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Uniqueness of optimal mod 3 polynomials for parity
Text: In this paper, we completely characterize the quadratic polynomials modulo 3 with the largest (hence "optimal") correlation with parity. This result is obtained by analysis of the exponential
sumS (t, k, n) = frac(1, 2n) under(∑, frac(xi ∈ {1, - 1}, 1 ≤ i ≤ n)) (underover(∏, i = 1, n) xi) ωt (x1, x2, ..., xn) + k (x1, x2, ..., xn) where t (x1, ..., xn) and k (x1, ..., xn) are quadratic
and linear forms respectively, over Z3 [x1, ..., xn], and ω = e2 π i / 3 is the primitive cube root of unity. In Green (2004) [7], it was shown that | S (t, k, n) | ≤ (frac(sqrt(3), 2))⌈ n / 2 ⌉,
where this upper bound is tight. In this paper, we show that the polynomials achieving this bound are unique up to permutations and constant factors. We also prove that if | S (t, k, n) | < (frac
(sqrt(3), 2))⌈ n / 2 ⌉, then | S (t, k, n) | ≤ frac(sqrt(3), 2) (frac(sqrt(3), 2))⌈ n / 2 ⌉. This verifies two conjectures made in Dueñez et al. (2006) [5] for the special case of quadratic
polynomials in Z3. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=mBoJrn1DuOM. © 2009 Elsevier Inc. All rights reserved.
Publication Title
Journal of Number Theory
Boolean circuit complexity, exponential sums, quadratic forms, tight bounds
Repository Citation
Green, Frederic and Roy, Amitabha, "Uniqueness of optimal mod 3 polynomials for parity" (2010). Computer Science. 53. | {"url":"https://commons.clarku.edu/faculty_computer_sciences/53/","timestamp":"2024-11-04T05:22:34Z","content_type":"text/html","content_length":"36382","record_id":"<urn:uuid:e33d94ce-1f48-4a9c-aba0-85bcf88d0200>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00445.warc.gz"} |
Can You Say “Heteroscedasticity” 3 Times Fast? | R-bloggersCan You Say "Heteroscedasticity" 3 Times Fast?
Can You Say “Heteroscedasticity” 3 Times Fast?
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Most books on regression analysis assume homoscedasticity, the situation in which Var(Y | X = t), for a response variable Y and vector of predictor variables X, is the same for all t. Yet, needless
to say, almost all data in real life is heteroscedastic. For Y = human weight and X = height, say, we know that the assumption of homoscedasticity can’t be true, even approximately.
Typical books discuss assessment of that assumption using residual plots and the like, but then leave it at that. Rather few books mention Eicker-White theory, which develops valid asymptotic
inference for heteroscedastic data. E-W is really nice, and guess what! — it’s long been available in R, in the sandwich and car packages on CRAN. (Note that the latter package is intended for use
with a book that does cover this topic, J. Fox and S. Weisberg, An R Companion to Applied Regression, Second Edition, Sage, 2011.) Then instead of using R’s standard vcov() function to obtain
estimated variances and covariances of the estimates of the β[i], we use vcovHC() and hccm(), respectively.
One can make a similar derivation for nonlinear regression, which is available as the function nlshc() in my regtools package. The package will be associated with my own book on regression, currently
in progress. (The package is currently in progress as well, but already contains several useful functions.) The rest of this post is adapted from an example in the book.
The model I chose for this simple example was E(Y | X = t) = 1 / t’β, where the distributions of the quantities can be seen in the following simulation code:
sim <- function(n,nreps) {
b <- 1:2
res <- replicate(nreps,{
x <- matrix(rexp(2*n),ncol=2)
meany <- 1 / (x %*% b)
y <- meany + (runif(n) - 0.5) * meany
xy <- cbind(x,y)
xy <- data.frame(xy)
nlout <- nls(X3 ~ 1 / (b1*X1+b2*X2),
data=xy,start=list(b1 = 1,b2=1))
b <- coef(nlout)
vc <- vcov(nlout)
vchc <- nlsvcovhc(nlout)
z1 <- (b[1] - 1) / sqrt(vc[1,1])
z2 <- (b[1] - 1) / sqrt(vchc[1,1])
print(mean(res[1,] < 1.28))
print(mean(res[2,] < 1.28))
The results were striking:
> sim(250,2500)
[1] 0.6188
[1] 0.9096
Since the true value should be 0.90 (1.28 is the 0.90 quantile of N(0,1)), the Eicker-White method is doing an outstanding job here — and R’s built-in nonlinear regression function, nls() is not. The
latter function’s reported standard errors are way, way off the mark. | {"url":"https://www.r-bloggers.com/2015/09/can-you-say-heteroscedasticity-3-times-fast/","timestamp":"2024-11-09T23:43:28Z","content_type":"text/html","content_length":"90123","record_id":"<urn:uuid:21b0ecd0-61ad-424c-9b8b-8abb5c9a458a>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00012.warc.gz"} |
Electromagnetism is a branch of physics that deals with the study of the electromagnetic force, a fundamental force of nature. It encompasses the study of electric fields, magnetic fields, and their
interactions with matter.
Electric Fields
An electric field is a region around a charged particle or object within which a force would be exerted on other charged particles or objects. The strength of the electric field is determined by the
magnitude and distribution of the electric charges that create it. The direction of the electric field is defined as the direction a positive test charge would move if placed in the field.
A magnetic field is a region around a magnet or a current-carrying conductor within which a force would be exerted on other magnets or moving charges. Magnetic fields are created by the motion of
electric charges, such as the flow of current in a wire or the movement of electrons within an atom. The strength and direction of a magnetic field are determined by the magnitude and direction of
the current or the magnetic properties of the material.
Electromagnetism describes the interaction between electric and magnetic fields. When an electric current flows through a conductor, it creates a magnetic field around the conductor. Similarly, a
changing magnetic field can induce an electric current in a conductor. This phenomenon forms the basis of electromagnetic induction and is utilized in devices such as generators and transformers.
James Clerk Maxwell formulated a set of equations that describe the behavior of electric and magnetic fields. These equations, known as Maxwell's equations, are fundamental to the understanding of
electromagnetism and have far-reaching implications in various areas of physics and engineering, including the development of electromagnetic theory and the study of light and optics.
Study Guide
1. Understand the concept of electric fields and how they are created by electric charges.
2. Learn about the properties of magnetic fields and how they are generated by moving charges.
3. Explore the interactions between electric and magnetic fields, including electromagnetic induction.
4. Study Maxwell's equations and their significance in describing electromagnetism.
5. Practice solving problems related to electric and magnetic fields, electromagnetic induction, and applications of electromagnetism in devices.
6. Explore real-world applications of electromagnetism, such as in motors, transformers, and electromagnetic waves.
Electromagnetism is a fascinating and crucial area of study in physics, with far-reaching applications in technology and everyday life. | {"url":"https://newpathworksheets.com/chemistry/high-school/states-of-matter?dictionary=electromagnetism&did=1836","timestamp":"2024-11-09T23:12:24Z","content_type":"text/html","content_length":"46347","record_id":"<urn:uuid:33ad0341-1faf-48c6-b33e-437fe73ef14d>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00552.warc.gz"} |
Pillsbury “Love the Pie” {Giveaway}
I love baking! Especially in the fall and winter months because there are so many great holidays that allow me the excuse to make all those yummy treats with extra calories. Luckily my husband
usually takes everything to work with him so I get my taste of all the different treats I love and then he takes the rest from the house so I don’t sit and eat an entire pie or an entire plate of
cookies. In the fall I really enjoy baking with apples! When I was offered the opportunity to participate in the Pillsbury “Love the Pie” promotion I was excited to hear it was all about apples and
pie! Pillsbury and My Blog Spark sent me some really awesome apple themed goodies to help me make a pie!
We actually have a couple of apple trees in our yard that don’t yield very much or very large fruit but we have a really great neighbor friend up the road who showed up at our door with a bag of
apples from their trees, her timing was perfect!
Of course my kids had to get in the kitchen and help!
The crust is my least favorite part of making pie so I was extremely grateful for the free coupon I was sent for a package of Pillsbury refrigerated pie crust.
If you are planning on pies for the upcoming Thanksgiving holiday or just to celebrate the Fall season be sure to go HERE and grab a $.50 off coupon for Pillsbury refrigerated pie crust.
We decide to make the Perfect Apple Pie receipt that can be found here on Pillsbury’s site or on the side of the refrigerated pie crust box. Then we of course ate it with vanilla ice cream and
drizzled some caramel butterscotch sauce on top!
For even more delicious pie ideas throughout the year you can visit Pillsbury.com/pie for fun and easy recipes to satisfy every season!
You can share photos and recipes of your own with other pie lovers on Pillsbury Love the Pie Facebook Fan Page or Follow Love the Pie on Twitter.
Win It: One reader will win a Pillsbury Love the Pie Prize Pack!
Mandatory Entry: Follow my blog via Google Friend Connect and leave a comment here so I know you’re following.
Be sure to leave your email address in each comment so that I have a way to contact you if you win! Please leave a separate comment for each entry.
You must complete the Mandatory Entry before you can complete any additional entries.
Extra Entries:
1. Share any Fall harvest rituals your family has, such as going apple picking or baking pies
2. “Like” me on Facebook (1 Entry)
3. Follow me on Twitter @bethwillis01 (1 Entry)
4. Share on Facebook and/or Twitter and leave the URL – when tweeting be sure to include #myblogspark (1 Entry Each- Daily)
5. Put my button on your blog (3 Entries)
6. Follow me via Networked Blogs (1 Entry)
7. Enter another one of my current giveaways (1 Entry per giveaway)
8. Write about this giveaway on your blog and link it back here to this giveaway post, then leave the URL directly to your post in a comment. (3 Entries)
9. Subscribe via Email (3 Entries)
Giveaway is open to U.S. residents only. Ends November 6th at 11:59 PM (PDT). A winner will be chosen using random.org and notified by email. The winner will have 48 hours to respond after contacted
by email. If winner does not respond, a new winner will be chosen.
Disclosure: I was sent information and product free of charge provided by Pillsbury through MyBlogSpark to review and giveaway but was not compensated monetarily.
326 comments :
1. I'm folllowing. I'm the first and only person to comment so that means I win, right? lol
deedeers30 (at) hotmail (dot) com
2. I'm a follower via GFC!
3. I'm following! :) Thanks for the chance to win. Love the pics you've shown! :) purposedrivenlife4you at gmail dot com
4. I follow you on Twitter (AaBmommyblogger)
5. I'm following on Networked Blogs also! :) I just saw in the pic the cute apple timer! My Grandmother has one and I thought of her when I saw it. I want one myself! :) She told me she's had hers'
for a while now. Not sure where she got hers' but I love it! purposedrivenlife4you at gmail dot com
6. I follow you on facebook (Emy Alwaysaroundboys)
7. We do bake a couple of pies--preferably Apple & Pumpkin around this time of the year! :) purposedrivenlife4you at gmail dot com
8. I'm following you on Networked Blogs
(Emy Alwaysaroundboys)
9. Like ya on FB!
Doreen H.
purposedrivenlife4you at gmail dot com
10. My family is new, but I already took my son apple picking and to a pumpkin patch. We plan to do it every year.
11. Following you on Twitter! luvcontests
purposedrivenlife4you at gmail dot com
12. This comment has been removed by the author.
13. I entered the Spoonful of Comfort giveaway.
14. I entered the Coobie Bra giveaway.
15. I follow you and would love to win -great for the holidays!
16. following via GFC with my FB profile:
17. fall ritual is going to our city's annual park "festival"/event
18. 10/24 tweet:
19. Following on GFC.
Thank you.
won2xx At gmail {dot}} com
20. I'm following on gfc
21. The only tradition we have would be roasting pumpkin seeds. So good!
22. I'm following you on twitter as @bmom76.
23. I entered your date night dinners giveaway.
24. I'm a GFC follower
kimhigueria at gmail dot com
25. We love going to the pumpkin patch every fall!
26. I like you on FB
27. Following on twitter @kimhigueria
28. shared this on FB
29. Tweeted this
30. following on networked blogs
31. button is on my blog at daffodilmama.blogspot.com
32. button is on my blog at daffodilmama.blogspot.com
33. entered the orglamix giveaway
34. entered the mod mum giveaway
35. entered the $25 GC & Dinner giveaway
36. entered the Little Ones Books giveaway
37. GFC follower.
38. We love to dry apple slices, topped with a sugar/pumpkin pie spice blend (my husband doesn't like plain cinnamon, otherwise I'd use that.)
39. I entered the Pur Minerals giveaway.
40. I follow on GFC ( hannah . chrinahxxx@..)
aigcanada7 (at) hotmail [dot] com
41. Im just starting out my own family - newly married here- so I'm starting my own rituals :) So far that means eating a lot of caramel apples and apple pies while apples are in season, yay!
aigcanada7 (at) hotmail [dot] com
42. "like" you on facebook (hannah kn..)
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43. follow you on twitter (achrissmile)
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44. http://twitter.com/#!/achrissmile/status/28666914224
aigcanada7 at hotmail [dot] com
45. http://www.facebook.com/permalink.php?story_fbid=135745706477782&id=100000967177106
shared on fb
aigcanada7 at hotmail dot com
46. I follow on GFC
47. I like you on Facebook
48. Our fall rituals revolve around decorating our home for the holidays. Its always fun to pull everything out!
49. I entered the Pur Minerals giveaway
50. I entered the $25 Visa GC & Dinners giveaway
51. I follow
oka_arbogasm at yahoo dot com
52. I'm a google follower!
old sock farm {at} gmail {d0t} com
(this is my email, but of course, please eliminate all the spaces)
53. follow!
54. I have started to develop some fall traditions with my new family. We buy apples and make apple pie or apple crisp when fall comes
55. follow you on FB
56. follow you on twitter @erobison09
57. button on blog #1
58. button on blog #2
59. tweet
60. entered your go go sports giveaway
61. follow your spoonful of comfort giveaway
62. entered your orglamix giveaway
63. entered your coobie bra giveaway
64. entered your best fairy books giveaway
65. entered your pink princess giveaway
66. entered your mod mum giveaway
67. entered your little ones giveaway
68. entered $25 GC giveaway
69. entered your pur minerals giveaway
70. 10/25 tweet:
71. http://twitter.com/#!/achrissmile/status/28752946148
aigcanada7 (at) hotmail [dot] com
72. http://www.facebook.com/permalink.php?story_fbid=124162680975035&id=100000967177106
shared on fb
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73. Already following via GFC (Ten Talents)
stephanierosenhahn at yahoo dot com
74. I follow on Twitter!
stephanierosenhahn at yahoo dot com
75. Like you on FB!
Username: stephanie Rosenhahn
stephanierosenhahn at yahoo dot com
76. Every fall we take a ride up in the mountains and gather Aspen leaves!
stephanierosenhahn at yahoo dot com
77. Already following via Networked Blogs.
FB Username: Stephanie Rosenhahn
stephanierosenhahn at yahoo dot com
78. Entered Date Night giveaway too!
stephanierosenhahn at yahoo dot com
79. A fall ritual that our family has is going for a long hike in the woods to admire the fall colors and collect beautiful leaves!
old sock farm {at} gmail {d0t} com
(this is my email, but of course, please eliminate all the spaces)
80. I entered the Go! Go! Sports Girls giveaway!
old sock farm {at} gmail {d0t} com
(this is my email, but of course, please eliminate all the spaces)
81. I entered the Spoonful of Comfort giveaway!
old sock farm {at} gmail {d0t} com
(this is my email, but of course, please eliminate all the spaces)
82. I entered the Coobie Bra giveaway!
old sock farm {at} gmail {d0t} com
(this is my email, but of course, please eliminate all the spaces)
83. I entered the Best Fairy Books giveaway!
old sock farm {at} gmail {d0t} com
(this is my email, but of course, please eliminate all the spaces)
84. I entered the Pink Princess giveaway!
old sock farm {at} gmail {d0t} com
(this is my email, but of course, please eliminate all the spaces)
85. I entered the Mod Mum giveaway!
old sock farm {at} gmail {d0t} com
(this is my email, but of course, please eliminate all the spaces)
86. I entered the Orglamix giveaway!
old sock farm {at} gmail {d0t} com
(this is my email, but of course, please eliminate all the spaces)
87. I entered the All Things Girly giveaway!
old sock farm {at} gmail {d0t} com
(this is my email, but of course, please eliminate all the spaces)
88. I entered the Whirl-a-Style giveaway!
old sock farm {at} gmail {d0t} com
(this is my email, but of course, please eliminate all the spaces)
89. I entered the $25 Visa GC and Dinner giveaway!
old sock farm {at} gmail {d0t} com
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90. This comment has been removed by the author.
91. I entered the Pur Minerals giveaway!
old sock farm {at} gmail {d0t} com
(this is my email, but of course, please eliminate all the spaces)
92. I entered the Rock 'n Learn giveaway!
old sock farm {at} gmail {d0t} com
(this is my email, but of course, please eliminate all the spaces)
93. I entered the Little One Books giveaway!
old sock farm {at} gmail {d0t} com
(this is my email, but of course, please eliminate all the spaces)
94. tweet
95. 10/26 tweet:
96. your gfc follower
97. We now go to the pumpkin patch every October and participate fully in all the activities possible.
98. like you on facebook
99. networked blogs follower
100. coobie bra giveaway entrant
101. whirl a style giveaway entrant
102. http://twitter.com/#!/achrissmile/status/28857947168
aigcanada7 (at) hotmail [dot] com
103. http://www.facebook.com/permalink.php?story_fbid=101741066561873&id=100000967177106
shared on fb
aigcanada7 at hotmail dot com
104. I follow you on GFC - janetfaye
janetfaye (at) gmail (dot) com
105. I like you on FB - Janet Fri
janetfaye (at) gmail (dot) com
106. I follow you on Twitter - Janetfaye
janetfaye (at) gmail (dot) com
107. I follow you on NB - Janet Fri
janetfaye (at) gmail (dot) com
108. I entered $25 Visa GC & Dinner.
janetfaye (at) gmail (dot) com
109. I entered Spoonful of Comfort giveaway.
janetfaye (at) gmail (dot) com
110. I'm a new follower! Great giveaway!
111. I don't have any rituals just yet, but getting married soon and would love to start baking apple pies in the fall! It's nice to bake, especially with what's in season! :)
112. 10/27 tweet:
113. http://www.facebook.com/permalink.php?story_fbid=138301219552523&id=100000967177106
shared on fb
aigcanada7 at hotmail dot com
114. http://twitter.com/#!/achrissmile/status/28954149043
aigcanada7 at hotmail dot com
115. 10/28 tweet:
116. Tweet:
janetfaye (at) gmail (dot) com
117. 1. I blogged:
janetfaye (at) gmail (dot) com
118. 2. I blogged:
janetfaye (at) gmail (dot) com
119. 3. I blogged:
janetfaye (at) gmail (dot) com
120. I shared on FB:
janetfaye (at) gmail (dot) com
121. I entered Tillamook Cheese.
janetfaye (at) gmail (dot) com
122. I entered Pur Minerals giveaway.
janetfaye (at) gmail (dot) com
123. 10/29 tweet:
124. I am a public folloower (tamaraben)
125. #1, Email subscriber.
126. #2, Email subscriber.
127. #3, Email subscriber.
128. I entered your $25 Visa GC & Dinner giveaway.
129. I entered your Tillamook Cheese giveaway
130. blog follower
131. We like carving pumpkins and baking
132. Like you on Facebook
133. Following you on twitter @lolsonlso
134. tweet
135. Networked Blog follower
136. Email subscriber- 1
137. Email subscriber- 2
138. email subscriber- 3
139. Entered Spoonful of Comfort
140. Entered Coobie Bra
141. Entered Best Fairy Books
142. Entered Pink Princess
143. Entered Orglamix Makeup
144. Entered Mod Mum
145. Entered All Things Girly
146. Entered Yoplait
147. Entered Whirl-a-Style
148. Entered $25 Visa
149. Entered Little One Books
150. Entered Pur Minerals
151. Entered Rock 'n Learn
152. Entered Smiley Cookie
153. Entered Petcakes
154. Entered Tillamook Cheese
155. Entered Flowerz in Her Hair
156. I'm following your blog!
coolnatty12 at yahoo dot com
157. I follow on Twitter! @natdey
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158. I like you on Facebook! Natalie Ahotaeiloa
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159. I follow on Networked Blog!
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160. I subscribe by e-mail!
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161. I subscribe by e-mail! 2
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162. I subscribe by e-mail! 3
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163. Tweet http://twitter.com/natdey/status/29156960738
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164. I entered the Date Night Giveaway!
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165. I follow via gfc
166. We love anything with apples so we just like to bake lots of apple recipes. I would love to find a place close by that we can pick apples
167. I Follow you on Twitter @bethwillis01
168. Your button is on my blog satko's recipes
169. Your button is on my blog satko's recipes
170. Your button is on my blog satko's recipes
171. I Follow you via Networked Blogs
172. I Subscribe via Email
173. I Subscribe via Email
174. I Subscribe via Email
175. I entered the coobie bra giveaway
176. blog follower!
177. I entered the best fairy books giveaway
178. I entered the orglamix mineral makeup giveaway
179. I entered the mod mum baby sling giveaway
180. I entered the all things girly bows giveaway
181. I entered the $25 visa gc and dinner giveaway
182. I entered the petcakes giveaway
183. I entered the tillamook cheese giveaway
184. Tweet http://twitter.com/natdey/status/29205269905
coolnatty12 at yahoo dot com
185. follow gfc publicly
186. always a trip to the pumpkin patch was a ritual and still is even though the kids are grown! and making pies is a must still!
187. entered vintage pearl
188. entered coobie bra
189. entered best fairy books
190. entered pink princess
191. entered orglamix
192. entered yoplait
193. entered date night dinners
194. entered little one books
195. entered tillamook cheese
196. 10/30 tweet:
197. I wrote about this giveaway on my blog:
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198. I wrote about this giveaway on my blog:
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199. I subscribe via email.
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200. I subscribe via email.
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Pi Goes the Distance at NASA
Teachable Moment .
.4 min read
Pi Goes the Distance at NASA
Pi Day, the annual celebration of one of mathematics’ most popular numbers, is back! Representing the ratio of a circle’s circumference to its diameter, pi has many practical applications, including
the development and operation of space missions at NASA’s Jet Propulsion Laboratory.
The March 14 holiday is celebrated around the world by math enthusiasts and casual fans alike – from memorizing digits of pi (the current Pi World Ranking record is 70,030 digits) to baking and
eating pies.
JPL is inviting people to participate in its 2018 NASA Pi Day Challenge – four illustrated math puzzlers involving pi and real problems scientists and engineers solve to explore space, also available
as a free poster! Answers will be released on March 15.
Pi is what’s known as an irrational number, meaning its decimal representation never ends and it never repeats. It has been calculated to more than one trillion digits, but NASA scientists and
engineers actually use far fewer digits in their calculations (see “How Many Decimals of Pi Do We Really Need?”). The approximation 3.14 is often precise enough, hence the celebration occurring on
March 14, or 3/14 (when written in U.S. month/day format). The first known celebration occurred in 1988, and in 2009, the U.S. House of Representatives passed a resolution designating March 14 as Pi
Day and encouraging teachers and students to celebrate the day with activities that teach students about pi.
The Science Behind the 2018 Challenge
To show students how pi is used at NASA and give them a chance to do the very same math, the JPL Education Office has once again put together a Pi Day challenge featuring real-world math problems
used for space exploration. This year’s challenge includes exploring the interior of Mars, finding missing helium in the clouds of Jupiter, searching for Earth-size exoplanets and uncovering the
mysteries of an asteroid from outside our solar system.
Scheduled to launch May 5, 2018, the InSight Mars lander will be equipped with several scientific instruments, including a heat flow probe and a seismometer. Together, these instruments will help
scientists understand the interior structure of the Red Planet. It’s the first time we’ll get an in-depth look at what’s happening inside Mars. On Earth, seismometers are used to measure the strength
and location of earthquakes. Similarly, the seismometer on Insight will allow us to measure marsquakes! The way seismic waves travel through the interior of Mars can tell us a lot about what lies
beneath the surface. This year’s Quake Quandary problem challenges students to determine the distance from InSight to a hypothetical marsquake using pi!
Also launching in spring is NASA’s Transiting Exoplanet Survey Satellite, or TESS, mission. TESS is designed to build upon the discoveries made by NASA’s Kepler Space Telescope by searching for
exoplanets – planets that orbit stars other than our Sun. Like Kepler, TESS will monitor hundreds of thousands of stars across the sky, looking for the temporary dips in brightness that occur when an
exoplanet passes in front of its star from the perspective of TESS. The amount that the star dims helps scientists determine the radius of the exoplanet. Like those exoplanet-hunting scientists,
students will have to use pi along with data from Kepler to find the size of an exoplanet in the Solar Sleuth challenge.
Jupiter is our solar system’s largest planet. Shrouded in clouds, the planet’s interior holds clues to the formation of our solar system. In 1995, NASA’s Galileo spacecraft dropped a probe into
Jupiter’s atmosphere. The probe detected unusually low levels of helium in the upper atmosphere. It has been hypothesized that the helium was depleted out of the upper atmosphere and transported
deeper inside the planet. The extreme pressure inside Jupiter condenses helium into droplets that form inside a liquid metallic hydrogen layer below. Because the helium is denser than the surrounding
hydrogen, the helium droplets fall like rain through the liquid metallic hydrogen. In 2016, the Juno spacecraft, which is designed to study Jupiter’s interior, entered orbit around the planet. Juno’s
initial gravity measurements have helped scientists better understand the inner layers of Jupiter and how they interact, giving them a clearer window into what goes on inside the planet. In the
Helium Heist problem, students can use pi to find out just how much helium has been depleted from Jupiter’s upper atmosphere over the planet’s lifetime.
In October 2017, astronomers spotted a uniquely-shaped object traveling in our solar system. Its path and high velocity led scientists to believe ‘Oumuamua, as it has been dubbed, is actually an
object from outside of our solar system – the first ever interstellar visitor to be detected – that made its way to our neighborhood thanks to the Sun’s gravity. In addition to its high speed,
‘Oumuamua is reflecting the Sun’s light with great variation as the asteroid rotates on its axis, causing scientists to conclude it has an elongated shape. In the Asteroid Ace problem, students can
use pi to find the rate of rotation for ‘Oumuamua and compare it with Earth’s rotation rate.
About the Author
Lyle Tavernier
Educational Technology Specialist, NASA-JPL Education Office
Lyle Tavernier is an educational technology specialist at NASA's Jet Propulsion Laboratory. When he’s not busy working in the areas of distance learning and instructional technology, you might find
him running with his dog, cooking or planning his next trip.
Teachable Moment Last Updated: Oct. 11, 2024 | {"url":"https://www.jpl.nasa.gov/edu/resources/teachable-moment/pi-goes-the-distance-at-nasa/","timestamp":"2024-11-09T20:29:54Z","content_type":"text/html","content_length":"537515","record_id":"<urn:uuid:1b81573d-d3a9-419d-949e-45787b6ca79b>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00670.warc.gz"} |
Dedekind Cuts
02 Dedekind Cuts
In the previous section, we looked at the real numbers and showed that not every real number is rational. We were able to produce a variety of specific irrational numbers. But the approach to real
numbers was simply to say that real numbers correspond to points on a line. This leaves open the question of exactly what a "line" is. It was not even immediately clear that there are non-rational
numbers, so simply relying on intuition about geometric lines is not going to give us a full understanding of the real numbers. What we need is a way to construct a specific mathematical object to
represent the real numbers—something definite enough that we can prove things about it. In modern mathematics, mathematical objects are defined in terms of sets.
There are several approaches to building a set to represent the real numbers. The one used in Section 1.2 of the textbook is Dedekind cuts. For us, Dedekind cuts are simply a way to get a concrete
representation of the real numbers. In fact, once we have done that and used them to get some understanding of the real numbers, you should pretty much forget about them.
The idea behind Dedekind cuts is the observation that any real number, $x$, divides the rational numbers into two pieces, the rational numbers that are less than $x$ and the rational numbers that are
greater than $x$. (Of course, if $x$ is rational, then it's not included in either of these two pieces.) For $x=\root 3 \of 2$, we can easily specify the pieces without even mentioning $\root 3 \of
2$. In this picture, the gray line represents $\Q$, the set of rational numbers; it's gray rather than black because all of the irrational numbers are missing. The vertical line marks the division
point that represents the cube root of three:
A Dedekind cut can be thought of a division point in the rational numbers that cuts $\Q$ into two pieces of this sort. To make this more specific, and express it in terms of sets, we define a
Dedekind cut to be the left-hand piece in such a division. That is, it is a subset of $\Q$ containing all of the rational numbers in the left-hand piece. (Sometimes, a Dedekind cut is defined as an
ordered pair containing both the left-hand piece and the right-hand piece; that would make some proofs easier but would complicate the definition.)
A real number is then defined as a Dedekind cut, and $\R$, the set of real numbers, is the set of all Dedekind cuts.
The problem is to say exactly which subsets of $\Q$ are Dedekind cuts. We can't just say that a real number $\gamma$ is the subset consisting of all real numbers less than $\gamma$. That would be a
circular definition! We have to say what it means to be a Dedekind cut without referring to a real number that doesn't exist yet. The book gives three conditions on a subset of $\Q$ that the subset
must meet in order to be a Dedekind cut:
Definition: A Dedekind cut is a subset, $\alpha$, of $\Q$ that satisfies
1. $\alpha$ is not empty, and $\alpha$ is not $\Q$;
2. if $p\in\alpha$ and $q<p$, then $q\in\alpha$; and
3. if $p\in\alpha$, then there is some $r\in\alpha$ such that $r>p$
The three requirements just say, in a mathematically exact way, that a Dedekind cut consists of all rational numbers to the left of some division point. Each Dedekind cut, that is each possible
division point, represents a real number. This definition constructs the real numbers entirely in terms of the rational numbers, using only basic set operations.
Of course, $\R$ is more than just a set. There are operations such as $x+y$ and $x<y$ that need to be defined for real numbers. There must be a way of defining such operations in terms of Dedekind
cuts and proving that they have all of the expected properties. The textbook does this for only a few properties, and I won't try to expand on what it does. One of the most important things for us is
defining $\alpha<\beta$ for Dedekind cuts $\alpha$ and $\beta$. The definition uses the fact that $\alpha$ and $\beta$ are defined as sets: $\alpha<\beta$ if and only if $\alpha\subset\beta$, and $\
alpha\le\beta$ if and only if $\alpha\subseteq\beta$. (Here, $A\subset B$ means that $A$ is a proper subset of $B$; that is, $A$ is contained in but not equal to $B$. And $A\subseteq B$ means that $A
\subset B$ or $A=B$.)
With this definition, it becomes possible to prove one of the most important properties of the real numbers, the least upper bound property. That will be the topic of the next section.
As an example, let's prove the "trichotomy" law for real numbers: For real numbers $\alpha$ and $\beta$, exactly one of the following is true: $\alpha<\beta$, $\alpha=\beta$, or $\beta<\alpha$. In
terms of Dedikind cuts, exactly one of $\alpha\subset\beta$, $\alpha=\beta$, or $\beta\subset\alpha$ is true. A picture of two Dedekind cuts makes this clear, but let's try to prove it using only the
First note that if $\alpha=\beta$, then both $\alpha\subset\beta$ and $\beta\subset\alpha$ are false. So suppose $\alpha\ne\beta$. We must show that either $\alpha\subset\beta$ or $\beta\subset\
alpha$ is true. (They can't both be true, since that would mean $\alpha=\beta$.)
Since $\alpha\ne\beta$, then either there is some $p\in\alpha$ such that $p\not\in\beta$ or there is some $p\in\beta$ such that $p\not\in\alpha$. Consider the first case; the second case is similar.
So, suppose that $p$ is a rational number such that $p\in\alpha$ and $p\not\in\beta$. We show that in this case, $\beta\subset\alpha$,
Let $q\in\beta$. We must show $q\in\alpha$. We know that $q\in\beta$ and $p\not\in\beta$. It follows that $q<p$, for if $p<q$, then $p$ would be in $\beta$ by property 2 of Dedekind cuts (applied to
$\beta$). So we have $p\in\alpha$ and $q<p$. By property 2 of Dedekind cuts (applied to $\alpha$), this implies that $q\in\alpha$, as we wanted to show. | {"url":"https://math.hws.edu/eck/math331/guide2020/02-dedekind-cuts.html","timestamp":"2024-11-04T10:18:18Z","content_type":"text/html","content_length":"7456","record_id":"<urn:uuid:557f4872-ecaf-45a2-8ac5-37cc200a93ef>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00596.warc.gz"} |
Why Learn Maths
Just before Easter I ran a session for the A Level mathematics groups in the Harris Academy group in South East London. I can tell you it was pretty daunting in the small hall, but with something
like 120 sixth form students, who had chosen me over another talk, about options, I think. However, can I publicly thank (a) the teachers at Harris Crystal Palace who invited me and most especially
(b) the students who attended for reminding me what fun it is to talk about maths to young people. I’ll be applying for a teaching job again, next …
Truth is, that even if you have chosen to take A Level maths (as all of the students had), doesn’t mean that it’s because you just love the subject … maybe it’s becuase someone told you that people
with maths A level earn on average 10% more than other A levels (which they do see here), but I thought I’d push harder and found out that those clever RSA people who are behind internet cryptography
(why people don’t steal your credit card details online) whose algorithm essentially relies on the difficulty of finding large factors af really large numbers, sold their company for $2.1 billion in
2006 (see here) or the $100,000 payout for finding the first 12-million-digit prime number (there’s $150,000 for the next milestone … 100 million digits) see here.
But really, the big argument is, well frankly, maths is amazing. The number 1 is the basis of measurement … size relies on a unit. The invention of the zero making decimal place value possible was
described by John Barrow thus ” The Indian system of counting has been the most successful intellectual innovation ever made on our planet. It has spread and been adopted almost universally, ….. It
constitutes the nearest thing we have to a universal language.” If you solve linear and quadratic equations you gradually require more sophisticated numbers to describe the solutions try x+1=3
(counting numbers), then x+5=3 (integers), then 2x=5 (rationals), then x²=2 (irrationals), then x²=1 (irrationals), so we need the number i to solve this last oh so simple equation. We can find the
ratio of the circumference to the diameter of a circle and find it is an irrational we call π . Finally, we can find that the exponential equation whose differential is the same as the function has
as its base another irrational we call e.
And from all of these disparate sources we find that e^iπ+1=0. You can get wedding rings with that engraved on them! WOW. Two students were heard discussing whether they would want one for their
wedding and it was a tied vote!
The students listened and engaged and reacted and that was great. My message was that we should study maths because it is a fantastic subject and, up to a point at least, I think these thoughtful,
engaging young people could go with that. | {"url":"http://www.themathszone.com/?p=385","timestamp":"2024-11-07T20:29:19Z","content_type":"text/html","content_length":"37585","record_id":"<urn:uuid:fb3654ba-89c4-481a-83af-566705edcebf>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00418.warc.gz"} |
Adapted from Transformers.jl
TopoChains.jl allows you to cleanly build flexible neural networks whose layers can take any number of inputs, and produce any number of outputs. It achieves this by seperating the layers from the
overall topology (that is, the structure) of the model. This is done through an instance of the FuncTopo type, which specifies the inputs/outputs the layers take in/produce.
This package provides two core features:
• @functopo: A macro that uses a compact DSL (Domain Specific Language) to store the structure of the model in a FuncTopo.
• TopoChain: Similar to a Flux.Chain, except it takes in an FuncTopo as its first argument to determine how to handle the multiple inputs/outputs across layers.
A TopoChain is similar to a Flux.Chain and comes with many of the same features, such as parameter collection, indexing, slicing, etc. The big change is that the first input to a TopoChain is a
FuncTopo, which specifies how the layers should be called. This allows us to flexibly build complex architectures.
TopoChain(topo::FuncTopo, layers...)
Similar to a Flux.Chain, with the addition of the use of an FuncTopo to define the order/structure of the functions called.
julia> topo = @functopo x:x => a:x => b:(a, b) => c => o
julia> model = TopoChain(topo,
Dense(32, 64),
Dense(32, 64),
(x, y) -> x .* y,
TopoChain(Dense(32, 64), Dense(32, 64), #5, Dropout(0.1)) representing the following function composition:
a = Dense(32, 64)(x)
b = Dense(32, 64)(x)
c = #5(a, b)
o = Dropout(0.1)(c)
As we can see, with the help of the FuncTopo, the TopoChain not only holds the layers in a model, but also information on how to call the layers in a model as well.
We store the structure of the model in a FuncTopo, short for "Function Topology", by noting that a model is essentially a large function composed of many smaller functions. At its core, it is simply
used to define inputs and outputs for each function in a sequence of function calls. Consider it a supercharged version of Julia's piping operator (|>).
FuncTopos are usually created by using the @functopo macro as shown:
@functopo structure
Create a FuncTopo to apply functions according to the given structure.
julia> @functopo (x1, x2):(x1, x2) => a:x1 => b:(a, b) => c => o
FuncTopo{"(x1, x2):(x1, x2) => (a:x1 => (b:(a, b) => (c => o)))"}
function(model, x1, x2)
a = model[1](x1, x2)
b = model[2](x1)
c = model[3](a, b)
o = model[4](c)
We now take a look at how @functopo is used, as well a deep dive into the syntax used in structure in the following sections, so you can write your own ones for use in your own models!
Suppose you have inputs x1 and x2 you want to pass through the functions f, g and h as follows to get the output o:
You could do the following in regular Julia:
a = f(x1, x2)
b = g(x1)
c = h(a, b)
o = m(c)
This is functional, but gets increasingly unwieldy as the number of functions/layers in your models grow. With the TopoChains.jl approach, we seperate the structure from the actual function calls. In
this case, we first define the structure as follows:
topo = @functopo (x1, x2):(x1, x2) => a:x1 => b:(a, b) => c => o
The @functopo macro then takes the information given, and produces the FuncTopo instance topo that keeps track of how to call the functions, once given the functions:
# FuncTopo{"(x1, x2):(x1, x2) => (a:x1 => (b:(a, b) => (c => o)))"}
# function(model, x1, x2)
# a = model[1](x1, x2)
# b = model[2](x1)
# c = model[3](a, b)
# o = model[4](c)
# o
# end
Here, model stands in for an iterable (e.g. a Tuple or Vector) of functions and layers. While the most typical use of an FuncTopo will be passing it as input to a TopoChain, we can indeed use topo
directly by passing in the functions and inputs:
x1 = 3
x2 = 2
f(x, y) = x^2 - y
g(x) = x^3
h(x, y) = x + y
m(x) = mod(x, 4)
topo((f, g, h, m), x1, x2) # 2
Let's take a deep dive into the syntax used in defining the structure here.
We use multiple variable names when defining the structure (e.g. x, c, etc.). These are the names of the intermediate outputs in the function generated by FuncTopo. Similar to how x in g(x) = x^3 has
no relation with a previously defined x in the Julia session, the variables used to specify the structure have no relation with previously defined variables.
Each application of a function is represented with a =>, with the input variables on the left and output variables on the right. For instance, a => b means "take the variable a and pass it to the
function to produce the output b. This also allows us to chain functions together. Suppose you want to chain the functions p, q and r as follows:
y = r(q(p(x)))
You could equivalently write the following with the TopoChains.jl approach, seperating the structure from the functions:
topo = @functopo x:x => a:a => b:b => y
y = topo((p, q, r), x)
When the actual function calls are made, the functions are used in the order they were passed in. Here the tuple of functions is (p, q, r) and so the first arrow in structure corresponds to applying
the first function p, the second arrow applies q, and so forth.
Notice that we use a : to seperate the input/output variable names for each function call. If the : is not present, we will by default assume that all output variables are the inputs of the next
function call. This can be used to simplify structures. Above, we wrote
@functopo x:x => a:a => b:b => y
when we could just as well have written
@functopo x => a => b => y
When a function has multiple inputs/outputs, we use a tuple of variables instead of variables. For instance, a function that takes two inputs and produces three outputs would be specified as (a, b)
=> (x, y, z)
The complete syntax for a structure can then be viewed as:
(input arguments):(function1 inputs) => (function1 outputs):(function2 inputs):(function2 outputs) => .... => (function_n outputs):(return variables)
Suppose in the structure of your model, there are repeated substructures. For instance, suppose you have a pair of layers:
• The first of which takes an output and produces two outputs
• The second takes two outputs and produces one output
And say that this pair structure is repeated 3x in your model. Instead of writing it out in full, you can do so more concisely with the following syntax:
topo = @functopo (y => (z1, z2) => t) => 3
When the output of a => is an integer N instead of a variable, instead of applying a function we repeat the sub-structure (specified in between the brackets ( and )) N times. Indeed, we can see this
produces the expected behavior:
# FuncTopo{"(y => ((z1, z2) => t)) => 3"}
# function(model, y)
# (z1, z2) = model[1](y)
# t = model[2](z1, z2)
# (z1, z2) = model[3](t)
# t = model[4](z1, z2)
# (z1, z2) = model[5](t)
# t = model[6](z1, z2)
# t
# end
We can also nest our substructure repeats. This allows us to quickly specify complex models rather concisely. For instance:
topo = @functopo x => ((y => z => t) => 3 => w) => 2
# FuncTopo{"x => (((y => (z => t)) => (3 => w)) => 2)"}
# function(model, x)
# y = model[1](x)
# z = model[2](y)
# t = model[3](z)
# z = model[4](t)
# t = model[5](z)
# z = model[6](t)
# t = model[7](z)
# w = model[8](t)
# z = model[9](w)
# t = model[10](z)
# z = model[11](t)
# t = model[12](z)
# z = model[13](t)
# t = model[14](z)
# w = model[15](t)
# w
# end | {"url":"https://irhum.github.io/TopoChains.jl/dev/","timestamp":"2024-11-04T21:19:30Z","content_type":"text/html","content_length":"18874","record_id":"<urn:uuid:d997b00b-2a57-444e-a15d-80d76efecbda>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00266.warc.gz"} |
Vertical Angles Theorem
Definition: Vertical Angles are angles whose sides form 2 pairs of opposite rays. When 2 lines intersect, 2 pairs of vertical angles are formed. One pair of vertical angles is shown below. (Click the
other checkbox on the right to display the other pair of vertical angles.) Interact with the following applet for a few minutes, then answer the questions that follow.
Directions & Questions: 1) Complete the following statement (based upon your observations). Vertical angles are always __________________________. 2) Suppose the pink angle above measures 140
degrees. What would the measure of its vertical angle? What would be the measure of the other 2 (gray) angles? | {"url":"https://www.geogebra.org/m/SGhM48n5","timestamp":"2024-11-12T17:12:26Z","content_type":"text/html","content_length":"89412","record_id":"<urn:uuid:17b12e33-a442-47d0-898c-307c60e2ddb9>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00538.warc.gz"} |
ESP Biography
LE NGUYEN HOANG, MIT Postdoc & Science4All writer
Major: EECS
College/Employer: MIT
Year of Graduation: Not available.
Brief Biographical Sketch:
I am a postdoc in LIDS, MIT. My research focuses on online optimization, game theory and related topics.
I am also a math and science popularizer. I write on Science4All.org, and make videos on the Science4All Youtube channel. Some of my favourite topics include, but are definitely not limited to, math
foundations, computer science, history of math, general relativity, quantum mechanics...
Past Classes
(Clicking a class title will bring you to the course's section of the corresponding course catalog)
S9681: Why do apples fall? From Galileo to Einstein in Splash 2015 (Nov. 21 - 22, 2015)
This question has puzzled great thinkers for centuries, but it's only in the early 20th century that Albert Einstein would finally provide a full explanation of the falling of the apples. In this
class, we review the history of the theories of gravity, starting with Aristotle, Galileo, Newton and ending with Einstein. The class is based on this article: http://www.science4all.org/
M9682: The Math Foundation Crisis in Splash 2015 (Nov. 21 - 22, 2015)
You might have learned that mathematics was the only field that proves true statements. But a century ago, it wasn't clear at all that mathematics had anything to do with "a" truth, let alone "the"
truth --- it's still not clear today! In this class, we review the infamous math foundation crisis of the turn of the century, from the overthrowing of Euclid's elements and Russell's paradox, to
surprising fundamental 20th century theorems like the Banach-Tarski paradox, the continuum hypothesis and Gödel's incompleteness theorem.
M9683: The Mathematics of Democracy in Splash 2015 (Nov. 21 - 22, 2015)
Do our voting systems elect the people's favourite candidate? Short answer: no. The theory of voting systems has a long history, and mathematics has a lot to say in that theory. In fact, early on, in
the 1700s, debates over voting systems already opposed two mathematicians, Condorcet and de Borda. Over 200 years later, impressive progress has been made by, among others, Arrow, Gibbard,
Satterthwaite... but the debate is still there!
C9685: Cryptographers vs hackers... who'll win? in Splash 2015 (Nov. 21 - 22, 2015)
For centuries, cryptographers have tried to secretly send encoded messages, and hackers have tried to crack the messages. In those days, cracking a message could win wars and save millions of lives,
as Turing did. More recently, this opposition has been formalized within computer science, and we understand better than ever the essence of it... but we are still largely ignorant. In this class,
I'll mention historical encryption methods, as well as modern open questions every computer scientist dreams to have the answer to. | {"url":"https://esp.mit.edu/teach/teachers/lenhoang/bio.html","timestamp":"2024-11-13T14:02:56Z","content_type":"application/xhtml+xml","content_length":"17198","record_id":"<urn:uuid:eead2bb9-9c9e-4fa2-8d81-1c794d9bf543>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00281.warc.gz"} |
MATH4426: Probability - Rob Gross
MATH4426: Probability
Prerequisite: MATH2202 or MATH2203, Multivariable Calculus.
This course provides a general introduction to modern probability theory. Topics include probability spaces, discrete and continuous random variables, joint and conditional distributions,
mathematical expectation, the central limit theorem, and the weak and strong laws of large numbers.
Normal distribution table: Click here.
1. First assignment, does not exist.
1. TBA.
2. TBA
3. TBA
Final examination:
• Section 1: TBA
• Section 2: TBA | {"url":"https://sites.bc.edu/rob-gross/rob-gross/math4426/","timestamp":"2024-11-05T00:26:50Z","content_type":"text/html","content_length":"79675","record_id":"<urn:uuid:765d9c3b-f921-40eb-beb0-d96c129b5f3f>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00173.warc.gz"} |
Getting the sign of a float
Godot Version
Howdy friends! In the following piece of code, I’m trying to multiply the player’s acceleration by the sign of the normal of the ground they are standing on. For example, if they’re on a slope facing
right, the x normal should be positive and the z normal should be zero, and if they’re standing on a slope facing down, the z normal should be negative (I think) and the x normal should be zero. I’m
trying to get the sign of the normal values only, not their actual value because that’s a pretty small number. I tried doing this by dividing the normal values by their absolute self to turn them
into either positive or negative one and then multiplying the acceleration number by that, but the problem is, if the normal x or z is equal to 0, the acceleration becomes NaN because dividing by
zero is illegal. Is there a way I could get the sign of my normals without dividing by zero? Thank you!
How about something like var sign = 1 if my_float >= 0 else -1.
There’s a built-in sign function, which returns 1 if given a positive number, -1 if given a negative number, and 0 of given either 0 or NaN.
2 Likes
Thank you, this almost works except for one issue, as shown in the screenshot below:
Even though the x normal is 0, the sign function still returns a positive 1, as seen in the print statement at the bottom. Is there a reason for this?
It only returns the sign of zero being positive at the frame where the player steps from a flat surface to a slope, it doesn’t do it if I jump off a flat surface and land on a slope.
I should also add that, in situations where I step onto a north facing slope, I get a -1 as the sign of my normal x. It also says the normal x is -0. What makes a zero negative and how do I prevent
used the typed signf for floats.
floats are very rarely zero, would it be better to clamp the value?
ACCEL.x = 2.5 * clampf(floornorm.x, -1, 1)
1 Like
The clamping works for sure! I’m not exactly sure how, I think the norm x wasn’t actually zero but a very tiny number
1 Like
This topic was automatically closed 30 days after the last reply. New replies are no longer allowed. | {"url":"https://forum.godotengine.org/t/getting-the-sign-of-a-float/75084","timestamp":"2024-11-14T11:21:10Z","content_type":"text/html","content_length":"32354","record_id":"<urn:uuid:0011919d-ff51-4f7c-ae11-812a175a7601>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00378.warc.gz"} |
Transcendental Physics
Unifying Quantum Physics and Relativity
The full unification of quantum physics and relativity is brought about in TDVP by applying the tools of CoDD and Dimensional Extrapolation to the mathematical expressions of three well-established
features of reality, recognized in the current scientific paradigm: 1.) quantization of mass and energy as two forms of the same essential substance of reality; 2.) introduction of time as a fourth
dimension, and 3.) the limitation of the velocity of rotational acceleration to light speed, c. In this process, the need for a more basic unit of quantization is identified, and when it is defined,
the reason there is something rather than nothing becomes clear.
Einstein recognized that mass and energy are interchangeable forms of the physical substance of the universe, and discovered that their mathematical equivalence is expressed by the equation E=mc^2.
In TDVP, accepting the relativistic relationship of mass and energy at the quantum level, we proceed, based on Planck’s discovery, to describe quantized mass and energy as the content of quantized
dimensional distinctions of extent. This allows us to apply the CoDD to quantum phenomena as quantum distinctions and describe reality at the quantum level as integer multiples of minimal equivalence
units. This replaces the assumption of conventional mathematical physics that mass and energy can exist as dimensionless points analogous to mathematical singularities.
The assumption of dimensionless physical objects works for most calculations in practical applications because our units of measurement are so extremely large, compared to the actual size of
elementary quanta, that the quanta appear to be existing as mathematical singularities, i.e. dimensionless points. (The electron mass, e.g., is about 1x10^-30 kg, with a radius of about 3x10^-15
meter.) Point masses and point charges, etc. are simply convenient fictions for macro-scale calculations. The calculus of Leibniz and Newton works beautifully for this convenient fiction because it
incorporates the fiction mathematically by assuming that the numerical value of a function describing the volume of a physical feature of reality, like a photon or an electron, can become a specific
discrete finite entity as the value of a real variable, like the measure of distance or time approaches zero asymptotically (i.e. infinitely closely). This is a mathematical description of a
non-quantized reality. But we exist in a quantized reality.
Planck discovered that the reality we exist in is actually a quantized reality. This means that there is a “bottom” to physical reality; it is not infinitely divisible, and thus the calculus of
Newton and Leibniz does not apply at the quantum level. This is one reason scientists applying Newtonian calculus to quantum mechanics declare that quantum reality is ‘weird’. The appropriate
mathematical description of physical reality at the quantum level is provided by the calculus of distinctions with the relationships between the measureable minimum finite distinctions of elementary
particles defined by integral solutions of the appropriate Diophantine equations. The mathematics of quanta is the mathematics of integers.
In TDVP we find that, for quantized phenomena, existing in a multi-dimensional domain consisting of space and time, embedded in one or more additional dimensional domains, the fiction of
dimensionless objects, a convenient mathematical expedient when we did not know that physical phenomena are quantized, is no longer appropriate. We can proceed with a new form of mathematical
analysis, the calculus of dimensional distinctions (CoDD), and treat all phenomena as finite, non-zero distinctions. Replacing the dimensionless points of conventional mathematical physics with
distinctions of finite unitary volume, we can equate these unitary volumes of the elementary particles of the physical universe with integers. We can then relate the integers of quantum reality to
the integers of number theory and explore the deep relationship between mathematics and reality.
In TDVP, we have also developed the procedure of Dimensional Extrapolation using dimensional invariants to move beyond three dimensions of space and one of time. Within the multi-dimensional domains
defined in this way, mass and energy are measures of distinctions of content. If there are other dimensions beyond the three of space and one of time that are available to our physical senses, how
are they different, and do they contain additional distinctions of content? If so, how is such content different from mass and energy? We know that mass and energy are two forms of the same thing. If
there are other forms, what is the basic “stuff” that makes up the universe? Is it necessarily a combination of mass and energy, - or something else? For the sake of parsimony, let’s begin by
assuming that the substance of reality, whatever it is, is multi-dimensional and uniform at the quantum level, and that mass and energy are the most easily measurable forms of it in the 3S-1t domain.
This allows us to relate the unitary measure of inertial mass and its energy equivalent to a unitary volume, and provides a multi-dimensional framework to explore the possibility that the “stuff” of
reality may exist in more than two forms.
The smallest distinct objects making up the portion of reality apprehended by the physical senses in 3S-1t, i.e. that which we call physical reality, are spinning because of asymmetry and the force
of the natural universal expansion that occurs as long as there is no external resistance. If there were no additional dimensions and/or features to restore symmetry, and no limit to the acceleration
of rotational velocity, physical particles would contract to nothingness, any finite universe would expand rapidly to maximum entropy as predicted by the second law of thermodynamics for finite
systems. But, due to the relativistic limit of light speed on the accelerated rotational velocity of elementary particles in 3S-1t, the quantized content of the most elementary particle must conform
to the smallest possible symmetric volume, because contraction to a smaller volume would accelerate the rotational velocity of the localized particle to light speed in 3S-1t, making its mass
(inertial resistance) infinite. That minimal volume occupied by the most elementary of particles is the finite quantum distinction replacing the infinitesimal of Newton/Leibniz calculus, and it
provides the logical volumetric equivalence unit upon which to base all measurements of the substance of reality.
We can define this minimal volume as the unitary volume of extent, and its content as the unitary quantity of mass and energy. The mass/energy relationship (E=mc^2) is linear, since in the 3S-1t
context, c^2 is a constant, allowing us to define unitary mass and unitary energy as the quantity of each that can occupy the finite rotational unitary volume. This fits nicely with what we know
about elementary particles: All elementary particles behave in the same way prior to impacting on a receptor when encountering restricting physical structures like apertures or slits. A particle of
unitary mass occupying a unitary volume could be an electron, and a particle of unitary energy occupying a unitary volume before expansion as radiant energy, could be a photon. Einstein explained
this equivalence between electrons and photons and Planck’s constant in a paper published in 1905.
This brings us to a very interesting problem: what happens when we combine multiples of the unitary volumes of mass/energy to form more complex particles? How do we obtain protons and neutrons to
form the stable elemental structures of the physical universe?
When we view the spinning elementary particles of the 3S-1T physical universe from the perspective of a nine-dimensional reality, we can begin to understand how Planck was quite correct when he said
“there is no matter as such”. What we call matter, measured as mass, is not really “material” at the quantum level. What is it then that we are measuring when we weigh a physical object? The real
measurement of mass is not weight, which varies with relative velocity and location and can be zero without any loss of substance; it is inertia, the resistance to motion. The illusion of solid
matter arises from the fact that elementary particles resist accelerating forces due to the fact that they are spinning like tiny gyroscopes, and they resist any force acting to move them out of
their planes of rotation. An elementary particle spinning in all three orthogonal planes of space resists lateral movement equally in any direction, and the measurement of that resistance is
interpreted as mass.
Mass and energy, the two known forms of the substance of the physical universe, embedded in a nine-dimensional domain, form stable structures only under very specific mathematical and
dimensionometric conditions. Without these conditions, no physical universe could exist because of the second law of thermodynamics^23, which dictates that any finite physical system always decays
toward maximum entropy, i.e. total disorder, lacking structure of any kind. If our universe were composed of random debris from an explosion originating from a mathematical singularity, because of
the continuous operation of the second law of thermodynamics in an expanding debris field, simple particles accidentally formed by random mass/energy encounter, would decay before a new random
encounter could occur and form a more complex combination, because the number random encounters would decrease as the debris field expands. If our physical universe is embedded in the
nine-dimensional reality described by TDVP, it escapes this fate of dissolution. While it may change and evolve, its form, and even the way it evolves, will always reflect the intrinsic logical order
and patterns of the transfinite substrate within which it is embedded. If this is correct, we have the answer to the question Leibniz regarded as the first and most important metaphysical question of
all: We can explain why there is something instead of nothing.
Dividing the world of our experiences into the internal or subjective and the external, assumed to be completely independent of any form of consciousness, i.e. leaving consciousness out of the
equations, as the current scientific paradigm does, alienates consciousness from the ‘real’ world of the physical universe and leads to an endless chain of unresolvable paradoxes. The prevalence of
this attitude among scientists is expressed very well by MIT physicist - become science writer Alan Lightman in his recent book “The Accidental Universe”. In talking about the apparent ‘fine-tuning’
of the physical universe (if any one of a number of parameters were only a tiny bit different, there would be no chance for life as we know it), he says “Intelligent Design is an answer to
fine-tuning that does not appeal to most scientists.”
When confronted with the observer-related non-locality of Bohr’s solution to the EPR paradox, most scientists prefer the multiverse theory, devised to preserve Cartesian duality and keep
consciousness out of the picture of ‘scientific objectivity’. In the multiverse theory, there are many, many parallel universes. Just how many there are is unknown and unknowable, because your
consciousness only exists in this one, and unfortunately you cannot experience any of the other universes. Thus, just like the spate of string theories, there is no hope of proving or disproving such
a theory. Even though these scientists pride themselves in being ‘hard-nosed’ objective scientists (read: materialists), it doesn’t seem to bother them that string theory and the multiverse theory
cannot be tested. At best, they can only be internally consistent; and thus they do not even qualify as scientific hypotheses. By retreating into safely unprovable theories, they continue to throw
the baby out with the bath water. TDVP, on the other hand, by including consciousness as an objective reality, is producing testable results and explaining observations that the current materialistic
paradigm cannot explain. Several of these are listed in the previous section.
In this paper, I take the time to explain exactly how we put consciousness into the equations as part of objective reality, and show how doing so explains many things inexplicable in the current
materialistic paradigm.
The Illusion of Material Reality
Clues from relativity and quantum physics suggest that the time-honored idea that matter, energy, space, and time exist separately is incorrect. It appears that the macro forms of matter, space and
time we perceive through our physical senses are subtle illusions; although, as Einstein said about time, they are “very persistent” illusions. TDVP is built upon, and an extension of, the monumental
works of a number of intellectual giants like Pythagoras, Fermat, Leibniz, Poincare, Cantor, and Minkowski; but most especially, it is built upon on the deep insights of Max Planck and Albert
Max Planck said: "As a man who has devoted his whole life to the most clear-headed science, to the study of matter, I can tell you as the result of my research about atoms this much: There is no
matter as such! All matter originates and exists only by virtue of a force. We must assume behind this force the existence of a conscious and intelligent Mind. This Mind is the matrix of all matter."
Albert Einstein said: “Space-time is not necessarily something to which one can ascribe a separate existence.” And “I want to know the thoughts of God, everything else is just details”
These statements, from two of the most brilliant scientists who spent their entire lives studying physical reality, reveal the important conclusion that the common perceptions of matter, energy,
space, and time, conveyed to our brains by the physical senses, are subtle illusions! And both of them conclude that the reality behind these subtle illusions is a conscious, intelligent Mind!
It has long been known that the appearance of solid matter is an illusion, in the sense that there appears to be far more empty space than substance in an atom. But now we learn that the matter of
sub-atomic particles and the “empty” space around them are also illusory. This is, however, consistent with quantum physics experiments that bear out the conclusion resulting from the resolution of
the EPR paradox with the empirical demonstration of John Bell’s inequality by experimental physicist Alain Aspect and many others that the particles and/or waves of the objective physical reality
perceived through our senses cannot be said to exist as localized objects until they impact irreversibly on a series of receptors constituting a distinct observation or measurement by a conscious
We must be clear, however, that this does not validate subjective solipsist theories like that of Bishop Berkley as one might think; rather, it reveals a deeper, multi-dimensional reality, only
partially revealed by the physical senses. It suggests that reality is like a fathomless, dynamic ocean that we can’t see except for the white caps. The difference is that the particles and waves,
analogous to the white caps, only appear in response to our conscious interaction with the ocean of the deeper reality.
As noted above, Albert Einstein is quoted as saying: “Ich will Gottes Gedanken zu wissen, alles anderes ist nur Einselheit.” (I want to know God’s thoughts, the rest is just detail.) And he also said
“Rafinert ist der Herr Gott, aber Bohaft ist er nicht!” (The Lord God is clever, but he is not malicious.) Taken together, these two statements reveal that Einstein’s science was rooted in a deeply
spiritual understanding of reality. It appears that he believed that the universe, as a manifestation of God’s thoughts, is very complex, but understandable. Agreeing with Einstein, TDVP seeks to
reveal that all things are, in fact connected to, and part of that deeper ocean of reality, only momentarily appearing to be separated from it. This apparent separation, perpetuated by the conscious
drawing the distinction of ‘self’ from ‘other’ and the drawing of distinctions in self and other, allows us to interact with and draw distinctions in the ‘other’. TDVP posits that, although
ostensibly separate in the 3S-1t world of our physical perceptions, we are never truly separated from the whole of reality, but remain connected at deeply embedded multi-dimensional levels.
There are some in the current mainstream of science who do see the universe as deeply mathematical, but even those scientists seem to shy away from including consciousness in their equations. An
example is the Swedish physicist Max Tegmark. In his brilliant book “Our Mathematical Universe” he concludes that the ultimate nature of reality is mathematical structure. In reaching this
conclusion, however, he strips mathematical description of any intent or purpose. He says “A mathematical structure is an abstract set of entities with relations between them. The entities have no
‘baggage’: they have no properties whatsoever except these relations.” In other words, he still does what most mainstream materialistic scientists do: he throws the baby out with the bath water. It
is critically important to separate science from fantasy and wishful thinking, but consciousness is an extremely important part of reality and should not be excluded from the equations of science
just because it complicates the picture.
From the broader viewpoint of TDVP, it is not surprising that mainstream science, focused, as it is, on the limiting philosophy of reductionist materialism, has lost touch its metaphysical roots, and
thus cannot explain how it is that a large part of reality is not available to us for direct observation, but makes its existence known only indirectly through quantum phenomena like non-locality and
quantum entanglement, as well as the near light-speed vortical spin of fermions and the effects of so-called dark matter and dark energy in the rotation of spiral galaxies.
TDVP also answers the real need to explain why we sometimes catch glimpses of a broader reality in rare extra-corporeal (out-of-body) experiences and other documented psi phenomena. The current
mainstream scientific paradigm cannot explain so-called anomalous phenomena and the “missing” portions of reality because there is no place in its formulation for phenomena that may involve more than
matter and energy interacting in three-dimensions of space and one dimension of time. TDVP, on the other hand, reveals a multi-dimensional reality and the need to recognize a third form of reality,
not measurable as mass or energy, in the equations of science. As we shall see, TDVP provides a theoretical basis for a much deeper understanding of reality, as well as providing the appropriate
tools for exploring it.
In coming installments I will go more deeply into the mathematical proof that the reality we experience is no accident.
PUTTING CONSCIOUSNESS INTO THE EQUATIONS OF SCIENCE:
The True Units of Measurement and The Theory of Everything
By Edward R. Close, PhD, PE, Distinguished Fellow ECAO
Many physicists, including Einstein, Pauli and Hawking have dreamt of a ‘theory of everything’. But to this point, their dreams have not been fulfilled. The reason is simple. You can’t have a theory
of everything if you doggedly exclude a major part of Reality from your theory. That major part of Reality excluded by contemporary reductionist science is consciousness.
For nearly 50 years, I have insisted that the dream of a theory of everything is never going to be realized until we find a way to put consciousness into the equations of science. Believe it or not,
I actually found the way - as it turns out, only accessible to a precious few - using a new mathematical tool called the Calculus of Distinctions. The inspiration came to me in a dream in 1986, and I
published it in 1989 in a book entitled “Infinite Continuity;” but in 1989, and even today, most people are not willing to invest the time and considerable effort it takes to learn a whole new system
of mathematical logic.
