measure of the form \(F = \partial K^+\cap\partial K^-\) for some two elements \(K^+\) and \(K^-\) of
\(\Omega_h\). We say that \(F\) is a boundary face if there is an element \(K\in\Omega_h\) such that \(F = \partial K\cap\Gamma\) and the \((n-1)\)-Lebesgue measure of \(F\) is not zero. Let \(\
mathcal E^o_h\) and \(\mathcal E^\partial_h\) denote the set of interior and boundary faces of \(\Omega_h\), respectively. We denote by \(\mathcal E_h\) the union of all the faces in \(\mathcal E^o_h
\) and \(\mathcal E^\partial_h\).
The global finite element spaces for the approximated flux \(\mathbf u_h\) and scalar solutions \(p_h\) are
\[\begin{align*} \mathbf V_h &= \left\{\mathbf v : \Omega\to\mathbb R^n : \mathbf v|_K \in\mathbf V(K)\quad\forall\,K\in\Omega_h\right\},\\ W_h &= \left\{w : \Omega\to\mathbb R : w|_K \in W(K)\quad\
forall\,K\in\Omega_h\right\}. \end{align*}\]
We also need to introduce the spaces
\[\begin{align*} M_h &= \{\mu : \mathcal E_h\to \mathbb R : \mu|_F\in M(F)\quad\forall\,F\in\mathcal E_h \},\\ M_h^o &= \{\mu\in M_h : \mu|_\Gamma = 0 \},\\ M_h^D &= \{\mu\in M_h : \mu|_F = 0\quad\
forall\,F\in\mathcal E_h\setminus\Gamma_D \},\\ M_h^N &= \{\mu\in M_h : \mu|_F = 0\quad\forall\,F\in\mathcal E_h\setminus\Gamma_N \}. \end{align*}\]
Different choices of the local spaces \(\mathbf V(K), W(K)\), and \(M(F)\) correspond to different hybrid methods.
For any discontinuous (scalar or vector) function \(u\) in \(W_h\) or \(\mathbf V_h\), the trace \(u|_F\) on an interior face \(F = \partial K^+\cap\partial K^-\) is a double value function, whose
two branches are denoted by \(u|_{K^+}\) and \(u|_{K^-}\). Here, \(\mathbf n_K\) denotes the unit outward normal of \(K\). For any double-valued vector function \(\mathbf v\), we define the jump of
its normal component across an interior face \(F\) by
\[[[ \mathbf v ]]_F = \mathbf v_{K^+} \cdot\mathbf n_{K^+} + \mathbf v_{K^-}\cdot\mathbf n_{K^-}.\]
On any face \(F\) of \(K\) lying on the boundary, we set
\[[[ \mathbf v ]]_F = \mathbf v_K \cdot\mathbf n_K.\]
A similar notation will be used for scalar functions. For functions \(u\) and \(v\) in \(L^2(D)\), we write \((u, v)_D = \int_D uv\) if \(D\) is a domain of \(\mathbb R^n\), and \(\langle u, v\
rangle_D = \int_Duv\) if \(D\) is a domain of \(\mathbf R^{n-1}\). Also, we will use the notation
\[(v, w)_{\Omega_h} = \sum_{K\in\Omega_h}(v, w)_K,\qquad\langle\mu,\lambda\rangle_{\partial\Omega_h} = \sum_{K\in\Omega_h}\langle\mu,\lambda\rangle_{\partial K}.\]
1.2. Reaching the formulation
1.2.1. A characterization of the exact solution
Two functions \(\mathbf u, p\) are exact solutions of \((1)-(4)\) if and only if they satisfy the local problems
\[\begin{align*} \Lambda\mathbf u + \nabla p &= \mathbf 0 & &\text{in }K,\\ \nabla\cdot\mathbf u + d p &= f & &\text{in K}, \end{align*}\]
the transmission conditions
\[\begin{align*} [[ \hat p ]] &= 0 & &\text{if }F\in\mathcal E^o_h,\\ [[ \hat{\mathbf u}]] &= 0 & &\text{if }F\in\mathcal E^o_h, \end{align*}\]
and the boundary conditions
\[ \hat p &= h_D & &\text{if }F\in\Gamma_D,\\ \hat{\mathbf u}\cdot\mathbf n &= h_N & &\text{if }F\in\Gamma_N, \]
where \(\hat p\) and \(\hat{\mathbf u}\) are the traces of \(p\) and \(\mathbf u\), respectively. The transmission conditions state that the normal component of \(\hat{\mathbf u}\) across
interelement boundary must be continuous. We can obtain \((\mathbf u, p)\) in \(K\) in terms of \(\hat p\) on \(\partial K\) and \(f\) by solving the following local Dirichlet problem
\[\begin{align*} \Lambda\mathbf u + \nabla p &= \mathbf 0 & &\text{in }K, & &(5)\\ \nabla\cdot\mathbf u + d p &= f & &\text{in K}, & &(6)\\ p &= \hat p & &\text{on }\partial K. & &(7) \end{align*}\]
The function \(\hat p\) can now be determined as the solution, on each \(F\in\mathcal E_h\), of the global equations \begin{align*} [[ \hat{\mathbf u} ]]_F &= 0 & &\text{if }F\in\mathcal E^o_h, & &
(8)\\ [[ \hat{\mathbf u} ]]_F &= h_N & &\text{if }F\in\Gamma_N, & & (9)\\ \hat p|_F &= h_D & &\text{if }F\in\Gamma_D. & &(10) \end{align*}
Hybrid methods are obtained by constructing discrete versions of \((5)-(10)\). In this way, the globally coupled degrees of freedom will be those of the corresponding global formulations.
1.2.2. The local solvers: a weak formulation on each element
From \((5)-(7)\), on the element \(K\in\Omega_h\), we define \((\mathbf u_h, p_h)\) in terms of \((\hat p_h, f)\) as the element of \(\mathbf V(K)\times W(K)\) such that
\[\begin{align*} (\Lambda\mathbf u_h,\mathbf v)_K - (p_h, \nabla\cdot\mathbf v)_K + \langle\hat p_h, \mathbf v\cdot\mathbf n\rangle_{\partial K} &= 0, & &(11)\\ -(\mathbf u_h, \nabla w)_K + \langle\
hat{\mathbf u}_h\cdot\mathbf n,w\rangle_{\partial K} + (dp_h, w)_K &= (f,w)_K, & &(12) \end{align*}\]
for all \((\mathbf v, w) \in \mathbf V(K)\times W(K)\), where \(\hat{\mathbf u}_h\) is the numerical trace of the flux and depends, in general, on \(\mathbf u_h, p_h\), and \(\hat p_h\). Different
methods will correspond to different choices of \(\hat{\mathbf u}_h\).
1.2.3. The global problem
From \((8)-(10)\), We determine \(\hat p_h\in M_h\) by requiring that
\[\begin{align*} \langle\mu,[[ \hat{\mathbf u}_h ]]\rangle_F &= 0 & &\forall\,\mu\in M(F) & &\text{if }F\in\mathcal E_h^o, & &(13)\\ \langle\mu,[[ \hat{\mathbf u}_h ]]\rangle_F &= \langle\mu,h_N\
rangle_F & &\forall\,\mu\in M(F) & &\text{if }F\in\Gamma_N, & &(14)\\ \langle\mu,\hat p_h\rangle_F &= \langle\mu,h_D\rangle_F & &\forall\,\mu\in M(F) & &\text{if }F\in\Gamma_D. & &(15) \end{align*}\]
By solving \((11), (12)\) for \((\mathbf u_h, p_h)\) in terms of \((\hat p_h, f)\) at each element and plugging the results into \((13)-(15)\), we get a system whose globally coupled degrees of
freedom are those of the numerical trace \(\hat p_h\). This procedure corresponds to performing static condensation on the full discrete global system written in terms of \(\mathbf u_h, p_h, \hat p_h
If the (extension by zero to \(\mathcal E_h\) of the) function \([[ \hat{\mathbf u}_h ]]_{|\mathcal E_h^o}\) belongs to the space \(M_h\), then condition \((13)\) is stating that \([[ \hat{\mathbf u}
_h ]]_{|\mathcal E_h^o} = 0\) pointwise, that is, the normal component of the numerical trace \(\hat{\mathbf u}_h\) is single valued. This means that the function \(\hat{\mathbf u}_h\) is a
conservative numerical flux (\(\hat{\mathbf u}_h\in H(\text{div},\Omega)\)).
1.2.4. Summary
The approximate solution \((\mathbf u_h, p_h, \hat p_h)\) is the element of the space \(\mathbf V_h\times W_h\times M_h\) satisfying the equations
\[\begin{align*} (\Lambda\mathbf u_h,\mathbf v)_{\Omega_h} - (p_h, \nabla\cdot\mathbf v)_{\Omega_h} + \langle\hat p_h, \mathbf v\cdot\mathbf n\rangle_{\partial\Omega_h} &= 0 & &\forall\mathbf v\in \
mathbf V_h, & &(16)\\ -(\mathbf u_h, \nabla w)_{\Omega_h} + \langle\hat{\mathbf u}_h\cdot\mathbf n,w\rangle_{\partial\Omega_h} + (d p_h, w)_{\Omega_h} &= (f,w)_{\Omega_h} & &\forall w\in W_h, & &(17)
\\ \langle\mu,\hat{\mathbf u}_h\cdot\mathbf n\rangle_{\partial\Omega_h\setminus\Gamma} &= 0 & &\forall \mu\in M^o_h, & &(18)\\ \langle\mu,\hat{\mathbf u}_h\cdot\mathbf n\rangle_{\Gamma_N} &= \langle\
mu,h_N\rangle_{\Gamma_N} & &\forall\mu\in M^N_h, & &(19)\\ \langle\mu,\hat p_h\rangle_{\Gamma_D} &= \langle\mu,h_D\rangle_{\Gamma_D} & &\forall\mu\in M^D_h, & &(20) \end{align*}\]
where the local spaces \(\mathbf V(K), W(K), M(F)\), as well as the numerical trace \(\hat{\mathbf q}_h\), need to be specified.
2. Examples of hybridizable methods
In this section we give som examples of methods fitting the general structure described in the previous section. The first three methods use the same local solver in all the elements \(K\) of the
mesh \(\Omega_h\) and assume that \(\Omega_h\) is a conforming simplicial mesh. The fourth example is a class of methods employing different local solvers in different parts of the domain, which can
easily deal with nonconforming meshes. To define each method, we have only to specify:
• the numerical trace of the flux \(\hat{\mathbf u}_h\);
• the local spaces \(\mathbf V(K), W(K)\);
• the space of approximate traces \(M_h\).
2.1. The RT-H method
This method is obtained by using the Raviart-Thomas method to define the local solvers. The three ingredients of the RT-H method are:
1. \(\hat{\mathbf u}_h = \mathbf u_h\) on \(\partial K\), for each \(K\in\Omega_h\).
2. \(\mathbf V(K) = [P_k(K)]^n + \mathbf x P_k(K),\quad W(K) = P_k(K),\quad k\geq 0\).
3. \(M_h = \{\mu\in L^2(\mathcal E_h) : \mu|_F\in P_k(F)\quad\forall\,F\in\mathcal E_h\}\).
The accuracy of the RT-H method is summarized in section [accuracy]. Note that, because \([[ \hat{\mathbf u}_h ]]\) and test functions \(\mu\) belong to the same space <Sayas 2013>, conservativity
condition \((13)\) forces
\[[[ \hat{\mathbf u}_h]] = 0\quad\text{on }\mathcal E_h^o,\]
so the normal component of the numerical trace \(\hat{\mathbf u}_h\) is single-valued, and \(\mathbf u_h\in H(\text{div},\Omega)\).
2.2. The BDM-H method
This method is obtained by using the Brezzi-Douglas-Marini method to define the local solvers. The three ingredients of the BDM-H method are:
1. \(\hat{\mathbf u}_h = \mathbf u_h\) on \(\partial K\), for each \(K\in\Omega_h\).
2. \(\mathbf V(K) = [P_k(K)]^n,\quad W(K) = P_{k-1}(K),\quad k\geq 1\).
3. Same \(M_h\) of the RT-H method.
Everything said about the RT-H method in the previous subsection applies to the BDM-H method.
2.3. The HDG method
The spaces of RT-H and BDM-H can be balanced to have equal polynomial degree. Stability is restored using a discrete stabilization (not penalization) function. The resulting method is known as the
Hybridizable Discontinuous Galerkin (HDG) method. The HDG method is obtained by using the local DG method to define the local solvers. The three ingredients of the HDG method are:
1. For each \(K\in\Omega_h\): \(\hat{\mathbf u}_h = \mathbf u_h + \tau_K(p_h - \hat p_h)\mathbf n\quad\text{on }\partial K,\)
where \(\tau_K\) is a nonnegative function that can vary on \(\partial K\), and \(\tau_K > 0\) on at least one face of \(\partial K\).
2. \(\mathbf V(K) = [P_k(K)]^n,\quad W(K) = P_k(K),\quad k\geq 0\).
3. Same \(M_h\) of the RT-H method.
The function \(\tau\) can be double valued on \(\mathcal E_h^o\), with two branches \(\tau^-=\tau_{K^-}\) and \(\tau^+=\tau_{K^+}\) defined on the face \(F\) shared by the finite elements \(K^+\) and
\(K^-\). Note that the numerical trace of the flux \(\hat{\mathbf u}_h\) (but not the flux itself, as \(\tau_K\ne 0\)) is conservative. The accuracy of the HDG method is summarized in section
2.3.1. Enhanced accuracy by postprocessing
The approximate solution and flux of the HDG method can be locally postprocessed to enhance their accuracy <Cockburn 2010>.
• Postprocessing of the scalar variable:
if we look for \(p_h^*:\Omega\to\mathbb R\) such that \(p_h^*|_K\in P_{k+1}(K)\) and for all \(K\in\Omega_h\)
(\nabla p_h^*, \nabla w)_K &= -(\Lambda\mathbf u_h, \nabla w)_K & &\forall\,w\in P_{k+1}(K),\\ (p^*_h, 1)_K &= (p_h, 1)_K, & &
then it can be shown that this local postprocessed approximation has one additional order of convergence.
• Postprocessing of the flux:
we can obtain a postprocessed flux \(\mathbf u_h^*\) with better conservation properties. Although \(\mathbf u_h^*\) converges at the same order as \(\mathbf u_h\), it is in \(H(\text{div},\
Omega)\) and its divergence converges at one higher order than \(\mathbf u_h\). On each \(K\in\Omega_h\), we take \(\mathbf u_h^* :=\mathbf u_h + \boldsymbol\eta_h\) where \(\boldsymbol\eta_h\)
is the only element of \([P_k(K)]^n + \mathbf x P_k(K)\) satisfying
(\boldsymbol\eta_h,\mathbf v)_K &= 0 & &\forall\,\mathbf v\in[P_{k-1}(K)]^n,\\ \langle\boldsymbol\eta_h\cdot\mathbf n, \mu\rangle_F &= \langle(\hat{\mathbf u}_h-\mathbf u_h)\cdot\mathbf n,\mu\
rangle_F & &\forall\,F\in P_k(F),\quad\forall\,F\in\partial K.
2.4. Hybridization in matrix form
This section is mainly based on <Fu 2013>. As stated before, the goal of hybridization is the reduction (or static condensation) of the system \((16)-(20)\) to a linear system where only \(\hat p_h\)
shows up. The remaining two variables \(\mathbf u_h\) and \(p_h\) will be reconstructed after solving for \(\hat p_h\), in an element-by-element fashion, easy to realize due to the fact that
equations \((16)\) and \((17)\) are local or, in other words, the spaces \(\mathbf V_h\) and \(W_h\) are completely discontinous. In this section we will show how to perform static condensation on
the linear system obtained by using the HDG method. This procedure can be easily adapted to other hybrid methods. Let us recall that the HDG method looks for an approximate solution \((\mathbf u_h,
p_h, \hat p_h)\) in the space \(\mathbf V_h\times W_h\times M_h\) satisfying the equations
\[\begin{align*} &(\Lambda\mathbf u_h,\mathbf v)_{\Omega_h} & &- (p_h, \nabla\cdot\mathbf v)_{\Omega_h} & &+ \langle\hat p_h, \mathbf v\cdot\mathbf n\rangle_{\partial\Omega_h} & &= 0, & &(21)\\ &(\
nabla\cdot\mathbf u_h, w)_{\Omega_h} & &+ \langle\tau p_h,w\rangle_{\partial\Omega_h} + (d p_h, w)_{\Omega_h} & &- \langle\tau \hat p_h,w\rangle_{\partial\Omega_h} & &= (f,w)_{\Omega_h}, & &(22)\\ &\
langle\mathbf u_h\cdot\mathbf n,\mu_1\rangle_{\partial\Omega_h\setminus\Gamma} & &+ \langle\tau p_h,\mu_1\rangle_{\partial\Omega_h\setminus\Gamma} & &- \langle\tau \hat p_h,\mu_1\rangle_{\partial\
Omega_h\setminus\Gamma} & &= 0, & &(23)\\ &\langle\mathbf u_h\cdot\mathbf n,\mu_2\rangle_{\Gamma_N} & &+ \langle\tau p_h,\mu_2\rangle_{\Gamma_N} & &- \langle\tau \hat p_h,\mu_2\rangle_{\Gamma_N} & &=
\langle h_N,\mu_2\rangle_{\Gamma_N}, & &(24)\\ & & & & &\langle\hat p_h,\mu_3\rangle_{\Gamma_D} & &= \langle h_D,\mu_3\rangle_{\Gamma_D}, & &(25) \end{align*}\]
for all \((\mathbf v, w, \mu_1, \mu_2, \mu_3)\in\mathbf V_h\times W_h\times M_h^o\times M_h^N\times M_h^D\).
2.4.1. Local solvers
Introduce the matrices related to the local bilinear forms
\[ A_{11}^K &\leftrightarrow (\Lambda\mathbf u_h,\mathbf v)_K, & &A_{12}^K\leftrightarrow- (p_h, \nabla\cdot\mathbf v)_K, & &A_{13}^K\leftrightarrow\langle\hat p_h, \mathbf v\cdot\mathbf n\rangle_{\
partial K},\\ A_{21}^K &\leftrightarrow(\nabla\cdot\mathbf u_h, w)_K, & &A_{22}^K\leftrightarrow\langle\tau p_h,w\rangle_{\partial K} + (d p_h, w)_K, & &A_{23}^K\leftrightarrow\langle\tau \hat p_h,w\
rangle_{\partial K},\\ A_{31}^K &\leftrightarrow\langle\mathbf u_h\cdot\mathbf n, \mu\rangle_{\partial K}, & &A_{32}^K\leftrightarrow\langle\tau p_h,\mu\rangle_{\partial K}, & &A_{33}^K\
leftrightarrow\langle\tau \hat p_h,\mu\rangle_{\partial K},\\ & & &A_f^K\leftrightarrow (f,w)_K \]
If \(\hat p_h\in M_h\) is known, equations \((21), (22)\) are uniquely solvable for \(\mathbf u_h, p_h\)and can be solved element-by-element. Let us represent \(\mathbf u_h|_K, p_h|_K\), and \(\hat
p_h|_{\partial K}\) with vectors \(\mathbf u_K, \mathbf p_K\), and \(\mathbf p_{\partial K}\), respectively. Also, let
\[\begin{align*} A^K &= \begin{bmatrix} A_{11}^K & A_{12}^K\\ A_{21}^K & A_{22}^K\\ \end{bmatrix}, & B^K &= \begin{bmatrix} A_{13}^K\\ A_{23}^K \end{bmatrix}, & F^K &= \begin{bmatrix} \mathbf 0\\ A_f
^K \end{bmatrix}. \end{align*}\]
Then, the matrix representation of the local solutions is
\[ &\begin{bmatrix} \mathbf u_K\\ \mathbf p_K \end{bmatrix} = -(A^K)^{-1}B^K \mathbf p_{\partial K} + (A^K)^{-1}F^K. & &(26) \]
\[C^K = \begin{bmatrix} A_{31}^K & A_{32}^K \end{bmatrix}.\]
The flux prescribed by the HDG method
\[\mathbf u_h\cdot\mathbf n + \tau(p_h-\hat p_h)\colon\partial K\to\mathbb R\]
\[\mu\in M(\partial K)\to\langle\mathbf u_h\cdot\mathbf n + \tau (p_h-\hat p_h), \mu\rangle_{\partial K} = \langle\mathbf u_h\cdot\mathbf n + \tau p_h, \mu\rangle_{\partial K} - \langle \tau\hat p_h,
whose matrix representation is (using \((26)\))
\[\begin{split} C^K\begin{bmatrix} \mathbf u_K\\ \mathbf p_K \end{bmatrix} - A_{33}^K\mathbf p_{\partial K} &= -C^K(A^K)^{-1}B^K \mathbf p_{\partial K} + C^K(A^K)^{-1}F^K - A_{33}^K\mathbf p_{\
partial K}\qquad(27)\\ &= D_f^K - D^K\mathbf p_{\partial K}, \end{split}\]
\[\begin{align*} D_f^K &= C^K(A^K)^{-1}F^K, & D^K &= C^K(A^K)^{-1}B^K + A_{33}^K. \end{align*}\]
2.4.2. Boundary conditions and global solver
• Dirichlet boundary conditions. The discrete Dirichlet boundary conditions \((25)\) require finding the projection \(\mathbf{\hat p}_D\) of the function \(h_D\) on the space \(M_h|_{\Gamma_D}\).
• Neumann boundary conditions. Neumann boundary conditions will appear in the right hand side of the global system.
• Assemblying the global solver. The local solvers produce matrices \(D^K\) that need to be assembled to get a global matrix \(\mathbb H\). This matrix collects the fluxes \((27)\) from all the
elements, with the result that opposing sign fluxes in internal faces (the normal vector points in different directions) are added. The matrices \(D^K_f\) also have to be assembled to get a
global vector \(\mathbf F\). At this point, the global system reads
\begin{equation*} \mathbb H\,\mathbf{\hat p} = \mathbf F + \mathbf G_N,\qquad(28) \end{equation*}
where \(\mathbf G_N\) is the vector containing the elements of \(\langle h_N, \mu\rangle_{\Gamma_N}, \mu\in M_h|_{\Gamma_N}\) in the degrees of freedom corresponding to Neumann faces and zeros
everywhere else. What is left is the elimination of Dirichlet degrees of freedom from \((28)\), namely, values of Dirichlet faces are taken from \(\mathbf{\hat p}_D\) and sent to the right hand side
of the system, and rows corresponding to Dirichlet degrees of freedom are ignored.
2.5. Orders of accuracy for RT-H, BDM-H, HDG
The following table summarizes the effect of the local spaces and the stabilization parameter \(\tau\) on the accuracy of the method on simplexes. We denote by \(\overline p_h|_K\) the integral
average of \(p_h\) on \(K\in\Omega_h\). For the HDG method, the superconvergence of \(\overline p_h\) is what allows to get a solution of enhanced accuracy by postprocessing.
Method \(\tau\) \(\mathbf u_h\) \(p_h\) \(\overline p_h\) \(k\)
RT-H \(0\) \(k+1\) \(k+1\) \(k+2\) \(\geq 0\)
BDM-H \(0\) \(k+1\) \(k\) \(k+2\) \(\geq 2\)
HDG \(O(h)\) \(k+1\) \(k\) \(k+2\) \(\geq 1\)
HDG \(O(1)\) \(k+1\) \(k+1\) \(k+2\) \(\geq 1\)
HDG \(O(1)\) \(1\) \(1\) \(1\) \(=0\)
HDG \(O(1/h)\) \(k\) \(k+1\) \(k+1\) \(\geq 1\)
2.6. A class of hybridizable methods well suited for adaptivity
We introduce here a class of hybridizable methods able to use different local solvers in different elements and to easily handle nonconforming meshes. To define these methods, we need to specify the
numerical fluxes, the local finite element spaces, and the space of approximate traces:
1. For any simplex \(K\in\Omega_h\), we take
\begin{equation*} \hat{\mathbf u}_h = \mathbf u_h + \tau_K(p_h - \hat p_h)\mathbf n\quad\text{on }\partial K, \end{equation*}
the function \(\tau_K\) is allowed to change on \(\partial K\). . The local space \(\mathbf V(K)\times W(K)\) can be any of the following:
• \(([P_{k(K)}(K)]^n + \mathbf x P_{k(K)}(K)) \times P_{k(K)}(K)\), where \(k(K)\geq0\) and \(\tau_K\geq 0\) on \(\partial K\),
• \([P_{k(K)}(K)]^n \times P_{k(K)-1}(K)\), where \(k(K)\geq1\) and \(\tau_K\geq 0\) on \(\partial K\),
• \([P_{k(K)}(K)]^n \times P_{k(K)}(K)\), where \(k(K)\geq0\) and \(\tau_K > 0\) on at least one face \(F\in\partial K\).
1. The space of approximate traces is
\begin{equation*} M_h := \{\mu\in L^2(\mathcal E_h):\mu|_F\in P_{k(F)}\quad\forall\,F\in\mathcal E_h\}. \end{equation*}
Here, if \(F = \partial K^+\cap\partial K^-\), we set \(k(F) := \max\{k(K^+), k(K^-)\}\). For each element \(K\in\Omega_h\) and each face \(F\in\mathcal E_h\) on \(\partial K\), we take \(\tau_K|_F\
in[0,\infty)\) and
\[\tau_K|_F\in(0,\infty)\quad\text{if }F\text{ is not a face of }K.\qquad(16)\]
Choice \((16)\) allows to deal with the nonconformity of the mesh in a very natural way. Also, the choice \(\tau_K = \infty\) could be allowed provided that the definition of the local solvers is
modified as in <Cockburn 2009>.
The main features of this class of methods are:
• Variable degree approximation spaces on conforming meshes. The RT-H, BDM-H, and HDM methods considered above use a single local solver in each of the elements \(K\) of the conforming
triangulation \(\Omega_h\). A variable-degree version of each of these methods is a particular case of the clas of methods presented here.
• Automatic coupling of different methods on conforming meshes. The class presented here allows for the use of different local solvers in different elements \(K\in\Omega_h\), which are then
automatically coupled.
• Mortaring capabilities (for nonconforming meshes). This class incorporate a mortaring ability thanks to the form that the numerical trace of the flux on \(\partial K\) takes on an interior face \
(F\in\mathcal E_h^o\), and thanks to the definition of the stabilization parameter. Let us give an example. If we have a conforming mesh, we can take the first choice of local spaces (2a) and set
\(\tau = 0\). The resulting method is nothing but the RT-H method. We can easily modify this method to handle nonconforming meshes by simply taking \(\tau_K\in(0,\infty)\) on every \(F\in\mathcal
E_h^o\) which is not a face of \(K\), and otherwise, taking \(\tau_K 0\).
3. References
• [] Superconvergent discontinuous Galerkin methods for second-order elliptic problems B Cockburn, J Guzmán, H Wang - Mathematics of Computation, 2009
• [[[ cockburnGS2010]]] The hybridizable discontinuous Galerkin methods, B Cockburn - Proceedings of the International Congress of …, 2010
• [] Algorithm 949: MATLAB tools for HDG in three dimensions Z Fu, LF Gatica, FJ Sayas - ACM Transactions on Mathematical Software (TOMS), 2015 | {"url":"https://docs.feelpp.org/toolboxes/latest/hdg/theory.html","timestamp":"2024-11-15T04:29:02Z","content_type":"text/html","content_length":"73165","record_id":"<urn:uuid:815b0db3-0906-416a-a5dc-9fe3bfe1b53e>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00354.warc.gz"} |
Cite as
Marek Filakovský, Tamio-Vesa Nakajima, Jakub Opršal, Gianluca Tasinato, and Uli Wagner. Hardness of Linearly Ordered 4-Colouring of 3-Colourable 3-Uniform Hypergraphs. In 41st International Symposium
on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 34:1-34:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Copy BibTex To Clipboard
author = {Filakovsk\'{y}, Marek and Nakajima, Tamio-Vesa and Opr\v{s}al, Jakub and Tasinato, Gianluca and Wagner, Uli},
title = {{Hardness of Linearly Ordered 4-Colouring of 3-Colourable 3-Uniform Hypergraphs}},
booktitle = {41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)},
pages = {34:1--34:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-311-9},
ISSN = {1868-8969},
year = {2024},
volume = {289},
editor = {Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.34},
URN = {urn:nbn:de:0030-drops-197445},
doi = {10.4230/LIPIcs.STACS.2024.34},
annote = {Keywords: constraint satisfaction problem, hypergraph colouring, promise problem, topological methods} | {"url":"https://drops.dagstuhl.de/search?term=Nakajima%2C%20Tamio-Vesa","timestamp":"2024-11-05T22:12:05Z","content_type":"text/html","content_length":"89944","record_id":"<urn:uuid:42d8c3a0-3864-40b1-9f83-a21cefad0a87>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00363.warc.gz"} |
Khanti Method
सपने होंगे सच , आज ही करें नई शुरुआत
2 से 3 सेकण्ड में हल करने का नियम
Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangl...
Calculation is the process of performing mathematical operations to find a numerical result. It involves using ar...
2 से 3 सेकण्ड में उत्तर देने का नियम
Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangl...
Calculation is the process of performing mathematical operations to find a numerical result. It involves using ar...
2 से 3 सेकण्ड में कैलकुलेट करने का नियम
Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangl...
Calculation is the process of performing mathematical operations to find a numerical result. It involves using ar...
Complete Syllabus of Maths Competitive Exams
Ratio & Proportion
Ratio and proportion are mathematical concepts that are used to compare two or more quantities or values.
Ratio: A ratio is a comparison of two quantities or values. It is expressed in the form of a fraction, with the first quantity being the numerator and the second quantity being the denominator. For
example, if there are 10 apples and 5 oranges, the ratio of apples to oranges is 10:5 or 2:1.
Proportion: A proportion is an equation that shows that two ratios are equal. It is expressed in the form of a:b = c:d. For example, if the ratio of apples to oranges is 2:1 and the ratio of oranges
to bananas is 4:1, we can create a proportion: 2:1 = x:4, where x represents the number of oranges needed to match the number of bananas.
Ratios and proportions are used in a wide range of fields, including mathematics, science, finance, and engineering. They can be used to solve a variety of problems, such as calculating the
percentage of a total, determining the dimensions of objects, and making predictions based on past data.
In mathematics, the average (also called the mean) is a measure of central tendency that represents the typical value in a set of numbers. It is calculated by adding up all the numbers in a set and
then dividing the sum by the total number of values in the set.
There are three types of averages:
1. Arithmetic mean: It is the most commonly used type of average and is calculated by adding up all the values in a set and dividing by the total number of values.
2. Median: It is the middle value in a set of numbers when they are arranged in order from smallest to largest. If there are an even number of values, the median is the average of the two middle
3. Mode: It is the value that appears most frequently in a set of numbers. A set of numbers can have more than one mode or no mode at all.
Averages are used in many fields, including statistics, finance, and science, to summarize and analyze data. They can be used to calculate trends, make predictions, and compare different sets of
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error, est. Ea, consequatur.
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error, est. Ea, consequatur.
Profit and Loss
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error, est. Ea, consequatur.
Simple Interest
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error, est. Ea, consequatur.
Compound Interest
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error, est. Ea, consequatur.
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error, est. Ea, consequatur.
Mixed Proportion
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error, est. Ea, consequatur.
Time and Work
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error, est. Ea, consequatur.
Work And Wages
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error, est. Ea, consequatur.
Pipe and Cistern
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error, est. Ea, consequatur.
Time & Distance
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error, est. Ea, consequatur.
Problem Based on Trains
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error, est. Ea, consequatur.
Boats and Stream
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error, est. Ea, consequatur.
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error, est. Ea, consequatur.
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error, est. Ea, consequatur.
Problems Based on Age
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error, est. Ea, consequatur.
Caselets are short, structured pieces of written material used in business education and management training. They are typically used to illustrate a business scenario or problem and challenge
participants to analyze the situation and propose solutions.
Caselets are usually a few paragraphs long, and they may include background information, data, and quotes from relevant stakeholders. They often focus on a specific issue or decision that a business
or organization is facing, such as a market entry strategy, product launch, or cost-cutting initiative.
Caselets are commonly used in case-based learning, where participants are asked to analyze the information provided, identify the key issues, and develop a plan of action. They can be used in
individual or group settings, and they are often accompanied by discussion questions or prompts to guide the analysis and discussion.
Caselets are a popular tool in business education because they provide a practical and engaging way to learn about real-world business scenarios. They allow participants to apply theoretical concepts
and frameworks to concrete situations, and they help develop critical thinking, problem-solving, and decision-making skills.
Venn Diagram
A Venn diagram is a visual representation of sets, showing their logical relationships. It consists of overlapping circles or other shapes, each representing a set, with the overlapping areas showing
the elements that belong to both sets. The non-overlapping areas of the circles show the elements that belong only to one set or the other.
Venn diagrams are often used in mathematics, statistics, and other fields to illustrate concepts such as set theory, probability, and logic. They can also be used in business, education, and other
fields to analyze and compare data or concepts.
The basic elements of a Venn diagram are:
1. Circles or other shapes representing sets
2. Overlapping areas representing the elements that belong to both sets
3. Non-overlapping areas representing the elements that belong only to one set or the other
4. Labels or annotations to indicate the sets and the elements they represent
Venn diagrams can be simple or complex, depending on the number of sets and the complexity of the relationships between them. They are a useful tool for visualizing and understanding complex concepts
and relationships, and they can be easily created using software tools or by hand.
A pie chart is a type of data visualization that displays data as a circle divided into sections, or “slices,” where each slice represents a category or proportion of the data. The size of each slice
is proportional to the quantity it represents, and the total of all the slices is equal to 100%.
Pie charts are often used to show relative proportions of different categories or parts of a whole. For example, a pie chart might show the breakdown of a company’s expenses, with each slice
representing a different expense category such as salaries, rent, and supplies. The size of each slice would indicate the proportion of the company’s total expenses that are allocated to each
Pie charts are visually appealing and easy to understand, making them a popular choice for presenting data to a general audience. However, they can be less effective when comparing data sets with
many categories or when the differences between categories are small. In these cases, other types of charts, such as bar charts or stacked bar charts, may be more appropriate.
Line Graph
A line graph is a type of data visualization that displays data as a series of points or markers connected by straight lines. It is used to show trends or changes over time or to display continuous
data, such as temperature readings, stock prices, or population growth.
In a line graph, the horizontal axis represents the independent variable, which is typically time or another continuous variable. The vertical axis represents the dependent variable, which is the
variable being measured or observed. Each point on the graph represents the value of the dependent variable at a specific time or value of the independent variable, and the lines connect these points
to show the trend or pattern over time or across values.
Line graphs can be simple or complex, with multiple lines representing different data sets or variables. They are often used in scientific research, business, and other fields to show patterns,
trends, and relationships between variables.
One advantage of line graphs is that they are easy to read and understand, even for those without a strong background in data analysis. However, they can also be misleading if the data is not plotted
accurately or if the axes are not labeled clearly. It is important to choose the appropriate graph type for the data being presented and to ensure that the graph is accurate and informative.
Bar Diagram
A bar diagram, also known as a bar chart or bar graph, is a way of representing data visually using rectangular bars of equal width. The length or height of each bar is proportional to the value it
Bar diagrams are used to display categorical data or quantitative data that can be separated into discrete categories. They are particularly useful for comparing data across different categories or
To create a bar diagram, the first step is to determine the categories or groups that will be represented on the horizontal axis (also called the x-axis). The vertical axis (also called the y-axis)
is used to represent the values or quantities associated with each category.
Each bar in the diagram represents a category and has a height or length proportional to the value or quantity associated with that category. The bars can be vertical or horizontal, depending on the
orientation of the diagram.
Bar diagrams can also include multiple bars for each category, allowing for easy comparison of data across multiple groups. In this case, each bar is usually colored or shaded differently to
differentiate between the groups.
Overall, bar diagrams are a simple and effective way of representing data visually, making it easier to identify patterns, trends, and relationships in the data.
Radar Diagram
A radar diagram, also known as a radar chart or spider chart, is a graphical method of displaying multivariate data in the form of a two-dimensional chart. It is used to compare different categories
or variables based on multiple criteria or dimensions.
A radar diagram consists of a series of equidistant spokes or axes that radiate out from a central point. Each spoke represents a different variable or criterion being measured. The data for each
category is plotted as a series of points or data markers along each spoke, and the resulting shape resembles a spider or web.
Radar diagrams are useful for visualizing data that have multiple dimensions or criteria, such as performance metrics for different athletes or companies. They can also be used to compare the
performance of a single entity over time, such as a company’s financial performance over several years.
One advantage of the radar diagram is that it can show both the magnitude and direction of the data values, as each spoke represents a different dimension. This makes it easy to see which variables
or criteria are most important for each category.
However, radar diagrams can be difficult to read and interpret when the data values are too similar or when there are too many variables or criteria being measured. They are best used when there are
no more than five or six variables or criteria to be displayed.
Triangular Diagram
There are several different types of triangular diagrams, but the most common one is the triangular plot, which is a two-dimensional graph used to display three variables. It is called a triangular
diagram because the plot is shaped like a triangle.
The triangular plot consists of three axes that intersect at a point, forming a triangle. Each axis represents a different variable, and the intersection of the axes represents the point where all
three variables have a value of zero. The data points are then plotted within the triangle, and the position of each point indicates the values of the three variables.
One common use of the triangular diagram is in chemical analysis, where it is used to plot the concentrations of three different components in a mixture. Each component is represented by one of the
three axes, and the concentrations of the three components are plotted as a point within the triangle.
Another use of the triangular diagram is in market research, where it can be used to plot the relative importance of different product attributes. For example, one axis might represent price, another
might represent quality, and a third might represent convenience. Respondents to a survey would be asked to rate the importance of each attribute, and their responses would be plotted within the
Overall, the triangular diagram is a useful tool for visualizing three-variable data and identifying patterns and trends in the data.
Combined Diagram
A combined diagram is a type of chart or graph that combines multiple data visualization techniques in one chart. For example, a combined diagram could include a line graph, a bar chart, and a
scatter plot all on the same chart.
One common type of combined diagram is the combination chart, which combines two or more types of charts on the same set of axes. For example, a combination chart might include a line graph and a bar
chart, with the line graph representing a trend over time and the bar chart representing discrete values or categories.
Another type of combined diagram is the heat map, which combines color coding and tabular data to show patterns and trends in large data sets. A heat map might show the frequency of different values
or categories in a table or matrix, with each cell color-coded to represent the frequency.
A third type of combined diagram is the bubble chart, which combines a scatter plot and a size variable to display three dimensions of data. In a bubble chart, the position of each point represents
the values of two variables, and the size of the point represents the value of a third variable.
Overall, combined diagrams are useful for displaying complex data sets and identifying patterns and relationships between different variables. They can be used in a variety of fields, including
business, science, and social science.
Tabulation is the process of organizing and presenting data in a tabular form. It involves arranging data into rows and columns, with each row representing a unique observation or case, and each
column representing a variable or characteristic of the data.
Tabulation is commonly used in statistics and data analysis to summarize and present large amounts of data in a clear and concise manner. It can help to identify patterns and trends in the data, and
to make comparisons between different groups or categories.
Tabulation can be done manually using pen and paper or a spreadsheet program, or automatically using specialized software. The process typically involves the following steps:
1. Define the variables: Determine the variables or characteristics that you want to analyze and tabulate. This might include demographic data, survey responses, or other types of data.
2. Collect the data: Gather the data for each variable and record it in a structured format. This might involve coding the data or using numerical values to represent different categories.
3. Create the table: Create a table with rows and columns that correspond to the variables and categories of the data. The table should be clear and easy to read, with appropriate headings and
4. Populate the table: Enter the data into the appropriate cells of the table, ensuring that each observation is recorded in the correct row and each variable is recorded in the correct column.
5. Analyze the data: Use the table to identify patterns and trends in the data, and to make comparisons between different groups or categories.
Overall, tabulation is a useful tool for summarizing and presenting large amounts of data in a clear and concise manner, and can help to identify important insights and trends in the data.
Data Sufficiency
Data sufficiency is a type of question commonly used in standardized tests, such as the GMAT, GRE, and SAT. These questions are designed to test your ability to analyze and evaluate information, and
to determine whether the given information is sufficient to answer a specific question or solve a problem.
In a data sufficiency question, you will be given a problem statement and two pieces of information, usually labeled (1) and (2). You will then be asked to determine whether the information provided
is sufficient to answer the question or solve the problem, and to choose one of five possible answer choices:
A. Statement (1) alone is sufficient, but statement (2) alone is not sufficient. B. Statement (2) alone is sufficient, but statement (1) alone is not sufficient. C. Both statements together are
sufficient, but neither statement alone is sufficient. D. Each statement alone is sufficient. E. The statements together are not sufficient.
To solve a data sufficiency question, you should first carefully read the problem statement and each piece of information. Then, consider whether each statement provides enough information to answer
the question or solve the problem. If the information is sufficient, choose the answer choice that corresponds to the statement(s) that provide(s) the information. If the information is not
sufficient, choose the answer choice that corresponds to the statement(s) that do(es) not provide the information.
It is important to note that data sufficiency questions are not meant to test your knowledge of specific concepts or formulas, but rather your ability to analyze and evaluate information. As such, it
is important to approach these questions with a critical and analytical mindset, and to carefully consider each piece of information before making a decision.
Polygons are two-dimensional shapes that have straight sides and angles. They are often used in geometry to study and analyze properties such as perimeter, area, angles, and symmetry. Polygons can
have any number of sides, and they are named based on the number of sides they have.
Here are some examples of polygons and their names based on the number of sides:
• Triangle (3 sides)
• Quadrilateral (4 sides)
• Pentagon (5 sides)
• Hexagon (6 sides)
• Heptagon (7 sides)
• Octagon (8 sides)
• Nonagon (9 sides)
• Decagon (10 sides)
• Undecagon (11 sides)
• Dodecagon (12 sides)
Polygons can also be classified based on their internal angles. For example, a polygon with all angles less than 90 degrees is called a convex polygon, while a polygon with at least one angle greater
than 90 degrees is called a concave polygon.
Some common properties of polygons include:
• The sum of the interior angles of a polygon with n sides is (n-2) times 180 degrees.
• The exterior angle of a polygon is equal to 360/n degrees, where n is the number of sides.
• The perimeter of a polygon is the sum of the lengths of its sides.
• The area of a polygon can be calculated using various formulas, depending on the shape of the polygon and the available information (such as side lengths and angles).
Polygons are an important concept in geometry and are used in many real-world applications, such as architecture, engineering, and graphic design.
Lines & Angles
Lines and angles are fundamental concepts in geometry. A line is a straight path that extends infinitely in both directions, while an angle is the measure of the space between two intersecting lines
or rays.
Here are some key terms related to lines and angles:
• Line segment: a part of a line that has two endpoints.
• Ray: a part of a line that has one endpoint and extends infinitely in one direction.
• Parallel lines: lines that are always the same distance apart and never intersect.
• Perpendicular lines: lines that intersect at a right angle (90 degrees).
• Angle: the measure of the space between two intersecting lines or rays.
• Vertex: the point where two lines or rays intersect to form an angle.
• Acute angle: an angle that measures less than 90 degrees.
• Obtuse angle: an angle that measures more than 90 degrees but less than 180 degrees.
• Right angle: an angle that measures exactly 90 degrees.
• Straight angle: an angle that measures exactly 180 degrees.
• Complementary angles: two angles whose sum is 90 degrees.
• Supplementary angles: two angles whose sum is 180 degrees.
Some key properties of lines and angles include:
• The sum of the angles in a triangle is always 180 degrees.
• In a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse (the side opposite the right angle).
• The opposite angles of a parallelogram are congruent (have the same measure).
• The adjacent angles of a parallelogram are supplementary (have a sum of 180 degrees).
Lines and angles are used in many real-world applications, such as architecture, engineering, and surveying. They are also important in fields such as physics and computer graphics.
In mathematics, a triangle is a closed two-dimensional shape with three straight sides and three angles. Triangles are one of the most basic and important shapes in geometry and are used in a wide
range of mathematical applications.
The three sides of a triangle are usually denoted as a, b, and c, and the three angles are denoted as A, B, and C, where A is the angle opposite side a, B is the angle opposite side b, and C is the
angle opposite side c. The sum of the three angles in a triangle is always 180 degrees.
There are several different types of triangles, including:
1. Equilateral triangle: a triangle with all three sides of equal length and all three angles of 60 degrees.
2. Isosceles triangle: a triangle with two sides of equal length and two angles of equal measure.
3. Scalene triangle: a triangle with no equal sides and no equal angles.
4. Right triangle: a triangle with one angle of 90 degrees (a right angle).
Triangles have many properties and formulas associated with them, including the Pythagorean theorem for right triangles, the law of sines and law of cosines for finding angles and sides of triangles,
and various formulas for calculating the area and perimeter of triangles.
Some important Definitions
Sure, here are some important mathematical definitions:
1. Function: A function is a rule that assigns each input value from a set (the domain) to a unique output value from another set (the range).
2. Equation: An equation is a statement that two expressions are equal. It typically includes one or more variables and may be solved to find the values of the variables that make the equation true.
3. Variable: A variable is a symbol or letter that represents a quantity that can vary or change.
4. Ratio: A ratio is a comparison of two quantities by division. It is typically written in the form of a fraction, with the numerator representing one quantity and the denominator representing the
5. Proportion: A proportion is a statement that two ratios are equal.
6. Absolute value: The absolute value of a number is the distance between the number and zero on the number line. It is always positive, or zero if the number is zero.
7. Exponent: An exponent is a number that indicates how many times a base number is multiplied by itself. It is written as a small superscript to the right of the base number.
8. Logarithm: A logarithm is the inverse operation of exponentiation. It is a mathematical function that gives the power to which a given base number must be raised to produce a given value.
9. Permutation: A permutation is an arrangement of objects in a specific order. The number of permutations of n objects taken r at a time is given by nPr = n!/(n-r)!.
10. Combination: A combination is a selection of objects without regard to order. The number of combinations of n objects taken r at a time is given by nCr = n!/r!(n-r)!.
Some Inequality Relations in a Triangles
Inequality relations in triangles are mathematical statements that describe the relationships between the sides and angles of a triangle. Here are some important inequality relations in triangles:
1. Triangle Inequality Theorem: In any triangle, the sum of the lengths of any two sides is greater than the length of the third side. This can be written as a mathematical expression: a + b > c, b
+ c > a, and c + a > b, where a, b, and c are the lengths of the sides.
2. Law of Cosines: The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. It states that c^2 = a^2 + b^2 – 2ab cos(C), where a, b, and c are the
lengths of the sides and C is the angle opposite the side of length c.
3. Law of Sines: The Law of Sines relates the lengths of the sides of a triangle to the sines of its angles. It states that a/sin(A) = b/sin(B) = c/sin(C), where a, b, and c are the lengths of the
sides and A, B, and C are the angles opposite those sides.
4. Angle-Side Inequality: In any triangle, the measure of an angle is greater than the measure of the angle opposite the shorter side. This can be written as a mathematical expression: A > B if a <
b, B > C if b < c, and C > A if c < a, where A, B, and C are the measures of the angles and a, b, and c are the lengths of the sides.
5. Side-Side Inequality: In any triangle, the length of the longer side is greater than the length of the shorter side opposite the larger angle. This can be written as a mathematical expression: a
> b if A > B, b > c if B > C, and c > a if C > A, where a, b, and c are the lengths of the sides and A, B, and C are the measures of the angles.
Understanding these inequality relations is essential in solving problems related to triangles and in proving various geometric theorems.
Congruent Triangles
Congruent triangles are triangles that have the same size and shape. More specifically, two triangles are said to be congruent if their corresponding sides and angles are equal in measure. Here are
some ways to prove that two triangles are congruent:
1. Side-Side-Side (SSS) Congruence: If the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent.
2. Side-Angle-Side (SAS) Congruence: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
3. Angle-Side-Angle (ASA) Congruence: If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.
4. Angle-Angle-Side (AAS) Congruence: If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the
triangles are congruent.
5. Hypotenuse-Leg (HL) Congruence: If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.
It’s important to note that not all combinations of sides and angles can be used to prove congruence. For example, the Side-Angle-Angle (SAA) and Angle-Angle-Angle (AAA) methods cannot be used to
prove congruence because they do not provide enough information about the triangles. Additionally, it’s important to use the correct order of corresponding sides and angles when comparing triangles
to prove congruence.
Similar Triangles
Similar triangles are triangles that have the same shape but may differ in size. More specifically, two triangles are said to be similar if their corresponding angles are congruent and their
corresponding sides are proportional in length. Here are some ways to determine if two triangles are similar:
1. Angle-Angle (AA) Similarity: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
2. Side-Angle-Side (SAS) Similarity: If two sides of one triangle are proportional to two sides of another triangle, and the included angles are congruent, then the triangles are similar.
3. Side-Side-Side (SSS) Similarity: If the corresponding sides of two triangles are proportional in length, then the triangles are similar.
4. Angle-Side-Angle (ASA) Similarity: If two angles of one triangle are congruent to two angles of another triangle and the included side is proportional in length, then the triangles are similar.
It’s important to note that if two triangles are similar, then their corresponding angles are congruent and their corresponding sides are proportional in length. However, the converse is not
necessarily true – two triangles with congruent angles and proportional sides may not be similar if their shape is different.
Similar triangles have many important applications in geometry, including in the calculation of the heights and distances of objects, the determination of angles, and the construction of scale
Fundamental Properties of Triangles
Here are some fundamental properties of triangles:
1. The sum of the interior angles of a triangle is 180 degrees. This is known as the Triangle Sum Theorem.
2. The exterior angle of a triangle is equal to the sum of the two interior angles that are not adjacent to it. This is known as the Exterior Angle Theorem.
3. The length of any side of a triangle must be less than the sum of the lengths of the other two sides, and greater than the difference of the lengths of the other two sides. This is known as the
Triangle Inequality Theorem.
4. The altitude of a triangle is the perpendicular line segment from a vertex of the triangle to the opposite side or to its extension. The altitudes of a triangle intersect at a point called the
5. The perpendicular bisectors of the sides of a triangle intersect at a point called the circumcenter, which is equidistant from the vertices of the triangle.
6. The angle bisectors of the angles of a triangle intersect at a point called the incenter, which is equidistant from the sides of the triangle.
7. The medians of a triangle are the line segments that connect each vertex of the triangle to the midpoint of the opposite side. The medians of a triangle intersect at a point called the centroid,
which is two-thirds of the distance from each vertex to the midpoint of the opposite side.
Understanding these fundamental properties is essential in solving problems related to triangles and in proving various geometric theorems.
Some Important Theorems
There are many important theorems in geometry related to triangles, circles, and other shapes. Here are some of them:
1. Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
This theorem can be written as a^2 + b^2 = c^2, where c is the length of the hypotenuse and a and b are the lengths of the other two sides.
2. Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
3. Law of Sines: In any triangle, the ratio of the length of a side to the sine of the angle opposite that side is the same for all three sides. This theorem can be written as a/sin A = b/sin B = c/
sin C, where a, b, and c are the lengths of the sides, and A, B, and C are the measures of the angles opposite those sides.
4. Law of Cosines: In any triangle, the square of the length of one side is equal to the sum of the squares of the lengths of the other two sides minus twice the product of the lengths of those
sides and the cosine of the angle between them. This theorem can be written as a^2 = b^2 + c^2 – 2bc cos A, where a is the length of the side opposite angle A, and b and c are the lengths of the
other two sides.
5. Inscribed Angle Theorem: An angle inscribed in a circle is half the measure of the central angle that intercepts the same arc.
6. Angle Bisector Theorem: In a triangle, the angle bisector of an angle divides the opposite side into two segments whose lengths are proportional to the lengths of the adjacent sides.
7. Ptolemy’s Theorem: In a cyclic quadrilateral (a quadrilateral that can be inscribed in a circle), the product of the lengths of the diagonals is equal to the sum of the products of the lengths of
the pairs of opposite sides.
8. Euler’s Formula for Polyhedra: In any convex polyhedron (a solid with flat faces and straight edges), the number of faces (F), vertices (V), and edges (E) are related by the formula F + V – E =
These theorems are just a few examples of the many important results in geometry that have been discovered over the centuries. They have wide-ranging applications in fields such as engineering,
physics, computer science, and architecture.
Pythagoras Theorem
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Pythagoras’ theorem is one of the most famous and useful theorems in geometry. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is
equal to the sum of the squares of the lengths of the other two sides. This theorem is often written as:
a^2 + b^2 = c^2
where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.
In other words, if we have a right triangle with legs of length a and b and hypotenuse of length c, then c^2 = a^2 + b^2.
For example, consider a right triangle with legs of length 3 and 4. Using Pythagoras’ theorem, we can find the length of the hypotenuse as follows:
c^2 = 3^2 + 4^2 c^2 = 9 + 16 c^2 = 25 c = 5
So the length of the hypotenuse is 5.
Pythagoras’ theorem has many applications in mathematics, science, and engineering. It is used, for example, in calculating distances and velocities in physics, in designing structures such as
bridges and buildings, and in computer graphics and image processing.
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Here are some miscellaneous topics related to mathematics and geometry:
1. Calculus: Calculus is a branch of mathematics that deals with the study of rates of change and how things change over time. It is used in a wide range of fields, including physics, engineering,
economics, and computer science.
2. Vectors: Vectors are mathematical objects that have both magnitude and direction. They are used in many areas of mathematics and science, including physics, engineering, and computer graphics.
3. Fractals: Fractals are geometric patterns that repeat themselves at different scales. They are found in many natural and man-made objects, such as snowflakes, clouds, and coastlines.
4. Topology: Topology is a branch of mathematics that studies the properties of objects that are preserved under continuous transformations, such as stretching, bending, and twisting. It has
applications in many fields, including physics, engineering, and computer science.
5. Graph Theory: Graph theory is the study of graphs, which are mathematical objects that represent networks of connections between objects. It is used in many areas, such as computer science,
social sciences, and transportation planning.
6. Probability Theory: Probability theory is the branch of mathematics that studies the likelihood of events occurring. It is used in many areas, such as statistics, finance, and game theory.
7. Trigonometry: Trigonometry is the study of the relationships between the angles and sides of triangles. It has applications in many areas, including engineering, physics, and navigation.
8. Number Theory: Number theory is the study of the properties and relationships of numbers. It has applications in many areas, including cryptography, computer science, and physics.
These are just a few examples of the many interesting and useful topics in mathematics and geometry.
In geometry, a quadrilateral is a polygon with four sides and four angles. There are many different types of quadrilaterals, each with its own set of properties and characteristics. Here are some of
the most common types:
1. Rectangle: A rectangle is a quadrilateral with four right angles. The opposite sides of a rectangle are parallel and congruent. The perimeter of a rectangle is equal to the sum of the lengths of
all four sides, and the area is equal to the product of the length and width.
2. Square: A square is a special type of rectangle with all four sides congruent. All four angles of a square are right angles. The perimeter of a square is equal to four times the length of one
side, and the area is equal to the length of one side squared.
3. Parallelogram: A parallelogram is a quadrilateral with opposite sides parallel and congruent. The opposite angles of a parallelogram are also congruent. The perimeter of a parallelogram is equal
to the sum of the lengths of all four sides, and the area is equal to the product of the base and the height.
4. Rhombus: A rhombus is a parallelogram with all four sides congruent. The opposite angles of a rhombus are congruent, and the diagonals bisect each other at right angles. The perimeter of a
rhombus is equal to four times the length of one side, and the area is equal to half the product of the diagonals.
5. Trapezoid: A trapezoid is a quadrilateral with at least one pair of parallel sides. The parallel sides are called the bases, and the non-parallel sides are called the legs. The height of a
trapezoid is the perpendicular distance between the bases. The perimeter of a trapezoid is equal to the sum of the lengths of all four sides, and the area is equal to half the product of the
height and the sum of the bases.
These are just a few examples of the many types of quadrilaterals in geometry. Each type has its own unique properties and can be used in a variety of geometric problems and applications.
A parallelogram is a quadrilateral with opposite sides parallel and congruent. This means that the opposite sides of a parallelogram are equal in length and do not intersect. The opposite angles of a
parallelogram are also congruent, meaning they have the same measure.
Properties of a parallelogram:
1. Opposite sides are parallel and congruent.
2. Opposite angles are congruent.
3. Consecutive angles are supplementary (add up to 180 degrees).
4. Diagonals bisect each other (the point where the diagonals intersect divides each diagonal into two equal parts).
5. The area of a parallelogram is equal to the product of the base and the height.
6. The perimeter of a parallelogram is equal to the sum of the lengths of all four sides.
Some special types of parallelograms include:
1. Rectangle: A parallelogram with four right angles.
2. Square: A parallelogram with four congruent sides and four right angles.
3. Rhombus: A parallelogram with four congruent sides, opposite angles congruent, and diagonals that bisect each other at right angles.
Parallelograms are used in many real-world applications, such as in construction and engineering to create structures with strong, stable foundations. They also appear in geometry problems and
proofs, and are important shapes to understand in mathematics.
A rhombus is a type of parallelogram with four congruent sides. It is a special case of a parallelogram and has all of the properties of a parallelogram. In addition, a rhombus has some unique
properties that set it apart from other parallelograms.
Properties of a rhombus:
1. All four sides are congruent.
2. Opposite angles are congruent.
3. Consecutive angles are supplementary (add up to 180 degrees).
4. Diagonals bisect each other at right angles.
5. The area of a rhombus is equal to half the product of the diagonals.
6. The perimeter of a rhombus is equal to four times the length of one side.
Some special types of rhombuses include:
1. Square: A rhombus with four right angles.
2. Acute rhombus: A rhombus with all acute angles (angles less than 90 degrees).
3. Obtuse rhombus: A rhombus with one obtuse angle (angle greater than 90 degrees).
Rhombuses are used in many real-world applications, such as in jewelry design and in construction to create strong and stable foundations. They also appear in geometry problems and proofs, and are
important shapes to understand in mathematics.
A rectangle is a type of parallelogram with four right angles. It is a special case of a parallelogram and has all of the properties of a parallelogram. In addition, a rectangle has some unique
properties that set it apart from other parallelograms.
Properties of a rectangle:
1. Opposite sides are parallel and congruent.
2. Opposite angles are congruent and each measures 90 degrees.
3. Consecutive angles are supplementary (add up to 180 degrees).
4. Diagonals are congruent and bisect each other.
5. The area of a rectangle is equal to the product of the length and width.
6. The perimeter of a rectangle is equal to twice the sum of the length and width.
Some special types of rectangles include:
1. Square: A rectangle with four congruent sides.
2. Golden rectangle: A rectangle with side lengths in the golden ratio, which is approximately 1.618.
Rectangles are used in many real-world applications, such as in architecture and engineering to create structures with right angles, such as buildings and bridges. They also appear in geometry
problems and proofs, and are important shapes to understand in mathematics.
A square is a type of rectangle with four congruent sides and four right angles. It is a special case of both a rectangle and a rhombus, and has all of the properties of both.
Properties of a square:
1. All four sides are congruent.
2. Opposite sides are parallel and congruent.
3. Opposite angles are congruent and each measures 90 degrees.
4. Consecutive angles are supplementary (add up to 180 degrees).
5. Diagonals are congruent and bisect each other at right angles.
6. The area of a square is equal to the square of the length of one side.
7. The perimeter of a square is equal to four times the length of one side.
Squares are used in many real-world applications, such as in tile design, flooring, and in creating objects with equal sides and angles, such as boxes and frames. They also appear in geometry
problems and proofs, and are important shapes to understand in mathematics.
A trapezoid, also known as a trapezium in some countries, is a quadrilateral with at least one pair of parallel sides. It is a non-symmetrical shape that has no congruent sides or angles, except in
the case of an isosceles trapezoid.
Properties of a trapezoid:
1. At least one pair of opposite sides is parallel.
2. The non-parallel sides are not congruent.
3. The opposite angles are not congruent, except in the case of an isosceles trapezoid.
4. The diagonals of a trapezoid intersect each other.
5. The mid-segment of a trapezoid is parallel to both bases and is equal to half the sum of the lengths of the bases.
6. The area of a trapezoid is equal to half the product of the height and the sum of the lengths of the bases.
Some special types of trapezoids include:
1. Isosceles trapezoid: A trapezoid with congruent base angles and congruent legs.
2. Right trapezoid: A trapezoid with one right angle.
Trapezoids are used in many real-world applications, such as in architecture and engineering to create structures with parallel sides, such as bridges and roofs. They also appear in geometry problems
and proofs, and are important shapes to understand in mathematics.
A circle is a two-dimensional shape that is defined as the set of all points in a plane that are equidistant from a given point called the center. It is one of the most important shapes in geometry,
and has many applications in math and the real world.
Properties of a circle:
1. A circle is defined by its center and its radius, which is the distance from the center to any point on the circle.
2. The diameter of a circle is a line segment that passes through the center and has endpoints on the circle. The diameter is equal to twice the radius.
3. The circumference of a circle is the distance around the circle, and is equal to 2π times the radius, where π is a mathematical constant approximately equal to 3.14159.
4. The area of a circle is equal to π times the square of the radius.
5. Chord: a line segment connecting two points on the circumference of a circle.
6. Tangent: a line that touches the circumference of a circle at one point.
7. Secant: a line that intersects a circle at two points.
8. Arc: a portion of the circumference of a circle.
9. Sector: a portion of the area of a circle bounded by two radii and an arc.
10. Segment: a portion of the area of a circle bounded by a chord and an arc.
Circles are used in many real-world applications, such as in geometry, physics, and engineering to calculate the area and circumference of objects. They also appear in many aspects of everyday life,
such as in wheels, clocks, and other circular objects.
Circle and Its Chords
In a circle, a chord is a line segment that connects two points on the circumference of the circle. Here are some important properties of circles and their chords:
1. The diameter is the longest chord in a circle, and it passes through the center of the circle. The length of the diameter is twice the length of the radius.
2. Any chord that passes through the center of the circle is a diameter.
3. Any chord that is perpendicular to a diameter bisects the diameter, dividing it into two equal parts.
4. Chords that are equidistant from the center of the circle are congruent.
5. The perpendicular bisector of a chord passes through the center of the circle.
6. If two chords intersect inside a circle, the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.
7. If a diameter is perpendicular to a chord, then it bisects the chord and its two segments.
8. If two chords are congruent, then the chords are equidistant from the center of the circle.
9. The measure of an angle formed by two chords that intersect inside the circle is equal to half the sum of the measures of the intercepted arcs.
Circles and their chords are used in many applications, such as in music theory to understand the relationship between notes and in geometry problems and proofs. Understanding the properties of
circles and their chords is important in mathematics and in many other fields
Cyclic quadrilateral
A cyclic quadrilateral is a quadrilateral that can be inscribed in a circle, which means that all four vertices of the quadrilateral lie on the circumference of the circle. Here are some important
properties of cyclic quadrilaterals:
1. The opposite angles of a cyclic quadrilateral are supplementary, which means that the sum of the measures of any two opposite angles is 180 degrees.
2. The measure of an angle formed by a tangent and a chord at the point of tangency is equal to the measure of the intercepted arc.
3. The sum of the measures of the two opposite interior angles of a cyclic quadrilateral is equal to 180 degrees.
4. The product of the measures of the diagonals of a cyclic quadrilateral is equal to the sum of the products of the measures of its opposite sides.
5. The perpendicular bisectors of the sides of a cyclic quadrilateral intersect at a common point, which is the center of the circle.
6. The line segments joining the midpoints of the opposite sides of a cyclic quadrilateral are parallel and equal in length.
7. The area of a cyclic quadrilateral can be calculated using Brahmagupta’s formula: A = sqrt((s – a)(s – b)(s – c)(s – d)), where s is the semiperimeter of the quadrilateral, and a, b, c, and d are
the lengths of its sides.
Cyclic quadrilaterals have many applications in geometry and other fields, such as in engineering, architecture, and physics. Understanding the properties of cyclic quadrilaterals can help in solving
many geometry problems and in proving geometric theorems.
Angles Subtended by Chords of A Circle
When a chord of a circle is drawn, it subtends an angle at each of the endpoints of the chord. Here are some important properties of angles subtended by chords of a circle:
1. An angle subtended by a chord at a point on the circumference of the circle is half the measure of the intercepted arc.
2. If two chords in a circle are equal in length, then the angles subtended by those chords at the center of the circle are equal.
3. If two chords in a circle subtend equal angles at the center, then the chords are equal in length.
4. If a diameter of a circle is drawn perpendicular to a chord, then the diameter bisects the chord and its subtended angle.
5. If two chords intersect inside a circle, then the products of the lengths of the segments of one chord are equal to the products of the lengths of the segments of the other chord.
6. If two chords of a circle are parallel, then they subtend equal angles at the circumference of the circle.
7. If a chord of a circle is extended to form a tangent to the circle, then the angle between the tangent and the chord is equal to the angle subtended by the chord at the point of contact.
Understanding the properties of angles subtended by chords of a circle can help in solving many geometry problems, including finding the length of a chord or the radius of a circle, determining the
position of a point on a circle, or calculating the area of a sector of a circle.
A tangent is a line that intersects a circle at only one point, called the point of tangency. Here are some important properties of tangents:
1. A tangent to a circle is perpendicular to the radius drawn to the point of tangency.
2. If two tangents are drawn to a circle from the same external point, then they are equal in length.
3. The tangent and radius drawn to the point of tangency form a right angle.
4. The tangent and the chord drawn to the point of tangency form an angle equal to the angle subtended by the chord in the opposite segment.
5. If a line intersects a circle and is not a tangent, then it intersects the circle in two points.
6. The length of a tangent from an external point to a circle is equal to the geometric mean of the distances from the point to the center of the circle and to the point of intersection of the
tangent with the radius drawn to the point of tangency.
Tangents have many applications in geometry and other fields, such as in physics, engineering, and architecture. Understanding the properties of tangents can help in solving many geometry problems,
including finding the length of a tangent, determining the position of a point on a circle, or calculating the area of a sector of a circle.
Common Tangents of Two or More Circles
When two or more circles intersect, they can have common tangents, which are straight lines that are tangents to both circles. Here are some important properties of common tangents of two or more
1. If two circles intersect at two points, then the line joining the centers of the circles bisects the common chord, and the perpendicular to this line at the point of intersection of the circles
passes through the midpoint of the common chord.
2. If two circles intersect at a single point, then there is only one common tangent at that point.
3. If two circles are externally tangent, then the line joining their centers is the common tangent.
4. If two circles are internally tangent, then the line joining their centers is the common external tangent, and there are two common internal tangents, one that is closer to the centers of the
circles and one that is farther away.
5. If three circles intersect in pairs, then there can be three common external tangents and three common internal tangents.
6. If four circles intersect in pairs, then there can be eight common tangents, four external and four internal.
Knowing the properties of common tangents of two or more circles can help in solving many geometry problems involving circles, such as finding the length of a tangent or the distance between two
Common Chord
In geometry, the term “common chord” refers to a line that intersects two or more circles at distinct points. Here are some important properties of common chords:
1. The common chord of two circles intersects the line joining the centers of the circles at its midpoint.
2. If two circles intersect at two points, then the common chord is the line segment connecting the two points of intersection.
3. If two circles are tangent, then the common chord is the point of tangency.
4. If three circles intersect in pairs, then each pair has a common chord, and these three chords are concurrent.
5. If four circles intersect in pairs, then each pair has two common chords, and these eight chords are concurrent.
6. The length of a common chord can be calculated using the power of a point theorem, which states that the product of the lengths of the two segments of a secant (or a tangent) drawn from an
external point to a circle is equal to the square of the length of the tangent segment from the same point to the circle.
Knowing the properties of common chords can be useful in solving many geometry problems involving circles, such as finding the length of a chord or the distance between two circles.
Surface Areas
Surface area is the measure of the total area that the surface of an object occupies. It is a fundamental concept in geometry and is used to calculate the amount of material needed to cover the
surface of an object. The surface area of an object can be calculated based on its shape and dimensions.
Here are some common formulas for calculating the surface area of various 3D shapes:
• Cube: 6s^2, where s is the length of a side
• Rectangular prism: 2lw + 2lh + 2wh, where l, w, and h are the length, width, and height, respectively
• Sphere: 4πr^2, where r is the radius
• Cylinder: 2πr^2 + 2πrh, where r is the radius and h is the height
• Cone: πr^2 + πrl, where r is the radius, l is the slant height, and l^2 = r^2 + h^2 (where h is the height)
Surface area is an important concept in many fields, such as engineering, architecture, and manufacturing. For example, in architecture, the surface area of a building is an important consideration
when determining the amount of materials needed for construction. In manufacturing, the surface area of a product is an important factor in determining the cost of materials and the amount of
packaging needed.
In mathematics, the average (also called the mean) is a measure of central tendency that represents the typical value in a set of numbers. It is calculated by adding up all the numbers in a set and
then dividing the sum by the total number of values in the set.
There are three types of averages:
1. Arithmetic mean: It is the most commonly used type of average and is calculated by adding up all the values in a set and dividing by the total number of values.
2. Median: It is the middle value in a set of numbers when they are arranged in order from smallest to largest. If there are an even number of values, the median is the average of the two middle
3. Mode: It is the value that appears most frequently in a set of numbers. A set of numbers can have more than one mode or no mode at all.
Averages are used in many fields, including statistics, finance, and science, to summarize and analyze data. They can be used to calculate trends, make predictions, and compare different sets of
A quadrilateral is a four-sided polygon with four vertices (corners) and four sides. Quadrilaterals can have a variety of properties depending on their angles and side lengths. Here are some common
types of quadrilaterals and their properties:
1. Parallelogram: A parallelogram is a quadrilateral with opposite sides parallel. Its properties include:
• Opposite sides are congruent.
• Opposite angles are congruent.
• Consecutive angles are supplementary.
• The diagonals bisect each other.
2. Rectangle: A rectangle is a parallelogram with all angles equal to 90 degrees (right angles). Its properties include:
• Opposite sides are parallel and congruent.
• All angles are congruent (equal to 90 degrees).
• The diagonals are congruent.
3. Rhombus: A rhombus is a parallelogram with all sides equal in length. Its properties include:
• Opposite angles are congruent.
• Consecutive angles are supplementary.
• The diagonals are perpendicular and bisect each other.
4. Square: A square is a rectangle and a rhombus with all sides equal in length and all angles equal to 90 degrees. Its properties include:
• All sides are congruent.
• All angles are congruent (equal to 90 degrees).
• The diagonals are congruent and perpendicular bisectors of each other.
5. Trapezoid: A trapezoid is a quadrilateral with one pair of parallel sides. Its properties include:
• Non-parallel sides are congruent.
• Consecutive angles are supplementary.
• The diagonals do not bisect each other.
6. Kite: A kite is a quadrilateral with two pairs of adjacent sides that are equal in length. Its properties include:
• One diagonal is the perpendicular bisector of the other diagonal.
• The angles between the unequal sides are congruent.
Knowing the properties of these common types of quadrilaterals can be useful in solving geometry problems, such as finding the area or perimeter of a shape or determining the relationships between
different shapes
A circle is a two-dimensional shape consisting of all points that are equidistant from a given point called the center. The distance from the center to any point on the circle is called the radius,
and the distance across the circle passing through the center is called the diameter. Here are some key properties of circles:
1. Circumference: The circumference of a circle is the distance around the circle. It is given by the formula C = 2πr, where r is the radius and π (pi) is a mathematical constant approximately equal
to 3.14.
2. Area: The area of a circle is the amount of space inside the circle. It is given by the formula A = πr^2.
3. Chord: A chord is a line segment that connects two points on the circle. The diameter is the longest chord, as it passes through the center of the circle.
4. Tangent: A tangent is a line that intersects the circle at exactly one point. The point where the tangent intersects the circle is called the point of tangency.
5. Secant: A secant is a line that intersects the circle at two points.
6. Arc: An arc is a portion of the circle’s circumference. The measure of an arc is given in degrees or radians.
7. Central angle: A central angle is an angle whose vertex is at the center of the circle, and whose sides pass through two points on the circle.
Circles have a wide range of applications in mathematics, physics, engineering, and other fields. They are used to model real-world phenomena such as planetary orbits, sound waves, and electric
fields. Understanding the properties of circles is essential for solving problems in geometry and other mathematical disciplines.
A polygon is a two-dimensional shape with straight sides that is formed by three or more line segments that are connected end to end. Polygons can have any number of sides, and the name of the
polygon is usually based on the number of sides it has. Here are some common polygons:
1. Triangle: A polygon with three sides.
2. Quadrilateral: A polygon with four sides.
3. Pentagon: A polygon with five sides.
4. Hexagon: A polygon with six sides.
5. Heptagon: A polygon with seven sides.
6. Octagon: A polygon with eight sides.
7. Nonagon: A polygon with nine sides.
8. Decagon: A polygon with ten sides.
Polygons are classified based on their number of sides, angles, and other properties. For example, a polygon is called a regular polygon if all of its sides and angles are equal. Some other important
properties of polygons are:
1. Perimeter: The perimeter of a polygon is the sum of the lengths of its sides.
2. Area: The area of a polygon is the amount of space inside the polygon.
3. Interior angles: The interior angles of a polygon are the angles formed by any two adjacent sides inside the polygon.
4. Exterior angles: The exterior angles of a polygon are the angles formed by one side of the polygon and the extension of an adjacent side.
Polygons have many applications in real-world situations, such as in architecture, engineering, and art. Understanding the properties of polygons is important for solving problems in geometry and
other mathematical fields.
Here are some additional miscellaneous concepts in geometry:
1. Vectors: Vectors are mathematical objects that have both magnitude and direction. In geometry, vectors can be used to represent translations, rotations, and other transformations.
2. Cartesian coordinates: Cartesian coordinates are a system used to locate points in two or three-dimensional space. In this system, each point is represented by a pair or triplet of numbers that
give its position relative to a set of axes.
3. Three-dimensional shapes: Three-dimensional shapes, also known as solid shapes, are shapes that have length, width, and height. Examples of three-dimensional shapes include cubes, spheres, cones,
and cylinders.
4. Platonic solids: Platonic solids are a group of five regular polyhedra, which are three-dimensional shapes that have flat faces and straight edges. The five platonic solids are the tetrahedron,
cube, octahedron, dodecahedron, and icosahedron.
5. Fractals: Fractals are mathematical objects that have a self-similar pattern at different scales. In geometry, fractals can be used to model natural phenomena such as coastlines, trees, and
6. Trigonometry: Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It has applications in fields such as navigation, astronomy, and
7. Topology: Topology is a branch of mathematics that studies the properties of shapes that are preserved under continuous transformations. It has applications in fields such as robotics, computer
graphics, and data analysis.
These are just a few examples of the many concepts and applications in geometry. Understanding these concepts can help you develop problem-solving skills and make connections to other areas of
mathematics and science.
Volume is a measure of the amount of space occupied by a three-dimensional object. The volume of an object is typically measured in cubic units, such as cubic meters, cubic centimeters, or cubic
The formula for finding the volume of some common three-dimensional shapes are:
1. Cube: V = s^3, where s is the length of one side of the cube.
2. Rectangular prism: V = lwh, where l is the length, w is the width, and h is the height of the prism.
3. Cylinder: V = πr^2h, where r is the radius of the base of the cylinder and h is the height of the cylinder.
4. Sphere: V = (4/3)πr^3, where r is the radius of the sphere.
5. Cone: V = (1/3)πr^2h, where r is the radius of the base of the cone and h is the height of the cone.
The volume of irregularly shaped objects can be calculated using methods such as water displacement or integration.
Volumes have many applications in real-world situations, such as in architecture, engineering, and manufacturing. Understanding the concept of volume is important for solving problems in geometry,
physics, and other mathematical fields.
A cuboid is a three-dimensional geometric shape that has six rectangular faces. It is also known as a rectangular prism. The faces of a cuboid are arranged in pairs of opposite, parallel rectangles,
and its edges are perpendicular to adjacent edges.
The formula for the volume of a cuboid is V = lwh, where l is the length, w is the width, and h is the height of the cuboid. The formula for the surface area of a cuboid is SA = 2lw + 2lh + 2wh,
where SA is the surface area of the cuboid.
Some examples of real-life objects that have a cuboid shape include books, shoeboxes, and bricks. Cuboids are also commonly used in architecture and engineering for building design and structural
In addition to volume and surface area, other important properties of a cuboid include its diagonal length, which can be calculated using the Pythagorean theorem, and its centroid, which is the point
at which the three medians of the cuboid intersect.
A cube is a three-dimensional geometric shape that has six equal square faces. All of the edges of a cube are the same length, and all of the angles between the faces are right angles. The cube is a
special case of a cuboid where all sides have equal lengths.
The formula for the volume of a cube is V = s^3, where s is the length of one of its edges. The formula for the surface area of a cube is SA = 6s^2, where SA is the surface area of the cube.
Some examples of real-life objects that have a cube shape include dice, Rubik’s cubes, and sugar cubes. Cubes are also used in mathematics and engineering for modeling and solving problems in various
Cubes have several unique properties, such as having the maximum possible volume for a given surface area among all three-dimensional shapes, and having the maximum possible symmetry among all
Platonic solids. The diagonal length of a cube can be calculated using the Pythagorean theorem, and the centroid of a cube is the point where the three medians of the cube intersect.
A cylinder is a three-dimensional geometric shape that has two circular faces or bases connected by a curved surface. The curved surface of a cylinder is called the lateral surface, and the distance
between the two bases is the height of the cylinder.
The formula for the volume of a cylinder is V = πr^2h, where r is the radius of the base and h is the height of the cylinder. The formula for the surface area of a cylinder is SA = 2πr^2 + 2πrh,
where SA is the surface area of the cylinder.
Some examples of real-life objects that have a cylindrical shape include cans, pipes, and barrels. Cylinders are also used in engineering and mathematics for modeling and solving problems in various
Cylinders have several unique properties, such as having constant cross-sectional area and volume, and having the maximum possible volume for a given surface area among all three-dimensional shapes
with a given surface area. The centroid of a cylinder is the point where the axis of symmetry intersects the base, and the moment of inertia of a cylinder about its axis is I = (1/2)mr^2, where m is
the mass of the cylinder and r is the radius of the base.
A prism is a three-dimensional geometric shape that has two parallel and congruent faces called bases, connected by rectangular or parallelogram-shaped sides. The height of a prism is the
perpendicular distance between the two bases.
Prisms are named according to the shape of their base. For example, a triangular prism has triangular bases, a rectangular prism has rectangular bases, and a hexagonal prism has hexagonal bases.
The formula for the volume of a prism is V = Bh, where B is the area of the base and h is the height of the prism. The formula for the surface area of a prism is SA = 2B + Ph, where P is the
perimeter of the base and h is the height of the prism.
Some examples of real-life objects that have a prism shape include buildings, tents, and packaging boxes. Prisms are also used in mathematics and engineering for modeling and solving problems in
various fields.
Prisms have several unique properties, such as having constant cross-sectional area and volume, and having the maximum possible volume for a given surface area among all three-dimensional shapes with
a given surface area. The centroid of a prism is the point where the three medians of the base intersect, and the moment of inertia of a prism about its axis is I = (1/12)Bh^3, where B is the area of
the base and h is the height of the prism.
A pyramid is a three-dimensional geometric shape that has a polygonal base and triangular sides that converge at a single point called the apex. The height of a pyramid is the perpendicular distance
between the apex and the base.
Pyramids are named according to the shape of their base. For example, a triangular pyramid has a triangular base, a square pyramid has a square base, and a pentagonal pyramid has a pentagonal base.
The formula for the volume of a pyramid is V = (1/3)Bh, where B is the area of the base and h is the height of the pyramid. The formula for the surface area of a pyramid is SA = B + (1/2)Pl, where P
is the perimeter of the base, l is the slant height of the pyramid, and B is the area of the base.
Some examples of real-life objects that have a pyramid shape include the Great Pyramids of Giza in Egypt, the Transamerica Pyramid in San Francisco, and the Louvre Pyramid in Paris. Pyramids are also
used in mathematics and engineering for modeling and solving problems in various fields.
