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You are looking at all articles with the topic "Numbers". We found 8 matches.
142857, the six repeating digits of 1/7, 0.142857, is the best-known cyclic number in base 10. If it is multiplied by 2, 3, 4, 5, or 6, the answer will be a cyclic permutation of itself, and will correspond to the repeating digits of 2/7, 3/7, 4/7, 5/7, or 6/7 respectively.
142,857 is a Kaprekar number and a Harshad number (in base 10).
"142857" | 2015-08-05 | 100 Upvotes 22 Comments
An illegal number is a number that represents information which is illegal to possess, utter, propagate, or otherwise transmit in some legal jurisdiction. Any piece of digital information is representable as a number; consequently, if communicating a specific set of information is illegal in some way, then the number may be illegal as well.
"Illegal number – Represents information which is illegal to possess" | 2021-06-16 | 39 Upvotes 26 Comments
"Illegal number" | 2019-01-11 | 6 Upvotes 10 Comments
"Illegal number - Wikipedia" | 2013-10-01 | 14 Upvotes 1 Comments
"Illegal Numbers" | 2012-10-28 | 184 Upvotes 95 Comments
In number theory, a narcissistic number (also known as a pluperfect digital invariant (PPDI), an Armstrong number (after Michael F. Armstrong) or a plus perfect number) in a given number base
{\displaystyle b}
is a number that is the sum of its own digits each raised to the power of the number of digits.
"Narcissistic Number" | 2015-09-07 | 27 Upvotes 2 Comments
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A bond is a type of loan contract between an issuer (the seller of the bond) and a holder (the purchaser of a bond). The issuer is essentially borrowing or incurring a debt that is to be repaid at "par value" entirely at maturity (i.e., when the contract ends). In the meantime, the holder of this debt receives interest payments (coupons) based on cash flow determined by an annuity formula. From the issuer's point of view, these cash payments are part of the cost of borrowing, while from the holder's point of view, it's a benefit that comes with purchasing a bond.
The present value (PV) of a bond represents the sum of all the future cash flow from that contract until it matures with full repayment of the par value. To determine this—in other words, the value of a bond today—for a fixed principal (par value) to be repaid in the future at any predetermined time—we can use a Microsoft Excel spreadsheet.
\begin{aligned} &\text{Bond Value} = \sum_{ p = 1 } ^ {n} \text{PVI}_n + \text{PVP} \\ &\textbf{where:} \\ &n = \text{Number of future interest payments} \\ &\text{PVI}_n = \text{Present value of future interest payments} \\ &\text{PVP} = \text{Par value of principal} \\ \end{aligned}
Bond Value=p=1∑nPVIn+PVPwhere:n=Number of future interest paymentsPVIn=Present value of future interest paymentsPVP=Par value of principal
We will discuss the calculation of the present value of a bond for the following:
A) Zero Coupon Bonds
B) Bonds with annual annuities
C) Bonds with bi-annual annuities
D) Bonds with continuous compounding
E) Bonds with dirty pricing
Generally, we need to know the amount of interest expected to be generated each year, the time horizon (how long until the bond matures), and the interest rate. The amount needed or desired at the end of the holding period is not necessary (we assume it to be the bond's face value).
Let's say we have a zero coupon bond (a bond which does not deliver any coupon payment during the life of the bond but sells at a discount from the par value) maturing in 20 years with a face value of $1,000. In this case, the bond's value has decreased after it was issued, leaving it to be bought today at a market discount rate of 5%. Here is an easy step to find the value of such a bond:
Here, "rate" corresponds to the interest rate that will be applied to the face value of the bond.
"Nper" is the number of periods the bond is compounded. Since our bond is maturing in 20 years, we have 20 periods.
"Pmt" is the amount of the coupon that will be paid for each period. Here we have 0.
"Fv" represents the face value of the bond to be repaid in its entirety at the maturity date.
The bond has a present value of $376.89.
B. Bonds with Annuities
Company 1 issues a bond with a principal of $1,000, an interest rate of 2.5% annually with maturity in 20 years and a discount rate of 4%.
The bond provides coupons annually and pays a coupon amount of 0.025 x 1000= $25.
Notice here that "Pmt" = $25 in the Function Arguments Box.
The present value of such a bond results in an outflow from the purchaser of the bond of -$796.14. Therefore, such a bond costs $796.14.
C. Bonds with Bi-annual Annuities
The bond provides coupons annually and pays a coupon amount of 0.025 x 1000 ÷ 2= $25 ÷ 2 = $12.50.
The semiannual coupon rate is 1.25% (= 2.5% ÷ 2).
Notice here in the Function Arguments Box that "Pmt" = $12.50 and "nper" = 40 as there are 40 periods of 6 months within 20 years. The present value of such a bond results in an outflow from the purchaser of the bond of -$794.83. Therefore, such a bond costs $794.83.
D. Bonds with Continuous Compounding
Example 5: Bonds with continuous compounding
Continuous compounding refers to interest being compounded constantly. As we saw above, we can have compounding that is based on an annual, bi-annual basis or any discrete number of periods we would like. However, continuous compounding has an infinite number of compounding periods. The cash flow is discounted by the exponential factor.
E. Dirty Pricing
The clean price of a bond does not include the accrued interest to maturity of the coupon payments. This is the price of a newly issued bond in the primary market. When a bond changes hands in the secondary market, its value should reflect the interest accrued previously since the last coupon payment. This is referred to as the dirty price of the bond.
Dirty Price of the Bond = Accrued Interest + Clean Price. The net present value of the cash flows of a bond added to the accrued interest provides the value of the Dirty Price. The Accrued Interest = ( Coupon Rate x elapsed days since last paid coupon ) ÷ Coupon Day Period.
Company 1 issues a bond with a principal of $1,000, paying interest at a rate of 5% annually with a maturity date in 20 years and a discount rate of 4%.
The coupon is paid semi-annually: Jan. 1 and July 1.
The bond is sold for $100 on April 30, 2011.
Since the last coupon was issued, there have been 119 days of accrued interest.
Thus the accrued interest = 5 x (119 ÷ (365 ÷ 2) ) = 3.2603.
Excel provides a very useful formula to price bonds. The PV function is flexible enough to provide the price of bonds without annuities or with different types of annuities, such as annual or bi-annual.
Microsoft. "PV Function."
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Incomplete Information Choice on Incumbents, Cognitive Ability and Personality
1Department of Economics, Federal University of Santa Catarina, Florianopolis, Brazil
\eta ={\beta }_{0}+{\beta }_{1}S+{\beta }_{2}\text{CRT}+{\beta }_{3}\text{H}+{\beta }_{4}\text{E}+{\beta }_{5}\text{X}+{\beta }_{6}\text{A}+{\beta }_{7}\text{C}+{\beta }_{8}\text{O}+\epsilon
\epsilon
is a random error. The transition probability from State 0 to State 1 can be estimated by a logistic model:
P\left(\eta \right)=\frac{\mathrm{exp}\eta }{1+\mathrm{exp}\eta }
Table 3 shows the estimation of coefficients
\beta
by the Fisher scoring method, which is equivalent to iteratively reweighted least squares.
\eta =13.2-0.6S-0.87CRT-1.02H+1.58E-1.48X-1.18C
This can be used to estimate the transition probability
P\left(\eta \right)
from State 0 (no switching from Laptop A) to State 1 (switching from Laptop A to Laptop B) for each individual participant in our experiment. Figure 4 shows
\eta
to participants who switched (in red) and to those who did not (in blue). The larger
\eta
, the bigger
P\left(\eta \right)
To evaluate the fit of our model to the data, we consider that if
P\left(\eta \right)>0.5
then one individual participant would switch from Laptop A to Laptop B; if
P\left(\eta \right)<0.5
she would not. Table 5 shows the model adjusts well in 63 percent of cases (that is, 0.41 + 0.22), which refer to the tails of the distribution in Figure 4. Of note,
P\left(\eta \right)
increases for greater
\eta
, and values of
\eta
approaching zero mean random switches, that is,
P\left(\eta \right)=0.5
Figure 4. Probability distribution of switching from the prior choice. It shows the linear predictor of switching,
\eta
, to participants who switched from Laptop A to Laptop B (in red) and to those who did not (in blue). The larger
\eta
P\left(\eta \right)
Da Silva, S., Matsushita, R. and Ramos, M. (2019) Incomplete Information Choice on Incumbents, Cognitive Ability and Personality. Open Access Library Journal, 6: e5476. https://doi.org/10.4236/oalib.1105476
1. Kardes, F.R., Posavac, S.S., Silvera, D., Cronley, M.L., Sanbonmatsu, D.M., Schertzer, S., Miller, F., Herr, P.M. and Chandrashekaran, M. (2006) Debiasing Omission Neglect. Journal of Business Research, 59, 786-792.
2. Kunda, Z. (1990) The Case for Motivated Reasoning. Psychological Bulletin, 108, 480-498. https://doi.org/10.1037//0033-2909.108.3.480
3. Pyszczynski, T. and Greenberg, J. (1987) Toward an Integration of Cognitive and Motivational Perspectives on Social Inference: A Biased Hypothesis Testing Model. Advances in Experimental Social Psychology, 20, 297-340.
4. Russo, J.E., Medvec, V.H. and Meloy, M.G. (1996) The Distortion of Information During Decisions. Organizational Behavior and Human Decision Processes, 66, 102-110. https://doi.org/10.1006/obhd.1996.0041
5. Irmak, C., Kramer, T. and Sen, S. (2017) Choice under Incomplete Information on Incumbents: Why Consumers with Stronger Preferences Are More Likely to Abandon their Prior Choices. Journal of Consumer Psychology, 27, 264-269.
6. Harmon-Jones, E. and Mills, J. (1999) Cognitive Dissonance: Progress on a Pivotal Theory in Social Psychology. American Psychological Association, Washington DC.
7. Frederick, S. (2005) Cognitive Reflection and Decision Making. Journal of Economic Perspectives, 19, 25-42. https://doi.org/10.1257/089533005775196732
8. De Vries, R.W. (2013) The 24-Item Brief HEXACO Inventory (BHI). Journal of Research in Personality, 47, 871-880. https://doi.org/10.1016/j.jrp.2013.09.003
9. Evans, J.S.B.T. (2003) In Two Minds: Dual-Process Accounts of Reasoning. Trends in Cognitive Sciences, 7, 454-459. https://doi.org/10.1016/j.tics.2003.08.012
10. Evans, J.S.B.T. (2008) Dual-Processing Accounts of Reasoning, Judgment, and Social Cognition. Annual Review of Psychology, 59, 255-278.
11. Goldberg, L.R. (1993) The Structure of Phenotypic Personality Traits. American Psychologist, 48, 26-34. https://doi.org/10.1037//0003-066X.48.1.26
12. McCrae, R.R. and Costa Jr, P.T. (2004) A Contemplated Revision of the NEO Five-Factor Inventory. Personality and Individual Differences, 36, 587-596.
13. Ashton, M.C., Lee, K., Perugini, M., Szarota, P., De Vries, R.E., Di Blas, L., Boies, K. and De Raad, B. (2004) A Six-Factor Structure of Personality-Descriptive Adjectives: Solutions from Psycholexical Studies in Seven Languages. Journal of Personality and Social Psychology, 86, 356-366. https://doi.org/10.1037/0022-3514.86.2.356
14. Toplak, M.E., West, R.F. and Stanovich, K.E. (2014) Assessing Miserly Information Processing: An Expansion of the Cognitive Reflection Test. Thinking & Reasoning, 20, 147-168. https://doi.org/10.1080/13546783.2013.844729
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Trade - C3IAM - IAMC-Documentation
Revision as of 16:56, 27 June 2021 by Qiao-Mei Liang (talk | contribs)
Taking foreign trade into account, we adopts Armington assumption, which assumes there is imperfect substitutability between imports and domestic output sold domestically. The degree to which domestic and imported goods differ is reflected by the elasticity of substitution between them. Changes in the relative shares of foreign and domestic goods in the composite are determined by changes in the relative prices of these goods at home and abroad, given the Armington substitution elasticity and the initial shares of these goods in the benchmark SAM (Social Accounting Matrix). The commodity that supplied domestically is composed of domestic and imported commodities following a CES function. Furthermore, domestic commodity is used to meet domestic demands and for exports. In C3IAM, we use a constant elasticity transformation (CET) function to allocate total domestic output between exports and domestic sales, shown in Equations. (1) and (2).
{\displaystyle X_{i,r}=A_{Ex,i,r}\centerdot [\alpha _{Ex,i,r}\centerdot E_{i}^{\rho _{Ex,i,r}}+(1-\alpha _{Ex,i,r})\centerdot D_{i}^{\rho _{Ex,i,r}}]{\frac {1}{\rho _{Ex,i,r}}}}
{\displaystyle {\frac {E_{i},t,r}{D_{i},t,r}}=[{\frac {1-\alpha _{E}X,i,r}{\alpha _{E}X,i,r}}\centerdot {\frac {PE_{i},t,r}{PD_{i},t,r}}]_{E}^{\alpha }X,i}
{\displaystyle E_{i,r}}
{\displaystyle D_{i,r}}
respectively represent exports and domestic sales of domestically produced good
{\displaystyle i}n region
{\displaystyle r}
{\displaystyle PE_{i,r}}
{\displaystyle PD_{i,r}}
respectively represent export price and domestic sale price of domestically produced good
{\displaystyle i}n region
{\displaystyle r}
{\displaystyle \alpha _{Ex,i}}
respectively represent the shift parameter and share parameter in transformation function;
{\displaystyle \rho _{Ex,i}}
{\displaystyle \sigma _{Ex,i}}
respectively represent the substitution parameter and substitution elasticity in CET function between export and domestic sales.
Retrieved from "https://www.iamcdocumentation.eu/index.php?title=Trade_-_C3IAM&oldid=14829"
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Battle Hall - Bulbapedia, the community-driven Pokémon encyclopedia
Battle Stage redirects here. For the second round of a Pokémon Contest in the anime, see Contest Battle.
バトルステージ Battle Stage
"Let Each Pokémon Seek No. 1"
The Battle Hall (Japanese: バトルステージ Battle Stage) is a facility located in the northwestern corner of the Battle Frontier in Generation IV. The stadium is a huge catwalk, in where people walk down the catwalk to the battleground, while fans take a number of photos and the spotlights shine all over the place. There is also a red carpet on the floor, all the way from the entrance to the stadium.