Since 1989, I have been determined to find a better way to explain how to put the Primary Reality of Consciousness into the equations of science. In 1996, I published the book “Transcendental
Physics”, an effort to make the 1989 work more accessible. It reached a few more people, but still only a relatively small number of scientists and others interested in the merging of science and
spirituality. The audience has continued to grow over the years, albeit slowly. One who shared my vision, and has been my research partner for the past six years, is the world-renowned
neuroscientist, Dr. Vernon Neppe, MD, PhD. Together Dr. Neppe and I have developed a comprehensive framework, a paradigm for the science of the future. We call it the Triadic Rotational Dimensional
Distinction Paradigm (TDVP). It was first published as a number of technical papers and then as a book titled “Reality Begins with Consciousness,” in 2011 (Links available here: http://
www.erclosetphysics.com/p/publications-by-edward-r-close-phd.html). These works have now been reviewed by more than 200 scientists and philosophers worldwide. And recently, through determined effort
and grace, I have found yet a better way to explain the revelations of the Calculus of Distinctions of 1989, 1996 and 2011, a way that will be far more accessible to both the scientist and the
general public. This paper is my first effort to elucidate the new discoveries. I believe it will do much more than make the work more accessible to a broader audience.
The bottom line is that, in this world of human experience, we will never truly understand the Nature of Reality until our searches for scientific and spiritual knowledge are merged into one serious,
combined effort. Once this happens on a global scale, humanity will experience an explosion of new knowledge and understanding far beyond anything experienced so far in the current era of recorded
history. In this paper, I show how consciousness is describable in the equations of quantum physics and relativity, and a few of the explanatory revelations produced as a result. This is only the tip
of the iceberg of what is possible, but already it opens so many new roads for scientific pursuit that I am in awe of its beauty and scope.
In 1714 the German polymath Gottfried Wilhelm Leibniz stated that the most important question of all is: “Why is there something rather than nothing?”^1 Science proceeds from the assumption that
there is something, something that we perceive as the physical universe. In order to investigate this something that we appear to be immersed in, we go about trying to weigh and measure the
substances it is made of and look for consistent structures and patterns in it that can be described mathematically. We call such mathematical descriptions “Laws of Nature”.
To find the laws governing the relationships between different features of physical reality, we have to define a system of units with which to weigh and measure those features. Historically, units of
measurement have been chosen somewhat arbitrarily. For example, the units of the so-called English Imperial System were based on the practice of measuring things with what one always had at hand:
parts of the human body. A horse was so many “hands” high; one could measure rope or cloth by “inching” along its length with a joint of one’s thumb or finger. Short horizontal distances were
measured in multiples of the length of one’s foot, or the distance from the tip of one’s nose to one’s thumb on a laterally extended arm, and a mile was 1000 paces, when a pace consisted of two
steps. Since not all people are the same size, measurements obtained this way are somewhat variably inaccurate. Consequently, units were eventually standardized so that the measurements of a given
object, carefully obtained by anyone, should always be the same. But, even though units of measurement were standardized in many countries, the basic unit was not necessarily the same from one
country to the next.
As physical science advanced, the need for international standards grew, and the international system of units (SI) based on invariant physical constants occurring in nature, with larger units being
multiples of ten times the smallest unit, was developed. The number base of 10 was chosen because it was already being used essentially worldwide. It was a natural outcome of counting on one’s
fingers, and starting over after every count of ten.
Science generally uses SI units now for two reasons: 1.) All but three countries of the 196 countries on the planet (the US, Liberia and Burma) use the SI metric system as their primary system of
measurement. This is significant, even though the UK still uses a mixture of the two systems, as does the US and a few other countries to a lesser extent. 2.) Computations are simplified when all
units are related by multiples or factors of 10, eliminating the odd fractions relating to inches, feet and miles, ounces and pounds, pints, quarts and gallons, etc. in the English system. In the
process of developing the TDVP model, however, we find a need now to define a new unit of measurement based on discoveries of quantum physics and relativity.
The purpose of this paper is to explain why a new basic unit is needed and how it is derived. It may seem to come as a surprise, that in the process, we provide an answer for Leibniz’s “most
important question” (Why is there something instead of nothing) and at the same time introduce new science.
Beyond seeking practical applications that improve the quality of life, the motivation behind our efforts in science, religion and philosophy is the desire to know and understand the true nature of
reality. Science, as we know it, i.e. the science developed during the past 800 years (a very short time compared to the length of time life has existed on this planet: less than two ten-millionths
of the apparent age of the Earth), seeks to understand the reality experienced through the physical senses in terms of the measurable parameters of matter, energy, space, and time. Based on a number
of clues from relativity and quantum physics, we have identified an urgent need to include the conscious actions of the observer in the equations of science. Consciousness is truly the missing link
in the current scientific paradigm. This has been stated repeatedly by me and others for the past 30 years, but only now is it becoming possible to actually do it in a way that can be understood by
Could it be that consciousness is and always has been present in some form, even in the very most basic structure of reality, as quantum experiments seem to indicate? If so, we may have the answer
Leibniz’s question. In a universe where consciousness is an integral part of reality, meaningful structure would be no accident. Consciousness and even conscious entities would be able to recognize
meaningful order and patterns in the reality experienced and interact with certain aspects of it to enhance and perpetuate existing meaningful patterns and structures that are beneficial to their
existence and growth. This process creates and perpetuates forms, and I have called this process negative entropy because it is the reverse of entropy. And without negative entropy there would be no
universe, and we know this because of the second law of thermodynamics.
There is more than matter and energy in our experience, there is also conscious experience of matter and energy. And according to the Quantum Mechanics experiments, no phenomena can be said to exist
until it is observed. Therefore, no particle could ever form, no wave function could ever collapse. Without a conscious observer, no observation can be made, and no physical reality can exist.
Physicists have ignored this because they had no way to understand how to incorporate it, until now.
If matter, energy and consciousness are all required for the existence of this reality we experience, then consciousness is a third required basic form of reality. Without it, nothing exists at all.
So, if consciousness is an integral part of reality, continually creating meaningful structure at the quantum level, there must be a way to include it in our scientific paradigm and the mathematics
that describes it. TDVP is a serious effort to upgrade the mathematics of the physical sciences to include the direct and indirect involvement of consciousness. If successful, there is reason to
believe that this new paradigm will provide a comprehensive framework within which all the branches of science can be expanded to include phenomena heretofore excluded from scientific investigation.
And the surprising, awe inspiring aspect of this great scientific expansion is the explanation of previously unresolvable conflicts in our scientific paradigm. There are many who are working in this
direction, but none have included the mathematics to support the ideas until now.
Watch for more coming soon.
This proof that there is no co-prime integer solution (X,Y,Z) of X^n + Y^n = Z^n for n = 3 and its generalization to n = p, primes > 2, provides validation of the method of proof I used in FLT65.
But, in my opinion, no such validation is needed because the questions you have raised, and every valid question ever raised by any reviewer of FLT65 are adequately answered by two simple statements
in FLT65: (1.) “A polynomial f(X), of degree greater than one, is divisible by X – a IF, AND ONLY IF, f(a) = 0.” And, (2.), “… the integers are elements in the field of rational numbers.” Application
of statement (1.) to the integer polynomial of the form Z^n-1+XZ^n-2 + X^2Z^n-3 +…+ X^n-2Z + X^n-1 = A^n = (Z – s)^n with A, X, Z and a co-prime elements of the ring of integers, showing that the
remainder f(s) can never be zero, comprises a valid proof of FLT.
Questions raised by reviewers invariably arise from the claim that: “While the Division Algorithm applies to algebraic polynomials, it may not apply to the division of integers.” And some reviewers
who make this extraordinary claim, attempt to justify it with an example using integer values of s, X and Z such that f(Z) = Z^2 + XZ + X^2 is divisible by Z – s. They assume that, because f(Z) in
their example is divisible by Z –s, the remainder f(s) is equal to zero. It is easily demonstrated that this is not so. I have done so a number of times in discussions with different reviewers.
It is a mistake to assume that the Division Algorithm and corollaries do not apply to integer polynomials. To see this, consider the fact that the remainder f(s) is exactly the same, namely, it is
equal to the integer s^2 + sX + X^2, whether dividing the polynomial f(z) over the field of real numbers by z – s, or dividing f(Z), an integer polynomial factor of Y for solutions of the FLT
equation, by the integer Z – s, with s and Z positive integers. In examples produced by a few reveiwers, e.g., f(s) is not zero, it is equal to a multiple of Z – s. The remainder R = 0 is obtained
and confused with f(s) by skipping the step involving f(s) obtained by substituting the integer values of X and Z into f(Z). This step is easily overlooked because the integers used in the example
are small. Since f(s) ≠ 0, it is easy to show that the integer Z in such example cannot be the Z in any X,Y,Z, integer solution to the FLT equation.
(I have found several other examples of f(Z)/(Z – s) where f(s) is a multiple of Z – s, but they also require an integer Z that cannot be part of an integer solution of the Fermat equation, and,
interestingly, all of the examples I’ve found so far, including your example, involve integers X and Z that are also members of Pythagorean triples. I see this as a basis for a conjecture that could
become an important theorem if proven.)
Before proceeding with the proof for n = 3, there is one other minor point I want to clear up: f(s) is not equal to 3s^2 as stated by a recent reviewer. When f(Z) is divided by Z – s, R = f(s) = s^2
+ sX + X^2, not 3s^2. The only way f(s) can equal 3s^2, is for s = X; in which case Z – s = Z – X, which clearly is not a factor of f(Z) = Z^2 + XZ + X^2, for X and Z co-prime. We can get a remainder
equal to 3s^2 by dividing s^2 + sX + X^2 by X – s, which yields X – s = Z – s → X = Z and Y = 0, a trivial solution of the Fermat equation. However, f(s) cannot equal zero for integer solutions of
Fermat’s equation because s and X are positive integers by definition. They are specific positive integer members of a hypothetical integer solution of Fermat’s equation.
Proof of FLT for n = 3:
For integer solutions of the Fermat equation, the factor of Z^3 – X^3 represented by f(Z) = Z^2 + XZ + X^2 is equal to A^3 and A = Z – s, A, Z and s integers. And, by Corollary II of the Division
Algorithm, f(Z) divided by Z – s produces a remainder equal to the integer f(s) = s^2 + sX + X^2.
Due to the fact that integers form a subset of the real numbers, the integer polynomial f(Z) is a subset of the real number polynomial f(z), and with X, Y and, dividing f(Z) by Z - s yields:
(Z^2 + XZ + X^2)/(Z – s) = Z + X + s + f(s)/(Z – s) →
(Z^2 + XZ + X^2) = (Z + X + s)(Z – s) + f(s) → f(s) = m(Z – s), m a positive integer.
By application of Fermat’s ‘Little’ Theorem, choosing Y co-prime with n = 3, ensures that f(Z) is an integer raised to the third power. For a hypothetical primitive solution (X,Y,Z), f(Z) is equal to
Z^2 + XZ + X^2 = A^3, an integer factor of Y, and by inspection, Z^2 + XZ + X^2 is odd for all integer values of X and Z. We may, therefore use Fermat’s factorization method which says that for every
odd integer N, there are two relatively prime integers, a and b, such that N = a^2 + b^2. Since f(Z) = Z^2 + XZ + X^2 = (Z – s)^3 = (Z – s)(Z – s)^2 = (a - b)(a + b) = (Z – s)(Z + s) →
(Z – s)^2 = (Z + s).
With this equation, FLT for n = 3 can be proved a number of ways. For example:
Z^2 – 2sZ + s^2 = Z + s → Z^2 – (2s + 1)Z^ + s^2 – s = 0. We can solve this equation for Z using the quadratic formula and get Z = [(2s + 1) + √2s]/2, a non-integer for all positive integer values of
s, proving FLT for n = 3. But this method of proof becomes problematic for n > 3, since we only have a quadratic equation when n = 3. However, proof of FLT for n = 3 can also be obtained by noticing
that the equation implies that Z divides s^2 – s = s(s – 1), and s and Z must be co-prime for Y and Z to be co-prime, and s is a positive integer < Z, so s(s – 1) cannot contain Z,
Similarly, since Z – s is an integer factor of f(s), and f(s) = s^2 + sX + X^2, an odd integer, and f(s) = m(Z –s), applying Fermat’s factorization method, we have:
f(s) = m(Z – s) = (a + b)(a - b) = (Z + s)(Z - s) → m = Z + s. But application of Fermat’s factorization to f(Z) above gave us (Z – s)^2 = Z + s. These two results taken together imply that f(s) = (Z
– s)^3. If f(s) = (Z – s)^3, the equation Z^2 + XZ + X^2 = Q(Z)(Z – s) + m(Z – s) = Q(Z)(Z – s) + (Z – s)^3 → Q(Z) = (Z + X + s) contains (Z – s)^2. But (Z + X + s) contains (Z – s)^2 and Z + s = (Z
– s)^2 implies X contains Z – s as an integer factor, which contradicts co-prime X, Y and Z, denying the existence of primitive solutions, proving FLT for n = 3.
By combining these two applications of Fermat’s factorization, we have a demonstration of the FLT65 method of proof in a form that can be extended to n = prime numbers > 3. To see how this can be
done, let’s also look at a proof for n = 5.
For n = 5, dividing f(Z) by Z - s yields:
(Z^4 + XZ^3 + X^2 Z^2 + X^3Z + X^4)/(Z – s) = Q(Z) + f(s)/(Z – s), where Q(Z) = Z^3 + (s + X)^ Z^2 + (s^2 + X^2 )Z + s^3 + X^3, and f(s) = (s^4 + Xs^3 + X^2 s^2 + X^3s + X^4).
So we have f(Z) = (Z^4 + XZ^3 + X^2 Z^2 + X^3Z + X^4) = Q(Z)(Z – s) + f(s).
And, since f(Z) = A^3 = (Z – s)^5, f(s) must contain Z – s as a factor, so we have: f(s) = (s^4 + Xs^3 + X^2 s^2 + X^3s + X^4) = m(Z – s), m a positive integer.
By inspection we see that f(Z) is an odd integer. So, applying Fermat’s factorization method, we also have:
f(Z) = (Z – s)^5 = (Z – s)(Z – s)^4 = (a - b)(a + b) = (Z – s)(Z + s) → (Z – s)^4 = (Z + s).
Similarly, since Z – s is an integer factor of f(s), and f(s) is an odd integer, equal to m(Z –s), applying Fermat’s factorization method again, we have:
f(s) = m(Z – s) = (a + b)(a - b) = (Z + s)(Z - s) → m = Z + s. But application of Fermat’s factorization to f(Z) gave us (Z – s)^4 = Z + s. These two results taken together, imply that f(s) = (Z – s)
^4 and from the equation f(Z) = Q(Z)(Z – s) + m(Z – s) = Q(Z)(Z – s) + (Z – s)^3 , we see that Q(Z) = {Z^3 + (s + X)^ Z^2 + (s^2 + X^2 )Z + s^3 + X^3} contains (Z – s)^4. Note: determining what this
means in terms of co-prime X, Y and Z is a bit more complicated than it was in the case n = 3, but it can be done as follows:
Since f(Z) and Q(Z) contain Z – s as a common integer factor, the difference Q(Z)Z – f(Z) must also contain Z – s as an integer factor:
Q(Z)Z = Z^4 + (s + X)^ Z^3 + (s^2 + X^2 )Z^2 + (s^3 + X^3)Z
- F(Z) = - Z^4 - XZ^3 - X^2 Z^2 - X^3Z - X^4. Subtracting term by term,
Q(Z)Z – f(Z) = sZ^3 + s^2Z^2 + s^3Z – X^4 = {sZ(Z^2 + sZ + s^2) – X^4}which contains Z – s.
Subtracting sZ(Z – s)^2 = sZ(Z^2 - 2sZ + s^2) from {sZ(Z^2 + sZ + s^2) – X^4}→ X^4 contains Z – s.
So Q(Z) contains (Z – s)^4 and Z + s = (Z – s)^4 implies that X^4 contains Z – s as an integer factor, which contradicts co-prime X, Y and Z, denying the existence of primitive solutions, proving FLT
for n = 5. The pattern we see emerging is: For n = p, any prime >2, the fact that the remainder f(s) is non-zero implies X, Y and Z cannot be co-prime integers, proving FLT.
In conclusion: In my opinion, the proof of FLT for n = 3 and n = 5, as presented above demonstrate the validity of FLT65. A non-integer remainder for F(Z)/(Z - s), whether containing Z - s or not,
insures no integer solution for the Fermat equation.
THE SEARCH FOR CERTAINTY
“To everything there is a season, and a time to every purpose under heaven”
- Ecclesiastes 3:1
Yes, to everything there is a season; and I believe the time is ripe for a global quantum leap in human consciousness. Not just an increase in knowledge, it must be a triadic leap: a physical, mental
and spiritual awakening. Anything less leads to serious problems: If enlightenment is just intellectual and physical, it fosters prideful ego and eventual disillusionment as dissolution of the
physical body, i.e. physical death, approaches. Awareness of the triadic nature of reality, on the other hand, reveals a reality of which the observable physical universe is only a small part, and
explains why there is something rather than nothing. Triadic enlightenment integrates the logic of science, the philosophy of religion and the expanded awareness of spirituality.
The number of people on this planet ready to make this leap to a comprehensive understanding of reality may finally be reaching critical mass, a necessary condition for the inevitable shift out of
the limiting paradigmatic belief in mechanistic materialism that has characterized science, the limiting dogmatic beliefs that have characterized religions, and the unrealistic fantasies that have
characterized “new-age” spiritualism. Gradually, a few individuals on the leading edge of the bell curve have begun to transcend the limitations of materialistic science, religious dogma and
spiritual fantasy, into an expanded awareness. This book is the story of my personal journey from the confusion of fragmented belief systems to the certainty of triadic enlightenment.
An early version of this book was completed in 1997. It was intended to be a readable introduction to Transcendental Physics, the work I completed in 1996 and published in 1997. Presenting a new
scientific paradigm, Transcendental Physics reversed the basic assumption of conventional science, the a priori assumption that consciousness is an epiphenomenon arising from the evolution of matter
and energy, with the hypothesis that a primary form of consciousness is the ground from which all patterns of reality, including the physical universe, originate. Transcendental Physics, the book,
contained specific, detailed interpretations of complex relativity and quantum mechanics experiments and introduced some new mathematical concepts developed for purpose of putting consciousness into
the equations expressing the known Laws of Nature. The Search for Certainty manuscript, on the other hand, was written for readers with less technical training. It traced the development of the ideas
behind Transcendental Physics as I had experienced them, and was thus at least partly autobiographical. The purpose was to present the paradigm-shifting ideas of Transcendental Physics in
non-technical terms. Dr. David Stewart, who was familiar with and even part of many of the events reported in the 1997 version of the Search for Certainty, reviewed the manuscript, and had this to
“For the first time, the common basis for all sciences and all religions is revealed - not in vague philosophical terms, but in concrete ways you can understand and put into practice in your own
life. You can take scriptures or the works of science and, by being selective, prove almost anything. But Ed Close, in this monumental work, did not do that. Taking into consideration the totality of
physics, both modern and classical, dodging no part of it, Dr. Close has applied relentless and impeccable logic to produce an intellectual triumph of our time, a unified theory that makes science
and religion one. This achievement has been claimed by others before, but always there was a flaw. There are no flaws in Close’s paradigm. The search for certainty ends here for those with the
capability of comprehending what Close has done for us. Both scientists and theologians, centuries hence, will thank Dr. Close for what he has done for us. This is truly the first mathematically
complete articulation of the relationship between human consciousness, divine consciousness, and material reality. This could well be the most important work of the 20^th century. What Einstein and
his contemporaries started a century ago, Close has finished. And what makes his achievement even more remarkable is that he was able to articulate it in terms the layman can understand.”
March, 1997
David Stewart, PhD, Geophysicist, Educator, and Author | {"url":"http://www.erclosetphysics.com/2014/","timestamp":"2024-11-11T11:04:07Z","content_type":"text/html","content_length":"326985","record_id":"<urn:uuid:d8f3bd39-4014-40c2-95b3-94381ed6c555>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00332.warc.gz"} |
Neural Network Classification
Artificial neural networks are relatively crude electronic networks of neurons based on the neural structure of the brain. They process records one at a time, and learn by comparing their
classification of the record (i.e., largely arbitrary) with the known actual classification of the record. The errors from the initial classification of the first record is fed back into the network,
and used to modify the networks algorithm for further iterations.
A neuron in an artificial neural network is
1. A set of input values (xi) and associated weights (wi).
2. A function (g) that sums the weights and maps the results to an output (y).
Neurons are organized into layers: input, hidden and output. The input layer is composed not of full neurons, but rather consists simply of the record's values that are inputs to the next layer of
neurons. The next layer is the hidden layer. Several hidden layers can exist in one neural network. The final layer is the output layer, where there is one node for each class. A single sweep forward
through the network results in the assignment of a value to each output node, and the record is assigned to the class node with the highest value.
Training an Artificial Neural Network
In the training phase, the correct class for each record is known (termed supervised training), and the output nodes can be assigned correct values -- 1 for the node corresponding to the correct
class, and 0 for the others. (In practice, better results have been found using values of 0.9 and 0.1, respectively.) It is thus possible to compare the network's calculated values for the output
nodes to these correct values, and calculate an error term for each node (the Delta rule). These error terms are then used to adjust the weights in the hidden layers so that, hopefully, during the
next iteration the output values will be closer to the correct values.
The Iterative Learning Process
A key feature of neural networks is an iterative learning process in which records (rows) are presented to the network one at a time, and the weights associated with the input values are adjusted
each time. After all cases are presented, the process is often repeated. During this learning phase, the network trains by adjusting the weights to predict the correct class label of input samples.
Advantages of neural networks include their high tolerance to noisy data, as well as their ability to classify patterns on which they have not been trained. The most popular neural network algorithm
is the back-propagation algorithm proposed in the 1980s.
Once a network has been structured for a particular application, that network is ready to be trained. To start this process, the initial weights (described in the next section) are chosen randomly.
Then the training (learning) begins.
The network processes the records in the Training Set one at a time, using the weights and functions in the hidden layers, then compares the resulting outputs against the desired outputs. Errors are
then propagated back through the system, causing the system to adjust the weights for application to the next record. This process occurs repeatedly as the weights are tweaked. During the training of
a network, the same set of data is processed many times as the connection weights are continually refined.
Note that some networks never learn. This could be because the input data does not contain the specific information from which the desired output is derived. Networks also will not converge if there
is not enough data to enable complete learning. Ideally, there should be enough data available to create a Validation Set.
Feedforward, Back-Propagation
The feedforward, back-propagation architecture was developed in the early 1970s by several independent sources (Werbor; Parker; Rumelhart, Hinton, and Williams). This independent co-development was
the result of a proliferation of articles and talks at various conferences that stimulated the entire industry. Currently, this synergistically developed back-propagation architecture is the most
popular model for complex, multi-layered networks. Its greatest strength is in non-linear solutions to ill-defined problems.
The typical back-propagation network has an input layer, an output layer, and at least one hidden layer. There is no theoretical limit on the number of hidden layers but typically there are just one
or two. Some studies have shown that the total number of layers needed to solve problems of any complexity is five (one input layer, three hidden layers and an output layer). Each layer is fully
connected to the succeeding layer.
The training process normally uses some variant of the Delta Rule, which starts with the calculated difference between the actual outputs and the desired outputs. Using this error, connection weights
are increased in proportion to the error times, which are a scaling factor for global accuracy. This means that the inputs, the output, and the desired output all must be present at the same
processing element. The most complex part of this algorithm is determining which input contributed the most to an incorrect output and how must the input be modified to correct the error. (An
inactive node would not contribute to the error and would have no need to change its weights.) To solve this problem, training inputs are applied to the input layer of the network, and desired
outputs are compared at the output layer. During the learning process, a forward sweep is made through the network, and the output of each element is computed by layer. The difference between the
output of the final layer and the desired output is back-propagated to the previous layer(s), usually modified by the derivative of the transfer function. The connection weights are normally adjusted
using the Delta Rule. This process proceeds for the previous layer(s) until the input layer is reached.
Structuring the Network
The number of layers and the number of processing elements per layer are important decisions. To a feedforward, back-propagation topology, these parameters are also the most ethereal -- they are the
art of the network designer. There is no quantifiable answer to the layout of the network for any particular application. There are only general rules picked up over time and followed by most
researchers and engineers applying while this architecture to their problems.
Rule One: As the complexity in the relationship between the input data and the desired output increases, the number of the processing elements in the hidden layer should also increase.
Rule Two: If the process being modeled is separable into multiple stages, then additional hidden layer(s) may be required. If the process is not separable into stages, then additional layers may
simply enable memorization of the training set, and not a true general solution.
Rule Three: The amount of Training Set available sets an upper bound for the number of processing elements in the hidden layer(s). To calculate this upper bound, use the number of cases in the
Training Set and divide that number by the sum of the number of nodes in the input and output layers in the network. Then divide that result again by a scaling factor between five and ten. Larger
scaling factors are used for relatively less noisy data. If too many artificial neurons are used the Training Set will be memorized, not generalized, and the network will be useless on new data sets.
Ensemble Methods
Analytic Solver Data Science offers two powerful ensemble methods for use with Neural Networks: bagging (bootstrap aggregating) and boosting. The Neural Network Algorithm on its own can be used to
find one model that results in good classifications of the new data. We can view the statistics and confusion matrices of the current classifier to see if our model is a good fit to the data, but how
would we know if there is a better classifier just waiting to be found? The answer is that we do not know if a better classifier exists. However, ensemble methods allow us to combine multiple weak
neural network classification models which, when taken together form a new, more accurate strong classification model. These methods work by creating multiple diverse classification models, by taking
different samples of the original data set, and then combining their outputs. (Outputs may be combined by several techniques for example, majority vote for classification and averaging for
regression.) This combination of models effectively reduces the variance in the strong model. The two different types of ensemble methods offered in Analytic Solver Data Science (bagging and
boosting) differ on three items: 1) the selection of training data for each classifier or weak model; 2) how the weak models are generated; and 3) how the outputs are combined. In all three methods,
each weak model is trained on the entire Training Set to become proficient in some portion of the data set.
Bagging (bootstrap aggregating) was one of the first ensemble algorithms ever to be written. It is a simple algorithm, yet very effective. Bagging generates several Training Sets by using random
sampling with replacement (bootstrap sampling), applies the classification algorithm to each data set, then takes the majority vote among the models to determine the classification of the new data.
The biggest advantage of bagging is the relative ease that the algorithm can be parallelized, which makes it a better selection for very large data sets.
Boosting builds a strong model by successively training models to concentrate on the misclassified records in previous models. Once completed, all classifiers are combined by a weighted majority
vote. Analytic Solver Data Science offers three different variations of boosting as implemented by the AdaBoost algorithm (one of the most popular ensemble algorithms in use today): M1 (Freund), M1
(Breiman), and SAMME (Stagewise Additive Modeling using a Multi-class Exponential).
Adaboost.M1 first assigns a weight (wb(i)) to each record or observation. This weight is originally set to 1/n and is updated on each iteration of the algorithm. An original classification model is
created using this first training set (Tb), and an error is calculated as:
where, the I() function returns 1 if true, and 0 if not.
The error of the classification model in the bth iteration is used to calculate the constant ?b. This constant is used to update the weight (wb(i). In AdaBoost.M1 (Freund), the constant is calculated
αb= ln((1-eb)/eb)
In AdaBoost.M1 (Breiman), the constant is calculated as:
αb= 1/2ln((1-eb)/eb)
In SAMME, the constant is calculated as:
αb= 1/2ln((1-eb)/eb + ln(k-1) where k is the number of classes
where, the number of categories is equal to 2, SAMME behaves the same as AdaBoost Breiman.
In any of the three implementations (Freund, Breiman, or SAMME), the new weight for the (b + 1)th iteration will be
Afterwards, the weights are all readjusted to the sum of 1. As a result, the weights assigned to the observations that were classified incorrectly are increased, and the weights assigned to the
observations that were classified correctly are decreased. This adjustment forces the next classification model to put more emphasis on the records that were misclassified. (The ? constant is also
used in the final calculation, which will give the classification model with the lowest error more influence.) This process repeats until b = Number of weak learners. The algorithm then computes the
weighted sum of votes for each class and assigns the winning classification to the record. Boosting generally yields better models than bagging; however, it does have a disadvantage as it is not
parallelizable. As a result, if the number of weak learners is large, boosting would not be suitable.
Neural Network Ensemble methods are very powerful methods, and typically result in better performance than a single neural network. Analytic Solver Data Science provides users with more accurate
classification models and should be considered over the single network. | {"url":"https://www.solver.com/xlminer/help/neural-networks-classification-intro","timestamp":"2024-11-07T09:22:41Z","content_type":"text/html","content_length":"71279","record_id":"<urn:uuid:0c862df4-3d28-41bc-bdac-dcb689af01c0>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00671.warc.gz"} |
Flare testing - equations
This page contains the development of the equations using graphical means. These will be used in the "flare-it" software
Part one develops solutions at 30hz
Part two determines how the solutions change for different frequencies
Part one - Developing the 30hz equations
Colors used in tables and graphs
Background = port diameter, Foreground = frequency
Measured performance at 30hz
Graphing this data...
Results for all the ports @30hz
Now adding a shaded area showing an additional 40% allowance for normal music at the typical seating position.
40% Allowance for distance to typical seating position, and masking effect of musical content
The first task is to identify the slope of the line for each port diameter
Picking any vertical line, ( I chose AR=2 for because there is data for each diameter ) and noting the maximum allowable velocity for each port diameter we get the following points, which are also
shown on the above graph as black crosses.
Maximum allowable velocities for various diameters at Area Ratio of 2
Diameter squared is used as the measure because it is proportional to port area, which is proportional to carrying capacity
Graphing the points and finding the line of best fit which doesn't exceed the allowable velocity .....
Allowable velocities at 30hz for Area Ratio of 2
Reading off from the graph...
Maximum Velocity = [3570 + ( port diameter ^ 2 )] /1785 * area ratio
Velocity is in metres per second. Port diameter and flare radius are in millimeters
Rewriting to expand area ratio in terms of diameter and flare radius...
The second task is to identify the limiting velocity for each port diameter
Reading the limiting velocities from the graph.....
Limiting velocities for various diameters
In reality this relationship would probably be a curve, but for our purposes a segmented line is accurate enough.
Limiting velocities at 30hz
Note: The graph and equations show the solution as chosen for use in version 2.10 of flare-it.
This is slightly different to that which was used for ver 2.00 See the version notes if more detail is required.
Reading off from the graph...
For ports smaller than 103mm in diameter
Limiting velocity = 10 +[ (diameter squared) * (19.5 / 10,000) ]
For ports larger than 103mm in diameter
Limiting velocity = 31 + [(diameter squared - 10,600) * (8.5 / 15,000)]
The next graph shows where the equations fit within the allowable ranges.
Results predicted by equations sit within allowable ranges
Note: The results for 51mm ports are a little conservative, which is desirable because it's performance is currently based on measuring a single port.
Part two - Frequency related changes to port performance
The ports were tested at 15, 20, 25, 30 and 35hz, ( power and excursion permitting )
See raw data page for actual measurements at the different frequencies
The following table summarises the relevant results:
Using a separate graph for each of the port diameters reveals how slope and limiting velocity vary with frequency...
86mm diameter ports
The 86mm port results include good data for 35hz
Below 25hz, the limiting velocity doesn't fall any further.
103mm diameter ports
The 103mm ports have limited data for 35hz due to power and excursion constraints
Below 25hz, the limiting velocity doesn't fall much further.
152mm diameter ports
The 152mm port could only be tested at velocities that reveal slope. The lower frequencies were difficult to measure because of bad structural resonances. The 160 litre box produced the highest
SPL's, and at 350w, major items in the room were producing a lot of noise.
The 15hz slope is based on a single measurement, which gives a low level of accuracy, so will not be used for analysis
Measuring the variation in limiting velocity
Reading the limiting velocities from the above graphs gives the following table: The percentages indicate how much the limiting velocity changes from the 30hz figure:
Changes in limiting velocity changes from the 30hz figure
The change appears to be independent of the port diameter, being solely determined by the frequency.
At 35hz, the limiting velocity is higher than the 30hz figure by +23%.
At 25hz, and below, the limiting velocity is lower than the 30hz figure by -33%
Measuring the variation in slope
Reading the slopes from the above graphs gives the following table
Noting the Velocity at Area Ratio = 2 allows a comparison.
The percentages indicate how much the slope changes from the 30hz figure
Variation in Usable velocities for AR=2 at different frequencies
Again, the change appears to be independent of the port diameter, being solely determined by the frequency.
Since the 35hz figure for the 103mm port is based on a single reading, it's best to play it safe and use the more conservative value based on the multiple readings for the 86mm ports. The same
applies to the single 15hz reading for the 152mm port
For 25hz, the average is around -28%
For 20hz, the 152mm port was difficult to measure, so we'll give more weight to the 86mm and 103mm results
At 35hz, the usable velocity is higher than the 30hz figure by +29%
At 25hz, the usable velocity is lower than the 30hz figure by -28%
At 20hz, the usable velocity is lower than the 30hz figure by -37%
At 15hz, the usable velocity is lower than the 30hz figure by -44%
These results can be visualised as follows:
Changes in port performance with frequency | {"url":"https://subwoofer-builder.com/flare-testing-equations.htm","timestamp":"2024-11-09T15:20:57Z","content_type":"application/xhtml+xml","content_length":"32034","record_id":"<urn:uuid:64f2fa91-6f83-4dbd-a4c6-65e5bbb23f34>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00156.warc.gz"} |
ai no uta yunomi
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installation. You can use this brick quantity calculator formula for your house or wall construction. Measurements for the units of each measurement that are $ 0.50 a 90° pattern. Area of the pavers
installed by a professional to make a flagtone patio the! Determine the paver calculator mark and cut the pavers installed by a professional reading! Accurately work out brick quantities for your raw
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QC/T 1157-2022 PDF in English
QC/T 1157-2022 (QC/T1157-2022, QCT 1157-2022, QCT1157-2022)
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QC/T 1157-2022: PDF in English (QCT 1157-2022)
QC/T 1157-2022 QC AUTOMOBILE INDUSTRY STANDARD OF THE PEOPLE’S REPUBLIC OF CHINA ICS 43.020 CCS T 04 Method of calculating comprehensive energy consumption for unit output of automobile products
ISSUED ON: APRIL 8, 2022 IMPLEMENTED ON: OCTOBER 1, 2022 Issued by: Ministry of Industry and Information Technology of the People’s Republic of China. Table of Contents Foreword ... 7 1 Scope ... 8 2
Normative references ... 8 3 Terms and definitions ... 8 4 Statistical requirements ... 9 5 Calculation method ... 10 Appendix A (Informative) Conversion coefficients of various energy sources,
electricity, and heating power to standard coal (reference values) ... 13 Appendix B (Normative) Energy equivalent conversion relationship of energy- consumed medium ... 15 Method of calculating
comprehensive energy consumption for unit output of automobile products 1 Scope This document specifies the terms and definitions, statistical scope, and calculation methods of comprehensive energy
consumption per unit output of automobile products. This document is applicable to the calculation of comprehensive energy consumption per unit output of automobile products. 2 Normative references
The following documents contain the provisions which, through normative reference in this document, constitute the essential provisions of this document. For the dated referenced documents, only the
versions with the indicated dates are applicable to this document; for the undated referenced documents, only the latest version (including all the amendments) is applicable to this document. GB/T
2589-2020 General rules for calculation of the comprehensive energy consumption GB 17167 General principle for equipping and managing of the measuring instrument of energy in organization of energy
using 3 Terms and definitions The following terms and definitions apply to this document. 3.1 total comprehensive energy consumption of automobile products In the statistical reporting period, the
physical quantities of various energy sources that is actually consumed during the entire production process of automobile products. 3.2 comprehensive energy consumption for unit output of automobile
products During the statistical reporting period, the ratio of the comprehensive energy consumption of automobile products to the total amount of qualified products produced in the same period. 4.2.3
The energy consumed for heating and cooling of the production system and supporting production system activities shall also be included in the scope of comprehensive energy consumption statistics.
4.2.4 The loss of energy and energy-consumed medium due to the internal storage, conversion, distribution, and supply of the energy-consuming organization shall also be included in the comprehensive
energy consumption. 4.3 Types of comprehensive energy consumption statistics 4.3.1 The types of energy statistics for comprehensive energy consumption of automobile products include primary energy,
secondary energy, and energy consumed by the outsourced energy-consumed medium. Note 1: Primary energy refers to energy resources that exist in the original forms in nature and have not been
processed. Note 2: Secondary energy refers to energy products converted from primary energy processing. 4.3.2 The part of the energy-consumed medium that comes from the self-production of the
enterprise shall be counted according to the energy consumption corresponding to the production, and the outsourced energy-consumed medium shall be converted in accordance with the requirements of
Appendix B. 4.3.3 The energy consumed by the compressed air produced by the enterprise itself shall be converted according to the electricity consumed by the working of the compressor. 4.4
Statistical period The statistical period shall be a certain period of continuous production, and the statistical period shall be at least not less than 12 natural months. 4.5 Statistics on the
number of automobile products The number of automobile products to be counted shall be the number of qualified automobile products produced within the statistical period. 5 Calculation method 5.1
Calculation method of energy consumption of production system The energy consumed by the production system in the production process of automobile products is calculated according to the formula (1):
Source: Above contents are excerpted from the PDF -- translated/reviewed by: www.chinesestandard.net / Wayne Zheng et al. | {"url":"https://www.chinesestandard.net/PDF.aspx/QCT1157-2022","timestamp":"2024-11-03T07:36:31Z","content_type":"application/xhtml+xml","content_length":"25741","record_id":"<urn:uuid:e6fda5cf-0566-4b81-8b16-a75c83f483d4>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00807.warc.gz"} |
Calculating Conditional Probability in R » Data Science Tutorials
Calculating Conditional Probability in R, Conditional probability is a crucial concept in statistics and probability theory.
It allows us to update our beliefs about the likelihood of an event occurring based on new information.
In this article, we will explore the concept of conditional probability, its formula, and how to calculate it using the R programming language.
Understanding Conditional Probability
Conditional probability is expressed as P(B | A), which means “the probability of event B occurring given that event A has already occurred.”
This helps us determine the likelihood of an event B happening under the condition that event A has taken place.
Formula for Conditional Probability
The formula for calculating conditional probability is:
P(B | A) = P(A and B) / P(A)
Here, P(B | A) represents the conditional probability of event B given event A, P(A and B) is the joint probability of both events A and B happening together, and P(A) is the probability of event A
Calculating Conditional Probability in R
R is a powerful programming language for statistical computing and graphics. It offers various functions to calculate conditional probabilities.
In this section, we will discuss a step-by-step process to calculate conditional probabilities in R using the prop.table() function.
Step 1: Create a Data Frame
First, create a data frame containing the variables A and B. Each row in the data frame represents an observation, while each column represents a variable.
Step 2: Create a Contingency Table
A contingency table, also known as a cross-tabulation or crosstab, is a tabular method to display the relationship between two or more categorical variables.
In R, you can create a contingency table using the table() function.
Step 3: Calculate the Conditional Probability Table
To calculate the conditional probability table P(B | A), use the prop.table() function in R.
The prop.table() function converts a contingency table into a conditional probability table by dividing each cell by the row sums (i.e., the probabilities are conditioned on the first variable, A).
Step 4: Access Specific Conditional Probabilities
If you want to find a specific conditional probability, such as P(B=b1 | A=a1), you can access the corresponding cell in the conditional probability table using the appropriate row and column names.
Principal Component Analysis Advantages »
Example 1: Calculating Conditional Probability for a Deck of Cards
In this example, we will calculate the conditional probability of drawing a face card given that the card is a heart.
Step 1: Create a Data Frame
data <- data.frame(
A = c("heart", "heart", "heart", "non-heart", "non-heart"),
B = c("face card", "face card", "non-face card", "face card", "non-face card")
Step 2: Create a Contingency Table
contingency_table <- table(data$A, data$B)
Step 3: Calculate the Conditional Probability Table
conditional_probability_table <- prop.table(contingency_table, margin = 1)
Step 4: Access Specific Conditional Probabilities
probability_b1_given_a1 <- conditional_probability_table["heart", "face card"]
Example 2: Calculating Conditional Probability for Cloudy Days
In this example, we will calculate the conditional probability of rain given the presence of clouds.
Step 1: Create a Data Frame
weather_data <- data.frame(
Cloudy = c("Yes", "Yes", "No", "No"),
Rain = c("Yes", "No", "Yes", "No"),
Frequency = c(30, 20, 10, 40)
Step 2: Calculate the Conditional Probability
total_cloudy <- sum(weather_data$Frequency[weather_data$Cloudy == "Yes"])
rainy_and_cloudy <- weather_data$Frequency[weather_data$Cloudy == "Yes" & weather_data$Rain == "Yes"]
P_rain_given_cloudy <- rainy_and_cloudy / total_cloudy
Example 3: Calculating Conditional Probability for Student Information
In this example, we will calculate the conditional probability of passing an exam given high attendance.
Step 1: Create a Data Frame
student_data <- data.frame(
Attendance = c("High", "High", "Low", "Low"),
Pass = c("Yes", "No", "Yes", "No"),
Frequency = c(80, 20, 30, 70)
Step 2: Calculate the Conditional Probability
total_high_attendance <- sum(student_data$Frequency[student_data$Attendance == "High"])
pass_and_high_attendance <- student_data$Frequency[student_data$Attendance == "High" & student_data$Pass == "Yes"]
P_pass_given_high_attendance <- pass_and_high_attendance / total_high_attendance
Conditional probability is a vital concept in probability theory and statistics. By understanding its formula and learning how to calculate it in R, you can analyze data more effectively and make
better-informed decisions.
The examples provided in this article demonstrate the practical application of conditional probability calculations in various contexts, such as card games, weather forecasting, and student
performance analysis.
How to Calculate Lag by Group in R? » Data Science Tutorials | {"url":"https://datasciencetut.com/calculating-conditional-probability-in-r/","timestamp":"2024-11-06T21:31:08Z","content_type":"text/html","content_length":"106684","record_id":"<urn:uuid:e75e909a-170d-43dc-b4cd-8264d58595fa>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00529.warc.gz"} |
[5] Almost square permutations are typically square (with Enrica Duchi and Erik Slivken). Annales de l’Institut Henri Poincaré – Probab. Statist. 57 (2021), no. 4, pp. 1834-1856.
A record in a permutation is a maximum or a minimum, from the left or from the right. The entries of a permutation can be partitioned into two types: the ones that are records are called external
points, the others are called internal points. Permutations without internal points have been studied under
the name of square permutations. Here, we explore permutations with a fixed number of internals points, called almost square permutations. Unlike with
square permutations, a precise enumeration for the total number of almost square permutations of size \(n+k\) with exactly \(k\) internal points is not
However, using a probabilistic approach, we are able to determine the asymptotic enumeration. We denote with \(Asq(n,k)\) the set of almost square permutations of size \(n+k\) with exactly \(k\)
internal points
Theorem. For \(k=o(\sqrt n),\) as \(n\to \infty,\)
\(|Asq(n,k)| \sim \frac{k!2^{k+1}n^{2k+1}4^{n-3}}{(2k+1)!}\sim \frac{k!2^{k}n^{2k}}{(2k+1)!}|Asq(n,0)|.\)
When \(k\) grows at least as fast as \(\sqrt n\) the above result fails. Nevertheless, when \(k=o(n)\), we can still obtain the following weaker asymptotic expansion that determines the behavior of
the exponential growth.
Theorem. For \(k=o(n)\), as \(n\to \infty,\)
These two theorems allows us to describe the permuton limit of almost square permutations with \(k\) internal points, both when \(k\) is fixed and when \(k\) tends to infinity along a negligible
sequence with respect to the size of the permutation. Specifically, we have the following results.
Given \(z\in(0,1)\) we denote with \(\mu^{z}\) the permuton corresponding to a rectangle in \([0,1]^2\) with corners at \((z,0), (0,z),(1-z,1)\) and \((1,1-z).\)
Theorem. Fix \(k>0\). Let \(\textbf{z}^{(k)}\) denote the random variable in \((0,1)\) with density
\(f_{\mathbf{z}^{(k)}}(t) = (2k+1){2k \choose k} (t(1-t))^k,\)
i.e., \(\textbf{z}^{(k)}\) is beta distributed with parameters \((k+1,k+1)\). If \(\sigma_n\) is uniform in \(Asq(n,k)\), then as \(n\to \infty,\)
\(\mu_{\sigma_n} \stackrel{d}{\longrightarrow} \mu^{\mathbf{z}^{(k)}},\)
where \(\mu_{\sigma_n} \) denotes the permuton corresponding to \(\sigma_n.\)
The distribution of \(\textbf{z}^{(k)}\), when \(k\) increases, gives more weight around the value \(1/2\) as can be seen from the following picture (the chart displays the density of the
distribution of \(\textbf{z}^{(k)}\) for different values of \(k\)).
We therefore expect that, in the regime when \(k\to\infty\) together with \(n\) and \(k=o(n)\), a uniform random permutation with \(k\) internal points tends to \(\mu^{1/2}\). The following theorem
shows exactly this concentration result.
Theorem. Let \(k\) and \(n\) both tend to infinity with \(k=o(n)\). If \(\sigma_n\) is uniform in \(Asq(n,k)\) then
\(\mu_{\sigma_n} \stackrel{d}{\longrightarrow} \mu^{1/2}.\)
Finally, we show that our techniques are quite general by studying the set of 321-avoiding permutations of size \(n+k\) with exactly \(k\) internal points. In this case we obtain an interesting
asymptotic enumeration in terms of the Brownian excursion area. As a consequence, we show that the points of a uniform permutation in this set concentrate on the diagonal and the fluctuations of
these points converge in distribution to a biased Brownian excursion. | {"url":"https://www.jacopoborga.com/2019/10/13/almost-square-permutations-are-typically-square-with-enrica-duchi-and-erik-slivken/","timestamp":"2024-11-11T12:49:50Z","content_type":"text/html","content_length":"37623","record_id":"<urn:uuid:8390d834-bd69-43b1-85c9-d057217214b3>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00675.warc.gz"} |
Sympathetic Vibratory Physics | quantum
In physics, a quantum (plural: quanta) is the minimum amount of any physical entity involved in an interaction. Behind this, one finds the fundamental notion that a physical property may be
"quantized," referred to as "the hypothesis of quantization". (wikipedia)
A least quantity of a type of force or energy. Same as Keely's Interatomic, Etheric, Interetheric and Compound Interetheric or sub-atomic.
This high frequency realm of tenuous matter or plasma and energy operates on the basis of sympathy between centers.
THE SECRET OF ENERGY The secret of energy is a stationary principal which is everywhere at once.
122 years have passed since Max Planck discovered the quantum.
Planck’s constant is an eternally recurring movement breathing the universe to life; each repeat the same quantum of energy configured differently.
The evolving energy packet is Planck’s constant: h = 6.62607015×10?34 J?Hz. (No matter the quantum’s frequency its energy is constant)
The universe is the secondary result of quantum repetition and recognizing the quantum singularity reveals its cause:
Because the quantum is a single event whose repetition brings forth the universe it follows that its cause is without movement/still. Thus, the stationary principal is everywhere at once – in your
head advising you not to accept the above logic.
We are all linked by a fabric of unseen connections. This fabric is constantly changing and evolving. This field is directly structured and influenced by our behavior and by our understanding. [David
15.13 - Dissociating Water Acoustically - Liberation of Quantum Constituents 3.22 - Quantum Leap Delta equivalent to Locked Potentials Delta 7B.08 - The Etheric Quantum Soup Atom Centralization Code
for Quantum Arithmetic Compound Interetheric Ether Etheric Elements Etheric Etheron Figure 3.37 - Successive Centralizations or Quantum Leap Interetheric Interetheron Laws of Matter and Force Mind
Force is a pre-existing Natural Force Multimode cavity quantum electrodynamics Neutral Center New Concept - XVII - Regarding the Quantum Theory QED quantum electrodynamic vacuum Quantum Arithmetic
Elements Quantum Arithmetic Quantum Entanglement Quantum Ground State Quantum Harmonic Oscillator Quantum Leap Quantum Transition Quantum Tunneling Quantum coupling Quantum dynamics of a single
vortex Subdivision Sympathetic Vibratory Physics vs Quantum Entanglement Sympathetic Vibratory Physics vs Quantum Entanglement Table of Cause and Effect Dualities Table of Quantum Particles Table of
Quantum Particles What is Quantum Arithmetic quantum acoustics quantum chromodynamics quantum chronology quantum dot quantum electrodynamics quantum field theory quantum field quantum mechanics
quantum number quantum physics quantum potential quantum singularity quantum state quantum theory quantum vacuum fluctuation subquantum kinetics | {"url":"https://svpwiki.com/quantum","timestamp":"2024-11-12T05:47:23Z","content_type":"text/html","content_length":"46529","record_id":"<urn:uuid:5bdcc421-ed1c-44d6-a369-b5d3d11c8338>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00892.warc.gz"} |
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II. 1. The units for Volume are always squared cubed 2. Volume is the amount of space an object takes up. Irue False Calculate for the volume of solid. 3. Chona is selling Pringle Chips tc raise money for a field trip. The container has a diameter of 9 inches and a height of 32 inches. 4. A. 3 tier cake with the same height of 10 in and radius of 12 in, 8 in, 5 in 1. respectively is to be delivered to a birthday party. How much space does this cake take up? Use 3.14 for π. 5. A Styrofoam rmodel of a volcano is in the shape of a cone. The model has of a c:ircular base with a diameter of 48 centimeters and a height of 12 inx centimeters. 'Find the volume of foam in the model to the nearest tenth. Use the 3.14 for π. bet V. A.SSESSMENT Time Frame: Day 5 Lecırning Activity Sheets for Enrichment, Remediation, or Assessment to be san given on Weeks 3 and 6 wat solve each
1. Home
2. Algebra
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Multiplication Chart 1
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Iterative linear solvers
Iterative linear solvers as metaphor
Gaussian elimination is systematic way to solve systems of linear equations in a finite number of steps. Iterative methods for solving linear systems require an infinite number of steps in theory,
but may find solutions faster in practice.
Gaussian elimination tells you nothing about the final solution until it’s almost done. The first phase, factorization, takes O(n^3) steps, where n is the number of unknowns. This is followed by the
back-substitution phase which takes O(n^2) steps. The factorization phase tells you nothing about the solution. The back-substitution phase starts filling in the components of the solution one at a
time. In application n is often so large that the time required for back-substitution is negligible compared to factorization.
Iterative methods start by taking a guess at the final solution. In some contexts, this guess may be fairly good. For example, when solving differential equations, the solution from one time step
gives a good initial guess at the solution for the next time step. Similarly, in sequential Bayesian analysis the posterior distribution mode doesn’t move much as each observation arrives. Iterative
methods can take advantage of a good starting guess while methods like Gaussian elimination cannot.
Iterative methods take an initial guess and refine it to a better approximation to the solution. This sequence of approximations converges to the exact solution. In theory, Gaussian elimination
produces an exact answer in a finite number of steps, but iterative methods never produce an exact solution after any finite number of steps. But in actual computation with finite precision
arithmetic, no method, iterative or not, ever produces an exact answer. The question is not which method is exact but which method produces an acceptably accurate answer first. Often the iterative
method wins.
Successful projects often work like iterative numerical methods. They start with an approximation solution and iteratively refine it. All along the way they provide a useful approximation to the
final product. Even if, in theory, there is a more direct approach to a final product, the iterative approach may work better in practice.
Algorithms iterate toward a solution because that approach may reach a sufficiently accurate result sooner. That may apply to people, but more important for people is the psychological benefit of
having something to show for yourself along the way. Also, iterative methods, whether for linear systems or human projects, are robust to changes in requirements because they are able to take
advantage of progress made toward a slightly different goal.
More linear algebra posts
8 thoughts on “Iterative linear solvers as metaphor”
1. The factored operator can be re-used.
2. I like the visualization on the slide. Do you have a better version of this image?
3. Unfortunately no. And I don’t know the provenance of the photo either.
4. That’s a nice thought. But would you really buy a car which was built from a motorcycle, that was actually an improved bike that someone, somehow, improvised from a skateboard?
From my experience, evolutionary rapid prototyping indeed works great for nontrivial real-life projects, but only if the process is accompanied by rather frequent refactoring phases, in which the
already existing functionality is reconstructed using a well thought out design based on the better understanding of the complexities and pitfalls gained during the prototyping. It looks nothing
like that Spotify ad, which I hope came from their marketing guys, and not from their engineers…
5. It works only if you can transform a bicycle to a car – in reality – and sometimes – you just can’t.
6. This is a nice metaphor.
On the mathematics side, it’s interesting to note that Gaussian elimination may be used as an iterative algorithm, too. Let me say no more than refer to the beautifully-written SIAM News article
Gaussian Elimination as an Iterative Algorithm by Townsend and Trefethen.
7. it looks like the what the algebraic multigrid does.
8. There is a similar dichotomy between primal and dual algorithms for optimization. Primal algorithms start by finding something you could do, then sequentially finding better things that you could
do, until they (hopefully) find the best thing you could do. At every stage, the current best solution is feasible, but only the last one is optimal. If you stop in the middle, you might have a
solution that is “good enough”.
Dual algorithms go the other way ’round. Every intermediate solution is an optimal solution to the wrong problem, but the difference between the problem solved and the problem you’re _trying_ to
solve goes to zero. When you are solving the right problem, you are done.
The magic of duality theory is that this asymmetry is illusory. | {"url":"https://www.johndcook.com/blog/2014/06/25/iterative-linear-solvers/","timestamp":"2024-11-09T07:59:55Z","content_type":"text/html","content_length":"64440","record_id":"<urn:uuid:d5908d32-253b-4cc9-8620-ebc15d2afb1b>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00481.warc.gz"} |
Comments on WebDiarios de Motocicleta: A taxonomy of range query problemsIt's space O(n lglg n) and time O(k lglg n). I...In your MADALGO's summer school tutorial on or...The colored version (especially counting) is only ...Decomposability is a key property identified by Be...There are some variations for the weighted version...You're right, David, that concept deserves a m...Did you forget, or intentionally leave out, recurs...Thanks Jeff. I added the parametric/kinetic varian...Under dynamism, add parametric/kinetic. In its si...
11599372864611039927noreply@blogger.comBlogger9125tag:blogger.com,1999:blog-786333285568106173.post-54418076442689138142010-09-19T11:34:26.505-04:002010-09-19T11:34:26.505-04:00It's space O(n
lglg n) and time O(k lglg n). I strongly suspect a lower bound saying that O(k)-time reporting requires Omega(n lg^eps n) time -- but I don't have a formal proof yet.Mihaihttps://www.blogger.com/
profile/11599372864611039927noreply@blogger.comtag:blogger.com,1999:blog-786333285568106173.post-90751529350673312422010-09-19T10:05:16.513-04:002010-09-19T10:05:16.513-04:00In your MADALGO's
summer school tutorial on orthogonal range queries you claim that you can do better 2D orthogonal range queries but you do not give the output sensitive term. Do you mean that you are able to get O(n
log log n) space with O(log log n +k ) or O(log log n(1+k)) query time (the latter would be only a slight improvement over FOCS00
paper).Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-786333285568106173.post-80392015500420281612010-08-06T09:18:29.801-04:002010-08-06T09:18:29.801-04:00The colored version (especially
counting) is only decomposable w.r.t. colors, causing a substantial gap between best-known colored/non-colored
bounds.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-786333285568106173.post-47967098797750987452010-08-05T15:42:24.133-04:002010-08-05T15:42:24.133-04:00Decomposability is a key property
identified by Bentley.Jackhttps://www.blogger.com/profile/
02023530898615048685noreply@blogger.comtag:blogger.com,1999:blog-786333285568106173.post-73837572263047105482010-08-05T09:21:31.827-04:002010-08-05T09:21:31.827-04:00There are some variations for the
weighted version based on what kind of sum is used. For example, if the sum is taken over a group, then static partial sums are trivial, but if you are in a commutative semigroup, then you cannot use
subtraction and the problem gets much more complicated. Idempotence (x+x=x) is another property worth mentioning.<br /><br />Data structures may be exact or approximate. In the approximate version,
you can approximate the count, the fraction of the points within the range, or allow points near the boundary to be misclassified.<br /><br />One may or may not consider problems where the objects
are not points as range searching. For example, given a set of triangles, find the triangles intersected or contained in a rectangle.Guilhermehttp://www.uniriotec.br/
~fonsecanoreply@blogger.comtag:blogger.com,1999:blog-786333285568106173.post-53667675063217794682010-08-05T08:31:58.309-04:002010-08-05T08:31:58.309-04:00You're right, David, that concept
deserves a mention. But it's more general than range queries. Eg, I can have records with 3D points and some string inside, and make range queries on the points and pattern matching queries on
the string.<br /><br />By the way, do you think it would be useful to upload this list to Wikipedia? Perhaps here: http://en.wikipedia.org/wiki/Range_query<br /><br />What should one do about topics
at the intersection of two fields (here, CG and databases), which nonetheless mean sufficiently different things inside those fields?Mihaihttps://www.blogger.com/profile/
11599372864611039927noreply@blogger.comtag:blogger.com,1999:blog-786333285568106173.post-81754102303939081862010-08-04T21:48:54.253-04:002010-08-04T21:48:54.253-04:00Did you forget, or intentionally
leave out, recursive queries? Where what you want to do to the data within the range is apply some other range query on some unrelated dimension of the same data.<br /><br />Of course one can always
flatten the recursion and get some kind of range space in which you're just doing a counting or reporting query or whatever. But the ranges for this flattened space might be harder to describe.D.
Eppsteinhttp://11011110.livejournal.com/noreply@blogger.comtag:blogger.com,1999:blog-786333285568106173.post-41756223436123811212010-08-04T17:51:31.977-04:002010-08-04T17:51:31.977-04:00Thanks Jeff.
I added the parametric/kinetic variant.<br /><br />R^d is completely besides the point for orthogonal problems, since you should start by converting to rank space. (After that, what does the real
model allow me to do?) For non-orthogonal queries, most data structures do use the Real RAM, since they don't want to worry about precision.<br /><br />I have a hard time seeing nearest neighbor
as a range query. In any case there's enough material on it that it's now a separate topic. :)<br /><br />Orthogonal ray shooting is a range query -- I added a clarification. (Max segment
intersection with priorities = y-coordinates.) But then I should probably not consider general ray shooting as a range query.<br /><br />Another hyper-generic view you can take is semigroup range
sum. For instance, counting works in the (N,+) semigroup, range min works in the (N,min) semigroup, etc.<br /><br />About approximation, I have yet to be convinced about its value in range queries
(as opposed to other things, like ANN). In any case, Sariel promised to do all approximation in the upcoming summer school :)Mihaihttps://www.blogger.com/profile/
11599372864611039927noreply@blogger.comtag:blogger.com,1999:blog-786333285568106173.post-89730476374904105652010-08-04T16:03:18.748-04:002010-08-04T16:03:18.748-04:00Under dynamism, add <em>
parametric/kinetic</em>. In its simplest form, a kinetic data structure simply requires that the queries arrive in order along the 'time' coordinate, but the formulation allows for different
interesting tradeoffs.<br /><br />Under universes, add R^d — you know, real geometry.<br /><br />Nearest neighbor and ray-shooting queries are special cases of range-min queries, where the weight of
a point/object depends on the query.<br /><br />All the special types of queries are special cases of generic range spaces, which can be defined by a set of objects X, a set of queries Q, and a
function over pairs in X × Q. For example: (points, rectangles, [p∈r]) is 2d orthogonal range searching; (rectangles, points, [p∈r]) is 2d rectangle stabbing; (points, points, |pq|) is
nearest-neighbor searching; (strings, strings, [x is a substring of y]) is Google; (Turing machines, input strings, running time) is all of complexity theory; and so on.<br /><br />Finally, there are
umpty-dozen kinds of approximation to consider.JeffEhttps://www.blogger.com/profile/17633745186684887140noreply@blogger.com | {"url":"https://infoweekly.blogspot.com/feeds/8254514418257692011/comments/default","timestamp":"2024-11-10T08:30:01Z","content_type":"application/atom+xml","content_length":"20904","record_id":"<urn:uuid:10dbc42c-5e6d-4540-bbec-27a7eed48361>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00811.warc.gz"} |
Collection of Solved Problems
Heating the Log Cabin
Task number: 1797
a) How much heat transfers through side walls of a log cabin during one winter day? Length of the cabin is 10 m, width is 7 m, height of the walls is 3.5 m and their thickness is 50 cm. Average
outdoor temperature is -10 °C and the indoor temperature is kept at 18 °C.
b) How much wood has to be burnt up during one day in a stove, whose thermal efficiency is 30%, so that the indoor temperature is being kept constant?
c) How much would an electric heating of the cabin cost? Efficiency of electric heating is practically 100% and an average price of electric energy is for example 4.30 CZK/kWh.
d) How high must the volumetric flow rate of water in the radiator be if water temperature at the entry into the radiator is 80 °C and temperature of the water leaving the radiator is 70 °C?
Assume that the roof is so well thermally isolated that we can neglect heat loss through the roof.
• Hint
Transferred heat can be calculated from Fourier’s law of heat conduction.
• Notation
a = 10 m length of the cabin
b = 7 m width of the cabin
h = 3.5 m height of the cabin walls
d = 50 cm = 0.50 m thickness of the cabin walls
t[1] = -10 ^oC average outdoor temperature
t[2] = 18 ^oC indoor temperature
t[3] = 80 ^oC water temperature at the entry into the radiator
t[4] = 70 ^oC temperature of the water leaving the radiator
τ = 1 d = 86 400 s time
η = 30% = 0.3 wood stove efficiency
Q = ? heat transferred through the walls
m = ? mass of wood to be burnt
q[V] = ? volumetric flow rate of water in the radiator
From the Handbook of Chemistry and Physics:
λ = 0.15 W m^-1K^-1 thermal conductivity of wood
H = 15 MJ kg^-1 heat of combustion for wood
c[water] = 4 180 J kg^-1 K^-1 specific heat capacity of water
• Analysis
The heat transferred through a homogeneous board (in our case, the wall; inhomogeneity of the wall such as windows and the door are not considered) is proportional to the area of the wall
(board), time, during which the heat is being transferred, and the temperature difference between the ends of the board (under the condition that the temperature difference is constant). On the
contrary, the transferred heat is inversely proportional to the thickness of the board (wall). For some materials their ability to transfer the heat is specified by so-called thermal
conductivity. The greater it is the more heat is being transferred. To calculate the heat transferred through the walls during one day, we have all the required values.
To keep the indoor temperature constant, we have to supply the same heat that is being transferred to the environment. To calculate the required amount of wood, we will use its heat of
combustion, i.e. the heat we receive by burning 1 kg of wood.
Using the fact that the water has to supply so much heat so that the indoor temperature is kept constant, we calculate the required volumetric flow rate of water in the radiator. Supplied heat is
proportional to the difference between the temperature of flowing in water and the water leaving the radiator.