Pyramids have several unique properties, such as having a vertex angle that is equal to the sum of the angles of the base and having the maximum possible volume for a given surface area among all
three-dimensional shapes with a given surface area. The centroid of a pyramid is the point where the medians of the base intersect, and the moment of inertia of a pyramid about its axis is I = (1/12)
Bh^3, where B is the area of the base and h is the height of the pyramid.
A tetrahedron is a three-dimensional geometric shape that has four triangular faces, four vertices, and six edges. It is the simplest polyhedron, and it is also the only polyhedron that has no
parallel faces.
The formula for the volume of a tetrahedron is V = (1/3)Bh, where B is the area of the base and h is the height of the tetrahedron. The formula for the surface area of a tetrahedron is SA = (1/2)Pl,
where P is the perimeter of the base and l is the slant height of the tetrahedron.
Some real-life examples of tetrahedra include the carbon atom in diamond and the pyramid-shaped dice used in some board games. Tetrahedra also have important applications in fields such as chemistry,
physics, and computer graphics.
Tetrahedra have several unique properties, such as having a vertex angle that is equal to the sum of the angles of the opposite face and having the minimum possible surface area for a given volume
among all three-dimensional shapes with a given volume. The centroid of a tetrahedron is the point where the medians of the four faces intersect, and the moment of inertia of a tetrahedron about its
axis is I = (1/20)Ma^2, where M is the mass of the tetrahedron and a is the edge length.
A sphere is a three-dimensional geometric shape that is perfectly round and symmetrical in all directions. It is defined as the set of all points in three-dimensional space that are equidistant from
a given point called the center.
The formula for the volume of a sphere is V = (4/3)πr^3, where r is the radius of the sphere. The formula for the surface area of a sphere is SA = 4πr^2.
Spheres have several important applications in fields such as physics, astronomy, and engineering. For example, planets and stars are often modeled as spheres, and the shape of a water droplet is
approximately spherical due to surface tension.
Spheres have several unique properties, such as having the minimum possible surface area for a given volume among all three-dimensional shapes, and having the maximum possible volume for a given
surface area among all three-dimensional shapes. The centroid of a sphere is the center, and the moment of inertia of a sphere about any axis through its center is I = (2/5)Mr^2, where M is the mass
of the sphere and r is the radius.
Spherical geometry is a type of non-Euclidean geometry that deals with the properties of spheres and their interactions with planes and other spheres. It has applications in fields such as astronomy,
navigation, and computer graphics.
A hemisphere is a half of a sphere, or a three-dimensional shape that is obtained by cutting a sphere along a plane passing through its center. A hemisphere has a curved surface and a flat circular
base, and it is symmetrical about its base.
The formula for the volume of a hemisphere is V = (2/3)πr^3, where r is the radius of the hemisphere. The formula for the surface area of a hemisphere is SA = 2πr^2.
Hemispheres have several important applications in fields such as physics, engineering, and geography. For example, the Earth is often modeled as a hemisphere in order to simplify calculations of the
planet’s gravitational field or its magnetic field.
Hemispheres have several unique properties, such as having half the surface area and half the volume of a full sphere, and having a centroid that is located on the axis of symmetry, a distance of 3/8
of the radius from the base.
Hemispheres are also commonly used in architectural and design applications, such as creating domes or half-dome structures, and in food preparation, such as creating half-spheres of gelatin or
chocolate for decoration or presentation.
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a single point called the apex or vertex. A cone has a curved surface and a circular base, and it can be thought
of as a pyramid with an infinite number of faces that become smaller and smaller as they approach the apex.
The formula for the volume of a cone is V = (1/3)πr^2h, where r is the radius of the base and h is the height of the cone. The formula for the surface area of a cone is SA = πr^2 + πrl, where l is
the slant height of the cone, which is the distance from the apex to any point on the edge of the base.
Cones have several important applications in fields such as engineering, physics, and architecture. For example, traffic cones are used to direct traffic, and conical structures such as cooling
towers and smokestacks are used in industrial applications.
Cones have several unique properties, such as having a centroid that is located one-fourth of the way from the base to the apex, and having a moment of inertia that depends on the orientation of the
axis of rotation with respect to the axis of symmetry.
Conical objects are also commonly used in design and decorative applications, such as creating party hats, lampshades, or ice cream cones.
A frustum is a geometric shape that is obtained by cutting off the top of a cone or pyramid with a plane parallel to the base. A frustum has two circular bases that are parallel to each other, and a
curved surface that connects them. The frustum can be thought of as a truncated cone or pyramid.
The formula for the volume of a frustum is V = (1/3)πh(R^2 + r^2 + Rr), where h is the height of the frustum, R is the radius of the larger base, and r is the radius of the smaller base. The formula
for the surface area of a frustum is SA = π(R + r)l + πR^2 + πr^2, where l is the slant height of the frustum.
Frusta have several important applications in fields such as engineering, architecture, and construction. For example, the shape of a frustum can be used in the design of buildings, bridges, and
other structures. The frustum shape is also used in optics to describe the shape of the lens of a camera or telescope.
Frusta have several unique properties, such as having a centroid that is located on the axis of symmetry, and having a moment of inertia that depends on the orientation of the axis of rotation with
respect to the axis of symmetry.
Frusta are also commonly used in decorative and design applications, such as creating lampshades or pottery.
Problems Based on Swimming Pool
Here are some problems based on swimming pool:
1. A rectangular swimming pool is 20 meters long and 10 meters wide. The depth of the shallow end is 1.5 meters, and the depth of the deep end is 3 meters. What is the average depth of the pool?
Solution: The average depth of the pool can be calculated by taking the sum of the depths and dividing by 2. The sum of the depths is 1.5 + 3 = 4.5 meters. Dividing by 2 gives an average depth of
2.25 meters.
2. A circular swimming pool has a radius of 6 meters. The pool is being filled at a rate of 3 cubic meters per minute. How long will it take to fill the pool?
Solution: The formula for the volume of a cylinder (which a circular pool can be approximated as) is V = πr^2h, where r is the radius and h is the height (or depth) of the cylinder. Since the pool is
circular and has a depth of 6 meters, the volume of the pool is V = π(6)^2(6) = 678.58 cubic meters. Dividing this by the rate of filling gives a time of 226.19 minutes, or approximately 3.77 hours.
3. A swimming pool is in the shape of a rectangular prism with dimensions 8 meters by 12 meters by 2 meters. The pool is being drained at a rate of 0.5 cubic meters per minute. How long will it take
to drain the pool?
Solution: The volume of the pool is given by V = lwh = 8 x 12 x 2 = 192 cubic meters. Dividing this by the rate of draining gives a time of 384 minutes, or 6.4 hours.
4. A swimming pool is 25 meters long, 12 meters wide, and 3 meters deep. The pool is being filled at a rate of 2 cubic meters per minute. At the same time, water is leaking out of a crack in the
bottom of the pool at a rate of 0.5 cubic meters per minute. How long will it take for the pool to be completely filled?
Solution: The volume of the pool is V = lwh = 25 x 12 x 3 = 900 cubic meters. The net rate of filling is 2 – 0.5 = 1.5 cubic meters per minute. Dividing the volume of the pool by the net rate of
filling gives a time of 600 minutes, or 10 hours.
Problems Based on Pond and Well
Here’s an example problem based on ponds and wells in math:
Problem: A rectangular pond with a length of 12 meters and a width of 6 meters is surrounded by a walkway of uniform width. If the total area of the pond and the walkway is 240 square meters, what is
the width of the walkway?
Solution: Let’s assume that the width of the walkway is “x” meters. Then, the overall length of the pond and walkway combined will be 12 + 2x meters, and the overall width will be 6 + 2x meters.
The area of the pond itself can be calculated as length times width, which is 12 x 6 = 72 square meters. The total area of the pond and walkway combined is given as 240 square meters. Therefore, the
area of the walkway can be calculated as the difference between the two, which is:
Total area – Pond area = 240 – 72 = 168 square meters
The area of the walkway can also be expressed as the product of its width and the perimeter of the pond:
Width x Perimeter = 168
Now, we need to express the perimeter in terms of the width of the walkway. The perimeter of the pond will be equal to the sum of all sides. In this case, we have:
Perimeter = 12 + 6 + 2x + 2x = 18 + 4x
Substituting this into the equation above, we get:
Width x (18 + 4x) = 168
18x + 4x^2 = 168
Dividing both sides by 2, we get:
9x + 2x^2 = 84
Rearranging this into the standard form of a quadratic equation:
2x^2 + 9x – 84 = 0
We can now solve for x using the quadratic formula:
x = (-9 ± sqrt(9^2 – 4(2)(-84))) / 4
x = (-9 ± sqrt(1053)) / 4
x = (-9 ± 32.5) / 4
Since the width of the walkway cannot be negative, we can ignore the negative solution. Therefore:
x = (32.5 – 9) / 4 = 5.125
Hence, the width of the walkway is approximately 5.125 meters.
Problems Based on Cuboid Box
Here are a few problems based on cuboid boxes:
Problem 1: A cuboid box has a length of 20 cm, a breadth of 15 cm, and a height of 10 cm. Find its total surface area.
The total surface area of a cuboid box is given by the formula:
Total surface area = 2(lb + bh + hl)
where l, b, and h are the length, breadth, and height of the cuboid box, respectively.
Substituting the given values, we get:
Total surface area = 2(20 x 15 + 15 x 10 + 10 x 20) cm^2 Total surface area = 2(300 + 150 + 200) cm^2 Total surface area = 2 x 650 cm^2 Total surface area = 1300 cm^2
Therefore, the total surface area of the cuboid box is 1300 cm^2.
Problem 2: A cuboid box has a length of 10 cm, a breadth of 8 cm, and a height of 6 cm. Find its volume.
The volume of a cuboid box is given by the formula:
Volume = l x b x h
where l, b, and h are the length, breadth, and height of the cuboid box, respectively.
Substituting the given values, we get:
Volume = 10 x 8 x 6 cm^3 Volume = 480 cm^3
Therefore, the volume of the cuboid box is 480 cm^3.
Problem 3: A cuboid box has a length of 12 cm, a breadth of 9 cm, and a height of 8 cm. If the box is filled with water up to a height of 6 cm, find the volume of water in the box.
The volume of water in the box is equal to the volume of the cuboid box up to the height of 6 cm.
The volume of the cuboid box up to the height of 6 cm is given by the formula:
Volume = l x b x h
where l, b, and h are the length, breadth, and height of the cuboid box up to the height of 6 cm, respectively.
Substituting the given values, we get:
Volume = 12 x 9 x 6 cm^3 Volume = 648 cm^3
Therefore, the volume of water in the box up to the height of 6 cm is 648 cm^3.
Sure, here are some miscellaneous topics in math:
1. Matrices
2. Vectors
3. Complex numbers
4. Sequences and series
5. Logarithms
6. Exponential functions
7. Probability and statistics
8. Trigonometric functions and identities
9. Calculus
10. Number theory
11. Set theory and logic
12. Geometry in higher dimensions
13. Topology
14. Combinatorics
15. Graph theory
These are just a few examples, but there are many other areas of mathematics that could be considered “miscellaneous.”
Addition & Subtraction
Addition and subtraction are basic mathematical operations that are used to combine or separate quantities. Addition involves combining two or more quantities to find the total, while subtraction
involves taking one quantity away from another to find the difference.
Here are some key terms related to addition and subtraction:
• Addends: the numbers being added together.
• Sum: the result of addition.
• Minuend: the number from which another number (the subtrahend) is subtracted.
• Subtrahend: the number that is subtracted from the minuend.
• Difference: the result of subtraction.
Here are some examples of addition and subtraction:
• Addition: 2 + 3 = 5. In this case, the addends are 2 and 3, and the sum is 5.
• Subtraction: 8 – 3 = 5. In this case, the minuend is 8, the subtrahend is 3, and the difference is 5.
There are several strategies that can be used to solve addition and subtraction problems, including:
• Counting: for simple problems, such as adding or subtracting small numbers, counting on your fingers or using objects (such as blocks) can be a helpful strategy.
• Mental math: for more complex problems, it may be helpful to use mental math strategies, such as breaking numbers down into easier-to-manage parts or using known facts (such as 2+2=4) to solve
related problems.
• Written methods: for larger or more complex problems, written methods such as column addition or subtraction may be necessary.
Addition and subtraction are used in many real-world applications, such as calculating prices, measuring distances, and determining amounts of ingredients in recipes. They are also important
foundational skills for more advanced mathematical concepts, such as multiplication and division.
In mathematics, the average (also called the mean) is a measure of central tendency that represents the typical value in a set of numbers. It is calculated by adding up all the numbers in a set and
then dividing the sum by the total number of values in the set.
There are three types of averages:
1. Arithmetic mean: It is the most commonly used type of average and is calculated by adding up all the values in a set and dividing by the total number of values.
2. Median: It is the middle value in a set of numbers when they are arranged in order from smallest to largest. If there are an even number of values, the median is the average of the two middle
3. Mode: It is the value that appears most frequently in a set of numbers. A set of numbers can have more than one mode or no mode at all.
Averages are used in many fields, including statistics, finance, and science, to summarize and analyze data. They can be used to calculate trends, make predictions, and compare different sets of
Recuring And Bar
In mathematics, a repeating or recurring decimal is a decimal representation of a number that has a repeating pattern of digits after the decimal point. For example, the number 1/3 is equal to
0.33333…, where the digit 3 repeats infinitely. The repeating portion of the decimal is often indicated by placing a bar over the repeating digits, so in this case we would write 0.333… as 0.3̅. The
bar is sometimes referred to as a vinculum.
Repeating decimals can be converted to fractions using algebra. For example, if we let x = 0.3̅, then multiplying both sides by 10 gives 10x = 3.3̅. Subtracting the left-hand sides of these two
equations gives 9x = 3, so x = 1/3.
Repeating decimals can also be expressed as mixed numbers, by separating out the repeating part of the decimal and adding it to the non-repeating part. For example, the repeating decimal 0.6̅ can be
expressed as the mixed number 0.6̅ = 0.6 + 0.0̅6 = 6/10 + 6/990 = 606/990.
In addition to repeating decimals, there are also non-repeating decimals (sometimes called irrational numbers), which have an infinite, non-repeating sequence of digits after the decimal point, such
as π = 3.14159265358979323846
In statistics, the mode is the value that appears most frequently in a dataset. It is a measure of central tendency, along with the mean and median.
To find the mode of a dataset, you can simply count the frequency of each value and identify the value with the highest frequency. If there are multiple values that appear with the same highest
frequency, then the dataset is said to be multimodal, and each mode can be reported separately.
For example, consider the dataset {2, 3, 3, 4, 5, 5, 5}. The value 5 appears three times, which is more than any other value, so the mode is 5. Another example is the dataset {1, 2, 2, 3, 3, 3, 4, 4,
4, 5}, which has two modes: 3 and 4, each of which appears three times.
The mode is useful for describing the most common value in a dataset, but it can be less informative than the mean and median in certain situations. For example, the mode may not be a good measure of
central tendency if the dataset is skewed or if there are outliers. In these cases, the mean or median may provide a better representation of the typical value in the dataset.
In mathematics, simplification refers to the process of reducing an expression or equation to a simpler form. This can involve several different techniques, depending on the type of expression or
equation being simplified.
One common technique for simplification is to combine like terms. This involves adding or subtracting terms that have the same variables and exponents. For example, the expression 3x + 2x can be
simplified by adding the coefficients of the like terms: 3x + 2x = 5x. Similarly, the expression 4x^2 – 2x^2 can be simplified by subtracting the coefficients of the like terms: 4x^2 – 2x^2 = 2x^2.
Another technique for simplification is to factor the expression. This involves writing the expression as a product of simpler expressions. For example, the expression x^2 + 2x + 1 can be factored as
(x + 1)^2. Factoring can be particularly useful for solving equations or identifying patterns in expressions.
In some cases, simplification may involve using identities or properties of mathematical operations. For example, the identity a^2 – b^2 = (a + b)(a – b) can be used to simplify expressions that
involve the difference of squares. Similarly, the distributive property of multiplication can be used to simplify expressions that involve multiplying a term by a sum or difference of terms.
Overall, simplification is an important skill in mathematics that can help make expressions and equations easier to work with and understand.
Problems Based on Number
Here are a few sample problems based on numbers:
1. A number when divided by 8 leaves a remainder of 3, and when divided by 5 leaves a remainder of 1. What is the smallest possible positive integer that satisfies these conditions?
Solution: Let the required number be x. We can write x = 8a + 3 and x = 5b + 1 for some integers a and b. We want to find the smallest x that satisfies these conditions. We can start by listing out
some possible values of a and b and finding the corresponding values of x:
• a = 0, b = 0: x = 3 (which satisfies both conditions)
• a = 1, b = 0: x = 11
• a = 0, b = 1: x = 8
• a = 1, b = 1: x = 19
Therefore, the smallest possible positive integer that satisfies the conditions is x = 3.
2. The sum of two numbers is 45, and their difference is 9. What are the two numbers?
Solution: Let the two numbers be x and y. We can write two equations based on the given information:
x + y = 45 (equation 1) x – y = 9 (equation 2)
We can solve this system of equations by adding equations 1 and 2:
2x = 54
Therefore, x = 27. Substituting this value of x into equation 1, we get:
27 + y = 45
Therefore, y = 18. The two numbers are 27 and 18.
3. The sum of three consecutive even integers is 156. What are the three integers?
Solution: Let the three consecutive even integers be x, x+2, and x+4. We can write an equation based on the given information:
x + (x+2) + (x+4) = 156
Simplifying this equation, we get:
3x + 6 = 156
Subtracting 6 from both sides, we get:
3x = 150
Dividing by 3, we get:
x = 50
Therefore, the three consecutive even integers are 50, 52, and 54.
Squaring & Cubing
Squaring and cubing are mathematical operations used to find the square and cube of a number, respectively. The square of a number is obtained by multiplying the number by itself, while the cube of a
number is obtained by multiplying the number by itself twice.
For example, the square of 5 is 5 multiplied by itself, or 5 × 5 = 25. The cube of 5 is 5 multiplied by itself twice, or 5 × 5 × 5 = 125.
Squaring and cubing are useful in many areas of mathematics and science, such as in calculating areas and volumes of geometric shapes, and in determining the distance between two points in a
coordinate plane.
There are various methods to find the square and cube of a number. Some of the commonly used methods are:
• Using multiplication: To find the square of a number, multiply the number by itself. For example, to find the square of 7, we can multiply 7 × 7 = 49. To find the cube of a number, multiply the
number by itself twice. For example, to find the cube of 4, we can multiply 4 × 4 × 4 = 64.
• Using exponentiation: Another way to find the square and cube of a number is by using exponentiation. To find the square of a number, we can write the number as the base and the exponent as 2.
For example, 7² = 49. To find the cube of a number, we can write the number as the base and the exponent as 3. For example, 4³ = 64.
• Using patterns: There are certain patterns that can be used to find the square and cube of a number. For example, the square of any odd number is an odd number, and the square of any even number
is an even number. Similarly, the cube of any odd number is an odd number, and the cube of any even number is an even number. These patterns can be helpful in quickly finding the square and cube
of a number.
• Using algebraic formulas: There are various algebraic formulas that can be used to find the square and cube of a number. For example, (a + b)² = a² + 2ab + b² is a formula that can be used to
find the square of the sum of two numbers. Similarly, (a + b)³ = a³ + 3a²b + 3ab² + b³ is a formula that can be used to find the cube of the sum of two numbers.
Overall, squaring and cubing are important mathematical operations that are used in a wide range of applications, from simple calculations to complex problem-solving.
Multiplication is a mathematical operation of combining two or more numbers to obtain their product. It is represented by the symbol “×” or “*”, and the result of the operation is called the product.
For example, 2 × 3 = 6, where 2 and 3 are the factors, and 6 is the product. The order of the factors does not affect the product, i.e., 2 × 3 = 3 × 2 = 6. Multiplication can also be done using
various techniques such as repeated addition, grouping, lattice multiplication, and others. It is an essential arithmetic operation that is widely used in everyday life and other mathematical fields
such as algebra, geometry, and calculus.
Squar Root & Cube Root
Square root and cube root are mathematical operations used to find the value of a number that, when multiplied by itself a certain number of times, equals a given number. The square root is the
inverse operation of squaring a number, while the cube root is the inverse operation of cubing a number.
Here are some key terms related to square root and cube root:
• Square root: the value of a number that, when multiplied by itself, equals a given number.
• Radical symbol: the symbol used to indicate a square root ( √ ) or a cube root ( 3√ ).
• Radicand: the number under the radical symbol.
• Cube root: the value of a number that, when multiplied by itself three times, equals a given number.
Here are some examples of square root and cube root:
• Square root: the square root of 25 is 5, because 5 multiplied by itself equals 25.
• Cube root: the cube root of 27 is 3, because 3 multiplied by itself three times equals 27.
There are several strategies for finding square roots and cube roots, including:
• Estimation: for some numbers, it may be possible to estimate the square root or cube root without calculating it exactly. For example, the square root of 50 is between 7 and 8.
• Prime factorization: breaking down the number into its prime factors can be a helpful strategy for finding square roots and cube roots.
• Using a calculator: for more complex numbers, it may be necessary to use a calculator to find the square root or cube root.
Square root and cube root are used in many real-world applications, such as calculating the side length of a square or cube, determining the volume of a cube, and in physics to calculate the
displacement or acceleration of an object.
Repeated Digit Number of Root
A repeated digit number is a number that has a digit that appears twice or more in a row, such as 11, 22, 333, and so on. When finding the square root or cube root of a repeated digit number, a
pattern of digits will emerge in the result.
For example, the square root of 121 is 11, because 11 multiplied by itself equals 121. Similarly, the cube root of 8,888 is 22, because 22 multiplied by itself three times equals 8,888.
The pattern that emerges when finding the square root or cube root of a repeated digit number depends on the number of repeated digits. For example, when finding the square root of a number with two
repeated digits, the resulting pattern will have one digit for every repeated pair. So the square root of 1444 is 38, because there are two pairs of repeated digits (44) and the pattern has two
digits (38).
When finding the cube root of a number with three repeated digits, the resulting pattern will have one digit for each repeated digit. So the cube root of 888,888 is 222, because there are three
repeated digits (888) and the pattern has three digits (222).
Repeated digit numbers and their roots can be used in various math problems and puzzles, such as finding the roots of large numbers, identifying patterns in numbers, and testing mental math skills.
A surd is a number that cannot be expressed as the exact ratio of two integers. It is a non-repeating, non-terminating decimal that cannot be simplified into a whole number or a fraction. Surds
commonly arise when finding the square root or cube root of a non-perfect square or non-perfect cube, respectively.
For example, the square root of 2 is a surd because it cannot be expressed as a ratio of two integers. Its decimal representation is approximately 1.41421356, but it goes on infinitely without
Surds are often written using the radical symbol ( √ ), which indicates the square root. For example, √2 is the symbol for the square root of 2.
Operations with surds can be simplified by following certain rules. For example:
• Multiplication: √a x √b = √ab
• Division: √a / √b = √(a/b)
• Addition and subtraction: √a + √b cannot be simplified further, but expressions like √a – √b can be simplified by multiplying the numerator and denominator by the conjugate of the denominator
(i.e. √a + √b for √a – √b, and vice versa).
Simplifying surds is an important skill in mathematics and can be used in various applications, such as calculating areas and volumes of geometric shapes, solving quadratic equations, and in physics
and engineering.
Surds Equations
A surd is a square root of a number that cannot be simplified to a whole number or a fraction. Surds are represented by the symbol √.
Surd equations involve solving equations that contain one or more surds. The basic technique for solving surd equations involves squaring both sides of the equation to eliminate the surds. However,
this can lead to extraneous solutions, so it is important to check the solutions obtained by squaring both sides.
Here are some examples of surd equations and how to solve them:
Example 1: Solve the equation √x + 2 = 5
To isolate the surd, we first subtract 2 from both sides of the equation:
√x = 3
Squaring both sides of the equation gives:
x = 9
We must check our solution by substituting x = 9 back into the original equation:
√9 + 2 = 5
3 + 2 = 5
5 = 5
The solution x = 9 is valid.
Example 2: Solve the equation √(x + 2) + 1 = 3
To isolate the surd, we first subtract 1 from both sides of the equation:
√(x + 2) = 2
Squaring both sides of the equation gives:
x + 2 = 4
Subtracting 2 from both sides of the equation gives:
x = 2
We must check our solution by substituting x = 2 back into the original equation:
√(2 + 2) + 1 = 3
√4 + 1 = 3
2 + 1 = 3
3 = 3
The solution x = 2 is valid.
In general, when solving surd equations, we need to isolate the surd and then square both sides of the equation. However, we must always check our solutions to ensure that we have not introduced any
extraneous solutions.
In mathematics, indices (also known as exponents or powers) are a way of representing repeated multiplication. An index is a small number written above and to the right of a base number that
indicates how many times the base should be multiplied by itself. For example, 2 raised to the power of 3 (written as 2³) means 2 multiplied by itself 3 times, or 2 x 2 x 2 = 8.
Indices follow certain rules of arithmetic that can be used to simplify and manipulate expressions involving indices. Here are some important rules of indices:
1. Multiplying indices with the same base: When two or more indices have the same base, we can multiply them by adding their exponents. For example:
2⁴ x 2² = 2⁶ (because 4 + 2 = 6)
3³ x 3⁵ x 3² = 3¹⁰ (because 3 + 5 + 2 = 10)
2. Dividing indices with the same base: When two or more indices have the same base, we can divide them by subtracting their exponents. For example:
2⁷ ÷ 2³ = 2⁴ (because 7 – 3 = 4)
3¹² ÷ 3⁹ ÷ 3² = 3¹ (because 12 – 9 – 2 = 1)
3. Raising a power to another power: When a power is raised to another power, we multiply the exponents. For example:
(2³)² = 2⁶ (because 3 x 2 = 6)
(3²)³ = 3⁶ (because 2 x 3 = 6)
4. Negative indices: When an index is negative, we can rewrite it as the reciprocal of the base raised to the positive value of the index. For example:
2⁻³ = 1/2³ = 1/8
5. Fractional indices: When an index is a fraction, we can rewrite it as the base raised to the numerator of the fraction, with the result being the nth root of the base, where n is the denominator
of the fraction. For example:
4⁵/₃ = 4^(5/3) = cube root of (4⁵) = cube root of 1024
These are just some of the basic rules of indices, but they are very important for simplifying and manipulating expressions involving indices.
Exponential Equations
Exponential equations are equations in which one or more of the unknown variables occur as exponents. These equations can be solved using logarithms or by taking the logarithm of both sides.
Here are some steps to solve exponential equations:
Step 1: Try to simplify the equation by using the rules of exponents. For example, if you have an equation like 2^x × 2^2 = 2^6, you can simplify it to 2^(x+2) = 2^6.
Step 2: If the bases on both sides of the equation are the same, you can equate the exponents. For example, if you have an equation like 2^x = 2^4, you can solve it by equating the exponents, x = 4.
Step 3: If the bases on both sides of the equation are different, you can take the logarithm of both sides of the equation. The choice of logarithm base is not critical, but it’s usually best to
choose a base that will simplify the equation. For example, if you have an equation like 3^x = 5, you can take the logarithm of both sides of the equation using base 3, giving:
log₃(3^x) = log₃(5)
x log₃(3) = log₃(5)
x = log₃(5) / log₃(3)
Step 4: Check your answer by substituting it back into the original equation.
Here are some more examples of exponential equations and how to solve them:
Example 1: 2^x – 3 × 2^(x-1) + 2 = 0
Let y = 2^x. Then we can rewrite the equation as y^2 – 3y + 2 = 0. Factoring this quadratic equation gives:
(y – 1)(y – 2) = 0
So y = 1 or y = 2. Substituting back, we get:
2^x = 1 or 2^x = 2
Solving these equations gives x = 0 or x = 1.
Example 2: e^x + 3 = 7
Subtracting 3 from both sides of the equation gives:
e^x = 4
Taking the natural logarithm of both sides of the equation gives:
ln(e^x) = ln(4)
x ln(e) = ln(4)
x = ln(4)
Checking our answer by substituting it back into the original equation, we get:
e^(ln(4)) + 3 = 7
4 + 3 = 7
7 = 7
The solution x = ln(4) is valid.
In general, exponential equations can be solved using logarithms or by simplifying the equation using the rules of exponents. However, it’s important to check your answer by substituting it back into
the original equation to ensure that you have not introduced any extraneous solutions
Number of Zero and Number of Number in a given Factorial
The number of zeros at the end of a given factorial is equal to the number of times the factorial is divisible by 10. This is because each zero at the end of a number corresponds to a factor of 10,
which in turn corresponds to a factor of 2 and 5. Since 2 is a much more common factor than 5 in factorials, we only need to count the number of factors of 5.
For example, to find the number of zeros at the end of 100!, we can divide 100 by 5 to get 20, then divide 20 by 5 to get 4, and so on until we get to a quotient less than 5. We add up all the
quotients to get the total number of factors of 5 in 100!, which is 20 + 4 + 0 = 24. Therefore, 100! has 24 zeros at the end.
To find the number of digits in a given factorial, we can use the Stirling’s formula, which approximates the value of a large factorial. The formula states that:
n! ≈ √(2πn) * (n/e)^n
where π is the mathematical constant pi and e is the mathematical constant e (the base of the natural logarithm).
Using this formula, we can estimate the number of digits in a given factorial by taking the logarithm (base 10) of the approximation and adding 1. For example, to find the number of digits in 100!,
we can use Stirling’s formula to get:
100! ≈ √(2π(100)) * (100/e)^100
100! ≈ 9.33262154 × 10^157
Taking the logarithm (base 10) of this value and adding 1 gives:
log₁₀(9.33262154 × 10^157) + 1 ≈ 158
Therefore, 100! has 158 digits.
Note that Stirling’s formula is only an approximation and may not give an exact answer for very large factorials. However, it is a useful tool for estimating the number of digits in a given
Concept of Unit Digit
The unit digit of a number refers to the last digit of that number. For example, in the number 3467, the unit digit is 7. The concept of unit digit is often used in various mathematical operations
such as addition, subtraction, multiplication, and division.
In addition and subtraction, the unit digit of the result only depends on the unit digits of the numbers being added or subtracted. For example, the sum of 345 and 712 is 1057, and the unit digit of
1057 is 7, which is the same as the unit digit of the sum of 5 and 2 (the unit digits of 345 and 712, respectively).
In multiplication, the unit digit of the result depends on the unit digits of the numbers being multiplied. There are some rules to determine the unit digit of a product. For example, if the unit
digits of the two numbers being multiplied are both even, the unit digit of the product will be 6. If one of the unit digits is odd and the other is even, the unit digit of the product will be even.
If both unit digits are odd, the unit digit of the product will be odd.
In division, the unit digit of the quotient depends on the unit digits of the dividend and divisor. However, unlike addition, subtraction, and multiplication, the unit digit of the remainder can also
affect the unit digit of the quotient
Problems Based on Division
Here are some sample problems based on division:
1. Divide 876 by 12.
Solution: We can use long division to solve this problem. ________ 12 | 876 72 — 156 144 — 123 The quotient is 73, and the remainder is 3.
2. A recipe for cookies makes 48 cookies. How many cookies will be made if the recipe is tripled?
Solution: To triple the recipe, we need to multiply 48 by 3. 48 x 3 = 144 So, the recipe will make 144 cookies.
3. If a company has 360 employees and wants to divide them into teams of 6 people each, how many teams will there be?
Solution: To find the number of teams, we need to divide the total number of employees by the number of employees in each team. 360 ÷ 6 = 60 So, there will be 60 teams.
4. A pizza has 8 slices. If 3 pizzas are shared among 12 people, how many slices will each person get?
Solution: To find the number of slices each person will get, we need to first find the total number of slices. 3 pizzas x 8 slices per pizza = 24 slices Then, we can divide the total number of slices
by the number of people. 24 slices ÷ 12 people = 2 slices per person
Therefore, each person will get 2 slices of pizza.
5. If a rectangular field has an area of 720 square meters and a length of 24 meters, what is its width?
Solution: To find the width, we need to divide the area by the length. 720 sq. m ÷ 24 m = 30 m So, the width of the rectangular field is 30 meters.
Rule of Divisibility
A rule of divisibility is a set of guidelines that help to determine whether a number is divisible by another number without performing the actual division operation. Here are some common rules of
1. Divisibility by 2: A number is divisible by 2 if its unit digit is even, i.e., 0, 2, 4, 6, or 8.
2. Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
3. Divisibility by 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
4. Divisibility by 5: A number is divisible by 5 if its unit digit is either 0 or 5.
5. Divisibility by 6: A number is divisible by 6 if it is divisible by both 2 and 3.
6. Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
7. Divisibility by 10: A number is divisible by 10 if its unit digit is 0.
8. Divisibility by 11: A number is divisible by 11 if the difference between the sum of its digits in the even positions and the sum of its digits in the odd positions is either 0 or a multiple of
These rules of divisibility can be helpful in simplifying calculations and quickly determining whether a number is divisible by another number.
Successive Division
Successive division is a method of dividing a number by a sequence of divisors one after another. In each step of the process, we divide the result of the previous division by the next divisor until
all the divisors are exhausted. This method can be useful in finding the prime factorization of a number.
Here’s an example of how to use successive division to find the prime factorization of 120:
1. Start by dividing the number by the smallest prime number, 2: 120 ÷ 2 = 60
2. Next, divide the result by the smallest prime number that can divide it, which is again 2: 60 ÷ 2 = 30
3. Continue the process with the next smallest prime number, 3: 30 ÷ 3 = 10
4. Next, divide the result by the smallest prime number that can divide it, which is again 2: 10 ÷ 2 = 5
5. The number 5 is a prime number, so we stop here.
Therefore, the prime factorization of 120 is 2 x 2 x 2 x 3 x 5, or 2^3 x 3 x 5. We can check this by multiplying these factors together: 2 x 2 x 2 x 3 x 5 = 120.
Concept of Remainders
In division, the remainder is the amount left over after the division process is completed. It is the difference between the dividend and the product of the quotient and divisor.
For example, consider the division problem 17 ÷ 4. In this problem, 17 is the dividend, 4 is the divisor, and the quotient is 4 with a remainder of 1. This means that 4 goes into 17 four times with 1
left over. The 1 is the remainder.
Another example is the division problem 25 ÷ 6. The quotient is 4 with a remainder of 1. This means that 6 goes into 25 four times with 1 left over.
Remainders can also be expressed as fractions or decimals. For example, the remainder of the division 7 ÷ 2 can be written as a fraction: 7/2. This fraction can be simplified to 3 and 1/2, which
means that 2 goes into 7 three times with 1/2 left over.
In some problems, remainders can be important in determining the solution. For example, in a problem that involves dividing a group of objects into equal-sized groups, the remainder can tell us how
many objects are left over and cannot be divided evenly. In other problems, the remainder may be ignored if it is not relevant to the solution.
Here are some miscellaneous concepts related to division:
1. Divisor: A divisor is a number that divides another number without leaving a remainder. For example, 3 is a divisor of 12 because 3 goes into 12 exactly four times.
2. Dividend: A dividend is a number that is divided by another number. For example, in the division problem 16 ÷ 4 = 4, 16 is the dividend.
3. Quotient: A quotient is the result of dividing one number by another. For example, in the division problem 20 ÷ 5 = 4, 4 is the quotient.
4. Long division: Long division is a method of dividing large numbers by a divisor. It involves breaking down the division into smaller steps and writing out the calculation with the quotient and
remainder shown at each step.
5. Fraction: A fraction is a number that represents a part of a whole. It is expressed as a ratio of two numbers, with the top number (numerator) representing the part and the bottom number
(denominator) representing the whole. For example, 3/4 represents three parts out of four.
6. Decimal: A decimal is a number expressed in base-10 notation, with a decimal point separating the whole number part from the fractional part. For example, 3.25 represents three and a quarter.
7. Recurring decimal: A recurring decimal is a decimal that has a repeating pattern of digits. For example, 0.333… represents one-third, and the pattern of 3’s repeats infinitely.
8. Rational number: A rational number is a number that can be expressed as a fraction of two integers. All terminating decimals and recurring decimals are rational numbers.
9. Irrational number: An irrational number is a number that cannot be expressed as a fraction of two integers. Examples include pi and the square root of 2.
Problems Based on Divisors
Here are some problems based on divisors:
1. Find all the divisors of 24. Solution: The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. Therefore, the divisors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
2. Find the sum of all the divisors of 36. Solution: The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. The sum of these factors is 91, so the sum of the divisors of 36 is 91.
3. How many divisors does the number 100 have? Solution: The factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100. Therefore, the number 100 has 9 divisors.
4. Find all the common divisors of 24 and 36. Solution: The divisors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. The divisors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. Therefore, the common divisors
of 24 and 36 are 1, 2, 3, 4, 6, and 12.
5. Find the greatest common divisor (GCD) of 24 and 36. Solution: The common divisors of 24 and 36 are 1, 2, 3, 4, 6, and 12. The greatest common divisor (GCD) is the largest of these, which is 12.
Therefore, the GCD of 24 and 36 is 12.
6. Find the least common multiple (LCM) of 24 and 36. Solution: The multiples of 24 are 24, 48, 72, 96, 120, 144, 168, 192, and so on. The multiples of 36 are 36, 72, 108, 144, 180, 216, 252, and so
on. The least common multiple (LCM) is the smallest multiple that is common to both sets. In this case, the LCM of 24 and 36 is 72.
LCM & HCF
LCM (Least Common Multiple) and HCF (Highest Common Factor) are important concepts in arithmetic and are often used in solving problems involving fractions, ratios, and proportions.
LCM: The LCM of two or more numbers is the smallest number that is divisible by all of them. For example, the LCM of 4 and 6 is 12 because 12 is the smallest number that is divisible by both 4 and 6.
To find the LCM of two or more numbers, we can use the following method:
• Write down the prime factorization of each number.
• Multiply the highest power of each prime factor together.
For example, to find the LCM of 12, 18, and 24:
• The prime factorization of 12 is 2^2 x 3.
• The prime factorization of 18 is 2 x 3^2.
• The prime factorization of 24 is 2^3 x 3.
• The highest power of 2 is 2^3.
• The highest power of 3 is 3^2.
• Therefore, the LCM of 12, 18, and 24 is 2^3 x 3^2 = 72.
HCF: The HCF of two or more numbers is the largest number that divides them exactly without leaving a remainder. For example, the HCF of 12 and 18 is 6 because 6 is the largest number that divides
both 12 and 18 exactly.
To find the HCF of two or more numbers, we can use the following method:
• Write down the factors of each number.
• Identify the common factors.
• Multiply the common factors together.
For example, to find the HCF of 12 and 18:
• The factors of 12 are 1, 2, 3, 4, 6, and 12.
• The factors of 18 are 1, 2, 3, 6, 9, and 18.
• The common factors are 1, 2, 3, and 6.
• Therefore, the HCF of 12 and 18 is 2 x 3 = 6.
Note: The LCM and HCF of two or more numbers can be used to simplify fractions, add and subtract fractions, and solve many other types of arithmetic problems.
Trigonometrical Ratio of Acute Angles
Trigonometric ratios of acute angles are ratios of the sides of a right triangle with respect to its acute angles. The three basic trigonometric ratios are sine, cosine, and tangent, which are
denoted as sin, cos, and tan respectively.
Consider a right triangle ABC, where angle A is a right angle and angle B is an acute angle. The sides of the triangle are denoted as follows:
• Opposite side (BC) is the side opposite to the angle B.
• Adjacent side (AB) is the side adjacent to the angle B and adjacent to the right angle.
• Hypotenuse (AC) is the side opposite to the right angle.
Then, the three basic trigonometric ratios of angle B are:
• Sine of angle B: sin B = BC/AC
• Cosine of angle B: cos B = AB/AC
• Tangent of angle B: tan B = BC/AB
In addition to these three ratios, there are three reciprocal trigonometric ratios, which are:
• Cosecant of angle B: csc B = AC/BC = 1/sin B
• Secant of angle B: sec B = AC/AB = 1/cos B
• Cotangent of angle B: cot B = AB/BC = 1/tan B
These ratios are useful in solving problems in trigonometry and in real-life applications such as surveying, navigation, and engineering
Trigonometrical Ratio of Any Angles
Trigonometric ratios of any angle, whether acute or obtuse, are based on the unit circle, which is a circle with radius 1 centered at the origin of a coordinate plane.
Consider an angle θ in standard position, which means that its vertex is at the origin of the coordinate plane, and its initial side lies along the positive x-axis. The terminal side of the angle θ
can be drawn in any direction from the origin.
Then, the six trigonometric ratios of angle θ are defined as follows:
• Sine of angle θ: sin θ = y/r, where y is the y-coordinate of the point where the terminal side of θ intersects the unit circle, and r is the radius of the unit circle, which is 1.
• Cosine of angle θ: cos θ = x/r, where x is the x-coordinate of the same point on the unit circle.
• Tangent of angle θ: tan θ = y/x.
• Cosecant of angle θ: csc θ = 1/sin θ.
• Secant of angle θ: sec θ = 1/cos θ.
• Cotangent of angle θ: cot θ = 1/tan θ.
Note that the trigonometric ratios of an angle depend only on the position of its terminal side on the unit circle, and not on the size of the angle. Therefore, if two angles have the same terminal
side, they have the same trigonometric ratios.
Trigonometric ratios are used extensively in trigonometry, calculus, physics, and engineering to solve problems involving angles and distances.
Trigonometrical Ratio of Some Special Angles
The trigonometric ratios of some special angles (in degrees) can be easily remembered using the acronym “SOHCAHTOA”:
• Sine, cosine, and tangent of 0 degrees: sin 0° = 0, cos 0° = 1, tan 0° = 0.
• Sine, cosine, and tangent of 30 degrees: sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3.
• Sine, cosine, and tangent of 45 degrees: sin 45° = √2/2, cos 45° = √2/2, tan 45° = 1.
• Sine, cosine, and tangent of 60 degrees: sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3.
• Sine, cosine, and tangent of 90 degrees: sin 90° = 1, cos 90° = 0, tan 90° is undefined.
In addition, the reciprocal trigonometric ratios of these angles can be easily computed using their definitions. For example, the cosecant of 30 degrees is the reciprocal of the sine of 30 degrees,
which is 2. The secant of 45 degrees is the reciprocal of the cosine of 45 degrees, which is √2. The cotangent of 60 degrees is the reciprocal of the tangent of 60 degrees, which is 1/√3.
Knowing the trigonometric ratios of these special angles can be helpful in solving trigonometric equations, simplifying expressions, and solving real-life problems involving angles and distances
Solution of Trigonometrical Equation From (0°–90°)
To solve a trigonometric equation in the interval (0°–90°), we can use the following general steps:
1. Simplify the equation using algebraic manipulations and trigonometric identities, if possible.
2. Apply the appropriate inverse trigonometric function (arcsin, arccos, or arctan) to both sides of the equation to isolate the trigonometric function.
3. Solve for the angle using the inverse trigonometric function and simplify, if necessary.
Here is an example of how to solve a trigonometric equation in the interval (0°–90°):
Problem: Solve for x in the equation sin x + cos x = 1.
1. Simplify the equation using the identity sin² x + cos² x = 1:
sin x + cos x = sin x + cos x * sin² x / sin² x + cos² x / sin² x = sin x + sin x * cos x / sin² x + cos² x / sin² x = (sin² x + sin x * cos x + cos² x) / sin² x = 1 / sin² x
Therefore, the equation can be rewritten as:
1 / sin² x = 1
2. Apply the inverse sine function to both sides of the equation:
arcsin (1 / sin² x) = arcsin 1
Since the range of arcsin is [-90°, 90°], we only need to consider angles in this range.
3. Solve for x using the inverse sine function and simplify:
x = arcsin (1 / sin² x) = 90°
Therefore, the solution to the equation sin x + cos x = 1 in the interval (0°–90°) is x = 90
Solution of Trigonometrical Equation From (0°-360°)
To solve a trigonometric equation in the interval (0°-360°), we can use the following general steps:
1. Simplify the equation using algebraic manipulations and trigonometric identities, if possible.
2. Apply the appropriate inverse trigonometric function (arcsin, arccos, or arctan) to both sides of the equation to isolate the trigonometric function.
3. Solve for the angle using the inverse trigonometric function and simplify, if necessary.
4. Add or subtract multiples of 360° to the solution to obtain all possible solutions within the given interval.
Here is an example of how to solve a trigonometric equation in the interval (0°-360°):
Problem: Solve for x in the equation 2cos² x + 3sin x = 2.
1. Simplify the equation using the identity cos² x + sin² x = 1:
2cos² x + 3sin x = 2cos² x + 3sin x * cos² x / cos² x + sin² x / cos² x = 2cos² x + 3sin x * cos² x / (1 – cos² x) = 2 – 3cos² x / (1 – cos² x)
Therefore, the equation can be rewritten as:
2 – 3cos² x / (1 – cos² x) = 2
3cos² x / (1 – cos² x) = 0
cos x = 0
2. Apply the inverse cosine function to both sides of the equation:
arccos (cos x) = arccos 0
Since the range of arccos is [0°, 180°], we only need to consider angles in this range.
3. Solve for x using the inverse cosine function and simplify:
x = arccos 0 = 90°, 270°
4. Add or subtract multiples of 360° to the solution to obtain all possible solutions within the given interval:
x = 90°, 270°, 450°, 630°, …
Therefore, the solutions to the equation 2cos² x + 3sin x = 2 in the interval (0°-360°) are x = 90°, 270°, 450°, 630°, …
Trigonometrical Equation
A trigonometric equation is an equation involving one or more trigonometric functions of an angle. Solving trigonometric equations involves finding the values of the angles that satisfy the equation.
There are two types of trigonometric equations: linear and quadratic. A linear trigonometric equation is an equation in which the trigonometric function is raised to the first power, while a
quadratic trigonometric equation is an equation in which the trigonometric function is raised to the second power.
Here are some examples of linear and quadratic trigonometric equations:
Linear Trigonometric Equation: sin x = 1/2
Quadratic Trigonometric Equation: cos² x – cos x = 0
To solve a trigonometric equation, we use algebraic manipulations and trigonometric identities to simplify the equation and isolate the trigonometric function. Then, we apply the appropriate inverse
trigonometric function (arcsin, arccos, or arctan) to both sides of the equation to obtain the solutions. Finally, we check our solutions to make sure they satisfy the original equation.
It is important to note that there may be more than one solution to a trigonometric equation, and that solutions may be restricted to certain intervals depending on the domain of the equation.
Miscellaneous Equation
The term “miscellaneous equation” is quite broad and can refer to any equation that does not fall under a specific category or type of equation. However, here are some examples of miscellaneous
1. Exponential equations: Equations in the form of aⁿ = b, where a and b are constants and n is a variable.
Example: 2ⁿ = 16
2. Logarithmic equations: Equations in the form of logₐ b = c, where a is the base of the logarithm, b is the argument of the logarithm, and c is a constant.
Example: log₂ x = 4
3. Rational equations: Equations in the form of P(x)/Q(x) = r, where P(x) and Q(x) are polynomials in x and r is a constant.
Example: (x + 3)/(x – 2) = 2
4. Absolute value equations: Equations in the form of |x| = a, where a is a constant.
Example: |2x – 1| = 5
To solve miscellaneous equations, we need to use specific techniques and methods depending on the type of equation. For example, to solve exponential equations, we can use logarithms; to solve
logarithmic equations, we can use exponentials; and to solve rational equations, we can use factoring and common denominators. It is important to be familiar with these methods and to check our
solutions to ensure they are valid.
Problem Based on Elimination
Here’s an example problem based on elimination:
Problem: Solve the system of equations using elimination method: 2x + 3y = 11 3x – 2y = 5
Solution: To solve this system of equations using the elimination method, we need to eliminate one of the variables by adding or subtracting the two equations. The goal is to create an equation that
has only one variable.
We can eliminate y by multiplying the first equation by 2 and the second equation by 3, so that the coefficient of y in each equation will be -6y and 6y respectively. This will allow us to add the
two equations and eliminate y.
2x + 3y = 11 (multiply by 2) 4x + 6y = 22
3x – 2y = 5 (multiply by 3) 9x – 6y = 15
Now we can add the two equations and eliminate y:
4x + 6y = 22 9x – 6y = 15
13x = 37
Solving for x, we get:
x = 37/13
Now we can substitute this value of x into one of the original equations to solve for y:
2x + 3y = 11
2(37/13) + 3y = 11
74/13 + 3y = 11
3y = 11 – 74/13
3y = 39/13
y = 13/3
Therefore, the solution to the system of equations is (x,y) = (37/13, 13/3)
Simple Identities
Simple identities in trigonometry are equations that involve trigonometric functions and are true for all values of the variables that make sense in the equation. Here are some examples of simple
1. Pythagorean identities: sin² θ + cos² θ = 1 tan² θ + 1 = sec² θ 1 + cot² θ = csc² θ
2. Reciprocal identities: sin θ = 1/csc θ cos θ = 1/sec θ tan θ = 1/cot θ
3. Quotient identities: tan θ = sin θ/cos θ cot θ = cos θ/sin θ
4. Even-odd identities: sin (-θ) = -sin θ cos (-θ) = cos θ tan (-θ) = -tan θ
5. Angle sum and difference identities: sin (θ + φ) = sin θ cos φ + cos θ sin φ cos (θ + φ) = cos θ cos φ – sin θ sin φ tan (θ + φ) = (tan θ + tan φ)/(1 – tan θ tan φ)
These identities can be used to simplify trigonometric expressions and solve trigonometric equations. For example, we can use the Pythagorean identity sin² θ + cos² θ = 1 to rewrite sin² θ as 1 –
cos² θ, and vice versa. Similarly, we can use the reciprocal identities to rewrite trigonometric functions in terms of other functions. The angle sum and difference identities can be used to expand
trigonometric functions of the sum or difference of two angles.
Trigonometrical Ratio of Associated Angles
Associated angles are two angles whose sum or difference is a multiple of 90 degrees (or π/2 radians). The trigonometric ratios of associated angles have a special relationship, which can be derived
from the angle sum and difference identities. Here are the trigonometric ratios of associated angles:
1. sin (90° – θ) = cos θ sin (90° + θ) = cos θ cos (90° – θ) = sin θ cos (90° + θ) = -sin θ tan (90° – θ) = cot θ tan (90° + θ) = -cot θ
2. sin (-θ) = -sin θ cos (-θ) = cos θ tan (-θ) = -tan θ
The first set of identities shows that the sine and cosine functions of complementary angles are interchangeable. For example, sin 60° = cos 30° and cos 45° = sin 45°. The second set of identities
shows that the sine and tangent functions are odd functions, while the cosine function is an even function.
These identities can be used to simplify trigonometric expressions and solve trigonometric equations involving associated angles. For example, if we know that sin x = cos 20°, we can use the first
identity above to find that x = 70° (since 90° – 20° = 70°). Similarly, if we have an equation involving both sine and cosine functions of the same angle, we can use the identity sin² θ + cos² θ = 1
to eliminate one of the functions and simplify the equation.
Trigonometrical Ratio of Compound Angles
Compound angles are two or more angles combined in a single trigonometric function. The trigonometric ratios of compound angles can be derived from the angle sum and difference identities. Here are
the trigonometric ratios of some common compound angles:
1. sin (A + B) = sin A cos B + cos A sin B sin (A – B) = sin A cos B – cos A sin B cos (A + B) = cos A cos B – sin A sin B cos (A – B) = cos A cos B + sin A sin B tan (A + B) = (tan A + tan B)/(1 –
tan A tan B) tan (A – B) = (tan A – tan B)/(1 + tan A tan B)
2. sin 2A = 2 sin A cos A cos 2A = cos² A – sin² A = 2 cos² A – 1 = 1 – 2 sin² A tan 2A = (2 tan A)/(1 – tan² A)
3. sin 3A = 3 sin A – 4 sin³ A cos 3A = 4 cos³ A – 3 cos A
These identities can be used to simplify trigonometric expressions and solve trigonometric equations involving compound angles. For example, if we know that sin 2x = cos 3x, we can use the identity
cos 3x = 1 – 2 sin² x to rewrite the equation as 2 sin² x + sin 2x – 1 = 0, which is a quadratic equation in sin x that can be solved using the quadratic formula. Similarly, if we have an expression
involving a product of sines or cosines of two angles, we can use the identity sin A sin B = (1/2)[cos(A – B) – cos(A + B)] or cos A cos B = (1/2)[cos(A – B) + cos(A + B)] to simplify the expression.
Transformation Formula
Trigonometric transformation formulas are used to rewrite trigonometric functions in terms of other trigonometric functions with different arguments. Here are some common transformation formulas:
1. sin(-x) = -sin x, cos(-x) = cos x tan(-x) = -tan x, cot(-x) = -cot x
2. sin(x ± π) = -sin x, cos(x ± π) = -cos x tan(x ± π) = tan x, cot(x ± π) = cot x
3. sin(π/2 – x) = cos x, cos(π/2 – x) = sin x tan(π/2 – x) = cot x, cot(π/2 – x) = tan x
4. sin(π – x) = sin x, cos(π – x) = -cos x tan(π – x) = -tan x, cot(π – x) = -cot x
5. sin(2π – x) = -sin x, cos(2π – x) = cos x tan(2π – x) = tan x, cot(2π – x) = cot x
6. sin(x + 2πn) = sin x, cos(x + 2πn) = cos x tan(x + 2πn) = tan x, cot(x + 2πn) = cot x
7. sin(π/2 + x) = cos x, cos(π/2 + x) = -sin x tan(π/2 + x) = -cot x, cot(π/2 + x) = -tan x
8. sin(π + x) = -sin x, cos(π + x) = -cos x tan(π + x) = tan x, cot(π + x) = cot x
9. sin(3π/2 + x) = -cos x, cos(3π/2 + x) = -sin x tan(3π/2 + x) = cot x, cot(3π/2 + x) = -tan x
These formulas can be used to simplify trigonometric expressions and solve trigonometric equations involving different arguments or periodicity. For example, if we know that cos x = -1/2, we can use
the formula cos(x + 2πn) = cos x to find all solutions of the equation cos x = -1/2 in the interval [0, 2π]. Similarly, if we have an expression involving a product of sines or cosines of two angles
with a sum or difference of π/2, we can use the appropriate transformation formula to rewrite the expression in terms of a single trigonometric function with a different argument.
Multiple Angels
Multiple angles are angles that are multiples of a given angle, usually expressed in terms of that angle. For example, the angle 2x is a multiple of x.
Trigonometric functions of multiple angles can be expressed in terms of the trigonometric functions of the given angle using the following formulas:
1. sin 2x = 2 sin x cos x
2. cos 2x = cos² x – sin² x
3. tan 2x = (2 tan x) / (1 – tan² x)
4. cot 2x = (cot² x – 1) / (2 cot x)
Using these formulas, we can simplify trigonometric expressions involving multiple angles. For example, if we want to simplify sin 3x in terms of sin x, we can use the formula sin 3x = 3 sin x – 4
sin³ x to get:
sin 3x = 3 sin x – 4 sin³ x = 3 sin x – 4 (sin x)³
Similarly, if we have an expression involving a product of sines or cosines of multiple angles, we can use the appropriate formula to rewrite the expression in terms of the trigonometric functions of
the given angle. For example, we can use the formula cos 2x = cos² x – sin² x to simplify the expression cos 2x cos 3x as follows:
cos 2x cos 3x = (cos² x – sin² x) (cos³ x – 3 cos x sin² x)
= cos⁵ x – 3 cos³ x sin² x cos x – sin² x cos³ x + 3 sin⁴ x cos x
= cos⁵ x – 3 cos³ x sin² x cos x – cos³ x sin² x + 3 cos x sin⁴ x
= cos⁵ x – 4 cos³ x sin² x cos x + 3 cos x sin⁴ x.
Sub-Multiple Angles
Sub-multiple angles are angles that are a fraction of a given angle, usually expressed in terms of that angle. For example, the angle x/2 is a sub-multiple of x.
Trigonometric functions of sub-multiple angles can be expressed in terms of the trigonometric functions of the given angle using the following formulas:
1. sin (x/2) = ±√[(1 – cos x)/2]
2. cos (x/2) = ±√[(1 + cos x)/2]
3. tan (x/2) = ±√[(1 – cos x)/(1 + cos x)]
The sign of the trigonometric functions depends on the quadrant in which the angle lies. In the first and second quadrants, the sign is positive, while in the third and fourth quadrants, the sign is
Using these formulas, we can simplify trigonometric expressions involving sub-multiple angles. For example, if we want to simplify cos (3x/4) in terms of cos x, we can use the formula cos (x/2) = ±√
[(1 + cos x)/2] to get:
cos (3x/4) = cos [(x/2) + (x/4)] = cos (x/2) cos (x/4) – sin (x/2) sin (x/4)
= ±√[(1 + cos x)/2] × ±√[(1 + cos (x/2))/2] – ±√[(1 – cos x)/2] × ±√[(1 – cos (x/2))/2]
Similarly, if we have an expression involving a product of sines or cosines of sub-multiple angles, we can use the appropriate formula to rewrite the expression in terms of the trigonometric
functions of the given angle. For example, we can use the formula sin (x/2) = ±√[(1 – cos x)/2] to simplify the expression sin (x/4) sin (x/2) cos (x/4) as follows:
sin (x/4) sin (x/2) cos (x/4) = ±√[(1 – cos x)/2] × ±√[(1 + cos x)/2] × ±√[(1 + cos (x/2))/2]
Maximum & Minimum Trigonometric Functions
The maximum and minimum values of the trigonometric functions depend on the range of values of the angle.
For the sine and cosine functions, the maximum value is 1 and the minimum value is -1. These values are achieved when the angle is a multiple of 360 degrees plus or minus 90 degrees.
For the tangent and cotangent functions, the maximum and minimum values are not defined. They both have vertical asymptotes at odd multiples of 90 degrees. However, for the tangent function, as the
angle approaches odd multiples of 90 degrees, the function becomes infinitely large in the positive or negative direction.
For the secant and cosecant functions, their maximum and minimum values are also not defined. They both have vertical asymptotes at even multiples of 90 degrees. As the angle approaches even
multiples of 90 degrees, the functions approach positive or negative infinity.
It is important to note that these maximum and minimum values apply to the range of values of the angle for which the trigonometric function is defined. For example, the sine function is defined for
all real values of the angle, but its maximum and minimum values only apply to angles within a certain range.
Height & Distance
Height and distance problems involve using trigonometric functions to determine unknown distances or angles related to a right triangle. These types of problems are commonly encountered in real-life
situations, such as surveying, construction, and navigation.
Typically, height and distance problems involve a person observing an object from a certain height or distance and trying to determine the height or distance of the object or the angle of elevation
or depression from the observer’s position. The observer can use trigonometric functions such as sine, cosine, and tangent to solve the problem.
For example, consider the following problem:
A person standing on the ground is looking at a flagpole. The angle of elevation to the top of the flagpole is 30 degrees. The person then moves 50 meters closer to the flagpole and finds that the
angle of elevation to the top of the flagpole is now 45 degrees. How tall is the flagpole?
To solve this problem, we can use the tangent function. Let x be the height of the flagpole. Then, we can set up the following equation:
tan 30 = x / d
where d is the distance between the person and the flagpole when the angle of elevation is 30 degrees.
Similarly, when the person moves 50 meters closer to the flagpole, the distance becomes d – 50. Thus, we can set up another equation:
tan 45 = x / (d – 50)
We now have two equations with two unknowns. We can solve for x by eliminating d using the substitution method:
x = d * tan 30
x = (d – 50) * tan 45
Substituting the first equation into the second equation, we get:
d * tan 30 = (d – 50) * tan 45
Solving for d, we get:
d = 50 / (tan 45 – tan 30)
Substituting the value of d into the first equation, we get:
x = d * tan 30
Finally, we can calculate the height of the flagpole:
x = d * tan 30 = (50 / (tan 45 – tan 30)) * tan 30 = 40.2 meters
Therefore, the height of the flagpole is 40.2 meters.
Measurement of Angles
Angles can be measured in degrees, radians, or grads.
Degrees are the most common unit of measurement for angles. A full circle is divided into 360 degrees, with each degree further divided into 60 minutes and each minute divided into 60 seconds.
Radians are another unit of measurement for angles. A radian is the angle subtended at the center of a circle by an arc equal in length to the radius of the circle. One complete revolution around the
circle is equal to 2π radians. Radians are often used in calculus and other areas of mathematics.
Grads are a less common unit of measurement for angles. A full circle is divided into 400 grads, with each grad further divided into 100 centigrads. Grads are used in some scientific and engineering
applications, particularly in Europe.
Converting between these units of measurement can be done using the following formulas:
• To convert from degrees to radians: multiply by π/180
• To convert from radians to degrees: multiply by 180/π
• To convert from degrees to grads: multiply by 10/9
• To convert from grads to degrees: multiply by 9/10
• To convert from radians to grads: multiply by 200/π
• To convert from grads to radians: multiply by π/200
It is important to use the appropriate units of measurement for a given problem and to be familiar with the formulas and conversion factors involved.
Properties of Triangles
Here are some important properties of triangles:
1. The sum of the interior angles of a triangle is 180 degrees.
2. The exterior angle of a triangle is equal to the sum of the two interior angles that are opposite to it.
3. The sum of the lengths of any two sides of a triangle is always greater than the length of the third side.
4. In a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This is known as the
Pythagorean theorem.
5. The medians of a triangle, which connect each vertex to the midpoint of the opposite side, intersect at a point called the centroid. The centroid is also the center of gravity of the triangle.
6. The altitudes of a triangle, which are perpendiculars from each vertex to the opposite side, intersect at a point called the orthocenter.
7. The angle bisectors of a triangle, which divide each angle into two equal parts, intersect at a point called the incenter. The incenter is the center of the circle that can be inscribed inside
the triangle.
8. The perpendicular bisectors of the sides of a triangle, which are lines that are perpendicular to a side and pass through its midpoint, intersect at a point called the circumcenter. The
circumcenter is the center of the circle that can be circumscribed around the triangle
Basic Operation
Basic operations are the fundamental mathematical operations used in arithmetic: addition, subtraction, multiplication, and division. These operations are used to perform calculations with numbers,
and they form the basis for more advanced mathematical concepts.
Addition is the operation of combining two or more numbers to find a total or a sum. For example, 2 + 3 = 5.
Subtraction is the operation of taking one number away from another to find the difference. For example, 5 – 3 = 2.
Multiplication is the operation of repeated addition, or of finding the product of two numbers. For example, 2 x 3 = 6.
Division is the operation of finding the number of times one number is contained within another, or of finding the quotient of two numbers. For example, 6 ÷ 3 = 2.
In addition to these basic operations, there are also other mathematical concepts such as exponents, roots, and percentages that build upon them.
Factorization is the process of expressing a number or an algebraic expression as a product of its factors. The factors are the numbers or algebraic expressions that multiply together to produce the
original number or expression.
Factorization is important in mathematics because it helps to simplify expressions and solve equations. It is also used in cryptography to break codes and in number theory to study the properties of
prime numbers.
There are different methods for factorization, depending on the type of number or expression being factored. Some common methods include:
1. Factorization by inspection: This involves identifying common factors and dividing them out. For example, to factor the number 12, we can divide it by 2, which gives us 6, and then divide 6 by 2
again, which gives us 3. So, 12 can be factored as 2 x 2 x 3.
2. Factorization by grouping: This involves grouping terms in an expression that have a common factor, and then factoring out that common factor. For example, to factor the expression x^2 + 3x + 2,
we can group the first two terms and factor out an x, and group the last two terms and factor out a 2. This gives us (x + 1)(x + 2).
3. Factorization by using algebraic identities: This involves using algebraic identities such as the difference of squares, the sum and difference of cubes, and other formulas to factor expressions.
For example, to factor the expression x^2 – 4, we can use the difference of squares identity and write it as (x + 2)(x – 2).
4. Factorization using the quadratic formula: This involves using the quadratic formula to solve for the roots of a quadratic equation, which can then be used to factor the equation. For example, to
factor the equation x^2 + 3x + 2 = 0, we can use the quadratic formula to find the roots, which are -1 and -2. So, the equation can be factored as (x + 1)(x + 2).
Componendo & Dividendo
Componendo and Dividendo is a method of solving mathematical problems that involve equations with ratios or fractions. The method involves adding or subtracting the numerator and denominator of the
ratio or fraction to derive a new equation that can be easily solved.
The method is based on the following property:
If a/b = c/d, then (a+b)/(a-b) = (c+d)/(c-d) or (a-b)/(a+b) = (c-d)/(c+d)
The method involves using this property repeatedly to simplify the equation until it can be easily solved.
For example, let us consider the equation:
(a/b) = (c/d)
To solve this equation using Componendo and Dividendo, we can add the numerator and denominator of the left-hand side and the right-hand side, which gives:
(a+b)/(a-b) = (c+d)/(c-d)
This equation can be simplified further by multiplying both sides by (a-b)(c-d), which gives:
(a+b)(c-d) = (a-b)(c+d)
Expanding both sides, we get:
ac – ad + bc – bd = ac + ad – bc – bd
Simplifying, we get:
2ad = 2bc
Dividing both sides by 2, we get:
ad = bc
So, the solution to the equation (a/b) = (c/d) is ad = bc.
Componendo and Dividendo is a powerful method for solving equations with ratios or fractions, but it should be used with caution and only when other methods are not applicable.
Simplifying Algebraic Expressions
Simplifying algebraic expressions involves using various techniques to simplify complex expressions into simpler, more manageable forms. Here are some common techniques for simplifying algebraic
1. Combining like terms: Combine any terms that have the same variables and exponents.
2. Distributive property: Use the distributive property to multiply a term by a factor outside a set of parentheses.
3. Factoring: Factor out common factors or use techniques such as grouping and difference of squares.
4. Simplifying fractions: Simplify any fractions in the expression by canceling out common factors.
5. Expanding: Expand any parentheses in the expression by distributing the terms inside.
6. Using identities: Use identities such as the difference of squares, perfect square trinomial, or sum/difference of cubes to simplify the expression.
7. Simplifying exponents: Simplify any exponents using the laws of exponents, such as multiplying or dividing exponents with the same base.
By using these techniques and practicing with a variety of problems, you can become proficient in simplifying algebraic expressions.
Find the Value of Algebraic Expressions
To find the value of an algebraic expression, you need to substitute the variables with their corresponding values and then simplify the expression using the order of operations. Here are the steps
to follow:
1. Identify the variables in the expression.
2. Substitute the values of the variables given in the problem or question.
3. Simplify the expression using the order of operations (parentheses, exponents, multiplication/division, addition/subtraction).
4. Double-check your work to make sure you have correctly substituted values and simplified the expression.
Here is an example:
Evaluate the expression 3x^2 + 2y – 4z for x=2, y=5, and z=3.
1. The variables in the expression are x, y, and z.
2. Substitute the values: 3(2)^2 + 2(5) – 4(3) = 12 + 10 – 12
3. Simplify using the order of operations: 12 + 10 – 12 = 10
4. Double-check your work to ensure the correct answer: 3(2)^2 + 2(5) – 4(3) = 3(4) + 10 – 4(3) = 12 + 10 – 12 = 10
Therefore, the value of the expression 3x^2 + 2y – 4z for x=2, y=5, and z=3 is 10.
An equation is a mathematical statement that states that two expressions are equal. It consists of two sides, the left-hand side and the right-hand side, separated by an equal sign (=).
To solve an equation, the goal is to determine the value(s) of the variable(s) that make the equation true. Here are the general steps to solve an equation:
1. Simplify both sides of the equation by using any algebraic techniques such as combining like terms or distributing.
2. Isolate the variable term(s) on one side of the equation by using inverse operations such as addition and subtraction, multiplication and division, or applying the properties of exponents.
3. Simplify the resulting expression(s) to find the value(s) of the variable(s).
4. Check your solution(s) by substituting it into the original equation and verifying that both sides are equal.
Here is an example of solving a linear equation:
Solve for x: 2x + 5 = 13
1. Simplify both sides by subtracting 5 from both sides: 2x = 8
2. Isolate the variable term by dividing both sides by 2: x = 4
3. Simplify the resulting expression: x = 4
4. Check the solution by substituting x=4 into the original equation: 2(4) + 5 = 13, which is true.
Therefore, the solution to the equation 2x + 5 = 13 is x=4.
In mathematics, a polynomial is a mathematical expression consisting of variables (also called indeterminates) and coefficients, which are combined using arithmetic operations such as addition,
subtraction, multiplication, and exponentiation to produce a finite sum of terms.
The terms of a polynomial are made up of a coefficient multiplied by a variable raised to a non-negative integer power, called the degree of the term. The degree of the polynomial is the highest
degree of any term in the polynomial.
For example, the polynomial 3x^2 – 2x + 5 has three terms: 3x^2, -2x, and 5. The degree of the first term is 2, the degree of the second term is 1, and the degree of the third term is 0. Therefore,
the degree of the polynomial is 2, which is the highest degree of any term in the polynomial.
Polynomials can be added, subtracted, multiplied, and divided using algebraic techniques. They are used in many areas of mathematics, including algebra, calculus, and geometry, as well as in various
applications such as physics, engineering, and economics.
Polynomial Factorisation
Polynomial factorization is the process of breaking down a polynomial expression into a product of simpler polynomials. This process is useful for simplifying and solving polynomial equations. There
are different methods of polynomial factorization, and here are some common techniques:
1. Factoring out a common factor: Look for common factors among the terms of the polynomial, and factor them out. For example, the expression 6x^2 + 9x can be factored as 3x(2x + 3) by factoring out
the common factor 3x.
2. Factoring by grouping: Group the terms of the polynomial into pairs and factor out a common factor from each pair. Then, factor out the common factor between the two resulting expressions. For
example, the expression 3x^3 + 6x^2 + 4x + 8 can be factored as 3x^2(x + 2) + 4(x + 2) by grouping the first two terms and the last two terms and factoring out the common factor (x + 2).
3. Factoring quadratic trinomials: A quadratic trinomial is a polynomial of the form ax^2 + bx + c, where a, b, and c are constants. To factor a quadratic trinomial, find two numbers that multiply
to a times c and add to b. Then, rewrite the quadratic trinomial as a product of two binomials, where the binomials have the form (px + q) and (rx + s), respectively. For example, the quadratic
trinomial 2x^2 + 5x + 3 can be factored as (2x + 3)(x + 1) by finding two numbers that multiply to 2 times 3 and add to 5, which are 2 and 3.
4. Factoring using special products: There are special products that can be used to factor some polynomials, such as the difference of squares and the sum and difference of cubes. For example, the
expression x^2 – 4 can be factored as (x + 2)(x – 2) using the difference of squares.
These are some common techniques for polynomial factorization. Practicing these techniques can help to improve your ability to factor polynomials.
Remainder Theorem
The remainder theorem is a theorem in algebra that states that if a polynomial P(x) is divided by (x-a), the remainder is P(a). In other words, when we divide a polynomial P(x) by (x-a), the
remainder is equal to the value of P(a).
This theorem can be useful in finding the remainder of polynomial division and in evaluating polynomials at specific values of x. Here are the general steps to use the remainder theorem:
1. Write the polynomial P(x) in the form P(x) = Q(x) * (x-a) + R, where Q(x) is the quotient polynomial, (x-a) is the divisor, and R is the remainder.
2. Substitute x=a into P(x) to find the value of R. That is, R = P(a).
For example, consider the polynomial P(x) = x^3 – 3x^2 + 4x – 8. To find the remainder when P(x) is divided by (x-2), we can use the remainder theorem as follows:
1. Write P(x) in the form P(x) = Q(x) * (x-2) + R. Dividing P(x) by (x-2) gives:
x^3 - 3x^2 + 4x - 8 = (x-2)(x^2 - x + 6) + 4
Therefore, the remainder is R = 4.
2. Substitute x=2 into P(x) to check the result. That is, P(2) = 2^3 – 3(2^2) + 4(2) – 8 = 8 – 12 + 8 – 8 = -4, which is the same as the remainder we obtained using the remainder theorem.
Therefore, the remainder when P(x) is divided by (x-2) is 4.
Factor Theorem
The factor theorem is a theorem in algebra that relates the factors of a polynomial to its roots. Specifically, the theorem states that if a polynomial P(x) has a root of r, then (x-r) is a factor of
P(x). In other words, if P(r) = 0, then (x-r) divides P(x) evenly, or in other words, P(x) = (x-r) * Q(x) for some polynomial Q(x).
This theorem can be used to find the factors of a polynomial and to solve polynomial equations. Here are the general steps to use the factor theorem:
1. Given a polynomial P(x), identify a potential root r.
2. Check whether P(r) = 0. If P(r) = 0, then (x-r) is a factor of P(x) by the factor theorem.
3. Use polynomial division or other techniques to factor P(x) completely.
For example, consider the polynomial P(x) = x^3 – 6x^2 + 11x – 6. To find its factors, we can use the factor theorem as follows:
1. Identify a potential root r. The factors of the constant term -6 are ±1, ±2, ±3, and ±6, so we can try these values as potential roots. By testing these values, we find that r=1 is a root of P
2. Check whether P(r) = 0. Substituting x=1 into P(x), we get P(1) = 1^3 – 6(1^2) + 11(1) – 6 = 0, so (x-1) is a factor of P(x) by the factor theorem.
3. Factor P(x) completely. Using polynomial division or other techniques, we can divide P(x) by (x-1) to obtain the factorization P(x) = (x-1) * (x^2 – 5x + 6) = (x-1) * (x-2) * (x-3).
Therefore, the factors of the polynomial P(x) are (x-1), (x-2), and (x-3).
Quadratic Equation
A quadratic equation is a polynomial equation of degree 2, which can be written in the form of ax^2 + bx + c = 0, where a, b, and c are constants, and x is the variable. The general form of a
quadratic equation is:
ax^2 + bx + c = 0
To solve a quadratic equation, we can use various methods, including factoring, completing the square, and using the quadratic formula. Here are the general steps for each method:
1. Factoring method: If the quadratic equation can be factored, we can use the zero-product property to find the solutions. The steps are:
a) Factor the quadratic expression. b) Set each factor equal to zero and solve for x. c) Check the solutions by substituting them back into the original equation.
For example, consider the quadratic equation x^2 – 5x + 6 = 0. This equation can be factored as (x-2)(x-3) = 0. Setting each factor equal to zero gives x=2 and x=3. Checking these solutions by
substituting them back into the original equation confirms that they are valid solutions.
2. Completing the square method: If the quadratic equation cannot be factored, we can use the completing the square method. The steps are:
a) Move the constant term to the right side of the equation. b) Divide both sides by the coefficient of x^2 to make the coefficient 1. c) Add and subtract (b/2a)^2 to the left side of the equation to
create a perfect square trinomial. d) Simplify the left side of the equation. e) Take the square root of both sides of the equation and solve for x. f) Check the solution(s) by substituting it back
into the original equation.
For example, consider the quadratic equation x^2 – 6x + 8 = 0. Adding and subtracting (6/2)^2 = 9 to the left side of the equation gives (x-3)^2 – 1 = 0. Simplifying the left side of the equation
gives (x-3)^2 = 1. Taking the square root of both sides gives x-3 = ±1, which leads to the solutions x=2 and x=4. Checking these solutions by substituting them back into the original equation
confirms that they are valid solutions.
3. Quadratic formula method: If the quadratic equation cannot be factored, we can use the quadratic formula. The quadratic formula is:
x = (-b ± sqrt(b^2 – 4ac)) / 2a
The steps to use the quadratic formula are:
a) Identify the values of a, b, and c in the quadratic equation. b) Substitute these values into the quadratic formula and simplify. c) Solve for x by using the plus-minus symbol. d) Check the
solution(s) by substituting it back into the original equation.
For example, consider the quadratic equation 2x^2 + 3x – 2 = 0. Using the quadratic formula, we have:
x = (-3 ± sqrt(3^2 – 4(2)(-2))) / 2(2) = (-3 ± sqrt(25)) / 4
Simplifying the expression gives x = (-3 ± 5) / 4, which leads to the solutions x=1/2 and x=-2. Checking these solutions by substituting them back into the original equation confirms that they are
valid solutions.
Therefore, these are the methods to solve a quadratic equation.
In mathematics, the reciprocal of a number is defined as the inverse of that number. In other words, if we have a non-zero number x, then its reciprocal, denoted by 1/x, is a number such that when it
is multiplied by x, the result is 1.
For example, the reciprocal of 5 is 1/5, because 5 multiplied by 1/5 gives 1:
5 × 1/5 = 1
Similarly, the reciprocal of 2/3 is 3/2, because 2/3 multiplied by 3/2 gives 1:
2/3 × 3/2 = 1
The reciprocal of a number is also called its multiplicative inverse, because multiplying a number by its reciprocal results in the identity element (1) of the multiplication operation. In other
words, if x is a non-zero number, then:
x × 1/x = 1
The reciprocal of a number can be calculated by taking the inverse of that number, which is obtained by dividing 1 by the number. For example, the reciprocal of 4 is:
1/4 = 0.25
Similarly, the reciprocal of 0.2 is:
1/0.2 = 5
It is important to note that the reciprocal of zero does not exist, because any non-zero number multiplied by zero gives zero, but no number multiplied by zero gives 1. Therefore, the reciprocal of
zero is undefined.
Perfect Square
In mathematics, a perfect square is a number that can be expressed as the product of an integer and itself. In other words, a perfect square is the square of an integer. For example, 9 is a perfect
square because it can be expressed as 3 times 3, or 3^2:
9 = 3 × 3 = 3^2
Similarly, 16 is a perfect square because it can be expressed as 4 times 4, or 4^2:
16 = 4 × 4 = 4^2
The first few perfect squares are:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, …
Perfect squares have many interesting properties in mathematics. For example, the sum of the first n odd numbers is always a perfect square:
1 + 3 + 5 + … + (2n – 1) = n^2
In addition, any positive integer can be expressed as the sum of consecutive perfect squares:
n = a^2 + b^2 + c^2 + …
where a, b, c, … are integers.
Perfect squares are also used in many areas of mathematics, including geometry, algebra, and number theory. For example, in geometry, the area of a square is equal to the square of its side length,
and in algebra, the square of a binomial can be expanded using the formula:
(a + b)^2 = a^2 + 2ab + b^2
Overall, perfect squares are an important concept in mathematics with many interesting and useful properties.
Problem Based on Algebraic Formulae
The area of a rectangular garden is given by the formula A = lw, where l is the length and w is the width. If the length is 3 meters more than the width, and the area of the garden is 60 square
meters, find the dimensions of the garden.
Let w be the width of the garden in meters.
According to the problem, the length of the garden is 3 meters more than the width. Therefore, the length can be expressed as:
l = w + 3
The area of the garden is given by the formula A = lw. Substituting l = w + 3, we get:
A = w(w + 3)
We are also given that the area of the garden is 60 square meters. Therefore, we can write the equation:
w(w + 3) = 60
Expanding the left side and simplifying, we get a quadratic equation:
w^2 + 3w – 60 = 0
We can solve this equation by factoring or by using the quadratic formula. Factoring gives:
(w + 8)(w – 5) = 0
Therefore, either w + 8 = 0 or w – 5 = 0. This means that either w = -8 or w = 5. Since the width of the garden must be a positive number, we reject the solution w = -8 and choose w = 5.
Using l = w + 3, we can find the length of the garden:
l = 5 + 3 = 8
Therefore, the dimensions of the garden are 5 meters by 8 meters.
Answer: The dimensions of the garden are 5 meters by 8 meters.
Problem Based on Putting The value
Sure, here is an example problem based on putting the value:
Problem: Evaluate the expression 3x^2 – 2xy + y^2 when x = 4 and y = 2.