1.1 Gaining fans
1.3 Hall Matron
The Battle Hall is unique from any other facility in the Battle Frontier, as there are 10 battles per round instead of the usual 7, and only one Pokémon can be used in a single battle, meaning the battles are one-on-one. Before entering the Battle Hall, the player will be asked to select one Pokémon level 30 or higher for entry. If it is a different Pokémon from last time, the attendant will warn the player that they are using a different Pokémon, as Pokémon are used in winning streaks, and if the player uses a different Pokémon, the streak will be lost. If doing a Double Battle challenge, the player must enter two of the same species of Pokémon.
Once the Pokémon is entered, the player will go down the runway and will have to choose from a list of types which type they want to battle. As the player uses only a single Pokémon against a variety of different opponents, their primary advantage comes from many of those opponents being significantly lower level than the player. If the Pokémon has a particularly disadvantageous type, there is a choice to be made about whether to get that type out of the way early, when the level difference is greatest and the player may be able to use it to overcome the type disadvantage, or to play other types to build up a streak without worrying as much about Pokémon of that type. As the only thing known about an opponent is one of the types of their Pokémon, sometimes that Pokémon's second type may counteract an advantage the player was expecting to have; for example, a Fighting-type Pokémon selecting a Dark-type opponent, only to find that the Ghost/Dark-type Spiritomb is immune to Fighting moves.
As the player wins against each type, that type raises in rank, causing further challenges against that type to increase in level. The player gets their Pokémon fully healed, has the option to suspend or retire the current streak as well as save the most recent battle to the Vs. Recorder, then returns to the grid selection and can challenge the same type again (unless Rank 10 of that type was just beaten) or a different type, until the round of 10 battles has been completed. Each type begins at Rank 1, and increases to a maximum of Rank 10, after which no further challenges are allowed against that type.
Opposing Pokémon's levels are determined by the formula
{\displaystyle L=\min \left(Level_{player},\left\lceil Level_{base}+{types \over 2}+(rank-1)\cdot increment\right\rceil \right)}
{\displaystyle Level_{player}}
the level of the player's Pokémon
{\displaystyle Level_{base}}
is the base level for opponents to start at, which is
{\displaystyle Level_{player}-(3\cdot {\sqrt {Level_{player}}})}
{\displaystyle types}
is the number of types, excluding the currently selected type, which have progressed to Rank 2 or higher
{\displaystyle rank}
is the rank of the currently selected type
{\displaystyle increment}
is the level increase per rank, which is
{\displaystyle {{\sqrt {Level_{player}}} \over 5}}
The effect of the player's level on the opponents' base level (
{\displaystyle Level_{base}}
) and increase per rank (
{\displaystyle increment}
) can be seen in the following chart.
Level Added
Trainers at the Battle Hall use Pokémon from a different roster than that of the other facilities in the Battle Frontier. Pokémon with base stat totals below 340 (Group 1) can appear in Rank 1-5, Pokémon with BST between 340 and 439 (Group 2) appear in Rank 3-8, Pokémon with BST between 440 and 499 (Group 3) appear in Rank 6-10, and Pokémon with BST 500 or higher (Group 4) are restricted to Rank 9-10. The same species of Pokémon cannot appear multiple times as an opponent within the same round of 10 battles unless all the possible Pokémon that can appear in a given type and rank have already been seen earlier in the round, but different stages of a single evolutionary line are allowed to appear in the same round.
See more: List of Battle Hall Pokémon
In the Battle Hall, the player will gain fans as their total record or their winning streak increases. If the player is a female, their major fan is Winston, and if the player is a male, their major fan is Serena. They can be found in the Battle Hall lobby in various locations with varying dialogue. The player will also get visitors cheering them on in the lobby. If the player's total record is over 500 in Pokémon Platinum, they will get the professor's assistant. If the record is over 1,000, the player will get Johanna, and over 10,000 will get Professor Oak or Jasmine. In Pokémon HeartGold and SoulSilver, if the record is over 1,000, the player will get the player character's mother, Ethan, or Lyra, and over 10,000 will get Professor Oak or Whitney.
At the Battle Hall, the staff member next to the monitor will keep track of the player's total record, which is how many successive wins the player has earned with all of their Pokémon. For example, if two different Pokémon have both won 10 times, then the total record is 20. The player earns BP at the end of each round of 10 battles based on how far they are into the current streak, and if that streak pushes the total record across certain milestones, bonus BP will be awarded for that as well.
BP (Single/Double)
BP (Wi-Fi)
BP (Multi)
10 1BP 6BP 12BP
50 20BP 8BP 12BP
70 3BP 10BP 12BP
100 4BP 12BP 12BP
160 10BP 20BP 12BP
180+ 12BP 23BP 12BP
Total record milestones
BP received
50/100/150/200/250/300/350/400/450 5BP
500/600/700/800/900/1000 10BP
1200/1400/1600/1800 30BP
2000, and every further multiple of 500 50BP
Argenta is the Frontier Brain for the Battle Hall. She can be challenged on the 50th battle of a streak, unlike all other Frontier Brains whose first challenge comes on the 21st battle. Argenta uses a Pokémon chosen from the same base stat total grouping that the player's Pokémon belongs to (339 or lower, 340-439, 440-499, 500 or higher), but otherwise chosen at random without regard for the player's selected type. Argenta's Pokémon will always match the level of the player's Pokémon, even for levels in which no other opponent's level is capable of getting that high. Once defeated, she will give away the silver commemorative print. Argenta can be challenged again after 170 consecutive battles, with the same rules; on her second challenge she will only use a Pokémon with a base stat total of 500 or greater, and will give away the gold commemorative print when defeated.
After the player defeats Argenta for the second time and extends a streak to 170 wins, the player is allowed to continue the streak with further rounds. The grid at that point resets to be equivalent to having Rank 9 cleared in every type, and the player must then choose ten different types to challenge at rank 10 during the following round. Once all types have been defeated again, the grid resets again to the same point, until the streak ends.
Outside the Battle Hall Inside the Battle Hall Before battle in the Battle Hall
The Battle Hall in Pokémon Adventures
The Battle Hall first appeared in Deprogramming Porygon-Z as a part of the Battle Frontier. In Dealing with Dragonite, Platinum was shown to have reached Argenta after 169 battles and two days without sleep. For her challenge, Platinum had chosen to use her Froslass, and in the final battle, she came face to face with Argenta's rental Dragonite. During the battle, Platinum mused to herself how, despite knowing everything about her Froslass, whom she had once faced as her opponent when she had been under Candice's ownership, only now, after going through 169 consecutive battles with her, she had come to truly understand her.
Despite Froslass having a major type advantage over Dragonite, she still took heavy damage from the Dragon Pokémon's Steel Wing. However, thanks to the Babiri Berry that Platinum had given Froslass to hold before the battle, the Snow Land Pokémon was able to survive the super effective hit and defeat Dragonite with Ice Shard, earning Platinum her fourth commemorative print.
This listing is of cards mentioning or featuring the Battle Hall in the Pokémon Trading Card Game.
Pokémon in Battle Hall
There are some Pokémon in the Battle Hall that know moves they cannot legitimately learn: a Totodile with Brine, a Roselia with Sludge, and an Anorith with Stone Edge.
Mandarin Chinese 對戰舞台 Duìzhàn Wǔtái
Canada Scène de Combat
Europe Scène de Combat
German Kampfsaal
Italian Palco Lotta
Korean 배틀스테이지 Battle Stage
Spanish Sala Batalla
Vietnamese Sân khấu giao đấu
Retrieved from "https://bulbapedia.bulbagarden.net/w/index.php?title=Battle_Hall&oldid=3499735"
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{\displaystyle {\varepsilon }_{0}=a}
{\displaystyle {\varepsilon }_{S}=b}
{\displaystyle {\varepsilon }_{T+S}=c}
{\displaystyle {\begin{aligned}a+bZ^{S}+cZ^{S+T}.\end{aligned}}}
{\displaystyle {\begin{aligned}g_{0}+g_{S}Z^{S}+g_{T}Z^{T}+g_{S+T}Z^{S+T}=g_{0}+abZ^{S}+bcZ^{T}+acZ^{S+T}.\end{aligned}}}
{\displaystyle g_{0}}
{\displaystyle {\begin{aligned}{1}+abZ^{S}+acZ^{T}+acZ^{S+T}.\end{aligned}}}
{\displaystyle g_{S}=a\ b\ Z^{s}}
{\displaystyle g_{T}=b\ c\ Z^{T}}
{\displaystyle g_{T+S}=a\ c\ Z^{S+T}}
{\displaystyle {\begin{aligned}{\frac {bZ^{S}+cZ^{S+T}}{{1}+abZ^{S}+bcZ^{T}+acZ^{S+T}}}.\end{aligned}}}
{\displaystyle S={2,}T={=5}}
{\displaystyle {\begin{aligned}{\frac {bZ^{2}+cZ^{7}}{{1}+abZ^{2}+bcZ^{5}+acZ^{7}}},\end{aligned}}}
{\displaystyle {\rm {shot}},0,b,0,-\left(ab^{2}\right),{\rm {0,}}a^{2}b^{3},c-b^{2}c,-\left(a^{3}b^{4}\right),2ab\left(-{1}+b^{2}\right)c,a^{4}b^{5},}
{\displaystyle 3a^{2}b^{2}\left(1-b^{2}\right)c{,\ }-\left(a^{5}b^{6}\right)-bc^{2}+b^{3}c^{2}{\ ,\ 4}a^{3}b^{3}\left(-{1+}b^{2}\right)c,}
{\displaystyle a\left(a^{5}b^{7}-c^{2}+{4}b^{2}c^{2}-2b^{2}c^{2}\right){\ ,\ 5}a^{4}b^{4}\left(1-b^{2}\right)c,}
{\displaystyle {\begin{aligned}a^{2}b\left(-\left(a^{5}b^{7}\right)+{3}c^{2}-9b^{2}c^{2}+{6}b^{4}c^{2}\right){\ ,\ }b^{2}\left(-{1+}b^{2}\right)c\left(6a^{5}b^{3}-c^{2}\right),\dots .\end{aligned}}}
{\displaystyle {\text{ – }}ab^{2}}
{\displaystyle a^{2}b^{3}}
{\displaystyle -\left(a^{3}b^{4}\right)}
{\displaystyle 2ab\left(-{1\ +}b^{2}\right)c}
{\displaystyle a\;=0.8;\;b=-0.4;\;c=0.7}
Retrieved from "https://wiki.seg.org/index.php?title=Examples/en&oldid=168892"
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Analysis of Intermediate Temperature Solid Oxide Fuel Cell Transport Processes and Performance | J. Heat Transfer | ASME Digital Collection
Lund Institute of Technology (LTH)
, Box 118, 22100 Lund, Sweden
J. Heat Transfer. Dec 2005, 127(12): 1380-1390 (11 pages)
Yuan, J., and Sundén, B. (March 2, 2005). "Analysis of Intermediate Temperature Solid Oxide Fuel Cell Transport Processes and Performance." ASME. J. Heat Transfer. December 2005; 127(12): 1380–1390. https://doi.org/10.1115/1.2098847
A new trend in recent years is to reduce the solid oxide fuel cell (SOFC) operating temperature to an intermediate range by employing either a thin electrolyte, or new materials for the electrolyte and electrodes. In this paper, a numerical investigation is presented with focus on modeling and analysis of transport processes in planar intermediate temperature (IT, between 600 and
800°C
) SOFCs. Various transport phenomena occurring in an anode duct of an ITSOFC have been analyzed by a fully three-dimensional calculation method. In addition, a general model to evaluate the stack performance has been developed for the purpose of optimal design and/or configuration based on specified electrical power or fuel supply rate.
solid oxide fuel cells, electrolytes, anodes, pipe flow, flow through porous media, convection, diffusion
Anodes, Ducts, Electrochemical reactions, Flow (Dynamics), Fuel cells, Intermediate temperature solid oxide fuel cells, Solid oxide fuel cells, Temperature, Water, Fuels, Transport processes, Heat, Electrolytes
,” FUELCELL2003–1721,
, New York), pp.
Three-Dimensional Computational Analysis of Gas and Heat Transport Phenomena in Ducts Relevant for Anode-Supported Solid Oxide Fuel Cells
Buesell
Fundamental Study on Biomass Fuelled Ceramic Fuel Cells
Brchewitz
Design of an SOFC System Combined to the Gasification of Biomass
Proc. the 4th European SOFC Forum
, ed., Lucerne, Switzerland, Vol.
Simulation of Fully Developed Laminar Heat and Mass Transfer in Fuel Cell Ducts with Different Cross Sections
An Extension of Darcy’s Law to Non-Stokes Flow in Porous Media
Enhancing Heat Transfer in Parallel-Plate Channels by Using Porous Inserts
Analytical Solution of Flow Coefficients for a Uniformly Distributed Porous Channel
Experimental Characterization of Flow Regimes in Various Porous Media-III: Limit of Darcy’s or Creeping Flow Regime for Newtonian and Purely Viscous Non-Newtonian Fluids
On the Limitations of the Brinkman-Forchheimer-Extended Darcy Equation
Bounedien
Analysis of Non-Darcian Effects on Temperature Differentials in Porous Media
Fluid Mechanics of the Interface Region between a Porous Medium and a Fluid Layer—An Exact Solution
Evaluation and Modeling of Performance of Anode-supported Solid Oxide Fuel Cell
Analysis of Chemically Reacting Transport Phenomena in an Anode Duct of Intermediate Temperature SOFCs
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You are given that the graph of the function
y=f\left(x\right)
consists of three line segments
a
b
c
Your task is to construct the graph of the equation
f\left(x\right)=f\left(y\right)
, which also consists of a number of line segments. For each line segment in this graph, enter the coordinates of its endpoints into the two answer boxes below. If your answer is correct, the line segment will be added to the graph below. You will be scored based on the number of attempts it takes you to complete the graph.
Add the following line segment
PQ
f\left(x\right)=f\left(y\right)
P=
Q=
So far you have constructed the following parts of the graph of
f\left(x\right)=f\left(y\right)
Description: given a piecewise linear graph y=f(x), determine the graph of f(x)=f(y). serveur web interactif avec des cours en ligne, des exercices interactifs en sciences et langues pour l'enseigment primaire, secondaire et universitaire, des calculatrices et traceurs en ligne
Keywords: math, interactif, exercice,qcm,cours,classe,biologie,chimie,langue,interactive mathematics, interactive math, server side interactivity,exercise,qcm,class, analysis, functions,graphing
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The radius of a cone calculator
The radius of a cone formula
Is the radius of a cone proportional to its height
Cone dimension amazingness
Are you stuck in a geometric problem and need the assistance of a radius of a cone calculator? You have come to the right place.
Our radius of a cone calculator will help you determine the radius of a cone using various dimensions and formulas of a cone.
You will also get to learn:
How to calculate the radius of a cone?; and
The radius of a cone formula.