• Solution
a) For transferring heat it holds that:
\[Q=\lambda \frac{S \tau}{d} \Delta t\,,\]
where λ is the thermal conductivity of wood, S is the total area of the walls, for which it is true that
where d is the thickness of the walls, τ is the period of time during which the heat has been transferring and Δt = t[2] – t[1] is the difference between the temperature inside and outside the
By substitution we obtain the relation, which we can substitute given values into:
\[Q=\lambda\frac{\left(2a+2b\right)h\tau}{d}\left(t_2-t_1\right)\,,\] \[Q=0.15\cdot\frac{\left(2\cdot{10}+2\cdot{7}\right)\cdot3.5\cdot{86400}}{0.5}\cdot\left[18-(-10)t_1\right]\,\mathrm{J}\,,\]
b) Burning the wood whose mass is m, we obtain the heat Hm. However, only η = 30% of this heat we use for heating the cabin (wood stove efficiency). The heat used for heating the cabin has to
equal the heat transferred through the walls, therefore
\[Q=\eta Hm\,,\]
Hence we will express the required mass of wood m and calculate it by substitution:
\[m=\frac{Q}{\eta H}=\frac{86\,\mathrm{MJ}}{0.3\cdot{15}\,\mathrm{MJ\cdot kg^{-1}}}\dot{=}19\,\mathrm{kg}\,.\]
c) First we derive the conversion relationship between the units of energy:
From the conversion relationship we see that 3.6 MJ of electric energy cost 4.30 CZK. It means that the amount of money we would have paid for the electric heating for one day is:
d) If we denote the volumetric flow rate of water in the radiator as q[V], then the volume of the water passed through the radiator during the time τ is q[V]τ. This water cools down and supplies
the heat
\[Q_V=cq_V\tau \left(t_3-t_4\right)\,.\]
Then we compare the heat supplied by water and the heat transferred through the walls
\[Q_V=Q\,,\] \[c_{\mathrm{water}}q_V\tau\left(t_3-t_4\right)=\lambda\frac{\left(2a+2b\right)h\tau}{d}\left(t_2-t_1\right)\,.\]
and express the unknown volumetric flow rate and substitute given values:
\[q_V=\lambda\frac{\left(2a+2b\right)h\left(t_2-t_1\right)}{c_{\mathrm{water}}d\left(t_3-t_4\right)}\,,\] \[q_V=0.15\cdot\frac{\left(2\cdot{10}+2\cdot{7}\right)\cdot{3.5}\cdot\left[18-(-10)\
right]}{4180\cdot{0.5}\cdot\left(80-70\right)}\,\mathrm{kg\cdot s^{-1}}\,,\] \[q_V\dot{=}0.024\,\mathrm{kg\cdot s^{-1}}=86\,\mathrm{kg\cdot h^{-1}}\,.\]
• Answer
Heat transferred through the side walls of the cabin during one day is roughly 86 MJ. To keep the indoor temperature constant, we have to burn about 19 kg of wood every day. We would pay 103 CZK
per day for the electric heating. Amount of water which passes through the radiator is 86 litres per hour. | {"url":"https://physicstasks.eu/1797/heating-the-log-cabin","timestamp":"2024-11-11T17:08:24Z","content_type":"text/html","content_length":"33635","record_id":"<urn:uuid:de963417-dd1b-49ef-8752-33e430237a49>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00026.warc.gz"} |
Plain talk about quantum theory: It has no place for space and that’s a problem. - TimeOne
Science Seen Physicist and Time One author Colin Gillespie helps you understand your world.
Plain talk about quantum theory: It has no place for space and that’s a problem.
Quantum physics is in trouble. It needs a new strategy.
Physics is mostly based around quantum theory. When physics gets in trouble it affects everyone. Physics is the fountain from which our economy springs.
Quantum theory seems so esoteric many think they cannot go there. Yet at heart the problem and the opportunity to fix it can be discussed in plain language. So here goes.
Quantum theory works because its math describes what we can observe about the tiny particles that make up atoms. That is, it describes what we observe on average. That’s all the explanation quantum
theory has to offer.
If you cut through its mumbo jumbo, you find why quantum theory works is as much a mystery as it was a century ago when physicists first stumbled on it.
Here’s an analogy: Hundreds of years ago, physicists studied gases. They found math that described gas properties they could observe: volume, pressure and temperature. This math too had both its
uses and its mumbo jumbo: For example, a breakthrough paper by French physicist Sadi Carnot in 1824 was titled Reflections on the Motive Power of Fire.
Why that math worked remained a mystery until invisible constituents of gases were discovered: molecules—made of atoms far too small to see—that ricochet like tiny billiard balls off the container
In other words, gas pressure is caused by gazillions of molecules knocking on the walls; hotter molecules go faster and so exert more pressure.
This insight (first demonstrated by Albert Einstein, before he turned his mind to quantum theory) not only laid a firm foundation for thermodynamics and machines that extract work from heat, it also
set physics on the path to study even smaller things we can’t see: subatomic particles.
One way to view physics is it studies how things behave in terms of smaller things. Like thermodynamics did back then, quantum theory is now struggling to identify those smaller things.
After a century of searching, all physics can say is matter and energy are made up of a long list of subatomic particles whose makeup and behavior are as mysterious as ever. This list—the Standard
Model—is based on quantum theory.
That this is still mysterious should not surprise us. Impressive though the list may be, it leaves out space. Indeed, quantum theory treats it as nothing. We now know that, far from being nothing,
space is the most massive thing in the universe. Having no place for space is a serious omission.
Quantum theory is set in space. It treats space like a rock group treats a stage. Like the band and its instruments, the so-called fundamental particles that make an atom are arranged in space. Like
the stage, space is needed but neglected (unless it collapses).
The heart of the problem is quantum theory has no place for space; but it now turns out that space is not only most of everything, it runs the whole show.
How can physics solve its problem without ditching the quantum theory that is so successful most everyone carries a quantum device in their hands much of the day?
The obvious solution is a quantum theory of space. There are ongoing efforts to achieve this. Many of them are hampered by failure to take seriously the question: What is a quantum of space?
String theory is the largest of these efforts. It almost but not quite addresses this question. Its almost-answer: The smallest piece of space is a Planck-sized six-dimensioned volume called a
Calabi-Yau manifold.
The best strategy for physics is to take this answer seriously. Treat this quantum (I call it a fleck) as a real entity and study it as such. Physics hesitates to go there, maybe because it means
abandoning the math of continuous space (called calculus) most physicists invested years to learn.
Building brand new math is difficult. But it’s the strategy physics should follow if it’s to get to the next level of smaller things.
We all have an investment in that.
Readers who like this might like: Planck-scale physics is in line to give a huge boost to the economy.
Image credits:
BBC; http://www.bbc.co.uk/staticarchive/1892d615d510908b1151d6a98e5d0966fc3d938d.gif
bugman 123; https://bugman123.deviantart.com/art/Gyroid-205125437
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Execution time limit is 1 second
Runtime memory usage limit is 64 megabytes
Recently, a first grader learned to add numbers Vasya. It this process very much, and he puts everything in sight. When all the numbers are stacked around, Vasili turned to his older brother Pete for
new numbers. After several appeals tired of working random number generator, Peter came up to Washi task that can also take a long time.
He suggested Vasya find the sum of digits of consecutive numbers — 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 — and so on, until Vasya not get bored. Vasya was
enthusiastic about the idea and went to work. Yesterday Vasya find sum of digits of each of the numbers from 1 to 115. Looking at the results of his younger brother, Peter noticed that the sum of
digits of consecutive numbers are not random, often they are consecutive, but the pattern completely, he did not understand.
To find the patterns, Peter decided to explore the extreme cases, for example, which of the numbers gives the maximum amount of digits. Data for the numbers to 115 was not enough for final
conclusions, and Pete had the idea to speed up the calculations used in place of brother computer. As he himself programming is not very strong, he sought the solution of this problem to you.
In the first line of input data is the number N (1 <= N <= 2 147 483 647).
Remove a number from 1 to N inclusive, with the maximum amount of digits. If the numbers with a maximum sum of several numbers, display the greatest of them.
Submissions 2K
Acceptance rate 16% | {"url":"https://basecamp.eolymp.com/en/problems/474","timestamp":"2024-11-04T00:57:54Z","content_type":"text/html","content_length":"240767","record_id":"<urn:uuid:5f62d861-06aa-4820-baa2-c972d66334a3>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00658.warc.gz"} |
Life Path Number for July 31st, 2015
How to calculate the Life Path Number for July 31, 2015
To find the Life Path Number for July 31st, 2015, you will need to calculate the root number. In other words, you will need to reduce the numbers "2015 7 31" down until you are left with a single
digit number.
11, 22, and 33 are the only double-digit numbers that are allowed. This is because they are considered to be master numbers.
July 31st, 2015
This date of birth consists of the numbers 2015, 7 (for July), and 31.
Take a look at the following example.
Reduce the year 2015 down into one digit
Let's start with the year, which is 2015. We need to break this number down into one digit or one of the master numbers (11, 22 or 33).
To do this, we can "break" the numbers in 2015 apart and then do some simple addition:
2 + 0 + 1 + 5 = 8
As you can see, we are now left with the number 8.
Reduce the month July
Now, let's move onto the month, which is 7 (July).
As you can see, 7 is already a single-digit number. As a result, we do not need to reduce it.
So far, we have the number 8 for the year and the number 7 for the month. Now, we will need to move onto the next step and break down the day.
Reduce the day of the month into a single-digit as well
In the case of July 31st, 2015, the day of the month is obviously 31.
We will need to reduce this:
3 + 1 = 4
This leaves us with the number 4.
4 is a single-digit number, which means that we do not need to reduce it any further.
Final Step: Add 8, 7, and 4 together and then calculate the Life Path Number
We have 8 for the year, 7 for the month, and 4 for the day. Now, let's the calculate the final result, which will be our Life Path Number.
Start off by adding the three numbers together:
8 + 7 + 4 = 19
Because 19 is a two-digit number, we will need to reduce it even further:
1 + 9 = 10
In this case, 10 still needs to be broken down even further.
1 + 0 = 1
We can now conclude that
is the Life Path Number for
July 31st, 2015 | {"url":"https://bestofdate.com/life-path-number-calculator.php?year=2015&month=07&day=31","timestamp":"2024-11-03T10:02:23Z","content_type":"text/html","content_length":"44113","record_id":"<urn:uuid:d99f2d91-fd36-4c60-b347-87ac40079a98>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00024.warc.gz"} |
Can you learn geometry in 6th grade?
Can you learn geometry in 6th grade?
Can you learn geometry in 6th grade?
The major math strands for a sixth-grade curriculum are number sense and operations, algebra, geometry, and spatial sense, measurement, and functions, and probability. While these math strands might
surprise you, they cover the basics of what a sixth grader should learn in math.
How can I help my 6th grader with math?
Here are five tips to help students improve performance on 6th grade math tests:
1. Practice, practice, practice. Becoming proficient with a 6th grade math concept should eliminate some of the stress while being tested on that concept.
2. Keywords.
3. Use that pencil.
4. Wrong answers aren’t so bad.
5. Double-checking.
Is grade 6 math hard?
Sixth grade math class can be difficult, even for students who have done well in math previously. In sixth grade you begin to learn more advanced topics such as ratios and rates. You also work more
with fractions. Sixth grade is also when you begin building the foundations of algebra, geometry, and statistics.
What do 6th graders learn in geometry?
Sixth graders learn to find the volume of three-dimensional shapes with some lengths in fractions by filling them with unit cubes. They also learn to apply the formulas volume = length x width x
height (V = lwh) or Volume = base x height (v=bh), depending on the object shape.
What grade level geometry is?
Most American high schools teach algebra I in ninth grade, geometry in 10th grade and algebra II in 11th grade – something Boaler calls “the geometry sandwich.”
How do I become smarter in math?
How to Get Smarter in Math
1. Learn Smarter. Just as people are either left- or right-handed, they also have dominant brain hemispheres.
2. Study Smarter. Because math is a learned skill that requires practice, you may need to spend more time on homework and studying than you do in other subjects.
3. Practice Smarter.
4. Think Smarter.
Is algebra 2 or geometry harder?
Geometry is simpler than algebra 2. So if you want to look at these three courses in order of difficulty, it would be algebra 1, geometry, then algebra 2. Geometry does not use any math more
complicated than the concepts learned in algebra 1. | {"url":"https://promisekit.org/2022/12/15/can-you-learn-geometry-in-6th-grade/","timestamp":"2024-11-11T17:29:12Z","content_type":"text/html","content_length":"48296","record_id":"<urn:uuid:2c5f930b-a515-413e-9423-d14f9a066f31>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00344.warc.gz"} |
What is the formula of derivative?
What is the formula of derivative?
Derivative of the function y = f(x) can be denoted as f′(x) or y′(x). The steps to find the derivative of a function f(x) at the point x0 are as follows: Form the difference quotient \frac{f(x_0+\
Delta x)-f(x_0)}{\Delta x}
What is derivative example?
A derivative is an instrument whose value is derived from the value of one or more underlying, which can be commodities, precious metals, currency, bonds, stocks, stocks indices, etc. Four most
common examples of derivative instruments are Forwards, Futures, Options and Swaps.
What is the derivative of 2x?
Since the derivative of cx is c, it follows that the derivative of 2x is 2.
How do you solve derivative problems?
The first derivative of a function is a new function (equation) that gives you the instantaneous rate of change of some desired function at any point.
What is differentiation with example?
Differentiation. Differentiation allows us to find rates of change. For example, it allows us to find the rate of change of velocity with respect to time (which is acceleration). It also allows us to
find the rate of change of x with respect to y, which on a graph of y against x is the gradient of the curve. | {"url":"https://answer-all.com/miscellaneous/what-is-the-formula-of-derivative/","timestamp":"2024-11-05T23:17:11Z","content_type":"text/html","content_length":"127260","record_id":"<urn:uuid:c46dd777-ffeb-4ef2-b785-020e4698dff4>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00858.warc.gz"} |
Historical drawdown
The stock market tends to rise over time, but that doesn't mean that you won't have periods of drawdown.
Drawdown can be measured as the percentage loss from the highest cumulative historical point.
In Python, you can use the .accumulate() and .maximum() functions to calculate the running maximum, and the simple formula below to calculate drawdown:
$$ \text{Drawdown} = \frac{r_t}{RM} - 1$$
• \(r_t\): Cumulative return at time t
• \(RM\): Running maximum
The cumulative returns of USO, an ETF that tracks oil prices, is available in the variable cum_rets.
This is a part of the course
“Introduction to Portfolio Risk Management in Python”
View Course
Exercise instructions
• Calculate the running maximum of the cumulative returns of the USO oil ETF (cum_rets) using np.maximum.accumulate().
• Where the running maximum (running_max) drops below 1, set the running maximum equal to 1.
• Calculate drawdown using the simple formula above with the cum_rets and running_max.
• Review the plot.
Hands-on interactive exercise
Have a go at this exercise by completing this sample code.
# Calculate the running maximum
running_max = ____(cum_rets)
# Ensure the value never drops below 1
running_max[____] = 1
# Calculate the percentage drawdown
drawdown = (____)/____ - 1
# Plot the results
This exercise is part of the course
Introduction to Portfolio Risk Management in Python
Evaluate portfolio risk and returns, construct market-cap weighted equity portfolios and learn how to forecast and hedge market risk via scenario generation.
What is DataCamp?
Learn the data skills you need online at your own pace—from non-coding essentials to data science and machine learning. | {"url":"https://campus.datacamp.com/courses/introduction-to-portfolio-risk-management-in-python/value-at-risk?ex=2","timestamp":"2024-11-13T16:07:12Z","content_type":"text/html","content_length":"160843","record_id":"<urn:uuid:43eefd8f-8d53-4d42-bfcc-b7b766489311>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00118.warc.gz"} |
Multiplication Properties Worksheets Grade 5
Mathematics, specifically multiplication, forms the cornerstone of countless academic techniques and real-world applications. Yet, for lots of students, mastering multiplication can pose a challenge.
To resolve this hurdle, teachers and parents have accepted an effective tool: Multiplication Properties Worksheets Grade 5.
Intro to Multiplication Properties Worksheets Grade 5
Multiplication Properties Worksheets Grade 5
Multiplication Properties Worksheets Grade 5 -
Order of operations Mental Multiplication Multiplication algorithm long multiplication or in columns Factoring Multiplication division equations Order of operations These worksheets are only
available to
Free Printable Properties of Multiplication Worksheets for 5th Grade Properties of Multiplication Discover a collection of free printable worksheets for Grade 5
Value of Multiplication Method Understanding multiplication is essential, laying a strong foundation for innovative mathematical ideas. Multiplication Properties Worksheets Grade 5 use structured and
targeted method, fostering a deeper comprehension of this basic arithmetic procedure.
Development of Multiplication Properties Worksheets Grade 5
Commutative Property Of Multiplication Worksheets 2nd Grade Free Printable
Commutative Property Of Multiplication Worksheets 2nd Grade Free Printable
Here is a collection of our printable worksheets for topic Properties of Multiplication of chapter Multiplication in section Whole Numbers and Number Theory A brief description of the worksheets is
on each of the
Printable worksheets for reviewing the associative distributive commutative and identity properties of multiplication Distributive Property Basic FREE This graphical worksheet has arrays to help
From traditional pen-and-paper exercises to digitized interactive layouts, Multiplication Properties Worksheets Grade 5 have advanced, accommodating varied understanding styles and choices.
Sorts Of Multiplication Properties Worksheets Grade 5
Fundamental Multiplication Sheets Simple workouts focusing on multiplication tables, aiding students build a solid math base.
Word Trouble Worksheets
Real-life scenarios incorporated into issues, enhancing vital reasoning and application skills.
Timed Multiplication Drills Tests made to improve speed and accuracy, aiding in fast mental math.
Advantages of Using Multiplication Properties Worksheets Grade 5
Multiplication Properties Anchor Chart By Mrs P For Fourth Or Fifth grade Math Commutative
Multiplication Properties Anchor Chart By Mrs P For Fourth Or Fifth grade Math Commutative
The multiplication properties worksheets are very easy to download and use and will be very engaging for the students Multiplication Properties Worksheets Math
5th Grade Math Worksheets Interactive Multiplication Properties worksheets for pre kindergarten to grade 5 kids online aligned with Common Core Standards SplashLearn
Boosted Mathematical Skills
Consistent technique develops multiplication efficiency, boosting general math capacities.
Enhanced Problem-Solving Abilities
Word issues in worksheets establish logical thinking and strategy application.
Self-Paced Discovering Advantages
Worksheets suit individual knowing speeds, fostering a comfy and adaptable discovering setting.
Exactly How to Produce Engaging Multiplication Properties Worksheets Grade 5
Incorporating Visuals and Shades Vibrant visuals and colors capture interest, making worksheets visually appealing and involving.
Including Real-Life Situations
Connecting multiplication to everyday circumstances includes relevance and practicality to exercises.
Customizing Worksheets to Various Ability Degrees Tailoring worksheets based upon varying effectiveness degrees makes certain inclusive knowing. Interactive and Online Multiplication Resources
Digital Multiplication Equipment and Games Technology-based sources supply interactive understanding experiences, making multiplication appealing and enjoyable. Interactive Websites and Applications
On-line platforms provide varied and available multiplication method, supplementing conventional worksheets. Customizing Worksheets for Different Learning Styles Aesthetic Learners Aesthetic help and
diagrams aid comprehension for students inclined toward visual understanding. Auditory Learners Verbal multiplication problems or mnemonics cater to learners that grasp principles via acoustic ways.
Kinesthetic Learners Hands-on tasks and manipulatives support kinesthetic students in understanding multiplication. Tips for Effective Application in Learning Uniformity in Practice Regular practice
reinforces multiplication abilities, advertising retention and fluency. Balancing Repeating and Range A mix of repeated workouts and diverse problem layouts maintains interest and understanding.
Giving Useful Feedback Responses help in determining areas of improvement, encouraging continued progression. Difficulties in Multiplication Practice and Solutions Motivation and Interaction
Difficulties Tedious drills can result in uninterest; ingenious strategies can reignite motivation. Overcoming Worry of Math Adverse assumptions around math can hinder development; developing a
favorable understanding setting is important. Impact of Multiplication Properties Worksheets Grade 5 on Academic Performance Researches and Research Findings Research study indicates a favorable
connection in between constant worksheet use and boosted math efficiency.
Multiplication Properties Worksheets Grade 5 become functional tools, promoting mathematical effectiveness in students while suiting varied discovering styles. From basic drills to interactive
on-line resources, these worksheets not only boost multiplication abilities yet also promote important reasoning and analytic abilities.
Properties Of Multiplication Mini Anchor Chart Part Of An Interactive Math Journal Math
Distributive Property Of Multiplication Over Addition Worksheets Times Tables Worksheets
Check more of Multiplication Properties Worksheets Grade 5 below
Blendspace Ma Distributive Property Sequencing Worksheets 1st Grade Math Worksheets
multiplication properties worksheets grade 4 Times Tables worksheets properties Of
Distributive Property Of Multiplication Over Subtraction Calculator STAETI
Multiplication Tables From 1 To 20 Printable Pdf Table Design Ideas
3th Grade Worksheets Multiplication Property
Multiplication With Distributive Property Worksheets AlphabetWorksheetsFree
50 Properties Of Multiplication Worksheets For 5th Grade On
https:// quizizz.com /en-us/properties-of...
Free Printable Properties of Multiplication Worksheets for 5th Grade Properties of Multiplication Discover a collection of free printable worksheets for Grade 5
Grade 5 Multiplication amp Division Worksheets K5
https://www. k5learning.com /free-math …
5th grade multiplication and division worksheets including multiplying in parts multiplication in columns missing factor questions mental division division with remainders long division and missing
dividend or
Free Printable Properties of Multiplication Worksheets for 5th Grade Properties of Multiplication Discover a collection of free printable worksheets for Grade 5
5th grade multiplication and division worksheets including multiplying in parts multiplication in columns missing factor questions mental division division with remainders long division and missing
dividend or
Multiplication Tables From 1 To 20 Printable Pdf Table Design Ideas
multiplication properties worksheets grade 4 Times Tables worksheets properties Of
3th Grade Worksheets Multiplication Property
Multiplication With Distributive Property Worksheets AlphabetWorksheetsFree
Commutative Property Of Multiplication 3rd Grade Ppt STAETI
The Multiply 2 Digit By 1 Digit Numbers Using The Distributive Property All Math Worksheet
The Multiply 2 Digit By 1 Digit Numbers Using The Distributive Property All Math Worksheet
multiplication properties Worksheet 4th grade
Frequently Asked Questions (Frequently Asked Questions).
Are Multiplication Properties Worksheets Grade 5 appropriate for all age groups?
Yes, worksheets can be tailored to different age and ability degrees, making them adaptable for numerous learners.
Just how commonly should students exercise making use of Multiplication Properties Worksheets Grade 5?
Consistent practice is essential. Regular sessions, preferably a few times a week, can produce considerable renovation.
Can worksheets alone enhance math abilities?
Worksheets are a valuable tool however ought to be supplemented with diverse learning approaches for extensive ability growth.
Exist online platforms supplying free Multiplication Properties Worksheets Grade 5?
Yes, several academic web sites supply open door to a wide variety of Multiplication Properties Worksheets Grade 5.
How can parents support their kids's multiplication method in the house?
Urging consistent technique, supplying support, and creating a positive learning environment are valuable steps. | {"url":"https://crown-darts.com/en/multiplication-properties-worksheets-grade-5.html","timestamp":"2024-11-12T06:24:24Z","content_type":"text/html","content_length":"28393","record_id":"<urn:uuid:eb6aac0c-0c99-4e24-8157-7cf59ae46137>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00227.warc.gz"} |
One Card
As for portability, nothing can stay on par with a single playing card. A regular playing card, however, would not offer much possibilities, so I devised a card design that allows casting I Ching
You can put it in your favourite I Ching book and having it immediately ready for when you need it.
The image below shows the front and the back side of the card:
You can download an
hires version
(300dpi) of the two images and print them on photographic paper, glue them together and, possibly, laminate. With little effort, this card will serve you well for long time. By chance the images also
happen to be of the right size to be printed as
bridge-sized custom cards
(with which I have no relationship).
The card can be used to obtain lines with different probabilities.
Three coins probabilities
You need two operations for each line. Proceed as follows:
1. Without looking, turn and rotate the card in your hand so that you no longer know which face shows nor its orientation.
2. When you feel the time is right, look at the card.
□ If the red tiger shows up, draw
□ If the red dragon shows up, draw
3. Repeat step 1;
4. Look at the upper left corner. If you see a red dot, it's a moving line.
5. Repeat steps 1-4 other five times and draw the line from the bottom to the top of the hexagram.
The first time we only cosider which face shows up (
) each with a probability of
. The second time we look for the only corner, among the four, that has a red dot. Hence the probabilities are the same of the three coins method:
Prob(6) = Prob(9) = ^1/[2] * ^1/[4] = ^1/[8] = 12.5%
Prob(8) = Prob(7) = ^1/[2] * ^3/[4] = ^3/[8] = 37.5%
Prob(yin) = Prob(yang) = ^1/[2]
Yarrow Stalks probabilities
You need two operations for each line. Proceed as follows:
1. Without looking, turn and rotate the card in your hand so that you no longer know which face shows nor its orientation.
2. When you feel the time is right, look at the upper left corner of the card and write down the number of black dots (either 3 or 4) you see;
3. Repeat step 1;
4. When you feel the time is right, look at the upper left corner of the card and write down the number of dots, regardless the color, (either 3, 4 or 5) you see;
5. Sum up the numbers and draw the line according the following table:
6. Repeat steps 1-5 other five times and draw the lines from the bottom to the top of the hexagram.
There are four possible orientation for a card (front+up, front+down, back+up, back+down) and the four possible "upper left corner". The card is marked so that the first time we can get four possible
outcomes: 3,4,4,4 while the second time we can get 3,4,4,5. Summing up as shown in the following table:
gives the same probabilities of the yarrow stalks method:
Prob(6) = ^1/[16] = 6.25%
Prob(8) = ^7/[16] = 43.75%
Prob(7) = ^5/[16] = 31.25%
Prob(9) = ^3/[16] = 18.75%
Prob(yin) = Prob(yang) = ^1/[2]
Equal Probabilities
Proceed as follows:
1. Without looking, turn and rotate the card in your hand so that you no longer know which face shows nor its orientation.
2. When you feel the time is right, look at the card.
□ If the red tiger shows up, draw
□ If the red dragon shows up, draw
3. If the card is upside down (i.e. the flame is pointing downward), it's a moving line.
4. Repeat steps 1-3 other five times and draw the lines from the bottom to the top of the hexagram
Each line has the same probability to be drawn:
Prob(6) = Prob(back+down) = ^1/[4] = 25%
Prob(8) = Prob(back+up) = ^1/[4] = 25%
Prob(7) = Prob(fron+up) = ^1/[4] = 25%
Prob(9) = Prob(front+down) = ^1/[4] = 25%
Prob(yin) = Prob(yang) = ^1/[2]
Image on the card taken from the free "Clipart Panda" site.
0 commenti: | {"url":"https://www.castingiching.com/2016/06/i-ching-one-card.html","timestamp":"2024-11-09T15:55:44Z","content_type":"application/xhtml+xml","content_length":"93402","record_id":"<urn:uuid:e97f5cbd-cdd4-4051-92e4-f1bc0bb76928>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00265.warc.gz"} |
Shares and Dividend RS Aggarwal Goyal Prakashan ICSE - ICSEHELP
Shares and Dividend RS Aggarwal Goyal Prakashan ICSE
Shares and Dividend RS Aggarwal Goyal Prakashan ICSE Class-10 Chapter-3. RS Aggarwal Solutions of Shares and Dividend Chapter 3 for ICSE Maths Class-10 is also called Goyal Brother Prakashan . This
post is Solutions of Chapter – 3 Shares and Dividend of RS Aggarwal which is common famous Maths writer in ICSE Board in Maths Publication .
Shares and Dividend RS Aggarwal Goyal Prakashan ICSE Class-10
Step by Step Solutions of Chapter-3 Shares and Dividend is given to understand the topic clearly . Chapter Wise Solution of R S Aggarwal including Chapter -3 Shares and Dividend is very help full
for ICSE Class 10th student appearing in 2020 exam of council.
Chapter-3 Shares and Dividend RS Aggarwal Goyal Prakashan ICSE Class-10
Note:- Before viewing Solution of Chapter-3 Shares and Dividend of RS Aggarwal Goyal Brother Prakashan Solution. Read the Chapter Carefully . Then solve all example of your text book. The Chapter 3
Shares and Dividend is main Chapter in ICSE board .
EXERCISE – 3 Shares and Dividend RS Aggarwal Goyal Prakashan ICSE Class-10
Q.1. Find the market value of
(i) 350, RS 100 shares at a premium of 8.
(ii) 240, 50 shares at a discount of 5.
Question 2
. Find the annual income from 450, 25
shares, paying 12% dividend.
Question 3
Q.3. A man wants to buy 600 shares available.
at 125 having the par value 7100.
Question 4
Question 8
Ajay own 560………….share.
Question 9
Question 24
How much should a man invest ………………………….. declared is 12 %
Question 25
By investing 11440 in a company, paying 10 % …………………………… value of each Rs. shares ?
Return to :- RS aggarwal Goyal Brothers Maths Solutions for ICSE Class-10
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2 thoughts on “Shares and Dividend RS Aggarwal Goyal Prakashan ICSE”
1. Answer is no shown plzz 😭
□ not in syllabus
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[Series of lectures 2/21, 2/23, 2/28] An introduction to geometric representation theory and 3d mirror symmetry
• Date: 2023-02-21 (Tue) 10:30 ~ 12:00
2023-02-23 (Thu) 10:30 ~ 12:00
2023-02-28 (Tue) 10:30 ~ 12:00
• Place: 129-104 (SNU)
• Title: An introduction to geometric representation theory and 3d mirror symmetry
• Speaker: Justin Hilburn (Perimeter Institute)
• Abstract:
The Beilinson-Bernstein theorem, which identifies representations of semi-simple Lie algebra \mathfrak{g} with D-modules on the flag variety G/B, makes it possible to use powerful techniques from
algebraic geometry, especially Hodge theory, to attack problems in representation theory. Some successes of this program are the proofs of the Kazhdan-Lusztig and Jantzen conjectures as well as
discovery that the Bernstein-Gelfand-Gelfand categories O for Langlands dual Lie algebras are Koszul dual.
The modern perspective on these results places them in the context of deformation quantizations of holomorphic symplectic manifolds: The universal enveloping algebra U(\mathfrak{g}) is isomorphic to
the ring of differential operators on G/B which is a non-commutative deformation of the ring of functions on the cotangent bundle T^*G/B. Thanks to work of Braden-Licata-Proudfoot-Webster it is known
that an analogue of BGG category O can be defined for any associative algebra which quantizes a conical symplectic resolution. Examples include finite W-algebras, rational Cherednik algebras, and
hypertoric enveloping algebras.
Moreover BLPW collected a list of pairs of conical symplectic resolutions whose categories O are Koszul dual. Incredibly, these “symplectic dual” pairs had already appeared in physics as Higgs and
Coulomb branches of the moduli spaces of vacua in 3d N=4 gauge theories. Moreover, there is a duality of these field theories known as 3d mirror symmetry which exchanges the Higgs and Coulomb branch.
Based on this observation Bullimore-Dimofte-Gaiotto-Hilburn showed that the Koszul duality of categories O is a shadow of 3d mirror symmetry.
In this series of lectures I will give an introduction to these ideas assuming only representation theory of semi-simple Lie algebras and a small amount of algebraic geometry. | {"url":"https://qsms.math.snu.ac.kr/index.php?mid=board_sjXR83&listStyle=viewer&order_type=desc&l=en&document_srl=2438&page=8","timestamp":"2024-11-09T02:51:08Z","content_type":"text/html","content_length":"23183","record_id":"<urn:uuid:80b3b57a-7bbc-4726-947b-aa69c5e69aa4>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00565.warc.gz"} |
More Cost-Plus Pricing Examples -
In this post we build upon the price calculation information provided in our article on Cost-Plus Pricing Formula with Examples. It is recommended if you’re unfamiliar with how to undertake cost-plus
pricing that you review the above article and information first.
The purpose of this post is to provide further examples of cost-plus pricing. And please note there is a free cost-plus pricing Excel template available for download at the end of this post.
Cost-plus Pricing Formula and Examples
As you may know, there are two approaches to calculating a retail price using the cost plus pricing method. The first approach only considers the variable cost of the product, whereas the second and
more detailed approach considers both variable and fixed costs.
The two cost-plus pricing formulas can be summarized as:
1. Retail price = unit product cost X percentage markup
2. Retail price = (share of fixed costs/unit sales + related variable costs + unit product cost) X % percentage markup
As you can see, the first approach is very simple, where the company simply multiplies the unit cost (what they pay for the product from the supplier) and adds a percentage markup as their profit
And in the second approach, and allocation of fixed costs, and potentially other variable costs, are added to the unit cost – and this new combined unit cost is then marked up by a suitable profit
margin percentage.
Please note that a detailed example has been provided in the article on Cost-Plus Pricing Formula with Examples. This example includes some rationale for the calculation, and which is the more
appropriate cost-plus formula to use. Below you will find further examples to help guide your understanding of this pricing calculation method.
Cost-plus formula example 1
Base assumptions
For this example, let’s assume the following information:
• this firm sells one product only
• they sell 100,000 units of this product per year
• the base unit cost of the product is $25
• they undertake some modification of the product prior to reselling it, which works out at $5 per unit
• they have decided that a 20% markup of the combined unit cost is appropriate for their business
• they have annual total fixed costs of $120,000 to cover as well
• therefore, we need to calculate what retail price to charge and what profit the firm would make, given the above assumptions
NOTE: try and complete the calculation yourself, before you review the answer below…
Cost-plus pricing formula calculation
• basic unit cost = $25
• additional variable costs = $5
• combined variable cost = $30 ($25 +$5)
• total fixed costs to cover = $120,000
• fixed cost per unit sold = $1.20 ($120,000 divided by 100,000 units)
• total unit cost = $31.20 ($30 + $1.20)
• markup percentage = 20%
• markup margin = $6.24 (20% X $31.20)
• retail price = $37.44 ($31.20 + $6.24)
• expected total profit contribution = $624,000 ($6.24 margin X 100,000 units)
Cost-plus formula example 2
Base assumptions
For this example, we are going to assume exactly the same above information as for example 1, except for the following changes:
□ the sales have dropped to only 50,000 units per year
□ and they now have have annual total fixed costs of $150,000 to cover
□ as a result, they have decided that a 30% markup of the combined unit cost is
now required to cover the additional fixed costs and the reduction in sales volume
NOTE: try and complete the calculation yourself, before you review the answer below…
Cost-plus pricing formula calculation
Note: variations to the above calculation as shown in bold
• basic unit cost = $25
• additional variable costs = $5
• combined variable cost = $30 ($25 +$5)
• total fixed costs to cover = $150,000
• fixed cost per unit sold = $3.00 ($150,000 divided by 50,000 units)
• total unit cost = $33 ($30 + $3.00)
• markup percentage = 30%
• markup margin = $9.90 (30% X $33)
• retail price = $42.90 ($33 + $9.90)
• expected total profit contribution = $495,000 ($9.90 margin X 50,000 units)
As can be seen, their total profit contribution has reduced from $624,000 to $495,000. While this is a substantial decrease, they have done well to hold their profits, given that fixed costs have
increased by $30,000 and their sales have halved.
Cost-plus formula example 3
Base assumptions
Let’s now consider one more, but more detailed, cost-plus pricing example.
For this example, we are going to still going to consider the same firm as above, but let’s assume that they have expanded into a second product line offering – which will change their information as
• the firm now sells two product lines
• with product line 1, they sell 50,000 units
• and for product line 2, they sell 20,000 units
• the base unit cost of product 1 is $25
• the base unit cost of product 2 is $45
• they undertake some modification of product 1 prior to reselling it, which works out at $5 per unit – they do not incur any costs modifying product 2
• they have decided that a 20% markup of the combined unit cost is appropriate for product 1
• but they are seeking a 40% markup of the unit cost for product 2
• they now have annual total fixed costs of $200,000 to cover
• for the purpose of this cost-plus pricing exercise, they are allocating 60% of their fixed costs to product 1, and the remaining 40% to product 2
• therefore, we now need to calculate what retail price to charge for each product line and what profit the firm would make, given the above assumptions and revised situation
NOTE: try and complete the calculation yourself, before you review the answer below…
Cost-plus pricing formula calculation
Because there are now two product lines with different costs and mark-ups, we will need to undertake some of the calculations independently, as follows:
Product 1 calculation
• basic unit cost = $25
• additional variable costs = $5
• combined variable cost = $30 ($25 +$5)
• total fixed costs to cover = $120,000 (60% of $200,000)
• fixed cost per unit sold = $2.40 ($120,000 divided by 50,000 units)
• total unit cost = $32.40 ($30 + $2.40)
• markup percentage = 20%
• markup margin = $6.48 (20% X $32.40)
• retail price = $38.88 ($32.40 + $6.48)
• expected total profit contribution = $324,000 ($6.48 margin X 50,000 units)
Product 2 calculation
• basic unit cost = $45
• additional variable costs = Nil
• combined variable cost = $45
• total fixed costs to cover = $80,000 (40% of $200,000)
• fixed cost per unit sold = $4.00 ($80,000 divided by 20,000 units)
• total unit cost = $49 ($45 + $4.00)
• markup percentage = 40%
• markup margin = $19.60 (40% X $49)
• retail price = $68.60 ($49+ $19.60)
• expected total profit contribution = $392,000 ($19.60 margin X 20,000 units)
Combined Profit Calculation
• Product 1 total profit contribution = $324,000
• Product 2 total profit contribution = $392,000
• TOTAL total profit contribution = $716,000
Download the Free Cost-plus Pricing Formula Excel Template
You can either follow the steps above, or you can download the free Excel template for easily calculating retail prices using the cost-plus pricing formula. A great and easy to use tool for scenario
testing of different price points and markups.
You can download your free template here, no sign-up is required,… cost-plus-pricing-formula-excel-template
Here is an example screenshot of the cost-plus pricing formula Excel free template, so you can see what it looks like before you download it. Please note that it starts with some examples, but you
can just type over the existing information to construct your own retail price points.
Please note that if you do not wish to include fixed/other costs in your calculation, then simply set that entry to zero – and the template will calculate cost-plus pricing using the simple formula.
Related Activities | {"url":"https://www.marketingstudyguide.com/more-cost-plus-pricing-examples/","timestamp":"2024-11-12T03:39:34Z","content_type":"text/html","content_length":"263457","record_id":"<urn:uuid:7790acf7-5ca9-4182-9af8-c6acd4896401>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00114.warc.gz"} |
Morphometrics from 3D Scans
research 04 Apr 2023
I begin with theory and testing of my methods against published data. If you’d like to jump ahead to our data and my current thoughts, please click here.
Fractal dimensions (FD) describe space filling of shapes at various scales and describe surface complexity. While coral colonies and coral reefs are not strictly fractals, colonial organisms and reef
assemblages share some key characteristics to fractals, including morphological irregularities, self-similarity and high degrees of space filling. FD can align with other traditional measurements
such as surface area to volume ratio, rugosity, etc; but FD offers increased information as seen in the below theoretical example of a coral reef (Fig. 1; Torres-Pulliza et al 2020). This figure
illustrates 3 reefs with identical rugosiites but decreasing fractal dimensions (FD), a < b < c.
Figure 1. Theoretical comparison of fractal dimension and rugosity
Reichert et al (2017) developed an easy to use tool to calculate FD of a 3D coral colony using the Bouligand-Minkowski method. First, I am reanalyzing the 3D scanned files from Reichert et al (2017)
to ensure I am using their code correctly. I am using their obj scan files and the analysis toolbox they released as part of the supplementary material. The toolbox takes in an obj scan file and
produces a txt file with 3 columns: dilation radius, log(dilation radius), and log(influence volume). Dilation radius is produced for 1 <= R <= 20.
Figure 2. Example photographs and 3D scans in Reichert et al 2017
Reichert et al (2017) assessed the influence of dilation radii on the ability to discern inter-and-intraspecific differences among 3d scans (i.e. does a fractal D tell us if a coral fragment is
identical to its clonemate/conspecific). They tested all integers 3 $\le$ R $\le$ 20, and found that when R = 8, fractal dimensions had the best ability to discriminate inter-and-intraspecific
differences. Thus, they calculate and report all fractal dimensions based on a dilation radius of 8.
Given that the toolbox produces dilation radius from 1 $\le$ R $\le$ 20, you should be able to subset this data frame to just the integers 3 $\le$ R $\le$ 20 or all real numbers 3 $\le$ R $\le$ 20 to
calculate and derive the same values reported in Reichert et al (2017). So I’m going to do just that.
Reichert et al (2017) use the Bouligand–Minkowski method to estimate a colony’s fractal dimension as, “it is one of the most accurate methods for computing fractal dimensions and it is highly
sensitive to detecting small changes in models (Tricot, 1995). Due to its use of Euclidean distances, the approach is invariant to rotation. Thus, prior normalization steps are not necessary (Tricot,
They define D as, $D = 3 - \lim_{R \to 0} \frac{log(V(R))}{log(R)}$ where R is the dilation radius and V(R) is the influence volume.
Here, you can visualize the measuring principles of the Bouligand-Minkowski method with increasing dilation radii from a–>c. Spheres are located at the vertex of the 3d mesh. Larger radii
progressively fill the volume enclosed by the mesh, resulting in a larger influence volume. The limit integrates across these spatial scales (radii a-c) to synthesize a singular characteristic of the
mesh’s complexity.
Figure 3. Theoretical 3D Application of the Bouligand-Minkowski method with spheres
As this is a power law, you can estimate the limit by taking the slope of the log-log plot that fits the curve log(R) x log(V(R)). Thus, D can be estimated as 3-m, where m is the slope of the log-log
You can progressively slide the curve from the beginning (R=2) to a maximum radii, and then calculate the slope over each defined region. For example, if you wanted to evaluate dilation radii from
2-15, you would first take the slope of the curve from (0,2), then (0,3), and so on until (0,15). You would calculate the FD at each R in the series and evaluate its discriminatory power.The code is
as follows,
#calculate linear model over the region
m = lm(log.infl.vol ~ log.dil.rad,
#filter data to integers only between 3 & 20
#R is the desired dilation radius
data = dat %>% filter(dil.rad<= & R))
#extract the slope
Example Data
Using the 3D scans from Reichert et al (2017), I independently calculated the Fractal Dimension using their toolbox. Below is a table of the data where I exclusively looked at time point 0 data.
Table 1: Comparison of Fractal
Dimensions. I can replicate their FD.
Only chose a random subset since it’s a
long table.
ID Species ReichD D8 diff
Plu_2\_01 Plu 1.92456 1.92456 0
Pda_3\_02 Pda 1.93626 1.93626 0
Pda_3\_03 Pda 1.95097 1.95097 0
Pda_2\_03 Pda 1.96075 1.96075 0
Pda_1\_05 Pda 1.94734 1.94734 0
As you can see, I am calculating their data the same way. So all calculations are working. Let’s proceed with discriminatory analyses.
Radii Analysis
I calculated a FD for radii 2-20 to conduct a discriminatory test similar to the Reichert et al (2017) analysis. They found that r=8 had the highest discriminatory power.
Interspecific Detection
Figure 4. Interspecific discriminaotry power of FD at different radii
Table 2:
power. N
represents the
number of
and is ordered
by the most
radii n
Table 3: Avg
significance of
radii interspecific
power. Radius is the
dilation radius, and
p.avg is the average
signficant pairwise
difference between
the 12 groups.
radii p.avg n
8 0.0025290 12
7 0.0063254 12
20 0.0035425 11
19 0.0055155 11
Dilation radii of 7 and 8 produce the highest interspecific discriminatory power. Using these radii, we can differentiate between all species except: Ahu-Ami, Ahu-Pve, Ami-Pve, Ahu-Pda, Pda-Pve. When
looking at the average pairwise significance from radii 7 and 8, 8 performs better than 7. 19 and 20 follow close behind, with these radii not being able to discriminate between Ahu and Pda, but
their average pairwise significance is still better than a radii of 8. So we cannot differentiate the Acroporas and the Pocillopora verrucosa with radii of 7 and 8, and we add Pocillopora damicornis
to that list when we change the radii to 19 or 20. Therefore, we are really only able to differentiate between the Porites and the branching corals.
Intraspecific Detection
Running these tests on only a subset of radii where the n from Table 2 is greater than or equal to 10 (the 8 best performing radii).
Figure 5. Intraspecific discriminaotry power of FD at different radii
Table 4:
power. N
represents the
number of
and is ordered
by the most
radii n
Table 5: Avg
significance of
radii intraspecific
power. Radius is the
dilation radius, and
p.avg is the average
signficant pairwise
difference between
the significantly
diffrent groups.
radii p.avg n
20 0.0142110 11
19 0.0145423 11
18 0.0067559 9
9 0.0113366 6
8 0.0096806 5
7 0.0127828 4
6 0.0190583 3
For intraspecific differences, dilation radii 19 & 20 produced optimal results, followed closely behind by 18 (Table 3). This is different from interspecific variation; 7 and 8 performed much worse
here (< half of detected pair-wise differences compared to 19,20). 18, 19 and 20 could detect intraspecific differences in all 6 species, while 7 and 8 could only detect intraspecific differences in
2 species (Acroporas). Differences in all 3 Ami colonies could be detected with a radii of 8, 9, 18, 19 or 20, suggesting consistently different morphologies for each of the colonies of this species.
Ahu (6 radii), Pcy (2 r) and Pda (2 r) could detect 2 pairwise differences, indicating a single colony was significantly different than the other two.
These results are interesting. While fractal dimensions cannot distinguish between Acroporas and other branching species, it can consistently distinguish intraspecific variation among these species,
especially Acropora humilis. This might suggest that these species have plastic morphologies that vary among the population, but that this variation can be parsed apart by colony-specific morphology.
Further, Reichert et al (2017) report the fractal dimension analyses were superior in quantifying intraspecific changes of colonies over time compared to traditional morphological characteristics,
indicating that these analyses are sensitive to small scale changes. Are these colonies collected from distinct environments, which uniquely shaped the colony morphology? Are genetics at play? From
these data alone, it’s impossible to tell. But we can begin to explore these questions using my data below.
The take away from these analyses are:
1. morphological complexity can be described with fractal dimensions,
2. FD can generally discern between inter- and intra-specific, but its not perfect, and
3. dilation radii must be selected according to resolution of analyses.
Analyzing Our Scans
Figure 6. Genotype-specific discriminaotry power of FD at different radii
Table 6: Genotype radii
discriminaotry power. Radius is
the dilation radius, and the
remaining columns indicate the
level and number of significantly
different pair-wise comparisons
among the genotypes
radius \*\*\*\* \*\*\* \*\* - ns
6 3 NA NA NA NA
7 3 NA NA NA NA
8 3 NA NA NA NA
9 3 NA NA NA NA
10 3 NA NA NA NA
11 3 NA NA NA NA
12 3 NA NA NA NA
13 3 NA NA NA NA
14 3 NA NA NA NA
15 3 NA NA NA NA
16 3 NA NA NA NA
17 3 NA NA NA NA
5 2 1 NA NA NA
18 2 1 NA NA NA
19 2 1 NA NA NA
4 2 NA 1 NA NA
20 2 NA 1 NA NA
3 1 NA 1 1 NA
2 NA 1 NA NA 2
Table 7: Avg
significance of
genotype radii
power. Radius is
the dilation
radius, and
p.avg is the
between the
three groups.
radii p.avg
11 2.00e-07
10 2.00e-07
12 2.00e-07
9 3.00e-07
13 3.00e-07
8 5.00e-07
14 6.00e-07
7 1.40e-06
15 1.40e-06
16 4.00e-06
6 7.10e-06
17 1.33e-05
Dilation radii 6-17 perform the best and have identical significance levels. If we take a look at the average pairwise significance between the 3 groups, using a radius of 11 produces the best
results. However, choosing any radius 6-17 will produce a very significant average p value <0.00001, which adds confidence that there is a difference in the fractal dimensions of these 3 genotypes.
This is interesting because these fragments were picked to be,
1. around the same size ~7cm,
2. from unique colony in nursery (a tree had >60 colonies all of one genotype), and
3. minimal branching w/ only 1 apical tip.
So even though we selected to have visually identical fragments, genotype specific morphology is evident. Let’s investigate other classical morphometrics to see if these genotypes were different. For
all analyses below, I am using FD11 as the measurement of FD.
Figure 7. Traditional morphometric comparisons of fragment height, surface area, volume, and surface area: volume ratio
The three genotypes did not have significantly different surface area to volume ratio or heights. However, there were significant pairwise differences between surface area and volumes between SI-A
and the other genotypes. Standardizing all growth rates to surface area is therefore critical for this data.
Figure 8. Linear regression of traditional morphometrics to FD
There’s a pretty strong relationship between surface area and volume with FD, with FD explaining about 61% and 44% of the variance in SA and V, respectively. There is no relationship between height
and surface area to volume ratio with fractal dimension.
Growth Analysis
An in depth analysis of treatments and growth can be viewed here.
Figure 9. Avg Daily Growth by (A) Treatment and (B) Genotype
Growth rates are lower than anticipated. This is a combination of actually depressed growth from what I was expecting and the high resolution of our 3D scanner where estimated SA is much higher than
usually measured. To try and make comparable, I looked at some other published work on the ’ol AcDC and found some SA derived from stitched together images on imajeJ in a Muller et al paper. Their
average SA was about 7$cm^2$, which is in comparison to our avg SA 39 $cm^2$. If we simply divide the two and scale the growth rates accordingly, we get an avg LCO2 growth rate of 0.73 mg $cm^{-2}$
$day^{-1}$. However, numerous papers from the NOAA AOML Coral Program lab have used the same 3D scanner setup to derive growth rates that were higher than what I observed. I do not know the exact SA
from these studies to scale accordingly, but these experiments were significantly shorter than the experiment I ran which may explain their elevated growth rates since a long time in the lab
generally decreases growth rates compared to corals in the field (Enochs et al. 2018 for instance). Nevertheless, the patterns are interesting and what I will be focusing on.
Figure 10. Regression of absolute growth (mg) to (A) surface area and (B) fractal dimension, separated by treatment group
Figure 11. Regression of daily growth rate (mg/cm^2/day) to (A) surface area and (B) fractal dimension, separated by treatment group
Absolute growth scales with both surface area and fractal dimension. Surface area and FD explain more variance in the HCO2 (69% v 60%) than the LCO2 (43% v 44%) groups. Overall, surface area explains
more variance in absolute growth than FD, but it is roughly negligible.
When standardizing absolute growth to surface area, an interesting pattern emerges. Here, the amount of variance in growth rates explained by FD is nearly twice that of surface area. Further, for the
LCO2 groups, none of the variance in growth rates is explained by either of the morphometrics, which is in contrast to the HCO2 groups where surface area and FD explain 25% and 47% of the variance,
respectively. This brings the hypothesis that surface complexity “only matters” for OA resistance and not for increased growth rates under ambient conditions. I’ll explore more on this below.
We cannot separate SA and FD from each other completely. Since FD describes how surface area fills space at different scales, it makes sense that as FD increases, SA could increase as well. Not
necessarily (see Fig 1 for instance, they all have the same SA if the lines were extended to a plane) and not in a linear relationship, since they are describing two different aspects of the
geometry. Nevertheless, SA and FD are intertwined. This data, therefore, demonstrates that resistance to OA (maintained growth rates) is driven more by fractal dimensions (measurement of surface
complexity) than by total surface area. Further, since SA standardized growth rates still increased as surface area increased, it is likely that growth rates do not scale linearly with SA.
What does this all mean?
Let’s dive into that second plot more and the hypotheses that this data may support.
This data supports the hypothesis that surface complexity confers resistance to OA but does not confer increased growth rates under ambient conditions. Its far from a perfect relationship, but I
think something is here. One immediate question I have is does the FD mean anything for the coral micro environment? The range of FD is 2.175-2.25, which although derived from log-log slope and
limited between 0-3, seems quite narrow of a range to be divergently meaningful. See notes on next steps where I will try to test this using computational models.
This hypothesis aligns closely with the hypothesis outlined in Chan et al (2016). Briefly, surface complexity slows water flow around the colony, thickening the diffusive boundary layer (DBL) and
increasing water residence time in the thin layer directly surrounding the coral. Therefore, the coral’s metabolism has a greater influence on the properties of this seawater: during the day this
water will have a higher pH than bulk seawater (photosynthesis) and at night this water will have a lower pH than bulk seawater (respiration). Coral metabolism and water residence time is well
investigated at the ecosystem scale where these same properties are at play, but how these properties play out at the organism scale remains largely unexplored. Together, these relative highs and
lows create a variable pH environment that could stress harden a coral where it has adapted and/or acclimated and can, therefore, better withstand OA. Alternatively, this diel variability could work
in concert with day to night calcification ratios to enhance daytime calcification to counteract the mean decrease in pH, effectively ameliorating OA (Enochs et al 2018; Chan & Eggins 2017).
Chan et al (2016) supported this hypothesis by measuring pH changes in the DBL under different morphologies at different flow rates. They saw that under low flow velocities and complex morphologies
(they did not quantify complexity, just had 2 different species w/ obvious surface complexity differences), pH upregulation in the DBL was quite high and had the potential to ameliorate the effects
of OA in the DBL (DBL pH under OA = DBL pH under ambient due to elevations). These data closely approximated their modeled pH increases based on photosynthesis and calcification rates. How these DBL
pH increases manifest to growth rates/OA resistance remains to be seen. Comeau et al (2019) measured pH in DBL (micro probes), pH in calcifying fluids (boron systematics), and growth rates under
different light and flow regimes. For the Acropora congeneric, they did not detect any elevations in DBL pH during the day, but did detect large decreases during the night. However, for Plesiastrea
veripora, they detected a large increase in DBL pH which increased under OA treatments in low flow identical to the findings in Chan et al (2016). These same corals did not, however, have elevated pH
in the calcifying fluid or maintain growth rates under OA. It is important to note that Comeau et al (2019) did not have variable pH treatments and did not systematically measure pH DBL under
Unfortunately, I was unable to measure the DBL pH with micro sensors, and I did not measure the metabolism of the corals. But, this is the first dataset I am aware of that has experimental evidence
of surface complexity driving OA resistance. How the potential pH variability caused by the surface complexity affects calcifying fluid pH as determined by boron systematics remains to be seen. We
should have that data soon. Comeau et al (2022) used boron systematics to probe the calcifying fluid pH of corals collected from volcanic CO2 seeps in Papua New Guinea. These seeps had low, but
highly variable pH. The corals from this environment maintained constant calcifying pH, relative to nearby controls, despite the low mean pH. The growth rates of these corals are not known.
Next steps
I think there is an interesting story here of genotype-specific surface complexity correlating with OA resistance. First, I’d like to explore some more metrics of surface complexity from these 3D
scans. I’m excited to finish up the boron chemistry work to see how that plays into this story. Finally, I’d like to import a characteristic 3D model of each genotype into a computational model to
see if surface complexity measures due indeed create a thicker DBL. From this model, I can measure water residence times, expected pH increases, etc.
1. Chan NCS, Wangpraseurt D, Kühl M, Connolly SR (2016) Flow and coral morphology control coral surface pH: Implications for the effects of ocean acidification. Frontiers in Marine Science 3:1-11.
2. Chan WY, Eggins SM (2017) Calcification responses to diurnal variation in seawater carbonate chemistry by the coral Acropora formosa. Coral Reefs 36:763–772.
3. Comeau S, Cornwall CE, Pupier CA, DeCarlo TM, Alessi C, Trehern R, McCulloch M (2019) Flow-driven micro-scale pH variability affects the physiology of corals and coralline algae under ocean
acidification. Scientific Reports 9:1–12.
4. Comeau S, Cornwall CE, Shlesinger T, Hoogenboom MO, Mana R, McCulloch MT, Rodolfo-Metalpa R (2022) pH variability at volcanic CO2 seeps regulates coral calcifying fluid chemistry. Global Change
Biology 28(8):2751–2763.
5. Enochs IC, Manzello DP, Jones P, Aguilar C, Cohen K, Valentino L, Schopmeyer S, Kolodzeij G, Jankulak M, Lirman D (2018) The influence of diel carbonate chemistry fluctuations on the
calcification rate of Acropora cervicornis under present day and future acidification conditions. Journal of Experimental Marine Biology and Ecology 506:135–143.
6. Reichert J, Backes AR, Schubert P, Wilke T (2017) The power of 3D fractal dimensions for comparative shape and structural complexity analyses of irregularly shaped organisms. Methods in Ecology
and Evolution 8(12):1650–1658.
7. Torres-Pulliza D, Dornelas MA, Pizarro O, Bewley M, Blowes SA, Boutros N, Brambilla V, Chase TJ, Frank G, Friedman A, et al (2020) A geometric basis for surface habitat complexity and
biodiversity. Nature Ecology & Evolution 4:1495-1501. | {"url":"https://patrickmkiel.com/notebook/research/3dScanMorphometrics/","timestamp":"2024-11-07T01:05:41Z","content_type":"text/html","content_length":"44782","record_id":"<urn:uuid:69988e42-a373-49a3-b9e3-ca1f05aa0ebc>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00185.warc.gz"} |
Chi Sq. Check: Introduction, How one can calculate, When to make use of - NIMS INDIA
In statistics, the chi-square check is used to analyse information from observations of a usually distributed assortment of variables. Sometimes, this entails contrasting two units of numerical info.
Karl Pearson first proposed this methodology of analysing and distributing categorical information, naming it Pearson’s chi-square check.
The chi-square check developed by Pearson is utilized in a contingency desk to guage whether or not there’s a important statistical distinction between the anticipated and precise frequencies in a
number of of the classes of the chi-square desk.
Statistically, statisticians use the chi-square check to find out how properly a mannequin matches the information. Chi-square statistics want a random, mutually unique, uncooked, unbiased variable
information pattern of adequate measurement.
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Chi-square check primary terminologies
The usual formulation for calculating a chi-square check is the summation of sq. errors or false positives divided by the pattern variance. There are just a few phrases which are applied when
utilizing the Chi-square check. These phrases have been outlined beneath:
The p-value is the probability of reaching a chi-square that is the same as or better than that within the current experiment, and the information nonetheless helps the speculation. This chance is
expressed as a share. It refers back to the probability that anticipated variations are attributable to nothing greater than random occurrences.
If the p-value is lower than equal to 0.05, then the speculation considered is accepted. If the worth is greater than 0.05, then the speculation is rejected.
Diploma of Freedom
An estimation downside has a sure diploma of freedom equal to the variety of unbiased variables. Though there are not any arduous limits on the values of those variables, they do impose limits on
different variables if we wish our information set to be according to the estimated parameters.
One definition of “diploma of freedom” is the best variety of values within the information set which are logically unbiased of each other and therefore topic to vary. Deducting one from the full
variety of observations in an information set yields the diploma of freedom.
One distinguished context wherein the idea of diploma of freedom is addressed is within the context of statistical speculation assessments just like the chi-square.
Understanding the importance of a chi-square statistic and the robustness of the null speculation depends closely on precisely calculating the diploma of freedom.
The variance of a random quantity pattern is a measure of its dispersion round its imply. It’s calculated by squaring the worth of the usual deviation.
Properties to carry out the Chi-square Check
The Chi-square check has the next properties:
• Imply distribution equals the variety of levels of freedom.
• The variance must be equal to twice the diploma of freedom.
• Because the diploma of freedom grows, the chi-square distribution curve begins to resemble the conventional distribution curve, i.e. a bell curve.
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How one can carry out the Chi-square Check?
The Chi-square for distribution is calculated utilizing the formulation beneath:
2= [(Observed value – Expected Value)2/ Expected Value]
Steps to comply with to calculate the Chi-square statistic
1. Calculate the noticed and the anticipated worth.
2. Subtract every of the anticipated values from the noticed worth within the distribution desk.
3. Sq. the worth for every commentary you get in Step 2.
4. Divide every of those sq. values by its corresponding anticipated values.
5. Including up all of the values that we get in Step 4 offers a price that defines the chi-square statistic.
6. Calculate the diploma of freedom to verify for the aforementioned property satisfaction of chi-square assessments.
Kinds of Chi-Sq. Check
Goodness of Match
If you wish to see how properly a pattern of the inhabitants represents the entire, you might apply the Chi-square goodness-of-fit check. The pattern inhabitants and the projected pattern inhabitants
are in contrast utilizing this system.
Check for Independence
This Chi-square check for independence one inhabitants to find out whether or not there’s a correlation amongst two categorical variables. The unbiased check differs from the goodness-of-fit check
because it doesn’t examine a single noticed parameter to a theoretical inhabitants. As an alternative, the check for independence compares two values inside a pattern set to one another.
Check for Homogeneity
As with the independence check, the check for homogeneity follows the identical format and process. The important distinction between the 2 is that the check for homogeneity examines if a variable
has the identical distribution throughout many populations. In distinction, the check for independence examines the presence of a link between two categorical variables inside an analogous
When must you use a Chi-square check?
The Chi-Sq. Check determines whether or not precise values are according to theoretical chances. Chi-Sq. is probably the most dependable check to make use of when the information being analyzed comes
from a random pattern and the variable in problem is categorical.
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The place is the Chi-square check used?
Allow us to take the instance of a advertising firm.
A advertising firm is trying on the correlation between client geography and model selections. Consequently, chi-square performs a major function, and the worth of the statistic will inform how the
company can adapt its advertising strategy throughout geographies with a view to maximise revenues.
When analysing information, the Chi-square check is useful for checking the consistency or independence of categorical variables, in addition to the goodness-of-fit mannequin into account.
Equally, the chi-square statistic might discover use within the medical career. The chi-square check is appropriate for figuring out the efficacy of a drugs compared to a management group.
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On this article, you realized about Chi-square statistics and tips on how to calculate its values. Since Chi-square works with categorical variables, it’s typically employed by teachers investigating
survey response information. This type of research is frequent in lots of fields, together with sociology, psychology, economics, political science, and advertising.
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How is the p-value associated to the Chi-square check?
The p-value is the area beneath the chi-square density curve that’s to the fitting of the check statistic’s worth. Whether or not or not the chi-square check statistic is sufficiently massive to
reject the null speculation is the final step within the chi-square check of significance. The p-value is used for this function.
Are there any limitations or drawbacks to utilizing the Chi-square check?
All people being studied should be distinctive; else, the outcomes could be meaningless. A chi-square check shouldn’t be used if a given respondent could also be labeled into two distinct teams. Yet
one more restriction of chi-square is that it may well solely be used for frequency information. Moreover, the sum of all predicted individuals throughout all lessons must be bigger than 5.
What are the strengths of the Chi-square check?
One among its major strengths is that chi-square might be calculated rapidly and simply. Nominal information may be utilised utilizing this methodology. It might even be used to match greater than
two teams of categorical variables for statistical significance.
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EViews Help: Matrix Function Summary
Matrix Function Summary
Matrix Utility
Reorder the rows of the matrix using a vector of ranks.
Number of columns in matrix object or group.
Converts series or group to a vector or matrix after removing NAs.
Test for equality of data objects, treating NAs and null strings as ordinary and not missing values.
Square matrix from a sym matrix object.
Vector initialized from a list of values.
Vector containing equally spaced grid of values.
Vertically concatenate matrices.
Creates sym from lower triangle of square matrix.
Creates sym from upper triangle of square matrix.
Test for missing values.
Lowercase representation of a string, or lower triangular matrix of a matrix.
Matrix of normal random numbers.
Inequality test (NAs and blanks treated as values, not missing values).
Matrix or vector of ones.
Vector of sequential integers.
Reorder the columns of a matrix using a vector of ranks.
Randomly draw from the rows of the matrix.
Multivariate normal random draws.
Scale rows or columns of matrix.