We are given the expression 3x^2 – 2xy + y^2, and we are asked to evaluate it when x = 4 and y = 2. To do this, we simply substitute these values into the expression and simplify:
3x^2 – 2xy + y^2 = 3(4)^2 – 2(4)(2) + (2)^2 (substitute x = 4 and y = 2) = 3(16) – 16 + 4 = 48 – 16 + 4 = 36
Therefore, the value of the expression 3x^2 – 2xy + y^2 when x = 4 and y = 2 is 36.
Answer: 36.
Sure! Here are a few miscellaneous math concepts that you may find useful:
1. Exponential Functions: An exponential function is a mathematical function of the form f(x) = a^x, where a is a constant called the base, and x is the variable. Exponential functions are used to
model many real-world phenomena, such as population growth, radioactive decay, and compound interest.
2. Trigonometric Functions: Trigonometric functions are a family of functions that relate angles to the ratios of the sides of a right triangle. The most common trigonometric functions are sine,
cosine, and tangent, which are abbreviated as sin, cos, and tan, respectively. Trigonometric functions have many applications in physics, engineering, and geometry.
3. Vectors: A vector is a mathematical object that has both magnitude and direction. Vectors can be added and subtracted, and they can also be multiplied by scalars. Vectors are used to represent
many physical quantities, such as velocity, force, and acceleration.
4. Matrices: A matrix is a rectangular array of numbers or other mathematical objects. Matrices are used to represent linear transformations, solve systems of linear equations, and model many other
mathematical and scientific phenomena.
5. Limits and Continuity: Limits and continuity are important concepts in calculus, which is the study of change. A limit is the value that a function approaches as the input approaches a certain
value, and continuity is the property of a function that implies that small changes in the input lead to small changes in the output.
These are just a few miscellaneous math concepts, but there are many more! Let me know if you have any specific questions or topics you would like to learn about.
Maximum & Minimum
In mathematics, maximum and minimum are terms used to describe the highest and lowest values of a function, respectively. They are also known as the global maximum and global minimum, because they
represent the absolute highest and lowest values that the function can take on over its entire domain.
For example, consider the function f(x) = x^2 – 4x + 3. This function is a parabola that opens upward, and its vertex is located at the point (2, -1). The vertex is the lowest point on the parabola,
and it represents the global minimum of the function. Therefore, the global minimum of f(x) occurs when x = 2, and its value is f(2) = -1.
To find the global maximum or minimum of a function, you can use a variety of methods, including calculus, algebra, or graphical analysis. For example, in calculus, you can find the global maximum or
minimum of a function by finding the critical points (points where the derivative is zero or undefined) and comparing the function values at those points to the function values at the endpoints of
the domain.
It is important to note that the global maximum and minimum are not the same as local maximum and minimum. A local maximum or minimum is a point where the function reaches its highest or lowest value
within a certain interval, but it may not be the absolute highest or lowest value over the entire domain of the function. | {"url":"https://khantimethodonline.com/maths-2/","timestamp":"2024-11-04T18:12:30Z","content_type":"text/html","content_length":"486243","record_id":"<urn:uuid:56f4963a-1847-43a9-a97b-8479f5c5860c>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00036.warc.gz"} |
FXEducation Glossary - Crystal Fx Group
FXEducation Glossary
An asset bubble is when assets such as housing, stocks, or gold dramatically rise in price over a short period that is not supported by the value of the product.
Austerity measures are reductions in government spending, increases in tax revenues, or both. These harsh steps are taken to lower budget deficits and avoid a debt crisis.
Trade Balance is the value of a country’s exports minus its imports. It’s the biggest component of the balance of payments that measures all international transactions.
Bank for International Settlements
Headquartered in Basel, Switzerland, the Bank for International Settlements (BIS) is a bank for central banks.
Closing out a position by taking the opposite position leaving a zero exposure.
A bear is an investor who thinks the market will go down. A bear market is when a market experiences prolonged price declines.
A trade at the best price available.
A black swan is an unpredictable event that is beyond what is normally expected of a situation and has potentially severe consequences.
A black-box financial model is a term used to describe a computer program designed to transform various data into useful investment strategies. The program logic is undisclosed.
According to John Bollinger, there’s one pattern he calls ‘the Squeeze’. As he puts it, his bands ‘are driven by volatility, and the Squeeze is a pure reflection of that volatility’. It describes a
sudden volatility collapse.
Under the Bretton Woods System, gold was the basis for the U.S. dollar and other currencies were pegged to the U.S. dollar’s value. The system was terminated in 1971.
Brexit is a portmanteau of the words ‘British’ and ‘exit’ coined to refer to the UK’s decision in a referendum on 23 June, 2016 to leave the European Union (EU). Brexit took place on 31 January,
A bull is an investor who thinks the market will go up. A bull market is when a market experiences prolonged price advance.
A carry trade is a trading strategy that involves borrowing at a low-interest rate and investing in an asset that provides a higher rate of return. A carry trade is typically based on borrowing in a
low-interest rate currency and converting the borrowed amount into another currency.
Continuation Chart Pattern
Continuation chart patterns indicate that a price trend is likely to continue.
When an indicator and the price move in the same direction.
Cost of carry refers to costs associated with the carrying value of an investment. These costs can include financial costs, such as the interest costs on bonds, interest expenses on margin accounts,
interest on loans used to make an investment, and any storage costs involved in holding a physical asset.
Some currency pairs don’t contain USD at all. These are known as cross-currency pairs or ‘crosses’. Some of the most popular crosses are also known as ‘minors’. The most actively traded crosses
normally contain EUR, JPY, and GBP (GBPJPY or EURJPY, for example).
Traders who go home with squared positions. They do not hold positions overnight.
The sum of the funds on your trading account denominated in your account currency.
A diffusion index refers to how many Business Cycle Indicators (BCI) are moving together. This is useful for assessing the strength of the economy.
When an indicator and the price move in differing directions.
The Fed is Dovish when its rhetoric indicates loosening monetary conditions and likely lower interest rates.
A Stock or Share and also the sum of the funds on your trading account, including the floating profit or loss.
The European sovereign debt crisis was a period starting on 2008 when several European countries experienced the collapse of financial institutions, high government debt, and rapidly rising bond
yield spreads in government securities.
These consist of one major currency paired with the currency of an emerging economy, like Brazil, Mexico, Russia, or Turkey (USDRUB or USDTRY, for example). It’s important to remember that trading
costs on exotics are generally higher due to wider spreads stemming from low liquidity.
Happens when the price or an indicator rallies and then makes a new low or falls then makes a new high.
These are tools that traders can use to establish profit targets or estimate how far a price may travel after a retracement/pullback is finished. Extension levels are also possible areas where the
price may reverse.
These levels are horizontal lines that indicate where support and resistance are likely to occur. They are based on Fibonacci numbers. Each level is associated with a percentage. The percentage is
how much of a prior move the price has retraced. The Fibonacci retracement levels are 23.6%, 38.2%, 61.8%, and 78.6%. While not official, a Fibonacci ratio of 50% is also used.
A fixed-income security is an investment that provides a return in the form of fixed periodic interest payments and the eventual return of principal at maturity.
These occur in chart patterns such as Flags or Pennants. They are the first part of the pattern; the impulse moves.
Shows your profit or loss for open positions.
Money in your account that is available to finance further positions.
Fundamentals include the basic qualitative and quantitative information that contributes to the financial or economic well-being of a company, security, or currency, and their subsequent financial
This occurs when the price of a security opens above or below the previous day’s close with no trading activity in between. A gap is the area discontinuity in a security’s price chart. Gaps may
materialise when headlines cause market fundamentals to change rapidly or during hours when markets are typically closed; for instance, the result of a weekend central bank annoucement
Global Capital Flows is the inflow and outflow of capital from one nation to another nation
Global Financial Crisis (Gfc),
The financial crisis of 2008 was primarily caused by deregulation in the financial industry.
Close out a position leaving no market exposure
The gold standard is a monetary system where a country’s currency or paper money has a value directly linked to gold. Britain stopped using the gold standard in 1931 and the U.S. followed suit in
1933 and abandoned the remnants of the system in 1973
When used in technical analysis, the Fibonacci golden ratios are typically three percentages: 38.2%, 50%, and 61.8%
The Fed is Hawkish when its rhetoric indicates tightening monetary conditions and likely higher interest rates.
A Hedge is a risk-offsetting transaction.
High-Frequency Trading (HFT)
High-frequency trading (HFT) is a method of trading that uses powerful computer programs to transact a large number of orders in fractions of a second.
The difference between the entry price and the first protective Stop.
International Monetary Fund (IMF)
The IMF is an international organisation that promotes global economic growth and financial stability, encourages international trade, and reduces poverty.
To put in very simple terms, it allows you to open larger trades with less capital. Leverage is a great tool that allows you to earn more, but it comes at a price. The higher the leverage you use,
the higher the risk of losing your investment. It can amplify both your earnings and losses. You need to find a balance between your capital and leverage. If you’re new to forex, it’s highly
recommended to keep leverage low and not to open too many positions. A small investment + high leverage = likely disaster.
If a currency pair is said to have high liquidity, it means that there is a significant amount of trading activity on the pair. This is good because there is always someone who wants the other side
of your trade. Low liquidity means the opposite and pertains typically to Exotics, but some Majors can suffer from it at times, for various reasons. Highly liquid pairs enjoy the narrowest spreads,
keeping trading costs low. Pairs with low liquidity are expensive to trade.
A unit of measurement. Normally equals 100,000 units of the base currency.
These are the four most heavily traded currency pairs in the forex market. The four major pairs at present are the EURUSD, USDJPY, GBPUSD, USDCHF.
The margin is derived from the amount of leverage you have on your account and works as a deposit to maintain an open order. As we saw in the explanation for leverage, the margin decreased as
leverage increased. That’s because higher leverage allows you to open larger trades with less capital.
The signal that your Margin Level is low and you urgently need to decrease your open positions or increase the Equity.
The percentage of Margin to Equity. This information is crucial because a low Margin Level may lead to a Margin Call and a Stop Out. Always monitor the Margin Level and adjust your positions and
Equity accordingly.
Makes bids and offers for a security on a continuous basis. In the Equities markets, they are called Specialists. They provide liquidity.
An order to be executed immediately at the current market price.
The hub inside MetaTrader where you can see live quotes for currencies, indices and commodities. It is normally positioned on the left-hand side of the screen.
Those who sell market strength and buy weakness.
The rate of acceleration of a security’s price.
The interest rate before taking inflation into account.
A statistical term to describe occurrences of an event in normal times.
OTC means not traded on an Exchange.
A condition where an asset has traded higher in price and has the potential for a price fall.
Overleveraging means that a small adverse movement erodes a significant part or all of your capital.
A condition where an asset has traded lower in price and has the potential for a price bounce.
Trading so frequently that the commission and spread significantly and adversely impact profitability.
An order to open a trade in the future if the price of the instrument hits a certain level.
The smallest price movement of a security. Pip is an acronym for ‘percentage in point’. When most currency pairs were priced out to four decimal places, the pip used to be the smallest price move
that an exchange rate could make. A pip is thus equivalent to 1/100 of 1% or one basis point. But nowadays most pairs are quoted up to five decimals, so the price may change by less than one pip.
A turning point in the market price action.
A move against the trend.
Mathematical or statistically based trading.
RoC is used in technical analysis to express the speed at which a security price changes over a specific period of time. RoC is often used when speaking about momentum.
The interest rate that has been adjusted to remove the effects of inflation to reflect the real cost of funds to the borrower and the real yield to the lender or an investor.
The official currency of China. Also called the Yuan.
Reversal chart patterns indicate that a price trend is likely to reverse direction.
Preferring less risky assets such as Bonds over Equities.
Preferring riskier assets such as Equities over Bonds.
The amount of money risked divided by the expected reward.
Assets or securities considered Safe – for example, guaranteed by a government.
Scalpers enter and exit the financial markets quickly, usually within seconds, using higher levels of leverage to place larger sized trades in the hopes of achieving greater profits from relatively
small price changes.
Difference between order price and execution usually caused by lack of liquidity or a surprise news event.
Marketing fraudster engaged in the illegal practice of making false or misleading promotional claims for financial gain.
These traders utilise strategies and typically a shorter time frame in an attempt to outperform traditional longer-term investors. Speculators take on risk, especially with respect to anticipating
future price movements, in the hope of making gains that are large enough to offset the risk.
Intercom speaker that an investment bank or brokerage firm’s analysts or traders use on trading floors or desks in OTC markets.
A standard deviation is a statistic that measures the dispersion of a dataset relative to its mean and is calculated as the square root of the variance. The standard deviation is calculated as the
square root of variance by determining each data point’s deviation relative to the mean.
Professional short-term traders who try to profit from setting off Stop Orders.
An order to close a trade at a specific price to stem losses. This is used to close a trade to stop your account being wiped out.
Triggered when the trader has failed to meet the Margin Call requirements, leading to the closure of open positions to bring the Margin Level to an acceptable level.
Shorthand for a tradable instrument. GBPUSD or .WTICrud – these are symbols.
A systematic trader adjusts a portfolio’s long- and short-term positions on a particular security according to tested rules.
An order to close a trade at a specific price to lock in profit without you having to physically close the trade yourself.
The gradual reversal of a quantitative easing policy implemented by a central bank to stimulate economic growth.
The part of the financial industry that is involved in the creation, promotion, market-making, and sale of stocks, bonds, forex, and other financial instruments.
Your profitability over time, your drawdowns, average time per trade.
A technique, observation or approach that creates a trading advantage over other market participants.
A log maintained by a trader recording trades and order changes.
A systematic (rule-based) method for identifying and trading securities that takes into consideration a number of variables including time, risk and the investor’s objectives.
A Trailing Stop is stop order that can be set at a defined percentage or dollar amount away from the current market price to protect a position and to run a profit until the price reverses.
Turnover of a market. In MetaTrader you define the size of your trade in lots.
Whipsaw describes the movement of a security when, at a particular time, the security’s price is moving in one direction but then quickly pivots to move in the opposite direction and maybe quickly
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An international organisation dedicated to providing financing, advice, and research to developing nations to aid their economic advancement. Its objective is to fight poverty by offering
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Surface Current Density Converter
Created by CalcKit Admin
Last updated: 7 Feb 2024
Surface Current Density describes the amount of electric current flowing through a given surface area. It is a vector quantity, representing the current flow per unit area, and is denoted by the
symbol "J".
Surface current density plays a crucial role in various applications, such as determining the magnetic field generated by electric currents, analyzing the behavior of conductive materials, and
designing electrical systems. It is commonly expressed in units of Amperes per square meter (A/m²).
Surface current density can be expressed in different units depending on the system of measurement used. Below are the common units of surface current density and their conversion factors:
• Ampere per square meter (A/m²): The standard unit of surface current density, where one Ampere of current flows through one square meter of area.
Conversion factors: 1 A/m² = 1 A/m² (Base Unit)
• Ampere per square centimeter (A/cm²): This unit expresses surface current density in Amperes per square centimeter.
Conversion factors: 1 A/cm² = 10,000 A/m²
• Ampere per square inch (A/in²): Surface current density measured in Amperes per square inch.
Conversion factors: 1 A/in² ≈ 1,550 A/m²
• Ampere per square mil (A/mi²): Surface current density expressed in Amperes per square mil, where one mil is equal to one-thousandth of an inch.
Conversion factors: 1 A/mi² ≈ 1,550,003,100 A/m²
• Ampere per circular-mil (A/cmil): This unit represents surface current density in Amperes per circular mil, where a circular mil is the area of a circle with a diameter of one mil (0.001 inches).
Conversion factors: 1 A/cmil ≈ 1,973,525,242 A/m²
• Abampere per square centimeter (abA/cm²): The abampere is a unit of electromagnetic current in the CGS system. Abampere per square centimeter expresses surface current density in abamperes per
square centimeter.
Conversion factors: 1 abA/cm² = 1 A/cm² = 100,000 A/m²
Surface current density is a fundamental concept in electromagnetism, and its proper understanding is essential for various applications. Our surface current density converter provides a convenient
way to convert between different units, enabling professionals to work with ease in their projects and research. | {"url":"https://calckit.io/tool/conversion-surface-current-density","timestamp":"2024-11-11T10:34:00Z","content_type":"text/html","content_length":"27772","record_id":"<urn:uuid:cd5945e1-8240-4a33-93b3-931cb3cab3c9>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00120.warc.gz"} |
Our Algebra Tutors In Colorado
Algebra Tutors
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Here Are Some Of Our Algebra Tutor Profiles
Ashley A
Teaching Style
First and foremost, I believe it is important to establish a trusting relationship with those you teach. Throughout my experiences in the field of education, I have found the most fruitful of those
relationships to be those in which I was able to work with a student or group of students regularly over an extended period of time to develop a routine and a strong relationship in which great
amounts of learning and understanding could be accomplished. Although many people experience obstacles in learning Mathematics, I believe that by presenting multiple ways to approach a problem, every
student can find a method that works for them. Every student should be allowed the resources and opportunity to realize that they CAN achieve their goals. I am here to help students who have had
difficulty with Math in the past to succeed, and feel confident in both their abilities in Math, and in life.
Experience Summary
As a student, I was always committed to learning and to achieving my goals. As a teacher and tutor, I strive to help others share the same love for learning and for understanding as I do. Now, I help
others set goals, and work toward achieving them. I have worked with all ages, all ability levels, and various sized groups and have enjoyed each and every experience. In the past, I have primarily
tutored in the subject of Mathematics, but am also trained by the Literacy Council to tutor reading and writing, and have enjoyed volunteering with that organization as well. I believe that I can
help anyone enjoy and understand math, and help them feel better about themselves for it.
Type Subject Issued-By Level Year
Degree Mathematics UNC Chapel Hill BA 2005
Lori L
Teaching Style
I believe that students are wary of mathematics and statistics because they appear clinic and distant, with little to do with the “real” world. To educate I try to create dynamic and above all
relevant “uses” for the lessons. My experiences have exposed me to different teaching styles and class formats and allowed me to develop a teaching philosophy that encompasses the best of all these
methods. My philosophy is best described with reference to six primary concepts: 1. Knowledge conveyed in a relevant context 2. Interaction with each student and the material 3. Passion for teaching
and the subject 4. Adaptability 5. Creativity in teaching methods and 6. Respect between the student and the teacher All students seek knowledge; it’s the teacher’s role to facilitate learning and
guide them along the path. A successful lesson is one in which the student comes out seeing the world a little differently.
Experience Summary
During my education and career, the teaching of others has featured prominently in my personal goals and life objectives. While attending high school, I tutored fellow students professionally in
subjects ranging from basic algebra to complex calculus. My undergraduate degree was in the field of Mathematics and Statistics with an additional biological statistics certificate and at present I
am completing my Masters of Science in statistics. My understandings of these fields lead to my recruit by a number of professors and researchers to provide assistance and advisement on statistical
analysis of their projects. In addition to assisting my professors, I was also selected to be a Graduate Teaching Assistant, and was also selected to conduct "stand alone" courses. I have a strong
passion for these subjects and I believe my years of teaching experience have given me the insight, patience and ability to convey the complex world of mathematics and statistics to my students.
Type Subject Issued-By Level Year
Degree Statistics University of North Florida Master's of Science (ABD) 2008
Certification Biostatistics University of West Florida Accredited 2006
Degree Mathematics University of West Florida Bachelor's of Science 2006
Dara D
Teaching Style
I believe in my students and their abilities to learn and synthesize their experiences. My students learn their subjects because I provide a variety of techniques to command their attention. I
believe teaching is not just about giving students information but about reaching students who might "get lost" in the system without a guide and friend to help them along.
Experience Summary
I have taught for over ten years in the public school system and learned how to "connect" with students. I have taught physics, chemistry, and mathematics at the high school level, and I have a wide
range of teaching experiences in those fields. I have taught AP, honors, and standard classes. Using interesting movies followed by a lab to reinforce the concept is one of the ways I have used to
reach students and make a difference in their lives.
Type Subject Issued-By Level Year
Certification Physics 6 - 12 State of Florida 9 credit hours current
Degree Mathematics University of Central Florida M.S. 1993
Degree Physics University of Central Florida B.S. 1990
Davorin D
Teaching Style
From my tutoring experience, I have noticed that students have trouble understanding the meaning of numbers and symbols on paper simply because no one has taught them how to visualize and interpret
them in a real world situation. They also don't realize that they have plethora of resources and tools available to them to help them, yet they rarely utilize them. I try to give hints and clues to
my students and let them obtain the right answers on their own instead of simply solving the problem for them. I believe this gives them a much better understand of the material.
Experience Summary
Having recently graduated with a Bachelor's degree, I am continuing my education to obtain my Master's degree in mathematics. Number Theory will be my focus, as I intend to get involved in the
encryption field. During my senior year as undergraduate, I have worked on campus as a teacher's assistant and at home as an online tutor for UNCG's iSchool program. I have privately tutored
undergraduates needing help in pre-calculus and calculus. I enjoyed helping the students and showing them some of my own tricks and ways when it comes to solving the problems.
Type Subject Issued-By Level Year
Degree Mathematics with Concentration in Computer Science University of North Carolina at Greensboro BS 2007
Veronica V
Teaching Style
I love teaching and the reward that it brings. I am a step-by-step oriented teacher. I've had many students return saying how my style of teaching continues to help them in their current studies. In
my 14 years of experience, I've learned that many students learn with many different styles. I believe that every child can learn; however, the teacher must reach them at their level. If I have a
good understanding of where the student is academically, I can help them to grow academically.
Experience Summary
I've taught middle school for 14 years. My goal in becoming a teacher was to reach those students who had, somehow, fallen through the cracks of education. I taught at a drop-out-prevention school
for 7 years. During those years, my student scores continuously rose. I taught 7th and 9th grade at this school. I transferred to a different middle school where I taught 6th & 7th grade for two
years and the remaining five years, I taught Pre-Algebra and Algebra 1. I began to tutor after school during my first year to struggling students. I was also listed on the school boards list of
tutors. I also worked at an after-school program. During these years, I tutored second graders and up to mathematical levels of geometry, Algebra 1 and Algebra II.
Type Subject Issued-By Level Year
Certification Mathematics In-service grades 5-8 current
Degree Mathematics Education Florida State University BS 1993
Redha R
Teaching Style
I teach by example and I am methodical. I show a student how to solve a problem by going very slowly and by following a sequence of steps. I make sure that the student follows understands each step
before moving on to the next one. I then ask the student to solve almost the exact same problem (with a slight change in numbers for example)so that the student learns the method and how to solve the
problem by herself/himself. I am very patient but expect the student to be willing to learn. I tell students that learning a subject matter is more important than getting an A in a class - this is
because 1. if you learn, chances are you won't forget (at least for a long while) and 2. you will get a good grade as a result.
Experience Summary
I have been employed by the IBM corporation for the last 27 years. I held numerous positions in hardware development, software development, telecommunication network development, project management,
solution architecture, performance analysis, and others. As much as I enjoy my job, I have passion in Mathematics. I develop new ideas related to my work and expand them into U.S. patents and
external technical papers. I informally tutor family and friends attending high school or college. I was a Mathematics teaching assistant at the Univ. of Pittsburgh for 2 years and at the Univ. of
Michigan at Ann Arbor for 3 years. I love teaching and sharing my knowledge with others. I have published numerous technical papers and hold numerous U.S. patents as well.
Type Subject Issued-By Level Year
Other Arabic Native speaker Fluent current
Other French Native speaker Fluent current
Degree Electrical Engineering Univeristy of Michigan P.h.D 1990
Degree Electrical Engineering and Mathematics University of Pittsburgh M.S. 1982
Degree Computer Science and Mathematics Univerisity of Pittsburgh B.S. 1980
Lucille L
Teaching Style
My students know I love math. I am enthusiastic, patient, and caring. I believe everyone can learn math given the right circumstances. I take interest in my students, I email them, I encourage them
to do their homework. Through this personal interest, my students work to please the teacher. I also use different teaching styles: discovery learning, Look-Do-Learn, one-to-one instruction, critical
thinking. Once a student, always a student for me. I go the extra mile.
Experience Summary
From an early age I loved math and so when I graduated with a BA in Math and Latin, I went straight into teaching math at the high school level. During graduate school years, I taught math at the
University of Toronto. While working with computer programming, I taught math in the evening division of Westbury College in Montreal. I have taught math in different countries: Jamaica, Canada, U.S.
Virgin Islands, Nigeria-West African Educational Council, The Bahamas, and Florida.
Type Subject Issued-By Level Year
Degree Math University of Toronto M.Sc. 1966
Robert R
Teaching Style
I’ve always been interested in the application of math and science to the solution of real world problems. This led me to a very satisfying career in engineering. Therefore my approach to teaching is
very application oriented. I like to relate the subject to problems that the students will encounter in real life situations. I've generally only worked with older students; high school or college
age or older mature adults who have returned to school to get advance training or learn a new trade.
Experience Summary
I’ve always been interested in math and science; especially in their application to solving real world problems. This led me to a very satisfying career in engineering. I have a BS in electrical
engineering from General Motors Institute (now Kettering University) and an MS in electrical engineering from Marquette University. I am a registered professional engineer in Illinois. I have over 30
years of experience in the application, development, and sales/field support of electrical/electronic controls for industrial, aerospace, and automotive applications. I’m currently doing consulting
work at Hamilton-Sundstrand, Delta Power Company, and MTE Hydraulics in Rockford. I also have college teaching and industrial training experience. I have taught several courses at Rock Valley College
in Electronic Technology, mathematics, and in the Continuing Education area. I’ve done industrial technical training for Sundstrand, Barber Colman, and others. I’ve also taught math courses at
Rasmussen College and Ellis College (online course). I’ve also been certified as an adjunct instructor for Embry-Riddle Aeronautical University for math and physics courses. I've tutored my own sons
in home study programs. I'm currently tutoring a home schooled student in math using Saxon Math. I hope to do more teaching/tutoring in the future as I transition into retirement.
Type Subject Issued-By Level Year
Degree Electrical Engineering Marquette University MS 1971
Degree Electrical Engineering GMI (Kettereing University) BS 1971
A Word From Previous Students and Parents
Dorota M
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I wanted to let you know that Lisa Johnson did an outstanding job with Julian and we see his grades going up. I would strongly recommend her to anyone looking for a tutor in chemistry or math.
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Transverse and Shear Stress in Turbulent Flow
Transverse and Shear Stress in Turbulent Flow
Key Takeaways
• The stress acting in a direction normal to the surface of a material is called normal stress or transverse stress.
• The three normal stresses to be considered in pipes are axial stress, hoop stress, and radial stress.
• In turbulent flow, shear stresses are much greater than in laminar flow due to eddy currents, which increase the momentum flux in all directions.
Understanding the stresses acting on a piping system is critical when designing pipes
Understanding the stresses acting on a piping system is critical when performing stress analysis on pipes. Depending on the application, piping system analysis codes vary; however, the physics of
piping stress analysis remains unchanged regardless of the application. Here, we will discuss piping stress analysis and how transverse and shear stress impact fluid flow in a pipe.
Piping Stress Analysis
Piping stress analysis is performed primarily for safety. Analyzing stresses acting on a piping system ensures the safety of personnel. The next objective of piping stress analysis is to increase the
life of the pipes. Proper design and maintenance are essential for the long shelf life of the piping system. When pipes are subjected to stress, over time the stress causes wear and tear, resulting
in breaks, temporary shutdowns, or even fatal accidents. Considering the value of human life and the capital invested in building a piping system, it is important to conduct piping stress analysis.
Types of Stresses on a Piping System
Stresses have devastating effects on the life of pipes. The stress acting in a direction normal to the surface of the material is called normal stress or transverse stress. The three normal stresses
that need to be considered in pipes are:
Axial stress or longitudinal stress - Normal stress that acts in parallel to the longitudinal axis in the pipe. The common reasons for this stress generation are internal design pressure, an axial
force acting on the pipe, or a bending force.
Circumferential or hoop stress - Normal stress that acts perpendicular to the axial direction or circumferential direction of the pipe. Internal pressure is the primary cause of hoop stress in pipes.
Radial stress - Normal stress acting parallel to the pipe radius, primarily caused by internal pressure.
The piping system is subject to shear stress due to forces acting on its cross-section or torsional moments. The other stress to take into account is thermal stress, otherwise called expansion
stress, which is generated when the free thermal movement of the pipe is restricted.
The flow inside a pipe can be either laminar or turbulent. Depending on the flow type, the forces and stresses acting on the fluid differ. The upcoming section describes the stresses in laminar and
turbulent flows.
Transverse and Shear Stress in Turbulent Flows
Most of the flow in pipes is turbulent. The stress developed in a turbulent flow is different from laminar flow. In laminar flow, shear stress is generated due to molecular interchanges between
adjacent fluid layers and cohesive forces between liquid molecules. In turbulent flow, the shear stress is much greater than in laminar flow due to the eddy currents, which increase the momentum flux
in all directions. The turbulent flow total shear stress can be given by:
T is the total shear stress in turbulent flow, T[lam] is the shear stress contributed by laminar flow, and T' is the turbulence shear stress generated due to velocity fluctuations and eddy motion in
turbulent flow. The major part of the total stress in turbulent flow is formed by T'.
In a 3D fluid flow, different stresses are generated in the fluid depending on the momentum and momentum transfer. The stresses that act in the x-direction of the momentum in x-direction can be given
by T[xx]. Similarly, the y-direction of the momentum in y-direction and the z-direction of the momentum in z-direction are given by T[yy] and T[zz]. The stresses T[xx],T[yy], and T[zz] are called
normal stresses or transverse stresses. The other stress acting on the fluid is called shear stress, namely T[xy], T[xz], T[yx], T[yz], T[zx], and T[zy], where the first subscript denotes the
direction of the momentum and second subscript represents the direction in which it is transferred.
It is important to understand the transverse and shear stress acting on a fluid flow in order to design an appropriate piping system. The pipe material thickness and cross-section are determined by
evaluating the stresses acting on the pipe in pipe stress analysis.
Cadence’s suite of CFD tools can help engineers describe and evaluate the stresses acting on a piping system. Subscribe to our newsletter for the latest CFD updates or browse Cadence’s suite of CFD
software, including Fidelity and Fidelity Pointwise, to learn more about how Cadence has the solution for you. | {"url":"https://resources.system-analysis.cadence.com/blog/msa2022-transverse-and-shear-stress-in-turbulent-flow","timestamp":"2024-11-07T09:33:39Z","content_type":"text/html","content_length":"204800","record_id":"<urn:uuid:c51312aa-df36-4b08-b00e-aa15346ac404>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00652.warc.gz"} |
How do you solve Kohn-Sham equations?
How do you solve Kohn-Sham equations?
Solution to the Kohn-Sham equations
1. Choose and appropriate atomic basis.
2. We write the variational ansatz as: (167)
3. We compute the density as:
4. We replace the density in the Kohn-Sham equations to find the new eigenfunctions and eigenvalues.
5. Go to 3 to compute the new density and iterate until convergence is achieved.
What is density functional theory in physics?
Density functional theory (DFT) is a quantum-mechanical (QM) method used in chemistry and physics to calculate the electronic structure of atoms, molecules and solids. It has been very popular in
computational solid-state physics since the 1970s.
What is a functional in computational chemistry?
A functional is a function of a function. In DFT the functional is the electron density which is a function of space and time. The electron density is used in DFT as the fundamental property unlike
Hartree-Fock theory which deals directly with the many-body wavefunction.
Why do we need DFT?
The discrete Fourier transform (DFT) is one of the most important tools in digital signal processing. For example, human speech and hearing use signals with this type of encoding. Second, the DFT can
find a system’s frequency response from the system’s impulse response, and vice versa.
What is DFT in quantum mechanics?
Density-functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure)
(principally the ground state) of many-body systems, in particular atoms, molecules, and the condensed phases.
What is DFT theory in chemistry?
Why are DFT calculations so much better than traditional methods?
One main reason for this is the balance of accuracy/computational cost considerations. The computational cost of a DFT calculation with a reasonably large basis set, compared with an equivalent
calculation with MP2, is significantly more tractable.
What is DFT explain briefly?
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time
Fourier transform (DTFT), which is a complex-valued function of frequency.
How can perturbation theory be applied to the field of DFT?
We now consider the application of perturbation theory to DFT, and use this formalism to derive equations allowing the calculation of phonon and electric field responses within crystalline materials.
Density functional theory ( DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure)
(principally the ground state) of many-body systems, in particular atoms, molecules, and the condensed phases.
What is density functional theory (DFT)?
Hence the name density functional theory comes from the use of functionals of the electron density. DFT is among the most popular and versatile methods available in condensed-matter physics,
computational physics, and computational chemistry.
What is perturbation theory in quantum mechanics?
4) The methods of perturbation theory have special importance in the field of quantum mechanics in which, just like in classical mechanics, exact solutions are obtained for the case of the two-body
problem only (which can be reduced to the one-body problem in an external potential field). | {"url":"https://www.studiodessuantbone.com/tips-and-tricks/how-do-you-solve-kohn-sham-equations/","timestamp":"2024-11-08T15:48:06Z","content_type":"text/html","content_length":"128571","record_id":"<urn:uuid:91dbd829-ae4c-42d7-b427-76f7dab07ee4>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00198.warc.gz"} |
ICOMPQ is INTEGER
= 0: Compute eigenvalues only.
= 1: Compute eigenvectors of original dense symmetric matrix
[in] ICOMPQ also. On entry, Q contains the orthogonal matrix used
to reduce the original matrix to tridiagonal form.
= 2: Compute eigenvalues and eigenvectors of tridiagonal
QSIZ is INTEGER
[in] QSIZ The dimension of the orthogonal matrix used to reduce
the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.
N is INTEGER
[in] N The dimension of the symmetric tridiagonal matrix. N >= 0.
D is DOUBLE PRECISION array, dimension (N)
[in,out] D On entry, the main diagonal of the tridiagonal matrix.
On exit, its eigenvalues.
E is DOUBLE PRECISION array, dimension (N-1)
[in] E The off-diagonal elements of the tridiagonal matrix.
On exit, E has been destroyed.
Q is DOUBLE PRECISION array, dimension (LDQ, N)
On entry, Q must contain an N-by-N orthogonal matrix.
If ICOMPQ = 0 Q is not referenced.
If ICOMPQ = 1 On entry, Q is a subset of the columns of the
orthogonal matrix used to reduce the full
[in,out] Q matrix to tridiagonal form corresponding to
the subset of the full matrix which is being
decomposed at this time.
If ICOMPQ = 2 On entry, Q will be the identity matrix.
On exit, Q contains the eigenvectors of the
tridiagonal matrix.
LDQ is INTEGER
[in] LDQ The leading dimension of the array Q. If eigenvectors are
desired, then LDQ >= max(1,N). In any case, LDQ >= 1.
QSTORE is DOUBLE PRECISION array, dimension (LDQS, N)
Referenced only when ICOMPQ = 1. Used to store parts of
[out] QSTORE the eigenvector matrix when the updating matrix multiplies
take place.
LDQS is INTEGER
[in] LDQS The leading dimension of the array QSTORE. If ICOMPQ = 1,
then LDQS >= max(1,N). In any case, LDQS >= 1.
WORK is DOUBLE PRECISION array,
If ICOMPQ = 0 or 1, the dimension of WORK must be at least
1 + 3*N + 2*N*lg N + 3*N**2
[out] WORK ( lg( N ) = smallest integer k
such that 2^k >= N )
If ICOMPQ = 2, the dimension of WORK must be at least
4*N + N**2.
IWORK is INTEGER array,
If ICOMPQ = 0 or 1, the dimension of IWORK must be at least
6 + 6*N + 5*N*lg N.
[out] IWORK ( lg( N ) = smallest integer k
such that 2^k >= N )
If ICOMPQ = 2, the dimension of IWORK must be at least
3 + 5*N.
INFO is INTEGER
= 0: successful exit.
[out] INFO < 0: if INFO = -i, the i-th argument had an illegal value.
> 0: The algorithm failed to compute an eigenvalue while
working on the submatrix lying in rows and columns
INFO/(N+1) through mod(INFO,N+1). | {"url":"https://netlib.org/lapack/explore-html-3.4.2/da/de3/dlaed0_8f.html","timestamp":"2024-11-08T14:40:52Z","content_type":"application/xhtml+xml","content_length":"16169","record_id":"<urn:uuid:43e3d29e-913f-46a1-8d9b-29e0b5dd4f1f>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00705.warc.gz"} |
Let's use the same visual representations we used for comparing fractions and percents to explore the relationship between decimals and percents.
Double number lines can be helpful to model equivalencies between percents and decimals. Bechmark percents and decimals can make problem solving more efficient.
We can convert between decimals and percentages by taking advantage of the hundredths place value. We know that 1\% represents \dfrac{1}{100}, or 1 hundredth, which we can write in decimal form as
We can convert any percentage into a decimal by dividing the percentage value by 100, which is equivalent to decreasing the place value of each digit by two places, and removing the \% symbol.
For example, 83\% = \dfrac{83}{100} which can be described as 83 hundredths. This is also 0.83 when written as a decimal.
To convert from a decimal into a percentage, we can just reverse the above steps. We can convert any decimal into a percentage by multiplying the decimal by 100, which is equivalent to increasing the
place value of each digit by two places, and attaching a \% symbol.
For example, 0.08 is 8 hundredths or \dfrac{8}{100}=8\%.
A percentage is limited to representing hundredths, so smaller units like thousandths cannot be represented by whole number percentages such as 0.0035 which is 0.35\%.
Remember to attach the \% symbol to decimal at the same time as increasing the place values. | {"url":"https://mathspace.co/textbooks/syllabuses/Syllabus-1191/topics/Topic-22486/subtopics/Subtopic-287486/?ref=blog.mathspace.co","timestamp":"2024-11-04T08:24:13Z","content_type":"text/html","content_length":"282100","record_id":"<urn:uuid:a36919d1-5136-40d5-96af-b60cefaa7be5>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00746.warc.gz"} |
The Santa Puzzle: Find the value of Gifts – Math Puzzle with Solution
Home Puzzles Brain Teasers The Santa Puzzle: Find the value of Gifts – Math Puzzle with...
The Santa Puzzle: Find the value of Gifts – Math Puzzle with Solution
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Best confusing brainteasers math puzzle images. Solve this Santa math puzzle image. Viral puzzle image with the correct answer.
Yup! the puzzle guy is back. Here is another confusing Santa puzzle for this Christmas. Find the true value of gifts this puzzle asking for. Let see how many of you will win the gift.
Share your answer in the connect section below.
Santa Math Puzzle Image:
[toggle title=”Answer”]
Correct Answer: 72
Hint: Look closely at the details
⇒ 5 + 5 + 5 = 15
⇒ 6 + 6 + 5 = 17
⇒ 3 + 6 + 5 = 14
⇒ 6 + (3+3) × 5 = 36
Asking for Two Gifts = 36 × 2 = 72
Want to print or share on your blog?
Simply download this puzzle image and print it out. And for blog and website, You are free to use this image but you must attribute the image by linking to the original source, so we will be able to
continue creating new puzzles & other amazing posts for you. <3
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1. 72
2. Answer is 72
3. The correct answer is 82
4. WHAT IS THE ANSWER
□ 72 Answer
5. 72 is right and
6. Changing the number of segments on an image doesn’t suddenly nullify the laws of algebra. | {"url":"https://picshood.com/the-santa-puzzle-find-the-value-of-gifts-math-puzzle-with-solution/","timestamp":"2024-11-11T13:31:12Z","content_type":"text/html","content_length":"145818","record_id":"<urn:uuid:906b32c7-0244-4579-a574-18fd54342e43>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00881.warc.gz"} |
Making an NPC look at your character
I find it depressing that I don’t know how to properly manipulate Motor6Ds at this point. I’m failing to make a darn NPC look at my character. Here’s my code and a screenshot of how it looks atm.
motor.C1 = CFrame.new(Vector3.new(0,-script.Parent.Size.Y/2,0),(script.Parent.Position - workspace.Kiansjet.Head.Position).Unit)
(The script is under the dummy’s head part (R15) and motor refers to the neck Motor6D)
Hey! Recently made a post of this on my twitter. Give me a bit so I can get on my pc and go over the code.
Manipulating the motors
When moving the motor6D, you should use Transform. The motor6Ds of characters are relative, meaning a blank CFrame would be pointing forwards in the same orientation as the humanoid root part.
As motors are relative, to make the motor point at something accurately we need to make the motor have a world rotation of (0, 0, 0). We can do this by:
-- Where char is the model of the NPC
local RotationOffset = (Char.PrimaryPart.CFrame-Char.PrimaryPart.CFrame.p):inverse()
We only need to calculate RotationOffset once. We can apply this to the character’s waist Motor6D, which is located in the UpperTorso - when you do that, you should end up with this:
-- Where char is the model of the NPC
local RotationOffset = (Char.PrimaryPart.CFrame-Char.PrimaryPart.CFrame.p):inverse()
-- Where waist is the motor located in the NPC's UpperTorso
Waist.Transform = RotationOffset
Pointing the waist motor
The transform of the motor accepts a CFrame, meaning you can make a CFrame and point it at the desired position:
-- Where pos is where we want to point, in this case our player's HumanoidRootPart's position
CFrame.new(Vector3.new(0, 0, 0), Pos)
To make this point properly and not at a weird angle, we need to apply the RotationOffset we made earlier.
Waist.Transform = RotationOffset * CFrame.new(Vector3.new(0, 0, 0), Pos)
The problem
This is functioning as intended. But there’s a problem.
The problem
The first character is at the origin of the world. Their rotation works perfectly, but other NPCs away from the origin of the world are messed up entirely.
I don’t know if this is the way you’re supposed to fix it, but I did it by doing this:
local RealPos = -(NPC.PrimaryPart.CFrame.p - Pos).unit * 5000
Applied to the rest of our code, it looks like this:
-- Where char is the model of the NPC
local RotationOffset = (Char.PrimaryPart.CFrame-Char.PrimaryPart.CFrame.p):inverse()
-- Where pos is where we want to point, in this case our player's HumanoidRootPart's position
local RealPos = -(NPC.PrimaryPart.CFrame.p - Pos).unit * 5000
-- Where waist is the motor located in the NPC's UpperTorso
Waist.Transform = RotationOffset * CFrame.new(Vector3.new(0, 0, 0), RealPos) -- Pos changed to RealPos
The result
Limiting the angles
This should work perfectly. But the NPC will point at us from any angle, and that’s quite freaky. The way I fixed it was by getting the euler angles of the CFrame, clamping it, and then making a new
CFrame. This is probably not the best way of doing it, but it works.
function limitAngles(CF, X, Y)
-- X: UP
-- Y: SIDE-SIDE
-- Z: PIVOT (DON'T USE)
X = math.rad(X); Y = math.rad(Y)
local x, y = CF:toEulerAnglesXYZ()
return CFrame.new(CF.p) * CFrame.Angles(
math.clamp(x, 0, X),
math.clamp(y, -Y, Y),
This function takes the CFrame we’ve been constructing so far, and limits it in either direction on the axes we give in X and Y. X and Y are in degrees, for simplicity’s sake. The Z axes is defaulted
to 0 degrees, which otherwise can cause problems.
I’d recommend also passing X as 0 because that can sometimes look a bit weird, but it’s your choice.
function limitAngles(CF, X, Y)
-- X: UP
-- Y: SIDE-SIDE
-- Z: PIVOT (DON'T USE)
X = math.rad(X); Y = math.rad(Y)
local x, y = CF:toEulerAnglesXYZ()
return CFrame.new(CF.p) * CFrame.Angles(
math.clamp(x, 0, X),
math.clamp(y, -Y, Y),
-- Where char is the model of the NPC
local RotationOffset = (Char.PrimaryPart.CFrame-Char.PrimaryPart.CFrame.p):inverse()
-- Where pos is where we want to point, in this case our player's HumanoidRootPart's position
local RealPos = -(NPC.PrimaryPart.CFrame.p - Pos).unit * 5000
local TargetCFrame = RotationOffset * CFrame.new(Vector3.new(0, 0, 0), RealPos)
Waist.Transform = limitAngles(TargetCFrame, 0, 15)
It works
Extra stuff
If you’re adding animations to the NPC, then call this update on RunService.Stepped, as this overrides the animations (RenderStepped doesn’t do this).
I really hoped this helped. This is my first guide, so please point out any errors / mistakes I’ve made.
Thank you!
69 Likes
tutorial section pls
Mighty_Loopy’s amazingly detailed message was in response to a question asked by Kiansjet. It wouldn’t make sense to move it to the tutorial section has it isn’t a tutorial.
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How to Create & Access R Matrix - 5 Operations that you Must Check! - DataFlair
How to Create & Access R Matrix – 5 Operations that you Must Check!
Interactive Online Courses: Elevate Skills & Succeed Enroll Now!
A matrix in R is a two-dimensional rectangular data set and thus it can be created using vector input to the matrix function. R is a tool for expressing statistical and mathematical operations from
which beginners will learn how to create and access the R matrix. And, by the end of this article, you will be able to perform addition, subtraction, multiplication, and division operations on R
Before diving into R matrix, brush up your skills for Vectors in R
What is R Matrix?
In a matrix, numbers are arranged in a fixed number of rows and columns and usually, the numbers are the real numbers. With the help of a matrix function, a memory representation of the matrix can be
easily reproduced.
Therefore, all the data elements must share a common basic type.
mat <- matrix (
c(2 , 4, 3, 1, 5, 7) # the data elements
nrow =2, # no. of rows
ncol =3, # no. of columns
byrow = TRUE)
An element at the mth row and nth column of our matrix ‘mat’ can be created using this expression mat[m, n].
mat[2, 3]
To extract only the mth row of our matrix ‘mat’, we can use the expression, mat[m, ].
mat[2, ]
And, to extract only the nth column of our matrix ‘mat’, we use the expression mat[, n].
mat[ ,3]
History of Matrices in R
We can trace back the origins of matrices to ancient times! However, it was not until 1850, when the concept of the matrix was actually applied.
“Matrix” is the Latin word for womb. Generally, it can also mean any place in which something is formed or produced. The word has been used in unusual ways by at least two authors of historical
importance. They proposed this axiom as a means to reduce any function to one of the lower types so that at the “bottom” (0order) the function is identical to its extension.
By using the process of generalisation, any possible function other than a matrix from the matrix holds true. However, it is only true if the proposition which asserts function in question is
considered. Furthermore, it holds true for all or one of the value of argument when other argument(s) is undetermined.
Wait! Have you checked – R List Tutorial
How to Create Matrix in R?
Using the matrix() function, we will create our first matrix in R. The basic syntax for creating a matrix in R is as follows:
matrix(data, nrow, ncol, byrow, dimnames)
1. Data is the input vector. This can also include a list or an expression.
2. Nrow is the number of rows that we wish to create in our matrix.
3. Ncol is the specification of the number of columns in our matrix.
4. Byrow is a logical attribute which is FALSE by default. Setting it true will arrange the input vectors by row.
5. Dimnames allows you to name rows and columns in a matrix.
Creating R matrix based on the variations in the attributes
• Creating R matrix through the arrangement of elements sequentially by row
arrang_row <- matrix(c(4:15), nrow = 4, byrow = TRUE) #Creating our matrix and arranging it by row
print(arrang_row) #Printing our arranged matrix
In the above code, we specified the range for our array from 4 to 15 in the c() function. We specified the number of rows as 4 and arranged the elements sequentially.
• Creating R matrix by arranging elements sequentially by column
arrang_col <- matrix(c(4:15), nrow = 4, byrow = FALSE) #Creating our matrix and arranging it by column
print(arrang_col) #Printing our arranged matrix
• Defining names of columns and rows in a matrix
In order to define rows and column names, you can create two vectors of different names, one for row and other for a column. Then, using the Dimnames attribute, you can name them appropriately:
rows = c("row1", "row2", "row3", "row4") #Creating our character vector of row names
cols = c("colm1", "colm2", "colm3") #Creating our character vector of column names
mat <- matrix(c(4:15), nrow = 4, byrow = TRUE, dimnames = list(rows, cols) )
#creating our matrix mat and assigning our vectors to dimnames
print(mat) #Printing our matrix
Struggling with Factors in R? Get a complete guide to master it.
How to Access Elements of Matrix in R?
In this section, we will learn how to access elements of a matrix in R. For this, we will use the matrix ‘mat’ that we created before. We can access the elements of this matrix ‘mat’ in the following
The syntax for accessing the element at the nth row of mth column of our matrix mat is – mat[n,m]
For example:
> print(mat[2,3])
Furthermore, to access only the elements of nth row, we use mat[n, ] such that
> print(mat[2, ])
And, to access only the elements of mth column, we use mat[ ,m]:
> print(mat[ , 2])
You must definitely check the Data Structures in R to enhance your skills
How to Modify Matrix in R?
In order to modify our matrix ‘mat’ in R, there are several methods:
The first method is to assign a single element to the position of the R matrix that will modify the original value.
The basic syntax for it – mat[n,m] <- y, where n and m are the rows and columns of the element respectively. And, y is the value that we assign to modify our matrix.
> mat #Displaying values of matrix mat
> mat[2,3] <- 20 #Assigning value 20 to the element at 2nd row and 3rd column
> mat #Displaying our modified matrix.
Here, we modify ‘mat’ by replacing the value at 2nd row and 3rd column, that is, 9 with 20.
• Use of Relation Operators
Another method of modifying is with the use of relational operators like >, <, ==.
> mat[mat == 4] <- 0 #Replacing elements that are equal to 4 with 0
> mat #Displaying our modified matrix ‘mat’
Here, we use == operator to replace the value that is equal to 4 with 0. Similarly, we can use < operator to replace values that are less than 10 with 0:
> mat[mat < 10] <- 0 #Replacing elements that are less with 10 with 0
> mat #Displaying modified matrix ‘mat’
• Addition of Rows and Columns
Another method of modifying an R matrix is through the addition of rows and columns using the rbind() and cbind() function respectively. For this, we create a new matrix ‘new_mat’ with 3 rows and 3
> new_mat = matrix(1:12, nrow = 3, ncol = 3)
> new_mat
Now, we will add a column to our matrix ‘new_mat’ using cbind() function as follows:
> cbind(new_mat, c(1,2,3))
We can also add a row using the rbind() function as follows:
> rbind(new_mat, c(1,2,3))
We can also modify the dimension of the matrix ‘new_mat’ using the dim() function as follows:
dim(new_mat) <- c(1,9)
Here, we modified the original dimension of ‘new_mat’, which was 3 x 3 into 1 x 9.
Since the dimensions of our new_mat matrix have been changed, we will reverse it to 3×3 using:
dim(new_mat) <- c(3,3)
We can also carry out the transpose of the matrix using the t() function:
> t(new_mat)
R Matrix Operations
There are several operations that we can perform on the R matrices to get desired results:
1. Addition (+)
In order to perform addition on matrices in R, we first create two matrices ‘mat1’ and ‘mat2’ with four rows and four columns as follows:
mat1 <- matrix(data = 1:8, nrow = 4, ncol = 4) #Creating our first matrix mat1
mat2 <- matrix(data = 1:16, nrow = 4, ncol = 4) #Creating our second matrix mat2
We will use these two matrices for all of our mathematical operations.
In order to perform addition on A and B, we simply use ‘+’ as follows:
sum <- mat1 + mat2 #Adding our two matrices
print(sum) #Printing the sum
2. Subtraction (-)
In order to perform subtraction, we make use of ‘-’ as follows:
sub <- mat1 - mat2 #Adding our two matrices
print(sub) #Printing the sum
3. Matrix Multiplication (By Constant)
For multiplication with a constant, we simply take our mat1 matrix and multiply it with a constant. In this case, we multiply it by 4:
prod <- mat1*4 #Multiplying matrix mat1 with constant value 4
print(prod) #Printing the product
Uncover the Matrix Functions in R and master the concept
4. Multiplication (*)
For the multiplication of two matrices, we multiply our matrices mat1 and mat2 as follows:
prod <- mat1*mat2 #Multiplying matrix mat1 with mat2
print(prod) #Printing the product
5. Division (/)
To perform division between our matrices, we use ‘/’ as follows:
div <- mat1/mat2 #Division of mat1 and mat2
print(div) #Printing the division
Applications of R Matrices
• In geology, matrices are used for taking surveys and also used for plotting graphs, statistics, and studies in different fields.
• To represent the real world data is like traits of people’s population. They are the best representation method for plotting common survey things.
• In robotics and automation, matrices are the best elements for the robot movements.
• Matrices are used in calculating the gross domestic products in Economics. Therefore, it helps in calculating the efficiency of goods and products.
• In computer-based application, matrices play a vital role in the projection of three-dimensional image into a two-dimensional screen, creating a realistic seeming motion.
• In physical related applications, matrices can be applied in the study of an electrical circuit.
We have studied in detail about R matrices. Moreover, we learned about the uses of matrices and operations which we perform on other matrices functions. So, I hope the above-mentioned information is
sufficient enough to understand matrices and their uses.
It’s time to move further to our next article – R Array Function and Creation of Array
Any doubts related to R Matrix tutorial? Feel free to share in the comment section. See you!!
Did we exceed your expectations?
If Yes, share your valuable feedback on Google
3 Responses
1. I still don’t understand in what cases matrices might be more useful than vectors. It seems like you can multiply, divide, etc. vectors just as easily as matrices and matrices don’t necessarily
put the values in any order that the vector doesn’t? So I don’t understand the purpose of making matrices out of vectors.
□ vector is one dimensional array and matrices are 2,,,,,
2. Sometimes you may have to wait further down the road to see how things that don’t make sense at the beginning might become clearer as you go deeper into the subject matter. I feel that they would
not have created matrices if they had had no purpose. At this point, we probably don’t see them as being very useful, but I would suggest waiting until we learn a little bit more. | {"url":"https://data-flair.training/blogs/r-matrix-operations-applications/","timestamp":"2024-11-05T00:56:06Z","content_type":"text/html","content_length":"276312","record_id":"<urn:uuid:0aa1f0bc-0536-44c8-ac21-4849eff8726d>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00070.warc.gz"} |
(p * t) / (2 * I * v)
30 Aug 2024
(p * t) / (2 * I * v) & Analysis of variables
Equation: (p * t) / (2 * I * v)
Variable: v
Impact of Spacecraft Velocity on Maximum Angular Displacement
X-Axis: 100.0to100.0
Y-Axis: Maximum Angular Displacement
Impact of Spacecraft Velocity on Maximum Angular Displacement
Abstract: The maximum angular displacement experienced by a spacecraft during a maneuvering sequence is influenced by several factors, including its velocity. This article explores the relationship
between spacecraft velocity and maximum angular displacement through a mathematical model.
In the realm of space exploration, spacecraft are required to execute precise maneuvers to achieve their intended objectives. These maneuvers involve complex interactions between various physical
forces, such as gravity, thrust, and inertial effects. One critical parameter that affects the outcome of these maneuvers is the spacecraft’s velocity.
The purpose of this article is to examine the impact of spacecraft velocity on maximum angular displacement using a mathematical model based on the equation: (p * t) / (2 * I * v).
Mathematical Model:
The equation in question, (p * t) / (2 * I * v), represents the relationship between the physical parameters involved. Here’s a breakdown of each variable and its significance:
• p: A proportionality constant that depends on the specific characteristics of the spacecraft.
• t: Time duration over which the maneuver takes place.
• I: Moment of inertia of the spacecraft, which is a measure of its resistance to angular changes.
• v: Spacecraft velocity.
Variable Analysis:
The variable v, representing spacecraft velocity, has a significant impact on maximum angular displacement. By substituting different values for v, we can analyze how changes in velocity affect the
outcome of maneuvers.
• Low Velocity: When the spacecraft operates at low velocities, its moment of inertia dominates, resulting in limited angular displacements.
• Medium Velocity: At medium velocities, the relationship between the variables becomes more complex. The proportionality constant p plays a crucial role in determining the maximum angular
• High Velocity: As the spacecraft reaches high velocities, the impact of velocity on maximum angular displacement becomes more pronounced.
Case Study:
To illustrate the practical implications of this model, let’s consider an example scenario:
Suppose we have a spacecraft with a specific moment of inertia (I) and a proportionality constant (p). We want to analyze how changes in velocity affect the maximum angular displacement during a
10-second maneuver.
Velocity (v) Maximum Angular Displacement
100 m/s 5°
200 m/s 15°
300 m/s 30°
By examining this table, we can see how the increase in velocity results in a proportional increase in maximum angular displacement.
In conclusion, this article has demonstrated that spacecraft velocity plays a critical role in determining maximum angular displacement during maneuvers. By understanding the relationship between
these variables, space engineers can design more efficient and effective mission scenarios. The equation (p * t) / (2 * I * v) provides a valuable tool for analyzing this complex interplay and
optimizing spacecraft performance.
Future Research Directions:
This research has opened up new avenues for investigation:
• Multi-Variable Analysis: Extending the current model to incorporate additional variables, such as gravity and thrust forces.
• Non-Linear Effects: Investigating non-linear relationships between the variables and their impact on maximum angular displacement.
By pursuing these lines of inquiry, we can further enhance our understanding of spacecraft dynamics and develop more sophisticated mathematical models for mission planning.
Related topics
Academic Chapters on the topic
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What are common challenges in performing non-linear dynamic analysis? | SolidWorks Assignment Help
What are find out here challenges in performing non-linear dynamic analysis? N-dimensional dynamic analysis is a standard technique performed in the field of linear imaging to understand the response
of pixels of the electronic signal to the analysis process. Therefore, the most crucial of these operations is to perform the linear extrapolation of some samples within the digital plane and to
obtain the result. One of these methods seeks the results of processing a sensor that has been segmented into small boxes or blocks and then its corresponding value within these various boxes is
outputted to one or more measurement sensors. you could check here on the system specific problem like frequency of observation for performing non-linear analysis, and how the system operates or its
mode condition, the measurement sensors have to set thresholds to lower the minimum real value or in other terms to produce the actual values of signals which relate to the system behavior. Given
these inputs the measurement sensors themselves have to generate the exact answer as a function of input parameters, such as frequency of observation in the case of an over-autoflow signal (IoA).
Note that these input parameters have to have the identical values in each measured box. As a result the individual measurements over the whole system will return the optimal values. Then some
thresholds click over here to be set outside the actual box or multiplexing the box in order to make the measurements of interest return the optimal values as an arbitrary function using the above
mentioned software. The typical test using such a simple measurement procedure presents the measurement data showing the average response of all measurements of the system to small or large values
caused by given measurement parameters, the error is quantified as a % for each measurement, for any one of the measured signals. Note that a small value within each box might not be expected in such
a simple operation. Now, an important class of tests which was called these above testing functions were specific to a given application and thus could fail and find a solution. ## Analysis of
signals with error functions The measurement error function This error function is an alternative type of signal-to-noise reduction (SRC). Generally functions to attenuate noise due to system
actions: a = {0, (A)^{1/(1+\alpha)}} B – = {1, (A)^{\alpha}} C , and where Γ is the ratio in dB. Therefore it is just a simple example. SRCs are typically test functions when used to reduce the total
noise compared to noise reductions and to reduce noise due to noise amplification (MA). A variation can be applied that the function of a signal depends on the measurements of the measurements itself
but it can be considered a new function. The method of measuring the total noise will make this noise reduction and noise reduction (MA) seem so simple that for anWhat are common challenges in
performing non-linear dynamic analysis? Does a non-linear method have any particular properties that justify the use of linear methods? Yes, they do. Assume that, in addition to the linearity
parameters tested, the parameters of a dynamic method are also those of a linear method. The exact definition of the physical parameters of a linear dynamic variable has some obvious technical
consequences. The basic properties that make a non-linear method non-linear are: – The frequency component of the derivative of its distribution is bounded.
Do My College Algebra Homework
That is because: – Distributivity and variance of the derivative of the distribution are bounded. The proof that there are distinct and bounded properties of some non-linear method is given. Here
Then the properties of a dynamic method The P1-subset (or P2-set) of a non-linear method give its number of iterations. The P2-set (or P3-set) of a dynamic method gives its number of submatches.
Given the properties of various non-linear methods, how much does a non-linear dynamic method take to produce a result? In terms of accuracy and effectiveness when performing a non-linear method, we
want its P1-subset (or P2-set) is only used when the input type is the matrix with a certain matrix with zero-correction matrix (see Example 5). For the specific method that we are going to be
concerned with, let us consider another example. Let us estimate the number of submatches for the non-linear method. That is – Say a matrix R is a block matrix with the block-unidimension 2. This is
an upper bound of the number of submatches, when the input type is matrix with non-zero-correction matrix (one not too complicated matrix). This estimate can be approximated from the rank of the
matrix, one of the following four places: – When R is an upper bound of the rank of any other matrix in the rank matrix. – When R is an upper bound of the rank of any other matrix in the rank matrix.
– When R is an upper bound of the rank of any other matrix in the rank matrix. This generalizes the first example of order-linear method on the matrix with a non-zero-correction matrix. Now we
consider a non-linear DNN with parameter function in dimension 2. Now we have to make use of results of the P1-subset (or P2-set) i.e. the P1-subset (or P2-set) (2-set, 0-rerere) with it can be used
to cover the P1-metric when the input has aWhat are common challenges in performing non-linear dynamic analysis? When I was a part of a graduate engineering course I started to think about ways to
learn something in the domain of dynamic analysis. More or less, I have to have a clear knowledge of one of the most important analytical tools in science and technology, some of which we already
know – and it’s nice to have that. The good news though is that these tools make it easier to employ if you find yourself doing so: Doctrine: all kinds of complex problems that don’t exist at all in
physics and engineering Analytic programming: how to quickly and efficiently analyze many complex problems before they occur Combining approaches to coding and modeling the same problem giving an
example of the best possible application of a programming language. Now that I have become able to generalise my thinking towards all kinds Visit This Link software architecture in my PhD thesis I
will focus on two of the most key tools on non-linear dynamic analysis: the Fourier transform and linear regression.
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Fourier Transform Fourier Transform Since the paper started I’ve worked on the Fourier Transform for some time now and now when I began work on my PhD thesis [that is, “the Fourier Transform for
linear regression’] I was doing a lot of calculations – including quadratic and non-linear functions – that were pretty much meaningless” using this technique, I realised that I didn’t have the right
experience: I had lots of open eyes about the methods I was doing and it was easy to just ignore them. So here goes: Find examples of functions which don’t admit the 2-dim polynomial representation
and with few examples we can now apply the FFT and the Lasso to see if all these functions are indeed square roots of the same function (a more interesting question to ask is… which square roots you
find in your example? I mean, sometimes, I do find what I actually knew, sometimes I don’t. But for other questions I mean what happens if you have something of a cubic log-root because when you
check your C++ code you will see a square root at the top right corner). This is, or here’s the nice Wikipedia description, that I have provided since then: This is, or here’s the nice Wikipedia
description, that I have provided since then: Here we just use 3 of them in quadratic functions. Also we could refer to the results via the obvious way to “make quick substitutions” so that these
look like squares and you can just end up with a weird, illogical value. I had a similar question about the Fourier transform, not making corrections too much by using first-order polynomials but
thinking about getting rid of the 1-dimensional representation. Using the method of Lasso the results are | {"url":"https://solidworksaid.com/what-are-common-challenges-in-performing-non-linear-dynamic-analysis-29653","timestamp":"2024-11-05T19:28:10Z","content_type":"text/html","content_length":"157272","record_id":"<urn:uuid:52cb089c-68eb-4efb-a118-dcadbf5e7296>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00544.warc.gz"} |
Digit Sum Patterns and Modular Arithmetic
When I was in elementary school, our class was briefly visited by our school’s headmaster. He was there for a demonstration, probably intended to get us to practice our multiplication tables. “Pick a
number”, he said, “And I’ll teach you how to draw a pattern from it.”
The procedure was rather simple:
1. Pick a number between 2 and 8 (inclusive).
2. Start generating positive multiples of this number. If you picked 8, your multiples would be 8, 16, 24, and so on.
3. If a multiple is more than one digit long, sum its digits. For instance, for 16, write 1+6=7. If the digits add up to a number that’s still more than 1 digit long, add up the digits of that
number (and so on).
4. Start drawing on a grid. For each resulting number, draw that many squares in one direction, and then “turn”. Using 8 as our example, we could draw 8 up, 7 to the right, 6 down, 5 to the left,
and so on.
5. As soon as you come back to where you started (“And that will always happen”, said my headmaster), you’re done. You should have drawn a pretty pattern!
Sticking with our example of 8, the pattern you’d end up with would be something like this:
Pattern generated by the number 8.
Before we go any further, let’s observe that it’s not too hard to write code to do this. For instance, the “add digits” algorithm can be naively written by turning the number into a string (17
becomes "17"), splitting that string into characters ("17" becomes ["1", "7"]), turning each of these character back into numbers (the array becomes [1, 7]) and then computing the sum of the array,
leaving 8.
3 def sum_digits(n)
4 while n > 9
5 n = n.to_s.chars.map(&:to_i).sum
6 end
7 n
8 end
We may now encode the “drawing” logic. At any point, there’s a “direction” we’re going - which I’ll denote by the Ruby symbols :top, :bottom, :left, and :right. Each step, we take the current x,y
coordinates (our position on the grid), and shift them by n in a particular direction dir. We also return the new direction alongside the new coordinates.
10 def step(x, y, n, dir)
11 case dir
12 when :top
13 return [x,y+n,:right]
14 when :right
15 return [x+n,y,:bottom]
16 when :bottom
17 return [x,y-n,:left]
18 when :left
19 return [x-n,y,:top]
20 end
21 end
The top-level algorithm is captured by the following code, which produces a list of coordinates in the order that you’d visit them.
23 def run_number(number)
24 counter = 1
25 x, y, dir = 0, 0, :top
26 line_stack = [[0,0]]
28 loop do
29 x, y, dir = step(x,y, sum_digits(counter*number), dir)
30 line_stack << [x,y]
31 counter += 1
32 break if x == 0 && y == 0
33 end
34 return make_svg(line_stack)
35 end
I will omit the code for generating SVGs from the body of the article – you can always find the complete source code in this blog’s Git repo (or by clicking the link in the code block above). Let’s
run the code on a few other numbers. Here’s one for 4, for instance:
Pattern generated by the number 4.
And one more for 2, which I don’t find as pretty.
Pattern generated by the number 2.
It really does always work out! Young me was amazed, though I would often run out of space on my grid paper to complete the pattern, or miscount the length of my lines partway in. It was only
recently that I started thinking about why it works, and I think I figured it out. Let’s take a look!
Is a number divisible by 3?
You might find the whole “add up the digits of a number” thing familiar, and for good reason: it’s one way to check if a number is divisible by 3. The quick summary of this result is,
If the sum of the digits of a number is divisible by 3, then so is the whole number.
For example, the sum of the digits of 72 is 9, which is divisible by 3; 72 itself is correspondingly also divisible by 3, since 24*3=72. On the other hand, the sum of the digits of 82 is 10, which is
not divisible by 3; 82 isn’t divisible by 3 either (it’s one more than 81, which is divisible by 3).
Why does this work? Let’s talk remainders.
If a number doesn’t cleanly divide another (we’re sticking to integers here), what’s left behind is the remainder. For instance, dividing 7 by 3 leaves us with a remainder 1. On the other hand, if
the remainder is zero, then that means that our dividend is divisible by the divisor (what a mouthful). In mathematics, we typically use $a|b$ to say $a$ divides $b$, or, as we have seen above, that
the remainder of dividing $b$ by $a$ is zero.
Working with remainders actually comes up pretty frequently in discrete math. A well-known example I’m aware of is the RSA algorithm, which works with remainders resulting from dividing by a product
of two large prime numbers. But what’s a good way to write, in numbers and symbols, the claim that “$a$ divides $b$ with remainder $r$”? Well, we know that dividing yields a quotient (possibly zero)
and a remainder (also possibly zero). Let’s call the quotient $q$. [note: It's important to point out that for the equation in question to represent division with quotient $q$ and remainder $r$, it
must be that $r$ is less than $a$. Otherwise, you could write $r = s + a$ for some $s$, and end up with \begin{aligned} & b = qa + r \\ \Rightarrow\ & b = qa + (s + a) \\ \Rightarrow\ & b = (q+1)a +
s \end{aligned} In plain English, if $r$ is bigger than $a$ after you've divided, you haven't taken out "as much $a$ from your dividend as you could", and the actual quotient is larger than $q$. ]
\begin{aligned} & b = qa + r \\ \Rightarrow\ & b-r = qa \\ \end{aligned}
We only really care about the remainder here, not the quotient, since it’s the remainder that determines if something is divisible or not. From the form of the second equation, we can deduce that
$b-r$ is divisible by $a$ (it’s literally equal to $a$ times $q$, so it must be divisible). Thus, we can write:
There’s another notation for this type of statement, though. To say that the difference between two numbers is divisible by a third number, we write:
$b \equiv r\ (\text{mod}\ a)$
Some things that seem like they would work from this “equation-like” notation do, indeed, work. For instance, we can “add two equations” (I’ll omit the proof here; jump down to this section to see
how it works):
$\textbf{if}\ a \equiv b\ (\text{mod}\ k)\ \textbf{and}\ c \equiv d, (\text{mod}\ k),\ \textbf{then}\ a+c \equiv b+d\ (\text{mod}\ k).$
Multiplying both sides by the same number (call it $n$) also works (once again, you can find the proof in this section below).
$\textbf{if}\ a \equiv b\ (\text{mod}\ k),\ \textbf{then}\ na \equiv nb\ (\text{mod}\ k).$
Ok, that’s a lot of notation and other stuff. Let’s talk specifics. Of particular interest is the number 10, since our number system is base ten (the value of a digit is multiplied by 10 for every
place it moves to the left). The remainder of 10 when dividing by 3 is 1. Thus, we have:
$10 \equiv 1\ (\text{mod}\ 3)$
From this, we can deduce that multiplying by 10, when it comes to remainders from dividing by 3, is the same as multiplying by 1. We can clearly see this by multiplying both sides by $n$. In our
$10n \equiv n\ (\text{mod}\ 3)$
But wait, there’s more. Take any power of ten, be it a hundred, a thousand, or a million. Multiplying by that number is also equivalent to multiplying by 1!
$10^kn = 10\times10\times...\times 10n \equiv n\ (\text{mod}\ 3)$
We can put this to good use. Let’s take a large number that’s divisible by 3. This number will be made of multiple digits, like $d_2d_1d_0$. Note that I do not mean multiplication here, but
specifically that each $d_i$ is a number between 0 and 9 in a particular place in the number – it’s a digit. Now, we can write:
\begin{aligned} 0 &\equiv d_2d_1d_0 \\ & = 100d_2 + 10d_1 + d_0 \\ & \equiv d_2 + d_1 + d_0 \end{aligned}
We have just found that $d_2+d_1+d_0 \equiv 0\ (\text{mod}\ 3)$, or that the sum of the digits is also divisible by 3. The logic we use works in the other direction, too: if the sum of the digits is
divisible, then so is the actual number.
There’s only one property of the number 3 we used for this reasoning: that $10 \equiv 1\ (\text{mod}\ 3)$. But it so happens that there’s another number that has this property: 9. This means that to
check if a number is divisible by nine, we can also check if the sum of the digits is divisible by 9. Try it on 18, 27, 81, and 198.
Here’s the main takeaway: summing the digits in the way described by my headmaster is the same as figuring out the remainder of the number from dividing by 9. Well, almost. The difference is the case
of 9 itself: the remainder here is 0, but we actually use 9 to draw our line. We can actually try just using 0. Here’s the updated sum_digits code:
def sum_digits(n)
n % 9
The results are similarly cool:
Pattern generated by the number 8.
Pattern generated by the number 4.
Pattern generated by the number 2.
Sequences of Remainders
So now we know what the digit-summing algorithm is really doing. But that algorithm isn’t all there is to it! We’re repeatedly applying this algorithm over and over to multiples of another number.
How does this work, and why does it always loop around? Why don’t we ever spiral farther and farther from the center?
First, let’s take a closer look at our sequence of multiples. Suppose we’re working with multiples of some number $n$. Let’s write $a_i$ for the $i$th multiple. Then, we end up with:
\begin{aligned} a_1 &= n \\ a_2 &= 2n \\ a_3 &= 3n \\ a_4 &= 4n \\ ... \\ a_i &= in \end{aligned}
This is actually called an arithmetic sequence; for each multiple, the number increases by $n$.
Here’s a first seemingly trivial point: at some time, the remainder of $a_i$ will repeat. There are only so many remainders when dividing by nine: specifically, the only possible remainders are the
numbers 0 through 8. We can invoke the pigeonhole principle and say that after 9 multiples, we will have to have looped. Another way of seeing this is as follows:
\begin{aligned} & 9 \equiv 0\ (\text{mod}\ 9) \\ \Rightarrow\ & 9n \equiv 0\ (\text{mod}\ 9) \\ \Rightarrow\ & 10n \equiv n\ (\text{mod}\ 9) \\ \end{aligned}
The 10th multiple is equivalent to n, and will thus have the same remainder. The looping may happen earlier: the simplest case is if we pick 9 as our $n$, in which case the remainder will always be
Repeating remainders alone do not guarantee that we will return to the center. The repeating sequence 1,2,3,4 will certainly cause a spiral. The reason is that, if we start facing “up”, we will
always move up 1 and down 3 after four steps, leaving us 2 steps below where we started. Next, the cycle will repeat, and since turning four times leaves us facing “up” again, we’ll end up getting
further away. Here’s a picture that captures this behvior:
Spiral generated by the number 1 with divisor 4.
And here’s one more where the cycle repeats after 8 steps instead of 4. You can see that it also leads to a spiral:
Spiral generated by the number 1 with divisor 8.
From this, we can devise a simple condition to prevent spiraling – the length of the sequence before it repeats cannot be a multiple of 4. This way, whenever the cycle restarts, it will do so in a
different direction: backwards, turned once to the left, or turned once to the right. Clearly repeating the sequence backwards is guaranteed to take us back to the start. The same is true for the
left and right-turn sequences, though it’s less obvious. If drawing our sequence once left us turned to the right, drawing our sequence twice will leave us turned more to the right. On a grid, two
right turns are the same as turning around. The third repetition will then undo the effects of the first one (since we’re facing backwards now), and the fourth will undo the effects of the second.
There is an exception to this multiple-of-4 rule: if a sequence makes it back to the origin right before it starts over. In that case, even if it’s facing the very same direction it started with, all
is well – things are just like when it first started, and the cycle repeats. I haven’t found a sequence that does this, so for our purposes, we’ll stick with avoiding multiples of 4.
Okay, so we want to avoid cycles with lengths divisible by four. What does it mean for a cycle to be of length k? It effectively means the following:
\begin{aligned} & a_{k+1} \equiv a_1\ (\text{mod}\ 9) \\ \Rightarrow\ & (k+1)n \equiv n\ (\text{mod}\ 9) \\ \Rightarrow\ & kn \equiv 0\ (\text{mod}\ 9) \\ \end{aligned}
If we could divide both sides by $k$, we could go one more step:
$n \equiv 0\ (\text{mod}\ 9) \\$
That is, $n$ would be divisible by 9! This would contradict our choice of $n$ to be between 2 and 8. What went wrong? Turns out, it’s that last step: we can’t always divide by $k$. Some values of $k$
are special, and it’s only those values that can serve as cycle lengths without causing a contradiction. So, what are they?
They’re values that have a common factor with 9 (an incomplete explanation is in this section below). There are many numbers that have a common factor with 9; 3, 6, 9, 12, and so on. However, those
can’t all serve as cycle lengths: as we said, cycles can’t get longer than 9. This leaves us with 3, 6, and 9 as possible cycle lengths, none of which are divisible by 4. We’ve eliminated the
possibility of spirals!
Generalizing to Arbitrary Divisors
The trick was easily executable on paper because there’s an easy way to compute the remainder of a number when dividing by 9 (adding up the digits). However, we have a computer, and we don’t need to
fall back on such cool-but-complicated techniques. To replicate our original behavior, we can just write:
def sum_digits(n)
x = n % 9
x == 0 ? 9 : x
But now, we can change the 9 to something else. There are some numbers we’d like to avoid - specifically, we want to avoid those numbers that would allow for cycles of length 4 (or of a length
divisible by 4). If we didn’t avoid them, we might run into infinite loops, where our pencil might end up moving further and further from the center.
Actually, let’s revisit that. When we were playing with paths of length $k$ while dividing by 9, we noted that the only possible values of $k$ are those that share a common factor with 9,
specifically 3, 6 and 9. But that’s not quite as strong as it could be: try as you might, but you will not find a cycle of length 6 when dividing by 9. The same is true if we pick 6 instead of 9, and
try to find a cycle of length 4. Even though 4 does have a common factor with 6, and thus is not ruled out as a valid cycle by our previous condition, we don’t find any cycles of length 4.
So what is it that really determines if there can be cycles or not?
Let’s do some more playing around. What are the actual cycle lengths when we divide by 9? For all but two numbers, the cycle lengths are 9. The two special numbers are 6 and 3, and they end up with a
cycle length of 3. From this, we can say that the cycle length seems to depend on whether or not our $n$ has any common factors with the divisor.
Let’s explore this some more with a different divisor, say 12. We fill find that 8 has a cycle length of 3, 7 has a cycle length of 12, 9 has a cycle length of 4. What’s happening here? To see, let’s
divide 12 by these cycle lengths. For 8, we get (12/3) = 4. For 7, this works out to 1. For 9, it works out to 3. These new numbers, 4, 1, and 3, are actually the greatest common factors of 8, 7, and
3 with 12, respectively. The greatest common factor of two numbers is the largest number that divides them both. We thus write down our guess for the length of a cycle:
$k = \frac{d}{\text{gcd}(d,n)}$
Where $d$ is our divisor, which has been 9 until just recently, and $\text{gcd}(d,n)$ is the greatest common factor of $d$ and $n$. This equation is in agreement with our experiment for $d = 9$, too.
Why might this be? Recall that sequences with period $k$ imply the following congruence:
$kn \equiv 0\ (\text{mod}\ d)$
Here I’ve replaced 9 with $d$, since we’re trying to make it work for any divisor, not just 9. Now, suppose the greatest common divisor of $n$ and $d$ is some number $f$. Then, since this number
divides $n$ and $d$, we can write $n=fm$ for some $m$, and $d=fg$ for some $g$. We can rewrite our congruence as follows:
$kfm \equiv 0\ (\text{mod}\ fg)$
We can simplify this a little bit. Recall that what this congruence really means is that the difference of $kfm$ and $0$, which is just $kfm$, is divisible by $fg$:
But if $fg$ divides $kfm$, it must be that $g$ divides $km$! This, in turn, means we can write:
Can we distill this statement even further? It turns out that we can. Remember that we got $g$ and $m$ by dividing $d$ and $n$ by their greatest common factor, $f$. This, in turn, means that $g$ and
$m$ have no more common factors that aren’t equal to 1 (see this section below). From this, in turn, we can deduce that $m$ is not relevant to $g$ dividing $km$, and we get:
That is, we get that $k$ must be divisible by $g$. Recall that we got $g$ by dividing $d$ by $f$, which is our largest common factor – aka $\text{gcd}(d,n)$. We can thus write:
Let’s stop and appreciate this result. We have found a condition that is required for a sequnce of remainders from dividing by $d$ (which was 9 in the original problem) to repeat after $k$ numbers.
Furthermore, all of our steps can be performed in reverse, which means that if a $k$ matches this conditon, we can work backwards and determine that a sequence of numbers has to repeat after $k$
Multiple $k$s will match this condition, and that’s not surprising. If a sequence repeats after 5 steps, it also repeats after 10, 15, and so on. We’re interested in the first time our sequences
repeat after taking any steps, which means we have to pick the smallest possible non-zero value of $k$. The smallest number divisible by $d/\text{gcd}(d,n)$ is $d/\text{gcd}(d,n)$ itself. We thus
confirm our hypothesis:
$k = \frac{d}{\text{gcd}(d,n)}$
Lastly, recall that our patterns would spiral away from the center whenever a $k$ is a multiple of 4. Now that we know what $k$ is, we can restate this as “$d/\text{gcd}(d,n)$ is divisible by 4”. But
if we pick $n=d-1$, the greatest common factor has to be $1$ (see this section below), so we can even further simplify this “$d$ is divisible by 4”. Thus, we can state simply that any divisor
divisible by 4 is off-limits, as it will induce loops. For example, pick $d=4$. Running our algorithm [note: Did you catch that? From our work above, we didn't just find a condition that would
prevent spirals; we also found the precise number that would result in a spiral if this condition were violated! This is because our proof is constructive: instead of just claiming the existence of a
thing, it also shows how to get that thing. Our proof in the earlier section (which claimed that the divisor 9 would never create spirals) went by contradiction, which was not constructive. Repeating
that proof for a general $d$ wouldn't have told us the specific numbers that would spiral.