The radius of a cone calculator is an efficient and time-saving tool. It calculates the radius of a cone, primarily using the height and slant height of a cone. The other dimensions that use the radius or height are surface area, volume, lateral surface area, and base area. Thus these dimensions can be used to estimate the radius of a cone as well.
The tool is simple to use. All you have to do is:
Input the height of the cone.
Input the slant height of the cone. You have the option to choose different units of measuring heights as well. The default is centimeters (cm).
The result is the radius of the right circular cone along with the other dimensions.
If you have the surface area of the cone and want to determine the radius, you may enter the surface area and the slant height, and the result would be the radius along with all the other dimensions of the cone.
Similarly, there are other dimensions present in the calculator that can be used to figure out the radius of a cone.
Continue reading, and you will understand how to use all of the other formulas.
The radius of a cone formula is simple, but there are multiple ways to calculate the radius of a cone.
Using height and slant height
\large{r = \sqrt{l^2 - h^2}}
r
- Radius;
l
- Slant height; and
h
- Height.
This is the simplest of all the other radius of cone formulas. It is also the primary formula used in our tool.
The next formula is to find radius of the cone using its volume and height. It looks like:
\large{r = \sqrt{\frac{3 \times V}{\pi \times h}}}
V
If you decide to use the volume to determine the radius, you may input the volume and height of the cone in the tool, and the result is the radius in centimeters.
Using lateral area
The radius of a cone plays a role in determining the lateral area, so when you shuffle the formula, you can obtain the radius given the lateral area of the cone.
\large{r = \frac{A_L}{\pi \times l}}
A_L
Using base area
The radius of a cone can also be calculated using the base area. The formula looks something like this:
\large{r = \frac {A_B}{\pi}}
A_B
Using surface area
You might have the total surface area of the cone, so you have the formula in the form of a quadratic equation:
\large{\pi r^2 + \pi r l - A = 0}
A
- Surface area.
So, if you were wondering how to calculate the radius of a cone, now you have the answer and so many options to calculate it, with the best option being to use the radius of the cone calculator, we made it just for you.
No, the height and radius of a cone are not proportional to each other.
When we need to use both the height and radius of a cone to determine any other cone dimension, it is when the radius and height are interrelated. Other than that, the radius and height of a cone do not depend on each other, and you may not be able to predict one based on the other.
For instance, to determine the volume, slant height, lateral area, and surface area, you need the radius and height.
The radius of a cone is one of the many dimensions of a cone, check out some other of our tools to get to know more about each of them.
Lateral area of a cone;
Slant height of a cone; and
How can I calculate the radius of a cone?
The simplest formula to calculate the radius of a cone is:
r = √(l² - h²)
r - Radius;
l - Slant height; and
So, to calculate the radius:
Square the lateral height.
Square the height.
Subtract height squared from lateral height squared.
Find the square root of the result from step three.
The result is the radius of the cone.
What is the radius of a cone with base area of 34 cm²?
The radius is 3.29 cm if a cone has a base area of 34 cm².
The formula to determine the radius of a cone with known base area is:
r = AB / π
AB - Base area;
π - Constant with value 3.14159; and
r - Radius.
So, to determine the radius from the base area:
Divide the base area by pi.
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Transform IIR lowpass filter to IIR bandpass filter - MATLAB iirlp2bp - MathWorks í•œêµ
Design a prototype real IIR lowpass elliptic filter using the ellip function. The filter has a gain of about –3 dB at 0.5Ï€ rad/sample.
Transform the prototype lowpass filter into a bandpass filter by placing the cutoff frequencies of the prototype filter at 0.25Ï€ and 0.75Ï€.
H\left(z\right)=\frac{B\left(z\right)}{A\left(z\right)}=\frac{{b}_{0}+{b}_{1}{z}^{â1}+\cdots +{b}_{n}{z}^{ân}}{{a}_{0}+{a}_{1}{z}^{â1}+\cdots +{a}_{n}{z}^{ân}},
b=\left[\begin{array}{ccccc}{b}_{01}& {b}_{11}& {b}_{21}& ...& {b}_{Q1}\\ {b}_{02}& {b}_{12}& {b}_{22}& ...& {b}_{Q2}\\ ⋮& ⋮& ⋮& â±& ⋮\\ {b}_{0P}& {b}_{1P}& {b}_{2P}& \cdots & {b}_{QP}\end{array}\right]
H\left(z\right)=\underset{k=1}{\overset{P}{â}}{H}_{k}\left(z\right)=\underset{k=1}{\overset{P}{â}}\frac{{b}_{0k}+{b}_{1k}{z}^{â1}+{b}_{2k}{z}^{â2}+\cdots +{b}_{Qk}{z}^{âQ}}{{a}_{0k}+{a}_{1k}{z}^{â1}+{a}_{2k}{z}^{â2}+\cdots +{a}_{Qk}{z}^{âQ}},
H\left(z\right)=\frac{B\left(z\right)}{A\left(z\right)}=\frac{{b}_{0}+{b}_{1}{z}^{â1}+\cdots +{b}_{n}{z}^{ân}}{{a}_{0}+{a}_{1}{z}^{â1}+\cdots +{a}_{n}{z}^{ân}},
a=\left[\begin{array}{ccccc}{a}_{01}& {a}_{11}& {a}_{21}& \cdots & {a}_{Q1}\\ {a}_{02}& {a}_{12}& {a}_{22}& \cdots & {a}_{Q2}\\ ⋮& ⋮& ⋮& â±& ⋮\\ {a}_{0P}& {a}_{1P}& {a}_{2P}& \cdots & {a}_{QP}\end{array}\right]
H\left(z\right)=\underset{k=1}{\overset{P}{â}}{H}_{k}\left(z\right)=\underset{k=1}{\overset{P}{â}}\frac{{b}_{0k}+{b}_{1k}{z}^{â1}+{b}_{2k}{z}^{â2}+\cdots +{b}_{Qk}{z}^{âQ}}{{a}_{0k}+{a}_{1k}{z}^{â1}+{a}_{2k}{z}^{â2}+\cdots +{a}_{Qk}{z}^{âQ}},
IIR lowpass to IIR bandpass transformation effectively places one feature of the original filter, located at frequency −wo, at the required target frequency location wt1, and the second feature, originally at +wo, at the new location wt2. It is assumed that wt2 is greater than wt1. This transformation implements the “DC Mobility,†meaning that the Nyquist feature stays at Nyquist, but the DC feature moves to a location dependent on the selection of wts.
[3] Constantinides, A.G. “Design of Bandpass Digital Filters.†Proceedings of the IEEE 57, no. 6 (1969): 1229–31. https://doi.org/10.1109/PROC.1969.7216.
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Abstract: The nature and the origin of the fine structure are described. Based on the vortex model and hydrodynamics, a comprehensible interpretation of the fine structure constant is developed. The vacuum considered to have superfluid characteristics and elementary particles such as the electron and Hydrogen molecule are irrotational vortices of this superfluid. In such a vortex, the angular rotation ω is maintained, and the larger the radius, the slower the rotational speed. The fine structure value is derived from the ratio of the rotational speed of the boundaries of the vortex to the speed of the vortex eye in its center. Since the angular rotation is constant, the same value was derived from the ratio between the radius of the constant vortex core and the radius of the hall vortex. Therefore, the constancy of alpha is an expression of the constancy relation in the vortex structure.
Keywords: Fine Structure Constant, Angular Rotation, Irrotational Vortex, Vortex Electron Structure, Hydrogen Atom Structure
\alpha ={e}^{\text{2}}/\text{4}\pi {\epsilon }_{0}\hslash c
\alpha =v/c={e}^{2}2/\hslash c
\omega =c{r}_{c}
\Gamma =2\pi {r}_{e}c
h=2\pi rcm
r=h/2\pi {m}_{o}c=3.86\times {10}^{-13}\text{\hspace{0.05em}}\text{m}
F={e}^{2}/4\pi {r}_{e}^{2}{\epsilon }_{0}
F={m}_{o}{v}^{2}/r={e}^{2}/4\pi {r}^{2}{\epsilon }_{0}
v={e}^{2}/4\pi r{m}_{o}v{\epsilon }_{0}
v={e}^{2}/2h{\epsilon }_{0}=2.1876913\times {10}^{6}\text{m}/\text{s}
2.1892212626\times {10}^{6}/3\times {10}^{8}=0.007292304333=1/137.13
{r}_{e}=h/2\pi vm=5.2895948\times {10}^{-11}\text{\hspace{0.05em}}\text{m}
{r}_{e}/{r}_{c}=3.86\times {10}^{-13}/5.2895948\times {10}^{-11}=0.007297345347=1/137.036
{r}_{c}=h/2\pi c{m}_{p}=2.103104894\times {10}^{-16}\text{\hspace{0.05em}}\text{m}
{v}_{e}=Z{e}^{2}/2h{\epsilon }_{0}=2.1876913\times {10}^{6}\text{m}/\text{s}
{r}_{p}=\omega /v=6.3082019259\times {10}^{-8}/2.1876913\times {10}^{6}=2.8834972859\times {10}^{-14}\text{m}
{r}_{c}/{r}_{p}=2.103104894\times {10}^{-16}/2.8834972859\times {10}^{-14}=0.007292304332=1/137.13
Cite this paper: Butto, N. (2020) A New Theory on the Origin and Nature of the Fine Structure Constant. Journal of High Energy Physics, Gravitation and Cosmology, 6, 579-589. doi: 10.4236/jhepgc.2020.64039.
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m
n
Matrix (or 2-dimensional Array), then it is assumed to contain
m
\mathrm{with}\left(\mathrm{SignalProcessing}\right):
\mathrm{audiofile}≔\mathrm{cat}\left(\mathrm{kernelopts}\left(\mathrm{datadir}\right),"/audio/stereo.wav"\right):
\mathrm{Spectrogram}\left(\mathrm{audiofile},\mathrm{compactplot}\right)
\mathrm{Spectrogram}\left(\mathrm{audiofile},\mathrm{channel}=1,\mathrm{includesignal}=[\mathrm{color}="Navy"],\mathrm{includepowerspectrum},\mathrm{colorscheme}=["Orange","SteelBlue","Navy"]\right)
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What will the community average log-score be after the 500th question? | Metaculus
What will the community average log-score be after the 500th question?
Consider some Metaculus question you know little about. This might be whether the star KIC 9832227 will go "red nova", whether the 2048-bit RSA cryptosystem will be broken before 256-bit Elliptic Curve Cryptography, or whether Piracetam is a more effective Alzheimer's treatment than Memantine. Your guess might be that such a difficult question might well be equally likely to resolve one way as the other.
You can expect the community to do much better. In fact, even though the community predictions are not always perfectly calibrated, you should expect it to predict an approximate average of 63% to those that resolve positively, and 37% to those that resolve negatively [1]. That's how much signal the Metaculus community are able to extract from what might seem like noise. Impressive right?
The Log score is a commonly-used scoring rule, which (relative to the Brier score) gives a larger penalty for being confident (i.e. predicting near 1 or near 99%) but being wrong. Currently (as of 06/11/18), 276 questions have been resolved and the community log score is 0.167. The lower the score, the more precise and well-calibrated the predictions are.
It also seems like Many Are Smarter Than the Few: the community log-score is currently lower (i.e. better) than the average log-score of 0.1694 of the current top 25 predictors in the rankings.
What will the community average log-score be after the resolution of the 500th question on Metaculus?
The log-score is computed as follows for a single forecast of probability
p
S=-\frac{1}{4}\log_2 p
if the event occurred, and
S=-\frac{1}{4}\log_2 (1-p)
if not. The scaling is chosen such that it matches the Brier score for a 50% prediction.
[1] This is a back-of-the-envelope estimate. I assume that the success rates are on average 5% removed from predictions made (imperfect calibration). Then, given a log score of 0.167 you would expect
(x-0.025) (-1/4 log_2(x)) + (1-x+0.025)(-1/4 log_2(x))=0.167
x≈0.63.
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Classification - Objectives and metrics | CatBoost
Classification: objectives and metrics
\displaystyle\frac{ - \sum\limits_{i=1}^N w_{i}\left(c_i \log(p_{i}) + (1-c_{i}) \log(1 - p_{i})\right)}{\sum\limits_{i = 1}^{N} w_{i}}
\displaystyle\frac{- \sum\limits_{i=1}^N w_{i} \left(t_{i} \log(p_{i}) + (1 - t_{i}) \log(1 - p_{i})\right)}{\sum\limits_{i = 1}^{N} w_{i}}
\frac{TP}{TP + FP}
\frac{TP}{TP+FN}
(1 + \beta^2) \cdot \frac{Precision * Recall}{(\beta^2 \cdot Precision) + Recall}
\beta
parameter of the F metric.
(0; +\infty)
Default: This parameter is obligatory (the default value is not defined)
2 \frac{Precision * Recall}{Precision + Recall}
\frac{1}{2} \left(\frac{TP}{P} + \frac{TN}{N} \right)
\frac{1}{2} \left( \displaystyle\frac{FP}{TN + FP} + \displaystyle\frac{FN}{FN + TP} \right)
\displaystyle\frac{TP * TN - FP * FN}{\sqrt{(TP + FP)(TP + FN)(TN + FP)(TN + FN)}}
\frac{TP + TN}{\sum\limits_{i=1}^{N} w_{i}}
\displaystyle\frac{\left(\sum\limits_{i = 1}^{N} w_{i} t_{i}/N\right)}{\left(\sum\limits_{i = 1}^{N} w_{i} p_{i} /N\right)}
\displaystyle\frac{\sum I(a_{i}, a_{j}) \cdot w_{i} \cdot w_{j}} {\sum w_{i} \cdot w_{j}}
(i,j)
t_{i} = 0
t_{j} = 1
I(x, y) = \begin{cases} 0 { , } & x < y \\ 0.5 { , } & x=y \\ 1 { , } & x>y \end{cases}
t
w
o_{1}
t \cdot w
o_{2}
(1 – t) \cdot w
\displaystyle\frac{\sum I(a_{i}, a_{j}) \cdot w_{i} \cdot w_{j}} {\sum w_{i} * w_{j}}
(i,j)
t_{i} < t_{j}
I(x, y) = \begin{cases} 0 { , } & x < y \\ 0.5 { , } & x=y \\ 1 { , } & x>y \end{cases}
\displaystyle\frac{ \sum_q \sum_{i, j \in q} \sum I(a_{i}, a_{j}) \cdot w_{i} \cdot w_{j}} { \sum_q \sum_{i, j \in q} \sum w_{i} \cdot w_{j}}
(i,j)
t_{i} = 0
t_{j} = 1
I(x, y) = \begin{cases} 0 { , } & x < y \\ 0.5 { , } & x=y \\ 1 { , } & x>y \end{cases}
t
w
o_{1}
t \cdot w
o_{2}
(1 – t) \cdot w
\displaystyle\frac{ \sum_q \sum_{i, j \in q} \sum I(a_{i}, a_{j}) \cdot w_{i} \cdot w_{j}} { \sum_q \sum_{i, j \in q} \sum w_{i} * w_{j}}
(i,j)
t_{i} < t_{j}
I(x, y) = \begin{cases} 0 { , } & x < y \\ 0.5 { , } & x=y \\ 1 { , } & x>y \end{cases}
See AUC.