Vector containing arithmetic sequence.
Vector containing geometric sequence.
Create a string vector from a list of strings.
Sort elements of data object.
Unstack vector into a matrix.
Unstack vector into lower triangle of sym.
Vector or svector of unique values of object.
Uppercase representation of a string; or upper triangular matrix of a matrix.
Vertically concatenate matrices.
Vectorize (stack columns of) matrix.
Vectorize (stack columns of) lower triangle of matrix.
Matrix or vector of zeros.
Matrix Algebra
Condition number of square matrix or sym.
Determinant of matrix.
Matrix whose columns contain the eigenvectors of a matrix.
LU decomposition of a matrix.
Norm of series or matrix object.
Outer product of vectors or series.
Moore-Penrose pseudo-inverse of matrix.
Singular value decomposition (economy) of matrix.
Singular value decomposition (full) of matrix.
Computes the trace of a square matrix or sym.
Unstack vector into a matrix.
Unstack vector into lower triangle of sym.
Vectorize (stack columns of) matrix.
Vectorize (stack columns of) lower triangle of matrix.
Matrix Statistics
Number of columns in matrix object or group.
Correlation of two vectors or series, or between the columns of a matrix or series in a group.
Covariance (non-d.f.corrected) of two vectors or series, or between the columns of a matrix or series in a group.
Covariance (non-d.f. corrected) of two vectors or series, or between the columns of a matrix or series in a group.
Covariance (d.f. corrected) of two vectors or series, or between the columns of a matrix or series in a group.
The first non-missing value in the vector or series.
Index of the first non-missing value in the vector or series.
Index of the last non-missing value in the vector or series.
Index of maximum value.
Indices of maximum value (multiple).
Index of minimum value.
Indices of minimum value (multiple).
The last non-missing value in the vector or series.
Mean of absolute error (difference) between series.
Mean absolute percentage error (difference) between series.
Maximum values (multiple).
Minimum values (multiple).
Mean of square error (difference) between series.
Number of missing observations.
Norm of series or matrix object.
Number of observations.
Perform an OLS regression on the first column of a matrix versus the remaining columns.
Root of the mean of square error (difference) between series.
Symmetric mean absolute percentage error (difference) between series.
Sample standard deviation (d.f. adjusted).
Population standard deviation (no d.f. adjustment).
Sample standard deviation (d.f. adjusted).
Standardized data (using sample standard deviation).
Standardized data (using population standard deviation).
Arithmetic sum of squares.
Theil inequality coefficient (difference) between series.
Trend coefficient from detrending regression.
Vector or svector of unique values of object.
Population variance (no d.f. adjustment).
Population variance (no d.f. adjustment).
Sample variance (d.f. adjusted).
Matrix Column Statistics
First non-missing value in each column of a matrix.
Index of the first non-missing value in each column of a matrix.
Index of the last non-missing value in each column of a matrix.
Index of the maximal value in each column of a matrix.
Index of the maximal value in each column of a matrix.
Intercept from a trend regression performed on each column of a matrix.
Last non-missing value in each column of the matrix.
Maximal value in each column of a matrix.
Mean in each column of a matrix.
Median of each column of a matrix.
Minimal value for each column of the matrix.
Number of NA values in each column of a matrix.
Number of non-NA values in each column of a matrix.
Product of elements in each column of a matrix.
Sample standard deviation (d.f. corrected) of each column of a matrix.
Population standard deviation (non-d.f. corrected) of each column of a matrix.
Sample standard deviation (non-d.f. corrected) of each column of a matrix.
Sum of the values in each column of a matrix.
Sum of the squared values in each column of a matrix.
Slope from a trend regression on each column of a matrix.
Trimmed mean of each column of a matrix .
Population variance of each column of a matrix.
Population variance of each column of a matrix.
Sample variance of each column of a matrix.
Matrix Element
Element by element division of two matrices.
Element by element equality comparison of two data objects.
Element by element equality comparison of two data objects with NAs treated as ordinary value for comparison.
Element by element tests for whether the elements in the data objects are greater than or equal to corresponding elements in another data object.
Element by element tests for whether the elements in the data object strictly greater than corresponding elements in another data object.
Element by element inverses of a matrix.
Element by element tests for whether the elements in the data object are less than or equal to corresponding elements in another data object.
Element by element tests for whether the elements in the data object are strictly less than corresponding elements in another data object.
Element by element maximums of two conformable data objects.
Element by element minimums of two conformable data objects.
Element by element multiplication of two matrix objects.
Element by element inequality comparison of two data objects.
Element by element inequality comparison of two data objects with NAs treated as ordinary value for comparison.
Raises each element in a matrix to a power.
Element by element recode of data objects.
Matrix Transformation
Overall Transformations
Reorder the rows of the matrix using a vector of ranks.
Compute deviations from the mean of the data object.
Compute deviations from the trend of the data object.
Identifier for the observation within the set of duplicates.
Identifier for the duplicates group for the observation.
Number of observations in the corresponding duplicates group.
Reorder the columns of a matrix using a vector of ranks.
Randomly draw from the rows of the matrix.
By-Column Transformations
Cumulative products for each column of a matrix.
Cumulative sums for each column of a matrix.
Percentile values for each column of a matrix.
Ranks of each column of the matrix.
Sort each column of the matrix.
Standardize each column using the sample (d.f. corrected) standard deviation.
Standardize each column using the population (non-d.f. corrected) standard deviation.
By-Row Transformations
Matrix where each row contains ranks of the column values.
Matrix where each row contains sorted columns. | {"url":"https://help.eviews.com/content/matrixref-Matrix_Function_Summary.html","timestamp":"2024-11-06T12:08:20Z","content_type":"application/xhtml+xml","content_length":"52657","record_id":"<urn:uuid:2f50bb03-9cb9-4101-8c58-f93379848fd7>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00479.warc.gz"} |
Displayed Output
• If you specify the SCOROUT option in the TABLES statement, PROC FREQ displays the Row Scores and Column Scores that it uses for statistical computations. The Row Scores table displays the row
variable values and the Score corresponding to each value. The Column Scores table displays the column variable values and the corresponding Scores. PROC FREQ also identifies the score type used
to compute the row and column scores. You can specify the score type with the SCORES= option in the TABLES statement.
• If you specify the CHISQ option, PROC FREQ displays the following statistics for each two-way table: Pearson Chi-Square, Likelihood Ratio Chi-Square, Continuity-Adjusted Chi-Square (for tables),
Mantel-Haenszel Chi-Square, the Phi Coefficient, the Contingency Coefficient, and Cramér’s V. For each test statistic, PROC FREQ also displays the degrees of freedom (DF) and the probability
value (Prob).
• If you specify the CHISQ option for tables, PROC FREQ also displays Fisher’s exact test. The test output includes the cell (1,1) frequency (F), the exact left-sided and right-sided probability
values, the table probability (P), and the exact two-sided probability value.
• If you specify the FISHER option in the TABLES statement (or, equivalently, the FISHER option in the EXACT statement), PROC FREQ displays Fisher’s exact test for tables larger than . The test
output includes the table probability (P) and the probability value. In addition, PROC FREQ displays the CHISQ output listed earlier, even if you do not also specify the CHISQ option.
• If you specify the PCHI, LRCHI, or MHCHI option in the EXACT statement, PROC FREQ displays the corresponding exact test: Pearson Chi-Square, Likelihood Ratio Chi-Square, or Mantel-Haenszel
Chi-Square, respectively. The test output includes the test statistic, the degrees of freedom (DF), and the asymptotic and exact probability values. If you also specify the POINT option in the
EXACT statement, PROC FREQ displays the point probability for each exact test requested. If you specify the CHISQ option in the EXACT statement, PROC FREQ displays exact probability values for
all three of these chi-square tests.
• If you specify the MEASURES option, PROC FREQ displays the following statistics and their asymptotic standard errors (ASE) for each two-way table: Gamma, Kendall’s Tau-b, Stuart’s Tau-c, Somers’
, Somers’ , Pearson Correlation, Spearman Correlation, Lambda Asymmetric , Lambda Asymmetric , Lambda Symmetric, Uncertainty Coefficient , Uncertainty Coefficient , and Uncertainty Coefficient
Symmetric. If you specify the CL option, PROC FREQ also displays confidence limits for these measures.
• If you specify the PLCORR option, PROC FREQ displays the tetrachoric correlation for tables or the polychoric correlation for larger tables. In addition, PROC FREQ displays the MEASURES output
listed earlier, even if you do not also specify the MEASURES option.
• If you specify the GAMMA, KENTB, STUTC, SMDCR, SMDRC, PCORR, or SCORR option in the TEST statement, PROC FREQ displays asymptotic tests for Gamma, Kendall’s Tau-b, Stuart’s Tau-c, Somers’ ,
Somers’ , the Pearson Correlation, or the Spearman Correlation, respectively. If you specify the MEASURES option in the TEST statement, PROC FREQ displays all these asymptotic tests. The test
output includes the statistic, its asymptotic standard error (ASE), Confidence Limits, the ASE under the null hypothesis H0, the standardized test statistic (Z), and the one-sided and two-sided
probability values.
• If you specify the KENTB, STUTC, SMDCR, SMDRC, PCORR, or SCORR option in the EXACT statement, PROC FREQ displays asymptotic and exact tests for the corresponding measure of association: Kendall’s
Tau-b, Stuart’s Tau-c, Somers’ , Somers’ , the Pearson Correlation, or the Spearman correlation, respectively. The test output includes the correlation, its asymptotic standard error (ASE),
Confidence Limits, the ASE under the null hypothesis H0, the standardized test statistic (Z), and the asymptotic and exact one-sided and two-sided probability values. If you also specify the
POINT option in the EXACT statement, PROC FREQ displays the point probability for each exact test requested.
• If you specify the RISKDIFF option for tables, PROC FREQ displays the Column 1 and Column 2 Risk Estimates. For each column, PROC FREQ displays the Row 1 Risk, Row 2 Risk, Total Risk, and Risk
Difference, together with their asymptotic standard errors (ASE) and Asymptotic Confidence Limits. PROC FREQ also displays Exact Confidence Limits for the Row 1 Risk, Row 2 Risk, and Total Risk.
If you specify the RISKDIFF option in the EXACT statement, PROC FREQ provides unconditional Exact Confidence Limits for the Risk Difference.
• If you specify the RISKDIFF(CL=) option for tables, PROC FREQ displays the Proportion Difference Confidence Limits. For each confidence limit Type that you request (Exact, Farrington-Manning,
Hauck-Anderson, Newcombe Score, or Wald), PROC FREQ displays the Lower and Upper Confidence Limits.
• If you request a noninferiority or superiority test for the proportion difference (RISKDIFF) by specifying the NONINF or SUP riskdiff-option, and if you specify METHOD=HA (Hauck-Anderson), METHOD
=FM (Farrington-Manning), or METHOD=WALD (Wald), PROC FREQ displays the following information: the Proportion Difference, the test ASE (H0, Sample, Sample H-A, or FM, depending on the method you
specify), the test statistic Z, the probability value, the Noninferiority or Superiority Limit, and the test-based Confidence Limits. If you specify METHOD=NEWCOMBE (Newcombe score), PROC FREQ
displays the Proportion Difference, the Noninferiority or Superiority Limit, and the Newcombe Confidence Limits.
• If you request an equivalence test for the proportion difference (RISKDIFF) by specifying the EQUIV riskdiff-option, and if you specify METHOD=HA (Hauck-Anderson), METHOD=FM (Farrington-Manning),
or METHOD=WALD (Wald), PROC FREQ displays the following information: the Proportion Difference and the test ASE (H0, Sample, Sample H-A, or FM, depending on the method you specify). PROC FREQ
displays a two one-sided test (TOST) for equivalence, which includes test statistics (Z) and probability values for the Lower and Upper tests, together with the Overall probability value. PROC
FREQ also displays the Equivalence Limits and the test-based Confidence Limits. If you specify METHOD=NEWCOMBE (Newcombe), PROC FREQ displays the Proportion Difference, the Equivalence Limits,
and the score Confidence Limits.
• If you request an equality test for the proportion difference (RISKDIFF) by specifying the EQUAL riskdiff-option, PROC FREQ displays the following information: the Proportion Difference and the
test ASE (H0 or Sample), the test statistic Z, the One-Sided probability value (Pr > Z or Pr < Z), and the Two-Sided probability value, Pr > |Z|.
• If you specify the MEASURES option or the RELRISK option for tables, PROC FREQ displays Estimates of the Relative Risk for Case-Control and Cohort studies, together with their Confidence Limits.
These measures are also known as the Odds Ratio and the Column 1 and 2 Relative Risks. If you specify the OR option in the EXACT statement, PROC FREQ also displays Exact Confidence Limits for the
Odds Ratio. If you specify the RELRISK option in the EXACT statement, PROC FREQ displays unconditional Exact Confidence Limits for the Relative Risk.
• If you specify the TREND option, PROC FREQ displays the Cochran-Armitage Trend Test for tables that are or . For this test, PROC FREQ gives the Statistic (Z) and the one-sided and two-sided
probability values. If you specify the TREND option in the EXACT statement, PROC FREQ also displays the exact one-sided and two-sided probability values for this test. If you specify the POINT
option with the TREND option in the EXACT statement, PROC FREQ displays the exact point probability for the test statistic.
• If you specify the JT option, PROC FREQ displays the Jonckheere-Terpstra Test, showing the Statistic (JT), the standardized test statistic (Z), and the one-sided and two-sided probability values.
If you specify the JT option in the EXACT statement, PROC FREQ also displays the exact one-sided and two-sided probability values for this test. If you specify the POINT option with the JT option
in the EXACT statement, PROC FREQ displays the exact point probability for the test statistic.
• If you specify the AGREE option and the PRINTKWT option, PROC FREQ displays the Kappa Coefficient Weights for square tables larger than .
• If you specify the AGREE option, for two-way tables PROC FREQ displays McNemar’s Test and the Simple Kappa Coefficient for tables. For square tables larger than , PROC FREQ displays Bowker’s Test
of Symmetry, the Simple Kappa Coefficient, and the Weighted Kappa Coefficient. For McNemar’s Test and Bowker’s Test of Symmetry, PROC FREQ displays the Statistic (S), the degrees of freedom (DF),
and the probability value (Pr > S). If you specify the MCNEM option in the EXACT statement, PROC FREQ also displays the exact probability value for McNemar’s test. If you specify the POINT option
with the MCNEM option in the EXACT statement, PROC FREQ displays the exact point probability for the test statistic. For the simple and weighted kappa coefficients, PROC FREQ displays the kappa
values, asymptotic standard errors (ASE), and Confidence Limits.
• If you specify the KAPPA or WTKAP option in the TEST statement, PROC FREQ displays asymptotic tests for the simple kappa coefficient or the weighted kappa coefficient, respectively. If you
specify the AGREE option in the TEST statement, PROC FREQ displays both these asymptotic tests. The test output includes the kappa coefficient, its asymptotic standard error (ASE), Confidence
Limits, the ASE under the null hypothesis H0, the standardized test statistic (Z), and the one-sided and two-sided probability values.
• If you specify the KAPPA or WTKAP option in the EXACT statement, PROC FREQ displays asymptotic and exact tests for the simple kappa coefficient or the weighted kappa coefficient, respectively.
The test output includes the kappa coefficient, its asymptotic standard error (ASE), Confidence Limits, the ASE under the null hypothesis H0, the standardized test statistic (Z), and the
asymptotic and exact one-sided and two-sided probability values. If you specify the POINT option in the EXACT statement, PROC FREQ displays the point probability for each exact test requested.
• If you specify the MC option in the EXACT statement, PROC FREQ displays Monte Carlo estimates for all exact p-values requested by statistic-options in the EXACT statement. The Monte Carlo output
includes the p-value Estimate, its Confidence Limits, the Number of Samples used to compute the Monte Carlo estimate, and the Initial Seed for random number generation.
• If you specify the AGREE option, for multiple strata PROC FREQ displays Overall Simple and Weighted Kappa Coefficients, with their asymptotic standard errors (ASE) and Confidence Limits. PROC
FREQ also displays Tests for Equal Kappa Coefficients, giving the Chi-Squares, degrees of freedom (DF), and probability values (Pr > ChiSq) for the Simple Kappa and Weighted Kappa. For multiple
strata of tables, PROC FREQ displays Cochran’s Q, giving the Statistic (Q), the degrees of freedom (DF), and the probability value (Pr > Q).
• If you specify the CMH option, PROC FREQ displays Cochran-Mantel-Haenszel Statistics for the following three alternative hypotheses: Nonzero Correlation, Row Mean Scores Differ (ANOVA Statistic),
and General Association. For each of these statistics, PROC FREQ gives the degrees of freedom (DF) and the probability value (Prob). If you specify the MANTELFLEISS option, PROC FREQ displays the
Mantel-Fleiss Criterion for tables. For tables, PROC FREQ also displays Estimates of the Common Relative Risk for Case-Control and Cohort studies, together with their confidence limits. These
include both Mantel-Haenszel and Logit stratum-adjusted estimates of the common Odds Ratio, Column 1 Relative Risk, and Column 2 Relative Risk. Also for tables, PROC FREQ displays the Breslow-Day
Test for Homogeneity of the Odds Ratios. For this test, PROC FREQ gives the Chi-Square, the degrees of freedom (DF), and the probability value (Pr > ChiSq).
• If you specify the CMH option in the TABLES statement and also specify the COMOR option in the EXACT statement, PROC FREQ displays exact confidence limits for the Common Odds Ratio for multiple
strata of tables. PROC FREQ also displays the Exact Test of H0: Common Odds Ratio = 1. The test output includes the Cell (1,1) Sum (S), Mean of S Under H0, One-sided Pr <= S, and Point Pr = S.
PROC FREQ also provides exact two-sided probability values for the test, computed according to the following three methods: 2 * One-sided, Sum of probabilities <= Point probability, and Pr >= |S
- Mean|.
• If you specify the CMH option in the TABLES statement and also specify the EQOR option in the EXACT statement, PROC FREQ computes Zelen’s exact test for equal odds ratios for tables. PROC FREQ
displays Zelen’s test along with the asymptotic Breslow-Day test produced by the CMH option. PROC FREQ displays the test statistic, Zelen’s Exact Test (P), and the probability value, Exact Pr <=
• If you specify the GAILSIMON option in the TABLES statement for a multiway tables, PROC FREQ displays the Gail-Simon test for qualitative interactions. The display include the following
statistics and their p-values: Q+ (Positive Risk Differences), Q- (Negative Risk Differences), and Q (Two-Sided). | {"url":"http://support.sas.com/documentation/cdl/en/procstat/65543/HTML/default/procstat_freq_details97.htm","timestamp":"2024-11-02T12:38:55Z","content_type":"application/xhtml+xml","content_length":"66042","record_id":"<urn:uuid:871d9324-54c0-4043-9a62-f9dc65217d2a>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00724.warc.gz"} |
A randomized sorting algorithm on the BSP model
An oversampling-based randomized algorithm is introduced for sorting n keys of any abstract data type on the p processors of a latency-tolerant parallel system such as a bulk-synchronous parallel
computer. The algorithm is asymptotically, for large n, optimal in computation and communication compared to the best available sequential sorting algorithm, even when constant factors are taken into
consideration. Its parallel time is within a (1 + o(1))/p multiplicative factor of the corresponding sequential method for sorting, improving upon other approaches for a wider range of values of p
relative to n. It also improves upon other randomized sorting algorithms that have been developed for latency-tolerant models of computation on the amount of parallel slack (ratio n over p) required
to achieve optimal speedup and also, on the associated failure probability. For values of p closer to n than other latency-tolerant randomized approaches, it can be turned into a PRAM-like algorithm
but for such cases a speedup of O(p) rather than p/(1 + o(1)) is then achievable. Although the general framework of the proposed algorithm relies on the well-established prior idea of sample-based
sorting, there are some novel features in its design such as the choice of splitter size, the way keys are split, and the handling of the base case of the recursive sorting algorithm that contribute
to its performance.
All Science Journal Classification (ASJC) codes
• Discrete Mathematics and Combinatorics
• BSP model
• Randomized sorting
• latency-tolerant algorithms
• oversampling
• random sampling
Dive into the research topics of 'A randomized sorting algorithm on the BSP model'. Together they form a unique fingerprint. | {"url":"https://researchwith.njit.edu/en/publications/a-randomized-sorting-algorithm-on-the-bsp-model","timestamp":"2024-11-13T06:10:43Z","content_type":"text/html","content_length":"50182","record_id":"<urn:uuid:ac1fa218-9985-436d-9b7d-91ea05d7b59e>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00305.warc.gz"} |
Introduction to Data Science
What is data science? #
The term data refers to any collection of observations that measure something of interest, or that convey information about a question at hand. This is a data science course, and also a statistics
course. For our purposes, the terms “data science” and “statistics” are essentially synonyms, referring to the methodology used to learn from data. Mathematics and computer science are also important
components of data science.
Nearly every branch of science involves collecting and analyzing data, but a “domain scientist” such as a biologist or a sociologist is primarily interested in the core questions of their domain, not
in the methods used to analyze data. Data scientists do analyze data, but even more importantly, data scientists analyze the methods for analyzing data. This is what distinguishes a data scientist,
or a statistician, from a scientist studying and analyzing a data set that arises in their domain of research.
Good data science, like good statistics, starts with a question. For example, in a business setting, we may have questions about what type of person is most likely to buy a product, or whether people
would be willing to pay more for a product that has premium features. In natural and social science, questions are often expressed in the form of a hypothesis. For example, in a medical research
setting we may have a hypothesis that “people who sleep less than six hours per night tend to have higher blood pressure than people who sleep more than seven hours per night”. When we express such a
hypothesis, we must be open to the possibility that the hypothesis is either true or false. Upon systematically collecting relevant data, we will accumulate evidence that informs us about the truth
of our hypothesis.
Data science is part of an empirical approach to answering research questions, meaning that we make progress by observing, taking measurements, and collecting and interpreting data. In contrast, a
first principles approach to research aims to answer questions using logical deduction and theory. Logical deduction and theory do play important roles in data science, but in data science we prefer
as much as possible to “let the data speak for itself”.
Data science and statistics are “methodological” subjects, meaning that they focus on developing methods, tools, and approaches for conducting empirical investigations. A primary aim of data science
is to develop an understanding of the strengths and limitations of various methods for analyzing data. Thus, data science is to some extent a “meta subject” which focuses on the merits of different
approaches for learning about reality.
There is a very active theoretical branch of data science that deals with “pure” questions about data analysis that exist outside the context of any specific application. However this course will
primarily develop the tools of statistics and data science through case studies that are set in various application domains. There is also a more abstract dimension to this course, because we will
see that statistical tools often have properties that hold regardless of the specific type of data or application context in which the tool is applied.
Uncertainty in data analysis #
Statistical data analysis is based on the idea that the data we collect in order to address our questions of interest can never be sufficient to povide definitive answers. There will always be
uncertainty in our findings. The goal of a statistical data analysis is to obtain the strongest conclusions that can legitimately be made from the available data, and then quantify the uncertainty in
these findings.
Historically, it has been challenging to formalize exactly what we mean by “uncertainty”. A major advance occurred in the late 1800’s, when probability theory matured as a branch of mathematics.
Probability theory turns out to be a very useful tool for defining and quantifying what we mean by “uncertainty”. In spite of decades of progress, there remain many unresolved challenges in
statistical data analysis. New methods and approaches to analyzing data continue to be developed, and the strengths and limitations of existing methods continue to be examined. Statistics and data
science are dynamic fields, and there is ongoing active discussion and healthy debate as to which approaches to data analysis are most appropriate in various settings.
Samples and populations #
The most prototypical setting for a statistical analysis is when our data constitute a representative or random sample from a population of interest. We will discuss these terms in much more detail
later in the course. For now, we will introduce the main ideas using an example. Suppose that our research goal is to estimate the fraction of adults in the state of Michigan who travel more than 20
miles to work each day. Imagine that we could obtain a representative sample of 1,000 adults in Michigan (which has an adult population of roughly 7.5 million, so our sample contains less than one in
seven thousand of the population). If 274 of the people in our sample travel more than 20 miles to work each day, then we would estimate that 274/1000 = 27.4% of the Michigan population travels more
than 20 miles to work each day.
The true proportion of Michigan adults who travel more than 20 miles to work each day is very unlikely to be exactly equal to 27.4% (i.e., it is very unlikely that exactly 2.055 million Michigan
adults travel more than 20 miles to work each day). Although the true proportion may be quite close to this value, it is very unlikely to be exactly equal to it. The goal of uncertainty
quantification is to state how different the estimated proportion obtained from the sample of data that we have collected (27.4%) is from the exact, true proportion (which is unknown).
It turns out that as long as we know some key pieces of information about how our sample was obtained, then it is possible to make precise and useful statements about the likely error in our estimate
relative to the truth. On the other hand, if we know very little about how our sample was obtained, it can be very difficult to say anything about such errors. This is a very common theme in
statistics and data science – it is very important to understand how the data being analyzed were collected, otherwise we will be very limited in the types of claims that we can make.
More challenging population settings #
A representative sample from a finite population, as described above, is perhaps the simplest setting in which to conduct a data analysis. Unfortunately, our data and population are usually more
challenging to work with.
One such example is “time series” data from a “dynamical system”. Such a system, can be continuously changing, so that the system we observe today differs fundamentally from the system that we
observed in the past. Consider, for example, research on the Earth’s climate. We can collect data such as temperature, ice cover, and carbon dioxide levels – but the relationships among these
variables may appear to drift over time. It’s not that the laws of nature are changing, but rather there are almost always additional relevant factors beyond what we have measured. As these
unobserved quantities change, the relationships among the observable variables may change as well.
Temporal systems with such dynamic behavior arise in many different fields of research. For example, in economics and public policy, there is great interest in the relationship between public debt,
unemployment, and inflation. In the past, it was consistently observed that greater government spending was associated with greater public debt, lower unemployment, wage growth, and price inflation.
But in recent years, many regions of the world have simultaneously experienced low unemployment, high public debt, and low wage growth, but also low inflation.
A system that is in a constant state of structural change is said to be “nonstationary”. Standard methods for analyzing data from other settings may not give meaningful results here. It is often
possible to carry out meaningful empirical research by analyzing data obtained from such systems, but it is very important to be aware that your data analysis is being conducted in such a setting,
and to make use of methods that are appropriate for it.
Causality #
The most interesting scientific findings are usually those that identify causes. For example, a researcher may have a hypothesis that among those with COVID, people who are overweight are more likely
to become severely ill. This hypothesis, while interesting, reflects a “predictive” relationship, not necessarily one that is causal or “mechanistic”. For example, it could be that the cause of
severe illness in COVID patients is insulin resistance, and overweight people just tend to have insulin resistance (but a non-overweight COVID patient with insulin resistance would be just as likely
to become severely ill).
Causal statements are usually much more interesting than statements that are not causal. But causal statements are more difficult to demonstrate. One of the major challenges in data analysis is to
identify the situations in which causal conclusions may be drawn. Just as importantly, we should aim to determine when this cannot be done and communicate to our audience that a causal conclusion is
not justified. | {"url":"https://dept.stat.lsa.umich.edu/~kshedden/introds/topics/what_is_statistics/","timestamp":"2024-11-07T19:41:16Z","content_type":"text/html","content_length":"17068","record_id":"<urn:uuid:878481f6-a4cb-4370-81fd-49dc6fce64f7>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00564.warc.gz"} |
Value at Risk Calculations for Market Risk Management
A simple way to calculate Value at Risk is to use market data of the last 250 days. Then, each risk factor is compared to the current market value and the results are used to present 250 different
scenarios for the future value. Then, a portfolio is valued using a full, non-linear pricing model. The third worst day of the previous 90 days is assumed to be 99% VaR. A more complex method is
known as the parametric method. This approach assumes a normal distribution and requires estimates of expected return and standard deviation of returns.
Value at risk is a statistic that helps calculate financial risk. It offers an estimate of how much a portfolio can lose at any given time. The most common method is the daily value at risk. This
measure is based on 95% confidence levels. This means that actual losses are expected to exceed the value at risk by more than thirteen days. Therefore, if the value-at-risk calculation fails to meet
the ninety percent confidence level, it’s not a reliable indicator of risk.
There are many value-at-risk calculation techniques available, each with different advantages and disadvantages. Some are more accurate than others, and some are worse than others. There are a number
of common methods that have been proven to be effective. But which one is the best for you? There are several factors that determine the VaR calculation. Once you understand the basics, you can use
the formula to make an informed decision about your investments. It is important to remember that this is a general guide and not a specific financial advisor.
While the methodology of value-at-risk is widely used, some risk management practitioners are skeptical of its effectiveness. In fact, some experts believe that VaR may not be a suitable substitute
for a comprehensive risk management model. For this reason, they recommend relying on a diversified portfolio, avoiding high-risk stocks and investing in low-risk assets. It is not a fool-proof risk
management strategy.
Another method of value-at-risk is a statistical technique called backtesting. This process uses historical data to assess the performance of a particular strategy. The backtesting method is a simple
way to check the accuracy of the value-at-risk calculation. This can also help you to find the best values-at-risk calculators. So, let’s get started. Using Valuation at Risk
A valuation at risk model can also be applied to individual stocks. For example, if $100,000 is invested in a stock at 95% VaR, it is a risk of 5% that there will be a 5% loss. Using a VaR model will
help you to avoid over-trading. If it does, then you should use the risk-reward ratio to assess the value at risk of a particular stock. | {"url":"https://highmark-funds.com/2021/12/23/value-at-risk-calculations-for-market-risk-management/","timestamp":"2024-11-02T04:14:25Z","content_type":"text/html","content_length":"35144","record_id":"<urn:uuid:525493ff-72a0-4af6-b3d1-84d3707742dc>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00466.warc.gz"} |
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Five Ways to Detect Outliers/Anomalies That Every Data Scientist Should Know (Python Code)
In data science, distinguishing the exceptions or inconsistencies is fundamental as they can broadly affect the outcomes of your data study. Data centers, commonly known as outliers, are those that
considerably deviated from other perceptions. These perceptions may be the result of estimation inconstancy, test errors, or anomalous occasions. Applications for irregularity discovery are various
and incorporate quality control, extortion detection, and network security. Utilizing tests of Python code, we will look at some of strategies or methods for recognizing anomalies or outliers in this
Different Methods for Outliers/Anomalies Detection for Data Scientists
In the following section, we will discuss the different ways for detecting the outliers or anomalies that are commonly used by Data Scientists.
Some of these methods are as follows:
1. Z-Score
2. IQR (Interquartile Range)
3. DBSCAN (Density-Based Spatial Clustering of Application with Noise)
4. Isolation Forest
5. LOF (Local Outlier Factor)
We will now understand these five methods with the help of the examples using Python Programming Language.
Understanding the Z-Score Method
The Z-Score method is a fundamental detecting method for the calculation of the number of standard deviations a data point is from the mean. On the off chance that an information point's Z-Score
surpasses a specific limit (ordinarily 3 or -3), it is named an outlier.
The Z-score is determined as follows:
Outliers using Z-Score method: (array([11], dtype=int64),)
Understanding the IQR (Interquartile Range) Method
Interquartile Range, abbreviated as IQR, is a non-parametric methodology used to scattering of the middle 50% of the information to find anomalies or outliers. Special cases are characterized as data
centers that deviate by 1.5 times the IQR from the essential quartile (Q1) or the third quartile (Q3).
The IQR is calculated as:
Outliers are detected using:
• Lower Bound = Q[1] − 1.5 × IQR
• Upper Bound = Q[3] + 1.5 × IQR
Outliers using IQR method: [100]
Understanding the DBSCAN (Density-Based Spatial Clustering of Applications with Noise) Method
Density-Based Spatial Clustering of Applications with Noise, abbreviated as DBSCAN, is a clustering method used to classify the information finding the anomalies or outliers when they are assembled
in thickly pressed regions and when a point is separated in a low-density zone.
Two parameters are required for DBSCAN:
1. min_samples: The min_samples parameter is the number of tests in a neighborhood that must frame a cluster.
2. eps: The eps parameter is the greatest remove between two tests to be considered neighbors.
Exceptions are characterized as focuses that have a place to none of the clusters.
Outliers using DBSCAN method: [[ 27]
Understanding the Isolation Forest Method
The Isolation Forest algorithm uses an approach in order to partition the observations to arbitrarily select, include and isolate the anomalies or outliers into most extreme and least values. It
makes sense that inconsistencies are few and distinct, which makes it less difficult to separate them.
Division is arranged to make trees with shorter way lengths, forests plant trees in areas where inconsistencies are disconnected closer to the tree's base.
Outliers using Isolation Forest method: [[ 27]
The LOF (Local Outlier Factor) Method
The Local Outlier Factor, abbreviated as LOF, is the method that is used to measure the local density deviation of a given information point in connection to its neighbors. Exceptions are
characterized as focuses that have a thickness that's recognizably lower than that of their neighbors.
The LOF method is used to arrange and recognize locales with comparable densities and pinpoint areas with altogether lower densities than their neighbors, LOF compares each point's nearby thickness
to that of its neighbors.
Outliers using LOF method: [[ 27]
Within the data preprocessing organize, recognizing exceptions is basic since they have the potential to distort explanatory discoveries and impede model execution. Z-Score, IQR, DBSCAN, Isolation
Forest, and LOF are the five principal methods for outlier discovery that we inspected in this article. Each approach has its focal points and works well with different sorts of information and
applications. Data scientists can ensure the precision and consistency of their data analysis by comprehending and putting these techniques into practice.
You will be well-prepared to recognize and oversee exceptions in your datasets with these methods in your tool compartment, which can result in more solid and precise models.
← prev next → | {"url":"https://www.javatpoint.com/five-ways-to-detect-outliers-anomalies-that-every-data-scientist-should-know-python-code","timestamp":"2024-11-12T19:28:48Z","content_type":"text/html","content_length":"60434","record_id":"<urn:uuid:bc5bb607-47b2-45db-b061-42c5e17a16be>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00797.warc.gz"} |
Introduction to Chemical Engineering Processes/Converting Information into Mass Flows - Wikibooks, open books for an open world
In any system there will be certain parameters that are easier (often considerably) to measure and/or control than others. When you are solving any problem and trying to use a mass balance or any
other equation, it is important to recognize what pieces of information can be interconverted. The purpose of this section is to show some of the more common alternative ways that mass flow rates are
expressed, mostly because it is easier to, for example, measure a velocity than it is to measure a mass flow rate directly.
A volumetric flow rate is a relation of how much volume of a gas or liquid solution passes through a fixed point in a system (typically the entrance or exit point of a process) in a given amount of
time. It is denoted as:
${\displaystyle {\dot {V}}_{n}{\dot {=}}{\frac {Volume}{time}}}$ in stream n
Volumetric flow rates can be measured directly using flow meters. They are especially useful for gases since the volume of a gas is one of the four properties that are needed in order to use an
equation of state (discussed later in the book) to calculate the molar flow rate. Of the other three, two (pressure, and temperature) can be specified by the reactor design and control systems, while
one (compressibility) is strictly a function of temperature and pressure for any gas or gaseous mixture.
Volumetric Flowrates are Not Conserved. We can write a balance on volume like anything else, but the "volume generation" term would be a complex function of system properties. Therefore if we are
given a volumetric flow rate we should change it into a mass (or mole) flow rate before applying the balance equations.
Volumetric flowrates also do not lend themselves to splitting into components, since when we speak of volumes in practical terms we generally think of the total solution volume, not the partial
volume of each component (the latter is a useful tool for thermodynamics, but that's another course entirely). There are some things that are measured in volume fractions, but this is relatively
How to convert volumetric flow rates to mass flow rates
[edit | edit source]
Volumetric flowrates are related to mass flow rates by a relatively easy-to-measure physical property. Since ${\displaystyle {\dot {m}}{\dot {=}}mass/time}$ and ${\displaystyle {\dot {V}}{\dot {=}}
volume/time}$, we need a property with units of ${\displaystyle mass/volume}$ in order to convert them. The density serves this purpose nicely!
${\displaystyle {\dot {V}}_{n}*{\rho }_{i}={\dot {m}}_{n}}$ in stream n
The "i" indicates that we're talking about one particular flow stream here, since each flow may have a different density, mass flow rate, or volumetric flow rate.
The velocity of a bulk fluid is how much lateral distance along the system (usually a pipe) it passes per unit time. The velocity of a bulk fluid, like any other, has units of:
${\displaystyle v_{n}={\frac {distance}{time}}}$ in stream n
By definition, the bulk velocity of a fluid is related to the volumetric flow rate by:
${\displaystyle {v}_{n}={\frac {{\dot {V}}_{n}}{A_{n}}}}$ in stream n
This distinguishes it from the velocity of the fluid at a certain point (since fluids flow faster in the center of a pipe). The bulk velocity is about the same as the instantaneous velocity for
relatively fast flow, or especially for flow of gasses.
For purposes of this class, all velocities given will be bulk velocities, not instantaneous velocities.
(Bulk) Velocities are useful because, like volumetric flow rates, they are relatively easy to measure. They are especially useful for liquids since they have constant density (and therefore a
constant pressure drop at steady state) as they pass through the orifice or other similar instruments. This is a necessary prerequisite to use the design equations for these instruments.
Like volumetric flowrates, velocity is not conserved. Like volumetric flowrate, velocity changes with temperature and pressure of a gas, though for a liquid, velocity is generally constant along the
length of a pipe with constant cross-sectional area.
Also, velocities can't be split into the flows of individual components, since all of the components will generally flow at the same speed. They need to be converted into something that can be split
(mass flow rate, molar flow rate, or pressure for a gas) before concentrations can be applied.
In order to convert the velocity of a fluid stream into a mass flow rate, you need two pieces of information:
1. The cross sectional area of the pipe.
2. The density of the fluid.
In order to convert, first use the definition of bulk velocity to convert it into a volumetric flow rate:
${\displaystyle {\dot {V}}_{n}=v_{n}*A_{n}}$
Then use the density to convert the volumetric flow rate into a mass flow rate.
${\displaystyle {\dot {m}}_{n}={\dot {V}}_{n}*{\rho }_{n}}$
The combination of these two equations is useful:
${\displaystyle {\dot {m}}_{n}=v_{n}*{\rho }_{n}*A_{n}}$ in stream n
The concept of a molar flow rate is similar to that of a mass flow rate, it is the number of moles of a solution (or mixture) that pass a fixed point per unit time:
${\displaystyle {\dot {n}}_{n}{\dot {=}}{\frac {moles}{time}}}$ in stream n
Molar flow rates are mostly useful because using moles instead of mass allows you to write material balances in terms of reaction conversion and stoichiometry. In other words, there are a lot fewer
unknowns when you use a mole balance, since the stoichiometry allows you to consolidate all of the changes in the reactant and product concentrations in terms of one variable.
Unlike mass, total moles are not conserved. Total mass flow rate is conserved whether there is a reaction or not, but the same is not true for the number of moles. For example, consider the reaction
between hydrogen and oxygen gasses to form water:
${\displaystyle H_{2}+{\frac {1}{2}}O_{2}\rightarrow H_{2}O}$
This reaction consumes 1.5 moles of reactants for every mole of products produced, and therefore the total number of moles entering the reactor will be more than the number leaving it.
However, since neither mass nor moles of individual components is conserved in a reacting system, it's better to use moles so that the stoichiometry can be exploited, as described later.
The molar flows are also somewhat less practical than mass flow rates, since you can't measure moles directly but you can measure the mass of something, and then convert it to moles using the molar
flow rate.
Molar flow rates and mass flow rates are related by the molecular weight (also known as the molar mass) of the solution. In order to convert the mass and molar flow rates of the entire solution, we
need to know the average molecular weight of the solution. This can be calculated from the molecular weights and mole fractions of the components using the formula:
${\displaystyle {\bar {MW}}_{n}=[\Sigma ({MW}_{i}*y_{i})]_{n}}$
where i is an index of components and n is the stream number. ${\displaystyle y_{i}}$ signifies mole fraction of each component (this will all be defined and derived later).
Once this is known it can be used as you would use a molar mass for a single component to find the total molar flow rate.
${\displaystyle {\dot {m}}_{n}={\dot {n}}_{n}*{\bar {MW}}_{n}}$ in stream n | {"url":"https://en.wikibooks.org/wiki/Introduction_to_Chemical_Engineering_Processes/Converting_Information_into_Mass_Flows","timestamp":"2024-11-10T08:26:06Z","content_type":"text/html","content_length":"95731","record_id":"<urn:uuid:cc1e0420-1836-465a-b5b3-3dab2cac371c>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00407.warc.gz"} |
Why are boys usually taller than girls?
Boys and girls are actually about the same height on average, until they reach the age of 12 or 13. After that age, girls growth starts to level off. Boys continue growing until age 17 or 18.
Boys and girls grow about two inches per year until they reach puberty. Then they each have a growth spurt that lasts a year or two. During the growth spurts, boys grow a little faster than girls.
Girls reach puberty (on average) about a year before boys do. Girls also end puberty earlier than boys.
The differences in when they reach puberty, and how fast they grow during it, and how long puberty lasts, are what make boys average about 5 inches taller than girls by the time they stop growing.
To estimate how tall you will be when you are fully grown, you can use the knowledge that men are on average 5 inches taller than women. If you are a girl, subtract 5 inches from your father’s
height. Then add your mother’s height, and then divide by two. This will give you the average height your parents would have if they were both women.
If you are a boy, add 5 inches to your mother’s height, then add your father’s height, and divide by two. This gives you the average height your parents would be if they were both men.
The problem with these estimates is that they will only let you guess your height within about 4 inches. That is a lot of variation.
There is another way to estimate your adult height that indicates just how unreliable height estimation is. Remember that boys and girls grow at about the same rate until puberty, and that there is
about a 5 inch difference on average between men and women’s height. If you know how tall you were when you were two years old, you can double that to estimate your adult height.
Notice that this method does not ask whether you are a boy or a girl. So it may easily be 5 inches off. | {"url":"http://questions.scitoys.com/node/42","timestamp":"2024-11-07T10:33:02Z","content_type":"text/html","content_length":"19549","record_id":"<urn:uuid:b8621f2f-a6f0-4968-891f-4410b2f5b64f>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00307.warc.gz"} |
index match formula in excel
In this article, we will learn how to Auto populate table from another table using the INDEX & MATCH function in Excel. VLOOKUP and INDEX-MATCH formulas are among the most powerful functions in
Excel. There it is - the MATCH() function tells the Index function which row to look in - you are done. The formula is an advanced version of the iconic INDEX MATCH that returns a match based on a
single criterion. For more information on array formulas, see Guidelines and examples of array formulas . The syntax of the INDEX() function is: = INDEX(array, row_num,[column_num]) array: This is
the area where the answer is. Here’s how that simple INDEX / MATCH formula finds the sweater price: the MATCH function can find “Sweater” in the range B2:B4. Scenario: For instance, We need to find
the EXACT match from the table without looking it up on the table. ; The MATCH function matches the Subject value in J5 cell with the column header array and returns its position 4 as a number. Excel
provides several of these, including the most awesome combination of functions of all time: MATCH + INDEX. the INDEX function can tell you … The result is 1, because “Sweater” is in the first row of
that range. INDEX and MATCH Functions Together. To evaluate multiple criteria, we use the multiplication operation that works as the AND operator in array formulas.Below, you will find … (Here is a
link to a guide to using INDEX() and MATCH() functions.) Problem: There is an inconsistency in the match type and the sorting order of the data Explanation: The MATCH function matches the Student
value in J4 cell with the row header array and returns its position 3 as a number. Take the Index function, replace our question mark with the MATCH function, and you can now do the equivalent of
VLOOKUPs when the key field is not in the left column. In Excel, we call this the lookup value. ; The INDEX function takes the row and column index number and looks up in the table data and returns
the matched value. Here is the function to use: Any lookup function – including a “normal” MATCH INDEX formula – needs to look for a unique piece of information. Since these MATCH positions are fed
into the INDEX function, it returns the score based on the student name and subject name. In this accelerated training, you'll learn how to use formulas to manipulate text, work with dates and times,
lookup values with VLOOKUP and INDEX & MATCH, count and sum with criteria, dynamically rank … Formulas are the key to getting things done in Excel. In this accelerated training, you'll learn how to
use formulas to manipulate text, work with dates and times, lookup values with VLOOKUP and INDEX & MATCH, count and sum with criteria, dynamically rank … Formulas are the key to getting things done
in Excel. Excel Formula Training. Lookup formulas come in handy whenever you want to have Excel automatically return the price, product ID, address, or some other associated value from a table based
on some lookup value. This formula is dynamic, which means that if you change the student name or the subject names, it would still work and fetch the correct data. Excel Formula Training. Excel
inserts curly brackets at the beginning and end of the formula for you. We need some fixed formula which helps in finding the exact match … row_num: How many rows it has to go down to find the
answer. This is a variation of the classic INDEX MATCH formula to which you add one more MATCH function in order to get both the row and column numbers: Excel INDEX MATCH MATCH formula. The most
popular way to do a two-way lookup in Excel is by using INDEX MATCH MATCH. How INDEX MATCH Formula Works. INDEX MATCH Function in Excel. Go back to the Summary tab and build the formula using the
INDEX-MATCH approach. While the VLOOKUP function is widely used, it is known to be resource intensive and slow due to its inherent tendency to scan data columns from left to right, while performing a
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Coxeter groups
Coxeter groups
i tried the following input (as i found it in a tutorial of christian stump):
but it does not work (name 'CoxeterGroup' is not defined)
What's the problem with this?
1 Answer
Sort by » oldest newest most voted
It looks like this might require the sage-combinat queue. Or, it could be an old version of that code?
sage: W = WeylGroup(['A',3])
sage: W
Weyl Group of type ['A', 3] (as a matrix group acting on the ambient space)
It looks like there is more of a hierarchy now, based on the outcome of
sage: CoxeterGroups?
edit flag offensive delete link more | {"url":"https://ask.sagemath.org/question/9481/coxeter-groups/?sort=latest","timestamp":"2024-11-13T02:03:55Z","content_type":"application/xhtml+xml","content_length":"51849","record_id":"<urn:uuid:9045cb38-02c0-4b0e-8e51-7f6006a165c0>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00739.warc.gz"} |
AB 1513 weeds - DatatechAB 1513 weeds | Datatech
Question: Does the 4% method allow the employer to credit breaks that are overpaid in one pay period against breaks that are underpaid in another pay period?
Answer: No. Yes. Not really. It depends on how you look at it?
The following is not legal advice, but an explanation of how the Safe Harbor Report works based on our best understanding of the law and the information we have received from the DIR. If you ask the
same question of your labor attorney you may get a different answer. If you need the Safe Harbor Report to work differently based on the legal advice you have received, please let us know so we can
make the needed modifications to the calculations (see the bottom line, below).
We posed this question to the DIR back in May:
“When using the 4% less breaks paid calculation method, how should situations be handled when the 4% amount is less than the amount already paid for breaks?
In a single pay period, an employee works 10 hours, earns $93.24, takes three 10-minute breaks, and was paid minimum wage ($4.50—.5 (30 minutes) x $9/00/hour) for those three breaks separate from the
piecework earnings.
4% of the wages is $3.73, and $3.73 – $4.50 = -.77.
Should the 77 cents be deducted from payments made for other pay periods where the employee is owed break time, or should the employer show a zero dollar amount adjustment payable for this pay
(Just to be clear, I did not make these amounts up, they are from a real check issued to a real employee)”
And we got the following response from an attorney in the DIR legal unit:
“Under the 4% method in subdivision (b)(1)(B), the statute contemplates one overall calculation for the designated period, not a pay period by pay period calculation. In other words, an equation
like this (assuming for purposes of this example that there were no payments for other nonproductive time):
[4% of gross earnings in the piece-rate pay periods for period of 7/1/2012 – 12/31/2015] – [total of amounts already paid to the employee, separate from piece-rate compensation, for rest and
recovery periods during the same time] = [total of payment]
It may be that in some pay periods, the amounts already paid separate from piece-rate compensation are greater than 4% of the gross earnings for that pay period. Presumably, in other pay periods
that would not be the case.
If you are asking how to show that on the statement that will accompany the payment, the statute is not specific about exactly how the calculations must be shown. The overall intent reflected in the
statute is simply that the employee be provided a statement from which he or she can reasonably determine how the payment was calculated. Here is one possible example:
Pay Periods In Which Work Was Performed On Gross Earnings for Amounts Paid in Pay Period, Separate From Piece-Rate Compensation, Amounts Paid in Pay Period, Separate From Piece-Rate Compensation,
A Piece-Rate Basis Pay Period for Rest and Recovery Periods for Other Nonproductive Time
7/1/2012 – 7/7/2012 $435.25 — —
7/8/2012 – 7/14/2012 $500.00 $13.00 $50.00
TOTALS FOR ALL PAY PERIODS $ Total gross $Total of separate payments for rest and recovery periods $Total of separate payments for other nonproductive time
$Total gross
CALCULATION OF PAYMENT MADE (4% OPTION) earnings x .04 = (less $ Total of separate payments for rest and recovery periods) (less $Total of separate payments for other nonproductive time
[capped at 1% of gross earnings]
= $___TOTAL PAYMENT MADE_______” (emphasis added)
(The employee statement that we ended up designing for the 4% method is very close to the example provided by the DIR. The main difference is the addition of a piecework wages column and a subject
wages column so that it is clear what pay periods are included in the payment calculation (pay periods that have only hourly wages must still be listed on the statement) and what the total wages
are used for the 4% calculation.)
Backing up a little, how does an employer end up overpaying an employee for breaks?
Backing up a little more, “overpaying” isn’t really the right word to use in this context. If you are using the “actual sums due” method, then it is more appropriate. That is because you are
determining how much the employee was actually due for breaks, and you know exactly how much you paid the employee. Thus, the employee was either paid correctly, overpaid, or underpaid. If you are
using the “actual sums due” method, then we believe it could be problematic to apply over-payments in one pay period to underpayments in another pay period.
In fact, if you run the Safe Harbor Report using one of the “actual sums due” methods and the amount due for a pay period is negative (indicating the employee was overpaid for breaks), the report
zeroes this out so that it is not credited against other pay periods. Using “actual sums due” method, the report is in fact designed to look at break wages on a pay period by pay period basis. When
a pay period is found to be underpaid, then that amount is added to the total due to the employee. But over-payments are not treated as “credits”.
But the 4% method not the same. Per the text of the law (and the DIR’s answer), a single calculation is performed after adding up all of the gross wages, breaks previously paid, and non-productive
time paid over the entire Safe Harbor Period. The 4% method is simply describing a formula to determine a payment amount that will provide the employer with the safe harbor protection. Employees
are not actually “due”4% of their gross wages for pay periods with piecework, there is not actually a pay period by pay period calculation involved in determining the final payment amount (again, as
noted in the DIR’s answer), and the law does not specify a cap on the amount of break wages that you can deduct either for the entire Safe Harbor period or for individual pay periods.
So how does the employer pay an employee an amount for breaks that is more than 4% of their wages? There are several ways:
Suppose an employee works 6.6667 hours at piecework and earns the minimum wage of $10 hour, for a total of $66.67 The employee is also paid separately for two breaks, .3333 @ $10/hour, or $3.33.
Under the 4% method, the employee is due $2.80 (4% of $70) for this time. However the employee was actually paid $3.33, .53 more than what the 4% method determines is due.
The rate of 4% was probably determined as a approximation of the amount of time that an employee would be due for breaks (e.g. 20 minutes of break time on an eight hour day is 4.167% of the total
time worked). Another way that employers may have paid more than the 4% due is when employees are allowed and paid for 15 minute breaks instead of 10 minute breaks. This changes the percentage of
time paid for breaks; 30 minutes of break time on an eight hour day is 6.25% of the total time.
Another possibility is that employees are paid for more breaks than required by law. In the question that we posed to the DIR, the employee took three breaks on a 10 hour day, when only two breaks
are required. In this case, the third break might have been a heat recovery period, or it could have simply been an extra break period allowed by the employer. In any case, the law does not set any
limit on how much break wages can be counted against the 4% (unlike non-productive wages, which are capped at 1% of gross wages for the Safe Harbor period).
Sometimes the 4% method works to the employer’s disadvantage, even when breaks were paid correctly. For instance, suppose an employee earns $91.80 at piecework wages working 4.75 hours, and is paid
$2.25 for one 15 minute break for that time at minimum wage (per the case law in effect at the time), .25 @ $9.00/hour. The total wages on this day are $94.05. The gross due (4% of $94.05) is
$3.76, meaning that the employee is due $1.51 after subtracting the break paid at $2.25.
Another example of when the law works to the employer’s disadvantage: the 4% calculation is performed on all types of wages in pay periods with piecework wages, including those wages where employees
are not due breaks. This include bonuses, sick pay, vacation pay, and holiday pay. (The 4% calculation does not exempt any type of wages; this was also confirmed by the DIR).
These are all reasons to not think in terms of the 4% method as paying the employee what is “actually due”. In some cases where the employer did everything correctly, the employee actually ends up
getting paid more than what they would actually be due under the “actual sums method”. In other cases it could be less.
The legislators that wrote AB 1513 surely must have known that there would be some cases where employers would have paid more than 4% in break wages (it is simple math, after all). The fact is that
they did not include a cap on the break wages while including a cap specifically for non-productive wages. This very well could indicate that they did not intend to cap the amount of break wages that
employers may credit for the breaks paid.
The 4% method might be better thought of as being more like a calculation that determines how damages are awarded in a class action lawsuit. The actual damages awarded to class members can depend on
a number of factors and may or may not equal their individual loss. (Keep in mind, I am not a lawyer, so this analogy may not be 100% accurate.)
Why does the Safe Harbor Report list a net due calculation for each day and pay period if the actual payment amount is going to be based on the total subject wages for the Safe Harbor period?
There are a couple of reasons for this. First, it helps in determining how the breaks previously paid amount is determined when the amount is broken down by day. It is much easier to compare the
break wages paid on the report on a daily basis to the breaks actually paid or the hourly wages on the check to double check the report’s calculations.
Second, internally the report has to keep track of the grower (for FLCs) and cost center to charge the safe harbor payments to. To do this, the report must look at each individual payroll check
detail line anyway. Who or what gets charged for the safe harbor payment and the amounts that get charged have to maintained at a more detailed level. It is also easier to see how these amounts get
divided up between growers and cost centers when the detail by day is included on the report.
For internal cost accounting purposes, or for billing purposes for an FLC, the program does apply negative amounts due on a grower/cost center basis. For instance, we have seen cases where employees
that worked for a particular grower were consistently overpaid for breaks, and in that case the grower’s liability for safe harbor payments is zero, while other growers end up with amounts due to
cover the total share of the safe harbor liability.
These internal calculations that allocate the total safe harbor wages cost do not have anything to do with the calculations that are made to determine each employee’s amount due, just as the
individual calculations per day or per pay period are not used to determine the amounts due to each employee.
The Bottom Line (tl;dr)
Because the 4% method calculation (as described in the law and explained by the DIR) is based on the total wages for the entire Safe Harbor period and there are no limits on the amount of break time
that you can deduct, it is in fact possible for break wages paid in excess of the 4% amount on any given pay period to reduce the overall payment to the employee.
Based on comments we heard at AB 1513 seminars, some employers are in fact limiting the deduction for breaks to 4% of the gross wages in each pay period so that break wages over 4% do not reduce the
payment to the employee. If you want to do this, we can modify the Safe Harbor Report to calculate the higher amounts owed. This would likely also require a modification to the employee statement,
as you may still need to show the total breaks paid as required by the law (and so the employee can match that amount up to their checks stubs) but also show the lower amount that you are taking as a
credit for each pay period. Please contact Brian as soon as possible if you want this change made.
Note that the default employee statement does not show a pay period by pay period calculation–since it is not required by the law–it simply shows the wage amounts, and performs the calculations on
the totals for these amounts. Whether you show the net amount due per pay period would also be left up to your discretion. The AB 1513 Employee Statement is customizable, so you can add this column
if you want to. | {"url":"https://datatechag.com/ab-1513-weeds/","timestamp":"2024-11-14T06:59:06Z","content_type":"text/html","content_length":"104600","record_id":"<urn:uuid:5c878220-e5a3-4788-9624-7ed2f9347b0f>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00222.warc.gz"} |
What is a k-d tree, and how is it used in spatial searches? | TutorChase
What is a k-d tree, and how is it used in spatial searches?
A k-d tree, or k-dimensional tree, is a data structure used for organising points in a k-dimensional space.
A k-d tree is a binary tree in which every node is a k-dimensional point. Every non-leaf node generates a splitting hyperplane that divides the space into two half-spaces. Points to the left of this
hyperplane are represented by the left subtree of that node and points to the right of the hyperplane are represented by the right subtree. This process is repeated on the subspaces until there are
no more points to split, resulting in a binary tree where each node represents a k-dimensional point.
The k-d tree is particularly useful in applications involving multidimensional keys, such as computational geometry and database applications. It is an efficient data structure for spatial searches,
which are queries related to the position of points in a space. For example, one might want to find all points within a certain distance of a given point, or find the point that is nearest to a given
The k-d tree is efficient for these types of queries because it organises the points in a way that allows for efficient searching. When a query is made, the tree is traversed from the root to the
leaf that represents the point. At each node, the algorithm checks if the point lies to the left or the right of the splitting hyperplane and follows the appropriate subtree. This process is repeated
until the leaf node is reached.
The efficiency of the k-d tree comes from the fact that it eliminates half of the points from consideration at each level of the tree. This means that the number of points that need to be checked is
reduced exponentially, making the search much faster than a brute force approach that checks every point.
In summary, a k-d tree is a powerful tool for organising and searching points in a k-dimensional space. It is particularly useful in applications that involve spatial searches, where its efficient
organisation of points can significantly speed up query times.
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How Much Home Can I Buy With My Income
How Much Home Can I Buy With My Income
Determining this comes down to the debt-to-income (DTI) ratio. DTI is the percentage of your total debt payments as a share of your pre-tax income. A common. Your debt-to-income ratio (DTI) compares
your monthly debt against your monthly gross income. As a rule of thumb, try to keep your DTI below 43% when you take. To get a rough estimate of what you can afford, most lenders suggest you spend
no more than 28% of your monthly income — before taxes are taken out — on your. The general rule is that you can afford a mortgage that is 2x to x your gross income. · Total monthly mortgage payments
are typically made up of four. Your PITI, combined with any existing monthly debts, should not exceed 43% of your monthly gross income — this is called your debt-to-income ratio (DTI). Your.
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house? Aim to buy a house that equals about three times your yearly income. If you have no other debts, you can. Mortgage affordability calculator. Get an estimated home price and monthly mortgage
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factors such as income, debt, down payment, or simply budget. Two criteria that mortgage lenders look at to understand how much you can afford are the housing expense ratio, known as the “front-end
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how much house you can afford by examining factors that impact affordability like income and monthly debts. Another general rule of thumb: All your monthly home payments should not exceed 36% of your
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how much house you can afford based on your monthly income, expenses and specified mortgage rate. This rule asserts that you do not want to spend more than 28% of your monthly income on
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afford with your budget. Most lenders base their home loan qualification on both your total monthly gross income and your monthly expenses. These monthly expenses include property. This does not
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Non-Lyapunov annealed decay for 1d Anderson eigenfunctions
In Exact dynamical decay rate for the almost Mathieu operator by Jitomirskaya et al. [Math. Res. Lett. 27(3), 789–808 (2020)], the authors analysed the dynamical decay in expectation for the
supercritical almost-Mathieu operator in function of the coupling parameter, showing that it is equal to the Lyapunov exponent of its transfer matrix cocycle, and asked whether the same is true for
the 1d Anderson model. We show that this is never true for bounded potentials when the disorder parameter is sufficiently large.
Consider the one-dimensional Anderson model, i.e., the operator
acting on a dense subset of
are i.i.d. random variables. We assume that the distribution of
is bounded and not concentrated at one point (in most of the discussion below, the first assumption can be relaxed to the existence of a finite fractional moment
). Carmona–Klein–Martinelli
showed that under these assumptions
exhibits Anderson localisation, i.e., almost surely
has pure point spectrum, and moreover
$P∀(λ,ψ)∈Elim supx→±∞1|x|log|ψ(x)|=−γ(λ)=1,$
is the collection of eigenpairs of
(the spectrum is almost surely simple, so |
| is well-defined), and
) is the Lyapunov exponent of
at energy
. Under more restrictive assumptions on
, the pure point nature of the spectrum was first proved by Goldsheid–Molchanov–Pastur
and by Kunz–Souillard;
the exponential decay of the eigenfunctions was first established by Molchanov.
While the proof of Ref. 5 employs multi-scale analysis, single-scale proofs have recently been found by Bucaj et al.,^4 Gorodetski–Kleptsyn,^9 and Jitomirskaya–Zhu.^11 Generalisations to models with
off-diagonal disorder and to matrix-valued potentials are studied in Refs. 13 and 16.
A stronger notion of Anderson localisation involves the notion of eigenfunction correlator, introduced by Aizenman.
where the supremum is taken over Borel functions. If
has pure point spectrum, the correlator takes the form
Then there exists
> 0 such that for any
$Plim supy→±∞1|y−x|logQ(x,y)⩽−γ=1.$
In fact, in the current setting
holds with
, where
) is the spectrum of
(a deterministic set)—see Ref.
. This strong form of
, as well as dynamical localisation, decay of the Fermi projection as well as other properties of relevance in quantum dynamics.
Ge and Zhao built on the work^11 and proved the following:
Theorem 1.1
For the operator H of (1) with V[0] bounded and not concentrated at one point, one has, for any $x∈Z$,
$γE=−lim supy→±∞1|y−x|logEQ(x,y)>0.$
In Sec. II we give another, arguably, simpler, proof of this result, adopting an argument from Ref. 6.
Jitomirskaya et al. ^10 studied the validity of (6) in the almost-periodic setting, namely, for the supercritical almost-Mathieu operator with Diophantine frequency, and showed that in that setting
$γE$ can be taken to be equal to γ[inf]. They asked whether the same is true for the Anderson model. We show that this is not the case. A first counterexample comes from the Anderson–Bernoulli model:
Theorem 1.2.
For a > 0, consider the operator H^a = H[0] + aV with V[x] being a bounded random variable having an atom at 0. Then $γE$ is bounded from above uniformly in a.
In particular, if V[x] is a Bernoulli random variable with parameter p, by a result of Martinelli and Micheli,^14 γ[inf] ⩾ cloga for sufficiently large a. Therefore, by the above theorem, $γE(Ha)
≠γI(Ha)$ for a large enough.
Furthermore, the above theorem remains true for any bounded random potential satisfying mild conditions, at sufficiently high disorder:
Theorem 1.3.
Let $V={Vi}i∈Z$ be a nondeterministic, bounded, i.i.d. random potential, and let H^a ≔ H[0] + aV.
Then, for any a large enough
II. PROOF OF THEOREM 1.1
, denote by
the restriction of
to [
] (with Dirichlet boundary conditions), and let
. Let
> 0,
⩾ 1. A site
is called (
)-nonresonant [
∉ Res(
)] if
is called (
)-resonant [
∈ Res(
)]. The proof of the theorem uses the following.
Claim 1.
Assume that V[0] is bounded and not concentrated at one point. Then for any τ
> 0
there exist C
> 0
such that
See Ref. 13, Proposition 2.1 for this formulation (in the more general case of matrix potentials) and Ref. 11, Theorem 4.1, for a similar statement in the pure one-dimensional case.
Next, we need a representation for the eigenfunction correlator as a singular integral [see (7.4) at p. 102 of Ref.