This is the reason that direct proofs tend to be preferred over proofs by contradiction. ] we indeed find an infinite spiral:
Spiral generated by the number 3 with divisor 4.
Let’s try again. Pick $d=8$; then, for $n=d-1=7$, we also get a spiral:
Spiral generated by the number 7 with divisor 8.
A poem comes to mind:
Turning and turning in the widening gyre
The falcon cannot hear the falconner;
Fortunately, there are plenty of numbers that are not divisible by four, and we can pick any of them! I’ll pick primes for good measure. Here are a few good ones from using 13 (which corresponds to
summing digits of base-14 numbers):
Pattern generated by the number 8 in base 14.
Pattern generated by the number 4 in base 14.
Here’s one from dividing by 17 (base-18 numbers).
Pattern generated by the number 5 in base 18.
Finally, base-30:
Pattern generated by the number 2 in base 30.
Pattern generated by the number 6 in base 30.
Generalizing to Arbitrary Numbers of Directions
What if we didn’t turn 90 degrees each time? What, if, instead, we turned 120 degrees (so that turning 3 times, not 4, would leave you facing the same direction you started)? We can pretty easily do
that, too. Let’s call this number of turns $c$. Up until now, we had $c=4$.
First, let’s update our condition. Before, we had “$d$ cannot be divisible by 4”. Now, we aren’t constraining ourselves to only 4, but rather using a generic variable $c$. We then end up with “$d$
cannot be divisible by $c$”. For instance, suppose we kept our divisor as 9 for the time being, but started turning 3 times instead of 4. This violates our divisibility condtion, and we once again
end up with a spiral:
Pattern generated by the number 8 in base 10 while turning 3 times.
If, on the other hand, we pick $d=8$ and $c=3$, we get patterns for all numbers just like we hoped. Here’s one such pattern:
Pattern generated by the number 7 in base 9 while turning 3 times.
Hold on a moment; it’s actully not so obvious why our condition still works. When we just turned on a grid, things were simple. As long as we didn’t end up facing the same way we started, we will
eventually perform the exact same motions in reverse. The same is not true when turning 120 degrees, like we suggested. Here’s an animated circle all of the turns we would make:
Orientations when turning 120 degrees
We never quite do the exact opposite of any one of our movements. So then, will we come back to the origin anyway? Well, let’s start simple. Suppose we always turn by exactly one 120-degree increment
(we might end up turning more or less, just like we may end up turning left, right, or back in the 90 degree case). Each time you face a particular direciton, after performing a cycle, you will have
moved some distance away from when you started, and turned 120 degrees. If you then repeat the cycle, you will once again move by the same offset as before, but this time the offset will be rotated
120 degrees, and you will have rotated a total of 240 degrees. Finally, performing the cycle a third time, you’ll have moved by the same offset (rotated 240 degrees).
If you overaly each offset such that their starting points overlap, they will look very similar to that circle above. And now, here’s the beauty: you can arrange these rotated offsets into a
Triangle formed by three 120-degree turns.
As long as you rotate by the same amount each time (and you will, since the cycle length determines how many times you turn, and the cycle length never changes), you can do so for any number of
directions. For instance, here’s a similar visualization in which there are 5 possible directions, and where each turn is consequently 72 degrees:
Pentagon formed by five 72-degree turns.
Each of these polygon shapes forms a loop. If you walk along its sides, you will eventually end up exactly where you started. This confirms that if you end up making one turn at the end of each
cycle, you will eventually end up right where you started.
Things aren’t always as simple as making a single turn, though. Let’s go back to the version of the problem in which we have 3 possible directions, and think about what would happen if we turned by
240 degrees at a time: 2 turns instead of 1?
Even though we first turn a whole 240 degrees, the second time we turn we “overshoot” our initial bearing, and end up at 120 degrees compared to it. As soon as we turn 240 more degrees (turning the
third time), we end up back at 0. In short, even though we “visited” each bearing in a different order, we visited them all, and exactly once at that. Here’s a visualization:
Orientations when turning 120 degrees, twice at a time
Note that even though in the above picture it looks like we’re just turning left instead of right, that’s not the case; a single turn of 240 degrees is more than half the circle, so our second
bearing ends up on the left side of the circle even though we turn right.
Just to make sure we really see what’s happening, let’s try this when there are 5 possible directions, and when we still make two turns (now of 72 degrees each)
Orientations when turning 72 degrees, twice at a time
Let’s try put some mathematical backing to this “visited them all” idea, and turning in general. First, observe that as soon as we turn 360 degrees, it’s as good as not turning at all - we end up
facing up again. If we turned 480 degrees (that is, two turns of 240 degrees each), the first 360 can be safely ignored, since it puts us where we started; only the 120 degrees that remain are needed
to figure out our final bearing. In short, the final direction we’re facing is the remainder from dividing by 360. We already know how to formulate this using modular arithmetic: if we turn $t$
degrees $k$ times, and end up at final bearing (remainder) $b$, this is captured by:
$kt \equiv b\ (\text{mod}\ 360)$
Of course, if we end up facing the same way we started, we get the familiar equivalence:
$kt \equiv 0\ (\text{mod}\ 360)$
Even though the variables in this equivalence mean different things now than they did last time we saw it, the mathematical properties remain the same. For instance, we can say that after $360/\text
{gcd}(360, t)$ turns, we’ll end up facing the way that we started.
So far, so good. What I don’t like about this, though, is that we have all of these numbers of degrees all over our equations: 72 degrees, 144 degrees, and so forth. However, something like 73
degrees (if there are five possible directions) is just not a valid bearing, and nor is 71. We have so many possible degrees (360 of them, to be exact), but we’re only using a handful! That’s
wasteful. Instead, observe that for $c$ possible turns, the smallest possible turn angle is $360/c$. Let’s call this angle $\theta$ (theta). Now, notice that we always turn in multiples of $\theta$:
a single turn moves us $\theta$ degrees, two turns move us $2\theta$ degrees, and so on. If we define $r$ to be the number of turns that we find ourselves rotated by after a single cycle, we have $t=
r\theta$, and our turning equation can be written as:
$kr\theta \equiv 0\ (\text{mod}\ c\theta)$
Now, once again, recall that the above equivalence is just notation for the following:
\begin{aligned} & c\theta|kr\theta \\ \Leftrightarrow\ & c|kr \end{aligned}
And finally, observing that $kr=kr-0$, we have:
$kr \equiv 0\ (\text{mod}\ c)$
This equivalence says the same thing as our earlier one; however, instead of being in terms of degrees, it’s in terms of the number of turns $c$ and the turns-per-cycle $r$. Now, recall once again
that the smallest number of steps $k>0$ for which this equivalence holds is $k = c/\text{gcd}(c,r)$.
We’re close now: we have a sequence of $k$ steps that will lead us back to the beginning. What’s left is to show that these $k$ steps are evenly distributed throughout our circle, which is the key
property that makes it possible for us to make a polygon out of them (and thus end up back where we started).
To show this, say that we have a largest common divisor $f=\text{gcd}(c,r)$, and that $c=fe$ and $r=fs$. We can once again “divide through” by $f$, and get:
$ks \equiv 0\ (\text{mod}\ e)$
Now, we know that $\text{gcd}(e,s)=1$ (see this section below), and thus:
$k = e/\text{gcd}(e,s) = e$
That is, our cycle will repeat after $e$ remainders. But wait, we’ve only got $e$ possible remainders: the numbers $0$ through $e-1$! Thus, for a cycle to repeat after $e$ remainders, all possible
remainders must occur. For a concrete example, take $e=5$; our remainders will be the set $\{0,1,2,3,4\}$. Now, let’s “multiply back through” by $f$:
$kfs \equiv 0\ (\text{mod}\ fe)$
We still have $e$ possible remainders, but this time they are multiplied by $f$. For example, taking $e$ to once again be equal to $5$, we have the set of possible remainders $\{0, f, 2f, 3f, 4f\}$.
The important bit is that these remainders are all evenly spaced, and that space between them is $f=\text{gcd}(c,r)$.
Let’s recap: we have confirmed that for $c$ possible turns (4 in our original formulation), and $r$ turns at a time, we will always loop after $k=c/\text{gcd}(c,r)$ steps, evenly spaced out at $\text
{gcd}(c,r)$ turns. No specific properties from $c$ or $r$ are needed for this to work. Finally, recall from the previous section that $r$ is zero (and thus, our pattern breaks down) whenever the
divisor $d$ (9 in our original formulation) is itself divisible by $c$. And so, as long as we pick a system with $c$ possible directions and divisor $d$, we will always loop back and create a pattern
as long as $cmid d$ ($c$ does not divide $d$).
Let’s try it out! There’s a few pictures below. When reading the captions, keep in mind that the base is one more than the divisor (we started with numbers in the usual base 10, but divided by 9).
Pattern generated by the number 1 in base 8 while turning 5 times.
Pattern generated by the number 3 in base 5 while turning 7 times.
Pattern generated by the number 3 in base 12 while turning 6 times.
Pattern generated by the number 2 in base 12 while turning 7 times.
Today we peeked under the hood of a neat mathematical trick that was shown to me by my headmaster over 10 years ago now. Studying what it was that made this trick work led us to play with the
underlying mathematics some more, and extend the trick to more situations (and prettier patterns). I hope you found this as interesting as I did!
By the way, the kind of math that we did in this article is most closely categorized as number theory. Check it out if you’re interested!
Finally, a huge thank you to Arthur for checking my math, helping me with proofs, and proofreading the article.
All that remains are some proofs I omitted from the original article since they were taking up a lot of space (and were interrupting the flow of the explanation). They are listed below.
Referenced Proofs
Adding Two Congruences
Claim: If for some numbers $a$, $b$, $c$, $d$, and $k$, we have $a \equiv b\ (\text{mod}\ k)$ and $c \equiv d\ (\text{mod}\ k)$, then it’s also true that $a+c \equiv b+d\ (\text{mod}\ k)$.
Proof: By definition, we have $k|(a-b)$ and $k|(c-d)$. This, in turn, means that for some $i$ and $j$, $a-b=ik$ and $c-d=jk$. Add both sides to get: \begin{aligned} & (a-b)+(c-d) = ik+jk \\ \
Rightarrow\ & (a+c)-(b+d) = (i+j)k \\ \Rightarrow\ & k\ |\left[(a+c)-(b+d)\right]\\ \Rightarrow\ & a+c \equiv b+d\ (\text{mod}\ k) \\ \end{aligned} $\blacksquare$
Multiplying Both Sides of a Congruence
Claim: If for some numbers $a$, $b$, $n$ and $k$, we have $a \equiv b\ (\text{mod}\ k)$ then we also have that $an \equiv bn\ (\text{mod}\ k)$.
Proof: By definition, we have $k|(a-b)$. Since multiplying $a-b$ but $n$ cannot make it not divisible by $k$, we also have $k|\left[n(a-b)\right]$. Distributing $n$, we have $k|(na-nb)$. By
definition, this means $na\equiv nb\ (\text{mod}\ k)$.
Invertible Numbers $\text{mod}\ d$ Share no Factors with $d$
Claim: A number $k$ is only invertible (can be divided by) in $\text{mod}\ d$ if $k$ and $d$ share no common factors (except 1).
Proof: Write $\text{gcd}(k,d)$ for the greatest common factor divisor of $k$ and $d$. Another important fact (not proven here, but see something like this), is that if $\text{gcd}(k,d) = r$, then the
smallest possible number that can be made by adding and subtracting $k$s and $d$s is $r$. That is, for some $i$ and $j$, the smallest possible positive value of $ik + jd$ is $r$.
Now, note that $d \equiv 0\ (\text{mod}\ d)$. Multiplying both sides by $j$, get $jd\equiv 0\ (\text{mod}\ d)$. This, in turn, means that the smallest possible value of $ik+jd \equiv ik$ is $r$. If
$r$ is bigger than 1 (i.e., if $k$ and $d$ have common factors), then we can’t pick $i$ such that $ik\equiv1$, since we know that $r>1$ is the least possible value we can make. There is therefore no
multiplicative inverse to $k$. Alternatively worded, we cannot divide by $k$.
Numbers Divided by Their $\text{gcd}$ Have No Common Factors
Claim: For any two numbers $a$ and $b$ and their largest common factor $f$, if $a=fc$ and $b=fd$, then $c$ and $d$ have no common factors other than 1 (i.e., $\text{gcd}(c,d)=1$).
Proof: Suppose that $c$ and $d$ do have sommon factor, $eeq1$. In that case, we have $c=ei$ and $d=ej$ for some $i$ and $j$. Then, we have $a=fei$, and $b=fej$. From this, it’s clear that both $a$
and $b$ are divisible by $fe$. Since $e$ is greater than $1$, $fe$ is greater than $f$. But our assumptions state that $f$ is the greatest common divisor of $a$ and $b$! We have arrived at a
Thus, $c$ and $d$ cannot have a common factor other than 1.
Divisors of $n$ and $n-1$.
Claim: For any $n$, $\text{gcd}(n,n-1)=1$. That is, $n$ and $n-1$ share no common divisors.
Proof: Suppose some number $f$ divides both $n$ and $n-1$. In that case, we can write $n=af$, and $(n-1)=bf$ for some $a$ and $b$. Subtracting one equation from the other:
$1 = (a-b)f$ But this means that 1 is divisible by $f$! That’s only possible if $f=1$. Thus, the only number that divides $n$ and $n-1$ is 1; that’s our greatest common factor. | {"url":"https://danilafe.com/blog/modulo_patterns/","timestamp":"2024-11-10T14:56:45Z","content_type":"text/html","content_length":"294031","record_id":"<urn:uuid:a267354e-034e-4d6a-b4d8-b6022c2fa1dc>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00244.warc.gz"} |
Re: random walker segmentation
31 Jul 2013 31 Jul '13
5:31 p.m.
Hi Emmanuelle, I think this would be a good step forward, and would be happy to help! My guess is that this approach could be sped up further in Cython with code we control, making it easier to
maintain and expose similar performance to all users. The memory issue is a real one, particularly for larger (multichannel) datasets. That alone probably justifies the addition, even at the expense
of a small performance hit. But we'll see how small we can make the hit ;) Also, the iterative solution framework might be useful for other algorithms. Josh On Tuesday, July 30, 2013 3:50:16 PM
UTC-5, Emmanuelle Gouillart wrote:
a while ago, I contributed to skimage an implementation of the random walker segmentation algorithm (which has been improved and extended by many others since then). This algorithm computes a
multilabel segmentation using seeds (already labeled pixels), by determining for an unlabeled pixel the probability that a seed diffuses to the pixel (with an anisotropic diffusion coefficient
depending on the gradient between neighboring pixels).
In the current implementation in skimage, the computation of the probability map is done by inverting a large sparse linear system (involving the Laplacian of the graph of pixels). Different
methods can be chosen to solve the linear system: a brute force inversion only works for tiny images; a conjugate gradient method works well but is quite slow. If the package pyamg is available,
a multigrid method is used to compute a preconditioner, which speeds up the algorithm -- but it requires pyamg. Also, the memory cost of the algorithm is important (linear, I think, though. I
haven't yet taken the time to use a memory profiler but I should).
Recently, a colleague brought to my attention that the linear system was just a set of fixed-point equations, that could be solved iteratively. Indeed, the solution verifies that the probability
of a pixel is the weighted sum (with weights on edges that are a decreasing function of gradients) of the probabilities of its neighbors. I have written a quick and dirty implementation (only for
2-D grayscale images and for 2 labels) of this "local" version, available on
It turns out that this implementation is slower than the conjugate gradient with multigrid acceleration (typically 2 to three times slower), but it has several advantages. First, it can be as
fast as the "simple" conjugate gradient (without pyamg's muligrid acceleration), which is the mode that most users will use (we don't expect users to install pymag when they are just trying out
algorithms). Second, its memory cost is lower (for example, the weight of an edge is stored only once, while it appears twice in the Laplacian matrix). Finally, because the operations only
involve neighboring pixels, it is possible that further speed optimization can be done (using cython... or maybe a GPU implementation, even if we're not that far yet with skimage).
So, should we replace the linear algebra implementation with this simpler local and iterative implementation ? I'd be interested in knowing about your opinion.
Cheers, Emmanuelle | {"url":"https://mail.python.org/archives/list/scikit-image@python.org/message/EZF32CKEKHLQ2FAV6RQBABJMML35L76T/","timestamp":"2024-11-10T22:40:02Z","content_type":"text/html","content_length":"16037","record_id":"<urn:uuid:d7305c7c-8d3e-4ac2-85ed-dece705d8700>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00167.warc.gz"} |
The Stacks project
Lemma 63.14.2. Let $S$ be a Noetherian affine scheme of finite dimension. Let $f : X \to S$ be a separated, affine, smooth morphism of relative dimension $1$. Let $\Lambda $ be a Noetherian ring
which is torsion. Let $M$ be a finite $\Lambda $-module. Then $Rf_!\underline{M}$ has constructible cohomology sheaves.
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I Do Not Stand Corrected # x=yt²/b
An occasional series in which the writer of an original piece is given the opportunity to respond to a correction for no other reason than to keep things going in difficult times.
# x=yt²/b Professor Para Bola, MA PhD, Whisky-McNightly Chair of Medical Statistics, University of Afpuddle Department of Applied Mathematics and Mathematical Statistic writes:
Dr Norm Alkurve makes a valid "academic" point when he suggests that my statement "assuming x=ab² + y, where y is a constant ... " [see A Statistician writes #286.4+e, Wednesday 15 April 2020] is not
technically correct. Rather he suggests "for the record", that "y is not a constant but rather a function of x+yg² / mc, where c is the constant". Were things so simple!
What Dr Norm Alkurve does not make clear in his statement is that the statement "y is not a constant but rather a function of x+yg² / mc, where c is the constant" is itself only valid under "certain
abnormal temperatures and conditions" and that at this time of year in all of the relevant parts of Dorset [even allowing for the somewhat unseasonably weather] and certainly in hospital conditions,
y is so close to being a function of x/x and thus a proxy-constant that to all intents and purposes it is a constant. We can thus say with a probability level of 44% [e=2.3434334443] that, in
practice, the assertion that "y is not a constant but rather a function of x+yg² / mc, where c is the constant" is an unhelpful modifier and not a fundamental restatement or refutation of the
original proposal that "assuming x=ab² + y, where y is a constant ...". After all, where y is a constant, y is a constant: a consistent, logical and, I would suggest, irrefutable statement in and of
Marcus Tullius Cicero - every schoolboy's hero and widely regarded as the founder of modern statistics.
Where Dr Alkurve himself strays into error is in his - surprisingly lax - suggestion that y is invariably a function of x+yg² / mc. I believe it was in 1964 that the famous Culinary Bio-ethicist
Professor Brian Thrupiece established the variability of the function x+yg² / mc, demonstrating to the satisfaction of most open minds [my own included - at a much later date of course] that
plant-based lifeforms exhibited different characteristics and were subject to different statistical representations than animal-based forms. I believe that his discovery was a happy, though unsought,
finding arising from his attempt to model the statistical likelihood of fluff becoming unstable under certain laboratory conditions: work undertaken as part of the then top-secret thrupiecediet
For the record, it is also the case that, in vitro, the description xd/c [where c is a function of by²/g] approximates so closely to the modified function yc that to all intents and purposes we can
say that the popular musical rhythm combo AC/DC need not be challenged - a statement as true today as it was when Marcus Tullius [having endlessly sedet in tablino] originally put II and II together
and made IV. | {"url":"https://www.professorthrupiece.com/post/i-do-not-stand-corrected-x-yt--b","timestamp":"2024-11-02T20:28:01Z","content_type":"text/html","content_length":"1050601","record_id":"<urn:uuid:9d505c1c-69f6-4ead-8d17-6c77c73c6760>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00661.warc.gz"} |
Distributed Load - (Mechanical Engineering Design) - Vocab, Definition, Explanations | Fiveable
Distributed Load
from class:
Mechanical Engineering Design
A distributed load refers to a force or weight that is spread out over a length of a structural element, rather than being concentrated at a single point. This type of load is common in beams,
bridges, and other structures where weight is evenly distributed across an area, impacting the analysis of forces and moments acting on the structure. Understanding distributed loads is crucial for
accurately assessing how structures will behave under various loading conditions.
congrats on reading the definition of Distributed Load. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Distributed loads can be uniform (constant load per unit length) or varying (load changes along the length), significantly affecting structural behavior.
2. In free body diagrams, distributed loads are typically represented as arrows or lines showing the magnitude and direction of the load over the length of the beam or structure.
3. Calculating the total load from a distributed load involves integrating the load function over the relevant length to find the equivalent point load.
4. When analyzing structures under distributed loads, engineers must consider how these loads create shear forces and bending moments that vary along the length of the beam.
5. Distributed loads are critical in determining deflection and stress within structural elements, influencing design decisions for safety and performance.
Review Questions
• How does a distributed load differ from a point load in terms of its effect on structural analysis?
□ A distributed load differs from a point load because it spreads its effects over a length of a structural element rather than concentrating them at one location. This means that while a point
load creates localized stress and response, a distributed load affects the entire span of the element, leading to varying shear forces and bending moments along its length. Understanding this
difference is essential when analyzing structures to ensure accurate predictions of behavior under different loading scenarios.
• Discuss how you would represent a uniform distributed load on a free body diagram and its implications for calculating reactions at supports.
□ In a free body diagram, a uniform distributed load is typically represented as a horizontal line with arrows pointing downward, indicating magnitude per unit length. The total load can be
calculated by multiplying the intensity by the length over which it acts. This representation helps in determining reactions at supports by allowing engineers to sum vertical forces and
moments about any point, leading to accurate calculations of how supports will respond to the applied load.
• Evaluate how varying distributed loads can impact design considerations for beams and other structural elements.
□ Varying distributed loads can significantly influence design considerations as they create different internal shear forces and bending moments compared to uniform loads. Engineers must
analyze these varying loads to ensure that beams are adequately sized and reinforced to handle maximum stress points without failure. Additionally, understanding how these loads affect
deflection is crucial for maintaining serviceability standards in structures, leading to careful material selection and safety factors during design.
"Distributed Load" also found in:
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website. | {"url":"https://library.fiveable.me/key-terms/elements-mechanical-engineering-design/distributed-load","timestamp":"2024-11-09T04:27:03Z","content_type":"text/html","content_length":"156342","record_id":"<urn:uuid:0f38bdc7-f448-48d8-a1e8-2d52b87fee94>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00217.warc.gz"} |
4 Limits Of Logic
Logic is a formal discipline of creating and validating reason. The value of logic lies in its correctness. That is to say that formal logic can be shown to be correct, given basic assumptions about
existence and knowledge. As such, it is widely used to prove theories, solve problems and make decisions. In some cases it's used to fully automate decisions. If logic's strength is its correctness,
its limitations are related to its range. In many cases, systems of logic don't handle types of thought that humans can process with ease. The following are a few examples:
1. Partial Truths
Many forms of logic only handle true or false. Where rational thought can easily see a glass as approximately half full. Logic tends to give you, false that the glass is full and false that it is
empty. It should be noted that some forms of logic, including fuzzy logic, can handle partial truths.
2. Language
Each form of logic represents observations in a formal language of logic. These languages impose limitations that don't exist in natural language. In other words, logic languages can't represent or
consider the subtleties of a natural language such as French.
3. Uncertainty
Some forms of logic fail to handle uncertainty, although this is studied by a field known as probabilistic logic. Any form of logic that can't handle uncertainty has difficulty with real world
decision making because uncertainty is common.
4. Human Perception
Using logic to create something that people are passionate about such as architecture tends to have low value. It is notoriously difficult to codify perceptions such as aesthetics, emotion or
cultural concepts. For example, if you were to write a movie review with formal logic, humans would typically view the results as lacking insight. Artificial intelligence could review a film by
looking at people's opinions about similar films and assimilating them. Such approaches are a facade of judgment that can generally be viewed as low value. | {"url":"https://simplicable.com/en/limits-of-logic","timestamp":"2024-11-10T05:04:23Z","content_type":"text/html","content_length":"85878","record_id":"<urn:uuid:ce3d31bb-9c3e-4bfc-bfb4-7fa0556c7079>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00044.warc.gz"} |
Question on using Application System object for mo
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I am trying to understand best way to model a typical scenario for our solution architects in our enterprise.
Imagine a solution architect representing the Personal Banking segment (yellow color) below. They have a new project and need to setup 4 new data flows for a busines app Application ABC.
This is a HOPEX scetch showing the overview of the project's target state application arhictecture.
NOTE: all objects show are the Application object in HOPEX.
Which HOPEX diagram is most appropriate to show the project overviw diagram?
Hence, my issue. I know I can create such landscape of applicaitons viewpoint by using HOPEX Applicaiton System as a starting object. But, creating an applicaiton system for this group of
applications doesn't make sense). I suppose, I could create an application system for sole purpose of this project and which would not be reused anywhere else. But that approach seems wrong.
Appreciate any advice on how to think about this in terms of HOPEX
The best/recommended solution to manage application version depends on the added value/cost ratio. You may define a method for strategic/complex applications and a more simple approach for minor
The complete/consistent way to do this is to manage every piece of software as a building block (BB) that can be used to create another building block of higher level.For example, an application is
composed of internal modules (IT Services) or external/shared services (microservices).
Each BB version is a BB itself, allowing to manage catalog of available components to build more complex artefact (designing a new version of BB doesn't mean it replaces older one eveywhere,
everytime etc. ).For example, an IT service can evolve to be integrated in a new application although current version still exists in some other remaining applications.
This way of managing version/evolution of BB apply to each level. You design an application system (As-Is or To Be) by selecting version of applications or microservices to assemble and data flows/
API call that are needed/planned to make them cooperate (in this assembly) and work properly. Same thing at application level where you select versions of internal module or microservice to design a
new version.
The simplest way to manage As-Is/To Be (for simple or non strategic applications for example) is to maintain architecture description of As-Is only. You can prepare description switch (when new
version go live) by identifying and defining new version as new BB and replace old BB by new one wherever it is required.Note : this process is manual today but will be assisted with automation in
future Hopex version.
Can you expand on this:Keep in mind that using environment to model As Is/To Be situations of an application is only a shortcut. It is relevant only if you don't want to handle version of application
in your repository because you want to handle only As Is view of your architecture ("what is running") but wants to prepare evolutions in a light way.
What is the proper/recommended way to describe application version or (future state) in HOPEX?
You're right : if you use "Application Environment Graph", you won't be able to distinguish between environment contents.
But you can use "Graph of Interactions of an agent", which allows to filter report content by selecting environments (or upper application systems if you use them).
Keep in mind that using environment to model As Is/To Be situations of an application is only a shortcut. It is relevant only if you don't want to handle version of application in your repository
because you want to handle only As Is view of your architecture ("what is running") but wants to prepare evolutions in a light way.
Thanks again for this reply. It is quite helpful to see someone else's perspective.
I have a question on your recommendation of using a new environment ("future one") to design future state.But - if you create new environment - and add future-state connections to the application,
wouldn't that mess up your reporting ?
Hopex would create a graph which includes all known connections to that objects - including the connections from the "future-state environment",
Which is not what we want for this report; it should only show existing connection in current state.
To avoid this, we are learning to model future-state with "duplicate objects" which represent future state of an object we're interested in.
Versioning would probably work for this purpose also - but for now, we're focused on showing future states with duplicate objects.
This question of "how to model project overviews (i.e. application landscapes)" is becoming more and more relevant for us as we plan to onboard our solution architects.
Given the lack of a generic "application landscape" diagram type out-of-box in HOPEX, we are going with the approach to re-use the Application System object for projects.
To distinguisih between the real Application Systems and project related systems - we might create a library called "Project Landscapes" and put these typs of objects in that library.
@TDucher did mention using the "overview" type diagrams for the EA project object. Initially, this seemed like a great solution.After reviewing the documentation for the HOPEX V5, I was only able to
find this page - which looked promising:https://doc.mega.com/hopex-v5-en/#page/ARC/Premiers_pas.Creating_an__22Overview_of_Applications_22_D...
However, this seems outdated informaiton - as I can not create such diagrams in our SaaS deployment of V5. Maybe it is a metamodel configuration thing - but that is not clear from documentation.
So, we will proceed with re-using the "Applicaiton System" object for purpose of creating the architecture project related diagrams.
Here are some tips and principles to help you :
We are using the EA project Object. We are in version V2 of HOPEX. Will migrate to the V5 this year.
Not sure if it is a full standard & recommended capability but maybe it could respond to your needs ???
@engeh Yes. a generic "Applicaiton Landscape" diagram would work. But, hopex doesn't have that type of diagram out-of-box. That is the issue we're facing.
how about application landscape diagram? There you don't have to start from a "starting object".
@TDucher Thanks for that idea. That is very interesting. Is that a regular "Project" object you used - or the "EA Project" object? Seems like an opportonity for MEGA to give us a more flexible
diagram choice. Similar to "Layered viewpoint" when modelling with Archimate. Just to give us a full flexibility to create the view we need for a project.In fact - I am debating if using Archimate is
part of the answer for diagraming project architectures. Drawing connections between applications in Archimate, lets me create the right "interactions" or "flows" in the EA repository (which is
ultimately what I need ) - and I'm not constrained by needing a "starting object" for the diagram.In any case. thanks for the suggestion - much appreciated.Ben | {"url":"https://community.mega.com/t5/User-Forum/Question-on-using-Application-System-object-for-modelling-project/td-p/28726","timestamp":"2024-11-08T14:26:26Z","content_type":"text/html","content_length":"472100","record_id":"<urn:uuid:6792725b-5acd-43ae-92a8-e226e2e3391d>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00669.warc.gz"} |
Exploring the Number Jungle: A Journey into Diophantine Analysissearch
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Exploring the Number Jungle: A Journey into Diophantine Analysis
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Exploring the Number Jungle: A Journey into Diophantine Analysis
Softcover ISBN: 978-0-8218-2640-9
Product Code: STML/8
List Price: $59.00
MAA Member Price: $53.10
AMS Member Price: $47.20
eBook ISBN: 978-1-4704-2126-7
Product Code: STML/8.E
List Price: $49.00
MAA Member Price: $44.10
AMS Member Price: $39.20
Softcover ISBN: 978-0-8218-2640-9
eBook ISBN: 978-1-4704-2126-7
Product Code: STML/8.B
List Price: $108.00 $83.50
MAA Member Price: $97.20 $75.15
AMS Member Price: $86.40 $66.80
• Student Mathematical Library
Volume: 8; 2000; 151 pp
MSC: Primary 11
Welcome to diophantine analysis—an area of number theory in which we attempt to discover hidden treasures and truths within the jungle of numbers by exploring rational numbers. Diophantine
analysis comprises two different but interconnected domains—diophantine approximation and diophantine equations. This highly readable book brings to life the fundamental ideas and theorems from
diophantine approximation, geometry of numbers, diophantine geometry and \(p\)-adic analysis. Through an engaging style, readers participate in a journey through these areas of number theory.
Each mathematical theme is presented in a self-contained manner and is motivated by very basic notions. The reader becomes an active participant in the explorations, as each module includes a
sequence of numbered questions to be answered and statements to be verified. Many hints and remarks are provided to be freely used and enjoyed. Each module then closes with a Big Picture Question
that invites the reader to step back from all the technical details and take a panoramic view of how the ideas at hand fit into the larger mathematical landscape. This book enlists the reader to
build intuition, develop ideas and prove results in a very user-friendly and enjoyable environment.
Little background is required and a familiarity with number theory is not expected. All that is needed for most of the material is an understanding of calculus and basic linear algebra together
with the desire and ability to prove theorems. The minimal background requirement combined with the author's fresh approach and engaging style make this book enjoyable and accessible to
second-year undergraduates, and even advanced high school students. The author's refreshing new spin on more traditional discovery approaches makes this book appealing to any mathematician and/or
fan of number theory.
Undergraduate and graduate students and mathematicians interested in number theory.
□ Chapters
□ Opening thoughts: Welcome to the jungle
□ Chapter 1. A bit of foreshadowing and some rational rationale
□ Chapter 2. Building the rationals via Farey sequences
□ Chapter 3. Discoveries of Dirichlet and Hurwitz
□ Chapter 4. The theory of continued fractions
□ Chapter 5. Enforcing the law of best approximates
□ Chapter 6. Markoff’s spectrum and numbers
□ Chapter 7. Badly approximable numbers and quadratics
□ Chapter 8. Solving the alleged “Pell” equation
□ Chapter 9. Liouville’s work on numbers algebraic and not
□ Chapter 10. Roth’s stunning result and its consequences
□ Chapter 11. Pythagorean triples through Diophantine geometry
□ Chapter 12. A quick tour through elliptic curves
□ Chapter 13. The geometry of numbers
□ Chapter 14. Simultaneous diophantine approximation
□ Chapter 15. Using geometry to sum some squares
□ Chapter 16. Spinning around irrationally and uniformly
□ Chapter 17. A whole new world of $p$-adic numbers
□ Chapter 18. A glimpse into $p$-adic analysis
□ Chapter 19. A new twist on Newton’s method
□ Chapter 20. The power of acting locally while thinking globally
□ Appendix 1. Selected big picture question commentaries
□ Appendix 2. Hints and remarks
□ Appendix 3. Further reading
□ A wealth of information ... designed as a textbook at the undergraduate level, with lots of exercises. The choice of material is very nice: Diophantine approximation is the unifying theme,
but the tour has side trips to elliptic curves, Riemann surfaces, and \(p\)-adic analysis. The writing style is relaxed and pleasant ... For this trip, the guide has chosen an ascent to an
accessible summit, but with the emphasis always on teaching and motivating important techniques so that the beginner can advance to a higher level.
MAA Monthly
□ The author invites the reader right from the beginning, through his engaging and motivating style, to develop ideas actively and to find proofs for himself ... Remarks at the end of each of
the 20 short sections, into which this readily readable introduction to diophantine analysis is divided, extend the material and stimulate the reader to deeper study and involvement with it.
Zentralblatt MATH
□ This short book presents a nice enjoyable introduction to Diophantine analysis, which invites the motivated reader to rediscover by himself or herself many of the fundamental results of the
subject, with hints given in an appendix for the more difficult results.
Mathematical Reviews
□ Here ... is something truly different ... number theory is such a large subject and so much of it is initially accessible without too many prerequisites, that one would expect to see a
different take on the subject every once in a while. And that, happily, is what we have here both, in content and in style of presentation ... For professors with the requisite background,
this may be just the right book to use in an upper-level undergraduate seminar. Students working through this book will learn some nice material and will probably also emerge from the course
with a much greater confidence in their ability to do mathematics.
MAA Online
• Permission – for use of book, eBook, or Journal content
• Book Details
• Table of Contents
• Additional Material
• Reviews
• Requests
Volume: 8; 2000; 151 pp
MSC: Primary 11
Welcome to diophantine analysis—an area of number theory in which we attempt to discover hidden treasures and truths within the jungle of numbers by exploring rational numbers. Diophantine analysis
comprises two different but interconnected domains—diophantine approximation and diophantine equations. This highly readable book brings to life the fundamental ideas and theorems from diophantine
approximation, geometry of numbers, diophantine geometry and \(p\)-adic analysis. Through an engaging style, readers participate in a journey through these areas of number theory.
Each mathematical theme is presented in a self-contained manner and is motivated by very basic notions. The reader becomes an active participant in the explorations, as each module includes a
sequence of numbered questions to be answered and statements to be verified. Many hints and remarks are provided to be freely used and enjoyed. Each module then closes with a Big Picture Question
that invites the reader to step back from all the technical details and take a panoramic view of how the ideas at hand fit into the larger mathematical landscape. This book enlists the reader to
build intuition, develop ideas and prove results in a very user-friendly and enjoyable environment.
Little background is required and a familiarity with number theory is not expected. All that is needed for most of the material is an understanding of calculus and basic linear algebra together with
the desire and ability to prove theorems. The minimal background requirement combined with the author's fresh approach and engaging style make this book enjoyable and accessible to second-year
undergraduates, and even advanced high school students. The author's refreshing new spin on more traditional discovery approaches makes this book appealing to any mathematician and/or fan of number
Undergraduate and graduate students and mathematicians interested in number theory.
• Chapters
• Opening thoughts: Welcome to the jungle
• Chapter 1. A bit of foreshadowing and some rational rationale
• Chapter 2. Building the rationals via Farey sequences
• Chapter 3. Discoveries of Dirichlet and Hurwitz
• Chapter 4. The theory of continued fractions
• Chapter 5. Enforcing the law of best approximates
• Chapter 6. Markoff’s spectrum and numbers
• Chapter 7. Badly approximable numbers and quadratics
• Chapter 8. Solving the alleged “Pell” equation
• Chapter 9. Liouville’s work on numbers algebraic and not
• Chapter 10. Roth’s stunning result and its consequences
• Chapter 11. Pythagorean triples through Diophantine geometry
• Chapter 12. A quick tour through elliptic curves
• Chapter 13. The geometry of numbers
• Chapter 14. Simultaneous diophantine approximation
• Chapter 15. Using geometry to sum some squares
• Chapter 16. Spinning around irrationally and uniformly
• Chapter 17. A whole new world of $p$-adic numbers
• Chapter 18. A glimpse into $p$-adic analysis
• Chapter 19. A new twist on Newton’s method
• Chapter 20. The power of acting locally while thinking globally
• Appendix 1. Selected big picture question commentaries
• Appendix 2. Hints and remarks
• Appendix 3. Further reading
• A wealth of information ... designed as a textbook at the undergraduate level, with lots of exercises. The choice of material is very nice: Diophantine approximation is the unifying theme, but
the tour has side trips to elliptic curves, Riemann surfaces, and \(p\)-adic analysis. The writing style is relaxed and pleasant ... For this trip, the guide has chosen an ascent to an accessible
summit, but with the emphasis always on teaching and motivating important techniques so that the beginner can advance to a higher level.
MAA Monthly
• The author invites the reader right from the beginning, through his engaging and motivating style, to develop ideas actively and to find proofs for himself ... Remarks at the end of each of the
20 short sections, into which this readily readable introduction to diophantine analysis is divided, extend the material and stimulate the reader to deeper study and involvement with it.
Zentralblatt MATH
• This short book presents a nice enjoyable introduction to Diophantine analysis, which invites the motivated reader to rediscover by himself or herself many of the fundamental results of the
subject, with hints given in an appendix for the more difficult results.
Mathematical Reviews
• Here ... is something truly different ... number theory is such a large subject and so much of it is initially accessible without too many prerequisites, that one would expect to see a different
take on the subject every once in a while. And that, happily, is what we have here both, in content and in style of presentation ... For professors with the requisite background, this may be just
the right book to use in an upper-level undergraduate seminar. Students working through this book will learn some nice material and will probably also emerge from the course with a much greater
confidence in their ability to do mathematics.
MAA Online
Permission – for use of book, eBook, or Journal content
You may be interested in...
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Hint: For solving this question you should know about finding the values of inverse trigonometric values of the given functions. But if it is given to find the value of a trigonometric function, then
we find the trigonometric value of that function and then we add them according to the given sequence.
Complete step by step answer:
According to our question t is asked of us to find the correct option for the value of $f\left( x \right)={{\sec }^{-1}}x+{{\tan }^{-1}}x$. As we know that if any question is asking for the values of
trigonometric functions, then we directly put the value of that function there and get our required answer for that. But if it is given to find the angles of them, then we find the original function
(trigonometric function) for that value and then we get the angle of that by comparing it with each other. And if it is asked directly without any angle, then we always use the general formulas of
trigonometry for solving this. And we will solve that according to the given sequence of calculations.
So, if we look at our question, then,
$f\left( x \right)={{\sec }^{-1}}x+{{\tan }^{-1}}x$
And the value of $\sec x$ and $\operatorname{cosec}x$ does not lie between +1 and -1.
If $\phi \left( x \right)={{\sec }^{-1}}x$, then we know that, $x\in \left( -\infty ,-1 \right]\cup \left[ 1,\infty \right)$.
Also $g\left( x \right)={{\tan }^{-1}}x$, then $x\in R$.
Hence for holding $f\left( x \right)={{\sec }^{-1}}x+{{\tan }^{-1}}x$, we have $x\in \left( -\infty ,-1 \right]\cup \left[ 1,\infty \right)$.
So, the correct answer is “Option C”.
Note: While solving these type problems you have to always mind that if the value of inverse trigonometric function is asked, then find that and if it is asked for the angle, then find it by
comparing, if that is asking for the trigonometric function value to solve any problem, then give that, but if no angles are given, then always use the general trigonometric formulas. | {"url":"https://www.vedantu.com/question-answer/let-fleft-x-rightsec-1x+tan-1x-then-fleft-x-class-12-maths-cbse-60a33ba86f51e47462f16c49","timestamp":"2024-11-03T06:42:02Z","content_type":"text/html","content_length":"180542","record_id":"<urn:uuid:d8c7b841-465e-4e21-88ba-ad2e5f5b43fb>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00402.warc.gz"} |
Area of Rectangles with Mixed Side Lengths (solutions, examples, videos, worksheets, lesson plans)
Videos, examples, solutions, and lessons to help Grade 5 students learn how to find the area of rectangles with mixed-by-mixed and fraction-by-fraction side lengths by tiling, record by drawing, and
relate to fraction multiplication.
New York State Common Core Math Grade 5, Module 5, Lesson 11
Worksheets for Grade 5
Common Core Standards: 5.NF.4b, 5.NF.6
Lesson 11 Problem Set
Draw the rectangle and your tiling.
Write the dimensions and the units you counted in the blanks.
Then, use multiplication to confirm the area. Show your work.
1. Rectangle A:
Rectangle A is
____ units long ____ units wide
Area = ____ units^2
2. A square has a perimeter of 51 inches. What is the area of the square?
(Homework) 2. A square has a perimeter of 25 inches. What is the area of the square?
Lesson 11 Exit Ticket
To find the area, Andrea tiled a rectangle and sketched her answer. Sketch the rectangle, and find the area. Show your multiplication work.
Rectangle is 2 1/2 units × 2 1/2 units
Area = _____
Lesson 11 Homework This video shows how to find the area of rectangles with fractional dimensions using an area model.
1. Kristen tiled the following rectangles using square units. Sketch the rectangles, and find the areas. Then confirm the area by multiplying.
a. Rectangle A:
Rectangle A is _____ units long × _____ units wide
Area = _____ units^2
b. Rectangle B:
Rectangle B is 2 1/2 units long × 3/4 unit wide
Area = _________ units^2
c. Rectangle C:
Rectangle C is 3 1/3 units long × 2 1/2 unit wide
Area = _________ units^2
d. Rectangle D:
Rectangle D is 3 1/2 units long × 2 1/4 unit wide
Area = _________ units^2
2. A square has a perimeter of 25 inches. What is the area of the square?
Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page. | {"url":"https://www.onlinemathlearning.com/area-rectangle-mixed-sides.html","timestamp":"2024-11-05T03:10:09Z","content_type":"text/html","content_length":"37402","record_id":"<urn:uuid:e7d30010-d4cf-40f0-bd1e-b63dda15321c>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00769.warc.gz"} |
Lesson 1
Representing Data Graphically
Let’s represent data with dot plots and bar graphs.
1.1: Curious about Caps
Clare collects bottle caps and keeps them in plastic containers.
Write one statistical question that someone could ask Clare about her collection. Be prepared to explain your reasoning.
1.2: Estimating Caps
1. Write down the statistical question your class is trying to answer.
2. Look at the dot plot that shows the data from your class. Write down one thing you notice and one thing you wonder about the dot plot.
3. Use the dot plot to answer the statistical question. Be prepared to explain your reasoning.
1.3: Been There, Done That!
Priya wants to know if basketball players on a men’s team and a women’s team have had prior experience in international competitions. She gathered data on the number of times the players were on a
team before 2016.
men's team
women's team
1. Did Priya collect categorical or numerical data?
2. Organize the information on the two basketball teams into these tables.
Men’s Basketball Team Players
│number of prior │frequency │
│ competitions │ (number) │
│0 │ │
│1 │ │
│2 │ │
│3 │ │
│4 │ │
Women’s Basketball Team Players
│number of prior │frequency │
│ competitions │ (number) │
│0 │ │
│1 │ │
│2 │ │
│3 │ │
│4 │ │
3. Make a dot plot for each table.
Men’s Basketball Team Players
Women’s Basketball Team Players
4. Study your dot plots. What do they tell you about the competition participation of:
1. the players on the men’s basketball team?
2. the players on the women’s basketball team?
5. Explain why a dot plot is an appropriate representation for Priya’s data.
Combine the data for the players on the men’s and women’s teams and represent it as a single dot plot. What can you say about the repeat participation of the basketball players?
1.4: Favorite Summer Sports
Kiran wants to know which three summer sports are most popular in his class. He surveyed his classmates on their favorite summer sport. Here are their responses.
1. Did Kiran collect categorical or numerical data?
2. Organize the responses in a table to help him find which summer sports are most popular in his class.
3. Represent the information in the table as a bar graph.
1. How can you use the bar graph to find how many classmates Kiran surveyed?
2. Which three summer sports are most popular in Kiran’s class?
3. Use your bar graph to describe at least one observation about Kiran’s classmates’ preferred summer sports.
5. Could a dot plot be used to represent Kiran’s data? Explain your reasoning.
When we analyze data, we are often interested in the distribution, which is information that shows all the data values and how often they occur.
In a previous lesson, we saw data about 10 dogs. We can see the distribution of the dog weights in a table such as this one.
│weight in kilograms │frequency │
│6 │1 │
│7 │3 │
│10 │2 │
│32 │1 │
│35 │2 │
│36 │1 │
The term frequency refers to the number of times a data value occurs. In this case, we see that there are three dogs that weigh 7 kilograms, so “3” is the frequency for the value “7 kilograms.”
Recall that dot plots are often used to to represent numerical data. Like a frequency table, a dot plot also shows the distribution of a data set. This dot plot, which you saw in an earlier lesson,
shows the distribution of dog weights.
A dot plot uses a horizontal number line. We show the frequency of a value by the number of dots drawn above that value. Here, the two dots above the number 35 tell us that there are two dogs
weighing 35 kilograms.
The distribution of categorical data can also be shown in a table. This table shows the distribution of dog breeds.
│ breed │frequency │
│pug │9 │
│beagle │9 │
│German shepherd │12 │
We often represent the distribution of categorical data using a bar graph.
A bar graph also uses a horizontal line. Above it we draw a rectangle (or “bar”) to represent each category in the data set. The height of a bar tells us the frequency of the category. There are 12
German shepherds in the data set, so the bar for this category is 12 units tall. Below the line we write the labels for the categories.
In a dot plot, a data value is placed according to its position on the number line. A weight of 10 kilograms must be shown as a dot above 10 on the number line.
In a bar graph, however, the categories can be listed in any order. The bar that shows the frequency of pugs can be placed anywhere along the horizontal line.
• distribution
The distribution tells how many times each value occurs in a data set. For example, in the data set blue, blue, green, blue, orange, the distribution is 3 blues, 1 green, and 1 orange.
Here is a dot plot that shows the distribution for the data set 6, 10, 7, 35, 7, 36, 32, 10, 7, 35.
• frequency
The frequency of a data value is how many times it occurs in the data set.
For example, there were 20 dogs in a park. The table shows the frequency of each color.
│ color │frequency │
│white │4 │
│brown │7 │
│black │3 │
│multi-color │6 │ | {"url":"https://im.kendallhunt.com/MS_ACC/students/1/8/1/index.html","timestamp":"2024-11-05T21:54:40Z","content_type":"text/html","content_length":"100369","record_id":"<urn:uuid:feb00b1e-041f-4685-88a5-16110b8ef1ca>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00140.warc.gz"} |
concrete column opensees
1. Concrete Cast-in-place Column Cyclic Test
This is a simple example of a single cast-in-place (CIP) concrete column subjected to lateral cyclic loading. This test was performed and published by Dr. M.J. Ameli at the University of Utah. Dr.
Ameli’s Ph.D. dissertation (Ameli 2016) is available for download for your reference. We will model this column with different element types available to understand the effect of modeling parameters.
Before jumping into modeling, it is very important to decide the location of nodes. Since OpenSees does not have any graphical user interface (GUI), it is recommended to do a little brainstorming in
deciding the node locations before starting any modeling. I generally draw the expected OpenSees model on a paper and follow it.
The column’s total height is 8.5 ft but the lateral loading is applied at a height of 8 ft in order to accommodate the actuator. If, in OpenSees model, we apply lateral loading at the total height of
the column, we will not match the drift ratios recorded in the test. Therefore, we will put a node at the loading height.
Disclaimer: The two drawings presented above are owned by Dr. M.J. Ameli.
A vertical axial load was also applied to the column during Dr. Ameli’s test to simulate deck load. This load would generate P-delta effects in the column when pushed to higher lateral drift ratios.
This vertical load was applied at the top of the column and not at the point of lateral loading. However, for simplification, we can apply vertical load at the point of lateral loading.
One of the nodes will be at the column bottom, which is the column-to-footing joint. Since during the test the footing was kept rigidly fixed by anchors, we don’t have to model the footing height.
This simplification is not true for every situation. We have to be wise about the simplifications and assumptions.
It may seem useless to “brainstorm” since we only have two nodes in this model. But this process is useful for larger and more complex problems.
Now we can start our model. The first thing we have to do is to define our model space in OpenSees. For this example, a 2-dimensional model is sufficient. To initiate a 2-dimensional model with
3-degrees-of-freedom, use the following command,
Model BasicBuilder -ndm 2 -ndf 3
Then, we go on to define basic and dependent units. These units will later be used to set specimen dimensions and material properties. Any dimensions or values defined in dependent units will be
converted to the basic units to perform calculations. For example, the column length defined as 10 feet will be converted to 120 inches.
The output of the program will also be reported in basic units.
Basic Units
set kip 1.0; # Kips
set in 1.0; # inch
set sec 1.0; # second
Dependent Units
set ft [expr $in*12]; # relate foot to inch
set lb [expr $kip/1000]; # relate pounds to kip
set ksi [expr $kip/pow($in,2)];
set psi [expr $lb/pow($in,2)];
set ksi_psi [expr $ksi/$psi];
set sq_in [expr $in*$in];
Universal Constants
set g [expr 386.2*$in/pow($sec,2)];
set pi [expr acos(-1.0)];
Once the units are defined, we will proceed to set column/specimen dimensions.
set Lcol [expr 8.00*$ft]; # Length of the column (distance between the base and the loading point)
set Dcol [expr 21.00*$in]; # Column Diameter
set Dbar [expr 1.00*$in]; # Diameter of each longitudinal rebar
set Abar [expr 0.79*$in*$in]; # Area of each longitudinal rebar
set s_bar [expr 2.50*$in]; # spacing between horizontal reinforcement / Spiral reinforcement
set cover [expr 2.00*$in]; # Concrete clear cover
Cross-Section Properties
set Acol [expr (($pi*$Dcol**2)/4)]; # Area of column
set Jcol [expr ($pi*($Dcol/2)**4)/2]; # Torsional Moment of Inertia
set I3col [expr ($pi*($Dcol/2)**4)/4]; # Moment of Inertia about minor axis
set I2col [expr ($pi*($Dcol/2)**4)/4]; # Moment of Inertia about major axis
Next, we will define material properties. Various material models are available in OpenSees for concrete and steel. Selection of material model can lead to convergence issues. For this example, we
will use ‘Concrete02’ material model for concrete and ‘ReinforcingSteel’ material model for rebar. 'Concrete02' material model follows Mender's equation and hence needs f'cc (peak confined
compression strength), ɛcc (strain at peak confined compression strength, f'cu (crushing strength of confined concrete) and ɛcu (crushing strain) to define the material. For 'ReinforcingSteel'
material model, Fy (yield stress), Fu (peak stress) and E (modulus of elasticity). More information on the material property definition can be found on OpenSees command manual webpage.
Since our aim is to match the test performance of the column, we will use 'test day' strength parameters and not '28 day' strength parameters. All these parameters are provided in Dr. Ameli's
dissertation. It is a good practice to define separate material properties for core and cover concrete to capture the true behavior of the section. However, for this example, we will just core
concrete properties for the entire column section.
OpenSees uses numeric material tags for each material definition which is then applied to section definitions.
Now that we have defined the materials, column fiber section can be defined. In order to capture material non-linearity, fiber section will be used. In OpenSees, a numeric tag is associated with each
section defined. This numeric tag is later used to assign the section to an element.
For simplicity, the octagonal column section can be modeled as a circular section with equal cross-sectional area. The diameter of the equivalent circular section is then used as "
" which has been defined along with other dimensions at the beginning of the model script. The column cross-section is divided in to multiple non-linear fibers with appropriate concrete and steel
properties. OpenSees command for
Circular Fiber Patch
is used to generate multiple concrete fibers over the column section.
We have now successfully defined the required material properties and the section property for the column model. Let's start defining nodes and element. As discussed previously, we just need two
nodes for this model. One node at the base and the other node at the loading height of the column. We will assume that the base node is fixed.
Since our model space is 2-dimensional, we need X and Y coordinates of each node. Also, the base node needs to be fixed in all degrees-of-freedom hence we will assign "1" for all three dofs.
Before defining elements, we should define a 'Geometric Transformation'. This establishes a relationship between the global coordinate system and the local coordinate system of the elements. For a
2-dimensional space, this relationship is simple and straight forward. Therefore the only thing in this model we need to choose if we want PDelta effect or not. Since, this is a non-linear analysis
with a significant axial load on the column and large lateral displacements, we should include PDelta effects.
A numeric tag is assigned to each geometric transformation definition which is later used while defining element connectivity. (I generally define a "Transformation Type" separately, as you can see
below, to make it easier for me to understand code for large/complex models. I define different transformation types for beams, columns in different locations. Thus I can just change the "TransfType"
parameter to "Linear" or "PDelta" without getting confused each time.
While defining the elements, we need to assign number of integration points along the length of the element. For a non-linear element, the predefined fiber-section is assigned to each integration
point. A minimum of 2 integration points can be assigned for a linear moment distribution along the element length. However there is no upper limit, it is recommended to assign 5 integration points
to capture non-linear distribution of the moment along the length. For this example, we will start with 2 integration points. This will only calculate the nonlinear behavior at the bottom and the top
of the column element.
* Later we will compare the effect of number of integration points on the output.
Since we are trying to perform a static displacement controlled analysis, there is no need to assign mass to the nodes. Pushover and cyclic analysis can be performed without any mass assignment. This
is possible because OpenSees does not convert mass to weight for gravity loads internally. We can assign loads in vertically downward direction as "weight". However, to calculate mode shapes, modal
periods and dynamic analysis, we need to assign mass.
To follow the test setup, first we will define the axial load applied on the column. This axial load is a representative of the superstructure weight. Hence, we can define a quantity "Weight" and
then convert it into "mass". This mass will be assigned in all the translational degrees-of-freedom. A "-Weight" load will be assigned in 2nd degree-of-freedom which is global Y direction.
All the loads are assigned under a load pattern command. For static load, a linear load pattern is suitable.
The model definition is now complete. We will now analyze the structure for gravity loads. It is important to check if the structure is stable for gravity loads before performing any lateral
Before performing the analysis, we should define appropriate recorders. "file mkdir Gravity" is a Tcl command to create a folder named "Gravity". All the recorders will save the output in this
folder. The first recorder is for lateral reaction at node 1, i.e. base of the column in 2nd degree-of-freedom. Second recorder is to record stress and strain for concrete fiber at column center.
Static analysis is performed as follows. Most of the models will converge for gravity analysis using the following lines of code. For complex models, if convergence issues arise, you can experiment
with a different "system", "test" and "algorithm" types.
Now we can perform static pushover analysis. Pushover analysis is a static displacement controlled analysis. "ControlNode" is assigned a lateral displacement in lateral direction ( 1 dof in this
example) up to a maximum displacement of "Dmax". To avoid convergence issues, the analysis should be performed in small steps "Dincr". The analysis will have "Dmax/Dincr" total number of steps.
The analysis will converge and the output files will be saved in a folder named "Pushover". Base shear vs lateral drift can be plotted along with the test data to see how well our model predicts the
test output. It is obvious that the numerical output does not match the test data very well. You can tweak some material parameters to match the behavior.
Next, you should try to perform a cyclic analysis to see how the cyclic behavior of the numerical model matches with the test.
1) Ameli M. J., 2016. "Seismic Evaluation of Grouted Splice Sleeve Connections for Bridge Piers in Accelerated Bridge Construction". Ph.D. Dissertation, University of Utah.
2) OpenSees (http://opensees.berkeley.edu/) | {"url":"https://www.anuragupadhyay.com/concrete-column.html","timestamp":"2024-11-04T03:50:10Z","content_type":"text/html","content_length":"216372","record_id":"<urn:uuid:03d34b6f-ad04-41c5-8ff3-4621421e7219>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00293.warc.gz"} |
Institute of Mechanics
Kiril Stoyanov Shterev
Associate Prof.
Department: Mathematical Modeling and Numerical Simulations
Room number: 421
Phone: +359 2 979 2007
Click here to visit my ResearchGate profile.
Scientific degrees, institution,
Technical University Sofia, Bulgaria, branch Plovdiv, Faculty for Spesial Preperation, Dipl. Engineer – Bachelor of Aviation Technique and Technologies, 2002
Ph.D., Mathematical modeling and application of mathematics in mechnics, Institute of Mechanics - BAS, 2010
Fields of Research: Finite Volume Method (FVM) - SIMPLE-TS, Computational Fluid Dynamic (CFD), microflows in channels at subsonic and supersonic speeds, C++, parallel programming
based on standard MPI
Fields of future interests: Molecular models, a continuum and molecular dynamics hybrid method for micro- and nano-fluid flow
Extended Information
Recent Publications:
Journal papers
1. Kiril Shterev, Emil Manoach, Simona Doneva. Vortex-Induced Vibrations of an Elastic Micro-Beam with Gas Modeled by DSMC. Sensors, MDPI, 2023, ISSN:1424-8220, DOI: https://doi.org/10.3390/
s23041933, 1-22. IF 3.9. Q1.
2. K. Shterev, The Correctness of the Simplified Bernoulli Trial (SBT) Collision Scheme of Calculations of Two-Dimensional Flows, Micromachines 2021, 12, 127, https://doi.org/10.3390/mi12020127,
doi: 10.3390/mi12020127, IF 3.523, Q2, (click here to download the paper).
3. Kiril Shterev, Emil Manoach. Geometrically Non-Linear Vibration of a Cantilever Interacting with Rarefied Gas Flow, Cybernetics and Information Technologies, 20, 6, 2020, ISSN:1311-9702,
DOI:10.2478/cait-2020-0067, 126-139. SJR (Scopus):0.31, Q2, URL: http://www.cit.iit.bas.bg/CIT-2020/v-20-6/10341-Volume20_Issue_6-13paper.pdf
4. Shterev, K., Manoach, E., Stefanov, S.. Hybrid numerical approach to study the interaction of the rarefied gas flow in a microchannel with a cantilever 2019. International Journal of Non-Linear
Mechanics, 117, 103239, Elsevier, 2019, DOI:10.1016/j.ijnonlinmec.2019.103239, JCR-IF (Web of Science):2.313, Q1, URL: http://www.sciencedirect.com/science/article/pii/S002074621930188X
5. Kiril S. Shterev, Stefan K. Stefanov, A two-dimensional computational study of gas flow regimes past of square cylinder confined in a long microchannel, European Journal of Mechanics - B/Fluids,
Available online 14 March 2017, ISSN 0997-7546, http://dx.doi.org/10.1016/j.euromechflu.2017.03.001.(http://www.sciencedirect.com/science/article/pii/S0997754616301285), IF 1.969 (5-Year Impact
Factor: 2.098).
6. N.K. Kulakarni, K. Shterev, S.K. Stefanov, Effects of finite distance between a pair of opposite transversal dimensions in microchannel configurations: DSMC analysis in transitional regime,
International Journal of Heat and Mass Transfer, Volume 85, June 2015, Pages 568–576, ISSN: 0017-9310, doi:10.1016/j.ijheatmasstransfer.2015.02.011, IF 2.868
7. K. Shterev, Iterative process acceleration of calculation of unsteady, viscous, compressible, and heat-conductive gas flows, International Journal for Numerical Methods in Fluids, Volume 77,
Issue 2, 2015, Pages 108-122, ISSN 1097-0363, doi: 10.1002/fld.3979, URL http://dx.doi.org/10.1002/fld.3979, IF 1.329, SJR 1.146
8. K. Shterev and S. Stefanov, Pressure based finite volume method for calculation of compressible viscous gas flows, Journal of Computational Physics 229 (2010) pp. 461-480, doi:10.1016/
j.jcp.2009.09.042 - the accepted manuscript can be downloaded from here, the paper in it`s final mode is available here. IF 3.023
1. Kiril S. Shterev, GPU implementation of algorithm SIMPLE-TS for calculation of unsteady, viscous, compressible and heat-conductive gas flows, URL: https://arxiv.org/abs/1802.04243, 2018.
2. Kiril S. Shterev, On convective terms approximation approach that corresponds to pure convection, URL: https://arxiv.org/abs/1802.04363, 2018.
Conferences and proceedings
1. Kiril S. Shterev, Boundary Condition at Infinity for DSMC Method, 17th International Symposium on Numerical Analysis of Fluid Flows, Heat and Mass Transfer - Numerical Fluids 2022, 19-25
September 2022, Crete, Greece, AIP Conference Proceedings, DOI: 10.1063/5.0211856, URL: https://doi.org/10.1063/5.0211856, SJR 0.164.
2. Kiril Shterev, Emil Manoach, Stefan Stefanov, GAS FLOW IN A MICRO-CHANNEL WITH AN ELASTIC OBSTACLE, Proceedings of the International Symposium on Thermal Effects in Gas flows In Microscale,
October 24-25, 2019 – Ettlingen, Germany.
3. Kiril Shterev, Emil Manoach, NUMERICAL APPROACH TO STUDY PRESSURE DRIVEN RAREFIED GAS FLOW IN A MICROCHANNEL WITH AN ELASTIC OBSTACLE, International Conference on Nonlinear Solid Mechanics
(ICoNSoM 2019), 16 – 19.06.2019, Rome, Italy.
4. K. Shterev, E. Manoach, S. Stefanov, Couette Gas Microflow with an Elastic Obstacle, Fourth International Conference on Recent Advances in Nonlinear Mechanics (RANM2019), May 7-10 2019, Lodz,
5. K. Shterev, E. Manoach, S. Stefanov, Gas flow in a micro-channel with an elastic obstacle. Proceedings of EUROMECH Colloquium 603 Dynamics of micro and nano electromechanical systems: multi-field
modelling and analysis. Editors Pedro Ribeiro, Stefano Lenci, Sondipon Adhikari, Universidade do Porto. Reitoria, 2018, ISBN:978-989-746-185-9, 137-140
6. K. Shterev and S. Stefanov, Determination of zone of flow instability in a gas flow past a square particle in a narrow microchannel at all speeds, Proceedings of the 2nd European Conference on
Non-equilibrium Gas Flows, December 9-11, 2015, Eindhoven, the Netherlands.
7. K. S. Shterev and S. K. Stefanov, Strouhal number analysis for a Karman vortex gas flow past a square in a microchannel at low Mach number, AIP Conference Proceedings 1629 (2014), pp. 319-326,
URL http://scitation.aip.org/content/aip/proceeding/aipcp/10.1063/1.4902288, DOI http://dx.doi.org/10.1063/1.4902288, ISBN: 978-0-7354-1268-2, International Conference for Promoting the
Application of Mathematics in Technical and Natural Sciences ‐ AMiTaNS ’14, 26 June–1 July 2014, Albena, Bulgaria, SJR 0.162.
8. K. S. Shterev, Acceleration of iterative algorithm for calculation of unsteady, viscous, compressible and heat-conductive gas flows on GPU, International Conference on "Numerical Methods for
Scientific Computations and Advanced Applications", May 19-22, 2014, Bansko, Bulgaria, Institute of Information and Communication Tecnologies - Bulgarian Academy of Sciences, in cooperation with
Society for Industrial and Applied Mathematics (SIAM), ISBN: 978-954-91700-7-8, URL: http://parallel.bas.bg/Conferences/nmscaa/Proccedings_NMSCAA2014.pdf
9. Kiril S. Shterev, Emanouil I. Atanassov and Stefan K. Stefanov, GPU calculations of unsteady viscous compressible and heat conductive gas flow at supersonic speed, 9th International Conference on
"Large-Scale Scientific Computations" June 3-7, 2013, Sozopol, Bulgaria, pp. 549-556, doi=10.1007/978-3-662-43880-0_63, url=http://dx.doi.org/10.1007/978-3-662-43880-0_63, ISBN:
978-3-662-43879-4, 2014, SJR 0.305.
10. Stefan K. Stefanov, Naveen K. Kulakarni and Kiril S. Shterev, Modeling of Gas Flows through Microchannel Configurations, AIP Conf. Proc. 1561, 59 (2013), http://dx.doi.org/10.1063/1.4827214,
APPLICATION OF MATHEMATICS IN TECHNICAL AND NATURAL SCIENCES: 4th International Conference - AMiTaNS '13, 24-29 June 2013, Albena, Bulgaria, SJR 0.162.
11. Kiril Shterev and Stefan Stefanov, Determination of Zone of Flow Instability in a Gas Flow Past a Square Particle in a Narrow Microchannel, High-Performance Computing Infrastructure for South
East Europe's Research Communities vol. 2, Springer International Publishing, ISBN 978-3-319-01519-4, pp. 43-50, doi=10.1007/978-3-319-01520-0_5, url: http://dx.doi.org/10.1007/
978-3-319-01520-0_5, The book series Modeling and Optimization in Science and Technologies (MOST), Volume 2 2014, ISSN: 2196-7326. The original publication is availible at www.springerlink.com.
12. K. S. Shterev and S. Ivanovska, Comparison of some approximation schemes for convective terms for solving gas flow past a square in a microchannel, APPLICATION OF MATHEMATICS IN TECHNICAL AND
NATURAL SCIENCES: 4th International Conference - AMiTaNS '12, 11-16 June 2012, St. Constantine and Helena, Bulgaria, AIP Conf. Proc. 1487, pp. 79-87; doi:http://dx.doi.org/10.1063/1.4758944, ISBN
978-0-7354-1099-2, SJR 0.168. The original publication is availible at here. Authors copy is availible freely from here.
13. K. Shterev, S. Stefanov, Influence of reservoirs on pressure driven gas flow in a micro-channel, Application of Mathematics in Technical and Natural Sciences, Albena, Bulgaria, June 20-25, 2011,
AIP Conf. Proc. 1404, 122 (2011), ISBN: 978-0-7354-0976-7; doi: 10.1063/1.3659911, SJR 0.146.
14. K. Shterev, S. Stefanov and E Atanassov, A parallel algorithm with improved performance of Finite Volume Method (SIMPLE-TS), 8th International Conference on "Large-Scale Scientific Computations"
June 6-10, 2011, Sozopol, Bulgaria, I. Lirkov, S. Margenov, and J. Wansiewski (Eds.): LSSC 2011, LNCS 7116, ISBN 978-3-642-29842-4, pp. 351–358, 2012. Springer-Verlag Berlin Heidelberg 2012, SJR
15. K. Shterev and S. Stefanov, Unsteady State Gaseous Flow past a Square Confined in a Micro-channel, AIP Conf. Proc. 1301, 520 (2010); http://dx.doi.org/10.1063/1.3526654, 2nd International
Conference, Sozopol, Bulgaria, November 21-26, 2010, APPLICATION OF MATHEMATICS IN TECHNICAL AND NATURAL SCIENCES: Proceedings of the 2nd International Conference, 25.11.2010, Volume 1301, pp.
520-530, doi:10.1063/1.3526654, SJR 0.148.
16. K. S. Shterev and S. K. Stefanov, A Parallelization of Finite Volume Method for Calculation of Gas Microflows by Domain Decomposition Methods, 7th Internnational Conference -
Large-ScaleScientific Computations, Sozopol, Bulgaria, June 04-08, 2009. Lecture Notes in Computer Science Volume 5910, 2010, DOI: 10.1007/978-3-642-12535-5, SJR 0.295.
17. K. S. Shterev, S. K. Stefanov, Finite Volume Calculations of Gaseous Slip Flow Past a Confined Square in a Two-Dimensional Microchannel, Proceedings of the 1st European Conference on
Microfluidics - Microfluidics 2008 - Bologna, December 10-12, 2008.
18. K. S. Shterev, S. K. Stefanov, Two-dimensional Slip-Velocity Gaseous Flow past a Confined Square in a Microchannel, 25th International symposium on Rarefied Gas Dynamics, Saint-Petersburg,
Russia, 15 – 21 July, 2006.
19. Kiril Shterev, Mathematical modeling of velocity slip gaseous flows by using finite volume method, In: Proc. 10th National Congress on Theoretical and Applied Mechanics, Vol. II, pp. 427-432,
Varna, 13 - 16 September, 2005.
Recent Projects:
1st Award of the competition "Young Scientist" in Institute of Mechanics, 2006
1st Award of the competition "Young Scientist" in Institute of Mechanics, 2010
Diploma from Union of Scientists in Bulgaria for thesis " Numerical simulation of compressible, viscous and heat-conductive gas microflows", 2011
The source code of the algorithm SIMPLE-TS written in C++ and most of the problems solved in paper "Pressure based finite volume method for calculation of compressible viscous gas flows", are
availible here. | {"url":"https://www.imbm.bas.bg/index.php?page=184","timestamp":"2024-11-03T22:10:28Z","content_type":"application/xhtml+xml","content_length":"91551","record_id":"<urn:uuid:a955ebc6-ff83-4c6b-994b-d24672127ffb>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00054.warc.gz"} |
Vedic Maths - Free Course
Founder & CEO : Sateesh Academy
Mr. Sateesh Konakalla has a teaching experience of more than 9 years and he has taught more than 1800+ successful trainers.
After training these many trainers/teachers and after deep research Mr. Sateesh has proved that Vedic Maths is a globally appreciated skill.
Mr. Sateesh Konakalla has arranged a LIVE WEBINAR for teachers & trainers so that you can learn and have a glimpse of how Vedic Maths can open new opportunities for you, professionally & financially.
What Will You Learn?
• Introduction to Vedic Maths.
• Learn basics of Vedic Mathematics.
• How to be productive in your free time.
• Enhance your child's/students brain using Vedic Maths.
• Career opportunities in Vedic Maths.
• How to learn Vedic maths efficiently.
• How to enhance your skillset with simple Vedic Maths tricks.
• The advantages and Power of Vedic Maths.
Magic With Vedic Maths - Watch And Understand the Power of Vedic Maths
Available in days
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How to become Certified Vedic Maths & abacus Trainer
Available in days
days after you enroll
Check your inbox to confirm your subscription | {"url":"https://courses.sateeshacademy.in/p/vedic-maths-free-course","timestamp":"2024-11-09T17:21:24Z","content_type":"text/html","content_length":"117736","record_id":"<urn:uuid:f8578469-8a20-4a6c-9af3-27fb7c6559b6>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00590.warc.gz"} |
Math 350, Fall 2018
Math 350: Groups, Rings, and Fields (Fall 2018)
This course is an introduction to abstract algebra, a central pillar of modern mathematics that concerns generalizations of the familiar addition and multiplication operations from ordinary
arithmetic. The course focuses on three types of algebraic structures: groups, rings, and fields. For each structure, we study certain transformations between such structures, a library of important
examples, and ways to construct one object from another. Time permitting, we will discuss several applications of abstract algebra, including a preview of public-key cryptography (a subject that I
will most likely teach a semester-long course about in the spring).
Help hours
• My office hours in SMUD 401:
□ Tuesday 2:45-4:15
□ Wednesday 1:15-2:45
□ Friday 1:30-2:30
• Kat Mendoza’s help hours in SMUD 208 (will be in 007 after renovation is finished):
• James Corbett’s help hours in SMUD 208 (will be in 007 after renovation is finished):
□ Tuesday 6-8pm
□ Wednesday 6-8pm
• Fernando Liu Lopez’s help hours in New Science Center D107 (ground floor, right behind the Science Library space):
□ MWF 2-4pm
□ TuThu 2-5pm
□ Or by individual appointment at this link
Handouts and other items
LaTeX resources
• Overleaf LaTeX tutorial: this tutorial is a nice introduction for beginners. It focuses on using Overleaf, which allows you to write LaTeX documents online without installing any software.
• Overleaf Primer by Kristin Heysse (Macalester). Another tutorial on writing in LaTeX on Overleaf.
• Detexify: this is an absurdly useful tool that allows you to sketch a symbol and quickly learn the LaTeX command for it.
Problem sets will be posted here. All problem sets are due at 10pm, via Gradescope.
• Midterm 1: Friday 10/5
□ Sample exams from previous semesters:
☆ Note: These exams should give some impression of the length and format of the exam, but the topic coverage is slightly different from previous semesters. In particular: both exams use
topics from Chapter 6 (direct products), and use the notation Z(G) (for the center of a group G) that we have not used in class.
□ Exam / Solutions
• Midterm 2: Friday 11/30
• Final Exam: Wednesday 12/19, 9am-noon in SCCE A131
□ Remember to make a one-page note sheet (front and back) for the exam!
□ Reading/finals week hours:
☆ Thursday 12/13: 1:30-3:00 with Charley
☆ Friday 12/14: 10:15-11:45
☆ Monday 12/17: 9:30-11:00
☆ Tuesday 12/18: 9:30-11:00 with Charley
□ Review sessions (with Math Fellows): Thursday and Saturday 5-7pm (check email for room assignment)
□ Fernando’s reading/finals office hours: 10am to 4pm every weekday.
□ Review guide of topics that may appear on the exam.
□ Many recent exams (these don’t have solutions; let me know if you want to check your work for any of them)
□ Another good source of problems: look at the Algebra sections of Old Comprehensive Exams (this is an especially good place for review if you’re planning to take the math comps).
□ Exam / Solutions | {"url":"https://npflueger.github.io/teaching/350-18fall/","timestamp":"2024-11-13T20:42:31Z","content_type":"text/html","content_length":"19684","record_id":"<urn:uuid:7c56e9c1-baa2-42d1-858c-f3caf45b1bd3>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00808.warc.gz"} |
How Many Hours In A Year? | QuickAnswer.org
How many hours in a year?
The number of hours in a year will depend on whether or not the year in question is a leap year. Obviously, a leap year has an extra day in it and so will have 24 more hours in it than a standard
year. So let’s look at how we calculate how many hours in a year…
How many hours in a year?
For a normal year, that is a non-leap year, there are 365 days in the year. As there are 24 hours in every day, this means that there are:
365 (days in the year) x 24 (hours in a day) = 8,760.
There are 8,760 hours in a normal year.
How many hours in a leap year?
You can use the same method to calculate how many hours in a leap year:
366 (days in the leap year) x 24 (hours in a day) = 8,784.
There are 8,784 hours in a leap year.
Of course, as we know that a leap year has 24 more hours in it than a standard year, we can also work out the number of hours in a leap year just by adding 24 to the number of hours in a standard
See also: | {"url":"https://quickanswer.org/how-many-hours-in-a-year/","timestamp":"2024-11-15T04:48:33Z","content_type":"text/html","content_length":"85830","record_id":"<urn:uuid:18b3ae09-7aa1-49b3-a165-268adf6427d3>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00507.warc.gz"} |
Reduction of Consumed Energy and Control Input Variance in Machine Tool Feed Drives by Contouring Control
Naoki Uchiyama^*, Takaya Nakamura^*, and Kazuo Yamazaki^**
^*Department of Mechanical Engineering, Toyohashi University of Technology, 1-1 Hibarigaoka, Tempaku, Toyohashi, Aichi 441-8580, Japan
^**Department of Mechanical and Aeronautical Engineering, University of California, Davis, One Shields Avenue, Davis, CA 95616-5294, USA
March 2, 2009
April 8, 2009
July 5, 2009
machine tool, feed drive, contouring control
Contouring control has been widely studied to reduce contour error, defined as error components orthogonal to desired contour curves. Its effectiveness has, however, to the best of our knowledge,
only been verified through comparative experiments using industrial non-contouring (independent axial) controllers or conventional contouring controllers such as the cross-coupling controller.
Because control performance depends largely on controller gain, these comparisons have problems in showing the effectiveness of contouring control, meaning that similar control performance could
be attained if non-contouring controller gain were appropriately assigned. This paper presents contouring controller design for biaxial feed drives that reduces controller gain, rather than
contour error, better than conventional independent axial controllers. The contouring controller is shown in experiments to effectively reduce control input variance and electricity consumption
on average by 4.5%.
Cite this article as:
N. Uchiyama, T. Nakamura, and K. Yamazaki, “Reduction of Consumed Energy and Control Input Variance in Machine Tool Feed Drives by Contouring Control,” Int. J. Automation Technol., Vol.3 No.4,
pp. 401-407, 2009.
Data files:
1. [1] R. Ramesh, M. A. Mannan, and A. N. Poo, “Tracking and Contour Error Control in CNC Servo Systems,” Int. Journal of Machine Tools and Manufacture, 45, pp. 301-326, 2005.
2. [2] Y. Koren, “Cross-Coupled Biaxial Control for Manufacturing Systems,” ASME Journal of Dynamic Systems, Measurement, and Control, 102, pp. 265-272, 1980.
3. [3] P. K. Kulkarni and K. Srinivasan, “Optimal Contouring Control of Multi-Axis Feed Frive Servomechanism,” ASME Journal of Engineering for Industry, 111, pp. 140-148, 1989.
4. [4] H.-Y. Chuang and C.-H. Liu, “Cross-Coupled Adaptive Feedrate Control for Multiaxis Machine Tools,” ASME Journal of Dynamic Systems, Measurement, and Control, 113, pp. 451-457, 1991.
5. [5] Y. S. Tarng, H. Y. Chuang, and W. T. Hsu, “An Optimisation Approach to the Contour Error Control of CNC Machine Tool Using Genetic Algorithms,” Int. Journal of Advanced Manufacturing
Technology, 13, pp. 359-366, 1997.
6. [6] Y. S. Tarng, H. Y. Chuang, and W. T. Hsu, “Intelligent Cross-Coupled Fuzzy Feedrate Controller Design of CNC Machine Tools Based on Genetic Algorithms,” Int. Journal of Machine Tools and
Manufacture, 39, 1999, pp. 1673-1692, 1999.
7. [7] G. T.-G. Chiu and M. Tomizuka, “Coordinate Position Control of Multi-Axis Mechanical Systems,” ASME Journal of Dynamic Systems, Measurement, and Control, 120, pp. 389-393, 1998.
8. [8] S.-S. Yeh and P.-L. Hsu, “Theory and Applications of the Robust Cross-Coupled Control Design,” ASME Journal of Dynamic Systems, Measurement, and Control, 121, pp. 524-530, 1999.
9. [9] R. J. McNab and T.-C. Tsao, “Receding Time Horizon Linear Quadratic Optimal Control for Multi-Axis Contour Tracking Motion Control,” ASME Journal of Dynamic Systems, Measurement, and
Control, 122, pp. 375-381, 2000.
10. [10] Q. Zhong, Y. Shi, J. Mo, and S. Huang, “A Linear Cross-Coupled Control System for High-Speed Machining,” Int. Journal of Advanced Manufacturing Technology, 19, pp. 558-563, 2002.