2 AUC - 1
\displaystyle\frac{\sum\limits_{i=1}^{N} w_{i}\left(p_{i} - t_{i} \right)^{2}}{\sum\limits_{i=1}^{N} w_{i}}
\displaystyle\frac{\sum\limits_{i=1}^{N} w_{i} max\{1 - t_{i} p_{i}, 0\}}{\sum\limits_{i=1}^{N} w_{i}} , t_{i} = \pm 1
\displaystyle\frac{\sum\limits_{i = 1}^{N} w_{i} [[p_{i} > 0.5] == t_{i}]]}{\sum\limits_{i=1}^{N} w_{i}}
1 - Accuracy
1 - \displaystyle\frac{1 - Accuracy}{1 - RAccuracy}
RAccuracy = \displaystyle\frac{(TN + FP) (TN + FN) + (FN + TP) (FP + TP)}{(\sum\limits_{i=1}^{N} w_{i})^{2}}
See the formula on page 3 of the A note on the linearly weighted kappa coefficient for ordinal scales paper.
The calculation consists of the following steps:
Define the sum of weights (
W
) and the mean target (
\bar{t}
W = \sum\limits_{i} w_{i}
\bar{t} = \frac{1}{W} \sum\limits_{i} t_{i} w_{i}
Denote log-likelihood of a constant prediction:
ll_0 = \sum\limits_{i} w_{i} (\bar{t} \cdot log(\bar{t}) + (1 - \bar{t}) \cdot log(1 - \bar{t}))
Calculate LogLikelihoodOfPrediction (
llp
), which reflects how the likelihood (
ll
) differs from the constant prediction:
llp = \displaystyle\frac{ll(t, w) - ll_0}{\sum\limits_{i} t_{i} w_{i}}
Logloss + +
CrossEntropy + +
Precision - +
Recall - +
F1 - +
BalancedAccuracy - -
BalancedErrorRate - -
MCC - +
Accuracy - +
CtrFactor - -
NormalizedGini - -
BrierScore - -
HingeLoss - -
HammingLoss - -
ZeroOneLoss - +
Kappa - -
WKappa - -
LogLikelihoodOfPrediction - -
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TB67S249FTG Stepper Motor Driver Carrier - Full Breakout Pololu 2973 – MakerSupplies Singapore
TB67S249FTG Stepper Motor Driver Carr...
TB67S249FTG Stepper Motor Driver Carrier - Full Breakout Pololu 2973
This breakout board makes it easy to use Toshiba’s TB67S249FTG microstepping bipolar stepper motor driver, which features adjustable current limiting and seven microstep resolutions (down to 1/32-step). In addition, it dynamically selects an optimal decay mode by monitoring the actual motor current, and it can automatically reduce the driving current below the full amount when the motor is lightly loaded to minimize power and heat. The TB67S249FTG has a wide operating voltage range of 10 V to 47 V and can deliver approximately 1.7 A per phase continuously without a heat sink or forced air flow (up to 4.5 A peak). It features built-in protection against under-voltage, over-current, and over-temperature conditions; our carrier board also adds reverse-voltage protection (up to 40 V).
This version uses a TB67S249FTG driver and can deliver approximately 1.7 A per phase continuously without a heat sink or forced air flow (up to 4.5 A peak). It can be distinguished by the marking “S249FTG” on the driver IC.
Another way to set the current limit is to measure the VREF voltage and calculate the resulting current limit. The VREF voltage is accessible on the VREFA and VREFB pins, which are tied together by default. The current limit relates to VREF as follows:
\text{current limit}=\text{VREF}×1.25 \frac{\text{A}}{\text{V}}
\text{current limit}=\text{VREF}×0.556 \frac{\text{A}}{\text{V}}
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Physics Letters. A (800)
Chinese Physics. B (438)
On the admissibility of some substitution sequences
Han, Zhujuan; Niu, Min, E-mail: niuminfly@sohu.com
[en] The relationship between some kinds of substitutions and admissible sequences is studied. Sufficient and necessary conditions for the admissibility of the sequences generated by non-constant length substitution and constant length substitution are investigated respectively
S0960-0779(13)00057-X; Available from http://dx.doi.org/10.1016/j.chaos.2013.03.011; Copyright (c) 2013 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
CHAOS THEORY, FRACTALS, LENGTH
DIMENSIONS, MATHEMATICS
Luo Chuanwen; Yi Chundi; Wang Gang; Li Longsuo; Wang Chuncheng, E-mail: lcw1234562000@yahoo.com.cn, E-mail: wangchuncheng@hit.edu.cn
[en] Uniform index is a conception that can describe the uniformity of a finite point set in a polyhedron, and is closely related to chaos. In order to study uniform index, the concept of contained uniform index is defined, which is similar to uniform index and has good mathematical properties. In this paper, we prove the convergence of the contained uniform index, and develop the base of proving the convergence of uniform index.
Chaos, Solitons and Fractals; ISSN 0960-0779; ; v. 42(5); p. 2748-2753
CHAOS THEORY, CONVERGENCE, INDEXES
DOCUMENT TYPES, MATHEMATICS
De-synchronization and chaos in two inductively coupled Van der Pol auto-generators
Beregov, R.Y.; Melkikh, A.V., E-mail: melkikh2008@rambler.ru
[en] In this article, we consider a system of autonomous inductively coupled Van der Pol generators. For two coupled generators, we establish the presence of metastable chaos, a strange non-chaotic attractor, and several stable limiting cycles. Areas of parametric dependence of different modes of synchronization are obtained
Chaos, Solitons and Fractals; ISSN 0960-0779; ; v. 73(Complete); p. 17-28
ATTRACTORS, CHAOS THEORY, SYNCHRONIZATION
Shifts, rotations and distributional chaos
Xu, Dongsheng; Xiang, Kaili; Liang, Shudi, E-mail: xudongshengmath@126.com, E-mail: xiangkl@swufe.edu.cn, E-mail: 1361417426@qq.com
[en] Let
{R}_{{r}_{0}},{R}_{{r}_{1}}:{\mathbb{S}}^{1}⟶{\mathbb{S}}^{1}
be rotations on the unit circle
{\mathbb{S}}^{1}
f:{\Sigma }_{2}×{\mathbb{S}}^{1}⟶{\Sigma }_{2}×{\mathbb{S}}^{1}
f\left(x,t\right)=\left(\sigma \left(x\right),{R}_{{r}_{{x}_{1}}}\left(t\right)\right),
x={x}_{1}{x}_{2}\cdots \in {\Sigma }_{2}:=\left\{0,1{\right\}}^{\mathbb{N}}
t\in {\mathbb{S}}^{1}
\sigma :{\Sigma }_{2}⟶{\Sigma }_{2}
is the shift, and
{r}_{0}
{r}_{1}
are rotational angles. It is first proved that the system
\left({\Sigma }_{2}×{\mathbb{S}}^{1},f\right)
exhibits maximal distributional chaos for any
{r}_{0},{r}_{1}\in \mathbb{R}
(no assumption of
{r}_{0},{r}_{1}\in \mathbb{R}\setminus \mathbb{Q}
), generalizing Theorem 1 in Wu and Chen (Topol. Appl. 162:91–99, 2014). It is also obtained that
\left({\Sigma }_{2}×{\mathbb{S}}^{1},f\right)
is cofinitely sensitive and
\left({\stackrel{^}{\mathcal{M}}}^{1},{\stackrel{^}{\mathcal{M}}}^{1}\right)
-sensitive and that
\left({\Sigma }_{2}×{\mathbb{S}}^{1},f\right)
is densely chaotic if and only if
{r}_{1}-{r}_{0}\in \mathbb{R}\setminus \mathbb{Q}
Copyright (c) 2019 The Author(s); Country of input: International Atomic Energy Agency (IAEA)
Advances in Difference Equations (Online); ISSN 1687-1847; ; v. 2019(1); p. 1-10
CHAOS THEORY, ROTATION, SENSITIVITY
MATHEMATICS, MOTION
The emergence of self-organization in complex systems–Preface
Paradisi, Paolo; Kaniadakis, Giorgio; Scarfone, Antonio Maria, E-mail: paolo.paradisi@isti.cnr.it
Chaos, Solitons and Fractals; ISSN 0960-0779; ; v. 81(Part B); p. 407-411
CHAOS THEORY, FRACTALS, ORGANIZING
The equal combination synchronization of a class of chaotic systems with discontinuous output
Luo, Runzi; Zeng, Yanhui
[en] This paper investigates the equal combination synchronization of a class of chaotic systems. The chaotic systems are assumed that only the output state variable is available and the output may be discontinuous state variable. By constructing proper observers, some novel criteria for the equal combination synchronization are proposed. The Lorenz chaotic system is taken as an example to demonstrate the efficiency of the proposed approach
Chaos (Woodbury, N. Y.); ISSN 1054-1500; ; CODEN CHAOEH; v. 25(11); p. 113102-113102.8
CHAOS THEORY, EFFICIENCY, SYNCHRONIZATION
A vast amount of various invariant tori in the Nosé-Hoover oscillator
Wang, Lei; Yang, Xiao-Song, E-mail: yangxs@hust.edu.cn
[en] This letter restudies the Nosé-Hoover oscillator. Some new averagely conservative regions are found, each of which is filled with different sequences of nested tori with various knot types. Especially, the dynamical behaviors near the border of “chaotic region” and conservative regions are studied showing that there exist more complicated and thinner invariant tori around the boundaries of conservative regions bounded by tori. Our results suggest an infinite number of island chains in a “chaotic sea” for the Nosé-Hoover oscillator
CHAOS THEORY, OSCILLATORS, TORI
ELECTRONIC EQUIPMENT, EQUIPMENT, MATHEMATICS
Anti-synchronization Between Lorenz and Liu Hyperchaotic Systems
Zheng Qiang; Zhang Xiaoping; Ren Zhongzhou, E-mail: qzhengnju@gmail.com
[en] Anti-synchronization between different hyperchaotic systems is presented using Lorenz and Liu systems. When the parameters of two systems are known, one can use active synchronization. When the parameters are unknown or uncertain, the adaptive synchronization is applied. The simulation results verify the effectiveness of the proposed two schemes for anti-synchronization between different hyperchaotic systems
CHAOS THEORY, SIMULATION, SYNCHRONIZATION
Chaotic sub-dynamics in coupled logistic maps
Lampart, Marek; Oprocha, Piotr, E-mail: marek.lampart@vsb.cz, E-mail: oprocha@agh.edu.pl
[en] We study the dynamics of Laplacian-type coupling induced by logistic family
{f}_{\mu }\left(x\right)=\mu x\left(1-x\right)
\mu \in \left[0,4\right]
, on a periodic lattice, that is the dynamics of maps of the form
F\left(x,y\right)=\left(\left(1-\epsilon \right){f}_{\mu }\left(x\right)+\epsilon {f}_{\mu }\left(y\right),\left(1-\epsilon \right){f}_{\mu }\left(y\right)+\epsilon {f}_{\mu }\left(x\right)\right)
\epsilon >0
determines strength of coupling. Our main objective is to analyze the structure of attractors in such systems and especially detect invariant regions with nontrivial dynamics outside the diagonal. In analytical way, we detect some regions of parameters for which a horseshoe is present; and using simulations global attractors and invariant sets are depicted.
S0167278916303116; Available from http://dx.doi.org/10.1016/j.physd.2016.06.010; Copyright (c) 2016 Elsevier B.V. All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
Physica D; ISSN 0167-2789; ; CODEN PDNPDT; v. 335; p. 45-53
ATTRACTORS, CHAOS THEORY, COUPLING
Conjugate Lorenz-type chaotic attractors
Xiong Xiaohua; Wang Junwei, E-mail: xhxiong8899@yahoo.com.cn
[en] Based on the generalized Lorenz system, a conjugate Lorenz-type system is introduced, which contains three different chaotic attractors, i.e., the conjugate Lorenz attractor, the conjugate Chen attractor and the conjugate Lue attractor. These new attractors are conjugate, respectively, to the Lorenz attractor, the Chen attractor and the Lue attractor in an algebraic sense. The conjugate attractors may be helpful for finally revealing the geometric structure of the Lorenz attractor.
ALGEBRA, ATTRACTORS, CHAOS THEORY
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What is ellipse circumference?
What is the formula for the circumference of the ellipse?
How do I use the ellipse circumference calculator?
Other ellipse calculators
The ellipse circumference calculator helps you find the total perimeter around the ellipse, which is also referred to as the circumference of the ellipse. The ellipse circumference depends on the lengths of the semi-major and semi-minor axes. Read on to know more about how to find the circumference of an ellipse, and the formula to calculate it.
The circumference of the ellipse is the total length of the boundary of the elliptical shape. In other words, we also refer to the perimeter of the ellipse as its circumference.
To calculate the ellipse circumference, we need to know the semi-major and semi-minor axes' lengths. Once we get these, we'd find the ellipse circumference using the following formula:
\small p \approx \pi (a+b) \left ( 1 + \frac {3h}{10 + \sqrt{4-3h}} \right )
a
- Semi-major axis of the ellipse; and
b
- Semi-minor axis of the ellipse.
h
in the above equation, we will use the following formula:
\small h = \frac{(a-b)^2}{(a+b)^2}
Thus, using the above equations for ellipse circumference, we can find its approximate value.
To find the ellipse circumference using our calculator, you need to do the following:
Enter the value of the semi-major axis (a).
Enter the value of the semi-minor axis (b).
Voila! The tool will perform all the heavy computing to calculate the ellipse circumference and will display it as the result!
If you found this tool useful, you may also want to check out our vast range of calculators related to ellipse:
Ellipse calculator;
Ellipse area calculator;
Center of ellipse calculator;
Ellipse perimeter calculator; and
Foci of an ellipse calculator.
How do I find the circumference of an ellipse?
To find the circumference of an ellipse, we need to use the ellipse circumference equation, which is governed by the following steps:
Find the values of the semi-major axis (a) and the semi-minor axis (b).
Calculate the value of the variable h using the formula h = (a - b)²/(a + b)².