Having these two ingredients, we argue as follows. Without loss of generality we can assume that
= 0. Set
, and consider the event
According to Claim 1,
. On the complement
, we have
, where
and an analogous bound can be deduced for
This proof can be extended to quasi-one-dimensional operator, such as the Anderson model on the strip of width
or the more general model studied in Ref.
. A slightly weaker version of
is still true in this case (see Ref.
and the argument above follows with minor modifications.
III. PROOF OF THEOREM 1.2
> 0 be a large numerical constant (independent of any parameters), to be specified later. For
> 0, consider the event
We shall prove the following: for any
> 0, one has on Ω
for sufficiently large
, this this would imply that
as claimed.
We now turn to the proof of
. Since the argument is uniform in
, we will use
. Observe that for any
> 0,
In fact,
be the free Laplacian [obtained by setting
≡ 0 in
], and let
be the restriction of
to the finite volume [−
, (
+ 1)
]. Then, by applying the resolvent identity and the reverse triangle inequality twice, we get
By the Combes–Thomas estimate (Ref.
, Theorem 10.5), we deduce that for
∈ (0, 1),
) =
) is by definition the square-summable solution to the equation
Plugging in the ansatz
) =
, we find that this is indeed a solution provided that
, where
> 0 small enough, as
in a neighbourhood of 0.
Having set c = c[2]δ and 2K + 1 = ⌈100 c[2]/c[1]⌉, we obtain (8).
A version of the Martinelli–Micheli bound the 1d Anderson model with absolutely continuous, bounded potential has been proven in 1983 by Avron et al. in Ref. 3.
Theorem 4.1
et al.^3
Let H^a
aV be a random Schrödinger operator where V is a bounded random potential with absolutely continuous density. Then the Lyapunov exponent γ[a] of H^a is such thatwhere K is a finite constant.
We will adapt the proof in Ref.
to any potential having finite first moment (not necessarily absolutely continuous).
et al.
’s proof relies on the Thouless’ formula for the Lyapunov exponent of a Schrödinger operator
, stating that
′) denotes the
integrated density of states
of the operator
. They proceed then to bound the negative part of the logarithm in 12 using the
Wegner estimate
: If
is a random Schrödinger operator with i.i.d. potential, then
which is unfortunately proven true only when the distribution
of the potential is absolutely continuous.
Fortunately, Shubin
et al.
proved in Ref.
a slightly weaker bound for the IDS of a random Schrödinger operator whose potential satisfies the conditions of Theorem 1.3: if
(condition that we have automatically since the distribution of
is bounded), then
for some
∈ (0, 1) and some constant
> 0. This bound is sufficient to let the Avron–Craig–Simon argument work in the present generality.
By applying the Thouless formula to
and splitting the logarithm into its positive and negative parts, we get that
We claim that
for some
> 0 uniform in
, and that
for some positive constant
The first bound
is proven by using inequality
We strongly believe that the main result of this section (and thus Theorem 1.3 as a whole) can be extended to the most general setting for which 1-d localisation has been proven (nondeterministic
potential with any finite fractional moment). However, generalising (13) to the case where one only has a generic fractional moment appears to be nasty.
An alternative to this approach could be extending the proof of the logarithmic divergence for the Anderson–Bernoulli model in Ref. 14 to the general case; however, even if this seems to be doable
and should not present major technical difficulties, additional estimates would be needed to make Martinelli’s and Micheli’s already six page long proof work for generic potentials, and many formulas
would get much longer and nastier.
In conclusion, the length of the present paper would likely get doubled by such an attempt, therefore we avoid it to keep the paper short and more readable while keeping the result reasonably
V. PROOF OF THEOREM 1.3: GREEN FUNCTION ESTIMATES
Since the above argument uses crucially the fact that a Bernoulli random variable is 0 with positive probability, one might suspect that the presence of an atom at zero is required for the annealed
dynamical decay to be non-Lyapunov. However, in this section we will use a simple trick to eliminate the atom at 0. The trick relies on the observation that it is possible to decompose any dilated
random variable aX having an atom as a sum of two (not necessarily independent) random variables, one of which is bounded in a, and the other has basically the same distribution as aX with the
difference that the atom has been subtracted some mass. If we subtract in this way mass from the atom at a sufficient rate, and control the error given by the bounded addend, then we can show that
the growth of the annealed decay rate in a is much slower than the logarithmic lower bound prescribed by the results of Avron–Craig–Simon and Shubin–Vakilian–Wolff.
In order to exploit the case of a potential with an atom at 0, we will make use of the following observation.
Observation 5.1.
Let X be a bounded random variable, and let $x̄∈supp(X)⊆[−R,R]$. Suppose that X is absolutely continuous in a neighborhood of $x̄$ and denote by $X̃ϵ$ the random variable having the same density as X,
except for the fact that it has an atom at $x̄$ of mass ϵ, suitably renormalised. Then there exists a bounded random variable $η̃$ (not necessarily independent on X) such thatfor some a
> 0.
This observation basically asserts that we can remove (or, by extension, subtract mass to) an atom from the distribution of a random variable at the cost of adding another (dependent) random variable
uniformly bounded in the coupling.
It follows by simply observing that if $η̃$ has the same distribution of X and $X̃ϵ$ is chosen to take the same values as $η̃$ (so that $X̃ϵ$ would retain its usual law and its atom at $x̄$, but becoming
totally dependent on $η̃$), then $aX̃ϵ+η̃=d(a+1)X$.
Observation 5.2.
Without loss of generality, we can take R = δ/10.
In fact, δ is by construction always positive and we can always multiply the potential by any finite constant and incorporate such constant into the disorder parameter a.
We now use these two observations to prove the general result. Take V such that supp(V) ⊆ [−R, R] and set R = δ/10. Apply Observation 5.1 to the potential V with ϵ = ϵ(a) ≫ a^−β for all β > 0, and
decompose $aV=aṼϵ(a)+η̃$, where $η̃=dV$, and $Ṽϵ(a)$ is a random variable having the same distribution as V except for having an atom at 0 of mass ϵ(a) (with the necessary renormalisation). Then $Ha=
TR+aṼϵ(a)$, where $TR=T0+η̃$.
Thus again, if
) ⩾ e
we get, as in
Furthermore, we call
the restriction of
to the box [−
, (
+ 1)
], analogously as before. We will shift
by −2·
so that the spectrum of the resulting operator lies below
/2. A double resolvent expansion analogue to the one performed in the Proof of Theorem 1.2 and the Combes-Thomas bound yield
on the event
Notice that this time we chose δ instead of iδ as a spectral parameter. The reason for this choice is that we need to use the negativity of the shifted Laplacian to compare its Green’s function to
that of the (negative) shifted operator.
Eventually, the only thing left to us to show is that
By writing down the Neumann series for
− 2·
], we get the following inequalities:
We can compute
− 2·
) explicitly via the same method used in the proof of 1.2. In this case, we get that
$|Gδ/2[T0−2⋅1](x,y)|⩾|α|e−ξ|x|, with α=2e−ξ+δ2−2−1,ξ=arccosh1+δ4.$
In particular, when
is small,
− 2·
) decays exponentially with rate of order
Setting, again,
large enough so that
, and setting
, we finally conclude that
This, combined with
and the Avron–Craig–Simon bound for general bounded potentials proven in Paragraph 4, implies the thesis.□
I am deeply indebted to my former Ph.D. advisor, Sasha Sodin, for suggesting the topic of this paper and for giving major contributions to its development. I believe he should have been a co-author
of this paper, but for reasons I do not understand he asked me to erase his name from it.
I am also grateful to Alexander Elgart for many useful comments on a preliminary version of this paper, and to the anonymous referee for useful suggestions and for pointing out a flaw in the original
This work has been supported by the Grant No. EPSRC EP∖T004290∖1.
This work was started when the author was a Ph.D. student at Roma Tre University.
Conflict of Interest
The author has no conflicts to disclose.
Author Contributions
Davide Macera: Conceptualization (equal); Investigation (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal).
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
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Published open access through an agreement with JISC Collections | {"url":"https://pubs.aip.org/aip/jmp/article/65/1/012103/2932932/Non-Lyapunov-annealed-decay-for-1d-Anderson","timestamp":"2024-11-11T07:10:27Z","content_type":"text/html","content_length":"251331","record_id":"<urn:uuid:f04db09f-38d8-4167-a502-83cc632bebd9>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00381.warc.gz"} |
Colloquium: Dr. Marlan Scully, Texas A&M, Princeton,& Baylor
Physics : 401
Date & Time
October 10, 2018, 3:30 pm – 4:30 pm
TITLE: From Special to General Relativity with Unruh and Hawking:
Light from atoms falling into a black hole
General relativity as originally developed by Einstein is based on the union of geometry and gravity. Half a century later the union of general relativity and thermodynamics was found to yield
surprising results such as Bekenstein-Hawking black hole entropy and Hawking radiation.
In their seminal works, Hawking, Unruh and others showed how quantum effects in curved space yield a blend of thermodynamics, quantum field theory and gravity which continues to intrigue and
stimulate. It has been shown [1] that virtual processes in which atoms jump to an excited state while emitting a photon is an alternative way to view Unruh acceleration radiation. The present work
[2] is an extension of that logic by considering what happens when atoms fall into a black hole. This problem also shows a new way to arrive at Einstein’s equivalence principle. Connection with the
“temperature as an imaginary time” paradigm of many-body theory is also illustrated by this problem. In general, the quantum optics – black hole physics interface is a rich field. | {"url":"https://physics.umbc.edu/home/events/?id=63902","timestamp":"2024-11-04T01:47:24Z","content_type":"text/html","content_length":"146346","record_id":"<urn:uuid:f0e0b0c7-39fb-476b-92be-96d4d8043874>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00642.warc.gz"} |
Thus, the DC voltage applied to the load resistor drops only by a small amount. Full-wave rectification converts both polarities of the input waveform to pulsating DC (direct current), and yields a
higher average output voltage. The blue plot on the waveform shows the result of using a 5.0uF smoothing capacitor across the rectifiers output. Unlike half wave rectifiers which uses only half wave
of the input AC cycle, full wave rectifiers utilize full wave. But we can improve this still by increasing the value of the smoothing capacitor as shown. This configuration results in each diode
conducting in turn when its anode terminal is positive with respect to the transformer centre point C producing an output during both half-cycles, twice that for the half wave rectifier so it is 100%
efficient as shown below. The full wave rectifier circuit consists of two power diodes connected to a single load resistance ( RL) with each diode taking it in turn to supply current to the load. it
has average output higher than that of half wave rectifier. The Best Way To Check A Start Capacitor Wikihow Capacitors In Loudspeakers Explained The Teufel Audio Blog How To Safely Discharge A
Capacitor Ifixit Repair Guide To discharge a capacitor the power source which was charging the capacitor is removed from the circuit so that only a capacitor and resistor can connected together in
series. Here we have increased the value of the smoothing capacitor ten-fold from 5uF to 50uF which has reduced the ripple increasing the minimum discharge voltage from the previous 3.6 volts to 7.9
volts. Advantages of Full Wave Rectifiers. For centre-tapped full wave rectifier, FF = 1.11. The full-wave rectifier can be designed by using with a minimum of two basic diodes or it can use four
diodes based on the topology suggested. Actually it alters completely and hence t… Half Wave Rectifier. Hence diode D 1 conducts and a current i 1 flows through the diode D 1 and load resistor R L as
shown in figure 1. But if the smoothing capacitor is sufficiently large enough (parallel capacitors can be used) and the load current is not too large, the output voltage will be almost as smooth as
pure DC. it rectifies both the positive and negative cycles in the waveform. The circuit which allows us to do this is called a Full Wave Rectifier. Whenever, point A of transformer is +ve w.r.t. Ask
Question Asked 3 years, 3 months ago. One method to improve on this is to use every half-cycle of the input voltage instead of every other half-cycle. The most common and widely used single-phase
rectifier is the bridge rectifier, but full-wave rectifiers and half-wave rectifiers can also be used. Full Wave Bridge Rectifier In Full Wave Bridge Rectifier, an ordinary transformer is used in
place of a center-tapped transformer.The circuit forms a bridge connecting the four diodes D 1, D 2, D 3, and D 4.The circuit diagram of the Full Wave Bridge Rectifier is shown below. About 9 or 10
parts on unregulated supplies 5_8-9-12-15V etc. The main advantage of a full-wave rectifier over half-wave rectifier is that such as the average output voltage is higher in full-wave rectifier, there
is less ripple produced in full-wave rectifier when compared to the half-wave rectifier. Smoothing or reservoir capacitors connected in parallel with the load across the output of the full wave
bridge rectifier circuit increases the average DC output level even higher as the capacitor acts like a storage device as shown below. Viewed 2k times 1 \$\begingroup\$ Where is the best to place
resistor in full wave rectifier circuit? Full-wave bridge rectifier: This is the most popular and most widely used circuit for rectification of AC voltage because this circuit doesn’t require any
transformer. Notify me of follow-up comments by email. The average (DC) output voltage is higher than for half wave, the output of the full wave rectifier has much less ripple than that of the half
wave rectifier producing a smoother output waveform. These circuits are called full-wave rectifiers. This type of low-pass filter consists of two smoothing capacitors, usually of the same value and a
choke or inductance across them to introduce a high impedance path to the alternating ripple component. Average DC output Voltage Vp/π 2Vp/π 2Vp/π 5. Briefly describe working principle of the
circuit. The single secondary winding is connected to one side of the diode bridge network and the load to the other side as shown below. A Schottky Diode is a metal-semiconductor diode with a low
forward voltage drop and a very fast [...], The Diode Clipper, also known as a Diode Limiter, is a wave shaping circuit that takes an [...]. Plz give me electronics, compenents explain. Function Of
Resistor In Full Wave Rectifier June 22, 2019 Get link; Facebook; Twitter; Pinterest; Email An alternating current has the property to change its state continuously. Above 100v should be done with a
discharge tool. The amount of ripple voltage that is superimposed on top of the DC supply voltage by the diodes can be virtually eliminated by adding a much improved π-filter (pi-filter) to the
output terminals of the bridge rectifier. We can improve the average DC output of the rectifier while at the same time reducing the AC variation of the rectified output by using smoothing capacitors
to filter the output waveform. But in full wave rectifier, both positive and negative half cycles of the input AC current will charge the capacitor. The main duty of the capacitor filter is to short
the ripples to the ground and blocks the pure DC (DC components), so that it flows through the alternate path and reaches output load resistor R L . Working of the Full Wave Rectifier Center Tapped
Transformer. The arrangement of diodes in the bridge network such that both positive and negative cycles of the input voltage can be rectified is known as a bridge rectifier. Power Diodes can be
connected together to form a full wave rectifier that convert AC voltage into pulsating DC voltage for use in power supplies. Uninterruptible Power Supply (UPS) circuits to convert AC to DC. Full
Wave Bridge Rectifiers are mostly used for the low cost of diodes because of being lightweight and highly efficient. We have already discussed the Full Wave Bridge Rectifier, which uses four diodes,
arranged as a bridge, to convert the input alternating current (AC) in both half cycles to direct current (DC). Abstract: How to build a full-wave rectifier of a bipolar input signal using the
MAX44267 single-supply, dual op amp. The other two connecting leads are for the input alternating voltage from a transformer secondary winding. I actually own a portable induction cooktop and i have
to say its saved me on many an occasion. The maximum ripple voltage present for a Full Wave Rectifier circuit is not only determined by the value of the smoothing capacitor but by the frequency and
load current, and is calculated as: Where: I is the DC load current in amps, ƒ is the frequency of the ripple or twice the input frequency in Hertz, and C is the capacitance in Farads. For making 100
amperes, 50 volts full wave rectifier, how do I calculate the circuit capacitance to avoid the ripple voltage? This is very easy to understand why centeraltap transformer is needed in a full wave
rectifier. The two voltage V 1 and V 2 fed to the two diodes are equal in magnitude but opposite in phase. As we all know the basic principle of the diode it can conduct the flow of current in one
single direction and the other is blocked. Type of Transformer Normal Center Tap Normal 3. In a Full Wave Rectifier circuit two diodes are now used, one for each half of the cycle. When point A of
the transformer is positive with respect to point C, diode D1 conducts in the forward direction as indicated by the arrows. Full-wave rectification can also be obtained by using a bridge rectifier
like the one shown in Figure 1. This cut-off corner indicates that the terminal nearest to the corner is the positive or +ve output terminal or lead with the opposite (diagonal) lead being the
negative or -ve output lead. This results in the capacitor discharging down to about 3.6 volts, in this example, maintaining the voltage across the load resistor until the capacitor re-charges once
again on the next positive slope of the DC pulse. Generally for DC power supply circuits the smoothing capacitor is an Aluminium Electrolytic type that has a capacitance value of 100uF or more with
repeated DC voltage pulses from the rectifier charging up the capacitor to peak voltage. This winding is split into two … Can you please clarify why the full wave rectifier is bi-phase while the full
wave bridge rectifier is single-phase? Another more practical and cheaper alternative is to use an off the shelf 3-terminal voltage regulator IC, such as a LM78xx (where “xx” stands for the output
voltage rating) for a positive output voltage or its inverse equivalent the LM79xx for a negative output voltage which can reduce the ripple by more than 70dB (Datasheet) while delivering a constant
output current of over 1 amp. the point C, diode D1 conducts in forward direction as shown with the help of arrows. 56.3 nV ; c. 21.3 mV ; d. 41.7 mV; 18. We previously explained diode-based
half-wave rectifier and full-wave rectifier circuit. The main advantages of a full-wave bridge rectifier is that it has a smaller AC ripple value for a given load and a smaller reservoir or smoothing
capacitor than an equivalent half-wave rectifier. Excellent conceptual explanation of subtle principles involved. The features of a center-tapping transformer are − 1. Mobile phones, laptops, charger
circuits. When point A of the transformer is positive with respect to point C, diode D1 conducts in the forward direction as indicated by the arrows. The main advantage of this bridge circuit is that
it does not require a special centre tapped transformer, thereby reducing its size and cost. How safe it depends on the voltage. In the case of centre-tap full wave rectifier, only two diodes are
used, and are connected to the opposite ends of a centre-tapped secondary transformer as shown in the figure below. I have seen such circuits with small unpolarized caps next to the diodes, 1 cap per
diode, so 4 diodes, 4 small caps, 1 (usually) electrolytic, and maybe 1 resistor. A full-wave rectifier converts the whole of the input waveform to one of constant polarity (positive or negative) at
its output. Half wave Rectifier: Center Tap Full wave Rectifier Bridge Full wave Rectifier 1. However, there are two important parameters to consider when choosing a suitable smoothing capacitor and
these are its Working Voltage, which must be higher than the no-load output value of the rectifier and its Capacitance Value, which determines the amount of ripple that will appear superimposed on
top of the DC voltage. a. Half-wave rectifier; b. Full-wave rectifier; c. Bridge rectifier; d. Impossible to say; 17. Here the 5uF capacitor is charged to the peak voltage of the output DC pulse, but
when it drops from its peak voltage back down to zero volts, the capacitor can not discharge as quickly due to the RC time constant of the circuit. We just want to show its output on an oscilloscope
and discuss some of the different factors that you should consider when you use a bridge full-wave rectifier. Induction cooker for travel . The tapping is done by drawing a lead at the mid-point on
the secondary winding. In a full wave rectifier circuit we use two diodes, one for each half of the wave. There are two types of full-wave rectifiers — the center-tapped full-wave rectifier, which
requires a center-tapped transformer, and the bridge rectifier, which does not need a center-tapped transformer. All contents are Copyright © 2020 by AspenCore, Inc. All rights reserved. In the full
wave rectifier circuit using a capacitor filter, the capacitor C is located across the RL load resistor. My Question is about unpolarized caps…what for? 21.3 pV ; b. They have low power loss because
no voltage signal is wasted in the rectification process. Construct full wave rectifier. In this tutorial, we're going to have a simple demo of the bridge full-wave rectifier. Question : state the
effect of connecting a single smoothing capacitor across the dc and RL. Like the half wave circuit, a full wave rectifier circuit produces an output voltage or current which is purely DC or has some
specified DC component. Half-Wave Rectifier Working A half-wave rectifier is an electrical circuit containing an AC source, a load resistor (RL), and a diode that permits only the positive half
cycles of the AC sine wave to pass, which creates pulsating DC. Full wave rectifiers have some fundamental advantages over their half wave rectifier counterparts. The average output of the bridge
rectifier is about 64% of the input voltage. a. The main disadvantage of this type of full wave rectifier circuit is that a larger transformer for a given power output is required with two separate
but identical secondary windings making this type of full wave rectifying circuit costly compared to the “Full Wave Bridge Rectifier” circuit equivalent. The full-wave rectifier circuit constitutes 2
power diodes connected to a load-resistance (Single RL) with the each diode taking it in turn to provide current to load. Ripple Frequency F in 2 F in 2 F in 6. Solved Tuskegee University Eeng 380l
Electrical Engineer, Full Wave Rectifier Basic Electronics Lab Assignment Docsity, Solved Use Piecewise Analysis To Study The Full Wave Rect, Half Wave Rectifier Positive And Negative Half Wave
Rectifier, Centre Tap Full Wave Rectifier Circuit Operation Working, Chapter 2 Diode Applications Ppt Video Online Download, Pre Lab Information Half Wave And Full Wave Rectifier, What Is The Need Of
Using Resistor In Half Wave Rectifier, Full Wave Rectifier And Bridge Rectifier Theory, What Is Bleeder Resistor Significance Functions Of, Chapter 6 Diode Applications Power Supplies Voltage, Half
Full Wave Rectifier Converting Ac To Dc Rectifier, All together the caps are a combined 22000uf at 180v. During the positive half-cycle of the source voltage (Figure 2(a)), diodes D2 and D3 are
forward biased and can therefore be replaced by a closed switch. In the previous Power Diodes tutorial we discussed ways of reducing the ripple or voltage variations on a direct DC voltage by
connecting smoothing capacitors across the load resistance. How to figure size? The only dissimilarity is half wave rectifier has just one-half cycles (positive or negative) whereas in full wave
rectifier has two cycles (positive and negative). Previously the load voltage followed the rectified output waveform down to zero volts. This type of single phase rectifier uses four individual
rectifying diodes connected in a closed loop “bridge” configuration to produce the desired output. Half Wave and Full Wave Rectifier In Half Wave Rectifier, when the AC supply is applied at the
input, a positive half cycle appears across the load, whereas the negative half cycle is suppressed.This can be done by using the semiconductor PN junction diode. The important uses of the full-wave
bridge rectifier are given below. High Quality Travel Cooking Appliances Induction Cookers The Best Portable Induction Cooktop Reviews By Wirecutter Travel Cooker Induction Cooktops Buy Travel Cooker
Im all about making this search as easy as possible for you so below you will find my best picks for the different types of inductions cooktops. The peak voltage of the output waveform is the same as
before for the half-wave rectifier provided each half of the transformer windings have the same rms voltage value. Applications of a Full-wave Bridge Rectifier. If we now run the Partsim Simulator
Circuit with different values of smoothing capacitor installed, we can see the effect it has on the rectified output waveform as shown. The full wave rectifier circuit consists of two power diodes
connected to a single load resistance (RL) with each diode taking it in turn to supply current to the load. The smoothing capacitor converts the full-wave rippled output of the rectifier into a more
smooth DC output voltage. How to find the voltage and how to plot the wave graph ?? We can see this affect quite clearly if we run the circuit in the Partsim Simulator Circuit with the smoothing
capacitor removed. You can also choose from non stick coating inner pot cooking time presetting and digital timer control. What is the difference in usage of the resistors in these two circuits? When
point B is positive (in the negative half of the cycle) with respect to point C, diode D2 conducts in the forward direction and the current flowing through resistor R is in the same direction for
both half-cycles. The four diodes labelled D1 to D4 are arranged in “series pairs” with only two diodes conducting current during each half cycle. But this rectification method can only be used if
the input voltage to the circuit is greater than the forward voltage of the diode which is typically 0.7V. Full wave rectifiers have higher rectifying efficiency than half-wave rectifiers. Since only
half of the wave is used in a half-wave rectifier circuit, more efficient power supplies have been developed to use both halves of the sine wave. The diode allows the current to flow only in one
direction.Thus, converts the AC voltage into DC voltage. A multiple winding transformer is used whose secondary winding is split equally into two halves with a common centre tapped connection, (C).
For the positive half, the upper part of the diode will be in forward bias that is in conducting mode. The full-wave bridge rectifier however, gives us a greater mean DC value (0.637 Vmax) with less
superimposed ripple while the output waveform is twice that of the frequency of the input supply frequency. Active 3 years, 3 months ago. If the load current is 5 mA and the filter capacitance is
1000uF, what is the peak-to-peak ripple out of a bridge rectifier? As the spaces between each half-wave developed by each diode is now being filled in by the other diode the average DC output voltage
across the load resistor is now double that of the single half-wave rectifier circuit and is about 0.637Vmax of the peak voltage, assuming no losses. Using this concept as the basis many rectifiers
are designed. As the output voltage across the resistor R is the phasor sum of the two waveforms combined, this type of full wave rectifier circuit is also known as a “bi-phase” circuit. Another type
of circuit that produces the same output waveform as the full wave rectifier circuit above, is that of the Full Wave Bridge Rectifier. The working of this rectifier is almost the same as a half wave
rectifier. The current flowing through the load is the same direction as before. Current is flowing through the load the same direction in each example, and both use the positive and negative cycles
to conduct. What is the type of the output signal from a rectifier circuits ? Diode rectifiers are simpler than the other types that use switching devices. In the previous article, we have discussed
a center-tapped full-wave rectifier, which requires a center-tapped transformer and the peak output of the rectifier is always half of the transformer secondary voltage.Where the bridge rectifier is
the full-wave rectifier with no such requirement and restriction. Portable Electric Cooktop Fixworkcover Com Induction Cooker Single Eurodib C1823 Induction Cooker Eurodib Eg13 Slim Portable The 6
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in parallel with resistor, impedance of resistor and capacitor in series, impedance resistor and capacitor in parallel, pn junction diode forward bias circuit diagram. 100Hz for a 50Hz supply or
120Hz for a 60Hz supply.). The image to the right shows a typical single phase bridge rectifier with one corner cut off. Mathematically, this corresponds to the absolute value function. A bi-phase
uncontrolled rectifier uses a single-phase center tapped ransformer and two diodes, one conducting per half-wave to supply a load, while a single-phase uncontrolled full-wave bridge rectifier uses
four diodes, two conducting per half-wave. PIV Rating of Diode Vp 2Vp Vp: 4. The effect of a supplying a heavy load with a single smoothing or reservoir capacitor can be reduced by the use of a
larger capacitor which stores more energy and discharges less between charging pulses. However, using the Partsim Simulator Circuit we have chosen a load of 1kΩ to obtain these values, but as the
load impedance decreases the load current increases causing the capacitor to discharge more rapidly between charging pulses. In other words, the capacitor only has time to discharge briefly before
the next DC pulse recharges it back up to the peak value. The former is therefore called a half-wave rectifier, as it only rectifies one half of the supply waveform, while the latter is called a
full-wave rectifier, as it rectifies both halves or the entirety of the waveform. full wave rectifier resistor place. Full-Wave Rectifier . Three basic types of rectifiers used in single-phase DC
power supplies are half-wave, full-wave, and full-wave bridge rectifiers. It raises in its positive direction goes to a peak positive value, reduces from there to normal and again goes to negative
portion and reaches the negative peak and again gets back to normal and goes on. of Diodes One Two Four 2. Where: VMAX is the maximum peak value in one half of the secondary winding and VRMS is the
rms value. Full wave rectifier rectifies the full cycle in the waveform i.e. Full-wave rectifier circuits are used for producing an output voltage or output current which is purely DC. While this
method may be suitable for low power applications it is unsuitable to applications which need a “steady and smooth” DC supply voltage. Although we can use four individual power diodes to make a full
wave bridge rectifier, pre-made bridge rectifier components are available “off-the-shelf” in a range of different voltage and current sizes that can be soldered directly into a PCB circuit board or
be connected by spade connectors. Discharge large capacitor you should have a rsistor parallel to your capacitor and this resistor is usually called the bleeder resistor since the stored energy is
being discharge to the resistor this in turn prevents voltage spike during start up. This is understood by observing the sine wave by which an alternating current is indicated. Do they balance, or
further smooth? The one reason i had for buying it is because the holidays can get pretty hectic in my kitchen and a portable cooktop has enabled me to have an extra burner that heats immediately
perhaps the biggest benefit of induction cooking. As the current flowing through the load is unidirectional, so the voltage developed across the load is also unidirectional the same as for the
previous two diode full-wave rectifier, therefore the average DC voltage across the load is 0.637Vmax. During the negative half cycle of the supply, diodes D3 and D4 conduct in series, but diodes D1
and D2 switch “OFF” as they are now reverse biased. This means that they convert AC to DC more efficiently. Try different values of smoothing capacitor and load resistance in your circuit to see the
effects on the output waveform. In the next tutorial about diodes, we will look at the Zener Diode which takes advantage of its reverse breakdown voltage characteristic to produce a constant and
fixed output voltage across itself. Full wave rectifier finds uses in the construction of constant dc voltage power supplies, especially in general power supplies. A multiple winding transformer is
used whose secondary winding is split equally into two halves with a common center tapped connection. What is the AC side input current in a full wave rectifier Give me an example the AC side current
drawn to charge 150ah battery. Full Wave Center Tapped Rectifier Working As the input applied to the circuit it gets equally split at the center that is positive half and the negative half. Figure 1
: Difference between outputs of half- and full- wave rectifiers Between the two types, the full-wave rectifier is more efficient as it uses the full cycle of the incoming waveform. There are 308
induction cooker travel suppliers mainly located in asia. Safety easy to clean. Vari. Full wave rectifier output Full Wave Rectifier Theory. To obtain a different DC voltage output different
transformer ratios can be used. This full-wave bridge rectifier uses four diodes. No. Why not test your knowledge about full wave rectifier circuits using the Partsim Simulator Tool today. I mean
which size of capacitor should I use? Introduction Implementing simple functions in a bipolar signal environment when working with single-supply op amps can be quite a challenge because, oftentimes,
additional op amps and/or other electronic components are required. As a general rule of thumb, we are looking to have a ripple voltage of less than 100mV peak to peak. We saw in the previous section
that the single phase half-wave rectifier produces an output wave every half cycle and that it was not practical to use this type of circuit to produce a steady DC supply. During the positive half
cycle of the supply, diodes D1 and D2 conduct in series while diodes D3 and D4 are reverse biased and the current flows through the load as shown below. Also explain about filter capacitor. Too low a
capacitance value and the capacitor has little effect on the output waveform. Therefore, the fundamental frequency of the ripple voltage is twice that of the AC supply frequency (100Hz) where for the
half-wave rectifier it is exactly equal to the supply frequency (50Hz). We have already seen the characteristics and working of Half Wave Rectifier.This Full wave rectifier has an advantage over the
half wave i.e. During its journey in the formation of wave, we can observe that the wave goes in positive and negative directions. Full wave rectifier is the semiconductor device which converts
complete cycle of AC into pulsating DC. The ripple frequency is now twice the supply frequency (e.g. During the first half cycle, as shown in figure 2, V 1 is positive. In a typical rectifier
circuit, we use diodes to rectify AC to DC. Let us assume that we have a simple transformer, and there are two diodes and the central wire coming out from the transformer is not present there which
is obvious since we … Discharge big capacitor . The transformer is center tapped here unlike the other cases. Ripple Factor: 1.21: 0.48: 0.48 7. Charging and rapidly discharging a 24 volt 4 farad
capacitor thats intended for a car audio system. A bridge rectifier with an efficient filter is ideal for any type of general power supply applications like charging a battery, powering a dc device
(like a motor, led etc) etc. “Full Wave Rectifier” during the academic year 2016-17 towards partial fulfillment of credit for the Physics Project evaluation of AISSCE 2017, and submitted working
model and satisfactory report, as compiled in the following pages, under my supervision. Features of a center-tapping transformer are − 1 timer control i have to say its saved me on many occasion...
Same direction in each example, and yields a higher average output of bridge! Forward bias that is in conducting mode to build a full-wave rectifier circuit two diodes are used! Capacitor thats
intended for a car audio system voltage into DC voltage power supplies improve on this is a... Improve on this is very easy to understand why centeraltap transformer is used whose secondary winding
connected. V 2 fed to the absolute value function and highly efficient out use of resistor in full wave rectifier center-tapping. Because no voltage signal is wasted in the waveform i.e one method to
improve on this is easy... That the wave above 100v should be done with a common center tapped.! F in 6 can improve this still by increasing the value of the will!, the upper part of the secondary
winding is split equally into two halves a... Most common and widely used single-phase rectifier is about 64 % of the input voltage instead of other... Are used for producing an output voltage or
output current which is purely DC AC into.: state the effect of connecting a single smoothing capacitor converts the AC voltage into voltage! Current has the property to change its state continuously
100 amperes, 50 full... Different transformer ratios can be used difference in usage of the full-wave bridge rectifiers applied to absolute. During the first half cycle the important uses of the
diode allows the current to flow only one... Single phase bridge rectifier in your circuit to see the effects on the waveform i.e a rule! Used for the positive and negative cycles in the Partsim
Simulator Tool today current which use of resistor in full wave rectifier! A capacitance value and the load voltage followed the rectified output waveform down to zero.! The sine wave by which an
alternating current has the property to its... Obtain a different DC voltage $ Where is the peak-to-peak ripple out of a bridge are. Device which converts complete cycle of AC into pulsating DC (
direct current ), and bridge. Across the DC and RL rippled output of the input AC cycle, wave. Best to place resistor in full wave rectifier bridge full wave rectifier circuit is in conducting mode
voltage from rectifier! Are for the input voltage into a more smooth DC output voltage located in asia Simulator Tool today single-phase. Making 100 amperes, 50 volts full wave rectifier is the type
of the bridge rectifier rectifies both the half... This tutorial, we use diodes to rectify AC to DC diode D1 in... I mean which size of capacitor should i use see the effects on output. Done with a
common center tapped connection the first half cycle, as with. Equally into two halves with a common center tapped connection than 100mV peak to peak center-tapping transformer are −.. Rapidly
discharging a 24 volt 4 farad capacitor thats intended for a car system. 100 amperes, 50 volts full wave rectifier: center Tap full wave is. F in 6 going to have a ripple voltage of wave, 're!
Aspencore, Inc. all rights reserved of a bridge rectifier is the type of the input alternating from. Also choose from non stick coating inner pot cooking time presetting and digital timer control
only by a small.. Rectifiers are mostly used for the positive half, the DC and RL or... How to build a full-wave rectifier circuits on the output signal from transformer... Do this is very easy to
understand why centeraltap transformer is used whose secondary.... Now used, one for each half of the secondary winding is split equally into two halves with a center.: state the effect of connecting
a single smoothing capacitor converts the bridge... Signal using the MAX44267 single-supply, dual op amp to convert AC to DC voltage from rectifier! Flowing through the load voltage followed the
rectified output waveform down to zero volts, and both use the and... Pairs ” with only two diodes conducting current during each half of the input alternating voltage from a secondary! 120Hz for a
60Hz supply. ) negative cycles in the rectification process and... Parts on unregulated supplies 5_8-9-12-15V etc of connecting a single smoothing capacitor converts the full-wave rippled output of
the bridge! A car audio system DC voltage applied to the right shows a typical rectifier circuit we use two are...: 0.48: 0.48: 0.48 7 positive or negative ) at its output for each half of the
capacitor. Is center tapped transformer as the basis many rectifiers are designed of capacitor! The first half cycle, full wave rectifiers have some fundamental advantages over their half wave
rectifier rectifies full! Going to have a ripple voltage of less than 100mV peak to peak saved me on many occasion. The voltage and how to build a full-wave rectifier of a bipolar input signal using
Partsim. F in 6 is to use every half-cycle of the diode will be in forward direction as shown the. As a half wave rectifiers have higher rectifying efficiency than half-wave rectifiers F in 6 the
rectifiers.... Supplies are half-wave, full-wave, and full-wave bridge rectifiers are mostly used for the input cycle! Output of the wave goes in positive and negative directions input voltage we
the. Capacitor as shown in Figure 2, V 1 is positive only diodes... To say its saved me on many an occasion have to say its saved me on many occasion. And negative cycles to conduct tapped here
unlike the other side as shown in 1. Rectifier and full-wave rectifier circuits about full wave rectifier at its output volts full wave rectifiers have higher efficiency. Capacitor should i use 5 mA
and the filter capacitance is 1000uF, what is the semiconductor device converts! In 2 F in 2 F in 2 F in 6 wave full! Bi-Phase while the full wave rectifier center tapped here unlike the other two
connecting leads are for the and. Is single-phase converts complete cycle of AC into pulsating DC ( direct current ), and bridge! Cut off positive or negative ) at its output pulsating DC ( current!
Car audio system 2k times 1 \ $ \begingroup\ $ Where is the ripple., diode D1 conducts in forward direction as shown below AC to more. Low power loss because no voltage signal is wasted in the
Partsim Simulator Tool today 120Hz for a 50Hz or... In one half of the input alternating voltage from a rectifier circuits are used for producing an output voltage output. Negative half cycles of the
input waveform to pulsating DC lead at the on... During its journey in the construction of constant polarity ( positive or negative ) at its output Question 3... This tutorial, we use diodes to
rectify AC to DC more efficiently fed to the absolute function...: 4 the two diodes are now used, one for each of... A typical single phase bridge rectifier are given below most common and widely
single-phase... Its saved me on many an occasion a car audio system, 50 full... ” with only two diodes conducting current during each half of the wave two connecting are... The working of this
rectifier is the rms value from non stick coating inner pot cooking time presetting digital! Waveform shows the result of using a 5.0uF smoothing capacitor and load in. Load the same direction as
shown below previously explained diode-based half-wave rectifier and full-wave rectifier two.. ) their half wave rectifier has an advantage over the half wave Rectifier.This full rectifier... During
its journey in the waveform shows the result of using a bridge rectifier how! Of diodes because of being lightweight and highly efficient bridge full wave bridge. Are looking to have a simple demo of
the input waveform to pulsating DC ( direct )! To build a full-wave rectifier circuit this affect quite clearly if we use of resistor in full wave rectifier the capacitance! Too low a capacitance
value and the capacitor the rectified use of resistor in full wave rectifier waveform down zero. Has little effect on the output waveform opposite in phase average output than! Frequency F in 2 F in
2 F in 2 F in 2 F in 2 in! We can observe that the wave as the basis many rectifiers are mostly used for an... Voltage instead of every other half-cycle output current which is purely DC into a more
smooth DC output.... Single-Supply, dual op amp output different transformer ratios can be used 60Hz supply ). Avoid the ripple frequency is now twice the supply frequency ( e.g flowing through the
load current is through! Both the positive and negative directions highly efficient portable induction cooktop and have! Will be in forward bias that is in conducting mode the first half cycle, full
wave rectifiers uses. Diode Vp 2Vp Vp: 4 of half wave rectifier is single-phase mV ; d. mV. Of using a 5.0uF smoothing capacitor removed do i calculate the circuit in the Partsim Simulator today.
Understood by observing the sine wave by which an alternating current is indicated full! Obtain a different DC voltage power supplies are simpler than the other side as shown below rectifier finds
uses the... To one side of the wave rectifier has an advantage over the half wave rectifier rectifies the full cycle the! Leads are for use of resistor in full wave rectifier input AC cycle, as shown
with the smoothing removed... Dc voltage power supplies rectifiers and half-wave rectifiers unlike half wave of the resistors in two... We 're going to have a ripple voltage of less than 100mV peak
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The 2018 South African Mathematics Olympiad — Problem 1
The final round of the South African Mathematics Olympiad will be taking place on Thursday, 28 July 2019. In the two weeks leading up to the contest, I plan to take a look at some of the problems
from the senior paper from 2018.
The first problem from the 2018 South African Mathematics Olympiad was
One hundred empty glasses are arranged in a $10 \times 10$ array. Now we pick $a$ of the rows and pour blue liquid into all glasses in these rows, so that they are half full. The remaining rows
are filled halfway with yellow liquid. Afterwards, we pick $b$ of the columns and fill them up with blue liquid. The remaining columns are filled with yellow liquid. The mixture of blue and
yellow liquid turns green. If both halves have the same colour, then that colour remains as is.
1. Determine all possible combinations of values for $a$ and $b$ so that exactly half of the glasses contain green liquid at the end.
2. Is it possible that precisely one quarter of the glasses contain green liquid at the end?
In order to find under what conditions half of the glasses contain green liquid at the end, it would be very helpful to know how many glasses (in terms of $a$ and $b$) contain green liquid, so that
we can set up an equation to solve for $a$ and $b$.
We note that glasses containing green liquid occur at the intersection of a blue row and a yellow column, or at the intersection of a yellow row and a blue column. To calculate the number of glasses
at the intersection of a blue row and a yellow column, we note that there are $a$ options for which row the glass is in, and $10 - b$ options for which column it is in. There are thus $a(10 - b)$
such glasses. Similarly, there are $b(10 - a)$ glasses at the intersection of a yellow row and a blue column. Since a glass must be in either a yellow row or a blue row, and can not be in both, we
see that this accounts for each of the green glasses exactly once. There are thus $a(10 - b) + b(10 - a)$ glasses of green liquid in total.
We see that to determine the combinations of $a$ and $b$ such that exactly half of the glasses contain green liquid, we want to solve the equation
$\displaystyle a(10 - b) + b(10 - a) = 50.$
This can be manipulated to become
$\displaystyle (a - 5)(b - 5) = 0$
and so we see that exactly half of the glasses contain green liquid if and only if either $a = 5$, or $b = 5$!
Does this make sense? Is filling half of the rows (or columns) with blue liquid really enough to ensure that half of the glasses will contain green liquid? Suppose that half of the rows are filled
with blue liquid. For each column, we note that if the column is filled with blue liquid, then a glass in that column will be green at the end precisely when it lies in a yellow row. By assumption,
this is true for half of the rows, and so half of this column will end up green. Similarly, if the column is filled with yellow liquid, then it is in precisely the $5$ rows in which we find blue
liquid that there will be a green glass at the end. Again we see that half of the column will end up green. Since half of every column turns out to be green, exactly half of the glasses will be
Let us consider the second question that was posed. Is it possible that exactly one quarter of the glasses contain green liquid at the end? In this case, we want to solve the equation
$\displaystyle a(10 - b) + b(10 - a) = 25$
which simplifies to
$\displaystyle 10a + 10b - 2ab = 25.$
This has no solutions in whole numbers, because for whole number values of $a$ and $b$, the left hand side of the equation will always be even, but $25$ is an odd number.
The reader may be interested to know that there is in fact a general method to solve equations of the type
$\displaystyle axy + bx + cy = d$
for given values $a, b, c, d$, and where we wish to find integer values for $x$ and $y$.
As we did for the first part of the SAMO problem, the general approach is to try to factorise the given expression in the form
$\displaystyle (px + q)(ry + t) = n.$
If we’re lucky, $n$ turns out to be $0$, and we can conclude that the only solutions are those where $x = -q/p$, or $y = -t/r$.
The case where $n eq 0$ does not actually pose a problem, however. We did not see it arise when looking at the SAMO problem, but it is far more common. In this case, we know that each of the terms on
the left hand side of the equation must be factors of $n$. The integer $n$ only has finitely many factors, and so we can find all solutions by setting $(px + q)$ to be equal to each one of these
factors in turn. We find that all solutions in this case are given by
$\displaystyle x = \frac{d - q}{p} \quad \text{and} \quad y = \frac{n - dt}{dr}$
where $d$ is one of the finitely many divisors of $n$.
Is it always possible to find a factorisation of the desired form? Yes, we can check that the given equation is in fact equivalent to
$\displaystyle (ax + c)(ay + b) = ad + bc.$
At this point, we can note that if $\gcd(a, b) \gcd(a, c)$ does not divide $ad + bc$, then the equation has no integer solutions. (I leave it to the reader to explain why $\gcd(a, c)$ is always a
divisor of $ad + bc$. This is why we consider the product of both of the $\gcd$‘s.) If we had applied this method to the second part of the SAMO problem, the factorised version of the equation would
have been
$\displaystyle (-2a + 10)(-2b + 10) = 50.$
Since $4$ is not a divisor of $50$, we again see that there are no solutions. The reader is cautioned, however, that there may be other obstacles to the existence of a solution. Even if $\gcd(a, b) \
gcd(a, c)$ is a divisor of $ad + bc$, not all factors of $n = ad + bc$ will yield integer solutions when applying the above method.
If the reader would like to put this method into practice, it can be applied to the first problem from the 2018 Putnam Mathematics Competition:
Find all ordered pairs $(a, b)$ of positive integers for which
$\displaystyle \frac{1}{a} + \frac{1}{b} = \frac{3}{2018}.$
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Newton's cannonball
Newton's cannonball was a thought experiment Isaac Newton used to hypothesize that the force of gravity was universal, and it was the key force for planetary motion. It appeared in his posthumously
published 1728 work De mundi systemate (also published in English as A Treatise of the System of the World). Newton's original plan for Philosophiæ Naturalis Principia Mathematica was that it should
consist of two books, the first analyzing basic laws of motion, and the second applying them to the Solar System. In order to include more material on motion in resisting media, the first book was
split into two; the succeeding (now third) book, originally written in a more popular style, was rewritten to be more mathematical. However, manuscripts of an earlier draft of this last book
survived, and a version of it was published in 1728 as De mundi systemate; an English translation was also published earlier in 1728 under the name A Treatise of the System of the World. The thought
experiment occurs near the start of this work. In this experiment from his book (pp. 5–8), Newton visualizes a stone (you could also use a cannonball) being projected on top of a very high mountain.
If there were no forces of gravitation or air resistance, the body should follow a straight line away from Earth, in the direction that it was projected. If a gravitational force acts on the
projectile, it will follow a different path depending on its initial velocity. If the speed is low, it will simply fall back on Earth. (A and B) for example horizontal speed of 0 to 7,000 m/s for
Earth. If the speed is the orbital speed at that altitude, it will go on circling around the Earth along a fixed circular orbit. (C) for example horizontal speed of at approximately 7,300 m/s for
Earth. If the speed is higher than the orbital velocity, but not high enough to leave Earth altogether (lower than the escape velocity), it will continue revolving around Earth along an elliptical
orbit. (D) for example horizontal speed of 7,300 to approximately 10,000 m/s for Earth. | {"url":"https://graphsearch.epfl.ch/en/concept/5646168","timestamp":"2024-11-03T19:43:21Z","content_type":"text/html","content_length":"93428","record_id":"<urn:uuid:680ea154-b3ff-4c85-a8b4-981972917c3c>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00292.warc.gz"} |
AC Power Analysis
Oh boi this is difficult RMS
Maximum Average Power Transfer
The maximum average power absorbed by the load occurs when
Active (Real) Power
The active/real power is constant and represents the portion of the power that is transformed from electric energy to non-electric energy (ex: heat).
is known as the Power Factor
Reactive Power
The reactive power of is also constant and represents the portion of the power that is NOT transformed into non-electric energy, but rather it is exchanged between circuit elements such as
capacitors, inductors, and sources.
Remember that the is written in terms of voltage. We know how Capacitors and Inductors behave (90 degrees between current and voltage), so you can make these final assumptions
Instantaneous Power
Average Power
It’s just the integral of the instantenous power divided by the time period.
→ Apparently, when solving circuits, the average power is the the real power .
Maximum Average Power Transfer
THe results found for DC Circuits for Maximum Power Transfer are extended.
Complex Power
The complex power for a general element is defined by
Complex power is the combination of Active Power and Reactive Power :
Apparent Power
Apparent power is just the norm of the Complex power.
title: Some Conventions for the Units of Power
In the exam, if you see different units, this is how to tell apart different powers.
| Type of Power | Unit |
| :---: | :---: |
| Active Power $P$ | Watt (W) |
| Reactive Power $Q$ | Volt-Ampere-Reactive (VAR) |
| Complex Power $S$| Volt-Ampere (VA) |
|Apparent Power $P_{app} = \vert S\vert$ | Volt-Ampere (VA) | | {"url":"https://stevengong.co/notes/AC-Power-Analysis","timestamp":"2024-11-14T17:09:42Z","content_type":"text/html","content_length":"93315","record_id":"<urn:uuid:d7740a22-a9f8-4a9a-acdd-8fde665d5a65>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00464.warc.gz"} |
Problem Solving Multiple Choice Topic Test
Here are 10 Problem Solving multiple choice questions written by people from around the world while using the main Pentransum activity. You can earn a Transum Trophy for answering at least 9 of them
1. If Dan and Don share $48 in the ratio 5:1 how much more than Don will Dan receive?
2. 30% of A = 30% of 30 + 30. Find A
This question was suggested by Nevin, Kerala
3. A snail climbs up a 12m wall. It climbs 3m each day, but slips back 2m each night. On what day will it reach the top of the wall?
This question was suggested by Gillian, New Zealand
4. If Bob has 44p and Bill has 22p how much does Bob have to give Bill so they have the same amount of money?
This question was suggested by Sophie Brown, Newcastle
5. How many buses will be needed to hold 476 people when a bus can hold 52 people.
This question was suggested by Finn Meyer, Christian-von-Dohm-Gymnasium Goslar, Germany
6. The perimeter of a rectangle is 28.6cm. One side of the rectangle is 5.1cm. What is the size of the longer side of the rectangle?
This question was suggested by Hockey Puck, Birmingham
7. If goldilocks and the three little pigs sat down at a table, how many legs would there be?
This question was suggested by Flossie Roberts, Portland, Dorset
8. I thought of a number, divided it by 6, added 52, doubled it and subtracted five. I ended up with 111. What number did I start with?
This question was suggested by Terry, Yorkshire
9. I'm thinking of a number: I add 6, divide by 4 and then times it by 5 and my answer is 35, what was my original number?
This question was suggested by Rebecca and Michelle, Leeds
10. Mr and Mrs Thomson have six children and the sum of their ages is 63. What was the sum of the ages of the Thomson children 7 years ago?
This question was suggested by Caitlyn Dawbin,
Please note that unlike other Transum online exercises, the check button for this multiple choice quiz can only be clicked once when you have answered all ten questions. Check your answers carefully
before clicking the button below. You teed to get at least 9 questions correct to be awarded a Transum Trophy. | {"url":"https://www.transum.org/Software/Pentransum/Topic_Test.asp?ID_Topic=31","timestamp":"2024-11-03T06:08:06Z","content_type":"text/html","content_length":"48715","record_id":"<urn:uuid:b21f3b99-0090-4f34-a87f-bb24748bdfea>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00531.warc.gz"} |
Multiple Frequencies
This blocks generates events at specific sample time of the simulation time. The sample time is given in the "Sample Time" field and the offset is given in the "Offset" field. This block has one
event input, the number of event outputs depends on the number of different sample time. For example if the vector of sample time is [1 1 2] and the vector of offset is [0 .5 0] then the block has 7
1. The first output is activated when the simulation time is equal to a multiple of the first sample time plus the first offset
2. The second output is activated when the simulation time is equal to a multiple of the second sample time plus the second offset.
3. The third output is activated when we have both cases, first case and second case.
4. The fourth output is activated when the simulation time is equal to a multiple of the third sample time plus the third offset.
5. The fifth output is activated when we have both cases, first case and forth case.
6. The sixth output is activated when we have both cases, second case and fourth case.
7. The seventh output is activated when we have both cases, third case and forth case.
8. etc...
So the number of outputs is equal to 2**number of different time values. Each of these time values is represented by a binary number associated to the output's number in decimal.
• Sample time
Vector of sample time values.
Properties : Type 'vec' of size -1.
• Offset
Vector of offset values. Must have the same size as the Sample time and each offset value must be less than its corresponding sample time.
Properties : Type 'vec' of size -1.
Default properties
• always active: no
• direct-feedthrough: no
• zero-crossing: no
• mode: no
• number/sizes of activation inputs: 1
• number/sizes of activation outputs: 3
• continuous-time state: no
• discrete-time state: no
• object discrete-time state: no
• name of computational function: m_frequ
Let us take the example where the sample time is equal to [1 1 2] and the
offset is equal to [0 .5 0]. Consider t=simulation time.
When t=0 , the fifth output is activated (001 + 100).
When t=0.5, the second output is activated (010).
When t=1 , the first output is activated (001).
When t=1.5, the second output is activated (010).
When t=2 , we loop back to 0.
Other example:
Interfacing function
• SCI/modules/scicos_blocks/macros/Events/M_freq.sci
Computational function
• SCI/modules/scicos_blocks/src/c/m_frequ.c (Type 4)
See also
• MFCLCK_f — triggered double clock with two output frequencies | {"url":"https://help.scilab.org/docs/2023.1.0/ru_RU/M_freq.html","timestamp":"2024-11-04T04:13:01Z","content_type":"text/html","content_length":"13885","record_id":"<urn:uuid:3da7cc52-b027-491a-845c-620877a5a4ee>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00544.warc.gz"} |
Absolute sum of elements in Numpy matrix
In this article, we have explained how to find the Absolute sum of elements in Numpy matrix with complete Python code example.
Table of contents:
1. Absolute sum of elements in Numpy matrix
2. Normal Sum
3. Conclusion
Absolute sum of elements in Numpy matrix
To find the Absolute sum of elements in Numpy matrix, we need to follow to steps:
• Get absolute value of elements in Numpy matrix (using np.abs)
• Find sum of elements of updated matrix (using np.nansum)
The code snippet is as follows:
abs_matrix = np.abs(matrix)
abs_sum = np.nansum(abs_matrix, dtype=np.float64)
• dtype is the data type of the sum output. We have set it to float 64 bits to avoid overflow.
• abs_matrix has the absolute value of all elements in the original matrix.
• nansum gives the sum of all elements in abs_matrix where there is no negative element. nansum will avoid nan values in the matrix.
The complete Python code example is as follows:
import numpy as np
matrix = np.array([[1.94353, -2.13254, 3.00845], [-4.3423, 5.5675, -6.01029]])
print ("Original matrix = ")
print (matrix)
abs_matrix = np.abs(matrix)
abs_sum = np.nansum(abs_matrix, dtype=np.float64)
print ("Absolute sum = ")
print (abs_sum)
Original matrix =
[[ 1.94353 -2.13254 3.00845]
[-4.3423 5.5675 -6.01029]]
Absolute sum =
Normal Sum
Now, the difference between absolute sum and normal sum is critical for several data as in normal sum, negative and positive numbers tend to cancel each other.
When you are using sum as a metric to compare Numpy data/ matrix, it is important to use absolute sum.
The complete Python code example is as follows:
import numpy as np
matrix = np.array([[1.94353, -2.13254, 3.00845], [-4.3423, 5.5675, -6.01029]])
print ("Original matrix = ")
print (matrix)
sum = np.nansum(matrix, dtype=np.float64)
print ("Sum = ")
print (sum)
Original matrix =
[[ 1.94353 -2.13254 3.00845]
[-4.3423 5.5675 -6.01029]]
Sum =
• The absolute sum is 23.00461
• The normal sum is -1.965650000000001
• The difference between the two values is significant.
With this article at OpenGenus, you must have the complete idea of how to get the absolute sum of a Numpy matrix easily with just two code lines. | {"url":"https://iq.opengenus.org/absolute-sum-of-elements-in-numpy-matrix/","timestamp":"2024-11-09T20:27:53Z","content_type":"text/html","content_length":"29742","record_id":"<urn:uuid:cae7f747-4c7f-4079-a406-8c1d190ceb92>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00369.warc.gz"} |
The Magic Cafe Forums - Does this math sound right?
Levi Bennett
I'd like to add Chameleon Silks to my walk around, but I don't like the small gimmick. There is one I've found that says it will hold 2 45cm, or 17.7 inch, silks. Using Bob's
Inner circle equivalency chart I'm thinking I could load 3 18 inch diamond cut silks in this.
1866 Posts Does that sound right?
Thank you,
The math looks right the way you put the problem (that is, putting 3 18-inch diamond silks in same space as 2 17.7 inch square silks):
17.7X17.7 = 313.3 square inches. So, two would would be 626.6 square inches of material.
An 18" diamond will act about the same as a 12X12 inch silk. So that is 12X12 = 144 square inches. Three of these would be 432 square inches of material.
432 is less than 626.6. So three of the 18" diamond cuts would take less space than two of the 17.7" silks (assuming the same density of material).
Inner circle
1084 Posts However, I am not sure that this is the way the problem should be viewed. With chameleon silks the gimmick needs to only contain two silks. So I am a little confused regarding that
Also, the challenge with chameleon silks is that the gimmick needs to be a certain diameter and also needs to be only partially filled to be manipulated in the appropriate way. I am not
sure a gimmick which can hold two 17.7" silks would be able to be manipulated in that way.
This constraint and possible alternatives were discussed here:
Levi Bennett
Inner circle
1866 Posts Cool, thanks Frank!
David Todd Levi, this improved chameleon silks gimmick may be something you would want to try?
Inner circle https://tomladshawmagic.com/chameleon-silk-gimmick-deluxe
2531 Posts
Personally, if I was going to do it, I would use the Palmo gimmick.
Levi Bennett 0
Inner circle David, that's actually the one I've been looking at. I'd like to start with "hands empty", produce a silk and then do 2 color changes. If possible. that's why I'm wondering about doing
1866 Posts a 3 silk load in that gimmick. Palmo is a little big for me, but anyway... I'm just kicking ideas around and seeing what the experts have to say.
Maybe 3 12 inch diamonds? I'm looking at doing this for walk around and table hopping.
Levi Bennett So, funny thing, after I posted this that exact gimmick was flying off the shelves. Weird timing I guess. But Tom Ladshaw had a few in stock and really went out of his way to work with
me on figuring out the load. That gimmick takes 3 18 inch diamond cut silks with no problem. It's smaller and easier to conceal than a Palmo. I'd say it's still pretty large though. Tom
Inner circle sent me the measurements. 2 3/8 by 1 3/8 inches at its widest.
1866 Posts The reveal of the large silks looks very nice. I still need to work on the handling, but so far I'm happy, I can see this working well. I am leaning toward using this as an opener at
tables with children. For the vanish I'll reach in my pocket to get my wand, a hot rod, and be reset while going into my next trick. Routine may change but that's what I'm working on
for now.
Tom Ladshaw is a pleasure to work with he definitely deserves our business.
David Todd On Aug 11, 2024, Levi Bennett wrote:
Inner circle Tom Ladshaw is a pleasure to work with he definitely deserves our business.
2531 Posts
Totally agree! Tom has great customer service, he will respond promptly to inquiries about products he sells with real information about the prop. He also ships out orders quickly.
Levi Bennett
Inner circle
1866 Posts Worked on the handling for a bit and tried it on the grandsons. Worked like a charm. Definitely a keeper.
The Magic Cafe Forum Index » » Smooth as silk » » Does this math sound right? (4 Likes) | {"url":"https://themagiccafe.com/forums/viewtopic.php?topic=772967#7","timestamp":"2024-11-13T01:03:09Z","content_type":"application/xhtml+xml","content_length":"20491","record_id":"<urn:uuid:3e7690b1-d3c3-4b56-a0f4-e3824c7dc519>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00567.warc.gz"} |
Algorithm to Calculate The Power Of n^xAlgorithm to Calculate The Power Of n^x | mr.wixXsid
Algorithm to Calculate The Power Of n^x
Q: Write an Algorithm to calculate the power of a number. Input will be n and x, and your algorithm should output the value of n^x.
base = input('Enter the base (n): ')
expo = input('Enter the exponent (x): ')
// expo = exponent (x)
float power(int base, int expo) {
if (expo is 0)
return 1
sid = power(base, expo/2)
if (expo % 2 == 0)
return sid * sid
if(expo > 0)
return sid * sid * base
return sid * sid / base
Complexity Analysis
Time Complexity: O(log n)
Space Complexity: O(log n)
Because of “recursive stack space“
0 Comments
You must be logged in to post a comment. | {"url":"https://blog.mrwixxsid.com/algorithm-to-calculate-the-power-of-nx/","timestamp":"2024-11-02T09:16:04Z","content_type":"text/html","content_length":"214761","record_id":"<urn:uuid:a4a6a47f-00d0-46a9-83a8-2daf8971789d>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00717.warc.gz"} |
Square & Square Root of 400- Methods, Calculation, Formula, How to find
Square & Square Root of 400
Square of 400
400² (400 × 400) = 160,000
To calculate the square of 400, you multiply 400 by itself:
400 × 400 = 160,000.
Therefore, the square of 400 is 160,000. Understanding how to find the square numbers is crucial in various mathematical fields including algebra, geometry, and data analysis, as it provides a
foundation for tackling more intricate problems and concepts.
Square Root of 400
√400 = 20
Thus, the square root of 400 is exactly 20. This calculation is important in various mathematical and practical contexts, such as geometry, where it can represent the side length of a square with an
area of 400 square units. It’s also significant in algebra and engineering, where understanding square roots helps in solving problems involving areas and other applications that require precise
Square Root of 400 : 20
Exponential Form : 400^½ or 400^0.5
Radical Form : √400
Is the Square Root of 400 Rational or Irrational?
The square root of 400 is rational.
This is because 400 is a perfect square, being 20 × 20 = 400. As a result, the square root of 400 is exactly 20, which can be expressed as a fraction 20/1, making it a rational number.
Methods to Find Value of Root 400
There are several methods to find the value of the square root of 400:
Prime Factorization Method:
Decompose 400 into its prime factors, which are 2 × 2 × 2 × 5 × 5. Pair up identical factors and take one from each pair to get 2 × 5 = 10. Therefore, the square root of 400 is 20.
Repeated Subtraction Method:
Start subtracting consecutive odd numbers from 400 until you reach 0. Count how many times you subtracted, and that count is the square root. For 400, this method will result in 20 subtractions,
indicating the square root is 20.
Estimation Method:
Since 20 is a known square root close to 400, you can estimate by finding a number close to 20 that squares to 400. By trial and error, you can quickly determine that 20 is indeed the square root of
Calculator: Use a calculator with a square root function to directly find the square root of 400, which is 20.
These methods offer different approaches to find the square root of 400, catering to various preferences and situations.
Square Root of 400 by Long Division Method
Let’s follow these steps to find the square root of 400 by long division:
Step 1: Group the digits into pairs by placing a bar over them. Since our number is 400, represent it inside the division symbol.
Step 2: Find the largest number such that when multiplied by itself, the product is less than or equal to 4. We know that 2 × 2 = 4. Now, divide 4 by 2.
Step 3: Bring down the next pair of numbers, which is 00. Multiply the quotient 2 by 2 and write it in the new divisor’s place, resulting in 4.
Step 4: Choose a number in the unit’s place for the new divisor such that its product with a number is less than or equal to 0. Since 4 is in the ten’s place and our product has to be 0, it means 40
× 0 = 0. The long division process stops here as the remainder is 0. Thus, the quotient 20 is the square root of 400.
By following these steps, we have successfully found that the square root of 400 is 20 using the long division method.
400 is Perfect Square Root or Not
Yes, 400 is a perfect square.
A perfect square is a number that can be expressed as the product of an integer multiplied by itself. In the case of 400, it can be expressed as 20 × 20, which equals 400. Therefore, 400 is indeed a
perfect square.
Is the square root of 400 a natural number?
Yes, the square root of 400 is a natural number.
A natural number is a positive integer, and the square root of 400 is exactly 20, which is a positive integer. Therefore, the square root of 400 qualifies as a natural number. | {"url":"https://www.examples.com/maths/square-and-square-root-of-400.html","timestamp":"2024-11-12T19:02:06Z","content_type":"text/html","content_length":"107779","record_id":"<urn:uuid:731dab4e-d159-42a1-a047-67b5e41b3319>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00634.warc.gz"} |
Online SSB OIR Free Test Quiz
Attempt the Online SSB OIR Free Test Quiz, the passing percentage is 75% with a limit of 10 minutes. The new feature is added to the quiz, if you get the question wrong, you will get the explanation
below. Do comment about this new feature. All the best!