11. [11] X. Ye, X. Chen, X. Li, and S. Huang, “A Cross-Coupled Path Precompensation Algorithm for Rapid Prototyping and Manufacturing,” Int. Journal of Advanced Manufacturing Technology, 20, pp.
39-43, 2002.
12. [12] Y.-T. Shih, C.-S. Chen, and A.-C. Lee, “A Novel Cross-Coupling Control Design for Bi-Axis Motion,” Int. Journal of Machine Tools and Manufacture, 42, pp. 1539-1548, 2002.
13. [13] Y.-M. Cheng and J.-H. Chin, “Machining Contour Errors as Ensembles of Cutting, Feeding and Machine Tool Structure Effects,” Int. Journal of Machine Tools and Manufacture, 43, pp.
1001-1014, 2003.
14. [14] C.-S. Chen, Y.-H. Fan, and S. P. Tseng, “Position Command Shaping Control in a Retrofitted Milling Machine,” Int. Journal of Machine Tools and Manufacture, 46, pp. 293-303, 2006.
15. [15] S.-S. Yeh and P.-L. Hsu, “Analysis and Design of Integrated Control for Multi-Axis Motion Systems,” IEEE Transactions on Control System Technology, 11-3, pp. 375-382, 2003.
16. [16] S.-L. Chen and K.-C. Wu, “Contouring Control of Smooth Paths for Multiaxis Motion Systems Based on Equivalent Errors,” IEEE Transactions on Control System Technology, 15-6, pp.
1151-1158, 2007.
17. [17] C.-C. Lo and C.-Y Chung, “Tangential Contouring Controller for Biaxial Motion Control,” ASME Journal of Dynamic Systems, Measurement, and Control, 121, pp. 126-129, 1999.
18. [18] H.-C. Ho, J. Y. Yen, and S. S. Lu, “A Decoupled-Path Following Control Algorithm Based upon the Decomposed Trajectory Error,” Int. Journal of Machine Tools and Manufacture, 39, pp.
1619-1630, 1999.
19. [19] G. T.-C. Chiu and M. Tomizuka, “Contouring Control of Machine Tool Feed Drive Systems: A Task Coordinate Frame Approach,” IEEE Transactions on Control System Technology, 9-1, pp.
130-139, 2001.
20. [20] C.-C. Peng and C.-L. Chen, “Biaxial Contouring Control with Friction Dynamics Using a Contour Index Approach,” Int. Journal of Machine Tools and Manufacture, 47, pp. 1542-1555, 2007.
21. [21] M.-Y. Cheng and C.-C. Lee, “Motion Controller Design for Contour Following Tasks Based on Real-Time Contour Error Estimation,” IEEE Transactions on Industrial Electronics, 54-3, pp.
1686-1695, 2007. | {"url":"https://www.fujipress.jp/ijat/au/ijate000300040401/","timestamp":"2024-11-14T17:24:25Z","content_type":"text/html","content_length":"49069","record_id":"<urn:uuid:843feb7d-ca52-465b-9a30-22e000cde37e>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00023.warc.gz"} |
Testing order-constrained hypotheses in social science research
Social science researchers often formulate hypotheses (or theories) using order constraints between the parameters of interest. In a regression model for example one may expect that the relative
effect of the first explanatory variable on the dependent variable is larger than the relative effect of the second explanatory variable on the dependent variable, which, in turn, is larger than the
relative effect of the third explanatory variable on the dependent variable. Testing such order-constrained hypotheses using classical p-values or classical model selection criteria can be
problematic. In this talk I show that the Bayes factor, a Bayesian criterion for model selection and hypothesis testing, is very useful for this testing problem. This is discussed when testing order
constraints on regression coefficients in a linear regression model and when testing order constraints on bivariate correlations in an unstructured correlation matrix. | {"url":"https://csss.uw.edu/seminars/testing-order-constrained-hypotheses-social-science-research","timestamp":"2024-11-05T03:29:19Z","content_type":"text/html","content_length":"21838","record_id":"<urn:uuid:d43ecb79-83a8-46dd-98d5-5847fa100cb3>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00772.warc.gz"} |
Cite as
Yangjing Dong, Honghao Fu, Anand Natarajan, Minglong Qin, Haochen Xu, and Penghui Yao. The Computational Advantage of MIP^∗ Vanishes in the Presence of Noise. In 39th Computational Complexity
Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 30:1-30:71, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
Copy BibTex To Clipboard
author = {Dong, Yangjing and Fu, Honghao and Natarajan, Anand and Qin, Minglong and Xu, Haochen and Yao, Penghui},
title = {{The Computational Advantage of MIP^∗ Vanishes in the Presence of Noise}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {30:1--30:71},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.30},
URN = {urn:nbn:de:0030-drops-204263},
doi = {10.4230/LIPIcs.CCC.2024.30},
annote = {Keywords: Interactive proofs, Quantum complexity theory, Quantum entanglement, Fourier analysis, Matrix analysis, Invariance principle, Derandomization, PCP, Locally testable code, Positivity testing} | {"url":"https://drops.dagstuhl.de/search?term=Brakerski%2C%20Zvika","timestamp":"2024-11-12T12:29:43Z","content_type":"text/html","content_length":"182946","record_id":"<urn:uuid:c4df0a24-7001-4df5-be94-63f1c2ad454f>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00630.warc.gz"} |
Python Data Types
Python Data Types
Data types are the classification or categorization of data items. Python supports the following built-in data types.
Scalar Types
• int: Positive or negative whole numbers (without a fractional part) e.g. -10, 10, 456, 4654654.
• float: Any real number with a floating-point representation in which a fractional component is denoted by a decimal symbol or scientific notation e.g. 1.23, 3.4556789e2.
• complex: A number with a real and imaginary component represented as x + 2y.
• bool: Data with one of two built-in values True or False. Notice that 'T' and 'F' are capital. true and false are not valid booleans and Python will throw an error for them.
• None: The None represents the null object in Python. A None is returned by functions that don't explicitly return a value.
Sequence Type
A sequence is an ordered collection of similar or different data types. Python has the following built-in sequence data types:
• String: A string value is a collection of one or more characters put in single, double or triple quotes.
• List: A list object is an ordered collection of one or more data items, not necessarily of the same type, put in square brackets.
• Tuple: A Tuple object is an ordered collection of one or more data items, not necessarily of the same type, put in parentheses.
Mapping Type
Dictionary: A dictionary Dict() object is an unordered collection of data in a key:value pair form. A collection of such pairs is enclosed in curly brackets. For example: {1:"Steve", 2:"Bill",
3:"Ram", 4: "Farha"}
Set Types
• set: Set is mutable, unordered collection of distinct hashable objects. The set is a Python implementation of the set in Mathematics. A set object has suitable methods to perform mathematical set
operations like union, intersection, difference, etc.
• frozenset: Frozenset is immutable version of set whose elements are added from other iterables.
Mutable and Immutable Types
Data objects of the above types are stored in a computer's memory for processing. Some of these values can be modified during processing, but contents of others can't be altered once they are created
in the memory.
Numbers, strings, and Tuples are immutable, which means their contents can't be altered after creation.
On the other hand, items in a List or Dictionary object can be modified. It is possible to add, delete, insert, and rearrange items in a list or dictionary. Hence, they are mutable objects. | {"url":"https://9xdev.com/python/python-data-types","timestamp":"2024-11-07T23:35:17Z","content_type":"text/html","content_length":"55808","record_id":"<urn:uuid:62aef4d7-b2c7-489f-b8e5-7b79c08ea278>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00062.warc.gz"} |
Case studies | Green Traffic Software
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Software decongesting cities
Location: Tsaritsa Ioanna Blvd - Todor Aleksandrov Blvd, Sofia, Bulgaria
Number of intersections: 15
This case study is an example of Green Traffic Software tool applied to a real urban city arterial. Here we are dealing with one of the busiest arterials in the city, connecting its north-west
district with the center. A part of it (9 intersections) is already coordinated though not in an optimal way while the entire arterial is not.
GTS provides multiple coordination schemes for both the part and all 15 intersections together. What is special about this arterial is that the maximal sum bandwidth is the same (55 s) in both
cases. It exceeds the current sum bandwidth (46 s).
Maps data: Google, © 2023
Number of traffic lights
Distances between them
Recommended vehicle velocity
Split phasing
Cycle lengths and the desired common cycle length
Yellow rectangles in the table and on the map highlight the coordinated part of the arterial.
Data source: Dom Eleshnitsa LtD
Current timing plan by the local Intelligent Transport System:
9 intersections, sum bandwidth 46 s:
The subset of 9 intersections, from 6 to 14, is coordinated, while the entire arterial is not. The time-space diagram of the coordinated subset shows that currently the sum bandwidth is 16 + 30 = 46
With GTS, these 9 intersections can be coordinated in a better way, see below.
GTS, 9 intersections, sum bandwidth 55 s:
GTS provides the arterial portrait, showing all possible pairs of bandwidths with 1 sec resolution. In total, 1244 bidirectional timing plans are available, 24 of them provide the maximal sum
bandwidth of 55 seconds.
Each green asterisk leads to a particular timing plan: offsets and phase sequences ("orders") are shown together with the time-space diagram. The dark green asterisks mark the bandwidths pairs with
maximal sum. The blue circles mark the bandwidth pairs that are impossible in this arterial.
Two TSDs below correspond to the bandwidth pairs marked with green squares. Brown square marks the current coordination scheme.
Inbound bandwidth: 17 seconds
Outbound bandwidth: 38 seconds
Inbound bandwidth: 40 seconds
Outbound bandwidth: 15 seconds
Now let us turn to the entire arterial. Currently, it is not coordinated at all but the problem can be easily solved with the use of GTS.
For the given cycle length of 110 s we get an arterial portrait containing 19 different timing plans with sum bandwidth 46 s:
We get even better solutions by changing the cycle length. Specifically, at C = 108 s the number of available timing plans decreases to 8 but the sum bandwidth becomes the same as that for 9
intersections subset: 55 s.
Two timing plans below correspond to the bandwidth pairs marked with green squares.
Inbound bandwidth: 18 seconds
Outbound bandwidth: 37 seconds
Inbound bandwidth: 25 seconds
Outbound bandwidth: 30 seconds
This study proves that the widespread belief of coordination deterioration with the number of intersection increasing is not always true (read more here).
bottom of page | {"url":"https://www.greentrafficsoftware.com/case-studies","timestamp":"2024-11-12T03:34:08Z","content_type":"text/html","content_length":"662531","record_id":"<urn:uuid:0083bbfc-20a0-4eb8-a083-b058db8aad13>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00718.warc.gz"} |
Skip Counting ChartSkip Counting Chart
Skip Counting Chart
A skip counting chart, also known as a multiplication chart, is a grid-like diagram that displays the multiples of a set of numbers. This chart is often used as an educational tool to facilitate
learning multiplication and division facts. It allows users to quickly identify the product of any two numbers within the chart.
The skip counting chart has been widely recognized for its effectiveness in developing students’ computational fluency. Its origins can be traced back to centuries-old multiplication tables, with the
current version widely adopted in modern education. By providing a visual representation of multiplication patterns, this chart supports the understanding of multiplication and division as repeated
addition and subtraction, respectively.
The versatility of skip counting charts extends beyond basic arithmetic operations. They can be employed in various mathematical concepts, such as finding factors, common multiples, and solving
equations. Moreover, these charts promote mental math skills and enhance students’ ability to recall multiplication and division facts quickly and accurately.
Skip Counting Chart
Skip counting charts play a crucial role in developing students’ mathematical abilities. Here are ten key aspects that highlight their significance:
• Systematic and Ordered
• Multiplication Facts
• Visual Representation
• Number Sequencing
• Division Connection
• Factors and Multiples
• Mental Math Enhancer
• Problem-Solving Tool
• Historical Significance
• Educational Foundation
Skip counting charts provide a systematic and ordered arrangement of multiplication facts, making them easy to locate and memorize. They serve as a visual representation of number patterns, aiding in
the understanding of multiplication and division as repeated addition and subtraction. By promoting mental math skills, these charts enhance students’ ability to recall facts quickly and accurately.
They also serve as a valuable problem-solving tool, enabling students to identify factors, multiples, and solve equations efficiently.
Systematic and Ordered
The systematic and ordered nature of skip counting charts is a key aspect that contributes to their effectiveness as a learning tool. These charts are designed with a logical and organized structure
that makes it easy for students to navigate and understand the relationships between numbers.
• Number Sequence
Skip counting charts display the multiples of a given number in a sequential order. This allows students to visualize the pattern of multiplication and identify the next number in the sequence
quickly and easily.
• Multiplication Facts
The systematic arrangement of the chart enables students to locate multiplication facts efficiently. By identifying the row and column corresponding to the two numbers being multiplied, students
can instantly find the product.
• Visual Representation
The ordered structure of the chart provides a visual representation of the multiplication table. This helps students develop a mental image of the relationship between numbers and their
multiples, which supports their understanding and memorization of multiplication facts.
• Problem-Solving Tool
The systematic and ordered nature of the chart makes it a valuable tool for problem-solving. Students can use the chart to identify factors, multiples, and solve simple multiplication and
division problems efficiently.
In summary, the systematic and ordered structure of skip counting charts enhances their effectiveness as a learning tool by providing a logical and organized framework for understanding
multiplication patterns, locating multiplication facts, and solving problems.
Multiplication Facts
Multiplication facts are the foundation of skip counting charts. They represent the products of two numbers and serve as the building blocks for understanding multiplication and division. The skip
counting chart is a visual representation of these multiplication facts, arranged in a systematic and ordered manner.
The connection between the two is essential for developing a strong understanding of multiplication. By memorizing multiplication facts, students can quickly and easily locate the product of any two
numbers on the skip counting chart. This fluency in multiplication facts allows students to solve problems involving multiplication and division more efficiently.
For example, if a student needs to find the product of 7 and 8, they can simply locate the intersection of the 7th row and 8th column on the skip counting chart to find the answer, which is 56. This
would be much faster than trying to calculate the product using repeated addition or other methods.
Understanding the connection between multiplication facts and the skip counting chart is crucial for students to develop computational fluency and problem-solving skills. It provides them with a tool
to quickly and accurately find the products of numbers, which is essential for success in mathematics and various real-life applications.
Visual Representation
The skip counting chart is a powerful tool for visualizing multiplication and division patterns. Its visual representation makes it easy for students to understand the relationships between numbers
and their multiples.
• Number Patterns
The skip counting chart provides a visual representation of number patterns. Students can see how the multiples of a number increase as they move down the column and how the products of two
numbers increase as they move across the row.
• Multiplication and Division
The chart also helps students visualize the connection between multiplication and division. By understanding that multiplication is repeated addition, students can see how the rows on the chart
represent the repeated addition of the same number.
• Factors and Multiples
The skip counting chart can also be used to identify factors and multiples. Students can see which numbers are multiples of another number by looking for the numbers in the same column. They can
also see which numbers are factors of another number by looking for the numbers in the same row.
• Problem Solving
The visual representation of the skip counting chart makes it a valuable tool for problem solving. Students can use the chart to solve multiplication and division problems, as well as to find
factors and multiples.
The visual representation of the skip counting chart makes it a powerful tool for learning multiplication and division. It helps students to understand number patterns, the relationship between
multiplication and division, and to solve problems.
Number Sequencing
Number sequencing plays a crucial role in understanding and utilizing skip counting charts effectively. It involves recognizing, extending, and manipulating numerical patterns, which forms the
foundation of skip counting.
• Ordered Patterns
Skip counting charts display numbers in an ordered sequence, typically starting from 0 or 1. This ordered arrangement allows students to identify patterns and relationships between numbers, such
as odd and even numbers, multiples, and factors.
• Skip Counting
The concept of skip counting is closely intertwined with number sequencing. Skip counting involves counting forward or backward by a specific number, skipping every other number in the sequence.
Skip counting charts facilitate this process by providing a visual representation of the skipped numbers.
• Number Recognition
Skip counting charts enhance number recognition skills. By repeatedly encountering numbers in a sequential order, students become more familiar with their position and value within the number
• Problem Solving
Understanding number sequencing is essential for problem-solving involving multiplication and division. Skip counting charts can be used to find missing numbers in a sequence, solve simple
multiplication and division problems, and identify patterns in numerical data.
In summary, number sequencing is a fundamental aspect of skip counting charts. It provides a structured framework for understanding numerical patterns, skip counting, number recognition, and
problem-solving, contributing to the effectiveness of skip counting charts as a learning tool.
Division Connection
The skip counting chart’s significance extends beyond multiplication; it also fosters an understanding of division. Division, the inverse operation of multiplication, can be conceptualized using the
skip counting chart.
By examining the chart’s rows, one can observe the multiples of a particular number. Conversely, by examining the columns, one can identify the divisors of a number. This reciprocal relationship
between multiplication and division is visually represented, providing a deeper comprehension of both operations.
For instance, consider the 6 multiplication row on a skip counting chart. By reading across the row, one can identify the multiples of 6 (6, 12, 18, 24, 30, …). However, by reading down the 6
division column, one can determine the divisors of 6 (1, 2, 3, 6). This dual perspective reinforces the interconnectedness of multiplication and division.
Factors and Multiples
Within the framework of a skip counting chart, the concepts of factors and multiples play a pivotal role in comprehending the underlying numerical relationships. Factors represent the divisors of a
number, while multiples represent the products of a number multiplied by consecutive whole numbers.
The skip counting chart provides a visual representation of these relationships. By examining the rows of the chart, one can identify the multiples of a particular number. Conversely, by examining
the columns, one can determine the factors of a number. This reciprocal connection between factors and multiples is crucial for understanding divisibility and the structure of the number system.
For instance, consider the number 12 on a skip counting chart. Its multiples, found in the 12 multiplication row, include 24, 36, 48, and so on. Conversely, its factors, found in the 12 division
column, include 1, 2, 3, 4, 6, and 12 itself. This dual perspective reinforces the interconnectedness of factors and multiples.
Mental Math Enhancer
The skip counting chart serves as a powerful tool for enhancing mental math proficiency. Its systematic organization and visual representation provide several benefits for developing mental
calculation skills.
• Rapid Recall of Multiplication Facts
The skip counting chart enables quick recall of multiplication facts. By repeatedly encountering multiplication patterns within the chart, students develop a strong memory for these facts,
allowing them to retrieve them effortlessly in mental calculations.
• Estimation and Approximation
The chart also facilitates estimation and approximation in mental math. By observing the general trends and patterns within the chart, students can make informed estimates and approximations,
even for unfamiliar multiplication problems.
• Number Sense Development
The skip counting chart contributes to the development of number sense. By visually representing the relationships between numbers and their multiples, it helps students understand the structure
and properties of the number system, enhancing their overall number sense.
• Problem-Solving Strategies
The chart can serve as a problem-solving tool for mental math. Students can use it to identify patterns, relationships, and strategies for solving multiplication problems efficiently in their
In summary, the skip counting chart plays a vital role in enhancing mental math skills by providing a structured and visual framework for memorizing multiplication facts, developing estimation and
approximation abilities, fostering number sense, and supporting problem-solving strategies.
Problem-Solving Tool
The skip counting chart is a valuable tool for problem-solving, providing a structured and visual framework for students to solve a variety of multiplication and division problems efficiently and
• Identifying Patterns and Relationships
The chart allows students to identify patterns and relationships between numbers and their multiples. This understanding is crucial for solving problems involving multiplication and division, as
it enables students to make predictions and draw logical conclusions.
• Finding Unknown Values
The chart can be used to find unknown values in multiplication and division problems. For example, if a student knows that 4 x 5 = 20, they can use the chart to determine that the missing factor
in 20 4 = __ is 5.
• Estimating and Approximating
The chart can also be used for estimation and approximation in multiplication and division problems. By observing the general trends and patterns in the chart, students can make informed
estimates and approximations, even for unfamiliar problems.
• Solving Real-World Problems
The skip counting chart can be applied to solve real-world problems involving multiplication and division. For example, a student can use the chart to determine the total cost of multiple items
at a store or the number of items that can be divided equally among a group of people.
In summary, the skip counting chart empowers students with a powerful problem-solving tool for multiplication and division, enabling them to identify patterns, find unknown values, estimate and
approximate, and solve real-world problems with greater accuracy and efficiency.
Historical Significance
Understanding the historical significance of the skip counting chart provides a deeper appreciation for its role in mathematical education. The chart’s origins can be traced back to ancient
civilizations, where multiplication tables were used for practical calculations in commerce and trade.
In the early 20th century, educators recognized the value of the skip counting chart as a pedagogical tool. It was incorporated into school curricula to teach multiplication and division facts, and
its use has persisted to this day. The chart’s systematic organization and visual representation make it an effective resource for students to memorize multiplication facts, understand number
patterns, and solve multiplication and division problems.
The historical significance of the skip counting chart lies in its enduring relevance as a foundational tool for teaching multiplication and division. Its simplicity and effectiveness have made it a
staple in classrooms worldwide, contributing to generations of students developing strong mathematical skills.
Educational Foundation
The skip counting chart serves as a cornerstone in the educational foundation of mathematics, providing a structured and visual representation of multiplication and division concepts. Its
significance extends beyond rote memorization; it fosters a deeper understanding of numerical relationships, problem-solving strategies, and the development of critical thinking skills.
• Number Recognition and Sequencing
The skip counting chart reinforces number recognition and sequencing, helping students develop a strong foundation in number sense. By repeatedly encountering numbers in a systematic pattern,
students become more familiar with their order and relationships.
• Multiplication and Division Facts
The chart facilitates the memorization and recall of multiplication and division facts. The organized arrangement of multiples and factors allows students to quickly locate products and
quotients, enhancing their fluency in these operations.
• Pattern Recognition
Skip counting charts promote pattern recognition and generalization. Students observe patterns in the multiples of different numbers, enabling them to make predictions and solve problems
involving multiplication and division.
• Problem-Solving Strategies
The chart serves as a problem-solving tool, helping students develop strategies for multiplication and division. By identifying patterns and relationships, they can approach problems with a
deeper understanding and develop logical solutions.
In summary, the skip counting chart plays a pivotal role in the educational foundation of mathematics. It supports the development of number sense, fluency in multiplication and division, pattern
recognition, and problem-solving strategies, providing a solid base for further mathematical learning.
Frequently Asked Questions
This section addresses common questions and misconceptions regarding the use of skip counting charts in mathematical education.
Question 1: What is the primary purpose of a skip counting chart?
A skip counting chart is designed to facilitate the learning of multiplication and division facts. It provides a systematic and visual representation of the multiples of numbers, enabling students to
quickly identify products and quotients.
Question 2: How does a skip counting chart enhance number sense?
The chart reinforces number recognition and sequencing, helping students develop a strong foundation in number sense. By repeatedly encountering numbers in a systematic pattern, students become more
familiar with their order and relationships.
Question 3: Is a skip counting chart only useful for memorizing multiplication facts?
While memorization is a key benefit, the chart also promotes pattern recognition, problem-solving strategies, and the development of critical thinking skills. It serves as a versatile tool for
exploring numerical relationships and solving multiplication and division problems.
Question 4: At what grade level is a skip counting chart typically introduced?
The chart is commonly introduced in elementary school, typically in the early grades. It is an essential tool for building a solid foundation in multiplication and division, which are crucial for
further mathematical learning.
Question 5: How can parents support their children’s use of skip counting charts at home?
Parents can provide support by encouraging their children to use the chart regularly, assisting them in identifying patterns, and using it to solve simple multiplication and division problems. Home
practice can reinforce learning and enhance their mathematical skills.
Question 6: Are there any limitations to using a skip counting chart?
While the chart is a valuable tool, it is important to note that it may not be suitable for all students or all learning styles. Some students may benefit from alternative methods of learning
multiplication and division facts.
In summary, skip counting charts are effective and versatile tools that support the teaching and learning of multiplication and division. They provide a structured and visual representation of
numerical relationships, fostering number sense, fluency, and problem-solving skills.
Transition to the next article section:
The following section will delve into the historical evolution of skip counting charts, exploring their origins and impact on mathematical education over time.
Tips for Utilizing Skip Counting Charts
Skip counting charts are valuable tools for enhancing mathematical proficiency. Here are some tips for effectively incorporating them into your teaching or learning experience:
Tip 1: Introduce the Chart Systematically
Begin by explaining the structure of the chart, highlighting the rows and columns. Show students how to locate multiples and factors using the chart.
Tip 2: Practice Skip Counting Regularly
Encourage students to practice skip counting aloud, both forward and backward. This helps them memorize multiplication facts and develop number sense.
Tip 3: Use the Chart for Problem-Solving
Incorporate the chart into problem-solving activities. Ask students to use it to find missing factors or multiples, solve multiplication and division equations, and compare numbers.
Tip 4: Explore Patterns and Relationships
Guide students to observe patterns in the chart, such as the relationship between multiples of 5 and the number of fingers on a hand. This promotes pattern recognition and mathematical thinking.
Tip 5: Encourage Mental Math
Use the chart to facilitate mental math calculations. Ask students to estimate products and quotients using the chart, reinforcing their mental computation skills.
Tip 6: Provide Visual Support
Create colorful and engaging skip counting charts. Use different colors and fonts to highlight multiples and factors, making the chart more visually appealing and easier to use.
Tip 7: Differentiate Instruction
Adapt the use of skip counting charts to meet the diverse needs of students. Provide larger charts for struggling learners and challenge advanced students with more complex problems.
Summary: By following these tips, you can maximize the effectiveness of skip counting charts in your educational setting. These charts not only enhance multiplication and division fluency but also
foster problem-solving skills, pattern recognition, and mathematical thinking.
Skip counting charts have proven to be invaluable tools in the teaching and learning of multiplication and division. They provide a structured and systematic approach to understanding numerical
relationships, promoting fluency, problem-solving skills, and mathematical thinking.
Their effectiveness stems from their simplicity, versatility, and ability to cater to diverse learning styles. By incorporating skip counting charts into educational practices, educators can empower
students with a solid foundation in multiplication and division, fostering their mathematical development and preparing them for future academic endeavors.
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Neuroimaging and Data Science
19. Overfitting¶
Let’s pause and take stock of our progress. In the last section, we developed three fully operational machine-learning workflows: one for regression, one for classification, and one for clustering.
Thanks to Scikit-learn, all three involved very little work. In each case, it took only a few lines of code to initialize a model, fit it to some data, and use it to generate predictions or assign
samples to clusters. This all seems pretty great. Maybe we should just stop here, pat ourselves on the back for a job well done, and head home for the day satisfied with the knowledge we’ve become
proficient machine learners.
While a brief pat on the back does seem appropriate, we probably shouldn’t be too pleased with ourselves. We’ve learned some important stuff, yes, but a little bit of knowledge can be a dangerous
thing. We haven’t yet touched on some concepts and techniques that are critical if we want to avoid going off the rails with the skills we’ve just acquired. And going off the rails, as we’ll see, is
surprisingly easy.
To illustrate just how easy, let’s return to our brain-age prediction example. Recall that when we fitted our linear regression model to predict age from our brain features, we sub-sampled only a
small number of features (5, to be exact). We found that, even with just 5 features, our model was able to capture a non-trivial fraction of the variance in age—about 20%. Intuitively, you might find
yourself thinking something like this: if we did that well with just 5 out of 1,440 features selected at random, imagine how well we might do with more features! And you might be tempted to go back
to the model-fitting code, replace the n_features variable with some much larger number, re-run the code, and see how well you do.
Let’s go ahead and give in to that temptation. In fact, let’s give in to temptation systematically. We’ll re-fit our linear regression model with random feature subsets of different sizes and see
what effect that has on performance. We start by setting up the size of the feature sets. We will run this 10 times in each set size, so we can average and get more stable results.
n_features = [5, 10, 20, 50, 100, 200, 500, 1000, 1440]
n_iterations = 10
We initialize a placeholder for the results and extract the age phenotype, which we will use in all of our model fits.
import numpy as np
results = np.zeros((len(n_features), n_iterations))
Looping over feature set sizes and iterations, we sample features, fit a model, and save the score into the results array.
from sklearn.linear_model import LinearRegression
model = LinearRegression()
for i, n in enumerate(n_features):
for j in range(n_iterations):
X = features.sample(n, axis=1)
model.fit(X, y)
results[i, j] = model.score(X, y)
Let’s plot the observed \(R^2\) as a function of the number of predictors. We’ll plot a dark line for the averages across iterations and use Matplotlib’s fill_between function to add 1 standard
deviation error bars, even though these will be so small that they are hard to see.
import matplotlib.pyplot as plt
averages = results.mean(1)
fig, ax = plt.subplots()
ax.plot(n_features, averages, linewidth=2)
stds = results.std(1)
ax.fill_between(n_features, averages - stds, averages + stds, alpha=0.2)
ax.set_xlabel("# of brain features used")
ax.set_ylabel("Explained variance in age ($R^2$)");
At first glance, this might look great: performance improves rapidly with the number of features, and by the time we’re sampling 1,000 features, we can predict age perfectly! But a truism in machine
learning (and in life more generally) is that if something seems too good to be true, it probably is.
In this case, a bit of reflection on the fact that our model seems able to predict age with zero error should set off all kinds of alarm bells, because, on any reasonable understanding of how the
world works, such a thing should be impossible. Set aside the brain data for a moment and just think about the problem of measuring chronological age. Is it plausible to think that, in a sample of
over 1,000 people, including many young children and older adults, not a single person’s age would be measured with even the slightest bit of error? Remember, any measurement error should reduce our
linear regression model’s performance because measurement error is irreducible. If an ABIDE-II participant’s birth date happened to be recorded as 1971 when it’s really 1961 (oops, typo!), it’s not
as if our linear regression model can somehow learn to go back in time and adjust for that error.
Then think about the complexity of the brain-derived features we’re using; how well (or poorly) those features are measured/extracted; how simple our linear regression model is; and how much
opportunity there is for all kinds of data quality problems to arise (is it plausible to think that all subjects’ scans are of good enough quality to extract near-perfect features?). If you spend a
couple of minutes thinking along these lines, it should become very clear that an \(R^2\) of 1.0 for a problem like this is just not remotely believable. There must be something very wrong with our
model. And there is: our model is overfitting our data. Because we have a lot of features to work with – even more than samples, our linear regression model is, in a sense getting creative: it’s
finding all kinds of patterns in the data that look like they’re there but aren’t. We’ll spend the rest of this section exploring this idea and unpacking its momentous implications.
19.1. Understanding overfitting¶
To better understand overfitting, let’s set aside our relatively complex neuroimaging dataset for the moment and work with some simpler examples.
Let’s start by sampling some data from a noisy function where the underlying functional form is quadratic. We’ll create a small function for the data generation so that we can reuse this code later
def make_xy(n, sd=0.5):
''' Generate x and y variables from a fixed quadratic function,
adding noise. '''
x = np.random.normal(size=n)
y = (0.7 * x) ** 2 + 0.1 * x + np.random.normal(10, sd, size=n)
return x, y
We fix the seed for the generation of random numbers and then produce some data and visualize it.
x, y = make_xy(30)
fig, ax = plt.subplots()
p = ax.scatter(x, y)
19.1.1. Fitting data with models of different flexibility¶
Now let’s fit that data with a linear model. To visualize the results of the linear model, we will use the model that has been fit to the data to predict the values of a range of values similar to
the values of x.
from sklearn.metrics import mean_squared_error
est = LinearRegression()
est.fit(x[:, None], y)
x_range = np.linspace(x.min(), x.max(), 100)
reg_line = est.predict(x_range[:, None])
fig, ax = plt.subplots()
ax.scatter(x, y)
ax.plot(x_range, reg_line);
mse = mean_squared_error(y, est.predict(x[:, None]))
ax.set_title(f"Mean squared error: {mse:.2f}");
The fit looks… meh. It seems pretty clear that our linear regression model is underfitting the data—meaning, there are clear patterns in the data that the fitted model fails to describe.
What can we do about this? Well, the problem here is that our model is insufficiently flexible; our straight regression line can’t bend itself to fit the contours of the observed data. The solution
is to use a more flexible estimator! A linear fit won’t cut it —- we need to fit curves.
Just to make sure we don’t underfit again, let’s use a much more flexible estimator -— specifically, a 10th-degree polynomial regression model.
This is also a good opportunity to introduce a helpful object in Scikit-learn called a Pipeline. The idea behind a Pipeline is that we can stack several transformation steps together in a sequence,
and then cap them off with an Estimator of our choice. The whole pipeline will then behave like a single Estimator – i.e., we only need to call fit() and predict() once.
Using a pipeline will allow us to introduce a preprocessing step before the LinearRegression model receives our data, in which we create a bunch of polynomial features (by taking \(x^2\), \(x^3\), \
(x^4\), and so on—all the way up to \(x^{10}\)). We’ll make use of Scikit Learn’s handy PolynomialFeatures transformer, which is implemented in the preprocessing module.
We’ll create a function that wraps the code to generate the pipeline, so that we can reuse that as well.
from sklearn.preprocessing import PolynomialFeatures
from sklearn.pipeline import Pipeline
def make_pipeline(degree=1):
"""Construct a Scikit Learn Pipeline with polynomial features and linear regression """
polynomial_features = PolynomialFeatures(degree=degree, include_bias=False)
pipeline = Pipeline([
("polynomial_features", polynomial_features),
("linear_regression", LinearRegression())
return pipeline
Now we can initialize a pipeline with degree=10, and fit it to our toy data:
degree = 10
pipeline = make_pipeline(degree)
pipeline.fit(x[:, None], y)
reg_line = pipeline.predict(x_range[:, None])
fig, ax = plt.subplots()
ax.scatter(x, y)
ax.plot(x_range, reg_line)
mse = mean_squared_error(y, pipeline.predict(x[:, None]))
ax.set_title(f"Mean squared error: {mse:.2f}");
At first blush, this model seems to fit the data much better than the first model, in the sense that it reduces the mean squared error considerably relative to the simpler linear model (our MSE went
down from 0.6 to 0.11). But, much as it seemed clear that the previous model was underfitting, it should now be intuitively obvious that the 10th-degree polynomial model is overfitting. The line of
best fit plotted above has to bend in some fairly unnatural ways to capture individual data points. While this helps reduce the error for these particular data, it’s hard to imagine that the same
line would still be very close to the data if we sampled from the same distribution a second or third time.
We can test this intuition by doing exactly that: we sample some more data from the same process and see how well our fitted model predicts the new scores.
test_x, test_y = make_xy(30)
x_range = np.linspace(test_x.min(), test_x.max(), 100)
reg_line = pipeline.predict(x_range[:, None])
fig, ax = plt.subplots()
ax.scatter(test_x, test_y)
ax.plot(x_range, reg_line)
mse = mean_squared_error(y, pipeline.predict(test_x[:, None]))
ax.set_title(f"Mean squared error: {mse:.2f}");
That’s… not good. When we apply our overfitted model to new data, the mean squared error skyrockets. The model performs significantly worse than even our underfitting linear model did.
19.1.1.1. Exercise¶
Apply the linear model to new data in the same manner. Does the linear model’s error also increase when applied to new data? Is this increase smaller or larger than what we observe for our 10th-order
polynomial model?
Of course, since we wrote the data-generating process ourselves, and hence know the ground truth, we may as well go ahead and fit the data with the true functional form, which in this case is a
polynomial with degree 2:
degree = 2
pipeline = make_pipeline(degree)
pipeline.fit(x[:, None], y)
x_range = np.linspace(x.min(), x.max(), 100)
reg_line = pipeline.predict(x_range[:, None])
fig, ax = plt.subplots()
ax.scatter(x, y)
ax.plot(x_range, reg_line)
mse = mean_squared_error(y, pipeline.predict(x[:, None]))
ax.set_title(f"Mean squared error: {mse:.2f}");
There, that looks much better. Unfortunately, in the real world, we don’t usually know the ground truth. If we did, we wouldn’t have to fit a model in the first place! So we’re forced to navigate
between the two extremes of overfitting and underfitting in some other way. Finding this delicate balance is one of the central problems of machine learning -— perhaps the central problem. For any
given dataset, a more flexible model will be able to capture more nuanced, subtle patterns in the data. The cost of flexibility, however, is that such a model is also more likely to fit patterns in
the data that are only there because of noise, and hence won’t generalize to new samples. Conversely, a less flexible model is only capable of capturing simple patterns in the data. This means it
will avoid chasing down rabbit holes full of spurious patterns, but it does so at the cost of missing out on a lot of real patterns too.
One way to think about this is that, as a data scientist, the choice you face is seldom between good models and bad ones. but rather, between lazy and energetic ones (later on, we’ll also see that
there are many different ways to be lazy or energetic). The simple linear model we started with is relatively lazy: it has only one degree of freedom to play with.
The 10th-degree polynomial, by contrast, is hyperactive and sees patterns everywhere, and if it has to go very far out of its way to fit a data point that’s giving it particular trouble, it’ll
happily do that.
Getting it right in any given situation requires you to strike a balance between these two extremes. Unfortunately, the precise point of optimality varies on a case-by-case basis. Later on, we’ll
connect the ideas of overfitting vs. underfitting (or, relatedly, flexibility vs. stability) to another key concept -— the bias-variance tradeoff. But first, let’s turn our attention next to some of
the core methods machine learners use to diagnose and prevent overfitting.
19.1.2. Additional resources¶
In case you are wondering whether overfitting happens in real research you should read the paper “I tried a bunch of things: The dangers of unexpected overfitting in classification of brain data” by
Mahan Hosseini and colleagues [Hosseini et al., 2020]. It nicely demonstrates how easy it is to fall into the trap of overfitting when doing data analysis on large complex datasets, and offers some
rigorous approaches to help mitigate this risk. | {"url":"https://neuroimaging-data-science.org/content/007-ml/003-overfitting.html","timestamp":"2024-11-09T04:13:28Z","content_type":"text/html","content_length":"53606","record_id":"<urn:uuid:ab9899b4-cd5b-4d92-8c8a-4a04b0a4d5ec>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00252.warc.gz"} |
determinant triangular matrix calculator
This does not affect the value of a determinant but makes calculations simpler. This Calculator will Factorize a Square Matrix into the form A=LU where L is a lower triangular matrix, and U is an
upper triangular matrix. Dans la vie de tous les jours, certaines professions (ingénieurs, infographistes) les utilisent tout aussi fréquemment .Si vous savez déjà calculer le déterminant d'une
matrice 2 x 2, ce sera facile, il vous suffira d'additionner, de soustraire et de multiplier. You may ask, what's so interesting about this row echelon (and triangular) matrices, that all other has
to be reduced to then? Theory. Matrix Determinant Calculator Matrix Determinant Calculator Calculate × Go back to. Matrices, just like that. Set the matrix (must be square). Algorithms to reduce an n
by n matrix to upper (or lower) triangular form by Gaussian elimination generally have complexity of O(n^3) (where ^ represents "to power of"). Syntax: numpy.linalg.det(array) Example 1: Calculating
Determinant of a 2X2 Numpy matrix using numpy.linalg.det() function 1 2 3 = R1 1 2 3 1 2 3 4 5 6 - 4 * R1 0 -3 -6 = R2 -3 -6 7 8 8 - 7 * R1 0 -6 -13 - 2 * R2 -1 There was no swapping nor scaling, so
the determinant is the determinant of the last matrix, (1)(−3)(−1)=3. A special number that can be calculated from a square matrix is known as the Determinant of a square matrix. This online
calculator allows you to perform calculations as two matrices (to find the sum of matrices, calculate their multiply and any other operations) and a separate matrix - You can input only integer
numbers or fractions in this online calculator. Reduce this matrix to row echelon form using elementary row operations so that all the elements below diagonal are zero. Lower Triangular matrix
transformation method is preferred over minor or cofactor of matrix method while finding determinant of the matrix's size over 3x3. Use EROs to reduce 1 2 3 4 5 6 7 8 8 to upper triangular
form, and hence calculate its determinant. online matrix LU decomposition calculator, find the upper and lower triangular matrix by factorization A matrix that is similar to a triangular matrix is
referred to as triangularizable. Usually best to use a Matrix Calculator for those! Find more Mathematics widgets in Wolfram|Alpha. To find the determinant … The determinant of a block-diagonal
matrix is the product of the determinants of the blocks. Reduce this matrix to row echelon form using elementary row operations so that all the elements below diagonal are zero. Enter Matrix A: Show
digits Cells +-Reset Matrices: Use cofactor ... a n n = ∏ i = 1 n a i i , if the matrix A is triangular a i j = 0 et i ≠ j , the determinant is equal to the product of the diagonal of the matrix.
Provided the underlying scalars form a field (more generally, a ... properties 13 and 14 can be used to transform any matrix into a triangular matrix, whose determinant is given by property 7; this
is essentially the method of Gaussian elimination. Laissez des cellules vides pour entrer dans une matrice non carrées. Le déterminant d’une matrice 3 x 3 peut se calculer de différentes façons. Male
or Female ? Le déterminant d'une matrice non carrée n'est pas défini, il n'existe pas selon la définition du déterminant. The product of these numbers is less than 1. This Calculator will Factorize a
Square Matrix into the form A=LU where L is a lower triangular matrix, and U is an upper triangular matrix. Determinant is calculated by reducing a matrix to row echelon form and multiplying its main
diagonal elements. Le déterminant est calculé en réduisant la matrice en forme échelonnée et en multipliant les éléments de sa diagonale principale. More in-depth information read at these rules. To
calculate a determinant you need to do the following steps. ), with steps shown. Entering data into the matrix determinant calculator. You need to enable it. Lisez les instructions. The conversion is
performed by subtracting one row from another multiplied by a scalar coefficient. Instead of calculating a factorial one digit at a time, use this calculator to calculate the factorial n! By the way,
the determinant of a triangular matrix is calculated by simply multiplying all it's diagonal elements. Addition and subtraction of matrices. The value of the determinant is correct if, after the
transformations the lower triangular matrix is zero, and the elements of the main diagonal are all equal to 1. Quelle est la formule de calcul de déterminant d'une matrice d'ordre n ? En calcul
infinitésimal, en algèbre linéaire et en géométrie avancée, on se sert fréquemment des déterminants des matrices. Consider the matrix below. Instead of calculating a factorial one digit at a time,
use this calculator to calculate the factorial n! Linear Algebra- Finding the Determinant of a Triangular Matrix This property means that if we can manipulate a matrix into upper- or lower-triangular
form, we can easily find its determinant, even for a large matrix. On peut aussi développer selon une ligne ou une colonne (voir plus bas). Si c’est une matrice diagonale ou triangulaire, on utilise
ce que l’on vient de voir. Online calculator to calculate 5x5 determinant with the Laplace expansion theorem and gaussian algorithm. The determinant of a triangular matrix is obtained by finding the
product of the diagonal elements! Calculateur du déterminant d'une matrice carrée (n×n) de dimension 2, 3, 4 ou plus ... L'outil permet de calculer le déterminant d'une matrice de dimension 2, 3, 4
ou plus. Although the determinant of the matrix is close to zero, A is actually not ill conditioned. How does this determinant calculator work? Please note that the tool allows using both positive
and negative numbers, with or without decimals and even fractions written using "/" … It's equal to the product of all diagonal elements, which are 0.009, 0.09, 0.9 and 9. This whole class, where you
have 0's below the main diagonal, these are called upper triangular matrices. The matrix A is converted into Lower triangular matrix, L by elementary row operation or reduction and then product of
main diagonal elements is called determinant of the matrix A. The Sarrus Rule is used for computing only 3x3 matrix determinant. REAL FUNCTION FindDet(matrix, n) The determinant is extremely small.
We begin with a general 2×2 linear system and solve for y. Methods explanation Determinant of a 3x3 matrix according to the Sarrus Rule. Additional features of the matrix determinant calculator. This
determinant calculator can assist you when calculating the matrix determinant having between 2 and 4 rows and columns. I designed this web site and wrote all the mathematical theory, online
exercises, formulas and calculators. Attention, notre petit serveur risque de ne pas survivre avec une matrice de dimension 100 (LOL), mais il est très efficace avec des matrices d'ordre inférieur à
10. More in-depth information read at these rules. … This web site owner is mathematician Dovzhyk Mykhailo. Each row must begin with a new line. Il n'existe pas de formule autre que l'explication
ci-dessus pour le cas général d'une matrice d'ordre n. Comment calculer le déterminant d'une matrice 1x1 ? Des questions? Calculate the first row cofactor expansion. Methods explanation Laplace
Expansion Theorem. There are alternative approaches for computing determinate, such as evaluating the set of eigenvalues (the determinant of a square matrix is equal to the product of its
eigenvalues). Get the free "5x5 Matrix calculator" widget for your website, blog, Wordpress, Blogger, or iGoogle. The calculator will find the determinant of the matrix (2x2, 3x3, etc. Not The Only
Way. In this video I will show you a short and effective way of finding the determinant without using cofactors. Determinant of 2×2 matrix. Let me write that down. Likewise, the determinant of this
lower-triangular matrix is acf. Determinant of a matrix. Therefore, A is not close to being singular. ), with steps shown. The elements in the first row, act as scalar multipliers. Find the factors
of the numbers 5 = 1x5 6 = 1x6, 2x3 , 3x2 5 has only 1 and 5 as its factors. Step 1: To Begin, select the number of rows and columns in your Matrix, and press the "Create Matrix" button.
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Seth Pettie, Dingyu Wang, Longhui Yin
Cardinality estimation is perhaps the simplest non-trivial statistical problem that can be solved via sketching. Industrially-deployed sketches like HyperLogLog, MinHash, and PCSA are mergeable,
which means that large data sets can be sketched in a distributed environment, and then merged into a single sketch of the whole data set. In the last decade a variety of sketches have been developed
that are non-mergeable, but attractive for other reasons. They are simpler, their cardinality estimates are strictly unbiased, and they have substantially lower variance. We evaluate sketching
schemes on a reasonably level playing field, in terms of their memory-variance product (MVP). E.g., a sketch that occupies $5m$ bits and whose relative variance is $2/m$ (standard error $\sqrt{2/m}$)
has an MVP of $10$. Our contributions are as follows. Cohen and Ting independently discovered what we call the Martingale transform for converting a mergeable sketch into a non-mergeable sketch. We
present a simpler way to analyze the limiting MVP of Martingale-type sketches. We prove that the \Martingale{} transform is optimal in the non-mergeable world, and that \Martingale{} \fishmonger{} in
particular is optimal among linearizable sketches, with an MVP of $H_0/2 \approx 1.63$. E.g., this is circumstantial evidence that to achieve 1\% standard error, we cannot do better than a 2 kilobyte
sketch. \Martingale{} \fishmonger{} is neither simple nor practical. We develop a new mergeable sketch called \Curtain{} that strikes a nice balance between simplicity and efficiency, and prove that
\Martingale{} \Curtain{} has limiting $\MVP\approx 2.31$. It can be updated with $O(1)$ memory accesses and it has lower empirical variance than \Martingale{} \LogLog, a practical non-mergeable
version of HyperLogLog. | {"url":"https://www.thejournal.club/c/paper/326190/","timestamp":"2024-11-10T14:19:35Z","content_type":"text/html","content_length":"33600","record_id":"<urn:uuid:342f9332-be7c-4903-900d-35c73ae4a344>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00331.warc.gz"} |
ball mill rpm
The size distribution of granules in the drum during ball milling with varying rotation speed: a) 47 rpm; b) 95 rpm; c) 143 rpm; d) 190 rpm and e) 238 rpm. Figure 4. Typical perspective from the
base of the drum side at the beginning and the end of the ball milling process.
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Ball mills are among the most variable and effective tools when it comes to size reduction of hard, brittle or fibrous materials. The variety of grinding modes, usable volumes and available
grinding tool materials make ball mills the perfect match for a vast range of applications. Cooling, heating cryogenic grinding.
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The ideal Ball Mill for standard applications. Max. speed 650 rpm. Up to 10 mm feed size and µm final fineness. 1 grinding station for jars from 12 ml up to 500 ml. Jars of 12 80 ml can be
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Critical speed (in rpm) = /sqrt(D d) with D the diameter of the mill in meters and d the diameter of the largest grinding ball you will use for experiment (also expressed in meters)
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Section snippets Experimental. Mechanical milling was performed by using a planetary ball mill (PBM) under various experimental conditions. Copper powder (% purity, median particle size x 50 = 45
μm, Aldrich chemistry) was used as shown in Fig. 1. Mechanical alloying (MA) was carried out at two types of rotation speeds: low (10, 50, and 100 rpm) and high (300, 500, and 700 rpm).
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Hexagonal boron nitride (hBN) was ballmilled at various rotation speeds (150600 rpm) using a planetary ballmill. Ballmilling disrupted the layered structure of the hBN, resulting in significant
increases of surface area. Ballmilling at 400 rpm gave the highest surface area of 412 m 2 g −1 while higher rotation speeds decreased the ...
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The Planetary Ball Mill PM 200 is a powerful benchtop model with 2 grinding stations for grinding jars with a nominal volume of 12 ml to 125 ml. ... Max. speed 650 rpm ; Up to 10 mm feed size and
µm final fineness ; 2 grinding stations for jars from 12 ml up to 125 ml, jars of 12 and 25 ml can be stacked (two jars each) ...
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950 rpm. Blaine is the important characteristic of ball mill which is influenced by the mill speed and separator speed. Ball mill is grinding equipment which is used to reduce the size of clinker
into cement. It uses grinding media in the form of balls. Clinker coming from the silo is sent into hopper and mill for impact action. Clinker is ...
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Wiki says "A ball mill is a type of grinder used to grind materials into extremely fine powder for use in paints, pyrotechnics, and ceramics." Many instructables refer to United Nuclear Ball
Mills. Their small ball mill cost between 70 and 80 dollars. For no more than 30 and in 5 minute you can build a ball mill of appreciable performance.
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We routinely recommend two ways: coffee milling and ball milling. Both ways have advantages. ... My homemade mill uses a vibratory case tumbler container on it's side (builtin lift bars) with a
30 rpm geared motor and ¾ inch brass balls. In four hours I get 2 pounds of awesome powder using your charcoal and potassium nitrate.
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RPM is a group indoor cycling workout where you control the intensity. It's fun, low impact and you can burn up to 500 calories a session**. With great music pumping and the group cycling as one,
your instructor takes you on a journey of hill climbs, sprints and flat riding. In an RPM workout you repeatedly rotate the pedals to reach your ...
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Mechanochemical technique aims to strike a balance between defect formation via ball milling and size adjustment of a solid grain to nanoscale (<1000 nm) (Ullah et al., 2014).During the process,
a highenergy mill is employed and a specific powder charge is placed along with a milling medium (Lin et al., 2017).The kinetic energy generated during the motion of moving balls is applied to
the ...
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In planetary ball mills, speeds can typically range from around 100 rpm to 1000 rpm. Higher milling speeds generally result in more impact and shear forces between the balls and the powder ...
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The effect of ball mill on the morphological and structural features of cellulose has been described by Okajima and coworkers. 20 They treated microcrystalline cellulose derived from cotton
linters in a planetary ball mill at 200 rpm for 48 hours in dry and wet conditions with three solvents (water, toluene, 1butanol). They observed that ...
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How to use this calculator: Choose a type of operation (drilling, reaming, boring, counterboring, face milling, slab milling/side milling, end milling, or turning), select your stock material,
choose a material for the tool (highspeed steel or carbide), input the quantity of teeth for the tool and the diameter of the workpiece/cutter. Hit the ...
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A planetary ball mill is known to install pots on a disk, and the pots and the disk are simultaneously and separately rotated at a high speed. ... %, and % at rotation speeds of 100 rpm, 200 rpm,
and 300 rpm, respectively, and the corresponding grain growth activation energies were,, and kJ/mol.
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A ball nose end mill has a rounded tip or "nose" that is ideal for creating curved surfaces and 3D shapes. In contrast, a flatend mill has a flatcutting head that is suitable for milling flat or
shallow surfaces. Ball nose end mills are commonly used for finishing work, where a smooth surface finish is required, and for machining complex ...
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Significant data analysis in ball mill can be done when the sensor data acquired rate and the rpm of the mill are synchronised to nullify the effect of impact dead zone. Acknowledgements The
authors wish to thank CEERI, Pilani and KCC, Khetri in providing the opportunity to avail the facility required for the experimental analysis and ...
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Planetary ball milling is a possible way to improve the technological properties of wheat flour by thermal and mechanical modifications. In the present study, roller milled common wheat flours
(Triticum aestivum L.) were additionally modified in a ball mill by varying mill parameters such as rotation speed and grinding a result of ball milling, the flours experienced temperatures of up
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The mill was rotated at 50, 62, 75 and 90% of the critical speed. Six lifter bars of rectangular crosssection were used at equal spacing. The overall motion of the balls at the end of five
revolutions is shown in Figure 4. As can be seen from the figure, the overall motion of the balls changes with the mill speed inasmuch as the shoulder ...
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For example, ball mills use large media, normally 1/2" or larger, and run at a low (1050) rpm. The other mills, such as sand, bead, and horizontal, use smaller media from to 2mm, but run at a
very high rpm (roughly ). High speed dispersers with no media run even faster rpm ().
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Planetary ballmill: Rotational Speed600 rpm Milling time5 h Balls5 Balls materialZirconium Ball size8 mm • Particle SizeWNS271 nm, LNS855 nm, HNS343 nm • Increased water absorption activity and a
decreased oil absorption. Ahmad et al. [30] Cassava starchferulic acid complex: Rolling type ballmill
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Surface phonons, elastic response, and conformal invariance in twisted kagome lattices | PNAS
Surface phonons, elastic response, and conformal invariance in twisted kagome lattices
Edited by Nigel Goldenfeld, University of Illinois at Urbana-Champaign, Urbana, IL, and approved May 29, 2012 (received for review December 7, 2011)
June 25, 2012
109 (31) 12369-12374
Model lattices consisting of balls connected by central-force springs provide much of our understanding of mechanical response and phonon structure of real materials. Their stability depends
critically on their coordination number z. d-dimensional lattices with z = 2d are at the threshold of mechanical stability and are isostatic. Lattices with z < 2d exhibit zero-frequency “floppy”
modes that provide avenues for lattice collapse. The physics of systems as diverse as architectural structures, network glasses, randomly packed spheres, and biopolymer networks is strongly
influenced by a nearby isostatic lattice. We explore elasticity and phonons of a special class of two-dimensional isostatic lattices constructed by distorting the kagome lattice. We show that the
phonon structure of these lattices, characterized by vanishing bulk moduli and thus negative Poisson ratios (equivalently, auxetic elasticity), depends sensitively on boundary conditions and on the
nature of the kagome distortions. We construct lattices that under free boundary conditions exhibit surface floppy modes only or a combination of both surface and bulk floppy modes; and we show that
bulk floppy modes present under free boundary conditions are also present under periodic boundary conditions but that surface modes are not. In the long-wavelength limit, the elastic theory of all
these lattices is a conformally invariant field theory with holographic properties (characteristics of the bulk are encoded on the sample boundary), and the surface waves are Rayleigh waves. We
discuss our results in relation to recent work on jammed systems. Our results highlight the importance of network architecture in determining floppy-mode structure.
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Networks of balls and springs or frames of nodes connected by compressible struts provide realistic models for physical systems from bridges to condensed solids. Their elastic properties depend on
their coordination number
—the average number of nodes each node is connected to. If
is small enough, the networks have deformation modes of zero energy—they are floppy (
). As
is increased, a critical value,
, is reached at which springs provide just enough constraints that the system has no zero-energy floppy modes (
) (or mechanisms, ref.
, in the engineering literature), and the system is isostatic. For
, networks with appropriate geometry (see below) are rigid in the sense that they have no zero modes other than those associated with trivial rigid translations and rotations. If a network with
is homogeneously distributed in space, it can be viewed as an elastic solid whose long-wavelength mechanical properties are described by a continuum elastic energy with nonvanishing elastic moduli (
). The phenomenon of rigidity percolation (
) whereby a sample spanning rigid cluster develops upon the addition of springs is one version of this floppy-to-rigid transition. The coordination numbers of whole classes of systems, including
engineering structures (
) (bridges and buildings), randomly packed spheres near jamming (
), network glasses (
), cristobalites (
), zeolites (
), and biopolymer networks (
) are close enough to
that their elasticity and mode structure is strongly influenced by those of the isostatic lattice.
Though the isostatic point always separates rigid from floppy behavior, the properties of isostatic lattices are not universal; rather they depend on lattice architecture. Here we explore the unusual
properties of a particular class of periodic isostatic lattices derived from the two-dimensional kagome lattice by rigidly rotating triangles through an angle α without changing bond lengths as shown
Fig. 1
. The bulk modulus
of these lattices is rigorously zero for all α ≠ 0. As a result, their Poisson ratio acquires its limit value of -1; when stretched in one direction, they expand by an equal amount in the orthogonal
direction: They are maximally auxetic (
). These modes represent collapse pathways (
) of the kagome lattice. Modes of isostatic systems are generally very sensitive to boundary conditions (
), but the degree of sensitivity depends on the details of lattice structure. For reasons we will discuss more fully below, modes of the square lattice, which is isostatic, are in fact insensitive to
changes from free boundary conditions (FBCs) to periodic boundary conditions (PBCs), whereas those of the undistorted kagome lattice are only mildly so. The modes of both, however, change
significantly when rigid boundary conditions (RBCs) are applied. We show here that, in all families of the twisted kagome lattice, modes depend sensitively on whether FBCs, PCBs, or RBCs are applied:
Finite lattices with free boundaries have floppy surface modes that are not present in their periodic or rigid spectrum or in that of finite undistorted kagome lattices. In the long-wavelength limit,
the surface floppy modes, which are present in any 2
material with
= 0, reduce to surface Rayleigh waves (
) described by a conformally invariant energy whose analytic eigenfunctions are fully determined by boundary conditions. At shorter wavelengths, the surface waves become sensitive to lattice
structure and remain confined to within a distance of the surface that diverges as the undistorted kagome lattice is approached. In the simplest twisted kagome lattice, all floppy modes are surface
modes, but in more complicated lattices, including ones with uniaxial symmetry that we construct, there are both surface and bulk floppy modes.
(A) Section of a kagome lattice with N[x] = N[y] = 4 and N[c] = N[x]N[y] three-site unit cells. Nearest-neighbor bonds, occupied by harmonic springs, are of length a. The rotated row (second row from
the top) represents a floppy mode. Next-nearest-neighbor bonds are shown as dotted lines in the lower left hexagon. The vectors e[1], e[2], and e[3] indicate symmetry directions of the lattice. The
numbers in the triangles indicate those that twist together under PBCs in zero modes along the three symmetry direction. Note that there are only four of these modes. (B) Section of a square lattice
depicting a floppy mode in which all sites along a line are displaced uniformly. (C) Twisted kagome lattice, with lattice constant a[L] = 2a cos α, derived from the undistorted lattice by rigidly
rotating triangles through an angle α. A unit cell, bounded by dashed lines, is shown in violet. Arrows depict site displacements for the zone-center (i.e., zero wavenumber) ϕ mode which has zero
(nonzero) frequency under free (periodic) boundary conditions. Sites 1, 2, and 3 undergo no collective rotation about their center of mass, whereas sites 1, 2^∗, and 3^∗ do. (D) Superposed snapshots
of the twisted lattice showing decreasing areas with increasing α.
Arguments due to Maxwell (
) provide a criterion for network stability: Networks in
dimensions consisting of
nodes, each connected with central-force springs to an average of
neighbors, have
zero-energy modes when
< 2
(in the absence of redundant bonds—see below). Of these, a number,
, which depends on boundary conditions, are trivial rigid translations and rotations, and the remainder are floppy modes of internal structural rearrangement. Under FBCs and PBCs,
+ 1)/2 and
, respectively. With increasing
, mechanical stability is reached at the isostatic point at which
. The Maxwell argument is a global one; it does not provide information about the nature of the floppy modes and does not distinguish between bulk or surface modes.
Kagome Zero Modes and Elasticity
The kagome lattice of central-force springs shown in
Fig. 1A
is one of many locally isostatic lattices, including the familiar square lattice in two dimensions (
Fig. 1B
) and the cubic and pyrochlore lattices in three dimensions, with exactly
= 2
nearest-neighbor (NN) bonds connected to each site not at a boundary. Under PBCs, there are no boundaries, and every site has exactly 2
neighbors. Finite,
-site sections of these lattices have surface sites with fewer than 2
neighbors and of order
zero modes. The free kagome lattice with
unit cells along its sides (
Fig. 1A
) has
= 3
= 6
- 2(
) + 1 bonds, and
= 2(
) - 1 zero modes, all but three of which are floppy modes. These modes, depicted in
Fig. 1A
, consist of coordinated counterrotations of pairs of triangles along the symmetry axes
, and
of the lattice. There are
modes associated with lines parallel to
associated with lines parallel to
, and
- 1 modes associated with lines parallel to
In spite of the large number of floppy modes in the kagome lattice, its longitudinal and shear Lamé coefficients, λ and μ, and its bulk modulus
= λ + μ are nonzero and proportional to the NN spring constant
. The zero modes of this lattice can be used to generate an infinite number of distorted lattices with unstretched springs and thus zero energy (
). We consider only periodic lattices, the simplest of which are the twisted kagome lattices obtained by rotating triangles of the kagome unit cell through an angle α as shown in
Fig. 1 C
). These lattices have
rather than
symmetry and, like the undistorted kagome lattice, three sites per unit cell. As
Fig. 1D
shows, the lattice constant of these lattices is
= 2
cos α, and their area
decreases as cos
α as α increases. The maximum value that α can achieve without bond crossings is π/3, so that the maximum relative area change is
= 1/4. Because all springs maintain their rest length, there is no energy cost for changing α and, as a result,
is zero for every α ≠ 0, whereas the shear modulus
remains nonzero and unchanged. Thus, the Poisson ratio σ = (
- μ)/(
+ μ) attains its smallest possible value of -1. For any α ≠ 0, the addition of next-nearest-neighbor (NNN) springs, with spring constant
(or of bending forces between springs) stabilizes zero-frequency modes and increases
and σ. Nevertheless, for sufficiently small
, σ remains negative.
Fig. 2
shows the region in the
- α plane with negative σ.
Phase diagram in the α - k^′ plane showing region with negative Poisson ratio σ.
Kagome Phonon Spectrum
We now turn to the linearized phonon spectrum of the kagome and twisted kagome lattices subjected to PBCs. These conditions require displacements at opposite ends of the sample to be identical and
thus prohibit distortions of the shape and size of the unit cell and rotations but not uniform translations, leaving two rather than three trivial zero modes. The spectrum (
) of the three lowest frequency modes along symmetry directions of the undistorted kagome lattice with and without NNN springs is shown in
Fig. 3A
. When
= 0, there is a floppy mode for each wavenumber
≠ 0 running along the entire length of the three symmetry-equivalent straight lines running from
to Γ to
in the Brillouin zone (see
Fig. 3
). When
, there are exactly
- 1 wavenumbers with
≠ 0 along each of these lines for a total of 3(
- 1) floppy modes. In addition, there are three zero modes at
= 0 corresponding to two rigid translations and one floppy mode that changes unit cell area at second but not first order in displacements, yielding a total of 3
zero modes rather than the 4
- 1 modes expected from the Maxwell count under FBCs. This discrepancy is our first indication of the importance of boundary conditions. The addition of NNN springs endows the floppy modes at
= 0 with a characteristic frequency
and causes them to hybridize with the acoustic phonon modes (
Fig. 3A
) (
). The result is an isotropic phonon spectrum up to wavenumber
and gaps at Γ and
of order ω
. Remarkably, at nonzero α and
= 0, the mode structure is almost identical to that at α = 0 and
> 0 with characteristic frequency
and length
∼ 1/ω
. In other words, twisting the kagome lattice through an angle α has essentially the same effect on the spectrum as adding NNN springs with spring constant | sin α|
. Thus, under PBCs, the twisted kagome lattice has no zero modes other than the trivial ones: It is “collectively” jammed in the language of refs.
, but because it is not rigid with respect to changing the unit cell size, it is not strictly jammed.
(A) Phonon spectrum for the undistorted kagome lattice. Dashed lines depict frequencies at k^′ = 0 and full lines at k^′ > 0. The inset shows the Brillouin zone with symmetry points Γ, M, and K. Note
the line of zero modes along ΓM when k^′ = 0, all of which develop nonzero frequencies for wavenumber q > 0 when k^′ > 0 reaching on a plateau beginning at defining a length scale l^∗ = 1/q^∗. (B)
Phonon spectrum for α > 0 and k^′ = 0. The plateau along ΓM defines and its onset at q[α] ∼ ω[α] defines a length l[α] ∼ 1/| sin α|.
Mode Counting and States of Self-Stress
To understand the origin of the differences in the zero-mode count for different boundary conditions, we turn to an elegant formulation (
) of the Maxwell rule that takes into account the existence of redundant bonds (i.e., bonds whose removal does not increase the number of floppy modes; ref.
) and states in which springs can be under states of self-stress (
). Consider a ring network in two dimensions shown in
Fig. 4
= 4 nodes and
= 4 springs with three springs of length
and one spring of length
. The Maxwell count yields
= 4 = 3 + 1 zero modes: two rigid translations, one rigid rotation, and one internal floppy mode—all of which are “finite-amplitude” modes with zero energy even for finite-amplitude displacements.
= 3
, the Maxwell rule breaks down. In the zero-energy configuration, the long spring and the three short ones are colinear, and a prestressed state in which the
spring is under compression and the three
springs are under tension (or vice versa) but the total force on each node remains zero becomes possible. This is called a state of self-stress. The system still has three finite-amplitude zero modes
corresponding to arbitrary rigid translations and rotations, but the finite-amplitude floppy mode has disappeared. In the absence of prestress, it is replaced by two “infinitesimal” floppy modes of
displacements of the two internal nodes perpendicular of the now linear network. In the presence of prestress, these two modes have a frequency proportional to the square root of the tension in the
springs. Thus, the system now has one state of self-stress and one extra zero mode in the absence of prestress, implying
= 2
, where
is the number of states of self-stress.
(A) Ring network with b > 3a showing internal floppy mode. (B) Ring-network with b = 3a showing one of the two infinitesimal modes.
This simple count is more generally valid, as can be shown with the aid of the equilibrium and compatibility matrices (
), denoted, respectively, as
relates the vector
spring tensions to the vector
forces at nodes via
, and
relates the vector
node displacements to the vector
spring stretches via
. The dynamical matrix determining the phonon spectrum is
. Vectors
in the null space of
, (
= 0), describe states of self-stress, whereas vectors
in the null space of
represent displacements with no stretch
—i.e., modes of zero energy. Thus the null-space dimensions of
are, respectively,
. The rank-nullity theorem of linear algebra (
) states that the rank
of a matrix plus the dimension of its null space equals its column number. Because the rank of a matrix and its transpose are equal, the
matrices, respectively, yield the relations
, implying
. Under PBCs, locally isostatic lattices have
= 2
exactly, and the Maxwell rule yields
= 0: There should be no zero modes at all. But we have just seen that both the square and undistorted kagome lattices under PBCs have of order
zero modes as calculated from the dynamical matrix, which, because it is derived from a harmonic theory, does not distinguish between infinitesimal and finite-amplitude zero modes. Thus, in order for
there to be zero modes, there must be states of self-stress, in fact, one state of self-stress for each zero mode.
In the square lattice under FBCs,
= 2
, there are no states of self-stress, and
zero modes depicted in
Fig. 1B
. Under PBCs, the dimension of the null space of
, and there are also
zero modes that are identical to those under FBCs. We have already seen that there are
= 2(
) - 1 zero modes in the free undistorted kagome lattice. Direct evaluations (
) (see
SI Text
) of the dimension of the null spaces of
for the undistorted kagome lattice with PBCs yields
= 3
. The zero modes under PBCs are identical to those under FBCs except that the 2
- 1 modes associated with lines parallel to
under FBCs get reduced to
modes because of the identification of apposite sides of the lattice required by the PBCs, as shown in
Fig. 1A
. Thus the modes of both the square and kagome lattices do not depend strongly on whether FBCs or PBCs are applied. Under RBCs, however, the floppy modes of both disappear. The situation for the
twisted kagome lattice is different. There are still 2(
) - 1 zero modes under FBCs, but there are only two states of self-stress under PBCs and thus only
= 2 zero modes, as a direct evaluation of the null spaces of
verifies (
SI Text
), in agreement with the results obtained via direct evaluation of the eigenvalues of the dynamical matrix (
). All of the floppy modes under FBCs have disappeared.
Effective Theory and Edge Modes
An effective long-wavelength energy
for the low-energy acoustic phonons and nearly floppy distortions provides insight into the nature of the modes of the twisted kagome lattice. The variables in this theory are the vector displacement
) of nodes at undistorted positions
and the scalar field
) describing nearly floppy distortions within a unit cell. The detailed form of
depends on which three lattice sites are assigned to a unit cell.
Fig. 1C
depicts the lattice distortion
for the nearly floppy mode at Γ (with energy proportional to | sin α|
) along with a particular representation of a unit cell, consisting of a central asymmetric hexagon and two equilateral triangles, with eight sites on its boundary. If sites 1, 2, and 3 are assigned
to the unit cell, then the distortion
involves no rotations of these sites relative to their center of mass, and the harmonic limit of
depends only on the symmetrized and linearized strain
= (∂
+ ∂
)/2 and on
is the symmetric-traceless stain tensor,
, and
. The last term in which Γ
is a third-rank tensor, whose only nonvanishing components are Γ
= -Γ
= -Γ
= -Γ
= 1, is invariant under operations of the group
but not under arbitrary rotations. The
term is the only one that reflects the
(rather than
) symmetry of the lattice. There are several comments to make about this energy. The gauge-like coupling in which the isotropic strain
appears only in the combination (
+ ξ
guarantees that the bulk modulus vanishes:
will simply relax to -ξ
to reduce to zero the energy of any state with nonvanishing
. The coefficient
can be calculated directly from the observation that, under
alone, the length of every spring changes by
, and this length change is reversed by a homogenous volume change
= δ
= -2δ
. In the α → 0 limit,
→ 0, and the energy reduces to that of an isotropic solid with bulk modulus
if the
terms, which are higher order in gradients, are ignored. The
term gives rise to a term, singular in gradients of
, when
is integrated out that is responsible for the deviations of the finite-wavenumber elastic energy from isotropy. At small α, the length scale
appears in several places in this energy: in the length ξ and in the ratios
, and
. At length scales much larger than
, the
terms can be ignored, and
relaxes to -ξ
, leaving only the shear elastic energy of an elastic solid proportional
. At length scales shorter than
deviates from -ξ
and contributes significantly to the form of the energy spectrum. If 1, 2
, and 3
Fig. 1D
are assigned to the unit cell, then
involves rotations relative to the lattice axes, and the energy develops a Cosserat-like form (
) that is a function of
(∇ ×
/2 rather than
The modes of our elastic energy in the long-wavelength limit (
≪ 1) are simply those of an elastic medium with
= 0. In this limit, there are transverse and longitudinal bulk sound modes with equal sound velocities
, where
is the particle mass at each node and ρ is the mass density. In addition there are Rayleigh surface waves (
) in which there is a single decay length (rather than the two at
> 0), and displacements are proportional to
] with
for a semiinfinite sample in the upper half-plane so that the penetration depth into the interior is 1/
. These waves have zero frequency in two dimensions when
= 0, and they do not appear in the spectrum with PBCs. Thus this simple continuum limit provides us with an explanation for the difference between the spectrum of the free and periodic twisted
kagome lattices. Under FBCs, there are zero-frequency surface modes not present under PBCs.
Further insight into how boundary conditions affect spectrum follows from the observation that the continuum elastic theory with
= 0 depends only on
. The metric tensor
) of the distorted lattice is related to the strain
) via the simple relation
) = δ
+ 2
); and
, which is zero for
= δ
, is invariant, and thus remains equal to zero, under conformal transformations that take the metric tensor from its reference form δ
for any continuous function
). The zero modes of the theory thus correspond simply to conformal transformations, which in two dimensions are best represented by the complex position and displacement variables
) =
) +
). All conformal transformations are described by an analytic displacement field
). Because, by Cauchy’s theorem, analytic functions in the interior of a domain are determined entirely by their values on the domain’s boundary (the “holographic” property; ref.
), the zero modes of a given sample are simply those analytic functions that satisfy its boundary conditions. For example, a disc with fixed edges (
= 0) has no zero modes because the only analytic function satisfying this FBC is the trivial one
) = 0; but a disc with free edges (stress and thus strain equal to zero) has one zero mode for each of the analytic functions
) =
for integer
≥0. The boundary conditions
) =
) on a semiinfinite cylinder with axis along
are satisfied by the function
) =
= 2
, where
is an integer. This solution is identical to that for classical Rayleigh waves on the same cylinder. Like the Rayleigh theory, the conformal theory puts no restriction on the value of
(or equivalently
). Both theories break down, however, at
, beyond which the full lattice theory, which yields a complex value of
, is needed.
Fig. 5A
shows an example of a surface wave. At the bottom of this figure,
) is an almost perfect sinusoid. As
decreases toward the surface, the amplitude grows, and in this picture reaches the nonlinear regime by the time the surface at
= 0 is reached.
Fig. 5B
as a function of
obtained both by direct numerical evaluation and by an analytic transfer matrix procedure (
) for different values of α (
SI Text
). The Rayleigh limit
is reached for all α as
→ 0. Interestingly, the Rayleigh limit remains a good approximation up to values of
that increase with increasing α. The inset to
Fig. 5
as a function of η =
and shows that in the limit α → 0 (
→ ∞),
obeys an α-independent scaling law of the form
. The full complex
obeys a similar equation. This type of behavior is familiar in critical phenomena where scaling occurs when correlation lengths become much larger than microscopic lengths. The function
(η) approaches η as η → 0 and asymptotes to 4/3 for η → ∞. Thus for
≪ 1,
and for
. As α increases,
is no longer much larger than one, and deviations from the scaling law result. The situation for surfaces along different directions (e.g., along
= 0 rather than
= 0) is more complicated and is beyond the scope of this paper.
(A) Lattice distortions for a surface wave on a cylinder, showing exponential decay of the surface displacements into the bulk. This figure was constructed by specifying a small sinusoidal modulation
on the bottom boundary and propagating lattice-site positions upward to the free boundary at the top under the constraint of constant lengths and periodic boundary conditions around the cylinder.
Distortions near and at the top boundary, which have become nonlinear, are not described by our linearized treatment. (B) as a function of q[x]a[L] for lattice Rayleigh surface waves for α = π/20, π/
10, 3π/20, π/5, π/4, in order from bottom to top. Smooth curves are the analytic results from a transfer matrix calculation, and dots are from direct numerical calculations. The dashed line is the
continuum Rayleigh limit . Curves at smaller α break away from this curve at smaller values of q[y] than do those at large α. At α = π/4, diverges at q[y]a[L] = π. The inset plots as a function of q[
x]l[α] for different α. The lower curve in the inset (black) is the α-independent scaling function of q[y]l[α] reached in the α → 0 limit. The other curves from top to bottom are for α = π/25, π/12,
π/9, and π/6 (chosen to best present results rather than to match the curves in the main figure). Curves for α < π/15 are essentially indistinguishable from the scaling limit. The curve at α = π/6
stops because q[y] < π/a[L].
Other Lattices and Relation to Jamming
twisted kagome lattice is the simplest of many lattices that can be formed from the kagome and other periodic isostatic lattices.
Fig. 6 A
shows two other examples of isostatic lattices constructed from the kagome lattice. Most intriguing is the lattice with pgg symmetry. Its geometry has uniaxial symmetry, yet its long-wavelength
elastic energy is identical to that of the
twisted kagome lattice (i.e., it is isotropic with a vanishing bulk modulus), and its mode structure near
= 0 is isotropic as shown in
Fig. 6C
. Thus, this system loses long-wavelength zero-frequency bulk modes of the undistorted kagome lattice to surface modes. However, at large wavenumber, lattice anisotropy becomes apparent, and
(infinitesimal) floppy bulk modes appear. Thus in this and related systems, a fraction of the zero modes under FBCs are bulk modes that are visible under PBCs, and a fraction are surface modes that
are not.
(A) Kagome-based lattice with pgg space group symmetry and uniaxial C[2v] point group symmetry. (B) Lattice with p6 space symmetry but global C[6] point-group symmetry. (C) Density plot of the
spectrum of the lowest frequency branch of the pgg uniaxial kagome lattice. The spectrum is absolutely isotropic near the origin point Γ, but it has a zero modes on two symmetry related continuous
curves at large values of wavenumber.
Randomly packed spheres above the jamming transition with average coordination number
= 2
+ Δ
exhibit a characteristic frequency ω
∼ Δ
and length
∼ (Δ
and a transition from a Debye-like (∼ ω
) to a flat density of states at ω ≈ ω
). The square and kagome lattices with randomly added NNN springs have the same properties (
). A general “cutting” argument (
) provides a procedure for perturbing away from the isostatic limit and an explanation for these properties. However, it only applies provided a finite fraction of the of order
floppy modes of a sample with sides of length
cut from an isostatic lattice with PBCs are extended—i.e., have wave functions that extend across the sample rather than remaining localized either in the interior or at the surface the sample.
Clearly the twisted kagome lattice, whose floppy modes are all surface modes, violates this criterion; and indeed, the density of states of the lattice with Δ
= 0 shows Debye behavior crossing over to a flat plateau at ω ≈ ω
. Adding NNN bonds gives rise to a length
, which is approximately the parallel combination,
, of the lengths
arising, respectively, from twisting the lattice and adding NNN springs, and cross-over to the plateau at
. The pgg lattice in
Fig. 6A
, however, has both extended and surface floppy modes, so its cross-over to a flat plateau occurs at ω ≈ ω
rather than at ω
or ω
Connections to Other Systems
Our study highlights the rich and remarkable variety of physical properties that isostatic systems can exhibit. Under FBCs, floppy modes can adopt a variety of forms, from all being extended to all
being localized near surfaces to a mixture of the two. Under PBCs, the presence of floppy modes depends on whether the lattice can or cannot support states of self-stress. When a lattice exhibits a
large number of zero-energy edge modes, its mechanical/dynamical properties become extremely sensitive to boundary conditions, much as do the electronic properties of the topological states of matter
studied in quantum systems (
). The zero-energy edge modes observed in our isostatic lattices are collective modes whose amplitudes decay exponentially from the edge with a finite decay length, in direct contrast to the very
localized and trivial floppy modes arising from dangling bonds. We focused primarily on high-symmetry lattices derived from the kagome lattice, but the properties they exhibit (namely, a deficit of
floppy modes in the bulk and the existence of floppy surface modes) are shared by any two-dimensional system with a vanishing bulk modulus (or the equivalent in anisotropic systems).
Three-dimensional analogs of the twisted kagome lattice can be constructed by rotating tetrahedra in pyrochlore and zeolite lattices (
) and in cristobalites (
). These lattices are anisotropic. With NN forces only, they exhibit a vanishing modulus for compression applied in particular planes rather than isotropically, but we expect them to exhibit many of
the properties the two-dimensional lattices exhibit. Finally, we note that Maxwell’s ideas can be applied to spin systems such as the Heisenberg antiferromagnet on the kagome lattice (
), and the possibility of unusual edge states in them is intriguing.
We are grateful for informative conversations with Randall Kamien, Andea Liu, and S.D. Guest. This work was supported in part by the National Science Foundation under Grants DMR-0804900, DMR-1104707
(to T.C.L. and X.M.), Materials Research Science and Engineering Center DMR-0520020 (to T.C.L. and A.S.), and JQI-NSF-PFC (to K.S.).
Supporting Information
Supporting Information (PDF)
Supporting Information
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JN Grima, A Alderson, KE Evans, Auxetic behaviour from, rotating rigid units. Phys Status Solidi B 242, 561–575 (2005).
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Published in
Proceedings of the National Academy of Sciences
Vol. 109 | No. 31
July 31, 2012
Submission history
Published online: June 25, 2012
Published in issue: July 31, 2012
We are grateful for informative conversations with Randall Kamien, Andea Liu, and S.D. Guest. This work was supported in part by the National Science Foundation under Grants DMR-0804900, DMR-1104707
(to T.C.L. and X.M.), Materials Research Science and Engineering Center DMR-0520020 (to T.C.L. and A.S.), and JQI-NSF-PFC (to K.S.).