Plug in the values of a, b and h in the formula for the circumference, given by this equation:
Circumference = π × (a + b)[1 + (3 × h/(10 + √(4 - 3h)))].
Tada! You now have the approximate value of the circumference of the ellipse!
Does ellipse have circumference?
Yes! We sometimes refer to the boundary of the ellipse as the circumference of the ellipse! Though the term circumference is usually associated with a circle, we also use it to refer to the ellipse's perimeter.
What's the Heron's area formula? What's the Heron's formula proof? Find out with this Heron's formula calculator!
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A small toy company makes only cars and trucks. The profit on cars is
\$2
each and the profit made on trucks is
\$3
each. To stay in business the company must make at least
\$500
profit each week. Write an inequality that represents possible combinations of toys that the company can make and remain in business. Be sure to define your variables.
This is similar to problem 10-73 from Lesson 10.2.4.
Use the same method to set up your inequality.
c=
t=
2c+3t\ge500
Click the link at right for the full version of the eTool: CCA 10-116 HW eTool
|
The Effect of Iso-Octane Addition on Combustion and Emission Characteristics of a HCCI Engine Fueled With n-Heptane | J. Eng. Gas Turbines Power | ASME Digital Collection
The Effect of Iso-Octane Addition on Combustion and Emission Characteristics of a HCCI Engine Fueled With n-Heptane
Institute for Chemical Process and Environmental Technology,
e-mail: cosmin.dumitrescu@nrc-cnrc.gc.ca
Wallace L. Chippior,
Trevor Connolly,
, Ottawa, ON, K1A 0H3, Canada
Lisa Graham,
Dumitrescu, C. E., Guo, H., Hosseini, V., Neill, W. S., Chippior, W. L., Connolly, T., Graham, L., and Li, H. (May 13, 2011). "The Effect of Iso-Octane Addition on Combustion and Emission Characteristics of a HCCI Engine Fueled With n-Heptane." ASME. J. Eng. Gas Turbines Power. November 2011; 133(11): 112801. https://doi.org/10.1115/1.4003640
This paper investigates the effects of iso-octane addition on the combustion and emission characteristics of a single-cylinder, variable compression ratio, homogeneous charge compression ignition (HCCI) engine fueled with n-heptane. The engine was operated with four fuel blends containing up to 50% iso-octane by liquid volume at 900 rpm, 50:1 air-to-fuel ratio, no exhaust gas recirculation, and an intake mixture temperature of
30°C
. A detailed analysis of the regulated and unregulated emissions was performed including validation of the experimental results using a multizone model with detailed fuel chemistry. The results show that iso-octane addition reduced HCCI combustion efficiency and retarded the combustion phasing. The range of engine compression ratios where satisfactory HCCI combustion occurred was found to narrow with increasing iso-octane percentage in the fuel.
NOx
emissions increased with iso-octane addition at advanced combustion phasing, but the influence of iso-octane addition was negligible once CA50 (crank angle position at which 50% heat is released) was close to or after top dead center. The total unburned hydrocarbons (THC) in the exhaust consisted primarily of alkanes, alkenes, and oxygenated hydrocarbons. The percentage of alkanes, the dominant class of THC emissions, was found to be relatively constant. The alkanes were composed primarily of unburned fuel compounds, and iso-octane addition monotonically increased and decreased the iso-octane and n-heptane percentages in the THC emissions, respectively. The percentage of alkenes in the THC was not significantly affected by iso-octane addition. Iso-octane addition increased the percentage of oxygenated hydrocarbons. Small quantities of cycloalkanes and aromatics were detected when the iso-octane percentage was increased beyond 30%.
air pollution, combustion, fuel, ignition, internal combustion engines, HCCI, n-heptane/iso-octane blends, regulated emissions, unregulated emissions
Combustion, Emissions, Engines, Fuels, Heptane, Homogeneous charge compression ignition engines, Exhaust systems, Computer simulation, Compression
Brassica Carinata as an Alternative Oil Crop for the Production of Biodiesel in Italy: Engine Performance and Regulated and Unregulated Exhaust Emissions
Vressner
Quantification of the Formaldehyde Emissions From Different HCCI Engines Running on a Range of Fuels
A Parametric Study of HCCI Combustion—The Sources of Emissions at Low Loads and the Effects of GDI Fuel Injection
Modelling Iso-Octane HCCI Using CFD With Multi-Zone Detailed Chemistry; Comparison to Detailed Speciation Data Over a Range of Lean Equivalence Ratios
Methodology Development of a Time-Resolved In-Cylinder Fuel Oxidation Analysis: Homogeneous Charge Compression Ignition Combustion Study Application
A Study on the Performance of Combustion in a HCCI Engine Using N-Heptane by a Multi-Zone Model
Universally Applicable Equation for the Instantaneous Heat Transfer Coefficient in the Internal Combustion Engine
Lawrence Livermore National Laboratory’ Primary Reference Fuels (PRF): Iso-Octane/N-Heptane Mixtures Detailed Chemical Kinetic Mechanism, Available Online at https://www-pls.llnl.gov/?url=science_and_technology-chemistry-combustion-prfhttps://www-pls.llnl.gov/?url=science_and_technology-chemistry-combustion-prf.
, 2010, Available Online at http://www.me.berkeley.edu/gri_mech/http://www.me.berkeley.edu/gri_mech/.
Kinetic Modeling of a Rich, Atmospheric Pressure, Premixed n-Heptane/O2/N2 Flame
Combustion Reactions of Paraffin Components in Liquid Transportation Fuels Using Generic Rates
Pepiot-Desjardins
Chemical Mechanism for High Temperature Combustion of Engine Relevant Fuels With Emphasis on Soot Precursors
Mounaïm-Rousselle
Modelling of Aromatics and Soot Formation From Large Fuel Molecules
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Introduction to Grammar and types of Grammar | DigitalBitHub
Introduction to Grammar and types of Grammar
Grammar is a finite set of rules for generating syntactically correct sentences or meaningful correct sentences. These rules are called Productions.
Each production rule is composed of two elements one is Variables and the other is Terminals.
Variables are the symbols that are used to generate the sentences, but not the part of the sentences. Variables are represented by capital letters. Variables are also known as Non-Terminals.
Terminals are the symbols that are the components of the sentences generated by the given grammar. Terminals are represented by small letters.
Productions are the rules of Grammar that specify the substitution of variables. Productions are in the form of
\alpha \to \beta
\alpha
is a Non-Terminal and
\beta
is a Terminal
Generally, Grammar G is represented by four tuples. G = (V, T, P, S)
Consider a Grammar G = (V, T, P, S)
V = {S, A, B} #set of variables
T = {a, b} #set of terminals
S\to AB
A\to a
B\to b
} #set of production rules
S = {S} #Start Symbol
According to Chomsky, there are four types of Grammar.
Type-0 Grammar
All the productions are in the form of
\alpha \to \beta
\alpha \in {\left(V+T\right)}^{+}
\beta \in {\left(V+T\right)}^{*}
Epsilon is not allowed in
\alpha
\alpha \to \beta
\alpha \in {\left(V+T\right)}^{+}
\beta \in {\left(V+T\right)}^{+}
\left|\alpha \left|\le \right|\beta \right|
Epsilon is allowed neither in
\alpha
\beta
and length of
\alpha
\beta
\alpha \to \beta
\alpha \in V
\beta \in {\left(V+T\right)}^{*}
\left|\alpha \right|=1
\alpha
\alpha \to \beta
\alpha \in V
\beta \in V{T}^{*} or {T}^{*} \left(Left Linear Grammar\right)
\in T* or {T}^{*}V \left( Right Linear Grammar\right)
\left|\alpha \right|=1
\alpha
must be 1. A grammar can only be Left Linear Grammar or Right Linear Grammar. Any type-3 grammar cannot be both LLG and RLG.
Note: Chomsky Hierarchy only works for grammar with no null production.
Chomsky HierarchyTypes of Grammar
Programming & Data StructureHashing - A Searching TechniqueFebruary 13, 2022
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FAQ | CatBoost
Why is the metric value on the validation dataset sometimes better than the one on the training dataset?
Why can metric values on the training dataset that are output during training, be different from ones output when using model predictions?
How should weights or baseline be specified for the validation dataset?
Why is it forbidden to use float values and nan values for categorical features?
None categorical feature
How to use GridSearchCV and RandomSearchCV from sklearn with categorical features?
How to understand which categorical feature combinations have been selected during the training?
How to overcome the Out of memoryerror when training on GPU?
How to reduce the size of the final model?
How to get the model with best parameters from the python cv function?
What are the differences between training on CPU and GPU?
Does CatBoost require preprocessing of missing values?
This happens because auto-generated numerical features that are based on categorical features are calculated differently for the training and validation datasets:
Training dataset: the feature is calculated differently for every object in the dataset. For each i-th object the feature is calculated based on data from the first i-1 objects (the first i-1 objects in some random permutation).
Validation dataset: the feature is calculated equally for every object in the dataset. For each object the feature is calculated using data from all objects of the training dataset.
When the feature is calculated on data from all objects of the training dataset it uses more information than the feature, that is calculated only on a part of the dataset. For this reason this feature is more powerful. A more powerful feature results in a better loss value.
Thus, the loss value on the validation dataset might be better than the loss value for the training dataset, because the validation dataset has more powerful features.
Details of the algorithm and the rationale behind this solution
Liudmila Prokhorenkova, Gleb Gusev, Aleksandr Vorobev, Anna Veronika Dorogush, Andrey Gulin. NeurIPS, 2018
NeurIPS 2018 paper with explanation of Ordered boosting principles and ordered categorical features statistics.
Anna Veronika Dorogush, Vasily Ershov, Andrey Gulin. Workshop on ML Systems at NIPS 2017
A paper explaining the CatBoost working principles: how it handles categorical features, how it fights overfitting, how GPU training and fast formula applier are implemented.
This happens because auto-generated numerical features that are based on categorical features are calculated differently when training and applying the model.
During the training the feature is calculated differently for every object in the training dataset. For each i-th object the feature is calculated based on data from the first i-1 objects (the first i-1 objects in some random permutation). During the prediction the same feature is calculated using data from all objects from the training dataset.
Thus, the loss value calculated during for the prediction might be better than the one that is printed out during the training even though the same dataset is used.
Use the Pool class.
An example of specifying weights:
data=[[1, 4, 5, 6],
weight=[0.1, 0.2, 0.3]
eval_data = Pool(
model.fit(X=train_data, eval_set=eval_data)
The algorithm should work identically regardless of the input data format (file or matrix). If the dataset is read from a file all values of categorical features are treated as strings. To treat it the same way when training from matrix, a unique string representation of each feature value is required. There is no unique string representation for floating point values and for nan values.
If floating point categorical features are allowed the following problem arises.
The feature f is categorical and takes values 1 and 2 .
A matrix is used for the training. The column that corresponds to the feature f contains values 1.0 and 2.0 .
Each categorical feature value is converted to a string during the training to calculate the corresponding hash value. 1.0 is converted to the string 1.0 , and 2.0 is converted to the string 2.0 .
After the training the prediction is performed on file.
The column with the feature f contains values 1 and 2 .
During the prediction, the hash value of the string 1 is calculated. This value is not equal to the hash value of the string 1.0 .
Thus, the model doesn't collate this value with the one in the training dataset, therefore the prediction is incorrect.
The feature f is categorical and takes the value None for some object Obj.
A matrix is used for the training. The column that contains the value of the feature f for the object Obj contains the value None .
Each categorical feature value is converted to a string during the training to calculate the corresponding hash value. The None value is converted to the string None .
After the training the prediction is performed on file. The column with the feature _f _ contains the value N/A , which would be parsed as None if it was read to a pandas.DataFrame before the training.
The hash value of the string N/A is calculated during the prediction. This value is not equal to the hash value of the string None .
Since it is not possible to guarantee that the string representation of floating point and None values are the same when reading data from a file or converting the value to a string in Python or any other language, it is required to use strings instead of floating point and None values.
Use the cat_featuresparameter when constructing the model (CatBoost, CatBoostRegressor or CatBoostClassifier).
model = catboost.CatBoostRegressor(cat_features=[0,1,2]) grid_search =
sklearn.model_selection.GridSearchCV(model,
param_grid)
Use the InternalFeatureImportance to familiarize with the resulting combinations. Generate this file from the command-line by setting the --fstr-type parameter to InternalFeatureImportance.
The format of the resulting file is described here.
The default feature importances are calculated in accordance with the following principles:
Importances of all numerical features are calculated. Some of the numerical features are auto-generated based on categorical features and feature combinations.
These importances are shared between initial features. If a numerical feature is auto-generated based on a feature combination, then the importance value is shared equally between the combination participants.
The file that is generate in the InternalFeatureImportance mode contains the description of initial numerical features and their importances.
How to overcome the Out of memory error when training on GPU?
Set the --boosting-type for the Command-line version parameter to Plain. It is set to Ordered by default for datasets with less then 50 thousand objects. TheOrdered scheme requires a lot of memory.
Set the --max-ctr-complexity for the Command-line version parameter to either 1 or 2 if the dataset has categorical features.
Decrease the value of the --gpu-ram-part for the Command-line version parameter.
Set the --gpu-cat-features-storage for the Command-line version parameter to CpuPinnedMemory.
Check that the dataset fits in GPU memory. The quantized version of the dataset is loaded into GPU memory. This version is much smaller than the initial dataset. But it can exceed the available memory if the dataset is large enough.
Decrease the depth value, if it is greater than 10. Each tree contains
2^{n}
leaves if the depth is set to
n
, because CatBoost builds full symmetric trees by default. The recommended depth is 6, which works well in most cases. In rare cases it's useful to increase the depth value up to 10.
If the dataset contains categorical features with many different values, the size of the resulting model may be huge. Try the following approaches to reduce the size of the resulting model:
Decrease the --max-ctr-complexity for the Command-line version to either 1 or 2
For training on CPU:
Increase the value of the --model-size-reg for the Command-line version parameter.
Set the value of the --ctr-leaf-count-limit for the Command-line version parameter. The number of different category values is not limited be default.
Decrease the value of the --iterations for the Command-line version parameter and increase the value of the --learning-rate for the Command-line version parameter.
Remove categorical features that have a small feature importance from the training dataset.
It is not possible. The CatBoost cv function is intended for cross-validation only, it can not be used for tuning parameter.
The dataset is split into N folds. N–1 folds are used for training and one fold is used for model performance estimation. At each iteration, the model is evaluated on all N folds independently. The average score with standard deviation is computed for each iteration.
The only parameter that can be selected based on cross-validation is the number of iterations. Select the best iteration based on the information of the cv results and train the final model with this number of iterations.