The Pattern of Quiz:
MON – Defence and Aviation Quiz
TUE – History/Geo/Polity
WED – SSB OIR
THU – SSB OIR
FRI – Static GK Quiz
SAT – Current Affairs Quiz
SUN – 50 Questions Test Series
1. Thanks for the quiz ✌️✌️
2. Its nice 😊. You start to give the explanation also thanks 🤠 | {"url":"https://defencedirecteducation.com/2021/07/29/online-ssb-oir-free-test-quiz/","timestamp":"2024-11-14T02:08:52Z","content_type":"text/html","content_length":"707150","record_id":"<urn:uuid:b392eea0-2891-4c51-a4d6-784f3678a024>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00471.warc.gz"} |
Video’s supporting the paper: 1D morpho stability
10.4121/13005323.v1 Ginger Egberts Ginger Egberts 0000-0003-3601-6496 4TU.ResearchData 2021 Dataset Applied Mathematics Numerical and Computational Mathematics Online Resource Morphoelasticity
Contraction Stability TU Delft, Delft Institute of Applied Mathematics 2021-07-21 1 CC BY-NC 4.0 These video files show the evolution of the components of the one-dimensional morphoelastic model for
contraction in burn injuries. In the corresponding paper, we analyse the stability of the model and we validate these stability constraints numerically. <br>This supplementary material is a
collection of videos that correspond to Figure 1, Figure 2 and Figure 4 in the paper. In the captions of these figures, we referred to these videos. unknown Dutch Burns Foundation under Project | {"url":"https://data.4tu.nl/export/datacite/datasets/8c93077e-7283-41d8-9ccc-15a88f6d76eb/1","timestamp":"2024-11-10T01:42:52Z","content_type":"application/xml","content_length":"2884","record_id":"<urn:uuid:4bddd256-749a-4eac-9560-997055d17e91>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00301.warc.gz"} |
Math Connections - Expanding Freshman Physics in Missouri
Mathematics Underlying the Physics First Curriculum
Click here for pdf (includes the document below, sample problems and a sample list of formulae used in Physics First)
Implications for 8th and 9th Grade Mathematics
by Dorina Mitrea
An ongoing collaboration between math and physics teachers is an integral part of a successful implementation of the Physics First curriculum. This summary is intended as the springboard for such a
collaboration. The list below contains some examples from the Physics First curriculum which you, the mathematics teacher, may incorporate into your lessons and/or homework assignments for 8th and
9th grade.
1. Notation
Students need to understand that in a physics course the quantities measured in an experiment dictate the notation used for the independent and dependent variables.
• the letters x and y are NOT the letters generally used to label the horizontal and vertical axes in a physics class
While in most mathematics textbooks the letter x is used for the independent variable (represented on the horizontal axis) and the letter y is used for the dependent variable (represented on the
vertical axis), this is not typically the case in the Physics First course. For example, in the study of uniform motion, time is an independent variable, denoted by t and represented on the
horizontal axis, while position is a dependent variable, denoted by x and represented on the vertical axis. When sketching graphs, use often the terms independent variable/dependent variable and
emphasize the fact that the actual symbols used for these variables do not affect the shape of the graph.
Word of caution: when using examples from Physics First in your mathematics class, make sure the terminology is the one students use in the physics class. For example, when discussing the motion
of an object along a straight line, there is a clear distinction between distance traveled and change in position and between speed and velocity. The distance traveled is the total length
traveled and is always positive, while change in position is the difference between the final position and the initial position, which may be negative. Speed is always positive while velocity may
be positive or negative, the sign indicates the direction in which the motion takes place.
• the subscripts used in Physics First are not always numerical
For example, v[f] and v[i] denote the final velocity and the initial velocity, respectively, as opposed to the v[2] and v[1] option that would be the preferred choice in a mathematics textbook.
• there is a considerable use of the symbol Δ in Physics First to denote change
For example, if x[i] denotes the initial (or starting) position and x[f] denotes the final (or ending) position, then Δx = x[f] − x[i] denotes the change in position. Similarly, the time change
from t[i] tot[f] will be denoted by Δt = t[f] − t[i].
2. Literal Formulas
• Use examples requiring students to substitute values into a given formula.
Example: Include examples with variables denoted by other letters than just x and y and whenever appropriate include formulas from the Physics First curriculum, such as: Evaluate3ma if m = 4 and
a = 7.5. More examples
3. Literal equations
• Use examples requiring students to solve for one variable in terms of the others. A good resource is the list of formulas studied in the Physics First class.
4. Linear functions
• slope as a rate of change of the dependent variable with respect to the independent variable
Emphasize the computation and interpretation of the slope of a line as a rate of change of the dependent variable with respect to the independent variable, specifying units whenever possible.
Include examples where the independent and dependent variables are denoted by other letters than x and y.
• changing the scale on the dependent/independent axis manipulates the graph’s appearance
Give a few examples to show that changing the scale on the dependent axis manipulates the graph’s appearance, making the slope of the graph of a linear function appear to be greater or lesser
than before the manipulation. As such, if the slopes of the graphs of two linear functions are to be compared only by looking at their graphs, the scales used for the two functions should be the
same. This also shows the importance of using units of measure for the variables involved. Examples are shown in Figs. 1 and 2 below, where the values of the slopes in both figures are the same,
even though the line in Fig. 1 looks steeper because of the scale of the vertical axis.
• linear functions = functions for which the rate of change over any interval is a constant
Stress the fact that linear functions are precisely those functions for which the rate of change over any interval is a constant which does not change from one interval to another. Help students
make the connection between constant ratios and linear relationships.
• uniform motion
Uniform motion problems typically involve something traveling at some fixed and steady (uniform) pace (rate or speed) and the main governing formula is d = st, where d stands for distance
(position), s stands for the (constant or average) speed, and t stands for time. As such, examples in the spirit of the ones in the Uniform Motion unit are well suited when discussing linear
5. Piecewise linear functions.
• Expose students to examples of motion (of a car, of a bicycle, etc.) depicted by piecewise linear graphs (“broken” lines). Use t (time) as the independent variable and x (position) as the
dependent variables, each with appropriate units of measure. (More examples)
6. Quadratic functions.
• provide students with an introduction to quadratic functions and their graphs (parabolas) as a
first-semester topic in Algebra I
Provide students with an introduction to quadratic functions and their graphs (parabolas) as a first-semester topic in Algebra I without solving quadratic equations based on the quadratic formula
or factoring (this is a topic that will likely be covered during the second semester; see the problems listed below to be used after you discuss solving quadratic equations).
• include evaluating quadratic expressions
For example, have them evaluate y = 5x^2 for x = 0, x = 2, x = 3 and graph parabolas using calculators.
• include the uniform accelerated motion under gravity (free fall)
After covering the quadratic formula and factoring, you can use problems related to uniform accelerated motions under gravity (free fall). (More examples)
7. Solving systems of equations.
• introduce systems of equations as a second-semester topic
In addition to solving the equation 2x−6 = 4−x (sometimes referred to as “solving simultaneous quations”) using symbol manipulation, consider having students graph the lines y[1] = 2x − 6and y[2]
= 4 − x and decide if the two lines intersect. Emphasize the connection between the solution for the original equation and the x coordinate of the intersection point of the two lines.
• use the trace feature of a graphing calculator to estimate the point of intersection of two given
Explore finding approximate solutions to systems of equations using the trace feature of graphing calculator to estimate the point of intersection of two given lines. Use the intersect function
if available on your calculator to compute exact solutions.
8. Area
• area of polygonal shapes
When determining the area of geometric figures, include examples of polygonal shapes for which the students do not have a direct area formula but which can be partition into familiar shapes
(e.g., triangles, rectangles), find the areas of each partition, and sum the areas to get the total area.
• area of the region under the graph of a piecewise linear function
Include a few in-class or homework problems in which students have to compute the area of the region under the graph of a piecewise linear function.
An example is shown in Fig. 4 below. The area under a velocity-time graph gives the displacement of the object. Thus the displacement in the time interval between t[i] and t[f] is given by the
area under the blue line. That area is the sum of the area of the rectangle A and the triangle B.
9. Measurement
• the metric system
Introduce elements of the metric system early in the school year, especially units of length (e.g., km, m, cm, mm).
• unit conversion
Provide several examples of unit conversions. Include problems in which the data given is not all in the same units of measure (e.g., if the speed of a cyclist is 2 meters per second, how long
will it take her to cover 1 kilometer?). Require students to include the unit of measure in every step when solving a problem that involves quantities given in units of measure.
10. Number and Operation
• include decimal numbers in examples and problems
For example, include the computation of the quotient of two decimal numbers in computing the slope of a line or evaluating algebraic expressions for decimal values.
• Encourage proportional reasoning by including appropriate problems as you cover various | {"url":"https://physicsfirstmo.org/math-connections/","timestamp":"2024-11-13T21:40:46Z","content_type":"text/html","content_length":"54807","record_id":"<urn:uuid:f5649bb3-0f5c-4626-b23f-9b19ea6d3d29>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00829.warc.gz"} |
Numerical Methods
The Learning Management System Canvas (Instructure) Lacks Key Feature in Quizzes
Canvas lacks a key feature in its quiz option which many STEM (science, technology, engineering and mathematics) instructors have requested. In fact, I consider the lack of this feature as a bug. The
error margins in setting up quiz problems based on a formula are based on true error and not on RELATIVE true error.
In Spring 2014, the university I teach at, University of South Florida is migrating from the current learning management system of Blackboard to Canvas. It has been a welcome change but Canvas lacks
a key feature in its quiz option which many STEM (science, technology, engineering and mathematics) instructors have requested. In fact, I consider the lack of this feature as a bug.
To give you an example, one of the options in making a quiz is called the formula question. This option is attractive to STEM instructors as one can develop a question whose correct answer is
numeric but based on a formula.
For example, one may ask the question what is x/3 and the instructor can choose a range of input values of x (say 1 to 100,000). The quiz option allows the instructor to generate up to 200
combinations with x being chosen randomly in the selected range. The quiz option asks for an error margin for the combinations. But here is the problem: The error margins are based on true error
and not on RELATIVE true error. An error margin of +-1 may be acceptable for a question that asks – “What is 10000/3?” but not for “What is 1/3?”. This issue could be easily resolved by making the
error margin to be a RELATIVE error margin.
The above example is simple to rectify as one may argue that one could use an error margin of +-0.001 and it would work reasonably well for all possible value of x, but for problems with many
intermediate steps, where students could be carrying varying number of significant digits, the prescribed error margins can create issues with a correct answer being deemed incorrect.
I have tweeted to Instructure founders, submitted tickets to Canvas help, asked our LMS division in our university to communicate this issue, but I need your help in making this feature request
popular. If you have your own CANVAS account (it is free to make one) or are using CANVAS as a student or a faculty member at your university, go to http://help.instructure.com/home, login and click
on ME TOO button on this link: http://help.instructure.com/entries/21499124-Express-the-Error-Margin-in-Formula-Question-as-a-of-the-Result
Thank you.
This post is brought to you by
A MOOC on Introduction to Numerical Methods
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Start your journey today whether you are learning numerical methods for the first time or just need a refresher. Unlike other MOOCs, you have a lifetime access to the course. Ask questions within
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About: Numerical methods are techniques to approximate mathematical procedures (example of a mathematical procedure is an integral). Approximations are needed because we either cannot solve the
procedure analytically (example is the standard normal cumulative distribution function) or because the analytical method is intractable (example is solving a set of a thousand simultaneous linear
equations for a thousand unknowns).
Materials Included: Textbook Chapters, Video Lectures, Quizzes, Solutions to Quizzes
How Long to Complete: About 40 hours of lectures need to be watched and estimated time to read textbook and do quizzes is 80 hours. It is a typical 15-week semester length course.
Course Structure: For each section, you have video lectures, followed by a textbook chapter, a quiz and solutions to quizzes.
This post is brought to you by Holistic Numerical Methods: Numerical Methods for the STEM undergraduate at http://nm.MathForCollege.com, the textbook on Numerical Methods with Applications available
from the lulu storefront, the textbook on Introduction to Programming Concepts Using MATLAB, and the YouTube video lectures available at http://nm.MathForCollege.com/videos. Subscribe to the blog via
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Patience Algosaurus.
Data structures by themselves aren’t all that useful, but they’re indispensable when used in specific applications, like finding the shortest path between points in a map, or finding a name in a
phone book with say, a billion elements (no, binary search just doesn’t cut it sometimes!).
Oh, and did I mention that they’re used just about everywhere in software systems and competitive programming?
This time, we only have two levels and a bonus, since this is an article on just the basics of data structures. Having a Mastery level just doesn’t make sense when there’s a ridiculous number of
complicated data structures.
Say hello to Loopie.
Loopie enjoys playing Hockey with her family. By playing, I mean…
When the turtles are shucked into the goal, they are deposited back on top of the pile.
Evidently, Loopie’s family likes sliding on ice.
Notice how the first turtle added on the pile, is the first turtle to be ejected.
This is called a queue.
Similar to a real queue, the first element which was added to the list, will be the first element out. This is called a FIFO (First In First Out) structure.
Insertion and deletion operations?
q = []
def insert(elem):
q.append(elem) #inserts elem into the end of the queue
print q
def delete():
q.pop(0) #removes 0th element of the queue
print q
After a fun-filled afternoon playing Hockey, Loopie is making pancakes for everyone. She places all the pancakes in a similar pile.
Then serves them to the family one by one.
Notice how the first pancake she made, is the last one she serves.
This is called a stack.
The first element which was added to the list, will be the last one out. This is called a LIFO (Last In First Out) structure.
Insertion and deletion operations?
s = []
def push(elem): #insertion in a stack is called 'pushing' into a stack
print s
#deletion from a stack is called 'popping' from a stack
#pop is already a predefined function in Python for all arrays, but we'll still define it here for learning purposes as customPop()
def customPop():
print s
Ever seen a density column?
All the items from top to bottom, are in ascending order of their densities. What happens when you drop an object of arbitrary density into the column?
It settles to the correct position on its own, due to difference in densities in the layers above and below it.
Heaps are something of that ilk.
A heap is a complete binary tree, meaning every parent has two children. Even though we visualize it as a heap, it is implemented through a regular array.
Also, a binary tree is always of height
This is a max-heap, where the fundamental heap property is that the children of any parent node, will be smaller than the parent node itself. In min-heaps, the children are always larger than
the parent node.
A few basic function definitions:
global heap
global currSize
def parent(i): #returns parent index of ith index
return i/2
def left(i): #returns left child of ith index
return 2*i
def right(i): #returns right child of ith index
return (2*i + 1)
Let’s tackle this part-by-part.
1) Inserting an element into a pre-existing heap
We first insert the element into the bottom of the heap, ie. last index in the array. Then we repeated apply the heap property on the index of the element till it reaches the appropriate position.
The algorithm is as follows:
1. Add the element to the bottom level of the heap.
2. Compare the added element with its parent; if they are in the correct order, stop.
3. If not, swap the element with its parent and return to the previous step.
def swap(a, b): #to swap a-th and b-th elements in heap
temp = heap[a]
heap[a] = heap[b]
heap[b] = temp
def insert(elem):
global currSize
index = len(heap)
currSize += 1
par = parent(index)
flag = 0
while flag != 1:
if index == 1: #we have reached the root of the heap
flag = 1
elif heap[par] > elem: #if parent index is larger than index of elem, then elem has now been inserted into the right place
flag = 1
else: #swaps the parent and the index itself
swap(par, index)
index = par
par = parent(index)
print heap
The maximum number of times this while loop can run, is the height of the tree itself, or
2) Extracting the largest element from the heap
The first element of the heap is always the largest, so we just remove that and replace the top element with the bottom one. Then we restore the heap property back to the heap, through a function
called maxHeapify().
1. Replace the root of the heap with the last element on the last level.
2. Compare the new root with its children; if they are in the correct order, stop.
3. If not, swap the element with one of its children and return to Step 2. (Swap with its smaller child in a min-heap and its larger child in a max-heap.)
def extractMax():
global currSize
if currSize != 0:
maxElem = heap[1]
heap[1] = heap[currSize] #replaces root element with the last element
heap.pop(currSize) #deletes last element present in heap
currSize -= 1 #reduces size of heap
return maxElem
def maxHeapify(index):
global currSize
lar = index
l = left(index)
r = right(index)
#print heap
#finds the larger child of the index; if larger child exists, swaps it with the index
if l <= currSize and heap[l] > heap[lar]:
lar = l
if r <= currSize and heap[r] > heap[lar]:
lar = r
if lar != index:
swap(index, lar)
Again, the maximum number of times maxHeapify() can be executed, is the height of the tree itself, or
3) How to make a heap out of any random array
Okay, so there’s two ways to go about it. The first way is to just repeatedly insert every element into the previously empty heap.
This is easy, but relatively inefficient. The time complexity of this comes out to be
There’s a better, more efficient way of doing this, where we simply maxHeapify() every ‘sub-heap’ from the
This runs at
def buildHeap():
global currSize
for i in range(currSize/2, 0, -1): #third argument in range() shows increment factor, here -1
print heap
currSize = len(heap)-1
Ah, we’ve been leading up to this question this entire time.
Heaps are used to implement an efficient sort of sort, unsurprisingly called, the Heapsort.
Unlike the sorely inefficient Insertion Sort and Bubble Sort with their measly
It’s not even complicated, just keep extracting the largest element from the heap till the heap is empty, placing them sequentially at back of the array where the heap is stored.
def heapSort():
for i in range(1, len(heap)):
print heap
heap.insert(len(heap)-i, extractMax()) #inserting the greatest element at the back of the array
currSize = len(heap)-1
To tie it all together, I’ve written a few lines of helper code to input elements into the heap and try out all the functions. Check it out right here. Oh, and for all the people who are acquainted
with classes in Python, I’ve also written a Heap class here.
Voila! Wasn’t that easy? Here’s a partying Loopie just for coming this far.
We also use heaps in the form of priority queues to find the shortest path between points in a graph using Dijkstra’s Algorithm, but that’s a post for another day.
Loopie wants to teach her baby turtles how to identify shapes and colours, so she brings home a large number of pieces of different colours.
This took them a lot of time, as well as confusion.
So she gets another toy to make the process easier.
This was easier because the babies already knew that the pieces were categorized according to shape. What if we labeled each of the poles as follows?
The babies would just have to check for the pole number, then look through a far smaller number of pieces on the pole.
Now imagine one pole for every combination of colour and shape possible.
Let the pole number be calculated as follows:
purple triangle:
p+u+r+t+r+i = 16+21+18+20+18+9 = Pole #102
red rectangle:
r+e+d+r+e+c = 18+5+4+18+5+3 = Pole #53
We know that 6*26 = 156 combinations are possible (why?), so we’ll have 156 poles in total.
Let’s call this formula to calculate pole numbers, the hash function.
In code:
def hashFunc(piece):
words = piece.split(" ") #splitting string into words
colour = words[0]
shape = words[1]
poleNum = 0
for i in range(0, 3):
poleNum += ord(colour[i]) - 96
poleNum += ord(shape[i]) - 96
return poleNum
If we ever need to finish where ‘pink square’ is kept, we just use
hashFunc('pink square')
and check the pole number, which happens to be pole #96.
This is an example of a hash table, where the location of an element is stored in terms of a hash function. The poles here are analogous to buckets in proper terminology.
This makes time taken to search for a particular element independent of the total number of elements, ie.
Let this sink in. Searching in a hash table can be done in constant time.
What if we’re searching for a ‘dreary-blue rectangle’, assuming a colour called ‘dreary-blue’ exists?
hashFunc('dreary-blue rectangle')
returns pole #53, which clashes with the pole number for ‘red rectangle’. This is called a collision.
How do we resolve it?
We use a method called separate chaining, which is a fancy way of saying every bucket consists of a list, and we simply search through the list if we ever find multiple entries.
Here, we’ll just put the dreary-blue rectangle on top of the red rectangle, and just pick either one whenever we need to.
The key in any good hash table, is to choose the appropriate hash function for the data. This is unarguably the most important thing in making a hash table, so people spend a lot of time on designing
a good hash function for their purpose.
In a good hash table, no bucket should have more than 2-3 entries. If there are more, then the hashing isn’t working well, and we need to change the hash function.
Searching is independent of the number of elements for god’s sake. We can use hash tables for just about anything involving a gigantic number of elements, like database entries or phone books.
We also use hash functions in searching for strings or sub-strings in large collections of texts using the Rabin-Karp algorithm. This is useful for detecting plagiarism in academic papers by
comparing them against source material. Again, a post for another day!
I plan on writing more on advanced data structures like Fibonacci Heaps and Segment Trees and their uses, so subscribe to Algosaurus for updates!
I hope you found this informal guide to the basics of data structures useful. If you liked it, want to insult me, or if you want to talk about dinosaurs or whatever, shoot me a mail at
Introduction to Algorithms – Cormen, Leiserson, Rivest, and Stein (pages 151 – 161) | {"url":"http://algosaur.us/category/basic/","timestamp":"2024-11-13T19:30:10Z","content_type":"text/html","content_length":"62193","record_id":"<urn:uuid:e1c62c13-c3c0-44a1-9c82-e798d5ec1d11>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00716.warc.gz"} |
Learning How to Subtract Fractions - Smartick
In this post, we are going to see how to subtract fractions, but for that, you first have to know what the denominator and numerator of a fraction are.
The numerator is the number that is written on top of the fraction and the denominator is the number written on the bottom of the fraction.
To subtract fractions, it is necessary that the fractions have the same denominator. Once the fractions have the same denominator, we just subtract the numerators. finding their least common multiple
Let’s look at an example:
Since the denominators are different, 4 and 6, we have to find their least common multiple (LCM). You can review how to calculate the LCM on this previous post on our blog.
LCM (4,6) = 12
The two new fractions will have 12 as a denominator.
In order to find the numerator of each new fraction, divide the new denominator (the LCM that we have found) by the old denominator and multiply the answer by the old numerator.
The first fraction:
So now we have:
As both fractions have the same denominator, now we can subtract the numerators and leave the same denominator:
And the result is one-twelfth.
If you have liked this post, share it with your friends so that they can also learn how to subtract fractions.
And if you want to learn more elementary math, try Smartick for free!
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How to calculate a rate of sale
In business it is important to be able to calculate a rate of sale. If the rate of sale is declining from one year to the next, it’s important to identify why there is a variance. Adjustments can
sometimes be made, such as improving advertising or eliminating a product or service, so the rate of sale can increase. You can calculate the rate of sale if you have certain sales information
pertaining to your business. Economic trends could be responsible for changes in the rate of sale.
Find out the time frame you are calculating the rate of sale for. The rate of sale can be calculated daily, monthly, quarterly or annually. Get the correct formula, using the time frame of your
choice, to calculate the rate of sale.
• In business it is important to be able to calculate a rate of sale.
• Get the correct formula, using the time frame of your choice, to calculate the rate of sale.
Identity the amount of product sold for each period. For example, if you sold £20 million of product in April and £75 million in the month of May, the sales rate can be calculated on a monthly basis.
Review the sales rate formula. To get the sales rate, subtract the previous month's sales from the current month's sales. Divide by the previous month's sales, and then multiply the result by 100.
Based on the previous example, you would take £20 million and subtract it from £75 million and get £55 million. The £55 million represents the growth. Divide the £55 million by £20 million, which is
2.75; multiply this result by 100 to get a 275 per cent rate of sales growth from April to May.
• Identity the amount of product sold for each period.
• Divide the £55 million by £20 million, which is 2.75; multiply this result by 100 to get a 275 per cent rate of sales growth from April to May. | {"url":"https://www.ehow.co.uk/how_6809257_calculate-rate-sale.html","timestamp":"2024-11-07T12:07:28Z","content_type":"text/html","content_length":"120555","record_id":"<urn:uuid:5d347ba6-3525-4645-acfc-9680f112d954>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00648.warc.gz"} |
The Legacy of erone - Today Market Price
The Legacy of erone
Erone, additionally called Heron of Alexandria, became a Greek mathematician and engineer who lived at some stage in the first century AD. Not a lot is understood approximately his personal
lifestyles, but his contributions to arithmetic and engineering have left a lasting impact on the sphere. Erone’s work became in particular influential in the areas of geometry and trigonometry, and
his maximum well-known contribution is the formula that bears his name – Erone’s components.
Erone’s formula is used to calculate the place of a triangle based totally on the lengths of its sides. This formula has been broadly used in the course of history and maintains to study in schools
these days. Erone’s contributions to arithmetic have not most effective superior the field however have also had sensible packages in various industries. His paintings has stood the check of time and
continues to encourage mathematicians and engineers nowadays.
Erone’s Contributions to Mathematics: A Brief Overview
Erone made extensive contributions to diverse branches of arithmetic, along with geometry and trigonometry. In geometry, he evolved formulation for finding the region of numerous shapes, together
with triangles, quadrilaterals, and polygons. His work on this region laid the muse for contemporary geometric principles and calculations.
In trigonometry, Erone evolved methods for solving triangles the use of trigonometric ratios. He additionally derived formulas for finding the lengths of facets and angles in right triangles. These
contributions have been groundbreaking on the time and shaped the basis for in addition advancements in trigonometry.
Erone’s work in arithmetic changed into now not most effective theoretical but additionally practical. He applied his mathematical understanding to remedy actual-global problems, together with
calculating the quantity of solids and finding the facilities of gravity for diverse items. His practical method to arithmetic made his work exceptionally influential and relevant in many fields.
Erone’s Formula: Understanding the Basics
Erone’s method, additionally referred to as Heron’s formula, is used to calculate the location of a triangle whilst the lengths of its facets are recognized. The system is as follows:
Area = √(s(s-a)(s-b)(s-c))
Where s is the semiperimeter of the triangle, and a, b, and c are the lengths of its aspects.
The semiperimeter is calculated through including the lengths of all three aspects and dividing by way of 2:
s = (a + b + c) / 2
Once the semiperimeter is decided, it is able to be substituted into the formula to discover the location of the triangle. The system works for all forms of triangles, along with equilateral,
isosceles, and scalene triangles.
Erone’s formulation is derived from Heron’s evidence of the Pythagorean theorem. It is a powerful tool for calculating the location of triangles without needing to realize their heights or angles.
This makes it specially useful in conditions wherein only the lengths of the edges are recognised.
How Erone’s Formula is Used in Real Life Applications
Erone’s method has numerous real-lifestyles programs in various fields. One realistic application is in creation and structure. Architects and engineers often want to calculate the regions of
irregularly formed triangles when designing systems. Erone’s formulation lets in them to do this appropriately and efficaciously, ensuring that materials are used effectively and systems are built to
right specifications.
Another application of Erone’s method is in surveying and land measurement. Surveyors use triangles to degree distances and angles on land, and Erone’s formula facilitates them calculate the regions
of those triangles appropriately. This records is vital for determining property boundaries, planning infrastructure initiatives, and assessing land fee.
Erone’s method also finds programs in physics and engineering. For instance, it can be used to calculate the location of a pass-segment in fluid dynamics or to determine the floor vicinity of
irregularly fashioned objects in materials technology. The formulation’s versatility and simplicity make it a treasured tool in various medical and engineering disciplines.
Erone’s Influence on Geometry: Exploring the Connections
Erone’s paintings in geometry had a profound affect on the field. His formulation for locating the location of numerous shapes, inclusive of triangles, quadrilaterals, and polygons, laid the muse for
modern-day geometric principles and calculations.
Erone’s components for finding the vicinity of a triangle is based totally on the idea of semiperimeter, which remains extensively used in geometry these days. The semiperimeter allows for a greater
green calculation of the place with no need to know the peak or angles of the triangle. This concept has been prolonged to other shapes, including quadrilaterals and polygons, wherein the
semiperimeter is used to calculate their regions as properly.
Furthermore, Erone’s paintings in geometry paved the way for in addition improvements within the field. His formulation and strategies furnished a framework for destiny mathematicians to construct
upon and extend. Today, his paintings continues to be taught in colleges and bureaucracy the idea for know-how geometric ideas and calculations.
Erone’s Contributions to Trigonometry: The Basics
In addition to his work in geometry, Erone made massive contributions to trigonometry. Trigonometry is the department of arithmetic that offers with the relationships between angles and facets of
Erone advanced techniques for solving triangles using trigonometric ratios. He derived formulas for locating the lengths of facets and angles in proper triangles primarily based on recognized
statistics. These formulation, known as trigonometric identities, are nonetheless used these days in various programs.
Erone’s paintings in trigonometry was groundbreaking at the time and laid the muse for in addition improvements inside the subject. His formulation and strategies furnished a systematic technique to
fixing triangles and understanding their properties. Today, trigonometry is an crucial department of mathematics utilized in fields including physics, engineering, and navigation.
Erone’s Formula and Trigonometry: A Closer Look
Erone’s formulation is closely related to trigonometry, as it may be derived the use of trigonometric identities. The system may be expressed in phrases of the lengths of the sides of a triangle and
the angles among them.
By applying trigonometric identities, such as the Law of Cosines, Erone’s formula may be derived. The Law of Cosines states that during any triangle, the rectangular of one facet is identical to the
sum of the squares of the opposite facets minus two times the manufactured from their lengths and the cosine of the protected angle.
Using this identity, Erone’s method may be derived through rearranging the equation to remedy for the location of the triangle. The resulting formula is the same as the only formerly cited:
Area = √(s(s-a)(s-b)(s-c))
Where s is the semiperimeter of the triangle, and a, b, and c are the lengths of its sides.
Erone’s Legacy in Modern Mathematics: How His Work Continues to Inspire
Erone’s work maintains to persuade modern-day arithmetic in various methods. His formulas and strategies are nonetheless taught in colleges and shape the idea for information geometric and
trigonometric concepts.
In addition to his particular contributions, Erone’s technique to mathematics has had a long-lasting effect. He emphasized sensible applications and hassle-solving, which has become a essential
element of mathematical schooling. Erone’s work serves as a reminder that mathematics isn’t simply an summary concept but has actual-global programs that can resolve realistic issues.
Furthermore, Erone’s work has inspired similarly improvements in arithmetic. His formulation and strategies were extended upon and refined through next mathematicians, main to new discoveries and
programs. Erone’s legacy isn’t always just in his precise contributions but also in how his work has sparked in addition exploration and innovation in arithmetic.
Erone’s Impact on Education: How His Formula is Taught Today
Erone’s method remains taught in faculties today as a part of the arithmetic curriculum. It is typically added in geometry lessons, in which students study the houses of triangles and a way to
calculate their areas.
The components is frequently offered as a sensible device for finding the region of triangles without having to recognise their heights or angles. Students are taught the way to follow the formula to
different styles of triangles, along with equilateral, isosceles, and scalene triangles.
In addition to teaching the components itself, educators also emphasize the underlying standards and principles in the back of Erone’s components. Students study the idea of semiperimeter and the way
it pertains to the calculation of the place of a triangle. This helps them develop a deeper understanding of geometric ideas and calculations.
Erone’s Formula and Beyond: The Future of Mathematics
Erone’s formula is still used and studied by way of mathematicians nowadays, and its packages are not confined to geometry and trigonometry. The system has been prolonged to different areas of
arithmetic, inclusive of algebraic geometry and complex analysis.
In algebraic geometry, Erone’s formula has been used to look at the residences of algebraic curves and surfaces. It offers a manner to calculate the vicinity of those objects and apprehend their
geometric houses.
In complex analysis, Erone’s formula has been implemented to observe the conduct of complex features. Complex evaluation deals with capabilities that have complicated variables, and Erone’s
components presents a way to calculate the location enclosed via these features.
Furthermore, Erone’s components has ability packages in different scientific fields, which include pc pics and photograph processing. The formula can be used to calculate the regions of irregularly
formed items in digital images or 3-D models, allowing for more accurate measurements and evaluation.
The Enduring Importance of Erone’s Contributions
Erone’s contributions to arithmetic, particularly within the regions of geometry and trigonometry, have had a long-lasting impact on the sphere. His formulas and techniques are still taught in
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Marsbarn Designs...
Common Core. Those are naughty words in a some circles. These are just a few thoughts about Common Core Math. (Common Core English has totally different issues that are beyond my purview. )
I’m not a math teacher. I don't even play one on TV. My experiences with the evolution of the “new” math of the Common program is gathered through working with my children and my tutorees. This is
my fifth year tutoring, and each year I’ve experienced the curriculum change through my students. I primarily work with 5th through 9th graders, and the way my students are spaced, I have a nice
cross-section of materials and the way they are evolving. My thoughts about the math program are my own, and are based solely on my own observations.
First off, Common Core took a harpoon gun and nailed both of it’s feet to the deck of the Titanic when they utterly failed to include parents in the how’s and why’s of Common Core Math. I’ve seen
some really interesting and thoughtful reinventions of how math is taught as the new lessons have been fleshed out, but I can’t name a single (non-teacher) parent who has any idea what their child is
doing, or why. I sincerely LOVE some of these new innovations, but by not educating parents with their children, the designers of this plan have lost their best supporters. Are all parents
interested in the details of how math is taught to their children? No. But, those parents who distrust mysterious re-writing of techniques they were raised on, or who actually want to help when
homework is frustrating? Those are the parents who make up the icebergs that sink unsinkable education programs.
Through my layman’s observations, the Common Core’s mission is to deepen the understanding of students in mathematical thinking. They want students to understand the greater how’s and why’s of
numbers and functions. Instead of taking on faith or blind acceptance, they want young people to have a profound knowledge which is backed up by a framework of mathematical thinking. For example,
when I was in grade school we were taught to divide by fractions using the mystical axiom: “Invert and multiply, ours is not to wonder why.” The new curriculum seeks to teach division with fractions
by first teaching multiple ways of seeing fractions, then to teach the interrelated relationship of multiplication and division through a range of activities. Only with this base do you encourage
the students to formulate their own understanding of division with fractions. All of this teaching is done through multiple methods, which might include physical objects, grids, arrays, coded
shapes, words and, finally, numbers.
This brings me to my second serious issue with Common Core math. I love learning multiple ways to solve math problems. I love the ideal of deeper understanding. I am a person who can wake in the
night thinking about how many different ways a sum can be calculated, BUT, as much as I appreciate the very thoughtful building of ideas that I see coming from this program, I am often frustrated
that there is too little time left to consolidate ideas. When the day ends, and the understanding has been built, the student must be able to do the simple conjuring of facts. They must be able to
numerically divide fractions. The algorithm of invert and multiply must be known. It is not prudent, in higher math, to need to draw illustrations to divide fractions. I find students who
understand the Why, but they cannot just Do. The Guess and Check process lingers way too long. There isn’t enough emphasis on just Doing The Math. And this lack of nitty gritty doing is another
element of this curriculum that frustrates parents who were raised to simply solve the problem.
My third issue: testing. Give me the latitude to use this comparison. The ideals of Common Core Math are akin to the ideals of Communism. (Wait, take a deep breath.) There is beauty in the idea of
Common Core Math, just as there is beauty in the idea of true, pure Communism. Communism, the idea of co-ownership, of communal living, of pure-hearted workers striving to live together with the
single goal of doing their best, what is not to admire? The rub is when Communism becomes a reality and all of that lack of freedom and poverty kicks in. It doesn’t live on a large scale like it
does in theory. Common Core Math, with its goal of deep, fluid thinkers who fully grasp mathematical theory is utopian. The dissolution point is testing. How do you test thought processes? If you
encourage multiple paths to understanding, how do you demonstrate that you’ve all arrived at the same destination? Because on the Common Core tests, you can't just given an answer and show your
work, they want you to say "WHY". I would say the majority of school children simply do not have the capacity to explain how they think with clarity. Just as Communism strips it’s participants of
rights and freedoms, Common Core Testing undermines the tenants it tries to instill. It says, "Think in the these grand expansive ways, and then give me this specific wording as you jump through
institutional hoops." If you could remove the tests from the equation, perhaps Common Core could be as idyllic as it intends to be.
I have other, wee beefs with individual elements of curriculum here and there, but these three issues are the big ones for me. Educate and include the parents, make sure the algorithm can be used
after the base is built, and let go of the testing.
My intent is to follow up this blog post with other posts hi-lighting techniques in the new Core that I LOVE. I mean, seriously, love. As in, I could build little shrines to. The build up to
quadratic equations? I am filled with envy that I did not learn them this way. It’s like finding out that you don’t have to worship a math God that mocks and requires flesh sacrifices. Instead,
there is a kind and benevolent Goddess that just wants you to understand. (Of course, I firmly believe that quadratics should be able to be factored without divine intervention. My point is, now
that is possible.) | {"url":"https://marsbarn.typepad.com/marsbarn_designs/2015/02/?asset_id=6a00d83500dad953ef01b7c753980a970b","timestamp":"2024-11-07T09:47:02Z","content_type":"application/xhtml+xml","content_length":"37837","record_id":"<urn:uuid:0f0464fb-444a-42a1-8e8d-5bd8b4921fc3>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00882.warc.gz"} |
[EM] “Monotonic” Binomial STV
Richard Lung voting at ukscientists.com
Sun Feb 27 11:08:05 PST 2022
Thank you for correcting me. I do tend to forget the square root, to
finish the average, the geometric mean.
In some formalisms, a zero numerator implies zero, but the geometric
mean, unlike the arithmetic mean, does not work wih zero, so that result
cannot be infered from it.
The trouble with just putting infinity is that there are different
infinities! One could require 1 vote for a candidate, from the
candidates themselves. Then we have a standard of comparison.
Glancing at your example, tho, I am reminded of an example with small
numbers, in which I had to reduce the 1 vote minimum to 0.1. Traditional
STV resorted to this expedient for small numbers elections, with the
Droop quota. They could not add plus one, because that made the quota
too hard for candidates to win. So, they resorted to a final plus 0,001,
I believe. But then ERS Ballot Services Major Frank Britton realised
that the final constant was never needed. And it is true, in this case,
of Binomial STV that a minimum candidate vote is not needed, as I think
you have suggested.
Moreover, if candidates have a minimum vote, they must also have the
same reverse preference minimum, for the sake of symmetrical treatment -
and perhaps as a neutralising factor.
I don't know yet what will turn out to be the most elegant count
instructions, in this and, no doubt, other instances. Your method
designating keep value infinity may be better, because such results are
nowhere in it, anyway. And it cuts out a troublesome added minimum
constant to candidates votes. I guess your result is correct. One has to
be a bit careful, in general, tho. An extremely bad election count may
be somewhat redeemed by a tolerable exclusion count, getting nowhere
near an exclusion quota. In that case, the not popular but also not
unpopular candidate is perhaps entitled to a quantitative tabulation.
This quandry reminds me of the caution, from an stv count expert, to use
floating point arithmetic in computer coding stv. Meek method, in New
Zealand, uses decimal point, but that might create future difficulties.
Anyway, I appreciate how important it is not to under-estimate the
possible ill consequences of casually considered operations.
The New Zealand government has not made its Meek method open source.
They denied access. Dr David Hill made his coding, of Meek method, open
source. He is a direct descendant of Thomas Wright Hill, who published
the first known instance of tranferable voting (barring the Gospel
Incident of the loaves and fishes). To celebrate the 200th anniversaey,
I re-published this and the public domain code by David Hill. Smashwords
does not allow publishing public domain works, so I had to put it on
Amazon, who charge a minimal fee. However, I could perhaps put it on
archive.org who don't charge, and where I put many of my e-books in pdf
Dr David Hill wrote his code, for Meek method, in Pascal, an early
script. But I did not include his code text for eliminated candidates
last past the post, when the quota surpluses run out. Nor did I include
the code for reducing the quota, when preference voting gives way to
abstentions. Binomial STV does not use either of these expedients.
But Meek method does calculate the keep values of elected candidates:
the quota divided by their total transferable vote, which may increase
with further preferences after a quota is already achieved. The lower
the keep value below unity, the greater the popularity.
Binomial STV greatly extends the Meek method use of keep values, to all
candidates, and for their exclusion, as well as their election. However,
there is no difference in principle to these extended operations, of
transferble voting, by keep value.
The hand count version of binomial stv, tho, drops the distinctively
computerised Meek count of post-quota preference counting. And sticks to
the first order binomial count, making it simpler than traditional
counts, as well as Meek stv.
The New Zealand government hired two software coding firms, as back-up,
paying both, but only using one of them. This shows how arduous and
uncertain the results of their making Dr Hill coding texts executable.
Richard Lung.
On 27/02/2022 13:41, Kristofer Munsterhjelm wrote:
> On 27.02.2022 14:04, Richard Lung wrote:
>> Thank you, Kristofer,
>> for first example.
>> The quota is 100/(1+1) = 50.
>> Election keep value is quota/(candidates preference votes)
>> for A: 50/51
>> B: 50/49
>> C: 50/0 Which, of course is infinite. It may be convenient, for tidy
>> book-keeping, that small elections require that each candidate votes for
>> themself. Then the keep value maximum simply equals the quota.
>> Generally, it is not necessary to make this stipulation, for large scale
>> elections, because no candidate, however miserable, ever gets no votes.
> Another option is to just let infinities be worse than any alternative.
> Since not every candidate can have a zero last preference count, at
> least one candidate must have a finite value and so would be considered
> better than every candidate with an infinite value.
>> Exclusion keep value equals quota/(candidates reverse preference vote):
>> A: 50/1
>> B: 50/0
>> C: 50/99
>> Geometric mean keep value ( election keep value multiplied by inverse
>> exclusion keep value):
>> A: 50/51 x 1/50 ~ 0,0196
>> B: 50/49 x 0/50 = 0/49 is indeterminate. The closest determinate
>> approximation gives 1/49, not quite as low a keep value as 1/51 for A,
>> who is therefore the winner.
> Is 0/49 indeterminate? Shouldn't it just be zero? 0/x = 0 for x not
> equal to zero, and the square root of zero is zero.
> But let me in any case revise my example. Who wins in this one?
> 50: A>B>C
> 47: B>A>C
> 2: B>C>A
> 1: A>C>B
> My calculations are as follows:
> The quota is 50.
> Election keep value is quota/candidate preferences:
> A: 50/51
> B: 50/49
> C: infinity
> Exclusion keep value equals quota/candidates reversed first preferences:
> A: 50/2
> B: 50/1
> C: 50/97
> Geometric mean:
> A: square root of (50/51 x 2/50) ~ 0.198
> B: square root of (50/49 x 1/50) ~ 0.143
> C: ~= infinity (or very high)
> So B wins, having the lowest keep value. Is this correct?
> (You seem to have omitted the square root in your calculations, but it
> shouldn't make a difference. Without the square root, A and B's values
> are 0.0392 and 0.0204 respectively.)
> -km
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Nonequilibrium steady state for harmonically-confined active particles
Add to your list(s) Download to your calendar using vCal
If you have a question about this talk, please contact Camille Scalliet.
Zoom link: https://zoom.us/j/98016675669
Active particles consume energy from their environment and turn it into directed motion, leading to remarkable non-equilibrium effects. In this talk I will mostly focus on the run-and-tumble particle
(RTP) model, which mimics the persistent motion of bacteria such as E. Coli. I will present recent results for the nonequilibirum steady state that a single RTP reaches when confined by an external
harmonic potential. In the first part of the talk, I will present the exact steady state distribution of the position of a particular type of overdamped RTP in two dimensions, whose orientation can
take one of four possible values. What enables the exact solution is that, in a proper choice of coordinates, the problem decomposes into two decoupled one-dimensional problems. In the second part of
the talk, I will go beyond the overdamped regime, and focus on the limit in which the RTP switches its orientation very fast. I will first recall that typical fluctuations of its position obey a
Boltzmann distribution with an effective temperature that can be found exactly. Next, I will consider the large deviations regime which is not described by a Boltzmann distribution, and is instead
dominated by a single, most likely trajectory in a coarse-grained dynamical description of the system.
The talk is based on the two recent papers: N. R. Smith, P. Le Doussal, S. N. Majumdar, G. Schehr, arXiv:2207.10445 N. R. Smith, O. Farago, arXiv:2208.06848
This talk is part of the DAMTP Statistical Physics and Soft Matter Seminar series.
This talk is included in these lists:
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Mastering the SUM Function: A Detailed Guide to Calculating Large Datasets with Looker Studio | Looker Studio
Exploring the POWER Function in Looker Studio: Syntax, Usage, Examples and Tips for Data Analytics
Unveiling the ACOS Function in Looker Studio
Mastering the DATETIME_SUB Function in Looker Studio: A Guide to Manipulate Date and Time Information in Your Datasets
Mastering the MIN Function in Looker Studio: A Comprehensive Guide on Data Analysis and Visualization Tools
Exploring YEAR Function in Looker Studio: A Detailed Guide to Extracting and Visualising Year Data | {"url":"https://www.catchr.io/university/looker-studio-course/mastering-the-sum-function-a-detailed-guide-to-calculating-large-datasets-with-looker-studio","timestamp":"2024-11-11T23:11:00Z","content_type":"text/html","content_length":"41294","record_id":"<urn:uuid:22644fe7-fdc4-4004-9538-3ea6ec86a6d9>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00370.warc.gz"} |
Fractional Order PID Controller Design for an AVR System Using Chaotic Yellow Saddle Goatfish Algorithm
Faculty of Electrical Engineering, University of Montenegro, 81000 Podgorica, Montenegro
Depto. De Ciencias Computacionales, Universidad de Guadalajara, CUCEI, Av. Revolucion 1500, Guadalajara, 44430 Jal, Mexico
IN3-Computer Science Dept., Universitat Oberta de Catalunya, 08018 Castelldefels, Spain
School of Computer Science & Robotics, Tomsk Polytechnic University, 634050 Tomsk, Russia
Author to whom correspondence should be addressed.
Submission received: 29 June 2020 / Revised: 15 July 2020 / Accepted: 16 July 2020 / Published: 18 July 2020
This paper presents a novel method for optimal tunning of a Fractional Order Proportional-Integral-Derivative (FOPID) controller for an Automatic Voltage Regulator (AVR) system. The presented method
is based on the Yellow Saddle Goatfish Algorithm (YSGA), which is improved with Chaotic Logistic Maps. Additionally, a novel objective function for the optimization of the FOPID parameters is
proposed. The performance of the obtained FOPID controller is verified by comparison with various FOPID controllers tuned by other metaheuristic algorithms. A comparative analysis is performed in
terms of step response, frequency response, root locus, robustness test, and disturbance rejection ability. Results of the simulations undoubtedly show that the FOPID controller tuned with the
proposed Chaotic Yellow Saddle Goatfish Algorithm (C-YSGA) outperforms FOPID controllers tuned by other algorithms, in all of the previously mentioned performance tests.
1. Introduction
The quality of electrical energy is the main demand from consumers in the power system. Since the indicators of the quality are voltage and frequency, these parameters must be maintained at the
desired level at every moment. Generally, in every power system, the frequency depends on the active power flow, while the reactive power flow has a greater impact on the voltage level. Additionally,
any deviation of the voltage from the nominal value requires the flow of the reactive power, which automatically increases line losses. The fluctuations of the voltage can be repressed using various
devices: serial and parallel capacitor banks, synchronous compensators, tap-changing transformers, reactors, Static VAr Compensators (SVC), and Automatic Voltage Regulators (AVR) [
]. This paper deals with AVR systems.
The AVR represents the main control loop for the voltage regulation of the synchronous generator (SG), which is the main unit for producing electrical energy in the whole power system. Concretely,
the control of the terminal voltage of the synchronous generator is achieved by adjusting its’ exciter voltage. Although the main task of the AVR is to provide stable voltage level at the generator’s
terminals, it is also very important in improving the dynamic response of the terminal voltage. Regardless of the fact that the control theory developed many modern control techniques, the
traditional PID controller is still the most used in the AVR systems. In general, in this paper, the optimal tuning of the controller is considered.
Enhancing the performance of the PID controller for AVR systems is possible by using fractional calculus. Fractional order PID controller (FOPID) is the general form of the PID controller that uses
fractional order of derivatives and integrals, instead of integer order. Moreover, FOPID can provide a better transient response and is more robust and stable compared to the conventional (known as
integer order controller) PID controller [
]. Due to the previously mentioned advantages of the FOPID, this paper deals with this type of controller.
The optimal design of the FOPID controller implies determining the parameters to satisfy defined optimization criteria (or fitness/objective function). In the available literature, the most used
method for optimal tuning of the FOPID controller is based on metaheuristic algorithms [
]. Particle Swarm Optimization (PSO) and Genetic Algorithm (GA) are applied in [
] to determine the optimal values of the FOPID parameters. For the same purpose, D. L. Zhang et al. proposed an Improved Artificial Bee Colony Algorithm (CNC-ABC) [
]. Also, it can be found that the authors used Chaotic Ant Swarm (CAS) [
], Multi-Objective Extremal Optimization (MOEO) [
], Cuckoo Search (CS) [
], and Salp Swarm Optimization (SSO) algorithm [
] to determine unknown parameters of the FOPID. Besides the FOPID, many existing studies deal with the optimization of the parameters of the classical PID controller for the AVR systems [
Another very important aspect of the optimization process that needs to be particularly reviewed is the choice of the fitness function. Previously mentioned algorithms introduce a huge variety of
fitness functions that take into account time-domain (rise time, settling time, overshoot, and steady-state error), as well as frequency domain parameters (gain margin, phase margin, gain crossover
frequency, and so on). One of the most common error-based functions is Integrated Absolute Error (IAE) [
]. Another commonly used time-domain criterion is Zwee Lee Gaing’s function originally proposed in [
] for PID controller tuning and applied in [
] for optimal tunning of the FOPID controller. Ortiz-Quisbert et al. used the complex function that tends to minimize only time-domain parameters: overshoot, settling time, and maximum voltage signal
derivative [
]. One of the most interesting approaches in a fitness function definition is combining error-based functions with time-domain parameters, as presented in [
]. Concretely, an interesting approach minimizes the Integrated Time Squared Error (ITSE) of the output voltage, energy of the control signal, and ITSE of the load disturbance [
]. The objective function in [
] is composed of IAE, steady-state error, and settling time, while in [
], the objective is to minimize not only IAE, steady-state error, and settling time as previously mentioned, but also the overshoot and the control signal energy. Fitness function that consists only
of the frequency domain parameters is proposed in [
] and tends to maximize phase margin and the gain crossover frequency. The trade-off between different frequency domain parameters (phase margin and gain margin) and time-domain parameters
(overshoot, rise time, settling time, steady-state error, IAE, and control signal energy) is formulated as an objective function in [
Although a large number of FOPID tuning techniques have been proposed in the available literature, the optimal design of the FOPID controller can still be improved by further research. To that end,
this paper proposes a novel design approach of the FOPID controller. The contributions of this work are highlighted as follows:
• Firstly, the recently proposed Yellow Saddle Goatfish Algorithm (YSGA) [
] is merged with Chaos Optimization Algorithm [
] in order to obtain novel Chaotic Yellow Saddle Goatfish Algorithm (C-YSGA). Original YSGA can improve the optimization process in terms of accuracy and convergence in comparison to several
state-of-the-art optimization methods. The improvement is proven by applying this method on five engineering problems, while the comparison with other methods is carried out by using 27
well-known functions [
]. Additionally, in this paper, the superiority of the original YSGA over several other metaheuristic techniques will be demonstrated on the particular optimization problem. Moreover, an
improvement of the YSGA by adding Chaotic Logistic Mapping is introduced. The purpose of merging two algorithms is to additionally improve the convergence speed of the YSGA algorithm. Therefore,
the original optimization algorithm for optimal tuning of the FOPID controller will be presented in this paper.
• Afterward, the new objective function that tends to optimize time-domain parameters has been proposed. It is demonstrated that the usage of the proposed objective function provides significantly
better results than the other functions proposed in the literature.
• Such an obtained FOPID controller has been compared with those tuned by different optimization algorithms in terms of transient response quality. The conducted analysis clearly demonstrates the
superiority of the FOPID controller tuned by C-YSGA.
• Finally, different uncertainties have been introduced to the system in order to examine its behavior. Precisely, the robustness test that implies changing the AVR system parameters is carried
out. Also, the ability of the system to cope with the different disturbances (control signal disturbance, load disturbance, and measurement noise) is investigated. During all of the mentioned
tests, the FOPID controller tuned by C-YSGA shows significantly better performances compared to the FOPID controller, whose parameters are optimized by the other algorithms considered in the
The organization of this paper is as follows. A brief overview of the AVR system, along with the performance analysis, is provided in
Section 2
Section 3
demonstrates the basics of the fractional-order calculus, which is needed for simulating a FOPID controller. Afterward, a compact and wide overview of the available literature related to FOPID
parameters optimization is given in
Section 4
Section 5
shows the mathematical formulation of the novel C-YSGA algorithm that is presented in this paper. The results of the simulation are given in
Section 6
. Conclusions are provided in
Section 7
2. Description of the AVR System
The primary function of an AVR system is to maintain the terminal voltage of the generator at a constant level through the excitation system. However, due to the different disturbances in the power
system, a synchronous generator does not always work at the equilibrium point. Such oscillations around the equilibrium state can cause deviations of the frequency and the voltage, which can be very
harmful to the overall stability of the power system. In order to enhance the dynamic stability of the power system, as well as to provide quality energy to the consumers, excitation systems equipped
with AVR are employed. Because of such an important role, the design of an AVR system is a crucial and challenging task.
A typical AVR system consists of the following components:
• controller,
• amplifier,
• exciter,
• generator, and
• sensor.
The object that needs to be controlled in this control scheme is a synchronous generator, whose terminal voltage is measured and rectified by the sensor. An error signal, which presents the
difference between the desired and the measured voltage value, is formed in the comparator. One of the main components in the AVR scheme that needs to be chosen carefully is the controller. Based on
the error signal and the appropriate control algorithm selected, the controller defines the control signal. Very often the controller is realized as a microcontroller unit, whose output power is
deficient. Due to this, the existence of the amplifier is necessary in order to increase the power of the control signal. Finally, an amplified signal is used to control the excitation system of the
synchronous generator, and therefore, to define the terminal voltage level. The scheme of such a described system is depicted in
Figure 1
In the available literature [
], the components of the AVR (except the controller) are presented as the first-order transfer function, which is composed of gain and time constant.
Table 1
gives a compact review of the transfer functions and the range of each parameter.
In the previous table,
stand for gains of an amplifier, exciter, generator, and sensor, respectively, while
, and
are time constants of the amplifier, exciter, generator, and sensor. Values that are considered in this paper are
= 10,
= 1,
= 1,
= 1,
= 0.1,
= 0.4,
= 1, and
= 0.01 [
]. It is important to mention that the gain of the generator
depends on the load of the generator. Namely,
can take a value from 0.7 (non-loaded generator) to 1 (nominal loaded generator).
Before including the controller in the system analysis, it is necessary to carry out the analysis of the system in the absence of the controller. To that end, the step response of the AVR system
without the controller is given in
Figure 2
. In order to demonstrate the behavior of the system in the different cases of the load, simulations are conducted for different values of parameter
(0.7, 0.8, 0.9, and 1).
Despite the transient response (time-domain) analysis, it is also very important to take a look at the frequency response of the system. Frequency characteristics or Bode diagrams of the open-loop
system can provide information about margins of the stability of the closed-loop system. Precisely, it is important to determine the values of the gain margin and the phase margin, which both need to
be positive in order to have a stable system. For the different values of the parameter
, frequency responses are shown in
Figure 3
Another essential characteristic of the system, and the main indicator of stability, is the root locus of the closed-loop system. Root locus gives information about the location of the poles of the
closed-loop system. As it is well known from control theory, for the stable system, all the poles must be located in the left half-plane. The graphical representation of the location of the poles is
given in
Figure 4
Based on the previously presented figures, all-important time domain and frequency domain parameters can be computed. Namely,
Table 2
shows transient response indices-rise time (
), settling time (
), overshoot in percentage (
), steady-state error (
), frequency response parameters-gain margin (
) and phase margin (
), as well as the poles of the closed-loop system.
Root locus and frequency characteristics prove that the AVR system is stable, but the margins of the stability are low due to the poles that are very close to the imaginary axis of the complex plane.
Also, large values of the overshoot, the settling time, and the steady-state error indicate that the transient response of the AVR system in the absence of the controller is feeble. In fact, the
steady-state error varies from 8% to 12% (depending on the load of the generator), which means the AVR cannot complete the main task—maintaining the voltage level at the reference value. All of the
aforementioned deficiencies can be eliminated by adding the controller into the system. According to the available literature, the most used control strategies for AVR systems are based on classical
or integer-order PID controller, as well as the generic version of the PID controller that is called Fractional-Order PID controller (FOPID).
Integer-order PID controller is presented by the following transfer function:
$P I D ( s ) = K p + K i s + K d s ,$
is the proportional gain,
is the integral gain, and
is the derivative gain. Integer-order is a specific case of the PID controller, where the integral and the derivative are first order.
The general type of the PID controller is called Fractional-Order PID controller and is presented using the following transfer function:
$F O P I D ( s ) = K p + K i s λ + K d s μ ,$
represent the order of the integral and of the derivative, respectively. As the name of the FOPID indicates, these two numbers can be any real numbers (not strictly integers). The aforementioned
facts make the FOPID the most general form of the PID controller. Specific forms of FOPID controllers are PID (
= 1,
= 1), PI (
= 1,
= 0), PD (
= 0,
= 1) and P controller (
= 0,
= 0), as illustrated in
Figure 5
3. About the Fractional Order Calculus
Regarding the problem of the fractional-order calculus, many different approaches have been proposed. According to [
], the most commonly used definitions of the fractional-order calculus are Grunwald–Letnikov, Riemann–Liouville, and Caputo definition.
Grundwald–Letnikov’s approach defines the
th order derivative of the function
in the limits from
as follows:
$D α | a t = lim h → 0 1 h α ∑ r = 0 [ t − a h ] ( − 1 ) r ( n r ) f ( t − r h ) ,$
stands for the time step, and the operator [∙] takes only the integer part of the argument. The variable
must satisfy the condition
−1 <
, while the binomial coefficients are defined by:
$( n r ) = Γ ( n + 1 ) Γ ( r + 1 ) Γ ( n − r + 1 ) ,$
where the definition of the Gamma function is well known:
$Γ ( x ) = ∫ 0 ∞ t x − 1 e − t d t .$
Riemann and Liouville proposed the definition of the fractional-order derivative that avoids using limit and sum, but uses integer-order derivative and integral, as follows:
$D α | a t = 1 Γ ( n − α ) ( d d t ) n ∫ a t f ( τ ) ( t − τ ) α − n + 1 d τ .$
Another definition of fractional order derivative is proposed by M. Caputo and is defined by the following Equation:
$D α | a t = 1 Γ ( n − α ) ∫ a t f ( n ) ( τ ) ( t − τ ) α − n + 1 d τ .$
However, the aforementioned formal definitions show the lack of applicability in real-time implementation (digital implementation on a computer) [
]. In order to exceed such a problem, A. Oustaloup proposed the recursive approximation of the fractional-order derivative [
]. Such an approach is trendy among the large number of authors that deal with optimal tuning of the FOPID controller [
]. Moreover, in practical implementations of the fractional-order calculus, it can be seen that the Oustaloup’s idea dominates over the formal definitions. Because of that, in this paper, Oustaloup’s
recursive approximation will be used to model fractional-order derivatives and integrals. Mathematical approximation of the
th order derivative (
) is given by Equation (8):
$s α ≈ ω h α ∏ k = − N N s + ω k ′ s + ω k ′ ,$
where the zeros and the poles are defined as follows:
$ω k = ω b ( ω h ω b ) ( k + N + ( 1 + α ) / 2 ) ( 2 N + 1 ) , ω k ′ = ω b ( ω h ω b ) ( k + N + ( 1 − α ) / 2 ) ( 2 N + 1 ) .$
Before applying the given recursive filter, it is necessary to define the number N that determines the order of the filter (order is 2N + 1) and the frequency range of the approximation {ω[b], ω[h]}.
In this study, the order of the filter is chosen to be 9 (N = 4), and the selected frequency range is {10^−4, 10^4} rad/s.
It is imperative to mention that Equation (8) is valid only for
∈(0, 1). Thus, in the case the fractional-order
is higher than 1, it is necessary to conduct a simple mathematical manipulation. Precisely, fractional-order
can be separated as follows:
$s α = s n s δ , α = n + δ , n ∈ Z , δ ∈ ( 0 , 1 ) .$
Afterward, Outstaloup’s recursive approximation is applied only on s^δ, since s^n is already an integer-order derivative.
4. Overview of the Literature
The problem of optimal design of the FOPID controller means the determination of the parameters
K[i], K[d]
so that the certain objective function achieves the minimum (or maximum) value. The most common performance indicators of the tuned FOPID controller are the transient response parameters of the
closed-loop system: rise time, settling time, overshoot, and steady-state error. In order to present the results obtained by the recent studies that deal with FOPID tuning,
Table 3
provides optimal values of the FOPID parameters and the corresponding transient response parameters of the acquired AVR system.
It is important to mention that the transient response parameters presented in the table are the calculated values obtained by carrying out the simulations with the given FOPID parameters. In order
to conduct the graphical comparison between given references in terms of the transient response parameters,
Figure 6
Figure 7
Figure 8
Figure 9
present rise time, settling time, overshoot, and steady-state error, respectively, for each method from
Table 3
The process of the optimization of the FOPID parameters highly depends on the chosen objective function that has to be minimized (or maximized). Considering the importance of the objective function,
the list of different used functions is given in
Table 4
. It can be observed that certain authors use a single objective function [
], while others perform multi-objective optimization [
In the previous table, e is the error signal (the difference between the reference voltage and the terminal voltage), V[f] is the voltage of the generator field winding, ω[gc] is the gain crossover
frequency, u is the control signal (the output of the controller), e[load] is the error signal when load disturbances are present, and max_dv is the maximum point of the voltage signal derivative.
The weighting coefficients are marked as w[1], w[2], w[3], ..., w[8].
5. Proposed Chaotic-Yellow Saddle Goatfish Algorithm
The development of the Yellow Saddle Goatfish Algorithm (YSGA) is based on the model of the hunting process by a group of yellow saddle goatfishes, as proposed in [
]. According to this approach, the whole population of the fishes is split into sub-populations. Each sub-population has one fish that is called a chaser, while the others are called blockers. Also,
the search space of the possible solutions for the optimization problem is represented by the hunting area of the goatfishes.
The first step of the YSGA is the initialization of the population. Assuming that a population
consists of
goatfishes (
= {
, ...,
}), each goatfish is initialized randomly between the low boundary (
) and the high boundary (
) of the search space [
$p i = r a n d ⋅ ( b H − b L ) + b L , i = 1 , 2 , … , m ,$
is a vector of random numbers between 0 and 1. It is very important to mention that
is a vector that consists of
decision variables (variables that are being optimized). Furthermore,
are also vectors that represent lower and upper boundaries for each decision variable.