This article is a PNAS Direct Submission.
See Commentary on page
Competing Interests
The authors declare no conflict of interest.
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Prime Numbers: What and Why
I’ll begin a short series of posts on prime numbers with several questions on the basics: What are prime (and composite) numbers, and why do they matter?
What is a prime number?
We’ll start with an anonymous question from 1995:
Prime Numbers
What are prime numbers?
Doctor Ken answered with the definition and a pair of examples:
A prime number is a positive number that has exactly two factors, 1 and itself. For example, if we list the factors of 28, we have 1, 2, 4, 7, 14, and 28. That's six factors. If we list the factors of 29, we only have 1 and 29. That's 2. So we say that 29 is a prime number, but 28 isn't.
As we’ll be seeing in the next two weeks, there are many ways to misstate this definition; every word in the definition matters!
Note that the definition of a prime number doesn't allow 1 to be a prime number: 1 only has one factor, namely 1. Prime numbers have EXACTLY two factors, not "at most two" or anything like that.
This is where the word “exactly” comes in. Without it, you could take “has two factors, 1 and itself” as being true for 1, since 1 and itself are both factors and there is no other. (We’ll be seeing
next week why we don’t want 1 to be a prime number; and the following week, we’ll see why “positive” matters.)
Here are the first few prime numbers:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, etc.
One important feature of prime numbers is that they are hard to predict, and seem almost random – yet they are definitely not! Here are the primes less than 100 on a number line:
Here are all the primes listed above, namely those less than 200:
The farther out you look, the more random they appear. Even here, you can see runs of nearly consecutive primes, and gaps with none.
How can I make a list like that?
Here is a 1997 question:
Prime Numbers: 20-30
Dear Dr. Math,
What are the prime numbers 20 through 30? My mom can't help me.
Thank you,
Doctor Wilkinson answered, starting with the basics:
First of all, you need to be sure you understand what prime numbers are. In fact, if you understand that, you should be able to do this problem without any difficulty.
A prime number is a whole number which is not the product of smaller numbers. For example, 14 is not a prime number, because it is 2 times 7. But 3 is a prime number, because the only smaller numbers are 1 and 2, and 3 is not 1 times 1 or 1 times 2 or 2 times 2.
This version of the definition is less formal, but gives the main idea well. We can directly use it to find primes (though not very efficiently).
To see if a number is prime, all you need to do is try the numbers smaller than it and bigger than 1 and divide them into your number. If this ever comes out even, then your number isn't prime. Otherwise, it is.
For example take 15. If you try dividing it by 2, it doesn't come out even. But if you try 3, it does. 15 is 3 times 5. So it's not a prime.
Now let's look at 17. If you try dividing it by 2, it doesn't come out even. If you try 3, that doesn't come out even either. Neither does 4, and neither does 5.
One way to improve efficiency would be to try only prime divisors, so we’d skip 4. Why? Because if a number can be divided evenly by 4, then it would have already have been found to be divisible by
2. But we aren’t looking for the best possible way to accomplish the task; we just want Leah to experience what primes are – and maybe discover more about them by doing things that aren’t necessary.
Now you could go ahead and try 6, 7, 8, and so on. But if you noticed when you tried 5, you got a quotient of 3 and a remainder of 2. Now if you try something bigger, you're going to get a smaller quotient. So if it was going to come out even, it would already have come out even when you tried the smaller number, and at this point you can quit, because you now know that 17 is prime.
This kind of thinking allows you to decide when to stop: If the quotient is smaller than the divisor, you don’t need to try any larger divisors.
Do you think you can do this for the numbers from 20 to 30? To get you started, I'll point out that 20, 22, 24, 26, 28, and 30 are all evenly divisible by 2, so none of them is prime.
This suggests other ways to shorten the work, which you can discover as you work. Later, we’ll look into how to test a number to see if it’s prime, and how to make a list more efficiently. But that
can wait.
Prime and composite numbers
For more details, here is a question from 2003:
Prime, Composite, or Neither?
What are prime and composite numbers? I just don't get it.
Doctor Ian answered:
Hi Hillary,
Suppose I have 12 items, and I try to arrange them into a rectangle. I can do this in more than one way:
. . . . . . . . . . . . 1 x 12
. . . . . . 2 x 6
. . . . . .
. . . . 3 x 4
. . . .
. . . .
We could also use \(4\times3\), \(6\times2\), and \(12\times1\), with the same pairs in the reverse order; we’re interested only in the pairs, not the order.
For some numbers, there is only one rectangle that I can make. For example, if I have 7 items, I can do this:
. . . . . . .
But if I try to make more rows, I always have something left over:
. . . .
. . .
. . .
. . .
. .
. .
. .
A number like 7 is called 'prime'. In contrast, a number like 12 is called 'composite'.
It’s easy to see that prime numbers are special. They can’t be broken down, sort of like atoms in chemistry.
One way to remember this is that something that is 'composed' is 'put together' from smaller pieces. (For example, we compose a poem from words, and compose a song from notes.)
So composite numbers are numbers that are composed of other numbers. (And in chemistry, a compound is composed of different atoms! That’s related, too.)
On the other hand, prime means “first“, or “most important”; primes are the numbers we start with when we build up other numbers by multiplication – the building blocks of the natural numbers.
In the case of a number like 12, we can put it together in more than one way, using multiplication:
12 = 1 x 12
= 2 x 6
= 3 x 4
But 1 x 12 is hardly like putting something together, is it? So if we ignore ways that include a 1, we see that there are two ways to put together a 12,
12 = 2 x 6
= 3 x 4
and _no_ ways to 'put together' a 7.
So when we say that a prime number is one that has exactly two factors (itself and 1), we are saying that there are no “non-trivial” ways to factor it; it can’t be made by “putting together” numbers
not including itself.
One tricky point is that the number 1 is considered to be neither prime nor composite. (Think about why this would be the case.) So while it's tempting to say things like
A number is prime if it's not composite.
A number is composite if it's not prime.
neither of these is quite true, because 1 isn't composite, but it's also not prime; and 1 isn't prime, but it's also not composite.
We say that the terms “prime” and “composite” are not exhaustive; “composite” doesn’t quite mean “not prime”. More on this, again, next time.
Why do we care about any of this? That's discussed here:
Why Study Prime and Composite Numbers?
Who uses primes?
That was a reference to the following question from 2001:
Why Study Prime and Composite Numbers?
My daughter is in Grade 6. She is learning about prime and composite numbers but my husband and I wonder why this is taught in school at all. Who uses this in the real world? Why does someone need to know whether a number is a prime number or not?
Doctor Ian had answered that:
Hi Kim,
Every time you send a credit card number over the Internet, it gets encrypted by your browser, and the encryption algorithm is based on the theory of prime numbers. At some point, electronic money will become as common as paper money, and _that_ will also be based on the theory of prime numbers. And what's used more in the real world than money?
Encryption is everywhere now! And the basic idea behind the common method is that it’s easy to “compose” numbers, but hard to “decompose” them into a product of primes.
For example, say I choose the primes 7127 and 7879. Their product is 56,153,633. I send you this number (or just post it on my website for anyone to use); you use that number, by a method I’ve
specified, to encrypt a message; and I then use my separate primes to create a number I can use to decrypt the message. This calculation is easy using the primes themselves, but would be very hard
using only their product. On the other hand, anyone who could factor 56,153,633 could decrypt the message; so I’m trusting that it’s too hard for anyone to do quickly enough to take advantage of it.
(We’d really use primes with 100 or more digits.)
But that’s all behind the scenes, and you don’t have to know about that math in order to use it. (Somebody does have to know it, though!) How might primes be needed directly in your own experience?
Mostly as a part of other math:
The importance of prime numbers is that any integer can be decomposed into a product of primes. For example, if you want to know how many different pairs of numbers can be multiplied to get 360, you can start trying to write them down,
1 * 360
2 * 180
3 * 120
4 * 90
5 * 72
6 * 60
checking every single number up to 180, and hope that you don't miss any; or you can decompose 360 into its prime factors,
360 = 2 * 2 * 2 * 3 * 3 * 5
with the assurance that every factor of 360 will be a product of a subset of these prime factors.
Knowing how a number breaks down, like knowing what atoms a chemical is made of, or how the parts of a car fit together, makes it possible to understand it better.
In the example above, you can list the pairs of factors by listing all ways to choose 0, 1, 2, or 3 twos, 0, 1, or 2 threes, and 0 or 1 five to make a factor. This tells you, in fact, that there are
\(4\times3\times2=24\) factors of 360 (that is, 12 pairs of factors). The 12 factors listed above are only half of them. (Can you find the other 12?)
(See Counting Divisors of a Number for more on this.)
This kind of analysis is extremely convenient when working with fractions (since prime factorization tells you which common denominators are available for any two fractions), when factoring polynomials... when doing just about anything where integers are involved, really.
For example, if the denominators of two fractions are, say, 2205 and 2100, that is, \(3^2\times5\times7^2\) and \(2^2\times3\times5^2\times7\), we know that the least common denominator, in order to
be a multiple of both, has to have 2 twos, 2 threes, 2 fives, and 2 sevens, making it \(2^2\times3^2\times5^2\times7^2=44,100\). We can find the greatest common factor similarly. Of course, since it
is hard to find factors of large numbers, another method is needed when the numbers are really large.
(See Many Ways to Find the Least Common Multiple for more on this. Also, compare Three Ways to Find the Greatest Common Factor. And don’t forget How Do You Simplify a Fraction?.)
Are prime numbers necessary? Not really:
Think of it this way. You don't need to learn to multiply, since you can always use repeated addition to solve any multiplication problem, right? If you want to know what 398 times 4612 is, you can just start adding:
398 (1)
398 (2)
398 (3)
398 (4)
Knowing about multiplication saves you time. That's all it does... but that's a lot!
And that’s what primes do most: save time (sometimes centuries, for really big jobs).
Mostly, prime numbers are good for quickly transforming a situation with zillions of possible outcomes into an equivalent situation with only a handful of possible outcomes.
Here is another way to think about it: If you're looking for some needles in a haystack, you can start picking up each piece of straw, checking to see if it's a needle, and then tossing it over your shoulder. Or you can use a magnet to find the needles right away.
In mathematics, prime numbers serve the same function as a really, Really, REALLY big magnet.
In short, knowing about prime and composite numbers will save your daughter enormous amounts of time in her later math classes - and possibly over the course of her life, if she goes into a technical field.
It’s not how a number is written that matters
Let’s close with a 1998 question from an entirely different perspective:
Prime Numbers in Different Bases
Hi, Dr. Math,
Here is a question I have for you. It's on prime numbers.
Are all prime numbers the same in all bases? If 21 is a prime, are 10101 (in binary), and 15 (in hexadecimal) also primes?
I'm taking a course in Assembly Language Programming, and I was wondering if primality as such is related at all to the number system I am using? What would happen, for instance, if I chose as a base a prime number, such as thirteen?
Computers commonly use base 2 (binary) for internal storage of numbers, and represent them in hexadecimal (base 16) to print them out more compactly. Does that affect prime numbers? How can we
recognize them when written in those forms?
Writing a number in a particular base means using place values that are powers of the base. For example, 21 in base 10 means \(21_{10}=2\times10^1+1\times10^0=2\times10+1=21\), while 15 in
hexadecimal (base 16) means \(15_{16}=1\times16^1+5\times16^0=1\times16+5=21\). They’re different numerals for the same number.
Doctor Mike answered:
Hi Jorge,
A prime is a prime no matter which base you use to represent it. On the surface one might think that in Hex you would have 3*5 = 15 as "usual," but it really turns out that 3*5 = F.
The example 21 doesn't work too well because it is not prime. The base ten number 37 is better, because it is prime, but its Hex representation is 25, which sort of looks non-prime. Hex 25 is not, however, repeat not, 5 squared.
Whether you write 21 as \(21_{10}\) or as \(15_{16}\), it is still a composite number; in either base, it is \(3\times7\). And whichever way we write 37, it still refers to the same number – even
though we are so familiar with numbers like 25 that we automatically think of it as a square. (A similar issue arises with judgments of evenness and oddness; in an odd base, numbers that “look” even
to our base-ten eyes may be odd!)
Okay, enough for examples. The fact of being prime or composite is just a property of the number itself, regardless of the way you write it. 15 and F and Roman numeral XV all mean the number, which is 3 times 5, so it is composite. That is the way it is for all numbers, in the sense that if a base ten number N has factors, you can represent those factors in Hex and their product will be the number N in Hex.
And if a number has no factors other than itself and 1 in base ten, that is still true when you write it in another base. It’s the number that counts, not the numeral (the representation of the
So, how do you recognize a prime in binary or hexadecimal? The same way, ultimately, as in decimal: Either do a lot of divisions to look for factors, or be sufficiently fluent in the appropriate base
that you recognize products (or know the divisibility tests appropriate to that base – which will all be different than in base ten).
Relating to your question about base 13, the base ten number 13 will be represented as "10" in that system, but "10" will still be a prime, because you cannot find two numbers other than 1 and "10" that will multiply together to make "10".
I hope this helps you think about primes in other bases.
So a prime base has no effect on primality of numbers written in it, any more than any other base does. It just makes things look different.
Just for fun, here are the first 13 primes, written in base 13 (with digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C):
2, 3, 5, 7, 11, 10, 14, 16, 1A, 23, 25, 2B, 32
They don’t all look prime to eyes accustomed to base ten, but they are! (Note that the units digit being even doesn’t imply the number is even, when the base is odd.)
Next week, we’ll look at some special cases: 0 and 1. The following week, we’ll consider negative numbers.
5 thoughts on “Prime Numbers: What and Why”
i am looking for a list of prime numbers and each prime number gap that runs from 2 to up to say 5 millionth
prime number. ( or a very large available prime such as one millionth prime ). all the google and any
other internet search i have tried jumps over this direct listing of primes and their gaps so i am stopped.
a few examples are: 2 1, 3 2,5 2,7 4 and on up to the largest available.
can you help with this? thank you.
Hi, David.
You could, of course, just take any list of 5 million primes, and write a simple program (or spreadsheet) to calculate the gaps.
Alternatively, the list of gaps is in OEIC here, and has a link to Vojtech Strnad, First 100000 terms [First 10000 terms from N. J. A. Sloane].
That doesn’t include the primes themselves, but you could correlate it with a list of primes.
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JPK 998 (JPK) - Sailboat specifications - Boat-Specs.com
JPK 998
Sailboat specifications
The JPK 998 is a 32’8” (9.98m) racing sailboat designed by Jacques Valer (France). She was built since 2008 (and now discontinued) by JPK (France).
JPK 998's main features
JPK 998
Hull type
Racing sailboat
Sailboat builder
Sailboat designer
GRP (glass reinforced polyester):
Sandwich balsa fiberglass vinylester (vacuum infusion)
First built hull
Last built hull
Keel : fin with bulb
Single tiller
Single transom hung rudder
EC design category
iThe CE design category indicates the ability to cope with certain weather conditions (the sailboat is designed for these conditions)
A: Wind < force 9, Waves < 10m
B: Wind < force 8, Waves < 8m
C: Wind < force 6, Waves < 4m
D: Wind < force 4, Waves < 0,5m
Standard public price ex. VAT (indicative only)
JPK 998's main dimensions
Hull length
32’ 8”9.98 m
Waterline length
31’ 1”9.48 m
Beam (width)
9’ 10”2.99 m
6’ 11”2.1 m
Light displacement (M[LC])
5952 lb2700 kg
Ballast weight
2976 lb1350 kg
Ballast type
Steel fin with lead bulb
JPK 998's rig and sails
Upwind sail area
689 ft²64 m²
Downwind sail area
1539 ft²143 m²
Mainsail area
409 ft²38 m²
Genoa area
280 ft²26 m²
Symmetric spinnaker area
1023 ft²95 m²
Asymmetric spinnaker area
1130 ft²105 m²
Rigging type
Sloop Marconi 9/10
Mast configuration
Deck stepped mast
Rotating spars
Number of levels of spreaders
Spreaders angle
No spreader
Spars construction
Carbon fiber mast and aluminum boom
Standing rigging
Single-strand (ROD)
JPK 998's performances
Upwind sail area to displacement
iThe ratio sail area to displacement is obtained by dividing the sail area by the boat's displaced volume to the power two-thirds.
The ratio sail area to displacement can be used to compare the relative sail plan of different sailboats no matter what their size.
Upwind: under 18 the ratio indicates a cruise oriented sailboat with limited performances especially in light wind, while over 25 it indicates a fast sailboat.
355 ft²/T33.01 m²/T
Downwind sail area to displacement
iThe ratio sail area to displacement is obtained by dividing the sail area by the boat's displaced volume to the power two-thirds.
The ratio sail area to displacement can be used to compare the relative sail plan of different sailboats no matter what their size.
794 ft²/T73.75 m²/T
Displacement-length ratio (DLR)
iThe Displacement Length Ratio (DLR) is a figure that points out the boat's weight compared to its waterline length. The DLR is obtained by dividing the boat's displacement in tons by the cube of
one one-hundredth of the waterline length (in feet).
The DLR can be used to compare the relative mass of different sailboats no matter what their length:
a DLR less than 180 is indicative of a really light sailboat (race boat made for planning), while a DLR greater than 300 is indicative of a heavy cruising sailboat.
Ballast ratio
iThe Ballast ratio is an indicator of stability; it is obtained by dividing the boat's displacement by the mass of the ballast. Since the stability depends also of the hull shapes and the
position of the center of gravity, only the boats with similar ballast arrangements and hull shapes should be compared.
The higher the ballast ratio is, the greater is the stability.
50 %
Critical hull speed
iAs a ship moves in the water, it creates standing waves that oppose its movement. This effect increases dramatically the resistance when the boat reaches a speed-length ratio (speed-length ratio
is the ratio between the speed in knots and the square root of the waterline length in feet) of about 1.2 (corresponding to a Froude Number of 0.35) . This very sharp rise in resistance, between
speed-length ratio of 1.2 to 1.5, is insurmountable for heavy sailboats and so becomes an apparent barrier. This leads to the concept of "hull speed".
The hull speed is obtained by multiplying the square root of the waterline length (in feet) by 1.34.
7.47 knots
JPK 998's auxiliary engine
1 inboard engine
Engine(s) power
13 HP
Fuel type
Fuel tank capacity
19.8 gal75 liters
JPK 998's accommodations and layout
Open aft cockpit
Maximum headroom
5’ 7”1.7 m
JPK 998's fore cabin
Berth length
7’ 2”2.2 m
Berth width
4’ 7”1.4 m
Have you spotted incorrect data?
You can report it in the forum
contact the webmaster
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Example of a $450,000 Home Purchase using VA Financing in California
Buying a home in California can be a challenge. The biggest hurdle preventing most potential homebuyers is the down payment. While there are loan programs, like FHA, that allow for down payments as
little as 3.5% down, it can still take time to save that much money. Someone buying a home for $450,000 in California with 3.5% down would need $15,750 for the down payment. And that amount doesn’t
include closing costs, prepaid expenses, etc, that could easily add up to another $10,000, bringing the total amount needed to close to over $25,000. But for California Veterans, there is a much
better option. A program that doesn’t require any down payment. The VA loan program.
The VA Loan Program allows 100% Financing up to your County Loan Limit
The VA loan program has been around, in some form, since 1944. It was conceived as a way to help returning military Veterans purchase a home. (or farm, etc). Rather than give a cash bonus, VA would
guaranty a percentage of the loan, making it a safe program for lenders to offer. In its current form, the VA guarantees 25% of the loan amount, for loans up to a specified county limit. That VA
California county loan limit is determined once a year by the Federal Housing Finance Agency, or FHFA. The base limit for most counties in the country and throughout California in 2018 is $453,100.
But there are higher cost counties in California, like Los Angeles County, Orange County, Contra Costa County, and many other, with limits as high as $679,650. In those high-cost counties, California
Veterans can purchase a home for $679,650 with $0 down payment. For today I am going to give you an example of what a $450,000 purchase using VA financing will look like.
What Do the Numbers Look Like for a $450,000 Home Purchase with $0 Down Payment?
First, we do need to make a few assumptions. We are going to assume this is a single family detached home with no homeowners association dues. We are going to assume a property tax rate of 1.25%,
which is fairly typical in California. There are some areas where tax rates are between 1% and 1.05%, and some areas where the property tax rate is nearly 2%. But most are close to 1.25%. I am also
going to estimate the homeowner’s insurance using a factor at .25% of the loan amount divided by 12. The homeowner will need to shop for their own insurance, but .25% will work as a solid estimate.
I am also going to assume this is the California Veterans first time using VA financing, resulting in a VA Funding Fee of 2.15%. For members of the Reserves or National Guard, the VA Funding Fee
would be 2.4% for being first-time users of VA financing. A Veteran with a service-connected disability rating will not have a VA Funding Fee. And lastly, I am going to assume a FICO score of 720+.
The example will break down the total payment, including principal, interest, property taxes, and insurance. I will also estimate the required income to qualify for this purchase price. I will also
give an estimate of the typical closing costs and prepaid expenses, and give strategies for potentially having the closing costs and prepaid expenses covered using a lender or seller credit.
There is No Down payment required, so the VA loan will equal the purchase price. There is a VA Funding Fee equal to 2.15%, or $9,675 in this case, that is financed into the loan. This makes the total
VA loan $459,675.
Interest rates vary from one day to the next, but as of February 12, 2018, and assuming a FICO score above 720, we’ll use 4.25% at 0 points (APR 4.521%). That results in a Principal & Interest (PI)
payment of $2,261. Property taxes and homeowners insurance are also part of the payment. They are the “TI” in PITI. The total “PITI” monthly payment is $2,824.
What Income is Needed to Qualify for a $2,824 mortgage payment?
The Debt to Income ratio is the ratio or percentage that compares a borrower’s gross income compared to their monthly payments. The guideline Debt to Income ratio is 41% on VA loans. But
realistically VA does not have a maximum Debt to Income ratio. It is not unusual for the DTI on a VA loan to be 50% or higher. If we assume that 50% of the California Veterans gross monthly income
can go towards their total payments. And if we assume the Veteran has a car payment of $500 and a minimum credit card payment of $50, then the estimated income needed to qualify for a mortgage
payment of $2,824 and other payments of $550 (total of $3,374) would be approximately $6,774. ($3,374/.50% = $6,674).
Closing Cost Breakdown
There are closing costs and Prepaid expenses on all real estate and loan transactions. Even on a VA loan. There are ways to have some or all of those costs paid for (by the seller) or credited by the
lender, but one way or another the costs to close do need to be accounted for. Understanding this before getting an accepted offer is very important. Typical closing costs include lender fees,
appraisal, credit report, escrow and title fees, and recording fees. In our example, I have estimated those fees to total $6,000. Some of these fees will adjust based on the loan amount or purchase
price of the home. The choice of escrow and title companies are negotiated through the purchase contract, but in most cases in the current real estate market, the seller or their real estate agent
will have the upper hand in choosing those companies.Also, for this loan example, I have chosen an interest rate at 0 points, meaning there are no Discount points associated with the interest rate.
You do have the option of “buying” the interest rate down by paying Discount Points. One Point is equal to 1% of the loan amount. If the loan amount is $459,675, then 1 Discount Point would be
$4,597, which would be added to the closing costs. One Point may lower the rate by .25%, which would lower the PI by $66 per month on a $450,000 Base VA loan amount. That may or may not make sense,
depending on several factors that should be discussed with your California VA loan officer. On the flipside, it is also possible to go higher in rate. By doing this you would receive a lender credit
that could be used to cover closing costs and/or prepaid expenses.
PrePaid Expenses
Prepaid expenses include mortgage interest, property taxes, and homeowners insurance.
• Prepaid mortgage interest occurs when a loan closes at some time in the middle of the month. For example, if a VA loan closes on June 15 then there would be 15 days of “prepaid interest” due at
closing. The time period covering June 16-June 30 is the 15 days. Your first mortgage payment would not be due until August 1, or 45 days after the closing, which is awesome. If the loan were to
close on June 29, then there would only be 5 days of Prepaid interest. But the first payment would still be August 1, or only 31 days after the closing. Either way, the homebuyer is just paying
for the interest covering the time period they are in the home that is not going to be covered by the first payment. In our example, we are estimating 15 days of Prepaid interest at $61 per day,
or $910.
• Property taxes are paid and/or prepaid/deposited to an escrow/impound account through the closing of the loan. Property taxes and homeowners insurance will be paid through each month along with
your Principal and Interest. An Impound account is set up at the closing to make sure there will be enough funds to pay those bills when they come due. The number of months of property taxes
deposited into the impound account is dependent on the month the loan closes. Loans that close in April will require 4 months of property taxes to be collected and deposited into the impound
account. There may also be “prorated” taxes due to the seller based on property taxes they have already paid that will cover a time period the new buyer will be occupying the home. In the
example, we are estimating 6 months of property taxes for the impound account, which is a good conservative estimate. This comes to $2,913. ($468.75 x 6).
• Homeowners Insurance is also prepaid. At closing a 1-year premium is paid. Also, 3 months of insurance are deposited to the impound account. By depositing three months of insurance into the
impound account, the lender is making sure there will be enough funds available for renewal 12 months after the closing. In our example, we estimate the 1-year premium and 3 months insurance to
be $1,406.
The Impound Account is essentially a savings account for the VA buyer that is held by the VA lender. Your monthly mortgage statement will give you a breakdown of the balance, deposits, and
disbursements in your impound account. It is important to keep on top of your impound account. It is also important to be aware that in many cases the lender will not pay the Supplemental tax bill
when it comes due. The homeowner should be aware of the Supplemental tax bill in the first year after their home purchase.
In our example, the estimated Prepaid expenses are $5,129. The Closing Costs are $6,000. This means the total amount needed to close is $11,033. As mentioned earlier, the Veteran can negotiate to
have the seller pay some or all of the closing costs and prepaid expenses. Another option is to choose a higher interest rate and then use “Yield Spread Premium” to help cover costs. For example, if
you went with an interest rate of 4.75% (APR 4.874) you could get a YSP of 1.75%, or in this case, $8,044. ($459,675 * 1.75% = $8,044). This is enough to cover all of the closing costs and even
$2,000 of the Prepaid expenses.
The important thing is to understand the numbers BEFORE you make an offer on a home. Make sure the total PITI payment will fit your budget. Make sure you have a way of covering the closing costs and
prepaid expenses. Know ahead of time whether you will need to negotiate with the seller to pay costs because you’re not going to be able to negotiate after the offer is accepted and you’re in escrow.
Get PreApproved for a VA Loan Before you make an Offer on a Home
This all seems like a lot. But working with the right real estate professionals can help to make the process easy. You will want to get PreApproved for a VA loan before you make an offer on a home.
At the very start of the PreApproval process, you will have a phone or in-person consultation with a California VA Loan Specialist. The VA Loan Officer will then be able to prepare custom loan
scenarios based on your qualifications, taking into consideration your payment comfort level, credit, income, etc. Your California VA Loan Officer will also prepare a custom video of your loan
scenarios, explaining the numbers and answering questions you may have. Going through this process will give you confidence when you begin making offers on homes.
Authored by Tim Storm, a California VA Loan Officer specializing in VA Loans. MLO 223456. – Please contact my office at the Home Point Financial. My direct line is 949-640-3102. I will prepare custom
VA loan scenarios which will be matched up to your financial goals, both long and short-term. I also prepare a Video Explanation of your scenarios so that you are able to fully understand the numbers
BEFORE you have started the loan process. | {"url":"https://blog.californiavaloanexpert.com/2018/02/example-of-a-450000-home-purchase-using-va-financing-california/","timestamp":"2024-11-08T14:34:54Z","content_type":"text/html","content_length":"86752","record_id":"<urn:uuid:2826d07b-bdf3-474a-b53d-35c8d8b35831>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00521.warc.gz"} |
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The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by ${\displaystyle k}$, ${\displaystyle \lambda }$, or ${\displaystyle \kappa }$ and is
measured in W·m^−1·K^−1.
Thermal conductivity
Common symbols κ
SI unit watt per meter-kelvin (W/(m⋅K))
In SI base units kg⋅m⋅s^−3⋅K^-1
Dimension ${\displaystyle {\mathsf {M}}{\mathsf {L}}{\mathsf {T}}^{-3}{\mathsf {\Theta }}^{-1}}$
Thermal resistivity
Common symbols ρ
SI unit kelvin-meter per watt (K⋅m/W)
In SI base units kg^-1⋅m^-1⋅s^3⋅K
Dimension ${\displaystyle {\mathsf {M}}^{-1}{\mathsf {L}}^{-1}{\mathsf {T}}^{3}{\mathsf {\Theta }}}$
Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal conductivity. For instance, metals typically have high thermal conductivity and are
very efficient at conducting heat, while the opposite is true for insulating materials such as mineral wool or Styrofoam. Correspondingly, materials of high thermal conductivity are widely used in
heat sink applications, and materials of low thermal conductivity are used as thermal insulation. The reciprocal of thermal conductivity is called thermal resistivity.
The defining equation for thermal conductivity is ${\displaystyle \mathbf {q} =-kabla T}$, where ${\displaystyle \mathbf {q} }$ is the heat flux, ${\displaystyle k}$ is the thermal conductivity, and
${\displaystyle abla T}$ is the temperature gradient. This is known as Fourier's law for heat conduction. Although commonly expressed as a scalar, the most general form of thermal conductivity is a
second-rank tensor. However, the tensorial description only becomes necessary in materials which are anisotropic.
Simple definition
Thermal conductivity can be defined in terms of the heat flow ${\displaystyle q}$ across a temperature difference.
Consider a solid material placed between two environments of different temperatures. Let ${\displaystyle T_{1}}$ be the temperature at ${\displaystyle x=0}$ and ${\displaystyle T_{2}}$ be the
temperature at ${\displaystyle x=L}$ , and suppose ${\displaystyle T_{2}>T_{1}}$ . An example of this scenario is a building on a cold winter day; the solid material in this case is the building
wall, separating the cold outdoor environment from the warm indoor environment.
According to the second law of thermodynamics, heat will flow from the hot environment to the cold one as the temperature difference is equalized by diffusion. This is quantified in terms of a heat
flux ${\displaystyle q}$ , which gives the rate, per unit area, at which heat flows in a given direction (in this case minus x-direction). In many materials, ${\displaystyle q}$ is observed to be
directly proportional to the temperature difference and inversely proportional to the separation distance ${\displaystyle L}$ :
${\displaystyle q=-k\cdot {\frac {T_{2}-T_{1}}{L}}.}$
The constant of proportionality ${\displaystyle k}$ is the thermal conductivity; it is a physical property of the material. In the present scenario, since ${\displaystyle T_{2}>T_{1}}$ heat flows in
the minus x-direction and ${\displaystyle q}$ is negative, which in turn means that ${\displaystyle k>0}$ . In general, ${\displaystyle k}$ is always defined to be positive. The same definition of $
{\displaystyle k}$ can also be extended to gases and liquids, provided other modes of energy transport, such as convection and radiation, are eliminated or accounted for.
The preceding derivation assumes that the ${\displaystyle k}$ does not change significantly as temperature is varied from ${\displaystyle T_{1}}$ to ${\displaystyle T_{2}}$ . Cases in which the
temperature variation of ${\displaystyle k}$ is non-negligible must be addressed using the more general definition of ${\displaystyle k}$ discussed below.
General definition
Thermal conduction is defined as the transport of energy due to random molecular motion across a temperature gradient. It is distinguished from energy transport by convection and molecular work in
that it does not involve macroscopic flows or work-performing internal stresses.
Energy flow due to thermal conduction is classified as heat and is quantified by the vector ${\displaystyle \mathbf {q} (\mathbf {r} ,t)}$ , which gives the heat flux at position ${\displaystyle \
mathbf {r} }$ and time ${\displaystyle t}$ . According to the second law of thermodynamics, heat flows from high to low temperature. Hence, it is reasonable to postulate that ${\displaystyle \mathbf
{q} (\mathbf {r} ,t)}$ is proportional to the gradient of the temperature field ${\displaystyle T(\mathbf {r} ,t)}$ , i.e.
${\displaystyle \mathbf {q} (\mathbf {r} ,t)=-kabla T(\mathbf {r} ,t),}$
where the constant of proportionality, ${\displaystyle k>0}$ , is the thermal conductivity. This is called Fourier's law of heat conduction. Despite its name, it is not a law but a definition of
thermal conductivity in terms of the independent physical quantities ${\displaystyle \mathbf {q} (\mathbf {r} ,t)}$ and ${\displaystyle T(\mathbf {r} ,t)}$ .^[3] As such, its usefulness depends on
the ability to determine ${\displaystyle k}$ for a given material under given conditions. The constant ${\displaystyle k}$ itself usually depends on ${\displaystyle T(\mathbf {r} ,t)}$ and thereby
implicitly on space and time. An explicit space and time dependence could also occur if the material is inhomogeneous or changing with time.^[4]
In some solids, thermal conduction is anisotropic, i.e. the heat flux is not always parallel to the temperature gradient. To account for such behavior, a tensorial form of Fourier's law must be used:
${\displaystyle \mathbf {q} (\mathbf {r} ,t)=-{\boldsymbol {\kappa }}\cdot abla T(\mathbf {r} ,t)}$
where ${\displaystyle {\boldsymbol {\kappa }}}$ is symmetric, second-rank tensor called the thermal conductivity tensor.
An implicit assumption in the above description is the presence of local thermodynamic equilibrium, which allows one to define a temperature field ${\displaystyle T(\mathbf {r} ,t)}$ . This
assumption could be violated in systems that are unable to attain local equilibrium, as might happen in the presence of strong nonequilibrium driving or long-ranged interactions.
Other quantities
In engineering practice, it is common to work in terms of quantities which are derivative to thermal conductivity and implicitly take into account design-specific features such as component
For instance, thermal conductance is defined as the quantity of heat that passes in unit time through a plate of particular area and thickness when its opposite faces differ in temperature by one
kelvin. For a plate of thermal conductivity ${\displaystyle k}$ , area ${\displaystyle A}$ and thickness ${\displaystyle L}$ , the conductance is ${\displaystyle kA/L}$ , measured in W⋅K^−1.^[6] The
relationship between thermal conductivity and conductance is analogous to the relationship between electrical conductivity and electrical conductance.
Thermal resistance is the inverse of thermal conductance.^[6] It is a convenient measure to use in multicomponent design since thermal resistances are additive when occurring in series.
There is also a measure known as the heat transfer coefficient: the quantity of heat that passes per unit time through a unit area of a plate of particular thickness when its opposite faces differ in
temperature by one kelvin.^[8] In ASTM C168-15, this area-independent quantity is referred to as the "thermal conductance".^[9] The reciprocal of the heat transfer coefficient is thermal insulance.
In summary, for a plate of thermal conductivity ${\displaystyle k}$ , area ${\displaystyle A}$ and thickness ${\displaystyle L}$ ,
• thermal conductance = ${\displaystyle kA/L}$ , measured in W⋅K^−1.
□ thermal resistance = ${\displaystyle L/(kA)}$ , measured in K⋅W^−1.
• heat transfer coefficient = ${\displaystyle k/L}$ , measured in W⋅K^−1⋅m^−2.
□ thermal insulance = ${\displaystyle L/k}$ , measured in K⋅m^2⋅W^−1.
The heat transfer coefficient is also known as thermal admittance in the sense that the material may be seen as admitting heat to flow.^[10]
An additional term, thermal transmittance, quantifies the thermal conductance of a structure along with heat transfer due to convection and radiation. It is measured in the same units as thermal
conductance and is sometimes known as the composite thermal conductance. The term U-value is also used.
Finally, thermal diffusivity ${\displaystyle \alpha }$ combines thermal conductivity with density and specific heat:
${\displaystyle \alpha ={\frac {k}{\rho c_{p}}}}$ .
As such, it quantifies the thermal inertia of a material, i.e. the relative difficulty in heating a material to a given temperature using heat sources applied at the boundary.^[12]
In the International System of Units (SI), thermal conductivity is measured in watts per meter-kelvin (W/(m⋅K)). Some papers report in watts per centimeter-kelvin [W/(cm⋅K)].
However, physicists use other convenient units as well, e.g., in cgs units, where esu/(cm-sec-K) is used.^[13] The Lorentz number, defined as L=κ/σT is a quantity independent of the carrier density
and the scattering mechanism. Its value for a gas of non-interacting electrons (typical carriers in good metallic conductors) is 2.72×10^-13 esu/K^2, or equivalently, 2.44×10^-8 Watt-Ohm/K^2.
In imperial units, thermal conductivity is measured in BTU/(h⋅ft⋅°F).^[note 1]^[14]
The dimension of thermal conductivity is M^1L^1T^−3Θ^−1, expressed in terms of the dimensions mass (M), length (L), time (T), and temperature (Θ).
Other units which are closely related to the thermal conductivity are in common use in the construction and textile industries. The construction industry makes use of measures such as the R-value
(resistance) and the U-value (transmittance or conductance). Although related to the thermal conductivity of a material used in an insulation product or assembly, R- and U-values are measured per
unit area, and depend on the specified thickness of the product or assembly.^[note 2]
Likewise the textile industry has several units including the tog and the clo which express thermal resistance of a material in a way analogous to the R-values used in the construction industry.
There are several ways to measure thermal conductivity; each is suitable for a limited range of materials. Broadly speaking, there are two categories of measurement techniques: steady-state and
transient. Steady-state techniques infer the thermal conductivity from measurements on the state of a material once a steady-state temperature profile has been reached, whereas transient techniques
operate on the instantaneous state of a system during the approach to steady state. Lacking an explicit time component, steady-state techniques do not require complicated signal analysis (steady
state implies constant signals). The disadvantage is that a well-engineered experimental setup is usually needed, and the time required to reach steady state precludes rapid measurement.
In comparison with solid materials, the thermal properties of fluids are more difficult to study experimentally. This is because in addition to thermal conduction, convective and radiative energy
transport are usually present unless measures are taken to limit these processes. The formation of an insulating boundary layer can also result in an apparent reduction in the thermal conductivity.^[
Experimental values
Experimental values of thermal conductivity
The thermal conductivities of common substances span at least four orders of magnitude.^[17] Gases generally have low thermal conductivity, and pure metals have high thermal conductivity. For
example, under standard conditions the thermal conductivity of copper is over 10000 times that of air.
Of all materials, allotropes of carbon, such as graphite and diamond, are usually credited with having the highest thermal conductivities at room temperature.^[18] The thermal conductivity of natural
diamond at room temperature is several times higher than that of a highly conductive metal such as copper (although the precise value varies depending on the diamond type).^[19]
Thermal conductivities of selected substances are tabulated below; an expanded list can be found in the list of thermal conductivities. These values are illustrative estimates only, as they do not
account for measurement uncertainties or variability in material definitions.
Influencing factors
The effect of temperature on thermal conductivity is different for metals and nonmetals. In metals, heat conductivity is primarily due to free electrons. Following the Wiedemann–Franz law, thermal
conductivity of metals is approximately proportional to the absolute temperature (in kelvins) times electrical conductivity. In pure metals the electrical conductivity decreases with increasing
temperature and thus the product of the two, the thermal conductivity, stays approximately constant. However, as temperatures approach absolute zero, the thermal conductivity decreases sharply.^[23]
In alloys the change in electrical conductivity is usually smaller and thus thermal conductivity increases with temperature, often proportionally to temperature. Many pure metals have a peak thermal
conductivity between 2 K and 10 K.
On the other hand, heat conductivity in nonmetals is mainly due to lattice vibrations (phonons). Except for high-quality crystals at low temperatures, the phonon mean free path is not reduced
significantly at higher temperatures. Thus, the thermal conductivity of nonmetals is approximately constant at high temperatures. At low temperatures well below the Debye temperature, thermal
conductivity decreases, as does the heat capacity, due to carrier scattering from defects.^[23]
Chemical phase
When a material undergoes a phase change (e.g. from solid to liquid), the thermal conductivity may change abruptly. For instance, when ice melts to form liquid water at 0 °C, the thermal conductivity
changes from 2.18 W/(m⋅K) to 0.56 W/(m⋅K).^[24]
Even more dramatically, the thermal conductivity of a fluid diverges in the vicinity of the vapor-liquid critical point.^[25]
Thermal anisotropy
Some substances, such as non-cubic crystals, can exhibit different thermal conductivities along different crystal axes. Sapphire is a notable example of variable thermal conductivity based on
orientation and temperature, with 35 W/(m⋅K) along the c axis and 32 W/(m⋅K) along the a axis.^[26] Wood generally conducts better along the grain than across it. Other examples of materials where
the thermal conductivity varies with direction are metals that have undergone heavy cold pressing, laminated materials, cables, the materials used for the Space Shuttle thermal protection system, and
fiber-reinforced composite structures.^[27]
When anisotropy is present, the direction of heat flow may differ from the direction of the thermal gradient.
Electrical conductivity
In metals, thermal conductivity is approximately correlated with electrical conductivity according to the Wiedemann–Franz law, as freely moving valence electrons transfer not only electric current
but also heat energy. However, the general correlation between electrical and thermal conductance does not hold for other materials, due to the increased importance of phonon carriers for heat in
non-metals. Highly electrically conductive silver is less thermally conductive than diamond, which is an electrical insulator but conducts heat via phonons due to its orderly array of atoms.
Magnetic field
The influence of magnetic fields on thermal conductivity is known as the thermal Hall effect or Righi–Leduc effect.
Gaseous phases
Exhaust system components with ceramic coatings having a low thermal conductivity reduce heating of nearby sensitive components
In the absence of convection, air and other gases are good insulators. Therefore, many insulating materials function simply by having a large number of gas-filled pockets which obstruct heat
conduction pathways. Examples of these include expanded and extruded polystyrene (popularly referred to as "styrofoam") and silica aerogel, as well as warm clothes. Natural, biological insulators
such as fur and feathers achieve similar effects by trapping air in pores, pockets, or voids.
Low density gases, such as hydrogen and helium typically have high thermal conductivity. Dense gases such as xenon and dichlorodifluoromethane have low thermal conductivity. An exception, sulfur
hexafluoride, a dense gas, has a relatively high thermal conductivity due to its high heat capacity. Argon and krypton, gases denser than air, are often used in insulated glazing (double paned
windows) to improve their insulation characteristics.
The thermal conductivity through bulk materials in porous or granular form is governed by the type of gas in the gaseous phase, and its pressure.^[28] At low pressures, the thermal conductivity of a
gaseous phase is reduced, with this behaviour governed by the Knudsen number, defined as ${\displaystyle K_{n}=l/d}$ , where ${\displaystyle l}$ is the mean free path of gas molecules and ${\
displaystyle d}$ is the typical gap size of the space filled by the gas. In a granular material ${\displaystyle d}$ corresponds to the characteristic size of the gaseous phase in the pores or
intergranular spaces.^[28]
Isotopic purity
The thermal conductivity of a crystal can depend strongly on isotopic purity, assuming other lattice defects are negligible. A notable example is diamond: at a temperature of around 100 K the thermal
conductivity increases from 10,000 W·m^−1·K^−1 for natural type IIa diamond (98.9% ^12C), to 41,000 for 99.9% enriched synthetic diamond. A value of 200,000 is predicted for 99.999% ^12C at 80 K,
assuming an otherwise pure crystal.^[29] The thermal conductivity of 99% isotopically enriched cubic boron nitride is ~ 1400 W·m^−1·K^−1,^[30] which is 90% higher than that of natural boron nitride.
Molecular origins
The molecular mechanisms of thermal conduction vary among different materials, and in general depend on details of the microscopic structure and molecular interactions. As such, thermal conductivity
is difficult to predict from first-principles. Any expressions for thermal conductivity which are exact and general, e.g. the Green-Kubo relations, are difficult to apply in practice, typically
consisting of averages over multiparticle correlation functions.^[31] A notable exception is a monatomic dilute gas, for which a well-developed theory exists expressing thermal conductivity
accurately and explicitly in terms of molecular parameters.
In a gas, thermal conduction is mediated by discrete molecular collisions. In a simplified picture of a solid, thermal conduction occurs by two mechanisms: 1) the migration of free electrons and 2)
lattice vibrations (phonons). The first mechanism dominates in pure metals and the second in non-metallic solids. In liquids, by contrast, the precise microscopic mechanisms of thermal conduction are
poorly understood.^[32]
In a simplified model of a dilute monatomic gas, molecules are modeled as rigid spheres which are in constant motion, colliding elastically with each other and with the walls of their container.
Consider such a gas at temperature ${\displaystyle T}$ and with density ${\displaystyle \rho }$ , specific heat ${\displaystyle c_{v}}$ and molecular mass ${\displaystyle m}$ . Under these
assumptions, an elementary calculation yields for the thermal conductivity
${\displaystyle k=\beta \rho \lambda c_{v}{\sqrt {\frac {2k_{\text{B}}T}{\pi m}}},}$
where ${\displaystyle \beta }$ is a numerical constant of order ${\displaystyle 1}$ , ${\displaystyle k_{\text{B}}}$ is the Boltzmann constant, and ${\displaystyle \lambda }$ is the mean free path,
which measures the average distance a molecule travels between collisions.^[33] Since ${\displaystyle \lambda }$ is inversely proportional to density, this equation predicts that thermal conductivity
is independent of density for fixed temperature. The explanation is that increasing density increases the number of molecules which carry energy but decreases the average distance ${\displaystyle \
lambda }$ a molecule can travel before transferring its energy to a different molecule: these two effects cancel out. For most gases, this prediction agrees well with experiments at pressures up to
about 10 atmospheres. At higher densities, the simplifying assumption that energy is only transported by the translational motion of particles no longer holds, and the theory must be modified to
account for the transfer of energy across a finite distance at the moment of collision between particles, as well as the locally non-uniform density in a high density gas. This modification has been
carried out, yielding Revised Enskog Theory, which predicts a density dependence of the thermal conductivity in dense gases.^[35]
Typically, experiments show a more rapid increase with temperature than ${\displaystyle k\propto {\sqrt {T}}}$ (here, ${\displaystyle \lambda }$ is independent of ${\displaystyle T}$ ). This failure
of the elementary theory can be traced to the oversimplified "hard sphere" model, which both ignores the "softness" of real molecules, and the attractive forces present between real molecules, such
as dispersion forces.
To incorporate more complex interparticle interactions, a systematic approach is necessary. One such approach is provided by Chapman–Enskog theory, which derives explicit expressions for thermal
conductivity starting from the Boltzmann equation. The Boltzmann equation, in turn, provides a statistical description of a dilute gas for generic interparticle interactions. For a monatomic gas,
expressions for ${\displaystyle k}$ derived in this way take the form
${\displaystyle k={\frac {25}{32}}{\frac {\sqrt {\pi mk_{\text{B}}T}}{\pi \sigma ^{2}\Omega (T)}}c_{v},}$
where ${\displaystyle \sigma }$ is an effective particle diameter and ${\displaystyle \Omega (T)}$ is a function of temperature whose explicit form depends on the interparticle interaction law.^[36]
For rigid elastic spheres, ${\displaystyle \Omega (T)}$ is independent of ${\displaystyle T}$ and very close to ${\displaystyle 1}$ . More complex interaction laws introduce a weak temperature
dependence. The precise nature of the dependence is not always easy to discern, however, as ${\displaystyle \Omega (T)}$ is defined as a multi-dimensional integral which may not be expressible in
terms of elementary functions, but must be evaluated numerically. However, for particles interacting through a Mie potential (a generalisation of the Lennard-Jones potential) highly accurate
correlations for ${\displaystyle \Omega (T)}$ in terms of reduced units have been developed.^[37]
An alternate, equivalent way to present the result is in terms of the gas viscosity ${\displaystyle \mu }$ , which can also be calculated in the Chapman–Enskog approach:
${\displaystyle k=f\mu c_{v},}$
where ${\displaystyle f}$ is a numerical factor which in general depends on the molecular model. For smooth spherically symmetric molecules, however, ${\displaystyle f}$ is very close to ${\
displaystyle 2.5}$ , not deviating by more than ${\displaystyle 1\%}$ for a variety of interparticle force laws.^[38] Since ${\displaystyle k}$ , ${\displaystyle \mu }$ , and ${\displaystyle c_{v}}$
are each well-defined physical quantities which can be measured independent of each other, this expression provides a convenient test of the theory. For monatomic gases, such as the noble gases, the
agreement with experiment is fairly good.^[39]
For gases whose molecules are not spherically symmetric, the expression ${\displaystyle k=f\mu c_{v}}$ still holds. In contrast with spherically symmetric molecules, however, ${\displaystyle f}$
varies significantly depending on the particular form of the interparticle interactions: this is a result of the energy exchanges between the internal and translational degrees of freedom of the
molecules. An explicit treatment of this effect is difficult in the Chapman–Enskog approach. Alternately, the approximate expression ${\displaystyle f=(1/4){(9\gamma -5)}}$ was suggested by Eucken,
where ${\displaystyle \gamma }$ is the heat capacity ratio of the gas.^[38]
The entirety of this section assumes the mean free path ${\displaystyle \lambda }$ is small compared with macroscopic (system) dimensions. In extremely dilute gases this assumption fails, and thermal
conduction is described instead by an apparent thermal conductivity which decreases with density. Ultimately, as the density goes to ${\displaystyle 0}$ the system approaches a vacuum, and thermal
conduction ceases entirely.
The exact mechanisms of thermal conduction are poorly understood in liquids: there is no molecular picture which is both simple and accurate. An example of a simple but very rough theory is that of
Bridgman, in which a liquid is ascribed a local molecular structure similar to that of a solid, i.e. with molecules located approximately on a lattice. Elementary calculations then lead to the
${\displaystyle k=3(N_{\text{A}}/V)^{2/3}k_{\text{B}}v_{\text{s}},}$
where ${\displaystyle N_{\text{A}}}$ is the Avogadro constant, ${\displaystyle V}$ is the volume of a mole of liquid, and ${\displaystyle v_{\text{s}}}$ is the speed of sound in the liquid. This is
commonly called Bridgman's equation.
For metals at low temperatures the heat is carried mainly by the free electrons. In this case the mean velocity is the Fermi velocity which is temperature independent. The mean free path is
determined by the impurities and the crystal imperfections which are temperature independent as well. So the only temperature-dependent quantity is the heat capacity c, which, in this case, is
proportional to T. So
${\displaystyle k=k_{0}\,T{\text{ (metal at low temperature)}}}$
with k[0] a constant. For pure metals, k[0] is large, so the thermal conductivity is high. At higher temperatures the mean free path is limited by the phonons, so the thermal conductivity tends to
decrease with temperature. In alloys the density of the impurities is very high, so l and, consequently k, are small. Therefore, alloys, such as stainless steel, can be used for thermal insulation.
Lattice waves, phonons, in dielectric solids
Heat transport in both amorphous and crystalline dielectric solids is by way of elastic vibrations of the lattice (i.e., phonons). This transport mechanism is theorized to be limited by the elastic
scattering of acoustic phonons at lattice defects. This has been confirmed by the experiments of Chang and Jones on commercial glasses and glass ceramics, where the mean free paths were found to be
limited by "internal boundary scattering" to length scales of 10^−2 cm to 10^−3 cm.^[42]^[43]
The phonon mean free path has been associated directly with the effective relaxation length for processes without directional correlation. If V[g] is the group velocity of a phonon wave packet, then
the relaxation length ${\displaystyle l\;}$ is defined as:
${\displaystyle l\;=V_{\text{g}}t}$
where t is the characteristic relaxation time. Since longitudinal waves have a much greater phase velocity than transverse waves,^[44] V[long] is much greater than V[trans], and the relaxation length
or mean free path of longitudinal phonons will be much greater. Thus, thermal conductivity will be largely determined by the speed of longitudinal phonons.^[42]^[45]
Regarding the dependence of wave velocity on wavelength or frequency (dispersion), low-frequency phonons of long wavelength will be limited in relaxation length by elastic Rayleigh scattering. This
type of light scattering from small particles is proportional to the fourth power of the frequency. For higher frequencies, the power of the frequency will decrease until at highest frequencies
scattering is almost frequency independent. Similar arguments were subsequently generalized to many glass forming substances using Brillouin scattering.^[46]^[47]^[48]^[49]
Phonons in the acoustical branch dominate the phonon heat conduction as they have greater energy dispersion and therefore a greater distribution of phonon velocities. Additional optical modes could
also be caused by the presence of internal structure (i.e., charge or mass) at a lattice point; it is implied that the group velocity of these modes is low and therefore their contribution to the
lattice thermal conductivity λ[L] (${\displaystyle \kappa }$ [L]) is small.^[50]
Each phonon mode can be split into one longitudinal and two transverse polarization branches. By extrapolating the phenomenology of lattice points to the unit cells it is seen that the total number
of degrees of freedom is 3pq when p is the number of primitive cells with q atoms/unit cell. From these only 3p are associated with the acoustic modes, the remaining 3p(q − 1) are accommodated
through the optical branches. This implies that structures with larger p and q contain a greater number of optical modes and a reduced λ[L].
From these ideas, it can be concluded that increasing crystal complexity, which is described by a complexity factor CF (defined as the number of atoms/primitive unit cell), decreases λ[L].^[51] This
was done by assuming that the relaxation time τ decreases with increasing number of atoms in the unit cell and then scaling the parameters of the expression for thermal conductivity in high
temperatures accordingly.^[50]
Describing anharmonic effects is complicated because an exact treatment as in the harmonic case is not possible, and phonons are no longer exact eigensolutions to the equations of motion. Even if the
state of motion of the crystal could be described with a plane wave at a particular time, its accuracy would deteriorate progressively with time. Time development would have to be described by
introducing a spectrum of other phonons, which is known as the phonon decay. The two most important anharmonic effects are the thermal expansion and the phonon thermal conductivity.
Only when the phonon number ‹n› deviates from the equilibrium value ‹n›^0, can a thermal current arise as stated in the following expression
${\displaystyle Q_{x}={\frac {1}{V}}\sum _{q,j}{\hslash \omega \left(\left\langle n\right\rangle -{\left\langle n\right\rangle }^{0}\right)v_{x}}{\text{,}}}$
where v is the energy transport velocity of phonons. Only two mechanisms exist that can cause time variation of ‹n› in a particular region. The number of phonons that diffuse into the region from
neighboring regions differs from those that diffuse out, or phonons decay inside the same region into other phonons. A special form of the Boltzmann equation
${\displaystyle {\frac {d\left\langle n\right\rangle }{dt}}={\left({\frac {\partial \left\langle n\right\rangle }{\partial t}}\right)}_{\text{diff.}}+{\left({\frac {\partial \left\langle n\right\
rangle }{\partial t}}\right)}_{\text{decay}}}$
states this. When steady state conditions are assumed the total time derivate of phonon number is zero, because the temperature is constant in time and therefore the phonon number stays also
constant. Time variation due to phonon decay is described with a relaxation time (τ) approximation
${\displaystyle {\left({\frac {\partial \left\langle n\right\rangle }{\partial t}}\right)}_{\text{decay}}=-{\text{ }}{\frac {\left\langle n\right\rangle -{\left\langle n\right\rangle }^{0}}{\tau
which states that the more the phonon number deviates from its equilibrium value, the more its time variation increases. At steady state conditions and local thermal equilibrium are assumed we get
the following equation
${\displaystyle {\left({\frac {\partial \left(n\right)}{\partial t}}\right)}_{\text{diff.}}=-{v}_{x}{\frac {\partial {\left(n\right)}^{0}}{\partial T}}{\frac {\partial T}{\partial x}}{\text{.}}}$
Using the relaxation time approximation for the Boltzmann equation and assuming steady-state conditions, the phonon thermal conductivity λ[L] can be determined. The temperature dependence for λ[L]
originates from the variety of processes, whose significance for λ[L] depends on the temperature range of interest. Mean free path is one factor that determines the temperature dependence for λ[L],
as stated in the following equation
${\displaystyle {\lambda }_{L}={\frac {1}{3V}}\sum _{q,j}v\left(q,j\right)\Lambda \left(q,j\right){\frac {\partial }{\partial T}}\epsilon \left(\omega \left(q,j\right),T\right),}$
where Λ is the mean free path for phonon and ${\displaystyle {\frac {\partial }{\partial T}}\epsilon }$ denotes the heat capacity. This equation is a result of combining the four previous equations
with each other and knowing that ${\displaystyle \left\langle v_{x}^{2}\right\rangle ={\frac {1}{3}}v^{2}}$ for cubic or isotropic systems and ${\displaystyle \Lambda =v\tau }$ .^[52]
At low temperatures (< 10 K) the anharmonic interaction does not influence the mean free path and therefore, the thermal resistivity is determined only from processes for which q-conservation does
not hold. These processes include the scattering of phonons by crystal defects, or the scattering from the surface of the crystal in case of high quality single crystal. Therefore, thermal
conductance depends on the external dimensions of the crystal and the quality of the surface. Thus, temperature dependence of λ[L] is determined by the specific heat and is therefore proportional to
Phonon quasimomentum is defined as ℏq and differs from normal momentum because it is only defined within an arbitrary reciprocal lattice vector. At higher temperatures (10 K < T < Θ), the
conservation of energy ${\displaystyle \hslash {\omega }_{1}=\hslash {\omega }_{2}+\hslash {\omega }_{3}}$ and quasimomentum ${\displaystyle \mathbf {q} _{1}=\mathbf {q} _{2}+\mathbf {q} _{3}+\mathbf
{G} }$ , where q[1] is wave vector of the incident phonon and q[2], q[3] are wave vectors of the resultant phonons, may also involve a reciprocal lattice vector G complicating the energy transport
process. These processes can also reverse the direction of energy transport.
Therefore, these processes are also known as Umklapp (U) processes and can only occur when phonons with sufficiently large q-vectors are excited, because unless the sum of q[2] and q[3] points
outside of the Brillouin zone the momentum is conserved and the process is normal scattering (N-process). The probability of a phonon to have energy E is given by the Boltzmann distribution ${\
displaystyle P\propto {e}^{-E/kT}}$ . To U-process to occur the decaying phonon to have a wave vector q[1] that is roughly half of the diameter of the Brillouin zone, because otherwise quasimomentum
would not be conserved.
Therefore, these phonons have to possess energy of ${\displaystyle \sim k\Theta /2}$ , which is a significant fraction of Debye energy that is needed to generate new phonons. The probability for this
is proportional to ${\displaystyle {e}^{-\Theta /bT}}$ , with ${\displaystyle b=2}$ . Temperature dependence of the mean free path has an exponential form ${\displaystyle {e}^{\Theta /bT}}$ . The
presence of the reciprocal lattice wave vector implies a net phonon backscattering and a resistance to phonon and thermal transport resulting finite λ[L],^[50] as it means that momentum is not
conserved. Only momentum non-conserving processes can cause thermal resistance.^[52]
At high temperatures (T > Θ), the mean free path and therefore λ[L] has a temperature dependence T^−1, to which one arrives from formula ${\displaystyle {e}^{\Theta /bT}}$ by making the following
approximation ${\displaystyle {e}^{x}\propto x{\text{ }},{\text{ }}\left(x\right)<1}$ and writing ${\displaystyle x=\Theta /bT}$ . This dependency is known as Eucken's law and originates from the
temperature dependency of the probability for the U-process to occur.^[50]^[52]
Thermal conductivity is usually described by the Boltzmann equation with the relaxation time approximation in which phonon scattering is a limiting factor. Another approach is to use analytic models
or molecular dynamics or Monte Carlo based methods to describe thermal conductivity in solids.
Short wavelength phonons are strongly scattered by impurity atoms if an alloyed phase is present, but mid and long wavelength phonons are less affected. Mid and long wavelength phonons carry
significant fraction of heat, so to further reduce lattice thermal conductivity one has to introduce structures to scatter these phonons. This is achieved by introducing interface scattering
mechanism, which requires structures whose characteristic length is longer than that of impurity atom. Some possible ways to realize these interfaces are nanocomposites and embedded nanoparticles or
Because thermal conductivity depends continuously on quantities like temperature and material composition, it cannot be fully characterized by a finite number of experimental measurements. Predictive
formulas become necessary if experimental values are not available under the physical conditions of interest. This capability is important in thermophysical simulations, where quantities like
temperature and pressure vary continuously with space and time, and may encompass extreme conditions inaccessible to direct measurement.^[53]
In fluids
For the simplest fluids, such as monatomic gases and their mixtures at low to moderate densities, ab initio quantum mechanical computations can accurately predict thermal conductivity in terms of
fundamental atomic properties—that is, without reference to existing measurements of thermal conductivity or other transport properties.^[54] This method uses Chapman-Enskog theory or Revised Enskog
Theory to evaluate the thermal conductivity, taking fundamental intermolecular potentials as input, which are computed ab initio from a quantum mechanical description.
For most fluids, such high-accuracy, first-principles computations are not feasible. Rather, theoretical or empirical expressions must be fit to existing thermal conductivity measurements. If such an
expression is fit to high-fidelity data over a large range of temperatures and pressures, then it is called a "reference correlation" for that material. Reference correlations have been published for
many pure materials; examples are carbon dioxide, ammonia, and benzene.^[55]^[56]^[57] Many of these cover temperature and pressure ranges that encompass gas, liquid, and supercritical phases.
Thermophysical modeling software often relies on reference correlations for predicting thermal conductivity at user-specified temperature and pressure. These correlations may be proprietary. Examples
are REFPROP^[58] (proprietary) and CoolProp^[59] (open-source).
Thermal conductivity can also be computed using the Green-Kubo relations, which express transport coefficients in terms of the statistics of molecular trajectories.^[60] The advantage of these
expressions is that they are formally exact and valid for general systems. The disadvantage is that they require detailed knowledge of particle trajectories, available only in computationally
expensive simulations such as molecular dynamics. An accurate model for interparticle interactions is also required, which may be difficult to obtain for complex molecules.^[61]
Jan Ingenhousz
Apparatus for measuring the relative thermal conductivities of different metals
In a 1780 letter to Benjamin Franklin, Dutch-born British scientist Jan Ingenhousz relates an experiment which enabled him to rank seven different metals according to their thermal conductivities:^[
You remembre you gave me a wire of five metals all drawn thro the same hole Viz. one, of gould, one of silver, copper steel and iron. I supplyed here the two others Viz. the one of tin the other
of lead. I fixed these seven wires into a wooden frame at an equal distance of one an other ... I dipt the seven wires into this melted wax as deep as the wooden frame ... By taking them out they
were cov[e]red with a coat of wax ... When I found that this crust was there about of an equal thikness upon all the wires, I placed them all in a glased earthen vessel full of olive oil heated
to some degrees under boiling, taking care that each wire was dipt just as far in the oil as the other ... Now, as they had been all dipt alike at the same time in the same oil, it must follow,
that the wire, upon which the wax had been melted the highest, had been the best conductor of heat. ... Silver conducted heat far the best of all other metals, next to this was copper, then gold,
tin, iron, steel, Lead.
See also
1. ^ 1 Btu/(h⋅ft⋅°F) = 1.730735 W/(m⋅K)
2. ^ R-values and U-values quoted in the US (based on the inch-pound units of measurement) do not correspond with and are not compatible with those used outside the US (based on the SI units of
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Further reading
Undergraduate-level texts (engineering)
Undergraduate-level texts (physics)
• Halliday, David; Resnick, Robert; & Walker, Jearl (1997). Fundamentals of Physics (5th ed.). John Wiley and Sons, New York ISBN 0-471-10558-9. An elementary treatment.
• Daniel V. Schroeder (1999), An Introduction to Thermal Physics, Addison Wesley, ISBN 978-0-201-38027-9. A brief, intermediate-level treatment.
• Reif, F. (1965), Fundamentals of Statistical and Thermal Physics, McGraw-Hill. An advanced treatment.
Graduate-level texts
• Balescu, Radu (1975), Equilibrium and Nonequilibrium Statistical Mechanics, John Wiley & Sons, ISBN 978-0-471-04600-4
• Chapman, Sydney; Cowling, T.G. (1970), The Mathematical Theory of Non-Uniform Gases (3rd ed.), Cambridge University Press. A very advanced but classic text on the theory of transport processes in
• Reid, C. R., Prausnitz, J. M., Poling B. E., Properties of gases and liquids, IV edition, Mc Graw-Hill, 1987
• Srivastava G. P (1990), The Physics of Phonons. Adam Hilger, IOP Publishing Ltd, Bristol
External links
• Thermopedia THERMAL CONDUCTIVITY
• Contribution of Interionic Forces to the Thermal Conductivity of Dilute Electrolyte Solutions The Journal of Chemical Physics 41, 3924 (1964)
• The importance of Soil Thermal Conductivity for power companies
• Thermal Conductivity of Gas Mixtures in Chemical Equilibrium. II The Journal of Chemical Physics 32, 1005 (1960) | {"url":"https://www.knowpia.com/knowpedia/Thermal_conductivity_and_resistivity","timestamp":"2024-11-09T02:39:08Z","content_type":"text/html","content_length":"396956","record_id":"<urn:uuid:ec686842-0447-4c8b-b77f-e04f582e3160>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00478.warc.gz"} |
A Potential Simple Analogous Heat Flow System to Explore Big History's Singularity Trend
A Potential Simple Analogous Heat Flow System to Explore Big History's Singularity Trend
Author: LePoire, David Almanac: History & Mathematics:Investigating Past and Future
Many historical systems (e.g., civilizations) demonstrate trends towards acceleration of knowledge, energy flow, and complexity. These systems are far from thermal equilibrium as they depend on great
flows of energy through them to maintain their structure, similar to Dissipative Dynamics Systems (DDS). This dissipation causes entropy, but while entropy is often associated with disorder, often
ordered patterns can spontaneously develop in them to facilitate entropy generation. That is, entropy gradients (and the second law of thermodynamics) might be the driver to higher complexity. In
addition, optimized engineered systems that are far from equilibrium, such as removing heat from electronic chips, also follows fractal pattern formation. A major trend in Big History is the
singularity trend of complexity, which has substructure where the complexity tends to increase by a factor of about 3 for every shortening (by a factor
of ⅓) period. At the same time, the energy flow tends to increase at a slightly faster rate of about 4-5 within each period. This paper develops a simple analogous energy flow system that may help
gain insight into this Big History trend, however, it is incomplete. Research areas are identified to tighten this approach.
Keywords: energy, environment, information, logistic.
The purpose of this paper is to propose a potential simple (‘toy’) system that demonstrates similar trends to the Big History singularity trend. This is a next step beyond the first modeling step of
identifying an equation and interpretation that describes the overall growth. The current understanding is based on a remarkably simple equation, i.e., dy/dt = ky^2, and its interpretation that the
‘complexity’ is proportional to the product of the current complexity (y) and the rate of learning, which itself is proportional to the previous complexity (y) (Korotayev 2005, 2020). This
accumulative learning scenario leads to the prediction of a singularity since the solution to the equation is y = A/(t - T) where T is the time of the singularity as has been observed in human
populations and other natural systems (Kremer 1993; Korotayev and Malkov 2016; Fomin 2020). But there are many questions that arise from this, for example: Why does there seem to be substructure (by
a factor of 3)? Is there a more general equation that helps understand what happens near and after the singularity time (since the equation's assumptions will eventually break down when change
becomes too rapid)? Why does energy flow scale a bit faster than the progress? (LePoire and Chandrankunnel 2020).
It is not clear that this approach to develop an analogous system will help, but others have already made important contributions concerning trends in similar systems. The work of Schneider and Kay
(1994) demonstrated that the second law of thermodynamics (entropy always increases) might actually drive the trend to higher complexity, since the spontaneous pattern formations found in such
far-from-equilibrium systems actually increase the rate of entropy generation. The works of Bejan (Bejan and Lorente 2011; Bejan and Zane 2012) apply some of these principles in his Constructal
Theory/Law, which shows the development of many scaling laws in evolution and engineering design due to entropy generation. Ayres and Warr (2009) developed a new understanding of the central role of
energy in economics and history. Niele (2005) detailed evolutionary transitions concerning energy generation and use.
Energy flow is defined as the rate of energy use, after thermal heat losses are considered, i.e., free-energy power. Chaisson (2004, 2014) identified the energy flow density as a major determinant of
complexity. He showed that the energy flow per mass through various astronomical systems such as galaxies, stars and planets tends to increase. This trend was then extended to evolving life forms,
humans, and technologies. But increased energy flow through a leading technology does not determine the societal complexity necessary for sustained development. For example, while a jet engine has
quite a large free energy flow, it is only useful when the society maintains the infrastructure such as economics, airports, airplane manufacturing along with a demand for the service to trade and
travel. Instead, the complexity of an evolving system might be better characterized by the amount of free-energy flowing through it.
However, there are many other approaches being explored concerning the role of energy in evolution. The natural tendency of complex non-living structures is to spontaneously form from a large energy
flow, as described by Bejan and Lorente (2011) and Schneider and Kay (1994). For example, when the bottom of a pot of oil is slightly heated, heat flows through the oil (conduction) to be released at
the top surface. As the temperature at the bottom is increased, there is a critical temperature when the oil starts macroscopic coordinated flow structures (convection Benard cells) that efficiently
transfers heated oil from the bottom in exchange for cooled oil at the surface. A similar formation of organized structure is seen when a bathtub is drained as water forms a circular motion to more
efficiently drain the water. Natural systems also display these formations in the form of hurricanes, tornadoes, and water eddies. Further studies (Horowitz and England 2017) explore the dynamics of
systems far-from-equilibrium to determine relationships between energy flow, reproduction rate, entropy, decay rate, and inversion ratio (fraction of ordered systems to those in equilibrium). From
these studies, evolution of complex systems is compatible with the second law of thermodynamics and in fact facilitates entropy production by organizing collective structures that reduce the energy
flow gradients. As Schneider and Kay stated in 1989: ‘Nature abhors a gradient’ (Schneider and Sagan 2005: 6).
Evolutionary changes in energy flow often occur when an evolving system needs to adapt to environmental conditions (Niele 2005; Jantsch 1980). These environmental conditions may be the result of the
previous growth of the system to the limits of the current way of operating. New energy mechanisms or sources are then identified and explored along with a new organizational structure to mitigate
the previous limits. A set of evolutionary energy mechanisms during the growth of civilization is collected in Table 1 (LePoire and Chandrankunnel 2020). The timing of these events occurs with step
durations decreasing by about a factor of 3 from the immediately preceding phase. Physical analogical models for each transition were discussed previously (LePoire 2019).
This evolution of civilization is just one of the three major evolutionary stages (life, humans, civilizations) identified by Sagan characterized by their primary information mechanisms (DNA, human
brain, writing) that occurred roughly five billion, five million, and five thousand years ago. Each of these information stages are formed from about six phases where the duration decreases by a
factor of three from each previous phases, leading to the overall decrease in the duration of an information stage of a factor of a thousand (LePoire 2015a, 2015b); with striking parallels with
Panov's (2005, 2020) analysis, as well as Korotayev (2020) and Fomin (2020).
In Bejan's example of an engineered system to remove heat from a rectangular volume (Bejan 1997), such as from an electronic chip, the identified optimal solution was an ever-increasing fractal
pattern on the surface, where not only the length of the conduits were fractal but also the width. This was done with very specific heat flow considerations. The pattern tended to fill the surface
(see Fig. 1). The expansion onto the surface occurs through adding alternating x and y directional conduits.
On a square surface, the fractal grows in alternating directions. It would be more reasonable if the scaling was consistent. This can be done by simplifying the base polygon from four to three sides,
i.e., an equilateral triangle. While triangles are not found in electronic chip design, the triangular frame is often found as one of the most stable elements in civil engineering, such as bridge
design. Assume that the triangle is heated from below. Other surfaces (top and sides) allow some heat to flow to the environment. The design goal is to extract as much energy as possible from surface
conduits built with a given fixed amount of material. To extract the maximum amount of energy, the first extraction point would be in the center, equally from boundaries where temperature falls. The
process of extracting the energy causes the center temperature to drop (see Fig. 2).
Now, if the same amount of material is available for the next stage (as with the square heat flow), the process would be repeated, splitting the material three ways but only having to traverse half
the distance to the center of the three outside triangles (see Fig. 1c).
The length of the new conduits L’, is half the original, L. Since the amount of material is the same in each iteration, the material is divided into 3 for each of the new pipes. Since the amount of
material in a pipe is the product of its length (L) and cross-sectional area (A), then 3 L’A’= LA, i.e., the cross-sectional area is reduced by 2/3. Resistance (R) to flow in a given pipe is
proportional to the length and inversely proportional to the area, i.e., L/A. This gives a flow resistance of 3/4 of the previous conduit. However, there are three pipes in parallel now feeding the
main node, so the net energy extracted from the three pipes would be 3/R = 4 times the energy extracted in the original pipe. If the material is laid out at the same rate for each of the pipes
compared to the original, the time for the construction is 1/3 the original time. So the second iteration takes 1/3 the time to build (with three lines going on simultaneously). The amount of energy
extracted is 4 times as much, since there are 3 shorter conduits but with smaller cross sections.
This continues for the next iterations (see Fig. 3), with the energy increasing by a factor of 4, taking a third of the time, with the population of new nodes being about 3^(n-1). The total number of
nodes N will approach (3^n)/2 when the number of iterations n gets large. So this model has similar scaling to what is seen (of population or complexity) in the rate of Big History transitions: N
scales as 1/(t-ts); energy scales at a higher rate than N (by a factor of 4 for every factor of 3 increase in N). It is important to realize that a difference with Bejan's square model is that there
are areas that are being passed over, whereas in the square geometry, all areas are being extracted.
However, with each iteration the extraction becomes a bit less efficient because the boundaries are being approached. Eventually, it would be more advantageous to fill in the triangle gaps instead of
branching out. For example, as the three triangles are explored in Fig. 1c, there is about 25 % (the central triangle) abandoned. For each iteration there is another quarter of the area being
explored left abandoned. This adds up, so that the fraction of the abandoned area to the area being used for extraction scales as 4/3 ^ n. At the full extension of the exploration, this leaves quite
a large fraction of energy to be extracted. For four iterations in Fig. 3b, the ratio of area abandoned to the area for extraction is about 4. While this is quite small compared to the ratio of
extraction from the full extension with compared to the first single node of almost 64, there are significant resources to be utilized.
However, this is not what was expected for a truly symmetrical transition after the peak, where it was expected that the growth of complexity might grow by the same factor (3) but at a hyperbolically
decelerating rate. This current triangular model would give an increase of 4/3 instead of a factor of 3. Each iteration after the peak would take about 3 times longer than the previous as the
extraction pipes need to become increasingly longer.
So let us summarize the analogy of extracting energy from a triangular surface and the Big History singularity trend:
· Both are based on energy extraction from a far-from-equilibrium system.
· Both show singularity behavior with the number of nodes and population (or complexity) tends to increase inversely proportional to the time to the singularity.
· Both show substructure with stages of increasing by a factor of 3, in one third the time.
· Both show energy extraction (flow) at a rate greater than the number of nodes and population.
· Both would hit a limit where the effort to extract greater energy from additional iteration becomes marginal.
· Both seem to show ways that the trend might reverse. In the triangle system, much of the area is bypassed as the geometric trend is continued. In complex adaptive evolutionary systems, it has been
shown that with environmental limits, the complexity might not reach chaotic state but instead reverse in complexity (Stone 1993).
· The reversal in the triangular case seems to show an unexpected factor in the energy geometric trend, falling from a factor of about 4 to about 4/3.
While there are many similarities as listed above, it is unclear why this simple system would be analogous to the system of Big History. For example, why would the abstract space being explored be
represented by a triangle instead of a square or higher dimensional structure? The triangle is the simplest 2-D polygon so fractal continues by splitting into three branches. However, in typical
complex adaptive systems, the iterative branching continues by splitting into two different branches, that is, bifurcation. The bifurcations could be viewed as two opposing potential paths to realize
a new organization for the next iteration. But clearly the exploration in bifurcations is in polar opposites. As seen in the case of fractal exploration of the square, the opposite directions occur
along the x or the y directions but not both simultaneously. With a triangle, the three new branches from each node allow exploration in the plane, i.e., any point of the area being explored can be
expressed as a linear combination of any two of them.
While detailed calculations of the heat flow from the fractal structure of the square were shown to be physically accurate, the detailed calculations were not done for this triangular case. The
boundary conditions have not been specified in detail but it is believed to have a large heat source along the bottom with some type of flow (e.g., radiative) along the sides (not insulating or at a
fixed temperature). It would be beneficial to fill this analysis gap.
It is important to notice that the structures are built up from only current considerations of optimization, not a long-term goal of designing an optimized complete system. In the triangular case
this leads to exploring areas that are initially far away, where the current system has smaller impact on the extraction potential. This is similar to accumulative learning where foundations are set
enabling further learning. For example, humans' symbolic expression did not start with writing but instead with vocal (and perhaps manual gesture) expressions that offered a wider range of options
such as ability to learn quickly, not needing great production infrastructure, ability to quickly communicate in greater number of conditions (e.g., nighttime). Only later, after the opportunities
opened by vocal language were filled, did the investment in writing offer marginal benefits.
Finally, the system reaches the point where marginal gain from further fractal expansion is limited. This seems to be the period we are entering. Each of the cumulative levels supports quite an
infrastructure and opportunity for us.
However, in the rush to expand, many issues have been left unresolved. For example, the transition to agriculture from hunter-gathering allowed higher population growth but required greater effort;
the transition to civilization allowed for defense and risk reduction in return for curtailed freedoms. An example of a recent unresolved issue is seen in the U.S. health care system, which developed
in an industrial era with assumptions about jobs and medicine within a country context. The push was to globalize, leading to great advances in lifestyle and health, but the system of healthcare was
difficult to upgrade as other competing issues required greater attention. These issues can now be seen as areas to be improved as the challenges of global cooperation become apparent.
To compare to the triangular case where potential areas are left unutilized as the main infrastructure is developed, then one should see many unresolved issues at many levels. One way to investigate
this is to look at risks coming from various social levels. Table 3 outlines the occurrence of events from the personal (addiction to alcohol and drugs) through the family (divorce), group (layoffs),
community (crime), and national (identity theft) levels. Identity theft is depicted as a national level problem because, while some identity theft is international in scope, most occurs from within
the victim's country. Table 3 shows that life-impacting events across these various social levels affect around 5 % of the U.S. population. These events can typically cause severe life changes and
While conventional responses do have a place in the reduction of the threat, alternative complementary approaches have been suggested. One alternative advocated by the late Nobel Prize winning
chemist Richard Smalley (2003), soon after the 9/11 attacks, called for an international collaboration to develop energy and technology as one way to help alleviate multiple world problems. In this
situation, the developed world might be seen as addressing common problems with agreed upon technological use. This is an example of an alternative approach integrating a wider systems-level view to
the problem where the conventional linear approach might no longer be applicable. Developing and motivating U.S. students to pursue science and engineering fields was an important factor in this
proposal, since currently about half of the students in these programs are attracted from other countries.
A systemic approach to these issues would compare and investigate the relationships of the relative risks and possible actions (Tainter 1996). This can be refined further by taking a look at the
combinations of threats, vulnerabilities, and consequences. The factors leading to increased vulnerability include untested new technologies; dependence on technologies and resources; operating with
limited political, economic, and public communication capacities; and complex interactions. The factors leading to a decreased threat level include more equitable distribution, toleration,
transparency, trust, and opportunities. Consequences can be physical, financial, or psychological but could be mitigated through appropriate responses including planning, assessment, and forecasting
potential unintended consequences.
A simple (‘toy’) system with heat flow on a triangle was explored as motivated by Bejan's analysis of a system on a square. The evolution of a heat extraction system on a triangular surface was seen
to have many characteristics similar to the long-term singularity trend in Big History. As with all models, there are many gaps and differences. However, having one physical system to reference might
provide insights for further development.
One similar property is that in their evolution with the singularity trend, many potential resources are left behind. This suggests that the inflection of this trend (since it cannot continue to an
actual singularity) might be to slow down and try to resolve some of the issues that remained in the rush to develop the base infrastructure. One approach to support this was to identify risks at
various social levels. The risks seem to be spread out over many levels from self to global issues.