The default value of the --border-count for the Command-line version parameter depends on the processing unit type and other parameters:
Training on CPU has the model_size_reg set by default. It decreases the size of models that have categorical features. This option is turned off for training on GPU.
The following parameters are not supported if training is performed on GPU: --model-size-reg for the Command-line version, --ctr-leaf-count-limit for the Command-line version, --monotone-constraints for the Command-line version.
The default value of the --leaf-estimation-method for the Quantile and MAE loss functions is Exact on CPU and GPU.
Combinations of categorical features are not supported for the following modes if training is performed on GPU: MultiClass and MultiClassOneVsAll. The default value of the --max-ctr-complexity for the Command-line version parameter for such cases is set to 1.
The default values for the following parameters depend on the processing unit type:
--bootstrap-type for the Command-line version:
When the objective parameter is QueryCrossEntropy, YetiRankPairwise, PairLogitPairwise and the bagging_temperature parameter is not set: Bernoulli with the subsample parameter set to 0.5
Not MultiClass and MultiClassOneVsAll, task_type = CPU and sampling_unit = Object: MVS with the subsample parameter set to 0.8.
Otherwise: Bayesian.
--boosting-type for the Command-line version:
Any number of objects, MultiClass or MultiClassOneVsAll mode: Plain
More than 50 thousand objects, any mode: Plain
Less than or equal to 50 thousand objects, any mode but MultiClass or MultiClassOneVsAll: Ordered
--model-size-reg for the Command-line version:
Feature combinations are regularized more aggressively on GPU.
The cost of a combination is equal to the number of different feature values in this combinations that are present in the training dataset.
The cost of a combination is equal to number of all possible different values of this combination. For example, if the combination contains two categorical features (c1 and c2), the cost is calculated as
number\_of\_categories\_in\_c1 \cdot number\_of\_categories\_in\_c2
, even though many of the values from this combination might not be present in the dataset.
Refer to the Model size regularization coefficient section for details on the calculation principles.
CatBoost can handle missing values internally. None values should be used for missing value representation.
If the dataset is read from a file, missing values can be represented as strings like N/A, NAN, None, empty string and the like.
Refer to the Missing values processing section for details.
|
Rydberg constant - Simple English Wikipedia, the free encyclopedia
In spectroscopy, the Rydberg constant is a physical constant relating to the electromagnetic spectra of an atom. Its symbol is
{\displaystyle R_{\infty }}
for heavy atoms or
{\displaystyle R_{\text{H}}}
for hydrogen. The constant is named after the Swedish physicist Johannes Rydberg. The constant first arose as an empirical fitting parameter in the Rydberg formula for the hydrogen spectral series. Niels Bohr later showed that its value could be calculated from more fundamental constants via his Bohr model. As of 2018[update],
{\displaystyle R_{\infty }}
and electron spin g-factor are the most accurately measured physical constants.[1]
The constant is expressed for either hydrogen as
{\displaystyle R_{\text{H}}}
, or at the limit of infinite nuclear mass as
{\displaystyle R_{\infty }}
. In either case, the constant is used to express the limiting value of the highest wavenumber (inverse wavelength) of any photon that can be emitted from an atom, or, alternatively, the wavenumber of the lowest-energy photon capable of ionizing an atom from its ground state. The hydrogen spectral series can be expressed simply in terms of the Rydberg constant for hydrogen
{\displaystyle R_{\text{H}}}
and the Rydberg formula.
In atomic physics, Rydberg unit of energy, symbol Ry, corresponds to the energy of the photon whose wavenumber is the Rydberg constant, i.e. the ionization energy of the hydrogen atom.[source?]
↑ Pohl, Randolf; Antognini, Aldo; Nez, François; Amaro, Fernando D.; Biraben, François; Cardoso, João M. R.; Covita, Daniel S.; Dax, Andreas; Dhawan, Satish; Fernandes, Luis M. P.; Giesen, Adolf; Graf, Thomas; Hänsch, Theodor W.; Indelicato, Paul; Julien, Lucile; Kao, Cheng-Yang; Knowles, Paul; Le Bigot, Eric-Olivier; Liu, Yi-Wei; Lopes, José A. M.; Ludhova, Livia; Monteiro, Cristina M. B.; Mulhauser, Françoise; Nebel, Tobias; Rabinowitz, Paul; Dos Santos, Joaquim M. F.; Schaller, Lukas A.; Schuhmann, Karsten; Schwob, Catherine; Taqqu, David (2010). "The size of the proton". Nature. 466 (7303): 213–216. Bibcode:2010Natur.466..213P. doi:10.1038/nature09250. PMID 20613837. S2CID 4424731.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Rydberg_constant&oldid=8076681"
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Reviewed by Davide Borchia
Angles of a trapezoid
Types of a trapezoid
How to calculate angles of a trapezoid?
Trapezoid calculators at Omni
Are you looking for a trapezoid angle calculator? Because you have come to the right place. Our trapezoid angle calculator is a tool specially designed for all you geometry lovers. It determines the angles of a trapezoid: whether it is an isosceles or right-angle trapezoid, we have got you covered. Worry not, as we will explain how to calculate the angles of a trapezoid.
There are four angles in a trapezoid:
Alpha α;
Beta β;
Gamma γ; and
Delta δ.
Like all other quadrangles, the sum of angles in a trapezoid is 360 degrees (or 2π radians).
Since they have a pair of parallel sides, the trapezoid has an additional condition. The pair of angles along one of the legs are supplementary angles, which means their sum must be equal to 180 degrees (or π radians).
α + β = 𝛾+ δ = 180°
Knowing the angles of a trapezoid comes in handy to identify its height, and the height helps identify the area of trapezoid.
Our trapezoid angle calculator is a convenient tool that lets you calculate the different angles between the sides of the trapezoid.
You may input the value of any angle and obtain the value of its supplementary partner. Yes, it is really that simple.
if you input,
α = 55°
then the tool determines,
β = 125°
γ = 95°
δ = 85°
In the tool, you can also select a different unit for angle conversion.
Now would be a good time to discuss the two most essential trapezoid types in terms of the angles.
Angle of isosceles trapezoid
A trapezoid in which the legs and both of the base angles are of equal measure is an isosceles trapezoid. The angles of isosceles trapezoids are independent of the shape and are calculated the same way as a regular trapezoid.
Angle of right trapezoid
A trapezoid whose one leg is perpendicular to the bases is a right trapezoid. It has at least one right angle.
🙋 Interestingly, if one of the legs is perpendicular to one of the bases, it is perpendicular to the other since the legs are parallel. So, chances are your right trapezoid has two right angles.
By now, we understand that the angles are supplementary and can calculate them in pairs.
\angle α +\angle β = \pi
\angle γ + \angle δ = \pi
Let's consider an example, you have ∠α = 75°, then to determine ∠β, subtract 75 from 180, and you have 105°.
The formula to calculate all four angles together is:
\angle α + \angle β + \angle γ + \angle δ = 2\pi
For instance, α = 75°, β =85°, and γ = 95°. To determine δ, follow the steps:
Sum the values of α, β, and 𝛾. You will obtain 255°
Next, subtract the summed value from 360 (2π). You will get 105°.
This is the value of angle δ.
What is the 4ᵗʰ angle of a right trapezoid, if first angle is 85°?
The value of the fourth angle is 95°.
A right trapezoid means a pair of its angles is 90°. This makes it easier to determine the angles. If two of the angles are 90° and 90°, and you know the third angle, you may subtract the value of the third angle from 180°.
How do I calculate the angles of a trapezoid?
To calculate the supplementary angles in pairs, the formula is:
∠α + ∠β = π
∠γ + ∠δ = π
π = 180°.
So, if you have α = 100°, then β is obtained by subtracting 100° from 180°, which gives β = 80°.
The same procedure is valid for the pair of angles γ and δ.
If α = 75°, how much is β?
If α = 75°, then β =105°.
Although a trapezoid is a quadrangle shape and the sum of its angles is 360°. There is a condition for trapezoid that the pair of angles along one of the legs are supplementary angles, which means their sum must be equal to 180 degrees.
So, you can determine one angle by knowing its adjacent partner.
|
Effects of Molybdenum Content and Heat Treatment on Mechanical and Tribological Properties of a Low-Carbon Stellite® Alloy | J. Eng. Mater. Technol. | ASME Digital Collection
Effects of Molybdenum Content and Heat Treatment on Mechanical and Tribological Properties of a Low-Carbon
Stellite®
, 1125 Colonel By Drive, Ottawa, ON, K1S 5B6, Canada
e-mail: rliu@mae.carleton.ca
, National Research Council Canada, Ottawa, ON, K1A 0R6, Canada
, Belleville, ON, K8N 5C4, Canada
Huang, P., Liu, R., Wu, X., and Yao, M. X. (December 13, 2006). "Effects of Molybdenum Content and Heat Treatment on Mechanical and Tribological Properties of a Low-Carbon
Stellite®
Alloy." ASME. J. Eng. Mater. Technol. October 2007; 129(4): 523–529. https://doi.org/10.1115/1.2744429
The chemical composition of
Stellite®
21 alloy was modified by doubling the molybdenum (Mo) content for enhanced corrosion and wear resistance. The specimens were fabricated using a casting technique. Half of the specimens experienced a heat treatment at
1050°C
for an hour. The microstructure and phase analyses of the specimens were conducted using electron scanning microscopy and X-ray diffraction. The mechanical properties of the specimens were determined in terms of the ASTM Standard Test Method for Tension Testing of Metallic Materials (E8-96). The mechanical behaviors of individual phases in the specimen materials were investigated using a nano-indentation technique. The wear resistance of the specimens was evaluated on a ball-on-disk tribometer. The experimental results revealed that the increased Mo content had significant effects on the mechanical and tribological properties of the low-carbon
Stellite®
alloy and the heat treatment also influenced these properties.
molybdenum alloys, cobalt alloys, corrosion resistance, wear resistance, chemical analysis, heat treatment, crystal microstructure, scanning electron microscopy, X-ray diffraction, indentation, Stellite alloy, heat treatment, molybdenum, carbides, Co solid solution, mechanical properties, hardness, wear
Alloys, Heat treating (Metalworking), Molybdenum, Tribology, Carbon, Mechanical properties, Wear resistance, Wear
Microstructural Effects on the Sliding Wear Resistance of a Cobalt-Based Alloy
Stellite as a Wear-Resistant Material
Microstructure and Property Relationships in Hipped Stellite Powders
Carbide Composition Change During Liquid Phase Sintering of a Wear Resistant Alloy
Cavitation Erosion of Cobalt Based Stellite Alloys, Cemented Carbides and Surface Treated Low Alloy Steel
Proc. 3rd International Conference on Wear of Materials
Properties of P/M Stellite Alloy No. 6
Prog. Powd. Metall.
Experimental Parameter Investigation on the Tribological Behavior of Stellite 6 in Liquid Sodium
Adhesive Wear Processes Occurring During Abrasion of Stellite Type Alloys
J. Aust. Inst. Met.
Recent Developments in Wear- and Corrosion-Resistant Alloys for the Oil Industry
Properties of Stellite Alloy No. 21 Made via Pliable Powder Technology
Met. Powd. Report.
Effect of Nitrogen Alloying on Stellite 21 Cobalt Chromium Molybdenum Biomedical Implant Alloy: Processing and Microstructure
The Influence of Phase Transformations and Oxidation on the Galling Resistance and Low Friction Behavior of a Laser Processed Co-Based Alloy
ASTM Standard Test Methods for Tension Testing of Metallic Materials. Designation: E8-96. American Association State, Highway and Transportation Officials Standard. AASHTO No: T 68.
Relationships Between Hardness, Young’s Modulus and Elastic Recovery in Hard Nanocomposite Coatings
ASTM Standard Test Methods for Notched Bar Impact Testing of Metallic Materials. Designation: E23-06. American Association State, Highway and Transportation Officials Standard. AASHTO No: T 267.
Microstructure and Phase Analyses of Stellite 6 Plus 6wt.% Mo Alloy
Thermodynamics Assessment of Heat Treatments for a Co‐Cr‐Mo Alloy
|
Multilabel Classification - Objectives and metrics | CatBoost
MultiLabel Classification: objectives and metrics
\displaystyle\frac{-\sum\limits_{j=0}^{M-1} \sum\limits_{i=1}^{N} w_{i} (c_{ij} \log p_{ij} + (1-c_{ij}) \log (1 - p_{ij}) )}{M\sum\limits_{i=1}^{N}w_{i}} { ,}
p_{ij} = \sigma(a_{ij}) = \frac{e^{a_{ij}}}{1 + e^{a_{ij}}}
c_{ij} \in {0, 1}
\displaystyle\frac{-\sum\limits_{j=0}^{M-1} \sum\limits_{i=1}^{N} w_{i} (t_{ij} \log p_{ij} + (1-t_{ij}) \log (1 - p_{ij}) )}{M\sum\limits_{i=1}^{N}w_{i}} { ,}
p_{ij} = \sigma(a_{ij}) = \frac{e^{a_{ij}}}{1 + e^{a_{ij}}}
t_{ij} \in [0, 1]
This function is calculated separately for each class k numbered from 0 to M – 1.
\frac{TP}{TP + FP}
\frac{TP}{TP+FN}
(1 + \beta^2) \cdot \frac{Precision * Recall}{(\beta^2 \cdot Precision) + Recall}
\beta
(0; +\infty)
2 \frac{Precision * Recall}{Precision + Recall}
The formula depends on the value of the
type
\displaystyle\frac{\sum\limits_{i=1}^{N}w_{i} \prod\limits_{j=0}^{M-1} [[p_{ij} > 0.5]==t_{ij}]}{\sum\limits_{i=1}^{N}w_{i}} { , }
p_{ij} = \sigma(a_{ij}) = \frac{e^{a_{ij}}}{1 + e^{a_{ij}}}
\frac{TP + TN}{\sum\limits_{i=1}^{N} w_{i}}
The type of calculated accuracy.
Possible values: Classic, PerClass.
\displaystyle\frac{\sum\limits_{j=0}^{M-1} \sum\limits_{i = 1}^{N} w_{i} [[p_{ij} > 0.5] == t_{ij}]]}{M \sum\limits_{i=1}^{N} w_{i}}
MultiLogloss + -
MultiCrossEntropy + -
Precision - -
Recall - -
Accuracy - -
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The Relationship between Global Solar Radiation and Sunshine Durations in Cameroon ()
1University of Douala, Faculty of Sciences, Douala, Cameroon.
2University of Yaoundé, Faculty of Sciences, Yaoundé, Cameroon.
3Nova Scotia Community College, Division of Applied Research, Springhill, Canada.