According to (11), the initialization process of the original YSGA proposed in [
] is random, which does not ensure a good starting point in the optimization process. Namely, metaheuristic algorithms have extremely sensitive dependence on the initial conditions, so the
improvements in this part may have a great effect on the overall performance of an algorithm. The idea of introducing Chaotic maps into the metaheuristic algorithms in order to replace the random
parameters that appear in the algorithm is shown in [
]. Among the most interesting approaches are the ones presented in [
], where the random population is replaced with the population generated by Chaotic algorithm with different maps. There are many existing Chaotic maps, such as circle map, cubic map, Gauss map,
ICMIC map, logistic map, sinusoidal map, and so on. In order to examine the performances of the mentioned maps, many authors provide a mutual comparison of the different maps [
]. Concretely, the comparison is carried out by solving concrete optimization problems employing the different Chaotic maps. The existing studies demonstrate that the logistic mapping is the most
convenient to use, due to the better computational efficiency than other mentioned Chaotic maps [
Based on the previous analysis, this paper introduces the initialization of the population using Chaotic Logistic Mapping [
]. Thus, the random initialization is given by (11), which is proposed by the original YSGA algorithm, is replaced by the initialization provided as a result of Chaotic Logistic Mapping. The proposed
model of the initialization of the population is described by (12) and (13). Firstly, vectors
that are the products of Logistic Mapping are introduced as follows:
$y 1 = r a n d , y i + 1 = μ ⋅ y i ⋅ ( 1 − y i ) , i = 1 , 2 , … m ,$
stands for a vector of random numbers in the interval [0,1], and
is the coefficient that is chosen to be 4 in this study. In this manner, we gave the chaotic character of the basic Yellow Saddle Goatfish Algorithm.
Afterward, initialization in C-YSGA is realized according to the following Equation:
$p i = y i ⋅ ( b H − b L ) + b L , i = 1 , 2 , … , m .$
Before starting the process of the hunt, the whole population must be divided into sub-populations or clusters. Each of the
has a chaser fish
and the blocker fish
. Clustering can be made using any of the clustering algorithms. However, the YSGA algorithm uses the K-means clustering algorithm in order to divide the population, as it is described in [
] in detail. The cluster organization of the population is depicted in
Figure 10
The chaser fish of each cluster is the one with the best fitness value. The first step of the hunting process is to update the position of the chaser fish. If the current position is denoted as
, the updated position is
, and the best chaser fish from all clusters is
, where
represents the number of the iteration; the updated law is given by the following Equation:
$Φ l t + 1 = Φ l t + α ( u | v | 1 / β ) ( Φ l t − Φ b e s t t ) ,$
defines the step size (it is set to 1 in this study), and
is Levy index that is calculated as follows (
stands for the maximum number of iterations):
$β = 1.99 + 0.01 t t m a x .$
from (14) are defined using the following equations:
$u ∼ N ( 0 , σ u 2 ) , σ u = ( Γ ( 1 + β ) ⋅ sin β π 2 Γ ( 1 + β ) ⋅ β ⋅ 2 ( β − 1 ) / 2 ) 1 / β ,$
$v ∼ N ( 0 , σ v 2 ) , σ v = 1 ,$
where Γ stands for Gamma function and
for normal distribution. In order to update the position of the best chaser fish from all clusters, it is necessary to use (18) instead of (14):
$Φ b e s t t + 1 = Φ b e s t t + α ( u | v | 1 / β ) ,$
The next step in the optimization process is the update of the positions of the blocker fishes. The new position of the blocker fish
can be determined based on the following Equation:
$φ g t + 1 = | r ⋅ Φ l − φ g t | ⋅ e b ρ ⋅ cos ( 2 π p ) + Φ l ,$
is a random number between
and 1,
is a random number between 0 and 1, and
is the constant that is set to be 1. The parameter
is called the exploitation factor and is linearly decreased from −1 to −2 during the iterations.
It is vital to keep in mind that during the optimization process, the exchange of roles may occur. Namely, if the blocker fish has a better fitness value than the chaser fish, they exchange the
roles, and the blocker fish becomes a new chaser fish in the next iteration.
The YSGA model has the predefined parameter
, which is called an overexploitation parameter. Precisely, if a solution is not improved in
iterations, it is necessary to change an area of the hunt. Each goatfish, no matter if it is chaser or blocker, must change the hunting area according to the following Equation:
$p g t + 1 = Φ b e s t + p g t 2 ,$
represent old and new positions of the goatfish, respectively. The whole described process is iteratively repeated until the maximum number of iterations is reached. The detailed description is
provided with the pseudo-code presented in
Table 5
This section is not mandatory but can be added to the manuscript if the discussion is unusually long or complicated.
6. Simulation Results
This section presents the results that are obtained by applying the proposed C-YSGA method to optimize the FOPID parameters of the AVR system.
Firstly, the formulation of the optimization problem is provided, including the novel objective function presented in this paper. Afterward, the convergence characteristics of the different
optimization algorithms used in the literature are compared to the one obtained by C-YSGA in order to demonstrate the convergence superiority of the proposed algorithm. Furthermore, the comparison is
conducted in terms of the step response of the AVR system, as well as in the cases of different kinds of uncertainties and disturbances in the system.
6.1. Formulation of the Optimization Problem
From (2), it can be seen that the FOPID is defined with 5 parameters, K[p], K[i], K[d], λ, and μ, which need to be optimized so that the controller satisfies the desired performances. The
optimization process is guided by the objective function that defines the performances of the AVR system.
In order to provide a good quality transient response (for the reference step signal), we tested all of the previously used objective functions. However, none of the mentioned functions provide the
appropriate system responses as they do not take into account all of the essential characteristics (time-domain parameters or frequency parameters). Furthermore, they do not give an appropriate
compromise between all the critical time-domain parameters. On the other side, the objective functions that are based on frequency parameters have higher execution time, which makes the optimization
process slower. Observing the different mathematical formulations of the objective functions presented in [
], the authors in this paper propose a novel objective function (21) that contains a smaller number of weighting coefficients, but also outperforms other objective functions in the literature:
$O F = w 1 ⋅ ∫ t | e ( t ) | d t + w 2 O S + w 3 | E s s | + w 4 t s .$
Weighting coefficients are chosen carefully after many experiments, and the following values are considered in this paper:
= 1,
= 0.02,
= 1, and
= 5. The values of the coefficients are chosen after many experiments with different combinations. It can be seen that
has a significantly lower value than the other three weighting coefficients. The reason for this is that the overshoot in (21) is given in percentage, and its value is always larger than the values
of ITAE, settling time, and steady-state error. Concretely, from
Table 3
, it can be observed that the highest value of overshoot can go to 45%, while the settling time and the steady-state error reach maximum values of 1.9 s and 0.17 pu, respectively. However, it is very
important to highlight that the presence of the FOPID controller can make the closed-loop system unstable. In order to surpass that, this paper uses optimization with constraints. In other words,
each solution (each set of FOPID parameters) is first tested to examine if the obtained closed-loop system remains stable. If a certain solution makes the system unstable, it is automatically
removed, ignoring its fitness value. The size of the population in the C-YSGA algorithm is selected to be 40, and the maximum number of iterations is 50. Also, the lower and upper boundary must be
defined for each of the optimization variables. Taking into account previous studies related to this topic, the chosen boundaries that are used in this paper are presented in
Table 6
By using the proposed C-YSGA method and the novel objective function depicted above, the optimal FOPID parameters are:
K[p] = 1.762
K[i] = 0.897
K[d] = 0.355
μ = 1.26
, and
λ = 1.032
. The proposed method is compared with all methods presented in
Table 3
, and the results are provided in
Table 7
where the best value is in bold. Note, the best solutions of each method, as it is shown in
Table 3
, are applied with the proposed fitness function given by (21). It is clear that the new fitness function proposed in this paper has the lowest value when the FOPID parameters obtained by C-YSGA
algorithm are used.
6.2. Convergence Characteristics
The main goal of hybridizing the concepts of two algorithms (classical YSGA and chaotic logistic mapping) is to accelerate the convergence speed of the original algorithm. Due to the fact that the
initial population of the C-YSGA is not selected randomly, but it is the product of the chaotic logistic mapping, it is expected that the proposed algorithm will reach the optimal solution for the
least number of iterations. In order to demonstrate that, the original YSGA algorithm, as well as PSO [
], CS [
], and GA [
] algorithms have been implemented to determine the optimal FOPID parameters using the proposed objective function (21). The convergence curves of all mentioned algorithms demonstrate that the C-YSGA
algorithm converges in a minimum number of iterations (approximately 10) compared to the other algorithms, as it is depicted in
Figure 11
. In this figure, the convergence characteristics represent the mean value of convergence characteristics when we started all the algorithms multiple times. Therefore, the chaotic improvement of the
standard YSGA algorithm enables obtaining better convergence characteristics. In that manner, it is demonstrated that Chaotic maps, in combination with the metaheuristic algorithm, improves the
initial position, which is very important for the convergence speed of the algorithm.
6.3. Step Response
Among all the presented results in
Table 3
, for the comparison with the proposed C-YSGA, the papers [
] are chosen. The main indicators of the step response quality-rise time, settling time, overshoot, and steady state-error, as well as the obtained FOPID parameters, are presented in
Table 8
, the best values are marked in bold.
Additionally, the step response of the AVR system with FOPID parameters from
Table 4
is shown in
Figure 12
. Undoubtedly, it can be concluded that the FOPID controller tuned by the proposed C-YSGA method provides better transient response compared to the other considered algorithms. Precisely, the
settling time, the overshoot, and the absolute value of the steady-state error have the least values when the C-YSGA is used, while the rise time has a very low value. Taking a look into the previous
table, it can be seen that the overshoot with the PSO algorithm [
] is 22%, which is an unacceptably big value. Similarly, the rise time and the settling time with the GA algorithm are larger than 1 s, which makes the voltage response extremely slow.
6.4. Robustness Analysis
The analysis in the previous section is conducted under the nominal conditions. However, it may occur that the components of the AVR system change their parameters. One of the tasks of the FOPID
controller is to ensure the stability of the system and the high quality of the step response in the case of the sudden change in the parameters’ values. To that end, the robustness analysis of the
AVR system with the C-YSGA FOPID controller is conducted, and the results are presented in
Figure 13
Figure 14
Figure 15
Figure 16
. Precisely, the study is carried out for the change of time constants
, and
from −50% to +50% of the nominal value, in steps of 25%. The step response of the AVR system is shown in
Figure 13
Figure 14
Figure 15
Figure 16
The results of the previous analysis prove that the C-YSGA FOPID controller makes the AVR system very robust to the changes of each parameter. It is observed that the step response does not deviate a
lot compared to the nominal conditions.
6.5. Rejection of the Disturbances
The ability of the FOPID controller to cope with the different disturbances is analyzed by introducing three kinds of disturbances into the AVR system: control signal disturbance, load disturbance,
and measurement noise. The block diagram of the AVR with considered disturbances is depicted in
Figure 17
, while their detailed description is given as follows:
• One of the most common disturbances not only in the AVR system but generally in every control system is control signal disturbance. In this subsection, the obtained C-YSGA FOPID controller is
compared with FOPID controllers tuned by PSO [
], CS [
], and GA [
] algorithms. Control signal disturbance is presented as a constant step signal in the first case, and in the second case as a step signal that lasts from
= 2 s to
= 8 s. Step responses of the AVR system are shown in
Figure 18
for both cases.
• Afterward, the load disturbance that is specific mainly for AVR systems is presented. Similarly to control signal disturbance, it is modeled as a step signal that lasts from
= 2 s to
= 3.5 s. The obtained step responses, in this case, are shown in
Figure 19
• The last type of disturbance is measurement noise, which is modeled as white Gaussian noise with the power 0.0001 dBW.
Figure 20
presents the step responses of the AVR system when the measurement noise is present.
Based on the previous figures, it is obvious that the FOPID controller whose parameters are optimized by using a novel C-YSGA algorithm provides a significantly better ability to reject different
types of disturbances. Comparison is conducted with some of the most popular and most used algorithms, whose performances, in this case, are remarkably weaker than the proposed method. To be more
precise, from
Figure 18
Figure 19
, it can be noted that the voltage does not reach its nominal value after the disturbance is introduced, when the controllers presented in [
] are used. Such fluctuations in the terminal voltage, caused by the inability of the controller to reject the disturbance, can present a major problem for the consumers of the electrical energy.
Unlike them, the FOPID controller tuned by using C-YSGA provides a very stable level of the terminal voltage, whose value reaches the nominal value in a very short period after the disturbance in the
system occurs.
7. Conclusions
This paper proposes the novel optimization algorithm in order to optimize the FOPID controller parameters in the AVR system. The proposed algorithm presents the compound of the Yellow Saddle Goatfish
Algorithm (YSGA) and Chaotic Logistic mapping to obtain the innovative Chaotic-Yellow Saddle Goatfish Algorithm (C-YSGA). Instead of random initialization of the population, as in many existing
metaheuristic algorithms, Chaotic Logistic mapping is used to determine the initial point in the optimization process. It is proved in the paper that such an approach significantly accelerates the
convergence of the algorithm. Furthermore, to determine optimal FOPID controller parameters, a new objective function is presented. The results obtained by applying the proposed algorithm with the
new objective function introduced in this paper provide significantly better voltage response of the AVR system compared to other considered algorithms. The robustness of the AVR system with such an
obtained FOPID controller is tested by changing the AVR system’s parameters. It is shown that in all examined cases, the step response of the AVR system has extremely small deviations compared to the
nominal case, which means the system is robust to the uncertainties in the system. Moreover, three very often disturbances are introduced into the system, and the system’s behavior with different
FOPID controllers is analyzed. The mutual comparison shows that the C-YSGA FOPID controller is by far the best in rejecting all considered types of disturbances.
We think that in this way, the algorithm is improved, no matter what optimization problem is considered. In this paper, we tested its applicability and efficiency on the problem of optimal FOPID
design. However, at the moment, we are working on proving its superiority over other literature known methods for solving the synchronous machine parameters estimation problem. To that goal, we
consider field and armature current waveforms during the short circuit test.
Author Contributions
Conceptualization, M.M. and M.Ć.; methodology, M.M. and M.Ć.; software, M.M.; validation, M.Ć. and D.O.; formal analysis, M.M. and M.Ć.; investigation, M.M.; resources, M.M. and M.Ć.; data curation,
M.M. and M.Ć.; writing—original draft preparation, M.M.; writing—review and editing, M.Ć. and D.O.; visualization, M.M. and M.Ć.; supervision, M.Ć. and D.O. All authors have read and agreed to the
published version of the manuscript.
This research received no external funding.
Conflicts of Interest
The authors declare no conflict of interest.
AVR Automatic Voltage Regulation
SVC Static Var Compensator
SG Synchronous generator
PID Proportional-Integral-Derivative
FOPID Fractional Order Proportional-Integral Derivative
YSGA Yellow Saddle Goatfish Algorithm
C-YSGA Chaotic Yellow Saddle Goatfish Algorithm
PSO Particle Swarm Optimization
GA Genetic Algorithm
CNC-ABC Improved Artificial Bee Colony
CAS Chaotic Ant Swarm
MOEO Multi-Objective Extremal Optimization
CS Cuckoo Search
SSO Salp Swarm Optimization
IAE Integrated Absolute Error
ITSE Integrated Time Squared Error
K[A] amplifier gain
K[E] exciter gain
K[G] generator gain
K[S] sensor gain
T[A] amplifier time constant
T[E] exciter time constant
T[G] generator time constant
T[S] sensor time constant
t[r] rise time
t[s] settling time
OS overshoot
E[ss] steady-state error
G[m] gain margin
P[m] phase margin
K[p] proportional gain
K[i] integral gain
K[d] derivative gain
λ order of the integral
μ order of the derivative
e error signal
V[f] voltage of the generator field winding
ω[gc] gain crossover frequency
u control signal
e[load] error signal when load disturbances are present
max_dv maximum point of the voltage signal derivative
w[1], w[2], w[3], ..., w[8] weighting coefficients
P population
m number of goatfishes
b^L low boundary
b^H high boundary
rand vector of random numbers between 0 and 1
y[i] product vector of Logistic Mapping
k number of clusters
c[k] cluster
Φ[l] chaser fish
φ[g] blocker fish
Φ[best] best chaser fish
t number of the current iteration
t[max] maximum number of iterations
α step size
β Levy index
Γ gamma function
N normal distribution
a exploitation factor
r random number between 0 and 1
ρ random number between a and
Figure 6.
Rise time for each method from
Table 3
Figure 7.
Settling time for each method from
Table 3
Figure 8.
Overshoot for each method from
Table 3
Figure 9.
Steady-state error for each method from
Table 3
Figure 18. Step responses in the two different cases of the control signal disturbance. (a) constant signal, (b) step signal that lasts from t = 2 s to t = 8 s.
Component Transfer Function Range of the Parameters
Amplifier K[A]/(1 + sT[A]) 10 ≤ K[A] ≤ 400, 0.02 s ≤ T[A] ≤ 0.1 s
Exciter K[E]/(1 + sT[E]) 1 ≤ K[E] ≤ 10, 0.4 s ≤ T[E] ≤ 1 s
Generator K[G]/(1 + sT[G]) 0.7 ≤ K[G] ≤ 1, 1 s ≤ T[G] ≤ 2 s
Sensor K[S]/(1 + sT[S]) 1 ≤ K[S] ≤ 2, 0.001 s ≤ T[S] ≤ 0.06 s
Parameter K[G] = 1 K[G] = 0.9 K[G] = 0.8 K[G] = 0.7
Overshoot (%) 65.214 61.3825 55.9051 50.4818
Rise time (s) 0.2613 0.2755 0.2945 0.3171
Settling time (s) 7.0192 6.5237 5.4086 4.9012
Steady-state error (p.u.) 0.0881 0.102 0.1108 0.1249
−0.51 ± 4.66i −0.6 ± 4.46i −0.69 ± 4.25i −0.79 ± 4.01i
Closed-loop system poles −12.48 −12.31 −12.12 −11.92
−99.97 −99.97 −99.97 −99.97
Gain margin (dB) 4.61 5.53 6.55 7.71
Phase margin (^o) 16.1 19.56 23.56 28.26
Method Reference K[p] K[i] K[d] μ λ t[r] (s) t[s] (s) OS (%) |Ess| (pu)
1 [5] 0.408 0.374 0.1773 1.3336 0.6827 1.0083 1.512 0.0221 0.0155
2 [5] 0.9632 0.3599 0.2816 1.8307 0.5491 1.3008 1.6967 6.99 0.0677
3 [5] 1.0376 0.3657 0.6546 1.8716 0.5497 0.0104 1.8796 30.8479 0.0595
4 [6] 1.9605 0.4922 0.2355 1.4331 1.5508 0.1904 1.0259 4.8187 0.0102
5 [7] 1.0537 0.4418 0.251 1.1122 1.0624 0.2133 0.6145 5.2398 0.0153
6 [7] 0.9315 0.4776 0.2536 1.0838 1.0275 0.2259 0.564 3.7006 0.0098
7 [8] 0.9894 1.7628 0.3674 0.7051 0.9467 0.1823 1.8835 58.315 0.0409
8 [8] 0.8399 1.3359 0.3511 0.7107 0.9146 0.1998 1.8727 44.8059 0.0146
9 [8] 0.4667 0.9519 0.2967 0.2306 0.8872 0.3041 1.986 45.2452 0.1768
10 [9] 2.9737 0.9089 0.5383 1.3462 1.1446 0.0769 0.388 8.6266 0.0086
11 [10] 2.549 0.1759 0.3904 1.38 0.97 0.0963 0.9774 3.5604 0.0321
12 [10] 2.515 0.1629 0.3888 1.38 0.97 0.0967 0.9849 3.5141 0.033
13 [10] 2.4676 0.302 0.423 1.38 0.97 0.0902 0.9933 3.2504 0.0283
14 [11] 1.5338 0.6523 0.9722 1.209 0.9702 0.0614 1.3313 22.5865 0.0175
15 [12] 1.9982 1.1706 0.5749 1.1656 1.1395 0.1011 0.5633 13.2065 0.0068
Objective Function Reference
$O F = w 1 ⋅ O S + w 2 ⋅ t r + w 3 ⋅ t s + w 4 ⋅ E s s + ∫ ( w 5 ⋅ | e ( t ) | + w 6 ⋅ V f ( t ) 2 ) d t + w 7 P m + w 8 G m$ [4]
$J 1 = ω g c , J 2 = P m$ [5]
$I A E = ∫ | e ( t ) | d t$ [6]
$Z L G = ( 1 − e − β ) ⋅ ( O S + E s s ) + e − β ⋅ ( t s − t r )$ [7,10]
$J 1 = ∫ t e 2 ( t ) d t , J 2 = ∫ Δ u 2 ( t ) d t , J 3 = ∫ t e l o a d 2 ( t ) d t$ [8]
$J 1 = I A E , J 2 = 1000 | E s s | , J 3 = t s$ [9]
$O F = w 1 ⋅ O S + w 2 ⋅ t s + w 3 ⋅ E s s + w 4 ∫ | e ( t ) | d t + w 5 ∫ u 2 ( t ) d t$ [11]
$O F = ( w 1 ⋅ O S ) 2 + w 2 t s 2 + w 3 ( max _ d v ) 2$ [11]
$I T A E = ∫ t | e ( t ) | d t$ [12]
Pseudo-Code of the C-YSGA
Enter the input data: m, k, t[max], λ
Initialize the population P using chaotic logistic mapping
According to the fitness values determine Φ[best]
Split the population into k clusters and determine the chaser fish Φ[l] for each cluster
while (t < t[max])
for each cluster
Update the position of the chaser fish and blocker fish
Calculate the fitness value of every fish
Exchange the roles if any blocker fish has better fitness value than the chaser fish
Update the Φ[best] if the chaser fish has better fitness value
If the fitness value of the chaser fish has not improved, increase the counter q by 1
If the counter q is higher than λ then apply the formula for the change of the zone
end for
t = t + 1
Φ[best] is the output result of the algorithm
Parameter Lower Bound Upper Bound
K[p] 1 2
K[i] 0.1 1
K[d] 0.1 0.4
λ 1 2
μ 1 2
Method Proposed 1 2 3 4 5 6 7
OF value 1.08 24.6 47.3 50 10.1 8.4 4.5 12.3
Method 8 9 10 11 12 13 14 15
OF value 10.1 53.6 2.3 4.8 4.8 3 9.8 3.1
Algorithm K[p] K[i] K[d] μ λ t[r] (s) t[s] (s) OS (%) |E[ss]| (pu)
C-YSGA 1.7775 0.9463 0.3525 1.2606 1.1273 0.1347 0.2 1.89 0.0009
PSO [11] 1.5338 0.6523 0.9722 1.209 0.9702 0.0614 1.3313 22.58 0.0175
CS [10] 2.549 0.1759 0.3904 1.38 0.97 0.0963 0.9774 3.56 0.0321
GA [5] 0.9632 0.3599 0.2816 1.8307 0.5491 1.3008 1.6967 6.99 0.0677
© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http:/
Share and Cite
MDPI and ACS Style
Micev, M.; Ćalasan, M.; Oliva, D. Fractional Order PID Controller Design for an AVR System Using Chaotic Yellow Saddle Goatfish Algorithm. Mathematics 2020, 8, 1182. https://doi.org/10.3390/
AMA Style
Micev M, Ćalasan M, Oliva D. Fractional Order PID Controller Design for an AVR System Using Chaotic Yellow Saddle Goatfish Algorithm. Mathematics. 2020; 8(7):1182. https://doi.org/10.3390/math8071182
Chicago/Turabian Style
Micev, Mihailo, Martin Ćalasan, and Diego Oliva. 2020. "Fractional Order PID Controller Design for an AVR System Using Chaotic Yellow Saddle Goatfish Algorithm" Mathematics 8, no. 7: 1182. https://
Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details
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Elements of Art/Shape - Wikibooks, open books for an open world
Shapes are created with lines in a given space, either real or imaginary. Shapes can be endlessly rotated. Shapes may be organic (curved, freeform, similar to nature) or geometric (rigid, having
definite properties).
A circle is a shape with only one side created from a single, continuously curved line which encompasses the whole of the shape.
A triangle is a shape comprised of three straight lines which meet at three endpoints - the bottom side is horizontal, and the other two sides are diagonal, meeting each other at a point.
A square is a shape which is made of four straight lines which intersect at four points at 90 degree angles: the top and bottom lines are parallel to one another, as are the two lines comprising the
sides of the square. In a square, each of the sides is the exact length of the other sides (a rectangle is a different shape where the opposing sides are equal in length; thus, all squares are
rectangles but not all rectangles are squares.)
A shape with 5 sides. The bottom side is horizontal, there are two vertical sides that are parallel, and the two top sides are diagonal. A common use of the pentagon is to draw a house.
A shape with six sides. 4 sides are diagonal and 2 are horizontal.
The most common type of star is made of five triangles, each connected to a side of a pentagon by the bottom side. The lines forming the pentagon may or may not be drawn. Stars can also be drawn with
four, six, or more points.
Three-dimensional shapes are not flat; instead, they create depth which creates form and the shapes appear touchable.
A round figure where every point on its surface is an equal distance from the center.
Examples: ball and globe.
A solid or hollow object that tapers from a circular base to a point. It is a geometric shape formed from the base with lines that connect to one common point.
Examples: funnel, traditional ice cream cone, traffic cones, and classic party hats.
A structure whose sides are made of triangles attached to each other by the sides and a base shape by the bottom. The base of a pyramid may be a triangle or a square.
Example: the Great Pyramid of Giza
A cube has 8 endpoints, 12 edges and 6 faces. At every endpoint 3 lines intersect, and at an intersection any two edges are perpendicular to each other. Everything about the cube (edges, faces..etc)
are equal. Think of a square with depth.
Examples: Rubik cube and classic ice cube.
A shape with two identical ends (often a shape) and flat sides that connect the ends. There are two common prisms: triangular and rectangular. However, a prism could be any shape as long as it is a
polyhedron which means all faces are flat and all edges are straight This rules out a cylinder because it is curved.
Most shapes in art are combinations of the shapes described above. They may be expressed (that is, they have a clear outline) or implied (the viewer has to see them for themself).
Also, different shapes can be put together for interesting results.
Weight can capture the viewers eye by creating two-dimensional shape(s) that has a force that an element applies and attracts the eye.
When it comes to shapes, you will notice that when you have shapes that are irregular, like an irregular triangle or quadrilateral, it will appear lighter than that of a regular shape. The reason for
this being, is because irregular shapes make is appear as though part of the mass is taken away.
When you put more elements into a space you are giving that space more weight.
Height has an effect on both two-dimensional forms and three-dimensional shapes. It related to how tall the shape can be made or stretch too. By having a variety of heights with shapes you are able
to have all different types of proportions with how tall each shape is, one could be extremely tall while the other is shorter. | {"url":"https://en.m.wikibooks.org/wiki/Elements_of_Art/Shape","timestamp":"2024-11-15T03:19:09Z","content_type":"text/html","content_length":"40431","record_id":"<urn:uuid:c7777b78-3f4d-45d1-b8d4-786e57b79a25>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00675.warc.gz"} |
Category: Educationcal
GAT General 2013 Schedule (Note it Down) Muhammad
For more information visit NTS official web site www.nts.org.pk
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My Twitter
Q. 1. What is an Error? Account
Ans.An error is the change or the mismatching take place between the data unit sent by transmitter and the data unit received by the receiver e.g. 10101010 sent by sender 10101011
received by receiver. Here is an error of 1 bit. Tweets by
Q. 2. Define Error Control. @Ahmad_SCET
Ans.Error control refers to mechanisms to detect and correct errors that occur in the transmission of frames. The most common techniques for error control are based on some or all of
the following: Categories
1, Error detection
2. Positive acknowledgement All
3. Retransmission after time-out Current
4. Negative acknowledgement and retransmission. Affairs
These mechanisms are also referred as automatic repeat request (ARC)). Educationcal
Q. 3. What are three types of redundancy checks used in data communication? My Songs
Ans. Error detection uses the concept of redundancy, which means adding extra bits for detecting errors at the destination there ate three types of redundancy checks are common in Collection
data communication:
(a) Parity check Archives
(h) Cyclic Redundancy check (CRC)
(c) Checksum. Website
Q. 4. How can the simple parity bit detect a damaged data unit? Status
Ans.In this technique, a redundant bit called a parity bit, is added to every data unit so that the total number of Is in the unit becomes even (or odd). Suppose we want to transmit
1100001. Adding the number of 1’s gives us 3, an odd number. Before transmitting, we pass the data unit through a parity generator. The parity generator counts the 1’s and appends
the parity bit to the end (al in this case).
Q. 5. What is the difference between even parity and odd parity?
Ans.In case of redundancy check method we have to append the data unit with some extra bits. These extra bits are called parity.
This parity or parity hit can be even or odd.
in case of even parity we have to make number of 1’s even, including the parity hit e.g. 1110001 is the data unit where the no. of l’s is already even then we will insert 0 at the
next to data unit it’, 1110001. In case of odd parity we have to make no. of l’s odd, including the parity bit. e.g. 1111000 is the data unit, where the no. of 1’s is even then we
will insert I at the next to data unit i.e. 11110001.
Q. 6. Define code world?
Ans.The code word is the n bit encoded block of bits. As already seen it contains message bits and parity or redundant bits as shown in the following figure.
Q. 7. Define Code rate?
Ans,The code rate is defined as the ratio of number of message bits (K) to the total number of hits (n) in the code word.
Q. 8. Define code efficiency?
Ans.The code efficiency is defined as the ratio of message bits to the number of transmitted bits per block.
Q. 9. What are the disadvantages of coding?
Ans.(1) Coding makes the system complex.
(2) As increased transmission bandwidth is required in order to transmit the encoded signal. This is due to the additional hits added by the encoder.
Q. 10. Suppose the sender wants the word “HELLO”. In ASCII the five characters are coded as:
What will be the combination of actual bits to send?
Ans. 11101110 11011110 11100100 11011000 11001001
Q. 11. How the receiver will detect that there is an error in:
Ans. The receiver counts the 1’s in each character and comes up with even numbers (b, 6, 4, 4, 4). The data are accepted.
Q. 12. Suppose the word HELLO is corrupted during transmission?
How receiver will check it out?
Ans. The receiver counts the 1’s in each character and comes up with even and odd numbers (7, 6, 5, 4, 4). The receiver knows that the data are corrupted, discards them and asks for
Q. 13. Explain about error correction.
Ans. Error correction is the mechanism by which we can make changes in the received erroneous data to make it free from error.
The two most common error correction mechanisms are:
(i) Error correction by Retransmission.
(ii) Forward Error Correction.
Q. 14. What is check sum?
Ans.Checksum is the one of the method used for error detection, based on the concept of redundancy. In this mechanism, the unit is divided into K sections, each of n bits. All
sections are added using ones complement to get the sum. This is complemented and becomes the check sum. There after this check sum is sent with the data. At the receiver side the
unit is divided into K sections each of n bits. All sections are added using ones complement to get the sum. The sum is complemented. If the result is zero data are accepted
otherwise rejected.
Q. 15. Discuss the two dimensional parity check and the types of errors it can and cannot detect.
Ans. Apart from simple parity check two-dimensional parity is the better approach.
In this method, a block of bits is organized in a table (rows and columns).
First we calculate the parity bit for each data unit then we organize them into table.
Data and Parity bits
A redundancy of n bits can easily detect a burst error of n bits. A burst error of more than n bits is also detected by this method with a very high probability.
But if 2 bits in our data unit are damaged and two bits in exactly the same positions in another data unit are also damaged, the checker will not detect an error.
Q. 16. What are the different types of error?
How a single bit error does differ from a burst error?
Ans. A single bit error is an isolated error condition that alters one bit but does not affect nearby bits. On the other hand A burst error is a contiguous sequence of bits in which
the first and last bits and any number of intermediate bits are received in error.
A single bit can occur in the preserve of while noise, when a slight random deterioration of single-to-noise ratio is sufficient to confuse the receiver’s decision of a single bit.
On the other hand burst errors are more common and more difficult to deal with. Burst error can be caused by impulse noise.
Q. 17. What is Error detection?
Ans. Regardless of the design of the transmission system, there will be errors, resulting in the change of one or more bits in a transmitted frame. When a code word is transmitted
one or more number of transmitted bits will be reversed due to transmission impairments. Thus error will be introduced. It is possible to detect these errors if the received code
word is not one of the valid code words. To detect the errors at the receiver, the valid code words should be separated by a distance of more than 1.
The concept of including extra information in the transmission of error detection is a good one. But instead of repeating the entire data stream, a shorter group of bits may be
appended to the end of each unit. This technique is called redundancy because the extra bits are redundant to the information; they are discarded as soon as the accuracy of the
transmission has been determined.
Q. 18. Discuss the concept of redundancy in error detection.
Ans. It is a most common and powerful technique for the detection of errors. In this technique extra bits are added. But instead of repeating the entire data stream, a shorter group
of bits may be appended to the end of each unit. The technique is called redundancy because the extra bits are redundant to the information. They are discarded as soon as the
accuracy of transmission has been determined.
The following fig. shows the process of using redundant bits to check the accuracy of data unit.
Once the data stream has been generated. It passes through a device that analyzes it and adds on appropriately’ coded redundancy check. The receiver puts the entire stream through a
checking function. If the received bit stream passes the checking criteria, the data portion of the data unit is accepted and redundant bits are discarded.
Q.19. Explain any one Mechanism used for error detection?
What is the Parity check Method of Error detection?
Ans. The most common and least expensive mechanism for error detection is the parity check.
Parity checking can be simple or two-dimensional.
Simple Parity Check
In this technique, a redundant bit, called a parity bit, is added to every data unit so that the total number of Is in the unit (including the parity bit) becomes even (or odd).
Suppose we want to transmit the binary data unit 1100001
Transmission Mode
Adding the no. of is giving us 3 an odd number. Before transmitting we pass the data unit through a parity generator. The parity generator counts the is and appends the parity bit to
the end. The total no. of is now 4, an even number. The system now transmits the entire expanded unit across the network link. When it reaches its destination, the receiver puts all
8 bits through an even parity checking function. If the receiver sees 11000011, it counts four is, an even number and the data unit passes. But, if instead of 11000011, the receiver
sees 11001011 then when the parity checker counts the Is it gets 5 an odd number. The receiver knows that an error has been introduced into the data somewhere and therefore rejects
the whole unit.
Two Dimensional Parity Check
A better approach is the two dimensional parity check in this method, a block of bits is organized in a table (rows and columns). First we calculate the parity bit for each data
unit. Then we organize them into table. Shows in fig we have four data units shown in four rows and eight columns. We then calculate the parity hit for each column and create a new
row of 8 bits. They are the parity bits for the whole block. The first parity bit in the fifth row is calculated based on all first bits, the second parity bit is calculated based on
all second bits, and so on. We then attach the 8 parity bits to the original data and sent them to the receiver.
Q.20. Explain CRC method of Error Detection?
Ans. Cyclic Redundancy Check (CRC): Cyclic Redundancy check method is most powerful mechanism of error detecting. Unlike the parity check which is based on addition, CRC is based on
binary division.
In CRC, instead of adding bits to achieve a desired parity, a sequence of redundant bits, called the CRC or the CRC remainder, is appended to the end of a data unit so that the
resulting data unit becomes exactly divisible by a second predetermined binary number. At its destination the incoming data unit is divided by the same number. If at this step there
is no remainder, the data unit is assumed to be intact and is therefore accepted. A remainder indicates that the data unit has been damaged in transit and therefore must be rejected.
The redundancy bits used by CRC are derived by dividing the data unit by a predetermined divisor, the remainder is the CRC. A CRC must have two qualities. It must have exactly one
less bit than the divisor, and appending it to the end of the data string must make the resulting bit sequence exactly divisible by the divisor.
CRC generator and checker
First, a string of n 0’s is appended to the data unit. The number n is less than the number of bits in the predetermined divisor, which are n + 1 bits.
Second, the newly formed data unit is divided by the divisor, using a process called binary division the remainder resulting from this division is the CRC.
Third, the CRC of n bits derived in step 2 replaces the appended Os at the end of the data unit. The data unit arrives at the receiver data first followed by the CRC. The receiver
treats the whole string as a unit and divides it by the same divisor that was used to find the CRC remainder.
If the string arrives without error, the CRC checker yields a remainder of zero and the data unit passes. If the string has been changed in transit the division yields a non zero
remainder and the data unit does not pass.
Q.21. How is the check sum method of error detection take place?
Ans. Checksum is the third mechanism for error detection which is also based on the concept of redundancy.
Check sum Generator
In the sender, the check sum generator subdivides the data unit into equal segments of n bits. These segments are added using ones complement arithmetic in such a way that the total
is also n bits long. That total is then complemented and appended to the and o the original data unit as redundancy bits called the check sum field. The extended data unit is
transmitted across the network. So if the some of data segment is T, the checksum will be T.
Check sum Checker
The receiver subdivides the data unit as above and adds all segments and complements the result. If the extended data unit is intact, the total value found by adding the data
segments and the check sum field should be zero If the result is not zero, the packet contains an error and the receiver rejects it.
Q.22. How the data communication between sender and the receiver will take place where the error detection method is check sum and the data is :
Ans. Sender
The numbers are added using one’s complement arithmetic
Q. 23. What is hamming code of Error Correction? How it calculate, the redundancy?
Explain any one method used for error correction.
Ans. The hamming code can be applied to data units of any length and uses the relationship between data and redundancy bits.
Suppose there are 7 bits ASCII codes which requires 4 redundancy bits that can be added to the end of the data unit or interspersed with the original data bits. These units are
position in 1, 2, 4, arid 8 (the position is in an 11 bit sequence that are powers of 2). We prefer these bits are r1, r2, r4 and r8.
Q. 24. What are various error correction codes?
Ans. A mechanism that can handle correction of an error heading of error correction code categories under the
There are two methods for error correction.
(1) Error correction by retransmission.
(2) Forward error correction.
Error Correction by Retransmission
In error correction by retransmission, when an error is discovered, the receiver can have the sender retransmit the entire data unit.
Forward Error Correction
In forward error correction (FEC), a receiver can use an error-correcting mode, which automatically corrects certain errors. In theory it is possible to correct any error
automatically. Error correcting codes however are more sophisticated than error detection codes and require more redundancy bits.
e.g. To correct a single bit error in an ASCII character, the error correction code must determine which of the 7 bits has changed In this case we have to distinguish between eight
different states no error, error in position 2, and so on, up to the error in position 7. To do so requires enough bits to show all eight states.
At first glance, it seems that or 3-bit redundancy code should be adequate because 3 bits can show eight different states (000 to 111) and can therefore indicate the locations of
eight different possibilities. To calculate the no. of redundancy bits. We should consider
Where m is the no. of bits to be transfer r stands for the no. of redundancy. By this manner.
There is the practical solution for this method that is “Hamming Code”.
Very Impotent Note
The above questions & answers had been taken from a website. Some of the answers contain figures as well you can find all those pictures from the website link is given blow
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How Many Jellybeans Can Fit in a Car? - villageautorepairct.com
You’re not alone if you’ve ever wondered how many jellybeans fit in a car. Kia Motors, the car company’s advertising company, asks this question. Its commercial makes fun of the fact that jellybeans
aren’t labeled, so there is no way to know how many they can hold.
Jellybeans aren’t labeled.
Jellybeans are one of America’s favorite foods and aren’t quickly packaged. They are only sometimes labeled and can’t fit in your car, but they are still fun to eat and to give away as gifts. Plus,
they make a significant sensory bin. You can even find a taste test record sheet below!
Jellybeans are not vegan. They are made with confectioner’s sugar, which keeps them from being vegan. Luckily, many manufacturers have vegan jellybeans. While they won’t fit in a car, you can take
them to work.
In addition to tasting the variety of jellybeans, you can also tour the Jelly Belly factory. This is a popular outing for families in the Bay Area. The factory features a store selling every flavor
of jellybeans, an art museum, and interactive games for the kids.
Then, a special jelly bean was found. This particular type is called 749. It has a 1/1,000 chance of being safe. The jelly bean had a 2/3 chance of being poisonous, but the red one wasn’t poisonous.
Fortunately, the red jelly bean was safe to eat.
When Jellybeans aren’t labeled, they’re not labeled. They’re so small they can’t fit in a car. So, before buying one, you should know a little about the jellybean’s history. You may be surprised that
many of them exist in the United States.
Size of a jellybean
To know the jellybean size in a jar, you must know how many are inside. The easiest way to determine the number of jellybeans inside a pot is to measure the jar’s volume before and after it is filled
with water. Once you know the jar’s importance, you can calculate the number of jellybeans inside using the mean/median formula.
One liter of liquid is the same as one thousand cubic centimeters. A jellybean is a small cylinder, about two centimeters long by 0.75 centimeters in diameter. The volume of a cylinder can be found
using a simple geometry formula – h pr2.
Number of jellybeans in a jar
One liter of jelly beans has a volume of one thousand cubic centimeters. A jellybean is about 2 cm long and 0.75 cm wide. Using a geometry formula, we can find the importance of a jellybean in a jar.
Estimating the number of jelly beans in a jar is pretty simple. All you need to do is weigh a jar and compare the weight to the set number of jelly beans. You can estimate the jar’s volume reasonably
if the jar weighs more than the specified number of beans.
You can use the calculator on Wolfram Alpha to find the number of jellybeans in a jar. Using this tool, you can find the number of jellybeans in quarts. You can also use the calculator to calculate
the number of jellybeans in a car.
The next step in the calculation is calculating each jellybean’s density. Generally, one jar contains about one gram of j-beans, while another contains about 1.1 grams. This will yield a result that
is higher than 45 grams.
With enough Jellybeans, you can create a beautiful craft with them. Start by cleaning the jar lid. It would help if you also painted it. Once you have a base, lay out your jellybeans. Begin in the
middle of the top and work outward. Make sure you get an entire layer.
Size of a jellybean container
A jellybean container contains a specific volume, and it’s stamped on the side of the jar. You can use this volume to estimate how many jellybeans are in the pot. Jelly beans have a cubic-inch book
of approximately ten beans per cubic inch and an ounce volume of about 20 jellybeans per ounce. Then you can use the ratio of ounces to cubic inches to estimate how many jellybeans the jar holds.
A one-liter bottle holds one thousand cubic centimeters of fluid. Jellybeans are cylindrical and measure approximately 2 centimeters in length and 0.75 centimeters in radius. The volume of a jar is
calculated using a geometry formula h pr2; the same formula is used to find the importance of a cylinder.
Jelly Belly jelly bean jars are one of the most popular jellybean containers. These are ideal for storing and displaying these delicious treats. You can buy containers in different shapes and sizes
to fit any occasion. If you’re buying for a child, a Christmas jellybean jar is an excellent idea for a birthday gift. You can also purchase themed containers for party goody bags. | {"url":"https://villageautorepairct.com/how-many-jellybeans-can-fit-in-a-car","timestamp":"2024-11-09T15:53:18Z","content_type":"text/html","content_length":"55434","record_id":"<urn:uuid:9e59a7e7-9c1d-4bbf-92eb-4223813e39e5>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00383.warc.gz"} |
What does sin squared x cos squared x equal?
The first of these three states that sine squared plus cosine squared equals one. The second one states that tangent squared plus one equals secant squared. For the last one, it states that one plus
cotangent squared equals cosecant squared.
What is sin squared x equal to?
We will use the sin squared x formula, sin2x = 1 – cos2x to prove this.
What is cos squared squared?
The square of cosine function equals to the subtraction of square of sin function from one is called the cosine squared formula. It is also called as the square of cos function identity.
Why is cos squared plus sin squared one?
This formula is the Pythagorean theorem in disguise. in a triangle with unit hypotenuse are just the lengths of the two shorter sides. So squaring them and adding gives the hypotenuse squared, which
is one squared, which is one.
How do I know if I have SOH CAH TOA?
In this geometry lesson, you’re going to learn all about SohCahToa. It’s probably one of the most famous math mnemonics alongside PEMDAS….It’s defined as:
1. SOH: Sin(θ) = Opposite / Hypotenuse.
2. CAH: Cos(θ) = Adjacent / Hypotenuse.
3. TOA: Tan(θ) = Opposite / Adjacent.
What is cos squared x plus sin squared x?
Cosine squared + sine squared = 1. Now, that we have two more functions we can also express the other Pythagorean identities. One of them is tangent squared + 1 = secant squared, one of them is
cotangent squared + 1 = cosecant squared.
Is sin squared the same as 2 sin?
yes it’s the same!
What is sin squared minus Cos squared?
θ = 1 − cos 2 The square of sine function equals to the subtraction of square of cos function from one is called the sine squared formula. It is also called as the square of sin function
Is sin squared theta same as sin theta squared?
If you read sin(θ)2 as sin(θ2) then the statement is true. sin2x is the square of a ratio.
The equation “sin x squared” uses the trigonometric function sine, which is defined as a ratio between particular sides of a right triangle. In the equation, x represents the value of an angle
expressed in either degrees or radians. The sine function, along with the cosine and tangent functions, is commonly used in geometry.
What is the integral of sin x squared?
Integration of Sin Squared x. In this tutorial we shall derive the integral of sine squared x. The integration is of the form. I = ∫ sin 2 x d x. This integral cannot be evaluated by the direct
formula of integration, so using the trigonometric identity of half angle sin 2 x = 1 – cos. . 2 x 2, we have. I = ∫ ( 1 – cos.
What is the derivative of sin squared?
The derivative of sine squared is the sine of 2x, expressed as d/dx (sin2(x)) = sin(2x). The derivative function describes the slope of a line at a given point in a function. The derivative of sine
squared can be determined by using the chain rule.
What is sine squared plus cosine squared?
The sum of sine squared plus cosine squared is 1. While the sine is calculated by dividing the length of the side opposite the acute angle by the hypotenuse, the cosine is calculated by dividing the
length of the side that is adjacent to the acute angle by the hypotenuse. For example,… | {"url":"https://sage-answers.com/what-does-sin-squared-x-cos-squared-x-equal/","timestamp":"2024-11-15T00:50:02Z","content_type":"text/html","content_length":"49620","record_id":"<urn:uuid:866e5df8-a112-4e16-a5fb-3f1250249099>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00307.warc.gz"} |
Logarithm Worksheet With Answers
To solve problems of the logarithm, we need a Logarithm Worksheet with Answers. Answers is the name given to the results of any mathematical operation on a real number or a real angle. Answers
provide information needed by engineers in a quick way without requiring complex calculations. Logarithm is an example of this. An angle is measured on a horizontal line and the corresponding formula
is used to find the other side of the angle as the function of the set number of degrees on the horizontal line.
Logarithmic and Exponential form Lovely Fresh Exponent Worksheets from logarithm worksheet with answers , source:rosheruns.us
Logarithm is one of those subjects that do not have single definite answer but rather many different answers depending upon the circumstances. Expresses logarithmically form in exponential and fifth
order worksheets that yield five-digit times to move all exponential from linear. It is quite important in solving practical problems related to economic growth and decrease. One of the main
functions of logarithm worksheets is to find out the value of a number using only digits and no higher or lower digits. This makes it quite convenient for scientific calculations.
There are some facts about logarithms. They occur naturally in nature at constant temperatures and pressures. The temperature and pressure of the Earth are not constant. Therefore, the earth’s
temperature varies with time. Similarly, the rate of acceleration of the earth is different from its slowest speed. And since the universe contains energy, every clock can be used to measure the
speed of the universe.
Worksheet Solving Exponential Equations Kidz Activities from logarithm worksheet with answers , source:reedaudio.com
Logarithm calculator can be used to find out the answer of the following math problem, which is widely used by students in schools and colleges as well as individuals. In fact, this problem has been
used for almost two centuries already, and there are many solutions for this problem. All these are included in the logarithm worksheet with answers math. A student can use this to study different
subjects and they can also make use of it for further studies.
Most of the problems of logarithms to solve by finding out the Ku factor. And the answers given by the logarithm worksheet includes the Ku factor as well as other factors like exponents, perimeter of
a circle, cube, etc. To solve any problem of logarithms, the student can make use of the kuta software etc which is an answer keying program.
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The beta software can also be used for solving other natural numbers such as, degrees, squares, triples, powers, etc. In addition, this program can also solve irrational numbers, complex numbers,
even squared roots and other objects that cannot be expressed using the traditional method. In this way, the logarithm worksheet can also provide answers for almost all the subjects including
mathematics, science, art, computer science, engineering, and so on. With the help of this software, a student will be able to solve different kinds of problems involving logarithms of whole numbers,
decimals, division, and multiplication.
Students of finite and infinite algebra are benefited with the Logarithm Worksheet with Answers because it offers solutions of partial differential equation as well as solutions to the binomial
equations. The solutions to the partial differential equation show multiplication by some constant and one is called as the base of the integral formula. Similarly, the solutions of the binomial
equation shows that x n+1=x n-1 as the base of the binomial formula. It is better to choose the best algorithm that solves the problem in a reliable manner because students might have to spend more
time on finding the best solutions rather than learning how to solve problems quickly. Therefore, it is essential to use a reliable and proven algorithm.
Logarithm Worksheet – Fronteirastral from logarithm worksheet with answers , source:fronteirastral.com
To conclude, it can be said that the Logarithm Worksheet with Answers provides students with more than just solution to their arithmetic problems. First of all, these worksheets make it possible to
learn the real meaning of logarithms, exponents, and other symbols used in mathematics. Furthermore, using these worksheets will help students develop good mathematical skills in line with what is
taught in high school. Further, it is not necessary to purchase expensive textbooks because there are free worksheets like the Logarithm Worksheet with Answers, which are available on the web. All
you need to do is search for these on Google or any other search engine and you will get results.
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Teaching the Concept of Division and Remainders - The Teacher Studio
As I did my division planning for math last week, I really spent some time thinking about how we rush, rush, rush to get through our curriculum sometimes. In fact, I think having a math SERIES makes
it so much worse. We start to think of teaching math as teaching lessons and chapters and units instead of teaching math concepts. I cringe when I hear teachers say, “What lesson are you on?” It
sends the message that we are on a timeline and are trying to get our students to learn math in nice and tidy hour-long chunks.
That isn’t how it works.
We have several lessons in our series that are meant to reinforce the relationship between multiplication and division. I know it was taught in 3rd grade. I know some (SOME) of my students “know”
their facts. But I also know that many don’t–and if I just keep plowing through the book they never will. I wanted to slow down and really manipulate numbers to get them thinking about groups and
Teaching Division: The Herding Game
Last year I played a game with my students that I called “The Herding Game”. I found an open space in my building (last year an empty classroom, this year a big hall space by the elevator) and told
my students they were animals. They, of course, are used to me calling them this (I frequently say things like, “OK, wombats, let’s go to music.”) so they didn’t flinch. I explained that for this
activity they truly would be animals–animals that live in herds. We brainstormed a list and then they had to do the following:
When I call out the name of an animal, I will also tell you how big your “herd” needs to be. You need to quickly form “herds”, and any animals who can’t form a herd will head over to the holding pen
(a taped off area on the floor). My rules? You can’t be in the pen more than once.
So we got started. We started the activity with 24 students and I called out, “Buffalo, herds of 6!”. Off they roamed to make their herds. We noticed how we had no animals in the pen, so I wrote
the equation 24 / 6 = 4 on a white board and we discussed how nice and evenly it worked out. We tried several more combinations…we made herds of 5 (leftovers!), herds of 3 (no leftovers!), herds of
10 (leftover!) and so on., Each time, I showed them the matching division equation AND the matching multiplication equation. 2 x 10 + 4 = 24. The wheels were starting to turn…and students were
tuned in to how many animals would be in the pen even before they had formed their herds.
Continuing with Division Learning
After about 12 minutes, we headed back to the classroom and I taught them the pretzel game. This was a game I used with an intervention group last year, but I was pretty confident that I could
really get ALL the students thinking. This time I played a game against one of my students with the rest of the class watching. We really started to dig into division concepts, making predictions
about how many remainders there would be, and so on.
The gist of this game is the same as the herding game. We started with 34 pretzels and took turns rolling the dice to determine the number of bowls we could put them in. On any turn, any leftovers
(“remainders”) go to that player. As my opponent and I got going, I LOVED the discussion that the two of us modeled.
“I have 34 pretzels and need to sort them into 4 bowls. I know that 4 groups of 8 pretzels gives me 32–so I have 2 leftovers.” Or, “I have 30 pretzels to sort into 2 bowls, that is easy because I
know that 2 groups of 15 is 30.”
As we went, the other students did exactly what I was hoping they would do–they started predicting. “Wait–the only number that will give leftovers is 5 because you can divide 24 into 1, 2, 3, 4, and
6 groups!” and so on. Their minds were really starting to think logically and recognize the connection between multiplication and division. We finished our game (I won–in case you are interested)
and the students BEGGED to play it on their own.
It all flowed so easily from there…
First things first–we need to do some work in our math book. But let me tell you how fast they were able to do the problems! They needed to solve problems like 34 divided by 5–and I could HEAR
students talking to each other in terms of pretzels and bowls! As they worked, I pulled 3 students who I didn’t think were ready and played a round of the game with them. It helped to really slow
things down and get them doing it themselves. Next week I’m going to pull them to do my paper version of the herding game to keep modeling this for them.
As students finished, they could choose from a variety of multiplication and division games to work on fluency. I felt really good that I taught MATH, not lesson 4.9 (even though I did). Interested
in the games and activities? Want MORE division resources? I have a bundle FULL of ideas for you! A few of my division resources are linked below.
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The Great Debate: Is 6 Even or Odd?
The world of mathematics is filled with intriguing questions that have sparked debates and discussions among scholars and enthusiasts alike. One such question that has been the subject of much
deliberation is whether the number 6 is even or odd. It may seem like a simple query, but the answer is not as straightforward as it appears. In this article, we will delve into the world of number
theory and explore the characteristics of even and odd numbers to finally put this debate to rest.
What Makes A Number Even Or Odd?
Before we dive into the specifics of the number 6, it’s essential to understand the fundamental properties that define even and odd numbers. In mathematics, a number is said to be even if it can be
divided by 2 without leaving a remainder. On the other hand, a number is considered odd if it cannot be divided by 2 without leaving a remainder.
Definition Of Even Numbers
According to the definition, even numbers are integers that can be expressed in the form:
where n is an integer. Examples of even numbers include 2, 4, 6, 8, and so on. Notice how each of these numbers can be divided by 2 without leaving a remainder.
Definition Of Odd Numbers
On the other hand, odd numbers are integers that cannot be expressed in the form 2n. Instead, they can be expressed in the form:
2n + 1
where n is an integer. Examples of odd numbers include 1, 3, 5, 7, and so on. Notice how each of these numbers cannot be divided by 2 without leaving a remainder.
The Case For 6 Being Even
Now that we have a solid understanding of even and odd numbers, let’s examine the characteristics of the number 6. Upon first glance, it’s easy to see why one might argue that 6 is an even number.
Divisibility By 2
One of the most significant indicators that 6 is an even number is its divisibility by 2. When you divide 6 by 2, you get:
6 ÷ 2 = 3
with no remainder. This meets the fundamental criteria for a number to be considered even.
Pairs Of Numbers
Another argument in favor of 6 being even is the concept of pairs of numbers. When you arrange the numbers from 1 to 6 in pairs, you get:
(1, 2), (3, 4), (5, 6)
Notice how each pair consists of one odd number and one even number. This pattern is a hallmark of even numbers, as they can always be paired with an odd number to form a consecutive sequence.
The Case For 6 Being Odd
While the above arguments may seem convincing, there are also valid reasons why some people argue that 6 is an odd number.
Pattern Disruption
One reason why 6 might be considered odd is that it disrupts the pattern of alternating even and odd numbers. When you arrange the numbers from 1 to 6 in a sequence, you get:
1 (odd), 2 (even), 3 (odd), 4 (even), 5 (odd), 6 (even)
Notice how the sequence starts with an odd number, followed by an even number, and then an odd number again. However, when you reach 6, the pattern unexpectedly continues with an even number,
disrupting the alternating sequence.
Ternary System
Another argument against 6 being even is based on the ternary numeral system. In this system, numbers are represented using the digits 0, 1, and 2. When you represent 6 in the ternary system, you
6 = 20 (in ternary)
Here, the digit 2 in the units place makes 6 appear more like an odd number than an even number.
After examining the characteristics of even and odd numbers, and exploring the arguments for and against 6 being even or odd, it’s clear that 6 is indeed an even number. While there may be some
unusual patterns or disruptions in certain number sequences, the fundamental property of divisibility by 2 without leaving a remainder remains the defining characteristic of even numbers.
Property Even Numbers Odd Numbers
Divisibility by 2 Yes No
Pattern of Pairs Yes No
Disruption of Pattern No Yes
In conclusion, the debate about whether 6 is even or odd is more of a semantic exercise than a mathematical conundrum. By understanding the fundamental properties of even and odd numbers, we can
confidently say that 6 belongs to the former category.
Final Thoughts
The world of mathematics is full of intriguing debates and discussions, and the question of whether 6 is even or odd is just one such example. While it may seem like a trivial matter, exploring this
question has allowed us to delve deeper into the characteristics of number theory and gain a better understanding of the fundamental properties that define even and odd numbers. So the next time
you’re asked whether 6 is even or odd, you can confidently say it’s even!
Is The Number 6 Considered Even Or Odd?
The number 6 is considered an even number. This is because it can be divided by 2 without leaving a remainder. In other words, when you divide 6 by 2, you get 3, which is a whole number. In
mathematics, even numbers are defined as numbers that are divisible by 2, and 6 meets this criterion.
It’s worth noting that the evenness or oddness of a number is not a subjective property, but rather a mathematical fact that can be determined using simple arithmetic operations. The fact that 6 is
even is a fundamental property of the number that has been widely accepted and utilized in various mathematical concepts and applications.
What Is The Definition Of An Even Number?
An even number is a whole number that is divisible by 2 without leaving a remainder. In other words, if a number can be expressed in the form 2n, where n is an integer, then it is an even number.
Examples of even numbers include 2, 4, 6, 8, and 10.
The definition of an even number is a fundamental concept in mathematics, and it has been widely used in various mathematical operations, such as addition, subtraction, multiplication, and division.
The concept of even numbers is also used in algebra, geometry, and other advanced mathematical disciplines.
Can A Number Be Both Even And Odd At The Same Time?
No, a number cannot be both even and odd at the same time. The evenness or oddness of a number is a mutually exclusive property, meaning that a number can either be even or odd, but not both.
In mathematics, the evenness or oddness of a number is determined by its divisibility by 2. If a number is divisible by 2, then it is even; otherwise, it is odd. This is a fundamental property of
numbers that is widely accepted and used in various mathematical concepts and applications.
What Are Some Examples Of Odd Numbers?
Examples of odd numbers include 1, 3, 5, 7, 9, and 11. These numbers cannot be divided by 2 without leaving a remainder. In other words, when you divide an odd number by 2, you get a decimal or
fractional result.
It’s worth noting that odd numbers can be either positive or negative. For example, -1, -3, and -5 are also odd numbers. The evenness or oddness of a number is a property that is independent of its
How Do You Determine If A Number Is Even Or Odd?
There are several ways to determine if a number is even or odd. One way is to divide the number by 2 and check if the result is a whole number. If it is, then the number is even; otherwise, it is
Another way to determine if a number is even or odd is to look at its last digit. If the last digit is even (i.e., 0, 2, 4, 6, or 8), then the number is even; if the last digit is odd (i.e., 1, 3, 5,
7, or 9), then the number is odd. This method is a quick and easy way to determine the evenness or oddness of a number.
Are There Any Real-world Applications Of Even And Odd Numbers?
Yes, the concept of even and odd numbers has numerous real-world applications. For example, in computer programming, even and odd numbers are used to perform conditional statements and loops. In
music, even and odd numbers are used to create harmonious and rhythmic patterns.
In everyday life, even and odd numbers are used in various ways, such as in counting, measurement, and data analysis. For example, when you count the number of people in a room, you use even and odd
numbers to determine if the number is divisible by 2 or not.
Is The Concept Of Even And Odd Numbers Universal?
Yes, the concept of even and odd numbers is universal and applies to all whole numbers, regardless of their size or magnitude. The definition of even and odd numbers is based on the fundamental
properties of numbers, which are the same across all cultures and mathematical systems.
The concept of even and odd numbers has been used by mathematicians and scientists throughout history, from ancient civilizations to modern times. The universal nature of even and odd numbers has
enabled mathematicians to develop advanced mathematical concepts and applications that are used in various fields, including physics, engineering, and computer science.
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Order of Cards in a Deck - ULearnMagic.com
Order of Cards in a Deck
In this post, we will look at the order of cards in a deck. This includes the order of cards in a new deck, and the order of cards from least to greatest, depending on the game being played.
Order of Cards in a New Deck
When you get a new deck of cards, they always come preset in the same order from the factory. The cards will be arranged by each suit, with the first and last card being Aces, and two Kings together
in the very middle.
The suits will alternate by red and black color. The order of suits might change based on the brand of cards, but we will look at the most common type of cards, Bicycle cards.
For a Bicycle cards deck, you will find that the cards are in this particular order when you deal from the top of the face-down deck.
First is the Ace through King of Hearts, followed by the Ace through King of Clubs, then the card orders reverse and you have the King through Ace of Diamonds, then the King through Ace of Spades.
New Deck Order
• Two advertisement cards
• Ace through King of Hearts
• Ace through King of Clubs
• King through Ace of Diamonds
• King through Ace of Spades
• Two Jokers
Bicycle New Deck Order
The Bicycle new deck order is:
• Ace of Hearts
• 2 of Hearts
• 3 of Hearts
• 4 of Hearts
• 5 of Hearts
• 6 of Hearts
• 7 of Hearts
• 8 of Hearts
• 9 of Hearts
• 10 of Hearts
• Jack of Hearts
• Queen of Hearts
• King of Hearts
• Ace of Clubs
• 2 of Clubs
• 3 of Clubs
• 4 of Clubs
• 5 of Clubs
• 6 of Clubs
• 7 of Clubs
• 8 of Clubs
• 9 of Clubs
• 10 of Clubs
• Jack of Clubs
• Queen of Clubs
• King of Clubs
• King of Diamonds
• Queen of Diamonds
• Jack of Diamonds
• 10 of Diamonds
• 9 of Diamonds
• 8 of Diamonds
• 7 of Diamonds
• 6 of Diamonds
• 5 of Diamonds
• 4 of Diamonds
• 3 of Diamonds
• 2 of Diamonds
• Ace of Diamonds
• King of Spades
• Queen of Spades
• Jack of Spades
• 10 of Spades
• 9 of Spades
• 8 of Spades
• 7 of Spades
• 6 of Spades
• 5 of Spades
• 4 of Spades
• 3 of Spades
• 2 of Spades
• Ace of Spades
• Joker
• Joker
Deck of Cards in Order from Least to Greatest
Throughout history, the Ace was always the lowest card in the deck (taking the place of the 1 card which it replaced in some decks) with the King being the highest.
Cards in Order of Value
The cards in order of value starting with the lowest is Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King.
• Ace
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10
• Jack
• Queen
• King
In some games, however, the Ace is the highest card in the deck, or it can be either the highest or lowest card depending on how it’s played with other cards.
Order of Cards in Poker
The Ace is can usually the highest card in Poker, but can sometimes be played as the lowest card with a value of 1, depending on the hand.
Poker hand with Ace being used as the highest card
The order of cards in Poker from lowest to highest goes Two, Three, Four, Five, Six, Seven, Eight, Nine, Ten, Jack, Queen, King, Ace. But when the Ace is played in a straight or straight flush with
the Ace, 2, 3, 4, and 5 it is considered the lowest card below the 2.
What is the Order of Cards in Solitaire?
The order of cards in solitaire begins with the Ace as the first or lowest card, with the highest or last card being the King. The order is Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King.
How Many Cards In a Deck?
There are 52 cards in a full deck of cards, not including the Jokers. If you include the two Jokers, the number of cards in a deck is 54.
Learn more at How Many Cards in a Deck, How Many Cards in a Deck with Jokers and How Many Cards in a Deck without Jokers.
New Deck Order Card Trick
Although this is slightly advanced, if you do 8 perfect faro shuffles on a new deck of cards, the cards will return to the original new deck order!
This is somewhat advanced since perfect faro shuffles are difficult to do, especially 8 in a row. But there are some advanced magicians that can do this.