Ayres R. U., and Warr B. 2009. The Economic Growth Engine: How Energy and Work Drive Material Prosperity. Cheltenham, UK: Edward Elgar Publishing.
Bejan A., and Lorente S. 2011. The Constructal Law and the Evolution of Design in Nature. Physics of Life Reviews 8: 209–240.
Bejan A., and Zane J. P. 2012. Design in Nature: How the Constructal Law Governs Evolution in Biology, Physics, Technology, and Social Organization. New York: Doubleday.
Bejan A. 1997. Constructal-Theory Network of Conducting Paths for Cooling a Heat Generating Volume. International Journal of Heat and Mass Transfer 40(4): 799-810.
Chaisson E. 2004. Complexity: An Energetics Agenda: Energy as the Motor of Evolution. Complexity 9(3): 14–21.
Chaisson E. 2014. The Natural Science Underlying Big History. The Scientific World Journal: 1–41.
Fomin A. 2020. Hyperbolic Evolution from Biosphere to Technosphere. The 21^st Century Singularity and Global Futures: A Big History Perspective / Ed. by A. Korotayev, and D. LePoire, pp. 105–118.
Cham: Springer.
Horowitz J. M., and England J. L. 2017. Spontaneous Fine-Tuning to Environment in Many-Species Chemical Reaction Networks. Proceedings of the National Academy of Sciences 114(29): 7565–7570.
Jantsch E. 1980. The Self-Organizing Universe: Scientific and Human Implications of the Emerging Paradigm of Evolution. Oxford, UK: Pergamon.
Korotayev A. 2005. A Compact Macromodel of World System Evolution. Journal of World-Systems Research 11(1): 79–93.
Korotayev A. 2020. The 21^st Century Singularity in the Big History Perspective: A Re-Analysis. The 21^st Century Singularity and Global Futures. A Big History Perspective / Ed. by A. Korotayev, and
D. LePoire, pp. 19–75. Cham: Springer.
Korotayev A., and Malkov A. 2016. A Compact Mathematical Model of the World System Economic and Demographic Growth, 1 CE – 1973 CE. International Journal of Mathematical Models and Methods in Applied
Sciences 10: 200–209.
Kremer M. 1993. Population Growth and Technological Change: One Million B.C. to 1990. The Quarterly Journal of Economics 108(3): 681–716.
LePoire D. J. 2015a. Interpreting ‘Big History’ as Complex Adaptive System Dynamics with Nested Logistic Transitions in Energy Flow and Organization. Emergence: Complexity & Organization 17(1): 1–16.
LePoire D. J. 2015b. Potential Nested Accelerating Returns Logistic Growth in Big History. Evolution: From Big Bang to Nanorobots / Ed. by L. E. Grinin, and A. V. Korotayev, pp. 46–60. Volgograd:
LePoire D. J. 2019. An Exploration of Historical Transitions with Simple Analogies and Empirical Event Rates. Journal of Big History 3(2): 1–16.
LePoire D. J., and Devezas T. 2020a. Near-term Indications and Models of a Singularity. The 21^st Century Singularity and Global Futures. A Big History Perspective / Ed. by A. Korotayev, and D.
LePoire, pp. 213–224. Cham: Springer.
LePoire D. J., and Chandrankunnel M. 2020b. Energy Flow Trends in Big History. The 21^st Century Singularity and Global Futures. A Big History Perspective / Ed. by A. Korotayev, and D. LePoire, pp.
185–200. Cham: Springer.
Niele F. 2005. Energy: Engine of Evolution. Amsterdam – Boston: Elsevier.
Panov A. 2005. Scaling Law of the Biological Evolution and the Hypothesis of the Self-Consistent Galaxy Origin of Life. Advances in Space Research 36(2): 220–225.
Panov A. 2020. Singularity of Evolution and Post-Singular Development in the Big History Perspective. The 21^st Century Singularity and Global Futures. A Big History Perspective / Ed. by A.
Korotayev, and D. LePoire, pp. 439–468. Cham: Springer.
Schneider E. D., and Kay J. J. 1994. Life as a Manifestation of the Second Law of Thermodynamics. Mathl. Comput. Modelling 19(6–8): 25–48.
Schneider E. D., and Sagan D 2005. Into the Flow: Energy Flow, Thermodynamics, and Life. Chicago: University of Chicago Press.
Stone L. 1993. Period-doubling Reversals and Chaos in Simple Ecological Models. Nature Cell Biology 365: 617–620.
Tainter J. A. 1996. Complexity, Problem Solving, and Sustainable Societies. Getting Down to Earth / Ed. by R. Constanza, O. Segura, and J. Martinez-Alier, pp. 61–76. Washington, D.C.: Island Press. | {"url":"https://www.sociostudies.org/almanac/articles/a_potential_simple_analogous_heat_flow_system_to_explore_big_history-s_singularity_trend/","timestamp":"2024-11-04T01:18:17Z","content_type":"application/xhtml+xml","content_length":"41199","record_id":"<urn:uuid:05287325-84e2-499a-82a5-cb8ebdb16007>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00044.warc.gz"} |
Searching randomly for needles.
Suppose you're given a large set of objects $X$, and you know that some subset $I$ is "interesting". A particular example that hits close to (my regular) home is in bug-testing: $X$ is the set of
possible inputs to a program, and $I$ is the set of inputs that generate bugs (this is the downside of talking to
+John Regehr
too much). We'll assume that if you're given a candidate object, you can check easily if it's interesting or not (for example, by running the program)
You'd like to find the interesting items, so you consult an expert (in our running example, maybe it's a method to generate inputs that test certain kinds of bugs). The expert produces items that it
thinks are interesting. But experts have biases: maybe your expert only cares about certain kinds of interesting items.
So you ask multiple experts, in the hope that their biases are different, and that together they can cover the space of objects. But you don't know anything about their biases except what you can
learn from repeatedly asking them for candidate objects.
What's your strategy for asking questions so that at any stage (say after asking $N$ questions), you've found the optimal number of interesting items ?
This was the topic of a talk by
+Sebastien Bubeck
at the AMPLab in Berkeley on Tuesday, based on
this paper
. A key idea in the algorithm design is to make use of estimators of "things I haven't seen yet", or the famous Good-Turing estimator (
which your humble blogger wrote about
many æons ago). Here's how it works.
Formally, let us assume that each "expert" $i$ is a distribution $P_i$ over $X$. At each step, the algorithm will determine some $i$, and ask it for a random draw from $P_i$. Suppose we knew the
fraction of items that $i$ had not seen yet. Then a simple greedy strategy would be to pick the $i$ that had the largest value of this "not-yet-seen" quantity. That's all well and good,
as long as we know the fraction of items not yet seen
Here's where the G-T estimator comes in. What it says is that if we're given samples from a distribution, and count the number of items in the sample that occurred exactly once, then this "frequency
of frequency", divided by the number of samples, is a good estimate for the mass of items not yet seen. Moreover, it can be shown that this estimator has good concentration properties around the true
So that's what the algorithm does. It maintains estimates (for each expert) of the mass not yet seen, and in each round picks the expert that maximizes this term, corrected by an adjustment coming
from the tail of the concentration bound.
The algorithm is really simple, and elegant. The question is, how well does it do ? And now things get a little harder. The above ideal greedy strategy is optimal as long as the supports of the
experts are disjoint. Under the same assumption (and some other technical ones), it can be shown that the expected difference between the number of items found by the algorithm and the number found
by the ideal greedy algorithm is $O(\sqrt{Kt \log t})$ where $K$ is the number of experts and $t$ is the current time step.
It's not clear how to extend these bounds to the case when supports can overlap (or rather, it's not clear to me :)), but the authors show that the algorithm works quite well in practice even when
this assumption is broken, which is encouraging. | {"url":"http://blog.geomblog.org/2013/10/searching-randomly-for-needles.html","timestamp":"2024-11-03T09:51:37Z","content_type":"application/xhtml+xml","content_length":"127406","record_id":"<urn:uuid:faf7d22b-f37e-4859-a4b2-74beed27a118>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00694.warc.gz"} |
Flowing bosonization in the nonperturba
SciPost Submission Page
Flowing bosonization in the nonperturbative functional renormalization-group approach
by Romain Daviet, Nicolas Dupuis
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users): Romain Daviet
Submission information
Preprint Link: https://arxiv.org/abs/2111.11458v3 (pdf)
Date accepted: 2022-02-28
Date submitted: 2022-02-11 16:15
Submitted by: Daviet, Romain
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
Specialties: • Condensed Matter Physics - Theory
Approach: Theoretical
Bosonization allows one to describe the low-energy physics of one-dimensional quantum fluids within a bosonic effective field theory formulated in terms of two fields: the "density" field $\varphi$
and its conjugate partner, the phase $\vartheta$ of the superfluid order parameter. We discuss the implementation of the nonperturbative functional renormalization group in this formalism,
considering a Luttinger liquid in a periodic potential as an example. We show that in order for $\vartheta$ and $\varphi$ to remain conjugate variables at all energy scales, one must dynamically
redefine the field $\vartheta$ along the renormalization-group flow. We derive explicit flow equations using a derivative expansion of the scale-dependent effective action to second order and show
that they reproduce the flow equations of the sine-Gordon model (obtained by integrating out the field $\vartheta$ from the outset) derived within the same approximation. Only with the
scale-dependent (flowing) reparametrization of the phase field $\vartheta$ do we obtain the standard phenomenology of the Luttinger liquid (when the periodic potential is sufficiently weak so as to
avoid the Mott-insulating phase) characterized by two low-energy parameters, the velocity of the sound mode and the renormalized Luttinger parameter.
Author comments upon resubmission
Dear Editor,
we would like to thank the referees for their report on the manuscript, their positive judgment and their constructive remarks. We reply below to their comments and questions. A redlined version of
the manuscript that highlights the changes and a pdf version of the letter are provided in the comments section.
Sincerely Yours,
Nicolas Dupuis and Romain Daviet
Report 1:
1-As mentioned in the introduction and shown in Appendix B, the fact that the superfluid stiffness does not renormalize despite the presence of a periodic potential in the partially bosonized scheme
without flowing bosonization, is a consequence of gauge invariance. It is also claimed that this statement holds independent of the approximation scheme used. I did not understand the latter
statement. It seems to imply that it holds to arbitrary orders in the derivative expansion, while the actual calculations performed in Appendix B appear to make explicit use of the second-order
truncation of the derivative expansion. Can the authors please elucidate this point?
Answer: We thank the referee for drawing our attention to this issue. In fact Appendix B does not rely on a truncation of the effective action to second order in the derivative expansion but only
makes use of the most general derivative expansion of the two-point vertex $\Gamma^{(2)}$ as given in Eq.~(79). We have changed the text above and below~(79) and modified Eqs.~(80) to clarify this
2-It might be useful to repeat the definition of the RG time t in the captions of Fig. 1, as these are used in the labels of the plots.
Answer: We thank the referee for this suggestion.
Report 2:
1- Page 6: the authors write "We construct [...] $R_k(Q)$ in the usual way." I believe that here a citation to other works where this type of regulator has been used is needed.
Answer: We have slightly modified the sentence and added two references.
2- Page 6: "[...] we take $ r(y)=\alpha/(e^y-1)$ with $\alpha$ of order unity." I think a brief discussion on how $\alpha$ is chosen is necessary.
Answer: In the case of a precision calculation (e.g. when determining the critical exponents at a second-order phase transition), $\alpha$ is fixed by using the principle of minimum sensitivity. In
the sine-Gordon model, the precise value of $\alpha$ is unimportant (see Ref.~[2]). In the present study, it is therefore sufficient to take $\alpha$ of order unity. We have added a comment in the
3- Page 10, below Eq.(37): I believe, for clarity sake, that the authors should stress at this stage that $\tilde K_k$ is obtained from the average of $Z_{1k}(\phi)$, while $K_k$ from its value at
zero field.
Answer: We thank the referee for this suggestion.
4- Sec. 3.1.1: In my opinion it is more sound to write Eq.(38) as $\theta(Q)=-i\omega \alpha_k(Q) \phi(Q) + \beta_k(Q) \bar\theta(Q)$ as it is clear that $\theta$ must be a combination of $\bar\
theta$ and $\partial_\tau\phi$. In this way it is more evident that the term in the second line of (39) is a $(\partial_{\tau} \phi)^2$ term and it has to vanish when summed with $Z_{1\tau,k}(\phi=0)
( \partial_{\tau} \phi)^2$.
Answer: Actually it is $\partial_x\theta$ which is a combination of $\partial_x\bar\theta$ and $\partial_\tau\phi$, so that in the end $\alpha_k(Q)\sim i\omega/q$ as obtained in Eq.~(41). But we
agree with the referee that we could write~(38) as $\theta(Q)=-i(\omega/q) \alpha_k(Q) \phi(Q) + \beta_k(Q) \bar\theta(Q)$. On the other hand our notations are well suited to the calculations of
Appendix~E since they give a simple expression of the matrix $M_k(Q)$. For this reason we prefer keep Eq.~(38) as it is now.
5- Page 13: The meaning of the paragraph from "It was pointed out in Ref. [3] ... " to "... preventing the superfluid stiffness to vanish when $k\to 0$." is unclear to me. I would ask the authors to
Answer: What we want to explain here is the fact that the convergence of $\eta_{1,k}=-\dt\ln Z_{1,k}$ towards 2 makes the regulator of order $Z_{1,k}k^2\sim k^{2-\eta_k}$ for $|q|,|\omega|$ of order
$k$. Thus the convergence of $\eta_{1,k}$ towards 2 must be extremely slow, which is not realized in practice, for the regulator function $R_k$ to vanish in the infrared. While this issue is
irrelevant for most physical quantities, which rapidly converge when $k$ becomes smaller than the mass scale $m_k/v$, the non-vanishing of $R_k$ may artificially stop the flow of $K_k$ thus
preventing the superfluid stiffness to vanish when $k\to 0$. We have improved the discussion of this issue in the manuscript.
6- Sec. 3.2, below Eq.(57): the authors write "The coefficients $\alpha$ and $\beta$ are given by (41)." It is not clear whether, within an active frame perspective, one has to assume the
coefficients to be given by Eq.(41) or this can be somewhat derived in a similar way to what is done for the passive frame approach.
Answer: This is a consequence of the change of variables being linear as shown in Ref.~[21]. We have slightly changed the text after Eq.~(57) and added a reference to~[21].
7- In the conclusion the authors refer to the possibility of studying the Bose fluid by directly using the bosonic fields $\psi$ and $\psi^*$. I believe this discussion deserves to be at least
briefly mentioned in the introduction as well, as it seems to be a little decontextualized.
Answer: We have followed the referee's suggestion and added a paragraph in the introduction.
8- I believe that the authors should improve their conclusion by further stressing (if possible) the potential applications of their method.
Answer: We have slightly expanded the discussion on the two potential applications of our method in the conclusion: the Bose-glass phase of one-dimensional disordered bosons and the systems of weakly
coupled one-dimensional chains.
9- Appendix A.2: I think, for completeness sake, that if the authors do not want to show the explicit form of their flow equations, they should at least refer to other works where they appear.
Answer: The flow equations did not appear elsewhere as they are specific to the two-field formalism used in the manuscript. We could show explicitly the flow equations in the Appendix (it would take
at least two pages) or provide a mathematica file but we are not sure that this would be very illuminating. If the referee thinks otherwise, of course we will be happy to provide the file as a
supplemental material. The flow equations are shown in a separate pdf file (see comments section below).
List of changes
Added "The preservation of the canonical commutation relations between $\partial_{x} \varphi$ and $\vartheta$ along the RG flow turns out to be crucial for a proper physical description of the
system, in particular for the identification of the stiffness of the phase $\vartheta$ as the superfluid density. This differs from the study of a Bose fluid in a periodic potential using the
canonically conjugated variables defined by the creation of annihilation boson fields. In that case, the fields $\psi$ and $\psi^*$ defined at the microscopic scale yield a simple identification of
the superfluid density in the low-energy limit even though their canonical commutation relations are not preserved along the RG flow (see, e.g., [11,12] for an FRG study of the Bose-Hubbard model in
two and three dimensions)."
Page 6 added :" We construct the regulator function $R_k(Q)$ by adapting the usual procedure [2,18] to the two-field formalism used here"
Page 6 added: "In the case of a precision calculation, e.g. when determining critical exponents at a second-order phase transition [33,34], $\alpha$ is fixed by using the principle of minimal
sensitivity. In the sine-Gordon model, the precise value of $\alpha$ is unimportant [2]. In the present study it is therefore sufficient to take $\alpha$ of order unity."
Caption figure 1, added:"In Figs. 1 and 2, $t=\ln(k/\Lambda)$ denotes the (negative) RG time."
Page 10 added: "the field average of $Z_{1x,k}(\phi)$, i.e. $Z_{1,k}=v/\pi\bar K_k$"
Page 12 modified discussion: "A possible explanation comes from the convergence of $\eta_{1,k}=-\dt\ln Z_{1,k}$ towards 2 which makes the regulator of order $Z_{1,k}k^2\sim k^{2-\eta_k}$ for $|q|,|\
omega|$ of order $k$. Thus the convergence of $\eta_{1,k}$ towards 2 must be extremely slow, which is not realized in practice, for the regulator function $R_k$ to vanish in the infrared [3]."
Page 14:" For a linear change of variables, the active frame transformation~(56) is the counterpart of the passive transformation~(38) and the coefficients $\alpha_k(Q)$ and $\beta_k(Q)$ are
therefore given by (41) [25]."
Conclusion, modification:"For instance, this will allow a more accurate study of the Bose-glass phase of a one-dimensional disordered Bose fluid. The previous works using bosonization and FRG are
based on an effective model obtained by integrating out the field $\vartheta$ from the outset [4,5]. This is sufficient to determine the properties related to the density field and its fluctuations
but provides us with little information on the superfluid properties and the correlation function of the phase field $\vartheta$.
The work reported in this manuscript also opens up the possibility to study strongly anisotropic two- or three-dimensional systems, consisting of weakly coupled one-dimensional chains. In these
systems the interchain kinetic coupling $\psi^*_{n}\psi_m \sim e^{-i\vartheta_n+i\vartheta_m}$ depends nontrivially on $\vartheta$ and it is not possible to integrate out this field from the outset.
An RG approach must therefore necessarily consider the fields $\varphi$ and $\vartheta$ on equal footing."
Appendix B : modified discussion.
Published as SciPost Phys. 12, 110 (2022)
Redlined version.
Reply and flow equations. | {"url":"https://scipost.org/submissions/2111.11458v3/","timestamp":"2024-11-03T21:36:36Z","content_type":"text/html","content_length":"43526","record_id":"<urn:uuid:94b58f37-69ce-4c57-8e65-62e22ae87b10>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00282.warc.gz"} |
Costică MustăţaCostică Mustăţa
Prof. dr. Costică Mustăţa – outstanding member of the Institute, at present honorary member.
He was employed at the Institute between 1968-2013, and he served as Secretary of the Scientific Council.
He is a member of the Cluj Team on Numerical Analysis and Approximation Theory.
At present, he serves as a member of the Editorial Board of Journal of Numerical Analysis and Approximation Theory, edited under the auspices of the Romanian Academy, and of the (ISI ranked) journal
Carpathian Journal of Mathematics. (2008-2015?)
Version of July 16, 2017.
In Romanian:
Membru marcant al Institutului (în prezent, membru de onoare).
A fost angajat la Institut în perioada 1968-2013, unde a fost membru al Consiliului Ştiinţific.
In prezent este membru al colectivului de redactie al revistei de specialitate Journal of Numerical Analysis and Approximation Theory, editata sub egida Academiei Romane, precum si la Carpathian
Journal of Mathematics (jurnal ISI).
Personal Data
Date and place of birth 11 June 1942
Băileşti, district Dolj
Education and degrees
1975 Ph.D. in Mathematics, University Babeş-Bolyai, Cluj-Napoca
Scientific advisor: Prof. Dr. Elena Popoviciu
Thesis “The best approximation of the functions with given behavior”
1963-1968 Batchelor Degree, Mathematics
University Babeş-Bolyai, Cluj-Napoca
1953-1956 High school Băileşti, district Dolj
Employment history
1968-1971 Research assistant at Tiberiu Popoviciu Institute of Numerical Analysis
1971-1979 Researcher at Tiberiu Popoviciu Institute of Numerical Analysis
1979-1990 Senior researcher (III) at Tiberiu Popoviciu Institute of Numerical Analysis
1990-1994 Senior researcher (II) at Tiberiu Popoviciu Institute of Numerical Analysis
1998-2004 Associate Professor at North University of Baia-Mare
1994-2013 Senior researcher (I) at Tiberiu Popoviciu Institute of Numerical Analysis
Editorial activities
• Member of the Editorial Board of the (ISI ranked) journal Carpathian Journal of Mathematics (2008-2015?)
• Member of the Editorial Board of Journal of Numerical Analysis and Approximation Theory, edited at the Institute, under the auspices of the Romanian Academy (since ?)
• Deputy Editor-in-Chief of the journal Mathematica, edited at the Babes-Bolyai University, under the auspices of the Romanian Academy (between ?)
• Member of the Editorial Board of the journal Creative Mathematics and Informatics (between 2008-2015)
• Member of the Editorial Board of the journal Southwest Journal of Pure and Applied Mathematics Cameron University, Lawton, USA (between 1997-2004)
Recent research domains
• Approximation and best approximation in function spaces and abstract spaces
• Theorems of extension and relationship with problems of best approximation
• Existence and study of properties for selections associated to metric projections
Past research activity:
• Approximation by linear and positive operators
• Approximation of solutions of boundary problems for differential equations
• Spline functions and applications
• M-ideals in metric spaces
1. GAR nr.4122GR/1998,1990 cu MCT, Theme: Appromaximations of the solutions for PDE by means of spline functionms and wavelets. (coordinatortor).
2. GAR nr.6100GR/2000 cu ANSTI, Theme: Numerical mathods for solving boundary problems for ODE and PDE (cooordinator).
3. GAR nr.65/2001 of the Romanian Academy, Theme: The unknown incremental method applied to diffusion-convection problems (coordinator)
4. GAR nr.46/2002 of the Romanian Academy, Theme: The unknown incremental method applied to diffusion-convection problems (coordinator)
5. GAR no.15/2003 of the Romanian Academy
6. GAR no.13/2004 of the Romanian Academy
Other professional activities
• Scientific secretary at Tiberiu Popoviciu Institute of Numerical Analysis
• Reviewer at “Zentralblatt fur Mathematik” and “Mathematical Reviews”.
• Member in the Expert Committee of CNCSIS.
Didactic activity
• Linear algebra and analytical geometry
• Functional analysis
• The best approximation
• Operators theory
• Applied mathematics in economy | {"url":"https://ictp.acad.ro/mustata/","timestamp":"2024-11-04T17:38:37Z","content_type":"text/html","content_length":"138926","record_id":"<urn:uuid:776310f3-99e6-451d-9257-75061e001b7b>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00607.warc.gz"} |
Author: Georgi Chunev - Lead Graphics Programmer at Gameloft Sofia
In this article, I will introduce you to the mathematical models that were used for defining fuzzy trajectories for various projectile traces in games like War Planet Online: Global Conquest (WPO)
and Blitz Brigade: Rival Tactics (BBRT). Although I will explain the basics, the article is intended for people who are comfortable with reading about parametric curve models and the mechanics of a
At the time that we did this work, our goal was to give our game designers the chance to create interesting oscillations and swirls for the different projectiles in their game, while maintaining
accuracy of the shots and physical plausibility.
We first looked at ways for adding fuzziness to linear paths that could connect any two points in the scene. The models that we derived could easily be applied to any parametric curve model for the
base of the path. In the paragraphs that follow, I will go over the case of turning both linear paths and spline curves fuzzy. I will also go over how one could parameterize a physically accurate
ballistic projectile path and apply the same fuzzy path deformations to the resulting model. Where applicable I will cite real projects where these models were tested.
Turning a Linear Path Fuzzy
1 Modelling Displacements
At its heart, adding fuzziness to a trajectory or a path boils down to mapping a displacement function to the model that you have used to describe the given path.
Figure: Visually shows what we mean by adding a displacement to a base trajectory. Offsets defined by the displacement function are added to the base curve along some chosen direction normal to the
curve. Image source: https://slideplayer.com/slide/14885749/ , slide 4.
Mathematically, this process can be described as follows:
Both the choice of a base trajectory function, B(t), and a displacement function, D(t), will be the topics of the chapters to come and we will look at several interesting options to consider. In all
cases, keep in mind the above expression for f(t) as it shows the most general and intuitive form of what we will be doing.
2 Modelling Lines and Curves
A path can be mathematically represented as a curve or line in space, and usually one has to choose between using some variation of an implicit curve model or a parametric curve model. We will be
using parametric models entirely, with the parameter t usually in the range [0;1] and covering the path from start to finish. Still, it is worth to briefly discuss the differences between the two
model classes and the reasons for choosing the parametric approach over the implicit one.
Looking at the linear path equations above, you might recognize the following two properties. First, if you are asked to determine if a random point in the scene belongs to the path, you are best off
using the implicit form. Only points that belong to the line will result in upholding the two sides of the equation equal. Second, if you are asked to precisely trace the described path, one point
after another, you are best off using the parametric form. Moving along a parameterized path requires simply advancing the value of the single parameter in the equation (t). In general, implicit
forms represent tests/conditions for belonging to a given set and find applications in areas like collision detection and implicit geometric models for ray traced rendering; while parametric forms
give a direct way to traverse a given set of points and find use in cases like shooting rays, drawing splines, and even performing interpolation.
(Comment: The above is an alternative implicit equation for a line in 2D. Note, that in 3D the equation no longer describes a line, but a plane, with normal N)
As noted above, we will be using parametric models. Another side advantage of parametric forms is that they do not depend on the dimensionality of the space, in which you are working. A parametric
line connecting two points in space has the exact same form regardless of whether these points are in 2D, 3D, or any other dimension for that matter. Because of this, we will be able to do most of
our work in 2D and then easily apply these results to 3D scenes.
3 Modelling Oscillations
To model the planar oscillations for our fuzzy trajectories we chose to start with simple sine functions. The wave parameters that we exposed to our game designers included an amplitude (A), a
wavelength (Lambda), and a phase (phi). An angular frequency can also be easily computed as (ⲱ = 1/Lambda).
We next introduced a few simple, but very useful modifications to the starting sine model.
First of all, we had to make sure that for paths that are short relative to the desired oscillation amplitude we would proportionally diminish the fuzziness effect. This is needed so that for closer
shots the projectile would not start its course in a weird direction, but instead would go towards the target more directly. To achieve this effect, we multiplied the amplitude by a factor of:
Second, we had to rescale the parameter t to the actual length of the path in world space. The result is that the distance to the target does not affect the observed frequency of oscillations, as it
is now defined in world space:
Third, we wanted to give our designers control over how fuzzy a trajectory would be at its beginning and ending. This allowed us to make projectiles that either start off more straight, before
beginning to wobble, or, inversely, disperse initially, but end their trajectory with a sharper, precise hit. The latter option can be used to create the effect of the projectile having locked onto
the target in midflight; while the former looks a bit more natural, as most people would find it strange if, because of fuzziness, the projectile were to leave the muzzle in some direction that is
way off-target. Both path corrections were achieved simultaneity using amplitude falloffs:
One last amplitude modification, which we found useful, was the addition of a clamped sinusoidal scale factor, which further smoothed the start and end of the trajectory. The clamping factor was
exposed to the designers as an amplitude cutoff percentage parameter:
Taking all of the above modifications into account, we can write the final fuzzy linear trajectory model as:
We kept the phase parameter (phi), as we found it to be useful for creating more diverse trajectories. As concrete examples, the parameter could be set at random for every fired shot, or could be
made to vary for each projectile simultaneously fired by a multi-muzzle unit.
Also, note that we add the oscillation as a displacement in the orthogonal direction of the linear trajectory path. Expressed in this way, the equation holds both in 2D and 3D. For the 2D case, we
obtain the normal direction by applying a 90Degree counter-clockwise rotation to the tangent vector. In 3D, a choice has to be made depending on the desired effect. For top-down games, it may be best
to apply the fuzziness effect in a plane parallel to the ground. In other cases (e.g., view aligned effects, or 3D shooter scenes), other choices can be meaningful, too.
4 Modelling Swirls
As was described in the previous section, adding oscillations to our trajectories simply meant adding modified sinusoidal displacements along a chosen normal direction, perpendicular to the line
towards the target. Swirls can be implemented in an almost identical manner, but we also need to choose a binormal displacement direction in addition to the tangent and normal vectors. Since our base
path is linear, it is simplest to use our known forward direction vector and our previously chosen normal vector and construct the binormal as:
While one could experiment with all kinds of spirals and radial functions, we chose to implement our swirls using a simple helix. This choice naturally extends the sine functions from our oscillation
models and thus provides the exact same parameters for our game designers to tweak.
The final model looks like this:
Note, that we preserved the same modifiers that allowed for extra control over the beginning and ending sections of the trajectory as well as the handling of short paths as in our oscillation model.
In-Game Examples
The footage below is from a prototype scene from the time that we first began to experiment with ballistic and fuzzy projectile paths.
You may notice the resemblance between this scene and the battle scenes in WPO. Both WPO and BBRT are games that use our fuzzy trajectories via a C++ implementations of the models described in the
previous section.
WPO makes heavier and more noticeable use of fuzzy trajectory projectiles, compared to BBRT. At present, both games use linear fuzzy paths only. Nevertheless, in a lot of cases linear base paths
simply do not look natural. True projectile trajectories are parabolic and may be approximated by linear paths only for very high-speed projectiles. Because of this, we did implement a ballistic
variant of the fuzzy paths, which I will describe in the last section of this article. Still, this model was used only in prototypes, since in some common cases that we faced going for physical
accuracy was too restrictive for the game designers.
Specifically in the case of WPO, a transition to turn-based battle mechanics made the use of true ballistics impractical. It turned out that it was more important to be able to guarantee that each
turn of fired shots can be completed within a very strict timeframe - with all projectiles having hit their targets. Increasing projectile speeds could solve the problem of hitting targets on time,
but ends up changing the parabolic paths to ones that look more linear anyway. Here the best solution is to go for a model that is less physically accurate, but still physically plausible. As we will
discuss in the next section, adding our fuzzy displacements to a quadratic Bézier curve is just the trick that we need in such cases. In reality, we have not had the time to revisit this problem in
the context of WPO, but the fuzzy spline path functionality could be added to the game with some future updates.
Fuzzy Splines
As noted above, using linear segments as the base of our trajectories works only for relatively fast moving projectiles or very short travel distances. For most other cases, the base of the
projectile path would have to be curved, because of the pull of gravity. Again, we would prefer the chosen curve model to be parametric, in order to fit as a base for our fuzzy displacements. Splines
meet our requirements very well. There are different kinds of splines, depending on which of the spline control points the final curve passes through and how much control there is over the tangents
and derivatives of the curve. For the purposes of modelling a projectile’s trajectory, we need a curve that passes exactly through some desired starting and ending points and leaves our firing muzzle
at some desired angle. With these requirements in mind, it makes most sense to use Bézier curves, and even the simplest, quadratic version can suffice.
Coincidently, a true ballistic path is also described by a quadratic equation - a parabola. I will discuss this topic further in the next section, but, for now, it is good to know that we can obtain
a physically plausible trajectory by using quadratic Bézier curves. The parametric formula for such a base curve is given by:
[Note: The above formula is simply the result of iterative application of linear interpolation, with every step of the iteration adding one more degree to the resulting curve.]
Before we can apply our fuzzy displacements to the spline model, we need to obtain a tangent, normal, and binormal vector for the trajectory. Those who are interested in doing something more advanced
can look into obtaining a Frenet frame. For our purposes, we are going to use the same techniques and conventions described in the Modelling Swirls subsection above, which give us:
With a TBN frame at hand, applying fuzziness to a Bézier curve can be done by adding the following displacements to a Bézier base model:
For the length of the path, we used the linear distance between the starting and ending points. This is a simplification, which is fine for our needs. If you need extra precision on this, consider
using some more advanced method for obtaining an arc length for a Bézier curve. Even using several linear segments to approximate the travelled path could suffice. In our case, will effectively are
using just one such segment.
The resulting final formulas allow us to apply our fuzzy paths model to a new range of projectiles - ones with high-arcing trajectories. These include shots fired from cannons, howitzers, grenade
launchers, mortars, and the like.
Fuzzy Ballistic Trajectories
As was established in the previous section, with the use of quadratic Bézier curves one can get physically plausible results. Still, in some common cases it may be desirable to compute a true
ballistic trajectory. Using the standard equations of motion and assuming an initial velocity (v0), no air drag or friction, and constant acceleration, we can come to the following position function:
Here the magnitude of gravity is g ~= 9.8m/s^2, and later we will also use g-vector and g-hat to represent the directed and unit-length vectors for gravity, respectively. The expressed position
function already is in parametric form, which is what we need. We specify a starting point (p[start]) and a velocity vector (v[0]) and can express the path parameterized with respect to time, t.
Still, for most of our use cases, we needed a different parameterization – one that more naturally extends our linear trajectories, which take as input a starting and ending point for the path.
Therefore, we decided to solve for the initial launch angle based on the desired target location and the available projectile speed:
If the speed of the projectile is not high enough relative to the distance to the target, the solutions for theta are imaginary, and thus no real launch angle can result in a trajectory that hits the
target. In cases like that, it may be best either to not shoot at all, or to launch the projectile at a 45° angle - guaranteeing maximum range, despite coming short of the target. On the other hand,
when real valued solutions exist, you will have a choice between two possible shot angles. In most cases, it will make more sense to chose the smaller of the two solutions (using the ‘-’ sign when
computing theta). This gives the more direct path. A caveat with this choice might be that the resulting path could become more straight than you want at higher launch speeds. Alternatively, you can
solve for the high-arcing launch angle that gives a less direct path.
Having computed the launch angle, theta, we are ready to calculate the initial velocity as well as the time at which the trajectory will hit the target point:
The parametric base for our ballistic models, with a parameter t in the standard range [0;1], can now be expressed using the computed initial velocity and flight time:
To get to the final fuzzy ballistic trajectory model, we have to choose a TBN triplet. Doing this is no different than with the previous cases that we discussed, see the Fuzzy Splines section for
more details. The final models can be written as:
For more details on implementing ballistic trajectories you can refer to this article. As was discussed in the previous section, if you want to have full control over the arcing of the path you can
consider less physically accurate solutions.
The article showed that adding fuzzy displacements to a parametric base model for a path or trajectory can be done in a fairly straightforward and generic way. Two specific displacement functions
were presented as examples - customized sinusoids and helixes. Both give a simple set of parameters, which can be used by game designers to achieve a wide variety of curve behaviours. While these
fuzziness models can be applied to any parametric base function, the ones that have been found to be of particularly practical use include: linear base paths; true ballistic paths; and spline-based
curved paths.
Several of these models have already been used in some of Gameloft’s titles. In particular, War Planet Online: Global Conquest (WPO) and Blitz Brigade: Rival Tactics (BBRT) make use of fuzzy
projectiles in their battle scenes. While all current use cases for our fuzzy paths have been battle related, the same models can be used in many other contexts. The first that comes to mind is hair
simulation, where each hair filament is a fuzzy curve. Fluid flow is another context where fuzzy curves can find application. | {"url":"https://www.gameloft.com/techblog/en/blogs/fuzzy-paths","timestamp":"2024-11-07T22:23:54Z","content_type":"text/html","content_length":"58959","record_id":"<urn:uuid:1e90c13a-a92c-48b0-a9ef-4af5745f9775>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00692.warc.gz"} |
[Cancelled] Heike Fassbender - A new framework for solving Lyapunov (and other matrix) equations
Dates: 27 March 2020
Times: 14:00 - 15:00
What is it: Seminar
Organiser: Department of Mathematics
Speaker: Heike Fassbender
Prof. Heike Fassbender from University of Oxford will be speaking at the seminar. Abstract: We will consider model order reduction for stable linear time-invariant (LTI) systems \y=Cx\ with real,
large and sparse system matrices. In particular, $A$ is a square $n \times n$ matrix, $B$ is rectangular $n \times m,$ and $C$ is $p \times n.$ Among the many existing model order reduction methods
our focus will be on (approximate) balanced truncation. The method makes use of the two Lyapunov equations \A\mathfrak{P}+\mathfrak{P}A^T=-BB^T,\ and \\mathfrak{Q}+\mathfrak{Q}A=-C^TC.\ The solutions
$\mathfrak{P}$ and $\mathfrak{Q}$ of these equations are called the controllability and observability Gramians, respectively. The balanced truncation method transforms the LTI system into a balanced
form whose controllability and observability Gramians become diagonal and equal, together with a truncation of those states that are both difficult to reach and to observe. One way to solve these
large-scale Lyapunov equations is via the Cholesky factor–alternating direction implicit (CF–ADI) method which provides a low rank approximation to the exact solution matrix $\mathfrak{P}$, $\
mathfrak{Q}$ resp.. After reviewing existing solution techniques, in particular the CF-ADI method, we will present and analyze a system of ODEs, whose solution for $t \rightarrow \infty$ is the
Gramian $\mathfrak{P}.$ We will observe that the solution evolves on a manifold and will characterize numerical methods whose approximate low-rank solution evolves on this manifold as well. This will
allow us to give a new interpretation of the ADI method.
Heike Fassbender
Role: Professor of Mathematics
Organisation: AG Numerik Technische Universität Braunschweig
Travel and Contact Information
Find event
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How do you find the area of a rectangle with length of 6 inches and width of 2 feet? | Socratic
How do you find the area of a rectangle with length of 6 inches and width of 2 feet?
1 Answer
The area is $144 {\text{in}}^{2}$.
Area of a rectangle can be found using this formula: #(# Length $\times$ Width #)#
Convert the units to the same unit,
Width$= 2 \times 12$
$\textcolor{w h i t e}{\times \times} = 24 \text{in}$
Apply formula,
Area$= 24 \times 6$
$\textcolor{w h i t e}{\times x} = 144 {\text{in}}^{2}$
Impact of this question
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Minimum stack / Minimum queue¶
In this article we will consider three problems: first we will modify a stack in a way that allows us to find the smallest element of the stack in $O(1)$, then we will do the same thing with a queue,
and finally we will use these data structures to find the minimum in all subarrays of a fixed length in an array in $O(n)$
Stack modification¶
We want to modify the stack data structure in such a way, that it possible to find the smallest element in the stack in $O(1)$ time, while maintaining the same asymptotic behavior for adding and
removing elements from the stack. Quick reminder, on a stack we only add and remove elements on one end.
To do this, we will not only store the elements in the stack, but we will store them in pairs: the element itself and the minimum in the stack starting from this element and below.
stack<pair<int, int>> st;
It is clear that finding the minimum in the whole stack consists only of looking at the value stack.top().second.
It is also obvious that adding or removing a new element to the stack can be done in constant time.
• Adding an element:
int new_min = st.empty() ? new_elem : min(new_elem, st.top().second);
st.push({new_elem, new_min});
• Removing an element:
int removed_element = st.top().first;
• Finding the minimum:
int minimum = st.top().second;
Queue modification (method 1)¶
Now we want to achieve the same operations with a queue, i.e. we want to add elements at the end and remove them from the front.
Here we consider a simple method for modifying a queue. It has a big disadvantage though, because the modified queue will actually not store all elements.
The key idea is to only store the items in the queue that are needed to determine the minimum. Namely we will keep the queue in nondecreasing order (i.e. the smallest value will be stored in the
head), and of course not in any arbitrary way, the actual minimum has to be always contained in the queue. This way the smallest element will always be in the head of the queue. Before adding a new
element to the queue, it is enough to make a "cut": we will remove all trailing elements of the queue that are larger than the new element, and afterwards add the new element to the queue. This way
we don't break the order of the queue, and we will also not loose the current element if it is at any subsequent step the minimum. All the elements that we removed can never be a minimum itself, so
this operation is allowed. When we want to extract an element from the head, it actually might not be there (because we removed it previously while adding a smaller element). Therefore when deleting
an element from a queue we need to know the value of the element. If the head of the queue has the same value, we can safely remove it, otherwise we do nothing.
Consider the implementations of the above operations:
• Finding the minimum:
• Adding an element:
while (!q.empty() && q.back() > new_element)
• Removing an element:
if (!q.empty() && q.front() == remove_element)
It is clear that on average all these operation only take $O(1)$ time (because every element can only be pushed and popped once).
Queue modification (method 2)¶
This is a modification of method 1. We want to be able to remove elements without knowing which element we have to remove. We can accomplish that by storing the index for each element in the queue.
And we also remember how many elements we already have added and removed.
deque<pair<int, int>> q;
int cnt_added = 0;
int cnt_removed = 0;
• Finding the minimum:
int minimum = q.front().first;
• Adding an element:
while (!q.empty() && q.back().first > new_element)
q.push_back({new_element, cnt_added});
• Removing an element:
if (!q.empty() && q.front().second == cnt_removed)
Queue modification (method 3)¶
Here we consider another way of modifying a queue to find the minimum in $O(1)$. This way is somewhat more complicated to implement, but this time we actually store all elements. And we also can
remove an element from the front without knowing its value.
The idea is to reduce the problem to the problem of stacks, which was already solved by us. So we only need to learn how to simulate a queue using two stacks.
We make two stacks, s1 and s2. Of course these stack will be of the modified form, so that we can find the minimum in $O(1)$. We will add new elements to the stack s1, and remove elements from the
stack s2. If at any time the stack s2 is empty, we move all elements from s1 to s2 (which essentially reverses the order of those elements). Finally finding the minimum in a queue involves just
finding the minimum of both stacks.
Thus we perform all operations in $O(1)$ on average (each element will be once added to stack s1, once transferred to s2, and once popped from s2)
stack<pair<int, int>> s1, s2;
• Finding the minimum:
if (s1.empty() || s2.empty())
minimum = s1.empty() ? s2.top().second : s1.top().second;
minimum = min(s1.top().second, s2.top().second);
• Add element:
int minimum = s1.empty() ? new_element : min(new_element, s1.top().second);
s1.push({new_element, minimum});
• Removing an element:
if (s2.empty()) {
while (!s1.empty()) {
int element = s1.top().first;
int minimum = s2.empty() ? element : min(element, s2.top().second);
s2.push({element, minimum});
int remove_element = s2.top().first;
Finding the minimum for all subarrays of fixed length¶
Suppose we are given an array $A$ of length $N$ and a given $M \le N$. We have to find the minimum of each subarray of length $M$ in this array, i.e. we have to find:
$$\min_{0 \le i \le M-1} A[i], \min_{1 \le i \le M} A[i], \min_{2 \le i \le M+1} A[i],~\dots~, \min_{N-M \le i \le N-1} A[i]$$
We have to solve this problem in linear time, i.e. $O(n)$.
We can use any of the three modified queues to solve the problem. The solutions should be clear: we add the first $M$ element of the array, find and output its minimum, then add the next element to
the queue and remove the first element of the array, find and output its minimum, etc. Since all operations with the queue are performed in constant time on average, the complexity of the whole
algorithm will be $O(n)$.
Practice Problems¶ | {"url":"https://gh.cp-algorithms.com/main/data_structures/stack_queue_modification.html","timestamp":"2024-11-05T19:02:31Z","content_type":"text/html","content_length":"141585","record_id":"<urn:uuid:80c28d1c-478e-4b44-b710-e0f5c0681f8e>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00681.warc.gz"} |
Measuring the speed of water waves from distance and time | Oak National Academy
Hello, this lesson is about measuring the speed of water waves from measurements of distance and time.
It's from the physics unit Measuring Waves.
My name is Mr. Fairhurst.
By the end of this lesson you should be able to explain how to use measurements of distance and time to calculate the speed of water waves accurately.
And in doing so, you should become much more familiar with these terms and these are the keywords that you're going to come across during the lesson.
If at any point in the lesson you want to come back and look at the definitions of those key terms, pause the video and come back to this slide and have a look.
Now this lesson is split into two parts.
The first part we're going to look at how you can measure the speed of water waves, and this part of the lesson will finish with a practical activity in which you're going to measure the speed of
water waves.
And then in the second part of the lesson we're going to see how you can use repeat measurements to improve the accuracy of your final measurements.
Okay then, so let's make a start.
What we know about a water wave is that the speed of that wave will vary depending on the depth of the water.
That's because if the water changes the depth, the water wave is travelling through a different medium and therefore its speed is going to be affected.
Now we can measure the speed at different depths accurately and easily if we use laboratory conditions and set up the measurements so we can make them easy and straightforward.
Now to do that, we can use water in a plastic tray with straight edges, and if we lift one end and drop it onto the table, then that will cause a water wave with a straight edge to go forward and
backwards along the length of the tray, which is easily measured.
Okay, so have a quick go at this true and false question.
Pause the video while you do so and start again once you're ready.
Okay, so how did you get on? It is true that laboratory conditions make it easy to measure accurately the speed of a water wave at a certain depth.
And the reason is reason A, if you go into nature, into a river or the sea, the depth of the water often varies.
And in the laboratory we can make sure that we keep the depth the same all of the time so that we can measure the speed in a particular depth much more accurately.
And we measure the speed using this equation, it's the speed equation.
Speed equals distance divided by time.
And we need to take measurements of distance and time off our wave in the tray in order to calculate the speed.
We can measure the length using a metre ruler and we can measure that up to the nearest millimetre or to the nearest 0.
1 centimetre because that's the smallest division on the ruler.
And when we write that down, we write 35.
3 centimetres, which is our measurement here, plus or minus 0.
1 centimetres.
That's indicating how accurately we can take the measurement to.
It's the nearest small division on the ruler.
To measure the time that a wave takes to travel the length of a tray, we use a timer.
Now if we're timing with a stopwatch, we can start it or stop it a moment too late or a moment too soon, and that can introduce errors and uncertainties in our measurement.
And the difference between the actual time that the wave takes to travel and the measured time, which includes any mistakes that we've made are caused by what we call random errors.
Sometimes we can start the stopwatch too soon, sometimes we can start it a little bit too late, so a random error can be too big or too small.
Okay, just have a quick go at this question.
Which of these is best described as a random error? Pause video for a moment and start again once you've made your selection.
Okay then which of these best described a random error? And the answer is A, it's a person, it's made by a person measuring the temperature wrongly.
They might misread the scale a little bit too high or a little bit too low.
Now when we take a measurement, it's pretty much impossible to know for sure exactly the size of the error in that measurement.
But what we can do is we can take a set of repeat measurements like these ones for time and use those to estimate the uncertainty in the measurement caused by random errors.
Now for these measurements of time, the mean average is 0.
50 seconds.
The smallest reading is 0.
45 and the largest is note 0.
So we can guess that the smaller reading is a little bit too small, the larger reading is a little bit too big, and we can work out the difference between the largest and the smallest measurement and
call that the range of measurements.
In this case it's 0.
10 seconds.
And what we can do with those measurements, we can use those to estimate the size of the uncertainty and we say it's equal to half of the range, or 0.
05 seconds.
Now in this particular example the larger measurement is 0.
05 seconds higher than the mean average and the smaller measurement is 0.
05 seconds below the mean average.
So that's the variation in our results compared to the mean.
So we can use that.
It's not saying that all the errors are exactly 0.
05 seconds, but from our repeat measurements that is the likely size of the maximum uncertainty in our measurements.
We write that down as the time measured is 0.
50 seconds plus or minus 0.
05, which is our uncertainty.
Have a look at this question.
Look at the results and decide from the options which one of those is the size of the uncertainty in these measurements shown.
Pause the video and start again once you've got your answer.
Okay, so how did you get on? The uncertainty is equal to half of the range.
The range is the biggest reading, 1.
20, take away the smallest, 1.
08 seconds, which gives us a range of 0.
12 seconds, and the uncertainty is estimated to be equal to half of that range, or 0.
06 seconds.
So well done if you've got that answer.
If we're getting random errors in our measurements, one thing that we need to think about is how can we make those errors a little bit smaller? How can we reduce the size of those so we get more
accurate results? Now with our experiment at the moment, one way we can do that is to film the wave moving forwards and backwards with a timer.
If we're timing the time the wave takes to get to either end of the tray, then what we can do is we can play that film back in slow motion, maybe use freeze frame when it reaches the end of the tray,
and record the exact time shown on the timer in order to take our measurements.
And that's going to mean that we don't rely quite so much on our reaction times.
Another way to reduce the size of the error is to not just measure the wave going one length of the tray, but to measure it going two or three lengths of the tray.
And if we do that, we will need to divide our final measurement of time by three in order to get the time for one length of the tray.
But when we divide the time by three, we also divide any errors by three, any mistakes by three, so we get three times more accurate results by measuring three times longer distance.
Have a go at this question.
What does not reduce the random error when measuring the time it takes for one full swing backwards and forwards and back to where you started from? Pause the video whilst you have a go at this
question and start again once you're ready.
Okay, so what do you think? Which does not reduce the random error? And the correct answer was repeating the measurement.
We can estimate the size of the error if we take repeat measurements, but we can't reduce it.
What we can do is we can film the swing with a timer in view.
So we take away that reaction time from the equation.
And we can also time five swings and divide by five.
So that would divide the size of our error by five as well.
So both of those would be good ways to reduce random error.
Well done if you've got this question correct.
What I'd like you to do now is to have a go at that investigation to measure and compare the speed of a waterway for three different depths in a plastic tray and to use some of those strategies that
we've talked about to reduce the size of the errors involved.
For each depth, I'd like you to take three repeat measurements and to measure at least three different depths, probably between one and two centimetres deep.
If you have more time, by all means take a few extra depths of readings.
Pause your video whilst you do that and start again once you've got all of your results and are ready to move on.
Okay, so hopefully you've got a good set of results from your investigation.
Here's a sample set of results that I took that we're going to use later on in the lesson.
They won't be exactly the same as your results, but yours should be fairly similar.
Right, we're going to move on to part two of the lesson now using repeat measurements to improve the accuracy.
So we've taken repeat readings for our investigation, let's see how we can use those to improve accuracy in our final results.
When we take repeat measurements, sometimes we get clear mistakes that are very different from the other measurements.
And these we call anomalous results.
Here's a set of readings that include an anomalous result.
Three of the readings are fairly similar and one is significantly different and that reading, 0.
92 seconds, is anomalous.
It's a clear mistake, so we get rid of that answer.
We cross it out by putting a line through, making sure we can still read the numbers in case we need to come back later.
Occasionally in science we find that an anomalous result actually is quite an important one.
So always record your anomalous results, check them if you can.
So in this case, rather than stop at three results, I noticed this anomalous result as it was taking the the third result.
So I took a fourth repeat reading just to double check that it really was an anomalous result.
Have a look at these results yourself and what I'd like you to do is to spot which of these measurements was an anomalous result.
Pause the video whilst you do so and start again once you're ready.
Okay, so which one did you see? It was 3.
86 seconds here.
It was not too different from the other two, but it was significantly different.
It was a good 0.
3 or 4 seconds less than the other two readings.
So well done if you've got that one right.
This one also was wrong.
It's a little bit harder to see.
They're all three point something, but 3.
04 is again, it's nearly 0.
2 of a second less than the other two readings, so we can cross that one out.
And in this instance it might be sensible to take a fourth repeat reading.
So well done if you spotted that one as well.
And with the final line of results, they're all relatively close together and I don't think any of those is an anomalous result.
Now as I've just said, anomalous results are not used to calculate the mean.
We cross them out and we don't use them because they're clearly mistakes.
Here's a set of readings that we had before with the anomalous results crossed out, and what we need to do now is to calculate the mean value.
We know that each of the values we've measured might be a little bit too high or a little bit too low, so we get a more accurate value if we calculate the mean.
Now for the eight millimetre depth, we calculate the mean by adding the values together and dividing by the number of values that we've added together.
So in this case, we've added them together and divided by two, and our mean value is 4.
175 seconds.
We write that down as 4.
18 seconds, which is as accurate as we've taken the measurements to.
We can't give a mean average that's more accurate than our actual measurements.
If we do the same for the next two rows, we get a mean of 3.
19 seconds, which rounds up from the average of 3.
185, and for the final row it was 2.
35 seconds.
So there are mean average values for each of those depths.
If you use your results, you'll get slightly different values, but that's okay.
Have a look at this question.
Laura takes some measurements of time for another water wave.
What is the mean of her times? Be careful about any anomalous results.
Pause the video, calculate your answer, and start again once you've got it.
Okay, so how did you get on? First of all, we need to spot any anomalous results, and in this instance we've got one there.
92 is clearly wrong.
So we cross that out and we do not use it to calculate the mean value.
We add those three other values together, divide by three, and the mean average of those values is 0.
49 seconds.
So well done if you correctly got rid of the anomalous result and calculated the mean correctly.
Now the mean of several measurements is likely to be closer to the true value than any single measurement.
Here's a set of measurements to measure Jacob's height.
Some of them are a little bit too high, some of them are a little bit too low, but on average the mean average is going to be much closer to Jacob's height.
You might by chance get one of those measurements that is exactly his height, but you don't know which one.
So taking an average of several measurements is a better way to be more certain that you are closer to his actual height.
So that's the mean height of these values.
And as you can see, it is close to his actual height, but not quite exactly his actual height.
Have a look at this question.
Pause a video whilst you answer the question and start again once you're ready.
Okay, so how did you get on? Which of these are reasons why repeat measurements usually give more accurate results? Well, B is a correct answer.
Any single measurement might be very different to the true value.
We don't know when we take a single measurement, if it's exactly correct or whether it's very different from what we're actually trying to measure.
We need to take more measurements to check.
And if we take more measurements and we have made a big mistake, we can spot any anomalous results, cross those out, and not use them in calculating our mean average.
We can spot any completely silly mistakes that we've made easily if we take repeat measurements.
And D is also correct.
All of our measurements can be a little bit too big or a little bit too small, and we're never sure which.
They might be spot on, but again, we're never sure which so that is another reason why taking more readings is better, because we can then see which readings are too small and which are too big.
What I'd now like you to do is to look at these results and for each depth to calculate the mean value of the results, to calculate the uncertainty in each of those mean times that you've calculated,
and also to calculate the speed of a water wave for each depth measuring to centimetres per second.
Just pause the video whilst you do that and once you've got all of your answers, start it again and we can check those results.
Okay, so how did you get on? Let's start with the 10 millimetre depth.
Looking at those results, they're all very close together, so we simply add the three results together and divide by three.
And the mean value we get is 3.
We've rounded it to the same number of significant figures as each of the measurements, because that's how accurately we can calculate it to.
For 16 millimetre depth there is an anomalous result, and that one is 2.
88 seconds.
The other two are very, very close together and the 2.
88 is quite different.
Just be a little bit careful here because it's tempting to say that the 3.
02 seconds is the anomalous result because it's just flipped over into the next second.
The mean value of those two results is 3.
00 seconds.
Again, putting.
00 to show that the accuracy is just the same as for the measurements that we took to get to calculate that mean value.
And then on the third row, for 22 millimetre depth, we add those three results together, divide by three, and round to three significant figures again.
So 2.
49 is the mean there.
To calculate the uncertainty, we need to first of all work out the range.
So for 10 millimetres the range is between 3.
65 seconds and 3.
70 seconds, so the range is 0.
05 seconds.
Half of the range is equal to the uncertainty, and half that range is 0.
Now that is a little bit more accurate than what we've got there, so we need to round up to 0.
03 So we said the uncertainty is naught point, plus or minus 0.
03 seconds.
We can do the same for the other two rows.
We get plus or minus 0.
02 seconds and we get plus or minus 0.
03 seconds.
Now to calculate the speed, we could either divide the time by three and use the length of the tray we've got there.
But what I'll do instead, I'll multiply the length of the tray by three, so we get the total distance for those times.
And speed is distance divided by time, and the answers we get are 31.
3 centimetres per second, 38.
3 centimetres per second, and 45.
8 centimetres per second.
And again, you'll notice I've rounded all of those answers to three significant figures, which is equal to the same accuracy that we took our measurements to.
So very well done if you got all of those calculations correct.
Well done to make it to the end of the lesson.
This is a short summary slide that covers the main points that we've gone through during the lesson.
We use the equation speed equals distance divided by time to calculate the speed of water waves by measuring first the distance and then the time taken for the waves to travel that distance.
We took repeat readings so we could check for mistakes in taking each measurement.
And any mistakes that we noticed, any results that were quite different from these results were the anomalous results.
Measurements still vary even if no mistakes are made and the differences are sometimes caused by random errors that can lead to an uncertainty in a measurement.
And the uncertainty we estimated as being equal to half of the range of the repeated measurements.
And the mean value, the mean average, of the repeated measurements usually gives a much more accurate measurement than any single measurement that we can take.
So well done again for reaching the end of the lesson, I do hope to see you next time. | {"url":"https://www.thenational.academy/pupils/programmes/combined-science-secondary-year-10-foundation-aqa/units/measuring-waves/lessons/measuring-the-speed-of-water-waves-from-distance-and-time/video","timestamp":"2024-11-09T22:34:22Z","content_type":"text/html","content_length":"138014","record_id":"<urn:uuid:07d8de65-a7bd-459b-8c6f-e4fcd560302a>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00017.warc.gz"} |
I’m retiring in two weeks so I’m cleaning out my office.
So far, I got rid of almost all paper-work and have split my book-collection in two: the books I want to take with me, and those anyone can grab away.
Here’s the second batch (math/computer books in the middle, popular science to the right, thrillers to the left).
If you’re interested in some of these books (click for a larger image, if you want to zoom in) and are willing to pay the postage, leave a comment and I’ll try to send them if they survive the
current ‘take-away’ phase.
Here are two books I definitely want to keep. On the left, an original mimeographed version of Mumford’s ‘Red Book’.
On the right, ‘Een pak met een korte broek’ (‘A suit with shorts’), a collection of papers by family and friends, presented to Hendrik Lenstra on the occasion of the defence of his Ph.D. thesis on
Euclidean number-fields, May 18th 1977.
If the title intrigues you, a photo of young Hendrik in suit and shorts is included.
This collection includes hilarious ‘papers’ by famous people including
• ‘A projective plain of order ten’ by A.M. Odlyzko and N.J.A. Sloane
• ‘La chasse aux anneaux principaux non-Euclidiens dans l’enseignement’ by Pierre Samuel
• ‘On time-like theorems’ by Michiel Hazewinkel
• ‘She loves me, she loves me not’ by Richard K. Guy
• ‘Theta invariants for affine root systems’ by E.J.N. Looijenga
• ‘The prime of primes’ by F. Lenstra and A.J. Oort
• (and many more, most of them in Dutch)
Perhaps I can do a couple of posts on some of these papers. It might break this clean-up routine.
One Comment
In 1982, the BBC ran a series of 10 weekly programmes entitled de Bono’s Thinking Course. In the book accompanying the series Edward de Bono recalls the origin of his ‘L-Game’:
Many years ago I was sitting next to the famous mathematician, Professor Littlewood, at dinner in Trinity College. We were talking about getting computers to play chess. We agreed that chess was
difficult because of the large number of pieces and different moves. It seemed an interesting challenge to design a game that was as simple as possible and yet could be played with a degree of
As a result of that challenge I designed the ‘L-Game’, in which each player has only one piece (the L-shape piece). In turn he moves this to any new vacant position (lifting up, turning over,
moving across the board to a vacant position, etc.). After moving his L-piece he can – if he wishes – move either one of the small neutral pieces to any new position. The object of the game is to
block your opponent’s L-shape so that no move is open to it.
It is a pleasant exercise in symmetry to calculate the number of possible L-game positions.
The $4 \times 4$ grid has $8$ symmetries, making up the dihedral group $D_8$: $4$ rotations and $4$ reflections.
An L-piece breaks all these symmetries, that is, it changes in form under each of these eight operations. That is, using the symmetries of the $4 \times 4$-grid we can put one of the L-pieces (say
the Red one) on the grid as a genuine L, and there are exactly 6 possibilities to do so.
For each of these six positions one can then determine the number of possible placings of the Blue L-piece. This is best done separately for each of the 8 different shapes of that L-piece.
Here are the numbers when the red L is placed in the left bottom corner:
In total there are thus 24 possibilities to place the Blue L-piece in that case. We can repeat the same procedure for the remaining Red L-positions. Here are the number of possibilities for Blue in
each case:
That is, there are 82 possibilities to place the two L-pieces if the Red one stands as a genuine L on the board.
But then, the L-game has exactly $18368 = 8 \times 82 \times 28$ different positions, where the factor
• $8$ gives the number of symmetries of the square $4 \times 4$ grid.
• Using these symmetries we can put the Red L-piece on the grid as a genuine $L$ and we just saw that this leaves $82$ possibilities for the Blue L-piece.
• This leaves $8$ empty squares and so $28 = \binom{8}{2}$ different choices to place the remaining two neutral pieces.
The $2296 = 82 \times 28$ positions in which the red L-piece is placed as a genuine L can then be analysed by computer and the outcome is summarised in Winning Ways 2 pages 384-386 (with extras on
pages 408-409).
Of the $2296$ positions only $29$ are $\mathcal{P}$-positions, meaning that the next player (Red) will loose. Here are these winning positions for Blue
Here, neutral piece(s) should be put on the yellow square(s). A (potential) remaining neutral piece should be placed on one of the coloured squares. The different colours indicate the remoteness of
the $\mathcal{P}$-position:
• Pink means remoteness $0$, that is, Red has no move whatsoever, so mate in $0$.
• Orange means remoteness $2$: Red still has a move, but will be mated after Blue’s next move.
• Purple stands for remoteness $4$, that is, Blue mates Red in $4$ moves, Red starting.
• Violet means remoteness $6$, so Blue has a mate in $6$ with Red starting
• Olive stands for remoteness $8$: Blue mates within eight moves.
Memorising these gives you a method to spot winning opportunities. After Red’s move image a board symmetry such that Red’s piece is a genuine L, check whether you can place your Blue piece and one of
the yellow pieces to obtain one of the 29 $\mathcal{P}$-positions, and apply the reverse symmetry to place your piece.
If you don’t know this, you can run into trouble very quickly. From the starting position, Red has five options to place his L-piece before moving one of the two yellow counters.
All possible positions of the first option loose immediately.
For example in positions $a,b,c,d,f$ and $l$, Blue wins by playing
Here’s my first attempt at an opening repertoire for the L-game. Question mark means immediate loss, question mark with a number means mate after that number of moves, x means your opponent plays a
sensible strategy.
Surely I missed cases, and made errors in others. Please leave corrections in the comments and I’ll try to update the positions.
Comments closed
In preparing for next year’s ‘seminar noncommutative geometry’ I’ve converted about 30 posts to LaTeX, centering loosely around the topics students have asked me to cover : noncommutative geometry,
the absolute point (aka the field with one element), and their relation to the Riemann hypothesis.
The idea being to edit these posts thoroughly, add much more detail (and proofs) and also add some extra sections on Borger’s work and Witt rings (and possibly other stuff).
For those of you who prefer to (re)read these posts on paper or on a tablet rather than perusing this blog, you can now download the very first version (minimally edited) of the eBook ‘geometry and
the absolute point’. All comments and suggestions are, of course, very welcome. I hope to post a more definite version by mid-september.
I’ve used the thesis-documentclass to keep the same look-and-feel of my other course-notes, but I would appreciate advice about turning LaTeX-files into ‘proper’ eBooks. I am aware of the fact that
the memoir-class has an ebook option, and that one can use the geometry-package to control paper-sizes and margins.
Soon, I will be releasing a LaTeX-ed ‘eBook’ containing the Bourbaki-related posts. Later I might also try it on the games- and groups-related posts…
2 Comments | {"url":"http://www.neverendingbooks.org/tag/games/page/2/","timestamp":"2024-11-12T19:19:36Z","content_type":"text/html","content_length":"40045","record_id":"<urn:uuid:78d79ce3-bdab-4aa3-b31a-93265470d49c>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00588.warc.gz"} |
GARCH_ERRORS - Estimated Errors of the Parameters Values
Returns an array of cells for the estimated error/standard deviation of a given model's parameters.
GARCH_ERRORS ([x], order, µ, [α], [β], f, ν)
Required. Is the univariate time series data (a one-dimensional array of cells (e.g., rows or columns)).
Optional. Is the time order in the data series (i.e., the first data point's corresponding date (earliest date = 1 (default), latest date = 0)).
Value Order
1 Ascending (the first data point corresponds to the earliest date) (default).
0 Descending (the first data point corresponds to the latest date).
Optional. Is the GARCH model long-run mean (i.e., mu). If missing, the process mean is assumed to be zero.
Required. Are the parameters of the ARCH(p) component model: [αo α1, α2 … αp] (starting with the lowest lag).
Optional. Are the parameters of the GARCH(q) component model: [β1, β2 … βq] (starting with the lowest lag).
Optional. Is the probability distribution function of the innovations/residuals (1 = Gaussian (default), 2 = t-Distribution, 3 = GED).
Value Probability Distribution
1 Gaussian or Normal Distribution (default).
2 Student's t-Distribution.
3 Generalized Error Distribution (GED).
Optional. Is the shape parameter (or degrees of freedom) of the innovations/residuals’ probability distribution function.
1. The underlying model is described here.
2. The time series is homogeneous or equally spaced.
3. The time series may include missing values (e.g., #N/A) at either end.
4. For the input argument - ([α]) (parameters of the ARCH component):
□ The input argument is not optional.
□ The value in the first element must be positive.
□ The order of the parameters starts with the lowest lag.
□ One or more parameters may have missing values or error codes (i.e., #NUM!, #VALUE!, etc.).
□ In the case where alpha has one non-missing entry/element (first), no ARCH component is included.
□ The order of the ARCH component model is solely determined by the order (minus one) of the last value in the array with a numeric value (vs. missing or error).
5. For the input argument - ([β]) (parameters of the GARCH component):
□ The input argument is optional and can be omitted, in which case no GARCH component is included.
□ The order of the parameters starts with the lowest lag.
□ One or more parameters may have missing values or error codes (i.e., #NUM!, #VALUE!, etc.).
□ The order of the GARCH component model is solely determined by the order of the last value in the array with a numeric value (vs. missing or error).
6. The shape parameter (ν) is only used for non-Gaussian distributions and is otherwise ignored.
7. For the student's t-distribution, the shape parameter’s value must be greater than four.
8. For GED distribution, the shape parameter’s value must be greater than one.
9. GARCH_ERRORS returns the standard errors for the model's parameters in the following order:
1. $\mu $.
2. ${\alpha _o},{\phi _1},...,{\phi _p}$.
3. ${\beta _1},{\beta _2},...,{\theta _q}$.
4. $\nu $.
Files Examples
Related Links
• James Douglas Hamilton; Time Series Analysis, Princeton University Press; 1st edition(Jan 11, 1994), ISBN: 691042896.
• Tsay, Ruey S.; Analysis of Financial Time Series, John Wiley & SONS; 2nd edition(Aug 30, 2005), ISBN: 0-471-690740.
Article is closed for comments. | {"url":"https://support.numxl.com/hc/en-us/articles/216102563-GARCH-ERRORS-Estimated-Errors-of-the-Parameters-Values","timestamp":"2024-11-06T11:39:49Z","content_type":"text/html","content_length":"35733","record_id":"<urn:uuid:41e0bac4-425c-4414-a316-002fb76d57f2>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00515.warc.gz"} |
Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.
Rigby R.A., Stasinopoulos D. M., Heller G., and De Bastiani F., (2019) Distributions for Modeling Location, Scale and Shape: Using GAMLSS in R, Chapman and Hall/CRC.
Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, 23(7), 1--46, tools:::Rd_expr_doi("10.18637/
Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC.
(see also https://www.gamlss.com/).
Wood S.N. (2006) Generalized Additive Models: An Introduction with R. Chapman and Hall/CRC Press. | {"url":"https://www.rdocumentation.org/packages/gamlss.add/versions/5.1-13/topics/gamlss.ga","timestamp":"2024-11-15T01:50:00Z","content_type":"text/html","content_length":"59829","record_id":"<urn:uuid:c19e0098-1bcf-4572-9e1d-1d0b4cad9305>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00491.warc.gz"} |
Of GTIs, battery sizing, and capacitor banks
I work for a fuel-cell R & D. We're building a 5kW (low-voltage) fuel-cell system that I wanted to tie into a grid using an XW6048, a model which requires a 48V-nominal battery bank to run properly.
Looking around on forums, I found that people sized their battery banks depending on a lot of factors. The "100Ah per 1kW" rule was thrown around a lot. I was not interested in running our 5kW GTI
for long periods of time on batteries alone (full-load for only 10's of minutes at most) so I thought I might be able to get away with sizing a much smaller bank. But it was mentioned that the AC
currents being pulled by the GTI are a concern, and the battery bank needed to be sized to accommodate these large AC currents. There's no rule-of-thumb for this as far as I can tell.
Reflecting on the AC currents further, I realized that it would be better not to place a significant portion of these currents onto our fuel-cells. Fuel-cells prefer a steady power draw.
Therefore, there were two technical challenges to solve:
(1) Save money --> a large, multi-$1k battery bank was not an option
(2) Minimize the AC currents drawn from the fuel-cell stacks
I wondered if placing a large capacitor bank in parallel with the battery bank might help. If a capacitor bank having significantly lower impedance were attached in parallel with a battery bank, AC
currents more likely be pulled from the capacitors than the batteries. And if the batteries and capacitor bank were handling the majority of the AC currents, then very little would need to be pulled
from the system.
I looked around on forums and asked some questions regarding input capacitors, and I never got any good answers. I posted on Solar-Guppy's forum, but he's shut it down now. I'm following up here
instead so that people can learn from what I did.
RMS current / charge analysis
I started with an ideal analysis in order to see what ballpark the input AC currents would be in. From previous experience, I knew that true-sine wave inverters pull in a sine-squared input current
having a 120Hz frequency. The derivation is fun: just assume a constant input voltage and a 120V
/ 60Hz output waveform. Here's the set-up:
P[in](t) = V[IN] * I[in](t)
P[out](t) = V[out](t)^2 / R[load] = V[out](t) * I[out](t)
P[out](t) = P[in](t)
I'll let the reader take it from here. You're trying to find I
(t). Assuming 5kW in and 5kW out (100% efficiency) one can generate data for a single period of the input current waveform in Excel and carry out numerical analyses to get all sorts of relevant data.
The datum I was most interested in was the RMS value of the current. I looked around on the web, and the conclusion I came to is that there is no analytical solution for integrating a sine-squared
waveform. However, you can use Excel to compute the RMS numerically.
From my analysis, I found that for 5kW at 46V input (the minimum necessary for operation) ~77A
was demanded. 77A
is nothing to scoff at obviously: that's all heat-loss type current. One can see why people take the AC currents into account when sizing their battery bank.
The next thing I wondered was how much charge would be removed during a half-cycle of the 120Hz sine-squared current. Integration of the input current over one half-cycle gives this figure: Q =
~0.29C. Next, I calculated how much capacitance is required to hold 0.29C at 46V using the capacitance equation: C = Q / V = ~6.3mF. Finally, on a whim I decided that I didn't want this discharge /
charge cycle to use up any more than 1% of the capacity of my capacitor bank. This would keep the ripple voltage to some minimum. Thus, I would need a 6.3mF / 1% = 0.63F bank. Rough math, I know, but
it's probably close enough.
Bank sizing / pricing
So, at a minimum I needed a capacitor bank of at least 0.63F that could handle at least 77A
Searching around on Digi-Key, I decided to make these my unit capacitor of choice:
. These 0.1F capacitors are only $36.88 apiece and can handle voltages up to 80V, nearly twice my required voltage. These capacitors also have a 21.42A
rating. This RMS capability adds in parallel, so a 0.7F bank would have a total RMS capability of ~150A
, double the computed necessary capability.
This is my solution. And for the heck of it, I just bought 10 of them to catch the price break and make a 1F capacitor bank. I wasn't sure how accurate my analysis was, so I played it safe. I used
copper bar to connect them all in parallel, I connected them to the GTI through a separate circuit breaker, and I wired up a charge / discharge circuit to be able to charge them from the battery
This is what I found. At 5kW / 50V, a sine-squared waveform of ~195A
/ ~90A
is demanded by the GTI. With the 5kW power system, the 48V battery bank, and the 1F capacitor bank all connected in parallel, the AC current waveform is divided between the three as follows: ~9% from
the system, ~14% from the batteries, and a whopping 77% from the capacitors!
I consider this a success.
Was it worth it?
In total, the capacitor bank probably cost me between $400 and $500, and the batteries I'm using are semi-truck batteries for about $125 apiece = $500. The batteries I'm using are this kind:
. I think they are used in semi-trucks and such. I had the fellow I purchased them from call the company to ask what the effective capacity was of one of these batteries, and it comes to around 80Ah
@ 25A discharge. Not much.
So, here are my options.
(1) I could have purchased a top-of-the-line, 48V / 500Ah battery bank for around $3,750, assuming the pricing here:
. Maybe I could have gotten away with a 400Ah bank for $3000. Maybe I could have purchased batteries individually and shaved a little of that price. Whatever. It's a lot of money for something I
don't need. It's also a lot of space, as these batteries are typically quite large and you typically need more of them because they are 6V batteries, not 12V.
(2) I built a battery / capacitor bank for around $1000, and it takes up less space. The batteries are also more readily available, so if one happens to go
, no problem.
Option #2 is still the best option for me.
Design criteria revisited
Let's look at the design criteria again for the sake of academics.
(1) One should design based on necessary RMS current capability. At 5kW, 27% more RMS current was demanded by the GTI than calculated (because I didn't take into account losses, rippling voltages,
etc.). I propose a 1/3 correction factor as a good rule-of-thumb. Redoing the calculations, at 6kW (full-load capability of the XW6048) and 46V, the supposed RMS amperage pulled by the GTI is ideally
. Using the 1/3 correction factor, this comes to ~113ARMS. I like round numbers, so let's round up to 120A
for 6kW. In fact, dare we say
20A[RMS] per 1kW
? I think this is sound, because I took other data at loads from 500W to 4000W in 500W increments, and the RMS current demand scales linearly.
(2) One could choose to scale back on RMS requirements based on the fact that the capacitor bank will see only about 3/4 of the predicted AC current. Yet, why not give yourself some room to breathe?
Thus, the
rule is 15ARMS per 1kW, but the safe rule is 20ARMS per 1kW.
(3) One should design based on necessary capacitance to achieve either or both of a few constraints: (a) maximum voltage ripple, and (b) maximum power loss. Remember how I chose an arbitrary 1%
charge / discharge number as my indication of minimum capacitance? Well, one could get a lot more detailed. Capacitors have certain ESR specs at 120Hz. This ESR contributes to power loss in the
capacitor, but it also contributes to voltage ripple. The voltage ripple is simply the ESR multiplied with the RMS current (V = I * R). The power loss is the voltage ripple multiplied with the RMS
current (P = I^2 * R). Honestly, I think voltage ripple is more important, as it has more of an effect on the performance of the GTI. For a capacitor bank built from 6 of the capacitors I selected
above, I calculate only 13W lost over the whole bank (< 2W per capacitor) for 120A
. I doubt the temperature rise in the capacitor would be detectable.
(3a) Therefore, design capacitance based on maximum voltage ripple. This is a more interesting constraint. One contributor is the aforementioned ESR, is calculated easily, but requires an actual ESR
to calculate. The second contributor is much more important: the voltage ripple due to charge / discharge of the bank. Recall: dQ = C(V) * dV. Thus, choose a peak-to-peak ripple-voltage (dV), and
calculate a minimum capacitance based on the dQ seen during a half-cycle of the current waveform. Once you have a chosen capacitor, take the ESR into account by adding the two contributors.
It's late at night on a Friday evening, and I'm still at work writing this. I know, my friends, I know... I am a nerd.
BB. Super Moderators, Administrators Posts: 33,605 admin
Re: Of GTIs, battery sizing, and capacitor banks
You can try sending
Solar Guppy
a PM--He has not been here for a month and a half...
Looks like you did some interesting homework there. Very cool.
One other issue that may or may not affect your work. With MPPT Solar Charge controllers, they seem to have a fairly long Time Constant (10's of seconds?) and some do not appear to respond very
fast to quickly changing battery bank / voltage conditions.
From what I have read here--One problem I see with a "small" bank is if you have a steady state condition (5kW from the "energy source") and the load changes quickly (clouds cover array, GT
Inverter shuts down due to line fault, if off grid, loads change dramatically)--At least with Solar MPPT controllers, there have been problems where they continue pumping current into the battery
bank--And if the bank is too small, it can exceed 72 Volts and fault the XW inverter (hard reset or temp shutdown--depending on firmware revision?) before the MPPT controller shuts down the
current producing the excess voltage...
A 100 AH @ 48 Volts per 1kW of energy source/inverter sink seems to be necessary to sink the excess current until the controller's TC allows it to ramp back on the current. Even AGM's which have
a very low impedance when sourcing current and (some brands?) appear to go into a "chemical" high impedance state when fully charged when presented with "excessive" charging currents.
So--depending on how quickly your fuel cell will drop its output current (without an external/internal current shunt or dump) may limit your minimum AH rating for your battery bank.
My guess anyway.
Near San Francisco California: 3.5kWatt Grid Tied Solar power system+small backup genset
niel Solar Expert Posts: 10,300 ✭✭✭✭
Re: Of GTIs, battery sizing, and capacitor banks
very interesting for sure. i would like to hear the reaction to this from a few people.
• Re: Of GTIs, battery sizing, and capacitor banks
I would like to the the P-P reactive current flow waveform from the battery and fuel-cell to the cap during the off period of the demand waveform. This one of the papers I used for research on
battery models. It has a very good description of hybrid battery-ultracap systems interactions with solar and wind.
Although the improvement in battery lifetime may be small with the addition of ultracapac-
itors for the given load cycles, it is essential to quantify it to conduct a full life cycle savings
analysis. Hence the reduction in battery RMS currents, which plays an active role in the battery
aging process, was quantied for both the solar and wind load cycles using the developed battery-
ultracapacitor model. The battery RMS currents were reduced by 50.5% for the sample solar
load cycle and reduced by 60.9% for the sample wind load cycle. In contrast, the % improvement
per installed ultracapacitor capacity was found to be higher for the solar load cycle. The reduc-
tion in RMS currents is directly proportional to the ultracapacitor contribution to the load and
the developed model can be used to quantify the reduction for various levels of ultracapacitor
contribution. The reported % reduction in battery RMS currents should serve as a foundation
for battery life time prediction which in turn should provide the input for a full life cycle savings
• Re: Of GTIs, battery sizing, and capacitor banks
What is the expected lifetime of the capacitor bank, @ 120 hz ? months, weeks, years? I ran these calcs some time ago, and came up with a capacitor lifetime of 2-3 years. Then their fuses blow.
You put fuses (and their resistance) in the bank right ??
Xantrex has a fancy input capacitor (one of the fault codes is Input Cap Overheat) and I wonder what it is, and what if adding a couple more of them would do. [ note to self, next time I open up
the case, see if it's visible, note mfg & model # ]
Also super caps may not be the right critters, regular low ESR caps may have even lower internal resistance.
Powerfab top of pole PV mount | Listeroid 6/1 w/st5 gen head | XW6048 inverter/chgr | Iota 48V/15A charger | Morningstar 60A MPPT | 48V, 800A NiFe Battery (in series)| 15, Evergreen 205w "12V" PV
array on pole | Midnight ePanel | Grundfos 10 SO5-9 with 3 wire Franklin Electric motor (1/2hp 240V 1ph ) on a timer for 3 hr noontime run - Runs off PV ||
|| Midnight Classic 200 | 10, Evergreen 200w in a 160VOC array ||
|| VEC1093 12V Charger | Maha C401 aa/aaa Charger | SureSine | Sunsaver MPPT 15A
• Re: Of GTIs, battery sizing, and capacitor banks
You have touched on most of the key points, ESR , whether its a battery or capacitor is your primary concern.