4Ecole Normale Supérieure d’Enseignement Technique (ENSET), Douala, Cameroon.
5Centre de Physique Atomique, Moléculaire Optique et Quantique (CEPAMOQ), Douala, Cameroon.
6Public Health of the San Diego University, San Diego, USA.
Based on the well-known modified Angstrom formula on the relationship between the sunshine duration and the global solar radiation, this paper aimed to estimate the value of the constant a and b in Cameroon. Only five cities (Maroua, Garoua, NGaoundéré, Yaoundé and Douala) had the both available in-situ data recorded during the period of eleven years (1996-2006) beside which four others cities (Dschang, Koundja, Yoko and Manfé) had only the in-situ sunshine duration available data recorded during the period of twenty years (1986-2006). The 9 cities were grouped in 3 different climate regions. Based on the data of the 5 first cities belonging the 3 regions, the follow constant values a1 = -0.05, a2 = -0.02, a3 = -0.14 and b1 = 0.94, b2 = 0.74, b3 = 1.12 were obtained. The Root Mean Square Error (RMSE) Mean Bias Error (MBE) and correlation coefficient (r) were also determined. Then we used these values to estimate the global solar radiation for the other four remain cities. The constants a and b obtained values are in accordance with those of the West Africa region which Cameroon belongs to. So they can be employed in estimating global solar radiation of location in Cameroon paying attention only to the geographical location information.
Solar Radiation, Sunshine Duration, Angstrom Constants, Climatic Region
Mbiaké, R. , Wakata, A. , Mfoumou, E. , Ndjeuna, E. , Fotso, L. , Tiekwe, E. , Djamen, J. and Bobda, C. (2018) The Relationship between Global Solar Radiation and Sunshine Durations in Cameroon. Open Journal of Air Pollution, 7, 107-119. doi: 10.4236/ojap.2018.72006.
{H}_{0}={G}_{sc}\times {\left(\frac{{R}_{0}}{R}\right)}^{2}\times \frac{24\times 3600}{\pi }\times \left[\mathrm{cos}\left(L\right)\mathrm{cos}\left(\delta \right)\mathrm{cos}\left(\omega \right)+\mathrm{sin}\left(L\right)\mathrm{sin}\left(\delta \right)\right]
\omega
\left(\delta \right)
\delta =23.45\mathrm{sin}\left[0.986\times \left(J+284\right)\right]
\stackrel{¯}{H}={\stackrel{¯}{H}}_{0}\left(a+b\frac{\stackrel{¯}{d}}{{\stackrel{¯}{d}}_{0}}\right)
\stackrel{¯}{d}
{\stackrel{¯}{d}}_{0}
{\stackrel{¯}{d}}_{0}=\frac{2}{15}{\mathrm{cos}}^{-1}\left(\mathrm{tan}\delta \times \mathrm{tan}L\right)
a=\frac{\left[\left({\displaystyle \sum \frac{\stackrel{¯}{H}}{{\stackrel{¯}{H}}_{0}}}\right)\left({\displaystyle \sum {\left(\frac{\stackrel{¯}{d}}{{\stackrel{¯}{d}}_{0}}\right)}^{2}}\right)\right]-\left[\left({\displaystyle \sum \frac{\stackrel{¯}{d}}{{\stackrel{¯}{d}}_{0}}}\right)\left({\displaystyle \sum \left(\frac{\stackrel{¯}{d}}{{\stackrel{¯}{d}}_{0}}\right)\left(\frac{\stackrel{¯}{H}}{{\stackrel{¯}{H}}_{0}}\right)}\right)\right]}{\left[M{\displaystyle \sum {\left(\frac{\stackrel{¯}{d}}{{\stackrel{¯}{d}}_{0}}\right)}^{2}-{\left({\displaystyle \sum \frac{\stackrel{¯}{d}}{{\stackrel{¯}{d}}_{0}}}\right)}^{2}}\right]}
b=\frac{M\left({\displaystyle \sum \left(\frac{\stackrel{¯}{d}}{{\stackrel{¯}{d}}_{0}}\right)\left(\frac{\stackrel{¯}{H}}{{\stackrel{¯}{H}}_{0}}\right)}\right)-\left[\left({\displaystyle \sum \frac{\stackrel{¯}{d}}{{\stackrel{¯}{d}}_{0}}}\right){\displaystyle \sum \left(\frac{\stackrel{¯}{H}}{{\stackrel{¯}{H}}_{0}}\right)}\right]}{\left[M{\displaystyle \sum {\left(\frac{\stackrel{¯}{d}}{{\stackrel{¯}{d}}_{0}}\right)}^{2}-{\left({\displaystyle \sum \frac{\stackrel{¯}{d}}{{\stackrel{¯}{d}}_{0}}}\right)}^{2}}\right]}
{\stackrel{¯}{H}}_{est}
{\stackrel{¯}{H}}_{mes}
MBE\left(\%\right)=100\left(\frac{1}{{\stackrel{¯}{H}}_{M}}\right)\left({\displaystyle \underset{i}{\sum }\frac{{E}_{i}}{M}}\right)
RMSE\left(\%\right)=100\left(\frac{1}{{\stackrel{¯}{H}}_{M}}\right)\sqrt{\left({\displaystyle \underset{i}{\sum }\frac{{E}_{i}^{2}}{M}}\right)}
{E}_{i}={\stackrel{¯}{H}}_{est}-{\stackrel{¯}{H}}_{mes}
r=\frac{{\displaystyle \sum \left({\stackrel{¯}{H}}_{est}-{\stackrel{¯}{H}}_{E}\right)\left({\stackrel{¯}{H}}_{mes}-{\stackrel{¯}{H}}_{M}\right)}}{\sqrt{\left({\displaystyle \sum {\left({\stackrel{¯}{H}}_{est}-{\stackrel{¯}{H}}_{E}\right)}^{2}{\displaystyle \sum {\left({\stackrel{¯}{H}}_{mes}-{\stackrel{¯}{H}}_{M}\right)}^{2}}}\right)}}
\frac{\stackrel{¯}{H}}{{\stackrel{¯}{H}}_{0}}=0.30+0.40\left(\frac{\stackrel{¯}{d}}{{\stackrel{¯}{d}}_{0}}\right)
\frac{\stackrel{¯}{H}}{{\stackrel{¯}{H}}_{0}}=0.18+0.62\left(\frac{\stackrel{¯}{d}}{{\stackrel{¯}{d}}_{0}}\right)
{\stackrel{¯}{d}}_{0}
\stackrel{¯}{d}
S=\frac{{\stackrel{¯}{H}}_{b}}{{\stackrel{¯}{H}}_{b,clear}}
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Overview - Training parameters | CatBoost
input_borders
output_borders
posterior_sampling
per_feature_ctr
--cd, --column-description
--learn-pairs
--test-pairs
--learn-group-weights
--test-group-weights
--learn-baseline
--test-baseline
--params-files
--nan-mode
roc_file
These parameters are for the Python package, R package and Command-line version.
For the Python package several parameters have aliases. For example, the --iterations parameter has the following synonyms: num_boost_round, n_estimators, num_trees. Simultaneous usage of different names of one parameter raises an error.
Command-line: --loss-function
Alias: objective
Command-line: --custom-metric
Command-line: --eval-metric
Command-line: -i, --iterations
Aliases: num_boost_round, n_estimators, num_trees
Command-line: -w, --learning-rate
Alias: eta
Command-line: -r, --random-seed
Alias:random_state
Command-line: --l2-leaf-reg, l2-leaf-regularizer
Alias: reg_lambda
Command-line: --bootstrap-type
Command-line: --bagging-temperature
Command-line: --subsample
Command-line: --sampling-frequency
Command-line: --sampling-unit
Command-line: --mvs-reg
\infty
Command-line: --random-strength
Command-line: --use-best-model
If this parameter is set, the number of trees that are saved in the resulting model is defined.
Command-line: --best-model-min-trees
Command-line: -n, --depth
Alias: max_depth
Command-line: --grow-policy
Command-line: --min-data-in-leaf
Alias: min_child_samples
Command-line: --max-leaves
Alias:num_leaves
Command-line: -I, --ignore-features
Feature indices or names to exclude from the training. It is assumed that all passed values are feature names if at least one of the passed values can not be converted to a number or a range of numbers. Otherwise, it is assumed that all passed values are feature indices.
The addition of a non-existing feature name raises an error.
Command-line: --one-hot-max-size
Command-line: --has-time
Command-line: --rsm
Alias:colsample_bylevel
Command-line: --nan-mode
Command-line: --input-borders-file
Load Custom quantization borders and missing value modes from a file (do not generate them).
Command-line: --output-borders-file
Save quantization borders for the current dataset to a file.
Command-line: --fold-permutation-block
Objects in the dataset are grouped in blocks before the random permutations. This parameter defines the size of the blocks.
Command-line: --leaf-estimation-method
Command-line: --leaf-estimation-iterations
Command-line: --leaf-estimation-backtracking
When the value of the leaf_estimation_iterations parameter is greater than 1, CatBoost makes several gradient or newton steps when calculating the resulting leaf values of a tree.
Command-line: --fold-len-multiplier
Command-line:--approx-on-full-history
Command-line: --class-weights
Command-line: --auto-class-weights
CW_k=\displaystyle\frac{max_{c=1}^K(\sum_{t_{i}=c}{w_i})}{\sum_{t_{i}=k}{w_{i}}}
CW_k=\sqrt{\displaystyle\frac{max_{c=1}^K(\sum_{t_i=c}{w_i})}{\sum_{t_i=k}{w_i}}}
The weight for class 1 in binary classification. The value is used as a multiplier for the weights of objects from class 1.
Command-line: --boosting-type
Command-line: --boost-from-average
Initialize approximate values by best constant value for the specified loss function.
Command-line: --langevin
Command-line: --diffusion-temperature
Command-line: --posterior-sampling
If this parameter is set several options are specified as follows and model parameters are checked to obtain uncertainty predictions with good theoretical properties.
Command-line: --allow-const-label
Command-line: --score-function
Command-line: --monotone-constraints
Command-line: --feature-weights
Command-line: --first-feature-use-penalties
Command-line: --penalties-coefficient
Command-line: --per-object-feature-penalties
Command-line: --model-shrink-rate
Command-line: model_shrink_mode
Per-feature quantization settings for categorical features.
A comma-separated list of input files that contain the validation dataset description (the format must be the same as used in the training dataset).
The path to the input file that contains the pairs description for the training dataset.
The path to the input file that contains the pairs description for the validation dataset.
The path to the input file that contains the weights of groups. Refer to the Group weights section for format details.
The path to the input file that contains the weights of groups for the validation dataset.
The path to the input file that contains baseline values for the training dataset.
The path to the input file that contains baseline values for the validation dataset.
Read the column names from the first line of the dataset description file if this parameter is set.
The path to the input JSON file that contains the training parameters, for example:
Command line: --logging-level
Command line: --metric-period
The frequency of iterations to calculate the values of objectives and metrics.
Command line: --verbose
Command line: --train-dir
Command line: --model-size-reg
This regularization is needed only for models with categorical features (other models are small).
Enable snapshotting for restoring the training progress after an interruption.
The name of the output file to save the ROC curve points to.
Command-line: --od-type
Command-line: --od-pval
The threshold for the IncToDec overfitting detector type.
Command-line: --od-wait
Command line: --task-type
Command line: --devices
These parameters are only for the Python package and Command-line version.
Command-line: --tokenizers
Command-line: --dictionaries
Command-line: --feature-calcers
Command-line: --text-processing
These parameters are only for the Python package.
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Estimation Bias - SAGE Research Methods
Estimation Bias | The SAGE Encyclopedia of Educational Research, Measurement, and Evaluation
Estimation bias, or simply bias, is a concept in statistical inference that relates to the accuracy of parameter estimation. The term bias was first introduced in the statistical context by English statistician Sir Arthur L. Bowley in 1897. This entry provides the formal definition of estimation bias along with the concept of error, its implications and uses in statistical inference, and relevance to other types of bias that may arise in the data collection process.
The Concept ofError in Statistical Inference
Suppose that we would like to estimate a population parameter θ (e.g., population mean). An estimator
\stackrel{^}{\theta }
is any sample statistic (e.g., sample mean) that is used to estimate θ. Because
\stackrel{^}{\theta }
is sample based, it does not perfectly agree with the true value of θ. ...
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Section 58.7 (03SF): Galois covers of connected schemes—The Stacks project
Section 58.7: Galois covers of connected schemes (cite)
58.7 Galois covers of connected schemes
Let $X$ be a connected scheme with geometric point $\overline{x}$. Since $F_{\overline{x}} : \textit{FÉt}_ X \to \textit{Sets}$ is a Galois category (Lemma 58.5.5) the material in Section 58.3 applies. In this section we explicity transfer some of the terminology and results to the setting of schemes and finite étale morphisms.
We will say a finite étale morphism $Y \to X$ is a Galois cover if $Y$ defines a Galois object of $\textit{FÉt}_ X$. For a finite étale morphism $Y \to X$ with $G = \text{Aut}_ X(Y)$ the following are equivalent
$Y$ is a Galois cover of $X$,
$Y$ is connected and $|G|$ is equal to the degree of $Y \to X$,
$Y$ is connected and $G$ acts transitively on $F_{\overline{x}}(Y)$, and
$Y$ is connected and $G$ acts simply transitively on $F_{\overline{x}}(Y)$.
This follows immediately from the discussion in Section 58.3.
For any finite étale morphism $f : Y \to X$ with $Y$ connected, there is a finite étale Galois cover $Y' \to X$ which dominates $Y$ (Lemma 58.3.8).
The Galois objects of $\textit{FÉt}_ X$ correspond, via the equivalence
\[ F_{\overline{x}} : \textit{FÉt}_ X \to \textit{Finite-}\pi _1(X, \overline{x})\textit{-Sets} \]
of Theorem 58.6.2, with the finite $\pi _1(X, \overline{x})\textit{-Sets}$ of the form $G = \pi _1(X, \overline{x})/H$ where $H$ is a normal open subgroup. Equivalently, if $G$ is a finite group and $\pi _1(X, \overline{x}) \to G$ is a continuous surjection, then $G$ viewed as a $\pi _1(X, \overline{x})$-set corresponds to a Galois covering.
If $Y_ i \to X$, $i = 1, 2$ are finite étale Galois covers with Galois groups $G_ i$, then there exists a finite étale Galois cover $Y \to X$ whose Galois group is a subgroup of $G_1 \times G_2$. Namely, take the corresponding continuous homomorphisms $\pi _1(X, \overline{x}) \to G_ i$ and let $G$ be the image of the induced continuous homomorphism $\pi _1(X, \overline{x}) \to G_1 \times G_2$.