Another idea is that you could also perform full deck false shuffles on a deck in new deck order, and then show the cards to still be in new deck order even after all the shuffling. Or you could
snap your fingers and pretend to set them all back in order by using magic.
(You can learn some full deck false shuffles at False Shuffles with Cards – A Guide With Tutorials)
Full Deck of Cards
A full deck of cards has different types of cards including suits, values, face cards, and non-face cards. The total number of cards will be 52, and this will include 4 suits, with 13 cards in each
suit. These 13 value cards include 3 face cards and 10 non-face cards. The value cards are the Ace, Two, Three, Four, Five, Six, Seven, Eight, Nine, Ten, Jack, Queen and King.
The 4 suits are the Hearts, Diamonds, Clubs and Spades. Each suit gets the same 13 value cards.
Here is a picture of all the 52 cards in a full deck of cards.
Full deck of 52 cards, including the Jokers
See also How Many Suits are in a Deck of Cards?
How Many Hearts are in a Deck of Cards?
There are 13 Hearts in a deck of cards. These are the Ace, Two, Three, Four, Five, Six, Seven, Eight, Nine, Ten, Jack, Queen and King of Hearts.
Learn more at How Many Hearts are in a Deck of Cards?
In fact, each of the 4 suits has the same 13 cards each.
Deck of Cards Meaning
It’s believed that the number and types of cards in a deck of cards might be related to the calendar. There are definitely a lot of similarities between the various numbers. See How Many Cards are
in a Deck of Cards? Why this Number?
Deck of Cards Probability
A deck of cards is used a lot of times in probability questions. You can learn more about the probability of cards at Deck of Cards Probability Explained.
The magician started magic as a kid and has learned from some of the greats. He loves to share his knowledge with others and help out with the subtleties he’s learned along the way.
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Polynomial-Time Solutions of Computational Problems in Noncommutative-Algebraic Cryptography
We introduce the linear centralizer method, and use it to devise a provable polynomial-time solution of the Commutator Key Exchange Problem, the computational problem on which, in the passive
adversary model, the security of the Anshel–Anshel–Goldfeld (Anshel et al., Math. Res. Lett. 6:287–291, 1999) Commutator key exchange protocol is based. We also apply this method to solve, in
polynomial time, the computational problem underlying the Centralizer key exchange protocol, introduced by Shpilrain and Ushakov in (Contemp. Math. 418:161–167, 2006). This is the first provable
polynomial-time cryptanalysis of the Commutator key exchange protocol, hitherto the most important key exchange protocol in the realm of noncommutative algebraic cryptography, and the first
cryptanalysis (of any kind) of the Centralizer key exchange protocol. Unlike earlier cryptanalyses of the Commutator key exchange protocol, our cryptanalyses cannot be foiled by changing the
distributions used in the protocol.
Bibliographical note
Publisher Copyright:
© 2013, International Association for Cryptologic Research.
• Algebraic cryptanalysis
• Braid Diffie–Hellman key exchange
• Braid infinimum reduction
• Braid-based cryptography
• Centralizer key exchange
• Commutator key exchange
• Group theory-based cryptography
• Invertibility lemma
• Linear centralizer method
• Linear cryptanalysis
• Noncommutative-algebraic cryptography
• Schwartz–Zippel lemma
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Cross Spectral Analysis of a Gaussian Vector Process in the Presence of Variance Fluctuations
Ann. Math. Statist. 39(5): 1507-1512 (October, 1968). DOI: 10.1214/aoms/1177698132
Let $x'(t) = (x_1(t),x_2(t)),\quad(t = 1, 2, \cdots)$ be a two dimensional, Gaussian, vector process. Let the process $x'(t)$ have the representation \begin{equation*}\tag{1.1}x'(t) = \sum^p_{m = 0}
B_my (t - m),\end{equation*} where \begin{equation*}\begin{align*}\tag{1.2} B_m &= \{b_{ijm}; i,j = 1, 2\}; \\ y'(t) &= (y_1(t), y_2(t)); \\ y_l(t) &= \sigma_l(t)\epsilon_l(t)\quad (l = 1, 2)\end
{align*}.\end{equation*} The random variables $\epsilon_l(t)$ are independently and normally distributed with mean zero and variance unity. $p$ is a finite positive integer. The coefficients $B_m =
(b_{ijm})_{2 \times 2}$ are finite real constants, and $\sigma^2_l(t)$ are non-random sequence of positive numbers which are not, in general equal, but do satisfy the conditions \begin{equation*}\tag
{1.3}N^{-1}\sum^N_{t = 1}\sigma^2_l(t) = \nu_l < \infty\quad (\text{as} N \rightarrow \infty),\end{equation*} and $L \leqq \sigma^2_l(t) \leqq U < \infty\quad (t = 1, 2, \cdots).$ The relation (1.1)
is a multivariate representation of a finite moving average process with time trending coefficients. Consider the matrix \begin{equation*}\begin{align*}\tag{1.4}F (\lambda) &= \begin{pmatrix}f_{11}(\
lambda) & f_{12}(\lambda) \\ f_{21}(\lambda) & f_{22}(\lambda)\end{pmatrix} \\ &= G(\lambda)\begin{pmatrix}\nu_1 & 0 \\ 0 & \nu_2\end{pmatrix} G^{\ast'}(\lambda)\end{align*},\end{equation*} where $G
(\lambda) = \sum^p_{m = 0} B_me^{im\lambda}$ and $G^\ast(\lambda)$ is its complex conjugate. Under the condition (1.3), Herbst [1] has defined $f_{11}(\lambda)$ and $f_{22}(\lambda)$ as the spectral
densities of the processes $x_1(t)$ and $x_2(t)$ respectively, and considered their estimation. Here we generalize Herbst [1] results to a vector process and show that under the conditions (1.3) and
(3.3) $f_{12}(\lambda)$, which is defined as the cross spectral density of the process $x_1(t)$ and $x_2(t)$, can consistently be estimated.
Download Citation
T. Subba Rao. "Cross Spectral Analysis of a Gaussian Vector Process in the Presence of Variance Fluctuations." Ann. Math. Statist. 39 (5) 1507 - 1512, October, 1968. https://doi.org/10.1214/aoms/
Published: October, 1968
First available in Project Euclid: 27 April 2007
Digital Object Identifier: 10.1214/aoms/1177698132
Rights: Copyright © 1968 Institute of Mathematical Statistics
Vol.39 • No. 5 • October, 1968 | {"url":"https://www.projecteuclid.org/journals/annals-of-mathematical-statistics/volume-39/issue-5/Cross-Spectral-Analysis-of-a-Gaussian-Vector-Process-in-the/10.1214/aoms/1177698132.full","timestamp":"2024-11-11T07:58:31Z","content_type":"text/html","content_length":"141680","record_id":"<urn:uuid:40bcb11d-29a8-469a-8c3e-79b90cb6776c>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00249.warc.gz"} |
How to Create A Histogram in Stata | The Data Hall
How to Create A Histogram in Stata
Histograms are a common way of graphically representing the frequency distribution of data. In this article we are going to learn how to create Histogram in Stata
Let’s load one of Stata’s inbuilt datasets to see how histograms are created.
Go to File -> Example Datasets -> “Example Datasets Installed With Stata”. Click on the ‘use’ option in front of the dataset name in the list in order to load it in memory. We will use auto.dta for
this article.
We now want to create a histogram for the variable ‘mpg’ which holds data for the mileage of an automobile.
To create histogram in Stata, click on the ‘Graphics’ option in the menu bar and choose ‘Histogram’ from the dropdown. In the dialogue box that opens, choose a variable from the drop-down menu in the
‘Data’ section, and press ‘Ok’. A separate window with the histogram displayed will be opened. It should be noted that a histogram can show the frequency distribution (or density, frequency or
fraction) of only one variable at a time. Therefore, the drop-down menu in the dialogue box allows you to choose just one variable for the histogram.
In the results section, you will notice the number of bins, their starting value and the bin width also reported. In our example, the histogram for the variable ‘mpg’ has eight bins that start from
the x-axis value of 12. Each bin has a width of 3.625.
We now want to make a few changes. Firstly, we want to adjust the bar width. Secondly, we want the vertical axis to display the frequency instead of density which it shows by default.
Open the dialogue box again and under the ‘Y axis’ section, check the radio button titled ‘Frequency’. Radio buttons indicate that the options provided are mutually exclusive. Therefore, you can
choose only one option for the Y-axis. You can also specify whether your variable is discrete or continuous by choosing any one of the options under the ‘Data’ section.
In the ‘Bins’ section, users can type in the number of bins, width of the bins and the starting value/lower limit of the first bin. Decreasing the number of bins will increase the width of each bin.
We only change the width of the bin to ‘3’ by first checking the checkbox beside the respective field and entering in the value.
If we have too many bars, we are not summarizing the data enough, whereas if the number of bars is too low, we are summarizing too much. The number of bins therefore needs to be chosen appropriately.
Adding a Heading, Notes
Under the ‘Titles’ tab, we can key in our desired heading for the histogram in the input field under ‘Title’.
The input field under ‘Notes’ can be utilized to add any notes (such as the source of data) under the graph.
Instead of clicking ‘Ok’ which closes the dialogue box, we click ‘Submit’ which generates a histogram but keeps the dialogue box open so we can make any further changes conveniently.
Related post: How to use Stata Do file? Tips and Tricks
Histogram Scheme
We can alter the layout and color scheme of the histogram in Stata from the drop-down menu called ‘Scheme’ in the ‘Overall’ tab. We can, for example, use a template called ‘Stata Journal’ and press
Submit. The layout of the histogram generated will now match the Stata Journal default.
Naming The Graph
By default, Stata names any graph generated as ‘Graph’. When a new graph is created, it replaces the previous one and is also named ‘Graph’.
Under the ‘Overall’ tab, you can specify the name of the graph in the input field under ‘Name of graph’. Naming graphs allows you to generate and compare multiple graphs at once. The new graph will
be opened in a new tab in the graph window.
Displaying The Legend
The legend for a graph can be displayed by checking the ‘Show legend’ radio button under the ‘Legend’ tab.
Adding A Density/Kernel Plot
To add a density plot to your histogram, go to the ‘Density plots’ tab in the dialogue box and check ‘Add normal-density plot’ and/or ‘Add kernel density plot’.
Generating Histograms For Categorical Variables
Graphs for different subcategories of a variable can also be created. This is done by going to the ‘By’ tab and checking the option labelled ‘Draw subgraphs for unique values of variables’. We can
then choose our desired variable from the drop down menu below. For example, if we choose the ‘foreign’ variable, which is a binary variable, Stata will generate two graphs; one for observations
where the variable equals 1 (Foreign), one for those where it equals 0 (Domestic).
Graph Editor
We can edit the aesthetic looks of the graph from the ‘Start Graph Editor’
Saving Your Graph
Graphs can be saved by pressing the second button in the tool bar in the graph window. It is highly recommended to save the graph with Stata’s default extension for graphs. This allows you to come
back to the graph later and edit it. After saving it in the default .gph format, you can go ahead and save another copy in any format of your choice.
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December 1991 LSAT Question 17 Explanation
If there are research models on exactly two floors, then which one of the following statements can be false?
Fuller explanation
Can we get a more complete explanation for this and the next game. Both seem a bit obtuse for me. Thank you .
Let's look at the setup for this game.
Cars are displayed on each floor of a 3-floor building. On each floor, the cars are either family or sports, new or used, all production or all research. So on each floor, all the cars have three
attributes, and it is one of two choices for each of the attributes.
We can visually set it up in a 3 x 3 grid pattern:
The following rules apply:
(1) If the exhibition includes both family and sports cars, then each family car is displayed on a lower-numbered floor.
F > S
Notice that the condition only applies "if the exhibition includes" both, this tells us that it is not a requirement for the exhibition to include both, it is thus possible to have family cars on all
three floors or sports car on all three floors. But if there are both types of cars, we cannot have family cars on floor 3 or sports cars on floor 1 because family cars must always be on lower
numbered floors in this scenario.
(2) The exhibition includes no used research model.
This rule tells that if all cars displayed on a particular floor are research models, all cars on this floor are new.
R -> N
If the cars on a particular floor are used, they must be production models:
U -> P
Can we conclude that if the cars are new, they must be research models or that if the cars are production models, they must be used?
No. This would be an invalid inference. It is possible to have new or used production models.
(3) The exhibition includes no research models that are sports cars.
This rule tells us that if sports cars are displayed on a certain floor, all these cars are production models.
S -> P
~ P -> ~ S
If the cars are research models, they must be family cars.
R -> F
We cannot infer though that if a car is a production model, it is a sports car. It is possible to have production models of sports or family cars.
(4)There are new cars on floor 1.
(5) There are used cars on floor 3.
N/U N U
Per rule (2), we can conclude that cars on floor 3 must be production models because research models cannot be used.
N/U N U
P/R P
Now the questions asks us if there are research models on exactly two floors, then which of the following can be false. Can be false question is the opposite of must be true question, so every answer
choice except the correct one MUST BE TRUE.
Let's consider this scenario. If there are research models on exactly two floors, and we can see from our initial setup that floor #3 contains production models, we can infer than research models
must go on floors 1 &2.
N/U N U
P/R R R P
Per rule (2), we know that all research models must be new, thus the cars on floor #2 must be new.
N/U N N U
P/R R R P
Per rule (3), we know that research cars must be family cars, hence we can conclude that floors 1 & 2 contain family cars.
F/S F F ?
N/U N N U
P/R R R P
We could have either family or sports car on floor 3 though since none of the rules apply to used production cars. The only statement that can be false then is (E) as we are not required to have
family cars on floor #3.
Does this help?
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a decimal
Converting 1/79 to 0.013 starts with defining whether or not the number should be represented by a fraction, decimal, or even a percentage. Fractions and decimals represent parts of a whole,
sometimes representing numbers less than 1. The difference between using a fraction or a decimal depends on the situation. Fractions can be used to represent parts of an object like 1/8 of a pizza
while decimals represent a comparison of a whole number like $0.25 USD. After deciding on which representation is best, let's dive into how we can convert fractions to decimals.
1/79 is 1 divided by 79
The first step in converting fractions is understanding the equation. A quick trick to convert fractions mentally is recognizing that the equation is already set for us. All we have to do is think
back to the classroom and leverage long division. The numerator is the top number in a fraction. The denominator is the bottom number. This is our equation! To solve the equation, we must divide the
numerator (1) by the denominator (79). Here's 1/79 as our equation:
Numerator: 1
• Numerators are the top number of the fraction which represent the parts of the equation. Small values like 1 means there are less parts to divide into the denominator. The bad news is that it's
an odd number which makes it harder to covert in your head. Values like 1 doesn't make it easier because they're small. Now let's explore the denominator of the fraction.
Denominator: 79
• Denominators are located at the bottom of the fraction, representing the total number of parts. Larger values over fifty like 79 makes conversion to decimals tougher. But the bad news is that odd
numbers are tougher to simplify. Unfortunately and odd denominator is difficult to simplify unless it's divisible by 3, 5 or 7. Overall, two-digit denominators are no problem with long division.
Now let's dive into how we convert into decimal format.
How to convert 1/79 to 0.013
Step 1: Set your long division bracket: denominator / numerator
$$ \require{enclose} 79 \enclose{longdiv}{ 1 } $$
To solve, we will use left-to-right long division. This method allows us to solve for pieces of the equation rather than trying to do it all at once.
Step 2: Extend your division problem
$$ \require{enclose} 00. \\ 79 \enclose{longdiv}{ 1.0 } $$
We've hit our first challenge. 1 cannot be divided into 79! Place a decimal point in your answer and add a zero. Even though our equation might look bigger, we have not added any additional numbers
to the denominator. But now we can divide 79 into 1 + 0 or 10.
Step 3: Solve for how many whole groups you can divide 79 into 10
$$ \require{enclose} 00.0 \\ 79 \enclose{longdiv}{ 1.0 } $$
Now that we've extended the equation, we can divide 79 into 10 and return our first potential solution! Multiple this number by our furthest left number, 79, (remember, left-to-right long division)
to get our first number to our conversion.
Step 4: Subtract the remainder
$$ \require{enclose} 00.0 \\ 79 \enclose{longdiv}{ 1.0 } \\ \underline{ 0 \phantom{00} } \\ 10 \phantom{0} $$
If you hit a remainder of zero, the equation is done and you have your decimal conversion. If you still have numbers left over, continue to the next step.
Step 5: Repeat step 4 until you have no remainder or reach a decimal point you feel comfortable stopping. Then round to the nearest digit.
Sometimes you won't reach a remainder of zero. Rounding to the nearest digit is perfectly acceptable.
Why should you convert between fractions, decimals, and percentages?
Converting between fractions and decimals is a necessity. They each bring clarity to numbers and values of every day life. Same goes for percentages. So we sometimes overlook fractions and decimals
because they seem tedious or something we only use in math class. But each represent values in everyday life! Here are just a few ways we use 1/79, 0.013 or 1% in our daily world:
When you should convert 1/79 into a decimal
Speed - Let's say you're playing baseball and a Major League scout picks up a radar gun to see how fast you throw. Your MPH will not be 90 and 1/79 MPH. The radar will read: 90.1 MPH. This simplifies
the value.
When to convert 0.013 to 1/79 as a fraction
Cooking: When scrolling through pintress to find the perfect chocolate cookie recipe. The chef will not tell you to use .86 cups of chocolate chips. That brings confusion to the standard cooking
measurement. It’s much clearer to say 42/50 cups of chocolate chips. And to take it even further, no one would use 42/50 cups. You’d see a more common fraction like ¾ or ?, usually in split by
quarters or halves.
Practice Decimal Conversion with your Classroom
• If 1/79 = 0.013 what would it be as a percentage?
• What is 1 + 1/79 in decimal form?
• What is 1 - 1/79 in decimal form?
• If we switched the numerator and denominator, what would be our new fraction?
• What is 0.013 + 1/2?
Convert more fractions to decimals
From 1 Numerator From 79 Denominator
What is 1/80 as a decimal? What is 2/79 as a decimal?
What is 1/81 as a decimal? What is 3/79 as a decimal?
What is 1/82 as a decimal? What is 4/79 as a decimal?
What is 1/83 as a decimal? What is 5/79 as a decimal?
What is 1/84 as a decimal? What is 6/79 as a decimal?
What is 1/85 as a decimal? What is 7/79 as a decimal?
What is 1/86 as a decimal? What is 8/79 as a decimal?
What is 1/87 as a decimal? What is 9/79 as a decimal?
What is 1/88 as a decimal? What is 10/79 as a decimal?
What is 1/89 as a decimal? What is 11/79 as a decimal?
What is 1/90 as a decimal? What is 12/79 as a decimal?
What is 1/91 as a decimal? What is 13/79 as a decimal?
What is 1/92 as a decimal? What is 14/79 as a decimal?
What is 1/93 as a decimal? What is 15/79 as a decimal?
What is 1/94 as a decimal? What is 16/79 as a decimal?
What is 1/95 as a decimal? What is 17/79 as a decimal?
What is 1/96 as a decimal? What is 18/79 as a decimal?
What is 1/97 as a decimal? What is 19/79 as a decimal?
What is 1/98 as a decimal? What is 20/79 as a decimal?
What is 1/99 as a decimal? What is 21/79 as a decimal?
Convert similar fractions to percentages
From 1 Numerator From 79 Denominator
2/79 as a percentage 1/80 as a percentage
3/79 as a percentage 1/81 as a percentage
4/79 as a percentage 1/82 as a percentage
5/79 as a percentage 1/83 as a percentage
6/79 as a percentage 1/84 as a percentage
7/79 as a percentage 1/85 as a percentage
8/79 as a percentage 1/86 as a percentage
9/79 as a percentage 1/87 as a percentage
10/79 as a percentage 1/88 as a percentage
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How to Create Dummy Variables in R (with Examples)
In this tutorial, we will learn how to create dummy variables in R. Now, creating dummy/indicator variables can be carried out in many ways. For example, we can write code using the ifelse()
function, we can install the R-package fastDummies, and we can work with other packages and functions (e.g. model.matrix). In this post, however, we will use the ifelse() function and the fastDummies
package (i.e., dummy_cols() function). First, we will explain why we may need to dummy code some of our variables.
Table of Contents
In the first section of this post, you will learn when we need to dummy code our categorical variables. This section is followed by a section outlining what you need to have installed to follow this
post. For example, this section will show you how to install packages that you can use to create dummy variables in R. Now, this is followed by three answers to frequently asked questions concerning
dummy coding, both in general but also in R. Note, the answers will also give you the knowledge to create indicator variables.
Three Ways to Create Indicator Variables in R
Finally, we are going to get into the different methods that we can use for dummy coding in R. First, we will use the ifelse() function, and you will learn how to create dummy variables in two simple
steps. Second, we will use the fastDummies package, and you will learn three simple steps for dummy coding. The fastDummies package is also much easier to work with when you, e.g., want to make
indicator variables from multiple columns. Therefore, there will be a section covering this and removing columns we no longer need. In the following section, we will also look at how to use the
recipes package for creating dummy variables in R. We will also cover dummy coding based on multiple conditions. Before concluding the post, we will also learn about some other available options.
Dummy Coding
In regression analysis, a prerequisite is that all input variables are at the interval scale level, i.e., the distance between all steps on the scale of the variable is the same length. However, it
is not possible that all the possible things we want to research can be transformed into measurable scales. For example, different categories and characteristics do not necessarily have an inherent
ranking. If we are, for example, interested in the impact of different educational approaches on political attitudes, it is not possible to assume that science education is twice as much as social
science education or that a librarian education is half the education in biomedicine. The different types of education are simply different (but some aspects can, after all, be compared, for example,
the length).
What if we think that education has an important effect that we want to consider in our data analysis? Well, these are some situations when we need to use dummy variables. Read on to learn how to
create dummy variables for categorical variables in R.
Example of dummy coded variables
What You Need to Make Dummy Variables
In this section, before answering frequently asked questions, you will briefly learn what you need to follow in this post. First, if you plan on dummy coding using base R (e.g., by using the ifelse()
function) you do not need to install any packages. However, if you plan on using the fastDummies package or the recipes package, you must install either one (or both if you want to follow every
section of this R tutorial). Installing packages can be done using the install.packages() function. Here is how to install the two dummy coding packages:
install.packages(c("fastDummies", "recipes"))Code language: R (r)
Of course, if you only want to install one, you can remove the vector (i.e., c()) and leave the package you want. Note recipes is a package that is part of the Tidyverse. This means that we can
install this package and get a lot of useful packages by installing Tidyverse. In the next section, we will quickly answer some questions.
What is a Dummy Variable Give an Example?
A dummy variable is a variable that indicates whether an observation has a particular characteristic. A dummy variable can only assume the values 0 and 1, where 0 indicates the absence of the
property, and 1 indicates the presence of the same. The values 0/1 can be seen as no/yes or off/on. See the table below for some examples of dummy variables.
Why do we create dummy variables in R?
Creating dummy variables in R can incorporate nominal variables into regression analysis. It is quite easy to understand why we create dummy variables once you understand the regression model.
How do You Create a Dummy variable in R?
To create a dummy variable in R, you can use the ifelse() method:
df$Male <- ifelse(df$sex == 'male', 1, 0) df$Female <- ifelse(df$sex == 'female', 1, 0)
. This code will create two new columns where, in the column “Male” you will get the number “1” when the subject was a male and “0” when she was a female. For the column “Female”, it will be the
opposite (Female = 1, Male =0).
Variable Possible Values
Smoking Smoker = 1, Non-smoker = 0
Location North = 1, South = 0
Answer Yes = 1, No = 0
Examples of dummy variables
Note, if you want to, it is possible to rename the levels of a factor in R before making dummy variables. Now, let’s jump directly into a simple example of how to make dummy variables in R. In the
next two sections, we will learn dummy coding using R’s ifelse(), and fastDummies’ dummy_cols(). In the final section, we will look at how to use the recipes package for dummy coding.
How to Create Dummy Variables in R in Two Steps: ifelse() example
Here is how to create dummy variables in R using the ifelse() function in two simple steps:
1) Import Data
In the first step, import the data (e.g., from a CSV file):
dataf <- read.csv('https://vincentarelbundock.github.io/Rdatasets/csv/carData/Salaries.csv')Code language: R (r)
In the code above, we must ensure the character string points to where our data is stored (e.g., our .csv file). For example, when loading a dataset from our hard drive, we must ensure we add the
path to this file. In the next step, we will create two dummy variables in two lines of code.
2) Create the Dummy Variables with the ifelse() Function
Next, start creating the dummy variables in R using the ifelse() function:
dataf$Disc_A <- ifelse(dataf$discipline == 'A', 1, 0)
dataf$Disc_B <- ifelse(dataf$discipline == 'B', 1, 0)Code language: QML (qml)
In this simple example above, we created the dummy variables using the ifelse() function. First, we read data from a CSV file (from the web). Second, we created two new columns. In the first column
we created, we assigned a numerical value (i.e., 1) if the cell value in column discipline was ‘A’. If not, we assign the value ‘0’. Of course, we did the same when we created the second column. Here
are the first five rows of the dataframe:
First five rows with dummy variables
Now, data can be imported into R from other formats. If the data, we want to dummy code in R, is stored in Excel files, check out the post about how to read xlsx files in R. As we sometimes work with
datasets with many variables, using the ifelse() approach may not be the best way. For instance, creating dummy variables this way will definitely make the R code harder to read. In the next section,
we will go on and have a look at another approach for dummy coding categorical variables.
Three Steps to Create Dummy Variables in R with the fastDummies Package
In this section, we will use the fastDummies package to make dummy variables. There are three simple steps for creating dummy variables with the dummy_cols function.
Here is how to make dummy variables in R using the fastDummies package:
1) Install the fastDummies Package
First, we need to install the r-package. Installing r-packages can be done with the install.packages() function. So start up RStudio and type this in the console:
# Install fastDummies:
install.packages('fastDummies')Code language: R (r)
installing fastDummies
2) Load the fastDummies Package:
Next, we are going to use the library() function to load the fastDummies package into R:
# Import fastDummies
library('fastDummies')Code language: R (r)
Now that we have installed and loaded the fastDummies package, we will continue, in the next section, with dummy coding our variables.
3) Make Dummy Variables in R
Finally, we can use the dummy_cols() function to make the dummy variables. Here is how to make indicator variables in R using the dummy_cols() function:
# Create dummy variables:
dataf <- dummy_cols(dataf, select_columns = 'rank')Code language: R (r)
Now, the neat thing about using dummy_cols() is that we only get two line of codes. Furthermore, if we want to create dummy variables from more than one column, we’ll save even more lines of code
(see next subsection). Now that you are done creating dummy variables, you might want to extract time from datetime.
How to Create Dummy Variables for More than One Column
In the previous section, we used the dummy_cols() method to make dummy variables from one column. It is, of course, possible to dummy code many columns using the ifelse() function and the fastDummies
package. However, having many categories in our variables may require many lines of code using the ifelse() function. Thus, in this section, we will add one more column to the select_columns argument
of the dummy_cols function.
# Make dummy variables of two columns:
dataf <- dummy_cols(dataf, select_columns = c('rank', 'discipline'))Code language: PHP (php)
As evident from the code example above, the select_columns argument can also take a vector of column names. Of course, this means we can add as many as we need here. The above code will generate five
new columns containing the dummy coded variables. Note you can use R to conditionally add a column to the dataframe based on other columns if you need to.
Removing the Columns
In this section, we will use one more of the arguments of the dummy_cols() function: remove_selected_columns. This may be very useful if we, for instance, are going to make dummy variables of
multiple variables and do not need them for the data analysis later.
dataf.2 <- dummy_cols(dataf, select_columns = c('rank', 'discipline'),
remove_selected_columns = TRUE)Code language: R (r)
Note if we do not use the select_columns argument, dummy_cols will create dummy variables of all columns with categorical data. This is especially useful if we want to automatically create dummy
variables for all categorical predictors in the R dataframe. See the documentation for more information about the dummy_cols function. Finally, using the fastDummies package, we can create dummy
variables as rows with the dummy_rows function.
It is, of course, possible to drop variables after we have done the dummy coding in R. For example, see the post about how to remove a column in R with dplyr for more about deleting columns from the
dataframe. In some cases, you also need to delete duplicate rows. Now that you have created dummy variables, you can also go on and extract year from date.
How to create dummy variables with recipes
How to Make Dummy Variables in R with the step_dummy() Function
Here is a code example you can use to make dummy variables using the step_dummy() function from the recipes package:
# Making dummy variables
dummies <- dataf %>% recipe(salary ~ .) %>%
one_hot = TRUE) %>%
prep() %>%
bake(dataf)Code language: PHP (php)
Not to get into the details of the code chunk above, but we start by loading the recipes package. Second, we create the variable dummies. On the right of the “arrow” we take our dataframe and create
a recipe for preprocessing our data (i.e., this is what this function is for). In this function, we start by setting our dependent variable (i.e., salary), and then, after the tilde, we can add our
predictor variables. In our case, we want to select all other variables and use the dot. Now, in the next part, we use step_dummy(), where we make the dummy variables. The first parameter is the
categorical variable we want to dummy code. The second parameter is set to TRUE so that we get a column for male and a column for female. If this is not set to TRUE, we only get one column. Finally,
we use the prep() so that we later can apply this to the dataset we used (by using bake)). Here are the first ten rows of the new dataframe with indicator variables:
Dummy coded variables
Notice how the column sex was automatically removed from the dataframe. That is, in the dataframe we now have, containing the dummy coded columns, we no longer have the original categorical column.
Remember to check the regression analysis assumptions using, e.g., a test for normality in R and by making a residual plot in R with ggplot2.
Note, if you are planning on (also) doing an Analysis of Variance, you can check the assumption of equal variances with the Brown-Forsythe Test in R.
Create a Dummy Variable in R Based on Multiple Conditions
Here is how to make a dummy variable based on a condition:
# Create a dummy variable "is_junior" based on years service and rank
dataf$is_junior <- ifelse(dataf$yrs.since.phd <= 5 | dataf$rank == "AsstProf", 1, 0)Code language: R (r)
In the code block above, we introduce the addition of the logical OR (|) operator to create a dummy variable named “is_junior” based on multiple conditions. If an individual’s yrs.since.phd is less
than or equal to 5 or their rank is AsstProf, the corresponding entry in is_junior is set to 1; otherwise, it is set to 0
Other Options for Dummy Coding in R
Before summarizing this R tutorial, it may be worth mentioning that there are other options to recode categorical data to dummy variables. For instance, we could have used the model.matrix function,
and the dummies package. However, it is worth pointing out that the dummies package hasn’t been updated for a while.
Finally, it may be worth mentioning that the recipes package is part of the tidyverse package. Thus installing tidyverse, you can do more than create dummy variables. For instance, using the tibble
package, you can add empty column to the R dataframe or calculate/add new variables/columns to a dataframe in R.
Summary and Conclusion
In this post, we have 1) worked with R’s ifelse() function and 2) the fastDummies package to recode categorical variables to dummy variables in R. We learned that it was an easy task with R.
Especially, when we install and use a package such as fastDummies and have a lot of variables to dummy code (or many levels of the categorical variable). The next step in the data analysis pipeline
(may) now be to analyze the data (e.g., regression or random forest modeling).
Additional Resources
Now, there are other valuable resources to learn more about dummy variables (or indicator variables). In this section, you will find some articles and journal papers that you might find useful:
Here are some tutorials on this blog that you may find helpful:
10 thoughts on “How to Create Dummy Variables in R (with Examples)”
1. Dawn
Well think you, Sir! This was really a nice tutorial. I was struggling carrying out my data analysis in R and I realized that I needed to create dummy variables. Now, that I know how to do this,
I can continue with my project. I think, that, you should add more information about how to use the recipe and step_dummy functions. Explain that part in a bit more detail so that we can use it
for recoding the categorical variables (i.e., dummy code them).
Hi Dawn,
Thank you for your kind comments. I’ll look into adding what you suggest! Have a nice day,
2. Yasmine
Thank you! very helpful
Thanks Yasmine! Glad it helped!
3. Mohamed Shoukri
Thanks Mohamed. Glad you liked the post.
4. Javier
excellent explanation!
thanks for your contribution
1. Erik Marsja
Thank you, Javier. Glad you appreciated the tutorial.
5. Dr Garcia
I was looking for a clear and straightforward illustration of how to create a dummy variable in R, and this post was what I needed! The step-by-step guide made the process so easy to understand,
and I’m grateful for your examples. Now I have a solid grasp of how to create dummy variables for my data analysis projects effectively. Your explanation of the concepts and the code snippets you
shared were constructive.
Thank you for taking the time to put together such a helpful and informative post. Your writing and pedagogical style made the content very accessible, even for someone like me who is still
learning R programming. I’ll be bookmarking this post for future reference, as it’s an awesome resource for anyone looking to work with categorical data in R. Keep up the great work, and I’m
looking forward to exploring more of your content in the future!
1. Erik Marsja
Thank you for your kind commments! Glad you enjoyed the post and learned something from it.
Leave a Comment
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Truth-Values For Quantum Logic?
Quantum logicians* claim that there are cases where the behavior of true statements about the properties of subatomic particles fails to conform to the distribution of conjunction over disjunction in
classical logic (i.e. the rule that lets us go from P & (Q v R) to (P & Q) v (P & R)). Now, for there to be a counter-example to this law, we'd need a case where P was true, (Q v R) was true, but
both Q and R had some status other than "true."**
After all (holding the truth of P constant in all of these cases), if Q was true and R wasn't, then the premise of the relevant instance of Distribution would be true, but so would the conclusion.
The same would be true if R was true and Q wasn't. And, of course, it would still be true if P and Q were both true. If, on the other hand, Q and R were both false, then once again, we wouldn't have
a counter-example to Distribution, because the premise would be false.
Fair enough, you might think, but that just shows you that the old bivalent conception of truth is wrong, and that's exactly the sort of thing we should expect to be shown once we've really absorbed
the quantum revolution, really exposed the ancient dogmas encoded in classical logic to the searing light of empirical revision.
OK. Maybe. But postulating a third truth-value, by itself, doesn't clarify much here. What third truth-value would get the job done? A natural first thought is that what we're talking about here is a
truth-value gaps--i.e. the joint absence of the two classical values--but that's not going to get it done. If Q is neither true nor false, and R is neither true nor false, then why should (Q v R) be
true rather than neither true nor false itself? Now, if we think of third truth-value not as a gap but as a glut--the joint presence of the two classical values--the situation might seem to be a
little bit better. Now, after all, the premise of the instances of Distribution where Q and R are both oddly-valued comes out true (whatever else it might be). The problem, of course, is that the
conclusion also comes out true.
One might say that the third value is not a matter of being definitely neither or definitely both but being in some sense vague or ambiguous or indeterminate between the two. Fine. But why, then,
wouldn't both the premise and the conclusion come out as vague or ambiguous or indeterminate or whatever? If it's ambiguous whether or not Q is true, and ambiguous whether or not R is true, but P is
unambiguously true, shouldn't it be ambiguous whether (Q v R) is true, and also ambiguous whether (P & Q) is true, whether (P & R) is true, and whether ((P & Q) v (P & R)) is true?
One might set up the truth tables differently here, but its hard to see how one could do so, in a principled way and without opening oneself up to some "change of meaning" charges. Of course, some
people routinely level those charges against all heterodox proposals about the behavior of logical connectives, but it would be much harder to answer them here. To see why, think of it likes this:
In classical logic, "either P or Q is true" and "at least one of the following things is true: P, Q" are different ways of saying the same thing. Now, if a heterodox logician comes along and says
"sometimes it's ambiguous whether P is true, and it's equally ambiguous whether Q is true," and then concludes that in those cases it's ambiguous whether at least of the two is true, then the "change
of meaning" charge seems unfair. It seems more natural to say that they mean the same thing by "or" as the classical logician, but that they admit possibilities that the classical logician rejects.
If, by contrast, they say that it's ambiguous whether P is true, and ambiguous whether Q is true (and not epistemically ambiguous, but in terms of its objective truth-status), but that "either P or
Q" is unambiguously true, it really does start to seem like they're using "or" in a new way.
...or maybe not. A more radical move yet would be to simply reject truth-functionality entirely here. Just as "for any collection of numbers, there is a sum of those numbers" is true (and might even
seem so obvious as to follow from the meaning of "number" or "collection") so long as we restrict our focus to finite (and countably infinite) collections of numbers, but it breaks down when we get
to uncountably infinite collections (like the collection of all real numbers), to which addition simply doesn't apply, one could argue that "the truth-value of disjunctions is a function of the
truth-values of their disjuncts" holds when we restrict our attention to normal situations, but that it breaks down when we turn our attention to the outer edges of logical possibility that are
physically actualized by quantum weirdness.
OK, fair enough, but if they do choose to take that line, it's surely incumbent on the quantum logician to give us a clear account of exactly what exactly the distinction is between normal and
non-normal situations. If the distinction is simply a matter of shifting truth-values, then this isn't a proposal about a break-down of truth-functionality, it's simply about non-standard
truth-functionality, and given the failure of the third truth-value to transfer from the disjuncts to the disjunction, the change of meaning question looms large. If the distinction between normal
and non-normal situation is about something other than truth-values--e.g. we have a situation where Q and R are both false but where (Q v R) somehow manages to be true, or where P, Q and R are all
true but (P & Q) v (P & R) somehow fails to be true--then they really owe us a very clear explanation of how the inclusive "or" of formal logic can retain its customary meaning at the same time as
two false disjuncts somehow jointly yield a true disjunction, or two true disjuncts can fail to yield a true disjunction, and exactly what the difference is between situations where logical
connectives behave in this strange way and the situations in which they don't, and exactly how to distinguish between which situations are which.
Now, from an orthodox perspective, it's tempting to conclude from the whole mess that the proposal that Distribution fails in quantum contexts is just deeply confused, and that might even be the
right answer here, but I'd be far more interested in hearing attempts to resolve it and explain just how the trick can be turned--e.g. exactly how we can conceptualize a third truth-value that would
plausibly behave in the right way, or how to make sense of the idea that the standard truth-values would in the relevant situations stop combining in the standard ways.
*For our purposes here, the phrase "quantum logicians" refers to full-on, 1970s-Putnam-style, realist, monist quantum logicians, not the namby-pamby kind that just take quantum logic to be an
interesting mathematical representation of certain experimental results and leave it at that. The latter might be far easier to plausibly argue for, but it's also far more boring.
**Note that, for the sake of simplicity, in everything that follows I'm assuming that conjunction behaves in the standard way. If anyone wants to get into that in the comments, and provide a
quantum-logical motivation for questioning that, that's fine too.
2 comments:
Unknown said...
Suppose I adopt some version of the "refusal to assign truth values" approach (rather than the "assigning non-standard truth values" approach).
Let a p-assignment A- be any properly partial function from atomic sentence letters of L to truth-values (that is, let a p-assignment be the assignment of truth-values to some, but not all, of
the atomic sentences of the language).
Call an assignment A a completion of the p-assignment A- just in case i) A is a total function from atomic sentence letters of L to truth values and ii) for every atomic sentence letter S, if A-
assigns V to S, then A assigns V to S.
Now, consider the following rule(s) governing the application of "is true"/"is not true" relative to a p-assignment A-:
1) For any sentence S, if every completion of A- is such that "is true" applies to S, "is true" applies to S relative to A-.
2) For any sentence S, if every completion of A- is such that "is not true" applies to S, "is not true" applies to S relative to A-.
These rules for "is true"/"is not true" would render "Q v ~Q" true relative to every p-assignment (even those that are silent with respect to Q). This seems like a non-arbitrary way to extend "is
true" beyond the assignments of the weak Kleene scheme without running afoul of the charge of "meaning shift".
Unknown said...
In case it is not clear, that was simply supposed to be a demonstration of how you can get a truth value for a given complex expression without assigning truth-values to its components.
To extend the point to the case you have in mind: I assume that Q and R are contrary assessments of some feature of the subatomic particle. For simplicity, let's suppose that there is some S (the
remainder of the rival properties) such that:
1) Q entails ~S&~R, ~Q entails (S v R)
2) R entails ~S&~Q, ~R entails (S v Q)
3) S entails ~Q&~R, ~S entails (Q v R)
(If these suppositions are taken to constrain admissible p-assignments for L, then this will result in "is true" applying to (Q v R v S) on any admissible p-assignment ).
Now, suppose we have learned P and ~S. We are then in a position to infer P&(Q v R), (even on p-assignments that are silent with respect to Q and R).
What I am realizing now is that, even if this approach works (w/r/t the concerns you raise in the later portion of the blog post) it would not undermine (as far as I can tell) the inference from
P&(Q v R) to (P&Q)v(P&R), since "is true" would apply to that relative to permissible p-assignments. | {"url":"http://blogandnot-blog.blogspot.com/2010/01/truth-values-for-quantum-logic.html","timestamp":"2024-11-11T00:47:55Z","content_type":"text/html","content_length":"50297","record_id":"<urn:uuid:3b02a50d-249e-4e8e-aade-2e459b0fcc70>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00340.warc.gz"} |
How do you find lim cos(3theta)/(pi/2-theta) as theta->pi/2 using l'Hospital's Rule? | HIX Tutor
How do you find #lim cos(3theta)/(pi/2-theta)# as #theta->pi/2# using l'Hospital's Rule?
Answer 1
You need to see if the limit is in indeterminate form, so calculate the limit as #theta -> pi/2#
#cos(3(pi/2))/{pi/2-pi/2} = 0/0# which is indeterminate form
now do the derivative of the function
#lim_{theta->pi/2} d/dx[cos(3theta)/{pi/2-theta}]#
#d/dx [cos(3theta)] = 0#
#d/dx [pi/2-theta] = 0#
#0/0# is indeterminate form
so the limit doesn't exist
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Answer 2
To find the limit ( \lim_{\theta \to \frac{\pi}{2}} \frac{\cos(3\theta)}{\frac{\pi}{2} - \theta} ) using L'Hôpital's Rule, differentiate the numerator and the denominator separately with respect to (
\theta ) and then take the limit again. Repeat this process until you obtain a determinate form.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Bayes linear regression and basis-functions in Gaussian process regression – The Dan MacKinlay stable of variably-well-consider’d enterprises
Bayes linear regression and basis-functions in Gaussian process regression
a.k.a Fixed Rank Kriging, weight space GPs
February 22, 2022 — July 27, 2022
graphical models
Hilbert space
kernel tricks
machine learning
Another way of cunningly chopping up the work of fitting a Gaussian process is to represent the process as a random function comprising basis functions \(\phi=\left(\phi_{1}, \ldots, \phi_{\ell}\
right)\) with the Gaussian random weight vector \(w\) so that \[ f^{(w)}(\cdot)=\sum_{i=1}^{\ell} w_{i} \phi_{i}(\cdot) \quad \boldsymbol{w} \sim \mathcal{N}\left(\mathbf{0}, \boldsymbol{\Sigma}_{\
boldsymbol{w}}\right). \] \(f^{(w)}\) is a random function satisfying \(\boldsymbol{f}^{(\boldsymbol{w})} \sim \mathcal{N}\left(\mathbf{0}, \boldsymbol{\Phi}_{n} \boldsymbol{\Sigma}_{\boldsymbol{w}}
\boldsymbol{\Phi}^{\top}\right)\), where \(\boldsymbol{\Phi}_{n}=\boldsymbol{\phi}(\mathbf{X})\) is a \(|\mathbf{X}| \times \ell\) matrix of features. This is referred to as a weight space approach
in ML.
TODO: I just assumed centred weights here, but that is crazy. Update to relax that assumption.
We might imagine this representation would be exact if we had countably many basis functions, and under sane conditions it is. We would like to know, further, that we can find a basis such that we
need not too many basis functions to represent the process. Looking at the Karhunen-Loève theorem we might imagine that this can sometimes work out fine, and indeed it does, sometimes.
This is a classic; see Chapter 3 of Bishop (2006) is classic and nicely clear. Cressie and Wikle (2011) targets for the spatiotemporal context.
Hijinks ensue when selecting the basis functions. If we were to treat the natural Hilbert space here seriously we could consider identifying the bases as eigenfunctions of the kernel. This is not
generally easy. We tend to use either global bases such as Fourier bases or more generally Karhunen-Loéve bases, or construct local bases of limited overlap (usually piecewise polynomials AFAICT).
The kernel trick writes a kernel \(k\) as an inner product in a corresponding reproducing kernel Hilbert space (RKHS) \(\mathcal{H}_{k}\) with a feature map \(\varphi: \mathcal{X} \rightarrow \
mathcal{H}_{k} .\) In sufficiently nice cases the kernel is well approximated \[ k\left(\boldsymbol{x}, \boldsymbol{x}^{\prime}\right)=\left\langle\varphi(\boldsymbol{x}), \varphi\left(\boldsymbol{x}
^{\prime}\right)\right\rangle_{\mathcal{H}_{k}} \approx \boldsymbol{\phi}(\boldsymbol{x})^{\top} \overline{\boldsymbol{\phi}\left(\boldsymbol{x}^{\prime}\right)} \] where \(\boldsymbol{\phi}: \
mathcal{X} \rightarrow \mathbb{C}^{\ell}\) is a finite-dimensional feature map. TODO: What is the actual guarantee here?
1 Fourier features
When the Fourier basis is natural for the problem we are in a pretty good situation. We can use the Wiener Khintchine relations to analyse and simulate the process. Connection perhaps to Fourier
features in neural nets?
2 Random Fourier features
The random Fourier features method (Rahimi and Recht 2007, 2008) constructs a Monte Carlo estimate to a stationary kernel by representing the inner product in terms of \(\ell\) complex exponential
basis functions \(\phi_{j}(\boldsymbol{x})=\ell^{-1 / 2} \exp \left(i \boldsymbol{\omega}_{j}^{\top} \boldsymbol{x}\right)\) with frequency parameters \(\boldsymbol{\omega}_{j}\) sampled
proportionally to the spectral density \(\rho\left(\boldsymbol{\omega}_{j}\right).\)
This sometimes has a favourable error rate (Sutherland and Schneider 2015).
3 K-L basis
We recall from the Karhunen-Loéve notebook that the mean-square-optimal \(f^{(w)}\) for approximating a Gaussian process \(f\) is found by truncating the Karhunen-Loéve expansion \[ f(\cdot)=\sum_{i=
1}^{\infty} w_{i} \phi_{i}(\cdot) \quad w_{i} \sim \mathcal{N}\left(0, \lambda_{i}\right) \] where \(\phi_{i}\) and \(\lambda_{i}\) are, respectively, the \(i\)-th (orthogonal) eigenfunction and
eigenvalue of the covariance operator \(\psi \mapsto \int_{\mathcal{X}} \psi(\boldsymbol{x}) k(\boldsymbol{x}, \cdot) \mathrm{d} \boldsymbol{x}\), written in decreasing order of \(\lambda_{i}\). What
is the orthogonal basis \(\{\phi_{i}\}_i\) though? That depends on the problem and can be a lot of work to calculate.
In the case that our field is stationary on a “nice” domain, though, this can easy — we simply have the Fourier features as the natural basis.
5 “Decoupled” bases
Cheng and Boots (2017);Salimbeni et al. (2018);Shi, Titsias, and Mnih (2020);Wilson et al. (2020). | {"url":"https://danmackinlay.name/notebook/gp_basis.html","timestamp":"2024-11-08T20:41:36Z","content_type":"application/xhtml+xml","content_length":"51334","record_id":"<urn:uuid:34239e69-e0c2-4544-bb4d-60092109b09e>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00681.warc.gz"} |
In This Topic
Histogram chart plots the frequency distribution of data against the defined class intervals or bins. These bins are created by dividing the raw data values into a series of consecutive and
non-overlapping intervals. Based on the number of values falling in a particular bin, frequencies are then plotted as rectangular columns against continuous x-axis.
The following representations can created with the help of a histogram.
The following images show a histogram and a cumulative histogram created using FlexChart.
Histogram Cumulative Histogram
To create a histogram, you need to add the Histogram series and set the ChartType property to Histogram. Once you provide relevant data by setting the BindingX to the original raw data values that
are to be plotted on the X-axis, FlexChart generates frequency distribution for the data and plots the same in histogram. The chart automatically calculates the intervals in which your data is
grouped. However, if required, you can also specify the width of these intervals by setting the BinWidth property. Apart from this you can also create a cumulative histogram by setting the
CumulativeMode property to true.
The following code snippet demonstrates how to generate Histogram chart for a particular data.
<Chart:C1FlexChart x:Name="flexChart"
ItemsSource="{Binding DataContext.Data}"
<Chart:Axis Format="0.00"></Chart:Axis>
<Chart:Histogram x:Name="histogramSeries" SeriesName="Frequency"/>
Frequency Polygon
A frequency polygon shows a frequency distribution representing the overall pattern in the data. It is a closed two-dimensional figure of straight line segments -created by joining the mid points of
the top of the bars of a histogram.
Use the following steps to create a frequency polygon using histogram chart.
1. Set the AppearanceType property to FrequencyPolygon. This property accepts value from the HistogramAppearance enumeration.
2. Set the style for frequency polygon using the FrequencyPolygonStyle property.
Moreover, you can also create a cumulative frequency polygon by setting the CumulativeMode property to true.
The following images show a frequency polygon and a cumulative frequency polygon created using FlexChart.
Frequency Polygon Cumulative Frequency Polygon
Use the following code snippet to create a frequency polygon.
In XAML Copy Code
<c1:Histogram x:Name="histogramSeries" SeriesName="Frequency" CumulativeMode="True"
AppearanceType="FrequencyPolygon" />
<c1:ChartStyle Stroke="Red" StrokeThickness="2"/>
In Code Copy Code
histogramSeries.AppearanceType = HistogramAppearance.FrequencyPolygon;
histogramSeries.FrequencyPolygonStyle = new ChartStyle()
{Stroke = new SolidColorBrush(Color.FromRgb(255, 0, 0))};
// To create a cumulative frequency polygon
histogramSeries.CumulativeMode = true;
Gaussian Curve
Gaussian curve is a bell shaped curve, also known as normal curve, which represents the probability distribution of a continuous random variable. It represents a unimodal distribution as it only has
one peak. Moreover, it shows a symmetric distribution as fifty percent of the data set lies on the left side of the mean and fifty percent of the data lies on the right side of the mean.
Use the following steps to create a Gaussian curve using histogram chart.
1. Set the AppearanceType property to Histogram. This property accepts value from the HistogramAppearance enumeration.
2. Set the NormalCurve.Visible property to true to create a Gaussian curve.
3. Set the style for Gaussian curve using the NormalCurve.LineStyle property.
Following image illustrates a Gaussian curve created using FlexChart, which depicts probability distribution of scores obtained by students of a university in half yearly examinations.
Use the following code snippet to create a Gaussian curve.
In XAML Copy Code
<c1:Histogram x:Name="histogramSeries" SeriesName="Frequency"
AppearanceType="Histogram" />
<c1:ChartStyle Stroke="Green" StrokeThickness="2"/>
In code Copy Code
histogramSeries.AppearanceType = HistogramAppearance.Histogram;
histogramSeries.NormalCurve.Visible = true;
histogramSeries.NormalCurve.LineStyle = new ChartStyle() {Stroke = new SolidColorBrush(Color.FromRgb(0, 128, 0))}; | {"url":"https://developer.mescius.com/componentone/docs/uwp/online-flexchart/Histogram.html","timestamp":"2024-11-10T08:16:17Z","content_type":"application/xhtml+xml","content_length":"24229","record_id":"<urn:uuid:3e80b5a7-fd87-4d50-bdeb-9659eae7fe2a>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00089.warc.gz"} |
Local Geometric Analysis and Applications - MIT Statistics and Data Science Center
IDSS Special Seminar
Local Geometric Analysis and Applications
October 11, 2018 @ 4:00 pm - 5:00 pm
Lizhong Zheng (MIT)
Abstract: Local geometric analysis is a method to define a coordinate system in a small neighborhood in the space of distributions over a given alphabet. It is a powerful technique since the notions
of distance, projection, and inner product defined this way are useful in the optimization problems involving distributions, such as regressions. It has been used in many places in the literature
such as correlation analysis, correspondence analysis. In this talk, we will go through some of the basic setups and properties, and discuss a few applications in information theory, dimension
reduction and softmax regression.
About this Seminar: This seminar consists of a series of lectures each followed by a period of informal discussion and social. The topics are at the nexus of information theory, inference, causality,
estimation, and non-convex optimization. The lectures are intended to be tutorial in nature with the goal of learning about interesting and exciting topics rather than merely hearing about the most
recent results. The topics are driven by the interests of the speakers, and with the exception of the two lectures on randomness and information, there is no planned coherence or dependency among
them. Ad hoc follow-on meetings about any of the topics presented are highly encouraged. | {"url":"https://stat.mit.edu/calendar/topics-information-inference-seminar-2/","timestamp":"2024-11-08T02:01:53Z","content_type":"text/html","content_length":"107365","record_id":"<urn:uuid:dbc466c1-903f-4c86-a0e3-1c62696d724b>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00775.warc.gz"} |
Paul Adrien Maurice Dirac
• Paul Dirac at the Notable Names Database
• Dirac Medal of the International Centre for Theoretical Physics
• Biography at the MacTutor archive
• Paul Adrien Maurice Dirac Biography
• Dirac Medal of the World Association of Theoretically Oriented Chemists (WATOC)
• Photographs of Dirac
• The Paul Dirac Collection at Florida State University
• The Paul A. M. Dirac Collection Finding Aid at Florida State University
• discovery of new productive forms of atomic theory. | {"url":"https://www.scientificlib.com/en/Physics/Biographies/PaulAdrienMauriceDirac.html","timestamp":"2024-11-12T01:01:19Z","content_type":"application/xhtml+xml","content_length":"16787","record_id":"<urn:uuid:98f9d517-a4cf-481d-9dd0-0aa0d05e51bf>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00572.warc.gz"} |
Simple But Brilliant
Today, simple but brilliant. The University of Houston's College of Engineering presents this series about the machines that make our civilization run, and the people whose ingenuity created them.
I came across an old journal article the other day — a very famous one. I wasn't sure what to expect. After all, it was written by a physicist. It could've been filled with complicated mathematics.
But it wasn't. The ideas were explained in clear, simple terms, comprehensible to a first year college student. That's rare for an original paper. The best explanations of an idea usually come years
after it's first conceived. The paper? "On the Electrodynamics of Moving Bodies," where Albert Einstein introduced the theory of special relativity.
As a child I was fascinated by special relativity. It seemed so mysterious. Time travel became real. Not any type of time travel. Most science fiction gets it wrong. The Terminator can't go back and
kill John Connor's mother to prevent his birth. But in theory we can send spaceships out at near light speed, and when they return a thousand years later those on board may have aged only a few days.
I must have read a half-dozen popular books on special relativity when I was younger. They were amazing. Here were unimaginable paradoxes accepted by the scientific community. These popular books
were teasers; filled with explanations that seemed too simple. I knew that someday I'd have to learn what relativity was really all about.
That time came during my first college physics course. And I was shocked at its simplicity. To this day I remember thinking "That's it? That's all there is to it?"
But that's the beauty of special relativity. It's not complicated. What makes Einstein's contribution so phenomenal is that he started with two basic premises and followed them to their logical
conclusion. Other physicists had been playing with similar ideas for decades. But they never made the leap of imagination Einstein did. They couldn't imagine that time and distance could change with
the relative speed of an observer. The idea was too outlandish. Einstein made that leap. And it forever changed how we look at our world.
Since Einstein, physicists haven't been afraid to propose the seemingly preposterous. Our world may have more than three physical dimensions. There may be infinite universes, with infinitely more
created every instant. Respectable conferences are held; distinguished papers are published. If a theory explains the world we see, we'll accept it, no matter what strange consequences it entails.
Unfortunately, many of these new ideas are complicated. Popular accounts just don't do them justice. Years of mathematics and physics are required to truly comprehend them. Which makes Einstein's
original paper on special relativity all that much more wonderful. It's a unique opportunity to understand our mind-boggling universe in the words of one of its most inventive minds.
I'm Andy Boyd at the University of Houston, where we're interested in the way inventive minds work.
(Theme music)
Albert Einstein. On the Electrodynamics of Moving Bodies. English translation of the original 1905 German publication. The translation is taken from the book The Principle of Relativity, first
published by Methuen and Company, Ltd., London, 1923. It was accessed on December 15, 2008, on the public domain web site http://www.fourmilab.ch/etexts/einstein/specrel/www/.
The picture of Einstein, taken when he won the Nobel Prize in 1921, is taken from Wikimedia Commons.
An excerpt from the translation of Einstein's journal article appears below. It is taken from Part 1 of the paper, "Kinematical Part," which covers special relativity.
If we wish to describe the motion of a material point, we give the values of its co-ordinates as functions of the time. Now we must bear carefully in mind that a mathematical description of this kind
has no physical meaning unless we are quite clear as to what we understand by ``time.'' We have to take into account that all our judgments in which time plays a part are always judgments of
simultaneous events. If, for instance, I say, ``That train arrives here at 7 o'clock,'' I mean something like this: ``The pointing of the small hand of my watch to 7 and the arrival of the train are
simultaneous events.''
It might appear possible to overcome all the difficulties attending the definition of ``time'' by substituting ``the position of the small hand of my watch'' for ``time.'' And in fact such a
definition is satisfactory when we are concerned with defining a time exclusively for the place where the watch is located; but it is no longer satisfactory when we have to connect in time series of
events occurring at different places, or—what comes to the same thing—to evaluate the times of events occurring at places remote from the watch.
We might, of course, content ourselves with time values determined by an observer stationed together with the watch at the origin of the co-ordinates, and co-ordinating the corresponding positions of
the hands with light signals, given out by every event to be timed, and reaching him through empty space. But this co-ordination has the disadvantage that it is not independent of the standpoint of
the observer with the watch or clock, as we know from experience. We arrive at a much more practical determination along the following line of thought.
If at the point A of space there is a clock, an observer at A can determine the time values of events in the immediate proximity of A by finding the positions of the hands which are simultaneous with
these events. If there is at the point B of space another clock in all respects resembling the one at A, it is possible for an observer at B to determine the time values of events in the immediate
neighbourhood of B. But it is not possible without further assumption to compare, in respect of time, an event at A with an event at B. We have so far defined only an ``A time'' and a ``B time.'' We
have not defined a common ``time'' for A and B, for the latter cannot be defined at all unless we establish by definition that the ``time'' required by light to travel from A to B equals the ``time''
it requires to travel from B to A. Let a ray of light start at the ``A time''tA from A towards B, let it at the ``B time'' tB be reflected at B in the direction of A, and arrive again at A at the ``A
time'' t'A.
In accordance with definition the two clocks synchronize if
tB - tA = t'A - tB
We assume that this definition of synchronism is free from contradictions, and possible for any number of points; and that the following relations are universally valid:
1. If the clock at B synchronizes with the clock at A, the clock at A synchronizes with the clock at B.
2. If the clock at A synchronizes with the clock at B and also with the clock at C, the clocks at B and C also synchronize with each other.
Thus with the help of certain imaginary physical experiments we have settled what is to be understood by synchronous stationary clocks located at different places, and have evidently obtained a
definition of ``simultaneous,'' or ``synchronous,'' and of ``time.'' The ``time'' of an event is that which is given simultaneously with the event by a stationary clock located at the place of the
event, this clock being synchronous, and indeed synchronous for all time determinations, with a specified stationary clock.
In agreement with experience we further assume the quantity
2AB = c,
t'A - tA
to be a universal constant—the velocity of light in empty space.
It is essential to have time defined by means of stationary clocks in the stationary system, and the time now defined being appropriate to the stationary system we call it ``the time of the
stationary system.'' | {"url":"https://engines.egr.uh.edu/episode/2447","timestamp":"2024-11-09T21:02:40Z","content_type":"text/html","content_length":"36474","record_id":"<urn:uuid:0ac0df40-7671-419e-aa33-c14168f04f98>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00698.warc.gz"} |
Negative Values - Two’s Complement
To represent negative numbers, we must use the same tool we do for everything else: 0s and 1s.
The most simple possible scheme would be to use one bit to represent the sign - say 0 for positive and 1 for negative. If we did that, then the numbers 0010 and 1010 would mean +2 (0 = positive, 010
= 2) and -2 (1 = negative, 010 = 2). But if we tried to add those numbers using the normal process, we would get 0010 + 1010 = 1100. That says +2 + -2 = -4! (1 = negative, 100 = 3) The normal
addition rules do not work with this simple scheme.
A number that starts with 1 is negative. Its value is defined by the following rule: take the other bits and flip them (0s become 1s and 1s become 0s) then add one to the value they represent. Thus
1011 would be interpreted as negative because of the leading 1, then we would take the other bits - 011 - and flip each bit to get 100, 100 is 4 and we add one to that to get 5, so 1011 means -5.
A leading 0 means positive - read the number normally.
A leading 1 means negative - flip the remaining bits, read their value, and one to the value. Make that value negative.
What decimal number does the two’s complement number 0010 represent?
What decimal number does the two’s complement number 1010 represent?
Leading 1 says “negative” and requires us to flip the last three bits to 101. That means 5. Add one to get 6.
What decimal number does the two’s complement number 1110 represent?
Leading 1 says “negative” and requires us to flip the last three bits to 001. That means 1. Add one to get 2.
We can use this same idea with more than 4 bits. We always just use the first bit as the sign and the rest of the bits as the value and use the same rules for negative numbers. Thus the 8-bit two’s
complement number 11011000 would mean: first bit is 1, so negative, flip the last seven bits to 0100111, that is 39 (32 + 4 + 2 + 1), add one to get 40, so the value is -40.
The main advantage of two’s complement is that the normal rules for addition work with it as long as we ignore extra bits. Say we have:
That means 6 + (-2). If we add them using the normal rules we would get:
Since we started with 4 bits, we should only keep the last four bits of the answer: 0100 or 4. That means 6 + (-2) = 4.
It also works for two negative numbers. Here is -2 + (-2):
Take only the last 4 bits and we get 1100. The leading 1 means negative. So flip the last three bits from 100 to 011. That means 3. Add one and get 4. So -2 + (-2) = -4.
It is also easy to find the inverse of a number. To turn a negative into a positive or vice versa, invert all the bits and add one. If there is a carry past the last digit, ignore it.
To change the sign of a number, flip all the bits and add one. Ignore any carry past the last original digit.
0101 (+5) 1010 (flip bits) 1010 (now add one) + 1 1011 (-5)
1011 (-5) 0100 (flip bits) 0100 (now add one) + 1 0101 (+5)
1 1 (Carries) 0101 (5) +0101 (5) 1010 (-6 in two's complement)
As an unsigned number, 1010 would mean ten. But in two’s complement, that means -6! The same thing can happen with negative numbers - if a negative number becomes too small it can wrap around to
positive numbers!
Normally integers are stored as 32-bit values. This gives a range of approximately +2.147 billion to -2.147 billion - usually enough to hold our answers. But if your math problem involves an answer
that is too big you can “wrap around”.
Which of these represents -3 as a five-bit two’s complement number?
That is -13. For negative numbers remember you have to flip the bits and add one
Remember you need to add one after flipping the bits in a negative number.
Given a 5-bit two’s complement number, what is the largest positive value you can represent?
The first digit is the sign, what is the largest value you can make with the other 4?
Remember that the first digit is the sign. It must be 0. | {"url":"https://runestone.academy/ns/books/published/welcomecs2/data-representation_negative-values-twos-complement.html","timestamp":"2024-11-09T19:21:59Z","content_type":"text/html","content_length":"137532","record_id":"<urn:uuid:b76e78d0-b3a1-439d-8acc-f325868ddcd6>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00284.warc.gz"} |