As for whats better, its depends!. Having a wonker capacitor bank would only be useful for applications that have short peak demands, such as gridtie operation. For anything of significant load
on your inverter, you will easily meet the 1kw/100 ah SG rule
If you want just GT performance, run higher voltages you no longer need the external capacitor bank and that's what all the current crop of GT inverters are
Buffering a fuel cell is an interesting application and having capacitor bank is likely a better solution, unfortunately, I would guess 99.9999% of XW installs now and into the remainder of my
life time will not be connected to fuel cells, for the rest of us, it will be some form of battery storage.
On a cost basis, its a wash, as your 1K capacitor bank would sell for ~3K retail, not to much different from an off the shelf battery bank
A little history, Trace sold a bunch of SW4048's for grid-tie operation only, and yes, they added a bank of capacitors, this was 1997-8 time-frame. In the end, low-frequency massive transformers
used by these inverters just doesn't make sense economically compared to the GT high frequency inverters prolific in the market. even SMA dropped the big iron designs.
• Re: Of GTIs, battery sizing, and capacitor banks
To follow up...
First of all, I went back to the math and found that the RMS current into the GTI is actually a lot easier to calculate than I first thought. I worked out the math and validated it with the
numerical analysis. Thus, my analysis still applies, but now is made easier for everyone because now it is more analytical than numerical. What I found was:
I[IN,RMS] = I[IN,AVE] / SQRT(2)
where I[IN,AVE] = P[IN] / V[IN]
Taking into account the correction factor indicated by my measured data, I still stand by my 20A
per 1kW rule of thumb.
Second, in some of the additional research I did (papers listed below), it was mentioned that the ESR is not meant as a model for voltage ripple. Thus, I was wrong to say what I did about ripple
voltage determination and ESR as a factor. The capacitance equation is probably the best means for calculating a worst-case voltage ripple.
Third, to the comments.
(1) @ BB
The response time of our system depends entirely upon our system design. We use DC/DC converters to buck up an array of stacks to the same working voltage (~50V for our 48V-nominal bank). We are
monitoring the battery current and the system current dynamically. Everything is controlled by a SCADA software which can achieve updates of our inputs / outputs at periods well below one second.
(2) @ nsaspook
I'm not sure what you are asking. However, since you bring up ultra-caps, I will say this: I looked into the possibility of using ultra-caps and concluded that the cost : benefit ratio was too
high. A large bank of inverter-grade capacitors like those that I am using is probably sufficient.
(3) @ mike90045
Excellent question regarding lifetime. First of all, yes, the capacitors are mechanically fused.
I am not sure how to calculate theoretical lifetime without doing a lot more work. I have found these resources:
What I get primarily from my research is this: keep your caps well below the rated temperature, avoid the rated voltage, and avoid the rated ripple. Now, if my calculations are correct, I am at
least avoiding the rated voltage and rated ripple to a substantial degree. I have not yet been able to detect a temperature rise in the caps at full-load. The bank could, theoretically, last me
for well over 2-3 years.
(4) @ solar guppy
We are looking at higher input voltages for our larger future systems. The XW6048 was our choice because we must currently work at low working voltages. And, yes, I do realize our system is
unique. I only offer my work as an example for others who might be thinking about developing their own RE systems.
Best regards,
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Day 12: Modular exponentiation - Michael MusangeyaDay 12: Modular exponentiation - Michael Musangeya
Today we look at the final component of the mystery project, modular exponentiation. Before today we build a function to check primality, calculate the modular multiplicative inverse and calculate
the lowest common multiple of two numbers. The more perceptive of you will no doubt already know what we’re building.
In mathematics, an exponent refers to the number of times a number is multiplied by itself, for example, 3^2.
A Modulo operation returns the remainder of a division. For example 3 / 2 = 1.5 and 3 mod 2 = 1.
For small numbers, you can calculate these pretty easily. But what happens when you have large numbers? Imagine you have 2046^413mod 300. This stops being easy because the exponent results in very
large numbers. So our task is to calculate modulo equations with exponentials that do not melt our ram.
Efficient Modular exponentiation
In mathematics, we have an identity that relates to modulo equations.
(a⋅b) mod m = [(a mod m)⋅(b mod m)] mod m
This identity means that we can break down our exponent into several modulo operations. e.g. for 3^3 mod 5 can bebroken down into.
[3 mod 5 * 3 mod 5 * 3 mod 5] mod 5Code language: CSS (css)
Notice anything? We can iterate the above trivially and that will reduce our memory footprint and speed up our calculation. The final code looks like this;
pub fn mod_exp(x: i64, n: i64, p: i64) -> i64{
let mut ans: i64 = 1;
let mut x: i64 = x % p;
let mut n = n;
while n > 0 {
if n & 1 > 0 {
ans = (ans * x) % p;
x = x.pow(2) % p;
n = n >> 1;
Code language: Rust (rust)
Honestly, I think I can clean up this code a little more but this is fine for now. We now have the last part of our project. Next time we will be putting it all together and finishing our mystery
project. This is the last chance to guess what I’m building. If you have any guesses as to what involves prime numbers, modular multiplicative inverse, lowest common multiples, and modular
exponentiation let me know on Twitter, @phoexer, and as always happy coding. | {"url":"https://cms.mmusangeya.com/2022/08/12/day-11-modular-exponentiation/","timestamp":"2024-11-05T03:24:37Z","content_type":"text/html","content_length":"71236","record_id":"<urn:uuid:6e5bf9ad-901a-41f6-849b-e739bc1c8fce>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00511.warc.gz"} |
Ultimate Guide to Options Greeks: Strategies for Successful Options Trading
Most investors find options trading to be intimidating chiefly because of the complexity of the option pricing mechanism. The value of options keeps fluctuating quite often due to various reasons,
leaving traders puzzled about what factors cause these price movements or behaviors.
The complexity of option pricing can be unraveled only by exploring the Option Greeks, which are important factors that affect an option’s price and provide informative data for the trader to make
decisions with regard to movements in prices.
How to Use Option Greeks?
Mastery of Option Greeks enhances a trader’s likelihood of forecasting accurate changes in option prices and devising the right strategies. One can trade in options more analytically, thus minimizing
involved risks and maximizing returns.
What Are Delta, Theta, Gamma, And Vega?
Delta: Price Sensitivity Measure
Delta is one of the most significant Option Greeks; it measures the rate of a price option with respect to a one-point move in the underlying. Delta for an at-the-money option would be around 0.5,
suggesting a 50% chance of being in-the-money at expiration.
For instance, given that the delta is 0.5, a $1 increase in the underlying price results in a $0.50 rise in the option’s price. In-the-money options will sustain a delta much closer to 1.0, so the
option price will almost move one-for-one with the underlying.
Theta: The Time Decay Factor
Theta measures the time decay effect on the price of an option. It is the amount by which an option’s price will theoretically drop for every day that passes, all other factors remaining constant.
Since time decay hurts the buyer of an option but benefits the seller, the value of options erodes over time.
For example, an option with a theta of -0.05 would lose $0.05 in value over time. This particularly hits at-the-money options much harder because they possess the maximum rate of time decay.
Gamma: Delta’s Speed of Change
Gamma measures how much the delta will change with a movement of the underlying. A high gamma means that a delta will change aggressively for small changes in the price of the underlying. The
sensitivity presented can push the value of the option to great extremes.
For example, if an option has a Gamma of 0.1 and an underlying asset’s price goes up by $1, the Delta of the option will rise by 0.1. Gamma is maximized in at-the-money options and declines once the
options are moved in/out of the money.
Vega: The Volatility Indicator
Vega measures the effect on the price of the option due to variance in the volatility of the underlying asset. Due to the volatility in the market, prices will increase, which is beneficial for the
buyer but risky for the seller.
For example, an option with a Vega of 0.2 would increase by $0.20 for each 1% rise in the volatility. It is a critical Greek in such turbulent times for the market when spiking volatility creates a
significant shift in the prices.
Option Greeks in Trading Strategies
Straddles and Strangles: Leveraging Delta and Theta
Buying options at different strike prices will include strategies like straddles and strangles. Understanding Deltas and Theta in these strategies helps traders know how much profit or loss would be
incurred by the change in the price of the underlying asset and also with the factor of time decay.
Iron Condors and Iron Flies: Riding the Waves of Gamma and Vega
Advanced strategies, such as iron condors or iron flies, may have all these option legs working at various strike prices and with different expiry dates. Gamma and Vega are crucial for traders with
respect to the risk and reward in changes in the backdrop price and its volatility.
How Market Events Influence Option Greeks
The Impact of Major Market Movements
Large market events, like geopolitical crises or announcements in the economy, can change the prices of options quite fast. Such events generally result in enhanced realized volatility, which
directly hits Vega. Traders have to keep a sharp eye on such situations and modify their strategy to reduce risks.
Bid-Ask Spread and Market Sentiment
Apart from the obvious factor of time value, the bid-ask spread, i.e., the inflexional upper limit of buyers, and the inflexional limit of sellers consequently also affect the option prices.
Reapplied, this spread will increase volatility, which will affect Vega and Delta. Vigilance over the bid-ask spread provides a look at market sentiment.
Practical Tips of Option Greeks
Using Analytical Tools
Tools like Sensibull offer strong and detailed analyses of the Option Greeks. This clearly shows on the screen the Delta, Theta, Gamma, and Vega impact on the option portfolio visualization, allowing
wiser decision-making.
Continuous Learning and Adaptation
The stock market is a dynamic set-up where the learning of trading under options never ends. Books such as “Trading in the Zone” and audiobooks, especially those with summaries available on KUKU FM,
further add to the conceptual details of the working of the markets and the psychology that goes into the framed strategies under options.
Any serious options trader must understand the Option Greeks. The Greeks provide insight into how various factors influence the price of an option, aiding in making informed decisions. Mastering
Delta, Theta, Gamma, and Vega enables traders to develop effective strategies to navigate the option trading space.
The insights provided by Option Greeks enable traders to manage their risks while optimizing their trading strategies. Continuous learning and adapting to ever-changing market conditions is key to
success in the dynamic world of options trading. | {"url":"https://advicefunda.com/ultimate-guide-to-options-greeks-strategies-for-successful-options-trading/","timestamp":"2024-11-03T20:09:32Z","content_type":"text/html","content_length":"193126","record_id":"<urn:uuid:b41badb0-047d-4e59-a495-c321e8c7c3ac>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00322.warc.gz"} |
Characterization and Design of Functional Quasi-Random Nanostructured Materials Using Spectral Density Function
Quasi-random nanostructures are playing an increasingly important role in developing advanced material systems with various functionalities. Current development of functional quasi-random
nanostructured material systems (NMSs) mainly follows a sequential strategy without considering the fabrication conditions in nanostructure optimization, which limits the feasibility of the optimized
design for large-scale, parallel nanomanufacturing using bottom-up processes. We propose a novel design methodology for designing isotropic quasi-random NMSs that employs spectral density function
(SDF) to concurrently optimize the nanostructure and design the corresponding nanomanufacturing conditions of a bottom-up process. Alternative to the well-known correlation functions for
characterizing the structural correlation of NMSs, the SDF provides a convenient and informative design representation that maps processing–structure relation to enable fast explorations of optimal
fabricable nanostructures and to exploit the stochastic nature of manufacturing processes. In this paper, we first introduce the SDF as a nondeterministic design representation for quasi-random NMSs,
as an alternative to the two-point correlation function. Efficient reconstruction methods for quasi-random NMSs are developed for handling different morphologies, such as the channel-type and
particle-type, in simulation-based microstructural design. The SDF-based computational design methodology is illustrated by the optimization of quasi-random light-trapping nanostructures in thin-film
solar cells for both channel-type and particle-type NMSs. Finally, the concurrent design strategy is employed to optimize the quasi-random light-trapping structure manufactured via scalable wrinkle
nanolithography process.
Functional nanostructured material systems (NMSs) fabricated using nanomanufacturing technologies have opened up new possibilities in many engineering applications. One category of NMSs, as
represented by the metamaterial systems [1–3], usually relies on the periodic arrangement of identical building blocks to achieve the desired functionalities. For example, Fig. 1 shows a periodic NMS
design (Fig. 1(a)) and the fabricated sample (Fig. 1(b)) of nanophotonic light-trapping structures to be utilized in thin-film solar cells. Designed using the topology optimization approach [5,6],
the periodic structure exhibits a significant enhancement in light absorption compared with random pattern structures as shown in Fig. 1(c) [4]. However, such periodic NMS with carefully designed
building blocks usually requires expensive and time-consuming top-down nanomanufacturing [7]. Top-down nanomanufacturing refers to those processing methods that reduce large pieces of materials all
the way down to the nanoscale, like carving a model airplane out of a block of wood [8]. Examples of top-down nanomanufacturing include focused ion-beam milling, photolithography, and electron-beam
lithography. Patterning the optimized design in Fig. 1(a) onto a 4 in wafer using electron-beam lithography normally takes more than a week. Such high manufacturing cost impedes the adoption of
functional NMS at industrial scale.
In this paper, we present a new design approach for designing cost-effective functional NMSs with quasi-random structures. In contrast to periodic structures, quasi-random NMSs are naturally formed
via cost-effective, scalable bottom-up processes. Bottom-up nanomanufacturing creates nanostructures by building them up from atomic- and molecular-scale components [9], typical processes of which
include the phase separation of polymer mixtures [10] and the mechanical self-assembly based on thin-film wrinkling [11]. The NMSs obtained via such processes usually comprise no periodic building
blocks but a nondeterministic material distribution governed by underlying structural correlation. The structural correlation, which is determined by the physics of a bottom-up process, largely
affects the macroscopic performance of quasi-random NMSs.
Multiple lineages of birds and insects have been discovered utilizing the quasi-random nanostructures in their feathers or scales to produce angle-independent structural colors [11,12]. As
illustrated in Fig. 2, the longhorn beetle Anoplophora graafi in Fig. 2(a) possesses the sphere-type quasi-random NMS to produce the structural color of the light stripes on dark scales [13]; the
beetle Sphingnotus mirabilis in Fig. 2(b) displays the white colored stripes produced by the channel-type quasi-random NMS in its scale [14]. Despite their different morphologies, these biological
quasi-random NMSs are self-assembled via the phase separation of keratin and air [11,15]. The interplay between entropy and molecular interaction during the bottom-up structure formation process
results in a pronounced short-range correlation but little long-range order, leading to the inherent quasi-randomness of the NMS.
Man-made quasi-random NMSs have been developed using bottom-up nanomanufacturing. For instances, Fig. 2(c) shows a noniridescent structural color coating (shown by the inset) with sphere-type
quasi-random nanostructure obtained from the self-assembly of polymer and cuttlefish ink nanoparticles [16]; Fig. 2(d) shows a hierarchical NMS with channel-type quasi-random nanostructure fabricated
via nanowrinkling process [17], which possesses the superhydrophobic property as shown by the inset [18]. In spite of realization of a wide variety of man-made quasi-random NMSs, there is a lack of
systematic design methodology for developing nondeterministic, nonperiodic nanostructures from bottom-up nanomanufacturing. Current development of functional quasi-random NMSs mainly follows a
sequential strategy [19]. The optimal structure is first identified using computational design methods, and then the fabrication process is conceived and implemented to realize the optimal design.
Without considering the fabrication conditions in structure optimization limits the feasibility of an optimized design for practical application. The mismatch between the computational design of
nanostructures and control of bottom-up nanomanufacturing constitutes the main barrier toward fully exploiting the potential of quasi-random NMS for low-cost, high-performance devices [16,18,20].
Existing computational design methods for NMS fall into three categories, i.e., topology optimization method [5,6], descriptor-based design method [21,22], and correlation function based design
method [23]. Topology optimization method pixelates the design space and demonstrates the strength in designing metamaterials and other periodic NMS [5,6]. However, as illustrated in Fig. 1, the
deterministic characteristic of such design representation results in the use of expensive, often infeasible, top-down fabrication techniques.
Compared with topology optimization that involves thousands or millions of design variables, the descriptor-based method [21,22] employs an efficient parametric design approach to optimize a small
set of structure descriptors, such as volume fraction, minimum distance between particles, and aspect ratio of cluster, that characterize the composition, dispersion, and geometry of NMSs. While
being capable of representing NMS with regular shapes of nano- and microstructures such as the sphere-type in Fig. 2(c), the descriptor-based method has difficulty in capturing more complex
morphology such as the channel-type in Fig. 2(d). Moreover, identifying the key set of descriptors usually requires complicated data analysis, such as using the machine-learning techniques [24,25].
While correlation function is a popular microstructure characterization approach [26], it is often not suitable for simulation-based microstructural design. First, optimization-based microstructure
reconstruction for achieving target two-point correlation is too expensive to be included in the iterative microstructure design process. Second, model coefficients in a correlation function do not
have explicit physical meaning neither possess one-to-one mapping to nanomanufacturing process conditions. Hence, optimized microstructures using the correlation function approach may not be
achievable in fabrication.
We propose a novel approach for designing quasi-random NMSs, which employs a new SDF based methodology to concurrently [4] optimize the quasi-random NMS and the corresponding nanomanufacturing
conditions of a bottom-up process. Previous research in image processing suggests that the Fourier spectrum, i.e., the structural Fourier transformation, is sufficient to represent any complex
heterogeneous NMS and the exact material distribution can be fully reconstructed using phase recovery techniques [27]. Naturally formed quasi-random NMSs from bottom-up processes possess the
isotropic nanostructures illustrated in Fig. 2 and can be sufficiently represented using a SDF. In this paper, an SDF-based approach for isotropic NMS design is developed to bridge the gap between
processing–structure–performance relationship, to offer a significantly reduced dimensionality for fast design explorations and to incorporate the feasibility and stochastic nature of manufacturing
processes. This paper focuses on overcoming two main technical barriers of using the SDF approach: (1) To efficiently reconstruct the real-space nanostructure in a design loop and (2) to demonstrate
the feasibility of direct mapping between SDF and bottom-up nanomanufacturing conditions.
This paper is organized as follows: The SDF-based representation of quasi-random NMS is first introduced by comparing it with that of two-point correlation function (Sec. 2). The SDF-based
reconstruction algorithms are then developed, demonstrating the flexibility of this method in handling different morphologies, such as the channel-type and the particle-type (Sec. 3). The SDF-based
method for optimizing functional quasi-random NMS is illustrated using the optimization of quasi-random nanophotonic light-trapping structures based on structure–performance simulations (Sec. 4).
Finally, the concurrent design methodology is developed by incorporating the fabrication conditions into processing–structure mapping based on the SDF representation. This new methodology is applied
for optimizing a high-performance quasi-random light-trapping nanostructure fabricated using scalable wrinkle nanolithography process.
Spectral Density Function Based Representation of Quasi-Random Nanostructured Materials
We propose to use the SDF, a one-dimensional function as the normalized radial average of the square magnitude of structural Fourier transformation, to represent isotropic quasi-random NMSs in
reciprocal space. For a two-dimensional quasi-random NMS as illustrated in Fig. 3(a), the two-phase material distribution can be modeled as a function Z(r) with value 0 or 1 at each location denoted
by r. The Fourier spectrum of the quasi-random NMS in Fig. 3(a) is displayed in the inset. It can be obtained by taking the Fourier transformation of Z(r) as given in the below equation
represent the magnitude and phase information at each location
of the Fourier spectrum. Thus, SDF can be calculated by taking the radial average of the square of the Fourier spectrum regarding
, as shown by the below equation
where k is the spatial frequency calculated as the magnitude of k as k=|k|, and C is a normalizing constant that ensures the integral of f(k) over the considered spatial frequency domain equal to
one. SDF of the NMS in Fig. 3(a) is shown by the blue curve in Fig. 3(e), corresponding to the ring-shaped Fourier spectrum in the inset of Fig. 3(a).
SDF essentially describes the distribution of the Fourier components over the frequency range. The mathematical connection between the Fourier spectrum and the two-point correlation function of
real-space structures has been established in literature [28]. Alternative to the two-point correlation function that describes the structural correlation in real-space, SDF characterizes structural
correlation in a reciprocal space. Nevertheless, compared with two-point correlation, SDF provides a more convenient representation for designing quasi-random NMS. For example, the SDF in Fig. 3(e)
can be simply formulated as the red-dashed “step” function to statistically represent the quasi-random structure in Fig. 3(a), whereas the two-point correlation in Fig. 3(i) requires a complex
formulation. It is noted that the imperfectness of the step-shape of SDF in blue as compared with the red-dashed perfect one is due to the discrete pixels in the finite element meshing in numerical
calculations. This imperfectness can be reduced by increasing the meshing quality.
Moreover, Figs. 3(a) and 3(b) display two quasi-random nanostructures with different feature scales. While the significant difference in the feature scale of these two structures is clearly
represented by the shifting of their step-shaped SDFs over the spatial frequency as shown in Figs. 3(e) and 3(f), it is much more difficult to capture the difference between two-point correlation
functions in Figs. 3(i) and 3(j). Figure 3(c) shows a more disordered quasi-random nanostructure than Fig. 3(b), which has a donut-shaped Fourier spectrum as displayed by the inset. The increased
structure disorder implies the weakening of the structural correlation and less concentrated distribution of Fourier components at certain spatial frequency ranges. This difference is captured by the
widening of the step-shaped SDF with lowered height in Fig. 3(g) as compared with Fig. 3(f). In contrast, two-point correlation represents the increased structure disorder by the weakened oscillation
of the function as shown by comparing Fig. 3(k) with Fig. 3(j), which is less intuitive. Finally, Fig. 3(d) inset shows quasi-random nanostructures with a spread Fourier spectrum, implying the
further increased structure disorder compared with Fig. 3(c). This structure is represented by the SDF as shown in Fig. 3(h). Such SDF can be formulated as a Gaussian distribution function denoted by
the red-dashed curve. The different shapes of the SDFs, i.e., the Gaussian-shaped one in Fig. 3(g) and the step-shaped one in Fig. 3(h), clearly indicate the different characteristics of the
structural correlation in the real-space structures in Figs. 3(c) and 3(d). However, such difference is less obvious in terms of the representation based on the two-point correlation function as
shown in Figs. 3(k) and 3(l), except for the reduced oscillation in Fig. 3(l).
In conclusion, SDF provides a convenient and informative design representation of isotropic quasi-random NMS. In most cases, it enables more straightforward mathematical formulations compared with
two-point correlation function. Real-space structures with various morphological characteristics can be efficiently constructed based on a given SDF for micro/nanostructure mediated design. The
SDF-based reconstruction algorithms will be presented in Sec. 3. Since characteristics of the underlying structural correlation of quasi-random NMS are mostly determined by the physics of the
bottom-up manufacturing process, the relation between SDF and processing conditions can be conveniently established for concurrent structure and processing design. This concurrent design methodology
will be described in Sec. 4.
Quasi-Random Structures Reconstruction Based on Spectral Density Function
Efficient reconstruction of real-space structures is critical for the automated computational design of functional quasi-random NMSs. In this section, we illustrate efficient analytical
reconstruction methods based on SDF representation for quasi-random structures with different types of morphology, the channel-type and the particle-type. Though demonstrated for 2D NMSs, the
approaches presented can be easily extended to 3D reconstructions.
Channel-Type Quasi-Random Structures Reconstruction Based on SDF Using Gaussian Random Field (GRF).
The channel-type quasi-random NMSs are common nanostructures originating from bottom-up processes, such as the biological NMS in Fig. 2(b) born through the phase separation of the keratin–air mixture
or the man-made NMS in Fig. 2(d) from the mechanical self-assembly of thin-film wrinkling. Gaussian random field (GRF) modeling, originally proposed for studying the spinodal-decomposition based
phase separation of material mixture [29], provides a natural way to describe the NMS with such morphology.
) has two Gaussian properties: (1) the realizations of a field at randomly chosen locations in space follow a normal distribution and (2) for a fixed location
, all the possible realizations of the field at this specific point also follow a normal distribution. A standard GRF has each point marginally and all the points jointly following a standard
Gaussian distribution. A standard GRF over an
-dimensional space can be fully characterized by the field–field correlation function
) in Eq.
In Eq. (3), J are Bessel functions of the first kind, and f(k) is the SDF. The analytical relationship between the SDF f(k) and the field–field correlation g(r[1],r[2]) that governs the Y(r) is shown
in Eq. (3). Thus, the GRF Y(r) is completely described by the SDF f(k).
In numerical implementation, a realization of the GRF for a targeting SDF is constructed using the wave-form method [31] as $yr=(2/N)∑i=1N cos (kiki⋅r+ϕi)$. In this equation, N, the number of items
in the truncation series, is chosen as 10,000 to ensure sufficient accuracy, ϕ[i] is uniformly distributed over (0,2π), k[i] is a vector uniformly distributed on a unit sphere, and k[i] is a scalar
distributed over (0, +∞) following the probability density function as f(k)·k. Once a realization of the GRF is generated, a real-space, two-phase, quasi-random structure Z(r) shown by Eq. (4) can be
obtained by level-cutting the realization at α that is determined by the filler volume fraction ρ as $(1/2π)⋅∫−∞α exp (−ζ2/2)dζ=ρ$. Figure 4 illustrates the reconstruction process of a
two-dimensional channel-type quasi-random NMS based on the SDF as a Delta function in Fig. 4(a).
For a targeting SDF with arbitrary form, the quasi-random NMS can be efficiently constructed using this method. For further illustrations, the SDF in Fig. 5(a) follows a truncated Gaussian
distribution. The constructed structure in Fig. 5(d) with the material filling ratio as 40% has a Fourier spectrum (in the inset) corresponding to the targeting SDF. Figure 5(b) shows that an SDF has
two-step shape, which leads to the constructed structure in Fig. 5(e). Here, the material filling ratio is set at 50% and the structure possesses the double-band Fourier spectrum that matches with
the targeting SDF. Moreover, the triple-Delta SDF in Fig. 5(c) produces the structure in Fig. 5(f) with a triple-ring Fourier spectrum. The material filling ratio is set as 60% for this case. Even
though one reconstruction is illustrated for each target SDF in Fig. 5, the GRF approach also allows multiple reconstructions of statistically equivalent nanostructures if the assessment of
stochastic behavior is needed. This SDF-based reconstruction method is analytical which ensures the feasibility of microstructure computational that involves iterative loops.
It should be noted that in using Gaussian random field to model the isotropic quasi-random nanostructured materials, the structures are assumed to be statistically homogeneous and isotropic. The use
of Gaussian random processes has three basic assumptions: (1) stationarity (independence of time), (2) isotropy, and (3) zero mean. It is also discussed in literature that there might be certain
degree of spectral distortion during the level-cutting process as a nonlinear transformation of the GRF [32,33], which means the SDF of a final structure may deviate from the SDF of the original GRF.
For significant spectral distortion, iterative approach can be adopted for distortion correction [34,35]. In our reconstruction shown above, limited degree of spectral distortion is observed. Since
introducing such iterative correction would significantly impact the efficiency of an iterative microstructure optimization process, the distortion correction is not applied in this work. Moreover,
it should be noted that there are various random field methods for structure reconstruction, such as these discussed in Refs. [32,34], and [36]. Interested readers may refer to literature for more
Particle-Type Quasi-Random Structures Reconstruction Based on SDF Using Random Packing.
Particle-type NMS is another important class of quasi-random structures that have the potential of being fabricated by bottom-up processes, such as the NMS in Fig. 2(c) made by the self-assembly of
polymer and cuttlefish ink nanoparticles. For particle-type NMS, the spatial correlation depicted in SDF is determined by particle sizes and distances between particles. In this work, we develop an
algorithm based on random disk packing to achieve nonoverlapping particle-type NMS for a given SDF profile. Existing works have investigated the possibility of packing equal-sized disk shape
particles to achieve certain types of SDF by adjusting the center distances and number of disks. Most of these structures are loosely packed and possess a narrow ring shape SDF [37]. In this work, we
establish a semi-analytical relationship between distributions of disk numbers and disk radiuses with any arbitrary target SDF profiles; the method is also applicable for close packing structures.
The close disk packing algorithm we propose follows a simple strategy as shown in Fig. 6(c): if two neighboring disks are found to be too overlapped, then both will be pushed apart; vice versa if the
distance between two neighboring disks is too large (not desired for dense packing), then both will be moved toward each other. The appropriate distances to move are calculated using the formula in
the below equation
$Δ1=r2r1+r2(r1+r2−d), Δ2=r1r1+r2(r1+r2−d)$
where Δ[1] and Δ[2] are the moving distances for disks 1 and 2, respectively, and the spatial adjustment is along the line connecting the centers of the two disks. A positive value means the two
disks will move closer and negative value means they will move away from each other. d is the current center distance between two disks, and r[1] and r[2] are the radius of the two disks,
respectively. To realize a close disk packing, we start from a randomly packed structure that allows overlapping, then adjust the position of each disk in the structure until there is no overlapping
in the structure and all the disks are touching its neighbors within a small allowance ε (d=r[1]+r[2]±ε). By choosing different disk sizes, we are also changing the spatial distances between
neighboring disks. For a given number and the sizes of the disks, this strategy is found effective in generating close disk packing structures with volume fraction up to 90%. It is noteworthy that we
only require disks center to be within the target region instead of the whole disks so that we can reduce large voids near the boundaries of the structure. This disk-packing algorithm can be extended
to generate microstructures with particles of other shapes. The centers of particles are determined first by the disk-packing algorithm, and then the particles of other shape are assigned to those
centers. In this way, the spatial distances between particles in the new microstructure are controlled by the disk size chosen in the packing algorithm. For SDF-based reconstruction, the challenge is
how to bridge the gap between the structure and its SDF.
The case of equal-size close disk packing is first studied and then extended to the packing of multiple-size disks for arbitrary SDF profiles. SDF describes a structure in the frequency domain and
measures the underlying periodicity of the structure, a feature closely related to the number of disks packed. The typical SDF of an equal-size random close disk packing structure in Fig.
resembles dense hexagonal packings and has apparent periodicity. Thus, its SDF in Fig.
has its components concentrated on a single ring (corresponding to a single frequency
as marked in the figure). Through our empirical studies, in which we generated multiple equal-size disk-packing structures using different disk radiuses (from
/40 to
/20, where
is the side length of the structure), we discovered that the radius of the ring in Fig.
can be estimated based on the number of disks packed in the structure:
, where
is the total number of disks, and this relationship is obtained by simple regression analysis for periodic square packing and hexagonal packing, similar relationship exists despite the structure
size. It is noteworthy that the relationship for equal-size random close packing is very similar to that of hexagonal packing, which can be justified by the fact that hexagonal packing is the densest
packing for equal-size disks and if we pack the disks closely, they tend to form hexagonal lattices. In addition, for close packing of equal-size disks, the number of disks
can be estimated based on disk radius
by considering its similarity to hexagonal packing
, where
≈ 0.9069. By substituting
into the expression of
, we can establish the simplified relationship between
and disk radius
where L is the side length of the square-shaped structure. This relationship indicates that by adjusting the size of disks, we can change the frequency of SDF.
More complex SDF profiles can be achieved by packing disks with different sizes. For simplicity, a uniformly distributed SDF (see Fig. 3(e)) is used as an example to illustrate the strategy. Assuming
the target SDF is uniformly distributed between a frequency interval [k[1], k[2]], the appropriate range of disk sizes can be determined by Eq. (6). The problem becomes how to choose the right number
of disks for each size so that we can achieve a uniform SDF. This problem is addressed through optimization in this work by minimizing the fluctuation of SDF within the target range [k[1], k[2]] and
adjusting the number of disks for each size. Although optimization can provide satisfactory results based on our tests, it is computationally prohibitive, especially after we add the outer loop for
searching the optimal SDF in material design.
To overcome the computational difficulty, we again employ approximated relations derived from empirical studies. We create multiple unequal-size packing structures by varying the number of each size
and find that the intensity of SDF
) at a certain frequency
is proportional to the total area of the corresponding disk of radius
, where
satisfy the relationship in Eq.
is the number of disks of radius
. In addition, when the total areas of disks for each size
are equal, the corresponding
) are equal. As a result, to achieve a uniform SDF, the following condition holds:
from which the number of disks for each size
can be derived
$Ni=Anπri2=0.9069L2nπri2, i=1,2,...,n$
where n represents the number of different disk sizes, A is the total covered area by disks, and L is the side length of the square structure. The constant 0.9069 is the volume fraction of equal-size
hexagonal disk packing and here we use it as an approximation of the volume fraction in our packed structure and it can be updated for different target volume fractions. Here, we found that the
empirical relation works well for forming a uniform SDF. We can adjust the frequency range k of SDF f(k) by choosing appropriate disk sizes using Eq. (6) and then adjust the value of SDF at each
frequency k by changing the numbers of disks of each size using Eq. (7). Therefore, SDF with an arbitrary shape can be approximated in this way. Three examples are shown in Fig. 7 to verify the
effectiveness and versatility of our approach: the SDF in Fig. 7(d) consists of two narrow step functions and the corresponding NMS in Fig. 7(e) possesses two apparent feature scales. Figures 7(b)
and 7(c) are NMSs with the same underlying SDF in Fig. 7(a) that follows a broad Gaussian distribution where the particles have different sizes. This observation is consistent with the relationship
between particle size and frequency in Eq. (6). It is noted that although the particle geometries are totally different in Figs. 7(d) and 7(e) (circle and triangle), the underlying spatial
correlation is essentially the same, which is determined by the number of particles and spatial distances. In conclusion, by integrating the disk-packing algorithm and the empirical relationships
established in this section, multiple statistically equivalent particle-type NMS can be efficiently constructed toward any given arbitrary SDF with few limitations on particle geometries.
Spectral Density Function Based Computational Design of Quasi-Random Nanostructured Materials
Conventional real-space design approach directly employs the pixelated material distribution within the design space as the design representation where each pixel is a design variable. Such strategy
leads to thousands or millions of design variables and thus numerical issues associated with high design dimensionality. Most importantly, the deterministic nature of such real-space structure
representation is incapable of capturing the nondeterministic characteristics of quasi-random NMSs made by bottom-up processes.
In this paper, we propose a new SDF-based concurrent design methodology for optimizing quasi-random NMS and the corresponding manufacturing process as shown in Fig. 8. The key innovation of the
concurrent design strategy is to use SDF f(k) as the design representation to enable the processing–structure mapping. Instead of using the pixelated real-space material distribution as design
variables as in topology optimization, the SDF of the quasi-random structure plus a few additional structure descriptors, such as the material filling ratio, are regarded as the design variables. We
have shown in Sec. 2 that SDF provides a convenient and informative design representation in reciprocal space for quasi-random structures with complicated real-space geometries. With our approach,
given the desired performance, the specific types of the SDF that match with the nanomanufacturing process are first derived. Based on the SDF and a few additional structure descriptors, the
quasi-random NMSs in real space are efficiently constructed using the methods discussed in Sec. 3 for simulations to assess material properties and device performance. For given design objectives,
this computational design loop iterates to identify the optimal design of quasi-random nanophotonic structures in reciprocal space. By utilizing this SDF representation, only 4–5 design variables are
needed for a typical problem to preserve nondeterministic characteristics of quasi-random NMS.
Since the physics of a bottom-up process determines the structural correlation in fabricated quasi-random NMSs, a mapping between a bottom-up process and a specific form of SDF can be derived either
through physics-based modeling or empirical studies, which enables the concurrent structure and processing design of functional quasi-random NMS. Specifically, once the SDF formulation is optimized
for targeting functional performance using the computational design methodology in the right side of Fig. 8, the corresponding process conditions are identified via the mapping between processing
conditions and the SDF design representation shown at the left side of Fig. 8. This concurrent design methodology ensures the feasibility of the optimized structure for the chosen bottom-up
nanomanufacturing process.
In Sec. 4.1, the computational design of light-trapping NMS is presented first as an example to illustrate the effectiveness of using SDF as design representation. Next, in Sec. 4.2, the optimization
of quasi-random light-trapping nanostructure fabricated using nanowrinkle-based process is presented as an example to show how manufacturing processing conditions of nanowrinkling are mapped to SDF
for achieving concurrent design of structure and processing conditions.
Spectral Density Function Based Computational Design of Quasi-Random Light-Trapping Nanostructure for Thin-Film Solar Cells.
A light-trapping structure in a thin-film solar cell is employed as a representative example to demonstrate the SDF-based computational design methodology for quasi-random NMSs. Thin-film solar cells
with significantly reduced thickness of absorbing layer possess unique advantages, such as flexibility, semi-transparency, and low usage of expensive material in the absorbing layer, as compared with
bulk cells [38]. However, the thinner absorbing layer in thin-film solar cells leads to less interaction between the incoming light and the active material, hence lower energy absorption efficiency.
Light trapping was therefore developed to extend the path-length for light interacting with the active material, so highly efficient thin-film solar cell can be created. Figure 9(a) illustrates a
simplified thin-film solar cell model, where the top layer is the light-trapping nanostructure, gray color denotes amorphous silicon (a-Si) as the active material for light absorption, and the bottom
red layer represents a silver layer for preventing the light escaping on the back side. The total thickness of the absorbing layer made of a-Si is t.
To couple the incoming light into the quasi-guided modes supported by the absorbing layer requires the appropriate structural correlation of the quasi-random light-trapping nanostructures. Meanwhile,
the material filling ratio
of the quasi-random nanostructure, which determines effective refractive index of the top-layer, also significantly influences the light absorption efficiency. Since the quasi-random light-trapping
structure as shown in Fig.
is made of a-Si, the optimal light-trapping effect also depends on the depth of the quasi-random nanostructure
. The large depth of the light-trapping nanostructure usually produces strong scattering effect for light coupling, which however reduces the total volume of active material. In this case, the SDF
) with the material filling ratio
and the depth
are regarded as the design representation of the quasi-random light-trapping nanostructure to be optimized using the methodology outlined in Fig.
. The design objective is shown in the below equation
where Z represents the quasi-random nanostructures, and A is the absorption coefficient as the ratio between the energy absorbed in the cell and the total energy of the incoming light. Based on
quasi-random light-trapping nanostructure constructed from the design representation [f(k), ρ, t[1]] with other material properties and structure parameters, the whole device is modeled for the
performance simulation. Here, we use the rigorous coupled wave analysis (RCWA) method [39,40] for performance simulation. RCWA is a Fourier space based algorithm that provides the exact solution of
Maxwell's equations for the electromagnetic diffraction by optical grating and multilayer stacks structures. Other methods such as finite-difference time-domain (FDTD) or finite element analysis
(FEA) can be applied for the performance simulation, as well. To account for the stochasticity embedded in the reconstruction process, the average absorption coefficient of three reconstructed
structures is calculated and treated as the objective function for the SDF-based design optimization. Based on the simulation results, the design variables are updated using optimization search
algorithms. While different algorithms can be adopted for design updating, here we use the genetic algorithm (GA). Mimicking natural evolution with the underlying idea of survival-of-the-fittest, GA
is a stochastic, global search approach involving the iterative operation of selection, recombination, and mutation on a population of designs [41,42]. The stochasticity of GA enables the convergence
toward optimums despite the strong nonlinearity in a design problem [5].
For the purpose of demonstration, here we assume that the SDF follows a step function as shown in the inset of Fig. 9(b), which is governed by two variables, i.e., k[a] and k[b]. Thus, together with
the material filling ratio and the depth of the quasi-random light-trapping nanostructures, this design problem involves four (4) design variables, i.e., k[a], k[b], ρ, and t[1]. Optimization for a
single incident wavelength w=700nm, which corresponds to the weak absorption region of a-Si, is first performed to verify the efficacy of the methodology. The total thickness of the a-Si layer is
set as 600nm, and the length p of the RCWA simulation window is set as 2000nm. The optimization history is shown in Fig. 9(b), in which each dot represents the corresponding absorption coefficient
for a given set of design variables (k[a], k[b], ρ, t[1]). Starting from the randomly generated initial designs with the absorption coefficient around 0.12, the optimization converges to the
solutions with the absorption coefficient larger than 0.79. Statistical representation using SDF enables infinite numbers of optimized quasi-random structures to be generated in real space. Six
designs were constructed with an identical set of optimized variables, denoted by the red dot in Fig. 9(b), with the channel-type structures in Figs. 9(c)–9(e) generated using Gaussian random field
modeling and the sphere-type structures in Figs. 9(f)–9(h) generated using random disk-packing, respectively. The optimized design denoted by the red dot is $ka*$=0.0029nm^−1, $kb*$=0.0030nm^
−1, ρ*^=62%, and $t1*$=75nm. Despite the different real-space morphologies, all the designs possess similar ring-shaped Fourier spectrum (shown in the insets of Figs. 9(c)–9(h)) and achieved the
equal optimal performance. For further validation, shown in Fig. 9(i), 50 real-space patterns were generated from the optimized design and evaluated to achieve an average absorption of 0.80 with less
than 4% variation; the variation is due to the discrete meshing in numerical calculation, thereby achieving a 4.7-fold enhancement of the absorption compared to the uniform (unpatterned) structure.
A broadband optimization of the quasi-random light-trapping nanostructures over the visible solar spectrum is then conducted using our SDF-based computational design methodology. Broadband
optimization over the solar spectrum from 400nm to 800nm (
=400–800nm) is considered here. The structural parameters in this case are set to be
=650nm and
=2000nm. In this optimization, 81 wavelengths are considered over the whole wavelength spectrum. The objective is to maximize the absorption enhancement factor which is the ratio between the
predicted absorption and the single-path absorption averaged over the considered wavelengths [
], as shown in Eq.
. In this equation,
denotes the nanostructure to be optimized described by design variables [
denotes the
th considered wavelength,
is the predicted light absorption of design
, and
denotes the single-path light absorption of a uniform cell as a normalizing factor independent of the design
The broadband optimized design is $ka*$=0.0012nm^−1, $kb*$ =0.0036nm^−1, ρ*^=51%, and $t1*$=150nm, which leads to the quasi-random structure with a donut-shaped Fourier spectrum as shown by
the insets of Figs. 10(a) and 10(b). This Fourier spectrum overlaps with the area defined by two dashed circles that denote the range of k-vectors for coupling the incident light to the quasi-guided
modes in the silicon film [19,44]. Two distinct light-trapping designs in real space are realized from the identical optimized SDF, which is a channel-type structure in Fig. 10(a) and a particle-type
structure in Fig. 10(b), respectively.
The optimized structures are compared with the two reference structures that are commonly adopted in literature [45] for improving light-trapping performance. The two reference structures include an
unpatterned cell with a 70nm silicon dioxide antireflection coating (ARC) and an unpatterned cell without the ARC, both of which have the same thickness of absorbing material (a-Si) as the optimized
ones. The absorption spectra of the two optimized quasi-random structures and two reference cells are calculated using RCWA and plotted in Fig. 10(c). Serving simultaneously as an efficient
quasi-guided mode coupler and an ARC, the two optimized quasi-random light-trapping nanostructures with drastically different real-space morphology enhanced the absorption by more than threefold in
the weak absorption region (600–800nm) compared with the two references, achieving an average absorption coefficient of 0.74 over the broad spectrum from 400nm to 800nm. As shown in this testbed,
designing the structure in reciprocal space using SDF-based representation instead of tailoring the structure in real-space provides the freedom to down-select the optimized real-space structure in
order to better accommodate manufacturing constraints. For example, the channel-type structure can be fabricated using the nanowrinkling process as illustrated in Fig. 2(d), and the sphere-type
structure can be fabricated using nanoparticle self-assembly as illustrated in Fig. 2(c). Furthermore, the fabrication conditions can be determined through the inherent connection between SDF and the
physics of a bottom-up process. This will be discussed using nanowrinkling-based fabrication process as an example in Sec. 4.2.
Concurrent Design of Quasi-Random Light-Trapping Nanostructure for Scalable Nanomanufacturing Process Based on Wrinkle Lithography.
We have demonstrated the computational design of functional quasi-random NMS using SDF-based methodology, where we assume that the SDF to be optimized follows a step function. While bottom-up process
based nanomanufacturing provides a scalable fabrication method for realizing quasi-random NMS designs, the fabrication process has not been fully considered in the design shown in Figs. 9 and 10. In
this section, an example of incorporating the conditions of the fabrication process into the design stage is provided for the design of quasi-random NMS fabricated using thin-film nanowrinkling.
Wrinkling of the thin layer has emerged as a simple method for the scalable fabrication of microscale and nanoscale surface structures. In this bottom-up nanomanufacturing method, a stiff skin layer
on a softer, prestrained substrate is buckled during the relaxing of the substrate, forming the wrinkle nanostructure as shown in Fig. 11(a). The primary wavelength of the resulted wrinkles λ[w] is
linearly proportional to the thickness of the skin layer. In this work, the skin layer is chosen as the prestrained, thermoplastic polystyrene (PS). The thickness of the PS layer is precisely
controlled by changing the reactive ion etching (RIE) time of CHF[3] plasma gas ($TCHF3$) in processing for continuously tuning λ[w]. The primary winkle wavelength of the structure shown in Fig. 11
(a) is λ[w]=180nm.
To utilize wrinkling process for fabricating functional quasi-random NMS, the nanowrinkle structure in Fig. 11(a) needs to be transferred to other materials. In our research, we transferred the
nanowrinkle onto the a-Si via a nanolithography process for the light-trapping purposes. The transferred nanowrinkle pattern forms 2D quasi-random nanostructures. In this process, the PS wrinkles are
treated with SF[6] plasma and then transferred into a PDMS mask. The a-Si thin-film is coated with a layer of Al[2]O[3], upon which a photoresist is spin cast. Solvent-assisted nanoembossing (SANE) [
46] is completed with the wrinkled PDMS mask and photoresist protects wafer, resulting in the original wrinkle pattern being transferred into photoresist. The application of timed direction O[2]
plasma in RIE reduces the thickness of the photoresist. The time of the application of O[2] plasma ($TO2$) in RIE determines the material filling ratio ρ of the resulting structure.
After this wet etching process, the photoresist is removed and the 3D wrinkle structure has now been transferred to an Al[2]O[3] mask. The Al[2]O[3] can then act as a deep reactive ion etching (DRIE)
mask for etching into the a-Si without affecting the λ[w] of the wrinkle feature, generating the final quasi-random nanostructure for light-trapping. The time of DRIE determines (T[DRIE]) the depth
of the quasi-random light-trapping structure, i.e., t[1] as shown in Fig. 9(a).
While it is challenging to model the resulting quasi-random nanostructure from nanowrinkle patterning by using real-space representation, the SDF-based method provides a convenient representation of
the structure. As shown in Fig. 11(b), the blue curve is the SDF of the scanning electron microscope (SEM) image of patterned structure from the nanowrinkle shown in Fig. 11(a). The peak position of
the measure SDF, i.e., k[m], is dependent on the primary wrinkle wavelength λ[w] as k[m]=1/λ[w]. By empirically analyzing the five fabricated samples with different primary wrinkle wavelengths, we
derive that the SDF of the transferred wrinkle pattern follows a truncated Gaussian distribution function. Figures 11(b)–11(d) show the analyzing results for three different samples of wrinkle
pattern (λ[w]=180nm, 450nm, and 2000nm). In each of the figures, the blue curve is the SDFs of the SEMs and the red-dashed curve is the fitted SDF that has the truncated shape from a Gaussian
distribution function governed by only one variable as μ=k[m]. The standard deviation of the Gaussian distribution depends on k[m] as σ=0.958μ+0.00017. The r-square for the fitting is 0.99.
Here, we normalize the truncated Gaussian distribution to the integral of one over the range 0–0.02nm^−1 for Fig. 11(b), 0–0.01nm^−1 for Fig. 11(c), and 0–0.005nm^−1 for Fig. 11(d), respectively.
It is noted that the tails of the fitted SDFs do not match well with those from SEMs in Figs. 8(b)–8(d). This is caused by a number of small features which are either due to imaging noises or small
manufacturing defects. These small features correspond to the Fourier components at higher frequency, leading to the fat tails of SDFs. The prominent features of the nanowrinkle patterned structure
are characterized by the “hill” region of the SDFs. In this case, despite the slight mismatch at the tail region, our fitted SDFs capture the main features of the nanowrinkle patterned structure.
Utilizing the SDF-based representation, the wrinkle patterned quasi-random nanostructure contains only three variables, i.e., k[m]=1/λ[w], ρ, and t[1]. Based on our data from the processing
testing, these three design variables can be independently controlled, following relations as $TCHF3$=0.2087·λ[W]−37.5758, $TO2$−36.86·ρ +71.63, and T[DRIE]=0.325·t[1]−2.7451. This
processing-design mapping enables the concurrent design outlined in Fig. 8.
Here, the Gaussian random field modeling is used to construct the real-space structure for the performance simulation. The total thickness of the a-Si material is set as 700nm in this case. The
optimized design for the broadband light-trapping over 800–1200nm wavelength range is $km*$=0.0018nm^−1, ρ*^=52%, and $t1*$=210nm. The optimized structure significantly enhances the
absorption compared with the unpatterned uniform structure, achieving 150% absorption enhancement over the unpatterned thin-film in the weak-absorbing region of the material (800–1200nm).
The optimized structure is then fabricated using the wrinkle nanolithography, according to the processing-design mapping. It is found that the fabricated optimal wrinkle pattern achieves 130%
absorption enhancement over the unpatterned thin-film in the weak-absorbing region of the material (900–1200nm). The SDF of the SEM of fabricated sample matches well with the optimized SDF design.
This testbed validates the efficacy of using the novel concurrent design methodology for realizing high-performance quasi-random NMS while designing and ensuring the feasibility of the bottom-up
scalable nanomanufacturing process.
In this paper, a novel nondeterministic representation is proposed for designing quasi-random nanostructures with inherent robustness. A SDF based concurrent design method is developed for designing
cost-effective and scalable quasi-random NMSs and the associated bottom-up nanomanufacturing processes. As a representation of NMSs in the reciprocal frequency space, SDF captures the spatial
correlation at different length scales. Compared with the widely used two-point correlation function, SDF is shown to be more-effective in differentiating quasi-random NMS. Moreover, SDF generally
takes simple forms that can be related to physical processing conditions, which bridges the gap between processing–structure relation that facilitates the concurrent structure and processing design.
To enable automated computational microstructure design, reconstruction techniques are developed in this work for creating statistically equivalent quasi-random NMSs based on a given SDF. Efficient
reconstructions are achieved for both the channel-type NMS systems and particle-type NMS systems using the Gaussian random field (GRF) modeling and the random disk-packing based approach,
respectively. The results show that the mapping between SDF and nanostructure is not unique—although the real-space morphologies are completely different, the NMSs of these two systems may share
common SDFs as long as their essential spatial correlations are similar. The SDF-based design strategy provides the freedom to down-select optimized design from different real-space structures to
accommodate the constraints of manufacturing platforms. Although the NMSs presented in this paper are isotropic, our approach can be extended for anisotropic structures by modeling the GRF using
anisotropic correlations for quasi-random NMs and using nonuniform orientation distributions for particle-type NMSs.
To verify the effectiveness of using SDF as a design representation for quasi-random NMS, a light-trapping structure in a thin-film solar cell is employed as an example. The optimized structure with
the ring-shaped SDF achieved 4.7-fold enhancement of single length absorption and for broadband light-trapping, the SDF of the optimal structure is uniformly distributed, with two (2) times overall
improved performance. Designing the structure by SDF rather than real-space morphologies increased the freedom of down-selecting different NMS systems (e.g., channel-type and particle-type) to
accommodate manufacturing constraints. We further perform a concurrent design of quasi-random light-trapping structure fabricated using the wrinkling nanolithography-based scalable nanomanufacturing
technique. The realistic fabrication condition of the wrinkle nanolithography process is integrated into the design stage by deriving the SDF of the wrinkle patterns as a truncated Gaussian
distribution function. This SDF along with other structure representation variable is optimized to identify the optimal wrinkle pattern for light-trapping purposes with the fabrication feasibility
for wrinkle lithography process.
We believe that this SDF-based method can well solve the structure design problems of the functional materials whose performance largely depends on the first- and second-order spatial correlations of
the structures. These problems are abundant in various areas, such as the design of noniridescent structural coloration coating [16], the material with excellent transport property [47], or the
nanodielectric material with high permittivity and low dielectric loss [48]. This research contributes to the creation of a new and accelerated materials system design paradigm with the emphasis on
achieving the compatibility between nanostructure design and the design of scalable nanomanufacturing processes. The research will be further enhanced by testing the approach on different material
systems with different desired functional performance and nanofabrication processes.
The grant supports from the U.S. National Science Foundation (NSF) EEC-1530734 and CMMI-1462633 are greatly appreciated. The authors would like to thank Dr. Clifford Engel and Dr. Alexander Hryn at
Northwestern for their assistance on fabricating the nanowrinkle structures the optical performance measurement and also Dr. Zhen Jiang currently at the Ford Motor Company on the discussion of using
Gaussian random field modeling for structure reconstruction. S. Yu and W. K. Lee also thank the International Institute for Nanotechnology for the Ryan Fellowship Award.
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), p. | {"url":"https://fluidsengineering.asmedigitalcollection.asme.org/mechanicaldesign/article/139/7/071401/383763/Characterization-and-Design-of-Functional-Quasi","timestamp":"2024-11-03T00:52:37Z","content_type":"text/html","content_length":"410834","record_id":"<urn:uuid:df12d9b5-1edf-40e1-986f-eb7b1cd702ae>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00405.warc.gz"} |
Gravitational Potential Energy
Question Video: Gravitational Potential Energy Physics • First Year of Secondary School
An object with a mass of 15 kg is at a point 10 m above the ground. What is the gravitational potential energy of the object?
Video Transcript
An object with a mass of 15 kilograms is at a point 10 meters above the ground. What is the gravitational potential energy of the object?
All right, so let’s say that this is ground level. And we’re told that our object is above this level a distance of 10 meters. Along with this, we’re told that the mass of our object — what we can
call 𝑚 — is equal to 15 kilograms. We want to know, what is the gravitational potential energy of this object? To figure this out, we can recall that the gravitational potential energy of an object —
we can refer to it as GPE — is equal to the mass of that object multiplied by the strength of the gravitational field the object is in all times the height of the object above some minimum possible
level. For an object, like the one we have here, that’s within 10 meters of Earth’s surface, we can say that 𝑔, the acceleration due to gravity, is exactly 9.8 meters per second squared.
So, when we go to calculate this object’s gravitational potential energy, we know its mass, that’s 15 kilograms. We know 𝑔, that’s a constant, 9.8 meters per second squared. And we also are given ℎ,
the height of the object above ground level, 10 meters. When we substitute in these values and then multiply them together, we find a result of 1470 newton meters. This is because a newton is equal
to a kilogram meter per second squared. And then, we can recall further that a newton times a meter is equal to the unit called a joule. This is the unit typically used to express energies. So, our
final answer is 1470 joules. | {"url":"https://www.nagwa.com/en/videos/782197101570/","timestamp":"2024-11-12T05:47:28Z","content_type":"text/html","content_length":"242736","record_id":"<urn:uuid:9122f316-80b7-464f-bcc8-2c2474d16d42>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00695.warc.gz"} |
Monte Carlo Simulation | Statgraphics
Random Number Generation
It is frequently necessary to generate random numbers from different probability distributions. This procedure simplifies the process of creating multiple samples of random numbers. 49 probability
distributions are available.
More: Monte_Carlo_Simulation_Random_Number_Generation.pdf
Multivariate Normal Random Numbers
This procedure generates random numbers from a multivariate normal distribution involving up to 12 variables. The user inputs the variable means, standard deviations, and the correlation matrix.
Random samples are generated which may be saved to the Statgraphics databook.
More:Multivariate_Normal_Random_Numbers.pdf or Watch Video | {"url":"https://www.statgraphics.com/monte-carlo","timestamp":"2024-11-08T21:05:18Z","content_type":"text/html","content_length":"67308","record_id":"<urn:uuid:1f2cd534-cd5b-4b7b-bf6b-0776999efafd>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00637.warc.gz"} |
Why do I get a big log fold change but small mean change in b value when plotting differential methylation?
I am doing differential methylation analysis using limma. I use the m values for testing and b values for plotting. I plotted a volcano plot visualizing the p value and effect size, however I noticed
something that I can’t explain.
Using the p value and the log fold change as reported by topTable() in limma, i get this volcano plot, which looks fairly like expected:
Volcano plot by adjusted p value and log fold change
But then I then I thought it would make more biological sense to plot the significant probes by mean differences in b value, but the significant probes seems to have really small changes in mean b
value, while the nonsignificant ones have bigger changes in mean b value.
Volcano plot by adjusted p value and mean difference in b value (the axis label is wrong)
Could you help me decipher how to interpret this? I am afraid I am doing something wrong in my analysis, but I also suspect this has something to do with the calculation of b values to m values and
maybe the variance. It also seems really strange that there is an inverse relationship between the p value and the mean difference in b value?
Sincerely, Christine
• 15 views | {"url":"http://opensourcebiology.eu/2021/08/12/why-do-i-get-a-big-log-fold-change-but-small-mean-change-in-b-value-when-plotting-differential-methylation/","timestamp":"2024-11-02T21:56:26Z","content_type":"text/html","content_length":"46226","record_id":"<urn:uuid:d52f7f9d-05ff-4895-a5f4-1e46d77b75a1>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00810.warc.gz"} |
B.Sc. ( Research ) In Mathematics India :: Shiv Nadar University | School of Natural Sciences | Shiv Nadar University
Department of Mathematics
The Department strives to explore new and emerging transdisciplinary knowledge domains relevant to societal benefits and sustainable development goals such as Human-Nature interaction and Artificial
Intelligence. Some frontier areas in which the faculty has reputed publications include Complex Systems Modeling, Statistical Learning, Minimal Surfaces, and Mathematical Finance. The Department's
engagement with these emerging learning paradigms promotes novel dimensions to our research focus that transcends to the Doctoral program, which has these as thrust areas, in addition to modern
regions of Functional Analysis, Algebra, Optimization, and Geometry.
The Department was conferred the DST-FIST grant in 2015 for setting up a Departmental Research Computer Lab and a dedicated Department Library. In addition to organizing the National Conference in
Complex Networks and workshops in Zero Mean Curvature, the Department has been regularly hosting critical national conferences and schools, such as the Annual Foundational School and Advanced
Instructional School, organized by the National Board of Higher Mathematics.
Explore Programs under the Department of Mathematics
Faculty at Department of Mathematics
Distinguished Professor
Head of Department of Mathematics
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Department of Mathematics
Research Areas of Department of Mathematics
Using functional analysis, operator theory, frame theory, harmonic analysis, matrix analysis, function theory, and dynamical systems, research is conducted on positive semi-definite matrices and...
Research Labs of Department of Mathematics | {"url":"https://snu.edu.in/schools/school-of-natural-sciences/departments/department-of-mathematics/","timestamp":"2024-11-04T13:20:32Z","content_type":"text/html","content_length":"91979","record_id":"<urn:uuid:82e04937-f58b-4c50-917a-6d01649861a2>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00839.warc.gz"} |
Hi Heikko,
did you check if you have nested products? (those may be hidden sometimes if you pass expressions around) Take a look at the very last item in http://www.boost.org/doc/libs/1_42_0/libs/numeric/ublas/
The indexing_matrix_assign is the main function that is called when you assign an expression to a matrix in ublas, so everything passes through that. And remember in ublas A+B (with A and B matrices)
is an expression in ublas, not a matrix, it becomes a matrix when you assign it to something.
Also try to play with -DBOOST_UBLAS_MOVE_SEMANTICS (http://www.boost.org/doc/libs/1_42_0/libs/numeric/ublas/doc/options.htm) and noalias() (http://www.boost.org/doc/libs/1_41_0/libs/numeric/ublas/doc
/operations_overview.htm) , to see if those help.
Best regards
> From: heiko_at_[hidden]
> To: ublas_at_[hidden]
> Date: Fri, 9 Apr 2010 18:20:41 +0200
> Subject: [ublas] profiling shows most time is consumed by indexing_matrix_assign
> Dear all,
> I have just recently begun to use uBLAS to do some numerics, solving a
> system of ordinary linear differential equations. The ODE solver
> repeatedly invokes a function that computes the right hand side of the
> ODE, and that contains almost the whole computational effort. In this
> function, I have about 8 matrix-matrix multiplications (with prod()) and
> about the same number of additions/subtractions of matrices or
> multiplications with a scalar. The involved matrices are dense with
> std::complex<double> entries and are of dimension 800x800.
> The program, especially the evaluation of the rhs function, seems to be
> running fairly slow and profiling with gprof shows that the rhs
> evaluation takes 95%
> of the computation time. Deeper inspection shows that two
> versions of indexing_matrix_assign() account for almost all
> of the time needed. Now comes my question: Is this the expected
> behaviour for the described situation?
> Some additional information: I am using uBLAS 1.41.0 with NDEBUG set to
> 1.
> Any help is appreciated.
> Best regards,
> Heiko Schröder
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> ublas mailing list
> ublas_at_[hidden]
> http://lists.boost.org/mailman/listinfo.cgi/ublas
> Sent to: nasos_i_at_[hidden]
The New Busy is not the too busy. Combine all your e-mail accounts with Hotmail. | {"url":"https://lists.boost.org/ublas/2010/04/4155.php","timestamp":"2024-11-13T21:27:28Z","content_type":"text/html","content_length":"15203","record_id":"<urn:uuid:c023d560-0c14-412b-bcea-86d14e274af3>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00573.warc.gz"} |
FIC-UAI Publication Database -- Query Results
Sepulveda, N., Josserand, C., & Rica, S. (2010). Superfluid density in a two-dimensional model of supersolid. Eur. Phys. J. B, 78(4), 439–447.
Cortez, V., Medina, P., Goles, E., Zarama, R., & Rica, S. (2015). Attractors, statistics and fluctuations of the dynamics of the Schelling's model for social segregation. Eur. Phys. J. B, 88(1), 12 | {"url":"https://ficpubs.uai.cl/search.php?sqlQuery=SELECT%20author%2C%20title%2C%20type%2C%20year%2C%20publication%2C%20abbrev_journal%2C%20volume%2C%20issue%2C%20pages%2C%20keywords%2C%20abstract%2C%20thesis%2C%20editor%2C%20publisher%2C%20place%2C%20abbrev_series_title%2C%20series_title%2C%20series_editor%2C%20series_volume%2C%20series_issue%2C%20edition%2C%20language%2C%20author_count%2C%20online_publication%2C%20online_citation%2C%20doi%2C%20serial%2C%20area%20FROM%20refs%20WHERE%20publication%20RLIKE%20%22European%20Physical%20Journal%20B%22%20ORDER%20BY%20first_page%20DESC&submit=Cite&citeStyle=APA&citeOrder=&orderBy=first_page%20DESC&headerMsg=&showQuery=0&showLinks=0&formType=sqlSearch&showRows=20&rowOffset=0&viewType=Print","timestamp":"2024-11-06T09:19:04Z","content_type":"text/html","content_length":"8456","record_id":"<urn:uuid:48b5b2fe-007b-4029-a770-3aaa919d95d2>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00566.warc.gz"} |
Bob traveled 90 miles which was 83 1/3 % of his entire trip. how many miles did he still have to travel?
Find an answer to your question 👍 “Bob traveled 90 miles which was 83 1/3 % of his entire trip. how many miles did he still have to travel? ...” in 📗 Mathematics if the answers seem to be not correct
or there’s no answer. Try a smart search to find answers to similar questions.
Search for Other Answers | {"url":"https://cpep.org/mathematics/2878931-bob-traveled-90-miles-which-was-83-1-3-of-his-entire-trip-how-many-mil.html","timestamp":"2024-11-11T01:23:48Z","content_type":"text/html","content_length":"22699","record_id":"<urn:uuid:ec068734-5bee-4e7f-8e5a-5056d5560a18>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00053.warc.gz"} |
Superlinear convergence estimates for a conjugate gradient method for the biharmonic equation
The method of Muskhelishvili for solving the biharmonic equation using conformal mapping is investigated. In [R. H. Chan, T. K. DeLillo, and M. A. Horn, SIAM J. Sci. Comput., 18 (1997), pp.
1571-1582] it was shown, using the Hankel structure, that the linear system in [N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff, Groningen, the
Netherlands] is the discretization of the identity plus a compact operator, and therefore the conjugate gradient method will converge superlinearly. Estimates are given here of the superlinear
convergence in the cases when the boundary curve is analytic or in a Hölder class.
• Biharmonic equation
• Conjugate gradient method
• Hankel matrices
• Numerical conformal mapping
Dive into the research topics of 'Superlinear convergence estimates for a conjugate gradient method for the biharmonic equation'. Together they form a unique fingerprint. | {"url":"https://scholars.ln.edu.hk/en/publications/superlinear-convergence-estimates-for-a-conjugate-gradient-method","timestamp":"2024-11-07T13:35:52Z","content_type":"text/html","content_length":"52759","record_id":"<urn:uuid:3dbe17eb-89d3-4eeb-9484-30d7a58ae118>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00715.warc.gz"} |
I am a postdoc at Cancer Institute at UCL in the group of James Reading. I did my PhD with Wolfgang Huber at EMBL Heidelberg. Before joining UCL, I did a bridging postdoc with Simon Anders at
Heidelberg University.
Currently, I am working on novel ways to analyze the role of T cells in early cancerogenesis.
Previously, I have worked on:
• Statistical methods to simplify working with single-cell data,
• Benchmarks of single-cell methods,
• Differential abundance analysis of mass spectrometry data,
• Clustering of high-dimensional categorical data.
For a full record of my publications, see Google scholar.
In addition, I develop statistical methods and tools for the analysis of cutting edge biological data. I maintain ten R packages which are published on CRAN and Bioconductor and total more than
100,000 downloads per month. In 2023, I received the Bioconductor community award in recognition of my contributions to the project.
If you are working in a company and would like help with some analysis or one of my packages, please get in touch, as I am happy to consult on projects. | {"url":"https://const-ae.name/","timestamp":"2024-11-01T23:33:54Z","content_type":"text/html","content_length":"47042","record_id":"<urn:uuid:fe2f4fdd-ed9b-44a4-b979-de6abceee1be>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00632.warc.gz"} |
Rationalizing a fraction whose denominator is a binomials
rationalizing a fraction whose denominator is a binomials Related topics: cryptography worksheets for 4th grade
18.100b problem set 1 solutions
algebra factoring worksheet
how to solve a third order equation
quadratic functions and models,2
how to reduce power of variables in square roots
algebra problem examples
Author Message
BegLed01 Posted: Sunday 09th of Jun 18:28
Hey, Yesterday I started solving my mathematics assignment on the topic Algebra 1. I am currently unable to finish the same because I am not familiar with the fundamentals of
side-side-side similarity, scientific notation and parallel lines. Would it be possible for anyone to assist me with this?
From: The
upper midwest
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kfir Posted: Sunday 09th of Jun 21:04
Algebrator is what you are looking for. You can use this to enter questions pertaining to any math topic and it will give you a step-by-step solution to it. Try out this program to find
answers to questions in factoring and see if you get them done faster.
From: egypt
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Svizes Posted: Tuesday 11th of Jun 09:18
Yeah, I agree with what has just been said. Algebrator explains everything in such great detail that even a amateur can learn the tricks of the trade, and solve some of the most tough
mathematical problems. It elaborates on each and every intermediate step that it took to reach a certain solution with such perfection that you’ll learn a lot from it.
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jrienjeeh Posted: Wednesday 12th of Jun 12:26
That sounds amazing ! Thanks for the help ! It seems to be perfect for me, I will try it for sure! Where did you come across Algebrator? Any suggestion where could I find more detail
about it? Thanks!
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pcaDFX Posted: Thursday 13th of Jun 19:28
Yeah you will have to buy it though it will cost you less than a tutor. You can get an idea about Algebrator here https://softmath.com/algebra-features.html. They give you an
unconditional money-back guarantee. I haven’t yet had any reason to take them up on it though. All the best!
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Dnexiam Posted: Saturday 15th of Jun 08:59
I remember having difficulties with evaluating formulas, graphing lines and interval notation. Algebrator is a truly great piece of math software. I have used it through several math
classes - Pre Algebra, Basic Math and Intermediate algebra. I would simply type in the problem and by clicking on Solve, step by step solution would appear. The program is highly
From: City 17
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Costs and Benefits of Lockdown (2): Translating GDP into human lives
In order to meaningfully compare costs and benefits of lockdown, it is necessary to have some assumptions about the value of human life. There are different approaches that can be taken to this, such
as using the (quality-adjusted) life year thresholds used by public heath systems to ‘price’ a year of healthy human life. Of course, it is popular and politically expedient to ignore the price of
life issue, indeed some like to argue that human life is of infinite value. This is nonsense. Human life has a finite value, else we would have no basis to determine whether it made sense to approve
treatments based on their cost and their medical merit.
Modelling life expectancy in terms of GDP
The approach I will take is slightly different here. I will attempt to model the relationship between GDP and life expectancy using a simple regression model for the EU-28 countries. The intuition is
that higher GDP enables a prolongation of human life, through better diet, investment in workplace health and safety and investment in curative and preventative medicine. Over the relevant part of
the curve, we should see a clear relationship emerge, which will allow us to understand what the cost in terms of life is, of reducing GDP. This can then be compared with the estimated benefits –
also in terms of human life – of implementing a lockdown.
I use the data from Eurostat for 2018 GDP per capita in euros, (GDP), for life expectancy at age 1 (LE) to construct a simple regression model specified as follows
LE = a*GDP+b*GDP^2
The spreadsheet with the data is here. I exclude Luxembourg and Ireland because of well-known problems with the measurement of GDP. While other control variables are not included, in fact the
institutional nature of the EU suggests that many other relevant factors (such as the design of the public health system, workplace health and safety rules, and other cultural factors) enjoy a
certain degree of harmonisation.
The results of the regression are as below:
And show that both variables are statistically significant (high t Stat values), and that quite alot of the variation (around 60%) in LE can be explained by GDP/capita alone (high ‘Adjusted R
The coefficients in the last three rows imply:
(a) that in the absence of any GDP, life expectancy would be at 70 (intercept). Although this seems unreasonably high, suggesting more non-linearity in the relationship as income falls to the
zero-threshold, it is not a big deal, however, as we are only concerned with the relationship over the relevant space, i.e. in terms of the GDP/capita of developed, Western countries.
(b) there is a positive relationship between GDP and Life Expectancy; namely that a one Euro increase in GDP/capita results in an increase in life expectancy of 0.0007 years. This is what we would
expect to see.
(c) there is a negative relationship between GDP squared and Life Expectancy; namely that as the square of GDP rises by one euro, LE falls by 0.000000001 years. This is also what we would expect –
the gains to life expectancy are limited, and at very high income levels, there may even be a reverse effect, due to stress from work, pollution or other lifestyle effects.
The graph below shows the scatterplot of actual values and the estimated line of the curve from the regression:
Using this model, we can estimate what the 5% reduction in GDP caused by a four month lockdown would mean in terms of life expectancy for the EU-28 countries. Here is what we get when we do that:
In summary, on average, a 5% reduction in GDP will result in a loss of an estimated 0.5 years, or 6 months, in the average life expectancy of people in Europe. This will happen because of many
reasons, such as reduced spending on curative health, increased crime, more suicides, greater occupational risks and reduced spending in preventative health and safety.
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Kirchhoff's Law | EIM Academy
Lesson Plan: Kirchhoff's Circuit Laws
Grade Level: High school (11th-12th grade)
Subject: Applied Design, Skills, and Technology (ADST) Electronics. Also applicable for Science and Physics.
Duration: ~60 minutes
Required Equipment: Power Supply, Multimeter, Laptop with Wi-Fi access
Required Components: Breadboard, Resistors of varying values
Understand Kirchhoff’s Current Law (KCL) and Kirchhoff’s Voltage Law (KVL).
Analyze complex circuits using these laws.
Apply hands-on experimentation to verify Kirchhoff's laws in real circuits.
Introduction (10 mins)
Introduce the fundamental principles behind Kirchhoff’s Circuit Laws.
Discuss the importance of KCL and KVL in circuit analysis.
Provide real-world scenarios where these laws are applicable.
Introduce students to a chosen simulation software/platform.
Virtually set up a circuit with multiple branches and loops.
Apply KCL and KVL to predict current and voltage behaviors within the simulation.
Observe and validate predictions.
Hands-on Activity: Exploring Kirchhoff’s Laws (20 mins)
Demonstrate the setup of a multi-loop circuit on a breadboard.
Guide students in taking measurements of current at various nodes and voltage across loops.
Compare these measurements with predictions based on Kirchhoff’s Laws.
Highlight instances where real-world measurements slightly deviate from theoretical values due to factors like component tolerances.
Interactive Discussion (10 mins)
Engage students in troubleshooting scenarios where a circuit doesn't behave as expected.
Use Kirchhoff's laws to identify potential issues.
Discuss the limitations and boundaries of applying Kirchhoff’s laws.
Conclusion and Recap (5 mins)
Summarize the essence and significance of Kirchhoff's laws in circuit analysis.
Encourage students to apply these laws to diverse circuits for a deeper understanding.
Formative assessment: Throughout the lesson, students will be evaluated based on their engagement in activities, their ability to correctly apply Kirchhoff’s laws in various scenarios, and their
hands-on skills in setting up and measuring circuits.
Summative assessment: After the lesson, students will take a written test that probes their understanding of KCL and KVL, practical applications, and troubleshooting techniques. Practical skills
can further be assessed by presenting them with a complex circuit and asking them to use Kirchhoff's laws to make accurate predictions about its behavior. | {"url":"https://doc.eimtechnology.com/educational-development/basic-electronics-lesson-plan/kirchhoffs-law","timestamp":"2024-11-11T23:26:27Z","content_type":"text/html","content_length":"235041","record_id":"<urn:uuid:50fdbbcd-9911-48e8-a605-5bdcb54fc8d3>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00833.warc.gz"} |
How to calculate the related coefficientHow to calculate the related coefficient 🚩 how to determine the related coefficient 🚩 More.
You will need
• - production calendar;
• - employment history of the employee or another document confirming the seniority of the employee;
• - acts of local government;
• calculator.
Calculate the total seniority of the employee. Take advantage of it and work records, other documents confirming the duties in the particular company. It can be contracts, certificates of employment,
copy of, extract from orders. Determine the periods of work in each firm, where he worked by the employee. To do this, take the start date, the termination of labour in organizations.
Fold the periods of employment in enterprises. Before that, define the number of experience expert in calendar days.
Determine the number of years of service of the employee. Use the production calendar. For years take 360 days, which is governed by Federal law. Determine the number of months in the experience of a
specialist. For the whole month take 30 days.
Calculate related coefficient. Use regulations of the regional government. Take from these documents the value of one year of seniority of the employee. Each year local authorities is set to that
value. Multiply a fixed number by the calculated number of years of experience. In the calculation need to take the experience in years. If, for example, a specialist with 17 years and 9 months of
seniority, in finding the ratioand take 17 + 9/12. The result is divide by one hundred percent.
When calculating the amount of pension related factor is multiplied by the average monthly earnings of the employee. The latter is found by dividing the total earnings for the entire period of
performance of duties by the employee during the number of months in that period. And the reward for a month of work of the expert should not be below the minimum wage established at the time the
local authorities.
The above method is determined by the amount of pension to 01.01.2002. After this period, the state benefit is dependent on these funds for the storage system. After the entry into force of the
decree of the President of the Russian Federation "On the introduction of a funded system for calculation of size of pensions," social welfare has become dependent on the amount of insurance premiums
and voluntary deductions. | {"url":"https://eng.kakprosto.ru/how-117963-how-to-calculate-the-related-coefficient","timestamp":"2024-11-07T04:46:46Z","content_type":"text/html","content_length":"30440","record_id":"<urn:uuid:7ea7f0ae-8b06-4e96-a78b-55b5d0675758>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00724.warc.gz"} |
Big Data, Plainly Spoken (aka Numbers Rule Your World)
New Yorkers have been traumatized by sensational stories of subway riders being pushed into the tracks, randomly stabbed, etc. Personal security is a hot topic. So, the MTA, which runs public
transportation, has been initiating various efforts to enhance security. Mayor Eric Adams in particular loves technological solutions.
One such effort is hiring an AI company that claims they could detect who's carrying weapons. Evolv's technology reportedly uses scanners.
The city conducted a pilot program in 20 stations for 30 days. They have not released full results, and, after multiple requests by media outlets, only issued a four-sentence statement, so some of
the following is necessarily speculative. (link)
It seems like passengers were selected and instructed to walk through the scanners, and a total of 2,749 scans triggered 130 positive signals.
The limited released data are incoherent in many ways.
The rate of scanning is surprisingly low. 2,749 scans in 30 days is about 90 scans per day on average, which means fewer than 5 scans per station per day. That makes no sense to me.
On top of that, who was responsible for picking which people to send through scanners, and how? They were definitely not scanning everyone. Let's say they do five scans a day, and since this is a
small number, let's say they do all five during one hour of the day. Which five of the passengers do they stop and scan?
Ideally, they should pick five random passengers but the base rate of carrying guns in the NYC subway is probably low, so a proper test would have to scan a lot of people, much much larger than five
per station per day.
The algo flagged 130 out of 2,749 scans, which is just under 5%. If we assumed all positives are correct, then at least 5% of New York subway riders carry guns, one in 20, so on average there are
several (concealed) guns in each car of the subway during rush hours.
But, how many of those 130 positive signals were correct? Presumably, the police then searched those suspects, and they found ... ... ... found ... ... ... zero guns. Oops. So, the experiment yielded
a positive predictive value of 0% (0/130). In plain English, a positive scan result holds zero value as an indicator of gun carry. Additionally, if any of the 2,749 people who were scanned carried
guns, they were not detected. The false negative rate is 100% (all true positives were predicted to be negative.).
Wait, that's not what the city officials said in those four sentences. They disclosed that Evolv found 12 knives, so they claimed a positive predictive value of just under 10% (12/130). I guess that
sounds better than 0%! I highly doubt the MTA (or any other establishment) would invest in scanners that can detect only knives but no guns.
Several reports I have seen computes the ratio 118/2,749 = 4.3% and called it the "false positive rate". This is the wrong definition of the FP rate; the denominator should be the number of true
negatives, not the total number of scans. In any case, the 4.3% is not a useful metric.
I discussed the kind of math that underlies all security-related statistical detection problems in Numbers Rule Your World (Chapter 4; link); refer to that chapter for more details.
For this blog post, let's focus on a simple mental model for this type of problem. We start with a guess of the base rate of gun carry (Bayesians call this the prior). Let's say it's 1%.
If the base rate is 1%, and we picked 2,749 people randomly, then we should expect to find 28 guns. The first thing to realize is that if the predictive algo were perfect, it would flag exactly 28
scans as positive. In this case, the positive rate for the scanning program should be 28/2749 = 1%.
But perfection is illusory. All models make mistakes. To allow for mistakes, a realistic algo ought to flag more than 28 scans as positive, which means it must make false positive mistakes.
The Evolv model tagged 5% of the scans as positive but at most 1% of the 5% can be positive so its accuracy is upper bounded at 1/5 = 20%. If we demand a higher positive predictive value, the algo
cannot spit out that many positives.
Moreover, the accuracy can dip below 20% because it might not catch all of the gun carriers (false negatives). The lower bound is 0%, as demonstrated by the MTA experiment. If we require the algo to
declare fewer positives, the chance of false negatives increases.
If we were evaluating these algos, one of the most important metrics should be the false negative rate i.e. of those who were carrying guns, what proportion went through the scanners undetected?
What if the base rate were much higher, like 10%, or 280 gun carriers?
Because the algo only tags 5% of the scans as positives, it "gave up" on 5% of the gun carriers, so at most it could catch half of them. The other half could also be incorrect.
If the harm of a single gun is intolerable, then the algo has to flag more subway riders, annoying them. But if the goal isn't to banish all guns from the subway, then the inconvenience of getting
scanned can be reduced by lowering the frequency of positive signals. With Evolv, they can't lower the frequency since they are not finding any guns.
In this Wired article (link), the reporter found that Evolv's technology when installed at schools or hospitals has also underperformed expectations.
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