Comment #4185 by Nicolas Müller on April 23, 2019 at 09:04
G = \text{Aut}(X/Y)
should be something like
G = \text{Aut}_X(Y)
. After item (4) there should not be ", and".
In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 03SF. Beware of the difference between the letter 'O' and the digit '0'.
The tag you filled in for the captcha is wrong. You need to write 03SF, in case you are confused.
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Foci of an Ellipse Calculator
Reviewed by Purnima Singh, PhD
How to find the foci of an ellipse
How to use this foci of an ellipse calculator
How to draw an ellipse using the foci of ellipse equation
This foci of an ellipse calculator will help you locate the foci of an ellipse, with respect to its center, given the values for its semi-major axis and the semi-minor axis. In this calculator, you will learn:
What foci of an ellipse are;
How to find the foci of an ellipse;
How to find the coordinates of the foci of an ellipse; and
How to draw an ellipse after finding the foci of an ellipse.
Keep on reading to start learning.
An ellipse's foci, the plural of focus, are the two reference points of the locus of points that form the ellipse that we know. An ellipse's foci lie within the area of the ellipse along its principal axis - which is the axis that cuts through the longer dimension of the ellipse.
To better understand how the foci act as an ellipse's reference points, imagine drawing an arbitrary point, P1, outside the principal axis and drawing two connecting lines |F1P1| (from the first focus, F1, to P1) and |P1F2| (from P1 to the second focus, F2), as shown in the image below:
Then, imagine drawing another pair of lines to another point, P2, such that the total length of this new pair of lines, |F1P2| + |P2F2|, is equal to the total length of our first pair of lines |F1P1| + |P1F2|.
If we continue this fashion to draw all the points P possible, we then form an ellipse, as shown in the illustration below:
In the illustration above, by considering point P3 along the principal axis, we can see that the total length of the pair of lines, |F1P3| + |P3F2|, equates to the major axis or the major diameter of our ellipse which is equal to a × 2, as we can better see illustrated below:
On the other hand, by considering point P2, which falls at the semi-minor axis at a distance b from the center, we form two right triangles. Since we already know that |F1P2| + |P2F2| is also equal to a × 2, we can say that |P2F2| = a, as shown in the illustration below:
With that in mind, we can go ahead and proceed to learn how to find the foci of an ellipse using the relationship between the distance to a focus from the ellipse's center, the semi-minor axis b, and semi-major axis a:
From the image we got from the previous section of this text and based on the Pythagorean theorem, we can formulate the foci of an ellipse equation:
F = \sqrt{a^2-b^2}
F
- Focal distance from an ellipse's center to one of its foci;
a
- Semi-major axis; and
b
- Semi-minor axis.
Using the focal distance, we can now determine the coordinates of the foci of an ellipse by following these notations:
For horizontal ellipses
F_1 = (c_1-F,\ c_2)
F_2 = (c_1+F,\ c_2)
For vertical ellipses
F_1 = (c_1,\ c_2-F)
F_2 = (c_1,\ c_2+F)
F_1
F_2
- Variables representing the foci of an ellipse;
c_1
- x-coordinate of the ellipse's center; and
c_2
- y-coordinate of the ellipse's center.
🙋 When the two foci of an ellipse intersect, we get a special type of ellipse we know as circle. That only happens when
and
b
are equal, and we calculate
F
to be 0. We can also say that the circle's focus is on its center.
To use our foci of an ellipse calculator to find the coordinates of the foci of an ellipse,
First enter the coordinates of your ellipse's center.
Then, input the values for
and
b
of your ellipse.
You can also enter the coordinates of the vertices of your ellipse by clicking on the Advanced mode button of our calculator.
Upon doing these steps, our foci of an ellipse calculator will then display the x and y coordinates of your ellipse's foci.
We can use the knowledge we learned from our foci of an ellipse calculator to draw any size of an ellipse we want. But first, we have to calculate the foci of a sample ellipse. Let's say we want to draw an ellipse with a 3 cm semi-minor axis and a 5 cm semi-major axis. By substituting these measurements to our foci of an ellipse equation, we can obtain:
\scriptsize \begin{align*} F &= \sqrt{a^2-b^2}\\ &= \sqrt{(5\ \text{cm})^2-(3\ \text{cm})^2}\\ &= \sqrt{25\ \text{cm}^2-9\ \text{cm}^2}\\ &= \sqrt{16\ \text{cm}^2}\\ &= 4\ \text{cm} \end{align*}
Now that we have our ellipse's focal distance, it's time to draw our ellipse. Here are the steps to follow:
Cut a string to make a loop with a perimeter equal to
2F + 2a
2\times4\ \text{cm} + 2\times5\ \text{cm} = 18\ \text{cm}
Push two pins on your drawing surface
2F
2\times4\ \text{cm} = 8\ \text{cm}
apart. The pins will represent our ellipse's foci.
Place the loop you made around the two pins and use a pencil to pull the loop taut.
Move the pencil around the pins while keeping the loop taut to draw our ellipse, as shown in the image below:
Want to learn more about ellipse? Here are our other ellipse-related tools you can check out:
How many foci does an ellipse have?
An ellipse has a maximum of 2 foci. On the other hand, we can draw multiple numbers of ellipses on any two specified foci. We can also have an ellipse with only one focus. In that particular case, we form a circle - which is a special type of ellipse.
How do I determine the foci of an ellipse?
To determine the foci on an ellipse with, let's say, 13 cm and 5 cm for their semi-major and semi-minor axes, respectively:
First take the difference between the squares of the semi-major axis and the semi-minor axis: (13 cm)² - (5 cm)² = 144 cm².
Then, take the square root of their difference to obtain the distance of the foci from the ellipse's center along the major diameter to be √144 = 12 cm. This means that one focus lies 12 cm to the left of the ellipse's center, and one focus lies 12 cm to the right.
x-coordinate, c₁
y-coordinate, c₂
📌 First focus F1
📌 Second focus F2
Use this diagonal of a rectangle calculator and find the answer to the questions: How to find the diagonal of a rectangle? What is the diagonal of a rectangle formula?
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Created by Adena Benn
How to use our general to standard form of a circle calculator:
What do we mean by equation of a circle?
What is the general form equation of a circle?
What is the equation of a circle in standard form?
How to convert from general to standard form of a circle?
Are you looking for a general to standard form of a circle calculator for mathematics? Then you have come to the right place. Our general to standard form of a circle calculator will help you convert from general to standard form with ease. Best of all, you do not need to know any formulas to use it. However, if you would like to understand how we went about converting from general to standard form or would like to do some conversions on your own, we have all the information you will need to get the job done. Continue reading to learn:
What the equation of a circle means;
What is the equation for the general form of a circle;
What is the equation for the standard form of a circle; and
The process used to convert from general to the standard form.
Our general to standard form of a circle calculator allows you to change from standard to general form by entering the values in the section marked standard form. Once you have don't this, the values for the general form of the circle will appear in real-time.
Have you been wondering what exactly we mean when we say the equation of a circle?
The equation of a circle is the formula we use to show the circle's position in the cartesian plane. The equation represents the points along the circumference of the circle.
There are several formulas that we may use to find the equation of the circle. However, this calculator will only look at the equations for the standard and general forms.
The general form of the equation of a circle is x² + y² + Dx + Ey + F.
In this equation, D, E, and F are real numbers. To find the center and radius from the general form, we need to convert this equation to its standard form.
The standard form of the equation of a circle is (x - h)² + (y - k)² = r².
This is the equation of a circle in standard form with center h, k, and radius r. Because this equation gives the coordinates and radius, it is excellent for drawing the circle in the Cartesian plane.
Here is what you do to convert from the general to the standard form of a circle:
Enter the values for D, E, and F into our general to standard form of a circle calculator;
The values in the standard form section are for the radius and the center coordinates.
Lastly, substitute the new values into the equations shown above the calculator.
Not satisfied with this? Are you trying to understand how to convert from the general to the standard form of a circle for school and need to learn the entire process? The following explanation is just what you need:
Let's take the equation for the general form of a circle:
\text{x}^2 + \text{y}^2 + \text{Dx} + \text{Ey} + \text{F} = 0
Before we proceed, let us replace
D
E
F
with numbers.
\text{x}^2 + \text{y}^2 + 22\text{x} - 12\text{y} - 8 = 0
So the first thing you need to do with this equation is, move the constant term to the other side of the equation. To move 8 to the other side, we need to add 8 to both sides. Having done that, you should have the following result:
\text{x}^2 + \text{y}^2 + 22\text{x} - 12\text{y} = 8
Next, group all the
x
terms and all the
y
terms. The resulting equation will be:
(\text{x}^2 + 22\text{x}) + (\text{y}^2 - 12\text{y}) = 8
Now, let's complete the square. We do this by halving the coefficient of
x
y
, then squaring them. Since the coefficient of
x
is 22, half of 22 would be 11. When we square 11, we get 121. So the first bracket should now look like this:
(\text{x}^2 + 22\text{x} + 121)
Since the coefficient of y is 12 and half of 12 is 6. When you square 6, you will get 36. The second bracket will be:
(\text{y}^2 - 12\text{y} + 36)
An equation must always be balanced. So, because we added 121 and 36 to one side of the equation, we need to also add those numbers to the other side of the equation.
Our resulting equation is:
(\text{x}^2 + 22\text{x} + 121) + (\text{y}^2 - 12\text{y} + 36) = 165
Next, you need to factorize this equation:
To do this, we go to the first bracket and take the square root of the first term, take the sign of the middle term, and take the square root of the last term. We repeat this process with the second bracket as well.
Your resulting equation should be:
(\text{x} + 11)^2 + (\text{y} - 6)^2 = 165
This final equation is the standard form of the circle.
Here is a list of related calculators that may interest you:
Equation of a circle calculator;
Center of a circle calculator;
General form of the equation of a circle calculator;
Standard equation of a circle calculator;
Equation of a circle with diameter endpoints calculator; and
Standard form to general form of a circle calculator.
The right trapezoid calculator will help you compute properties of the most special trapezoid - the one with two right angles!
Use our sin calculator to find out the sine value for chosen angle.
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The angles of a pyramid
How to calculate the pyramid angles
How to calculate the angles of a square pyramid
How to use our pyramid angle calculator
Use our pyramid angle calculator to find all the possible angles in a pyramid - we'll ask you a few parameters, and do all the math for you!
The types of angles in a pyramid;
How to calculate the angles of a pyramid;
An example of how to calculate the square pyramid angles; and
How to use our pyramid angle calculator.
A pyramid is a solid figure with a polygonal base. A triangular face corresponds to each side of the base, joining at the apex.
The base can be of any shape; however, regular pyramids are easier to study. The base of a regular pyramid is a regular polygon. We can find, for example:
Triangular pyramids, with an equilateral triangle as the base;
Square pyramids, where the base is, of course, a square;
Hexagonal pyramids; and
Well, infinitely more.
If the apex lies above the centroid of the base (the geometric mean point of the polygon), we call the pyramid a right pyramid; otherwise, we have an oblique pyramid. Here we will analyze only right pyramids.
💡 We have many pyramid calculators here at Omni: check out our pyramid volume calculator, our right rectangular pyramid calculator, and our triangular pyramid volume calculator!
There are a lot of angles in a pyramid, but luckily in a regular pyramid, most of them are identical. Let's identify them!
We can identify the angle between the faces' vertical medians and the base at the apex. It defines how "slender" a pyramid is. We call this angle
\alpha
The angle between the edge and the base
\beta
\alpha
in pyramids with a convex base: the corner of the base is farther from the centroid than the center of its sides.
The angles of a hexagonal pyramid. You can identify analogue angles in other regular pyramids
We can identify other angles, also related to the pyramid's height. They lie on each face, and since in a regular pyramid a face is an isosceles triangle, the two angles at the bottom are identical. We identify them with the letter
\gamma
. The angle on the top, at the apex, gets smaller for tall and slender pyramids. We call it
\delta
We highlighted
\alpha
\beta
in a hexagonal pyramid (because hexagons are the bestagons).
Now you know which angles to search for. It's time to learn how to calculate the angles in a pyramid!
To find the angles we need to use a bit of trigonometry - in particular the theorems to find the elements of a right triangle.
Let's start with the angle
\alpha
, between the median of a face and the base. Let's take a look at the diagram, now marked with some relevant points.
\alpha
corresponds to the angle
\text{C}\widehat{\text{M}}\text{O}
. We compute it noting that the catheti of the triangles are
\text{OC}
(the height of the pyramid) and
\text{CM}
(the segment from the side's midpoint to the centroid). We calculate the angle
\alpha
\footnotesize \tan{\alpha} = \frac{\text{OC}}{\text{CM}} \ \rightarrow \ \alpha = \arctan{\left(\frac{\text{OC}}{\text{CM}}\right)}
Let's proceed with the angle
\beta
. In this case, we need to use the segment
\text{AC}
to compute the tangent:
\footnotesize \tan{\beta} = \frac{\text{OC}}{\text{AC}}\ \rightarrow\ \beta= \arctan{\left(\frac{\text{OC}}{\text{AC}}\right)}
🔎 Note that
\text{AC}>\text{CM}
is always true: this explains why
\beta<\alpha
It's time to compute the angle on a pyramid's face. Using the old and reliable Pythagorean theorem, we find the length of the slanted side of the pyramid:
🙋 To analyze these angles, we move to the other side of the base! Take another look at the diagram if you have trouble visualizing the equations.
\text{PO}=\sqrt{\text{OC}^2+\text{PC}^2}
On each face, we can identify a pair of right triangles created by the median (in this case
\text{ON}
). The segment
\text{PN}
(half the length of the base's side) and the slanted side of the pyramid
\text{ON}
isolate the angle
\gamma
\text{O}\widehat{\text{P}}\text{N}
). We use another right triangle identity to calculate it:
\footnotesize \cos{\gamma}=\frac{\text{PN}}{\text{ON}}\ \rightarrow \ \gamma=\arccos{\left(\frac{\text{PN}}{\text{ON}}\right)}
The last angle,
\delta
, can be computed using trigonometry again, or if laziness is allowed, by considering that the sum of the internal angles of a triangle is
180\degree
\delta = 180\degree - 2\times\gamma
We will guide you step by step in calculating the angles of a right square pyramid. In fact, not of a general square pyramid - let's calculate the angles of the Great Pyramid of Giza!
The Great Pyramid of Giza is a great pyramid in Giza.
We need to take some measurements:
The original height of the pyramid is
146.7\ \text{m}
The side measures
230.6\ \text{m}
Let's calculate the segment connecting the midpoint of the side to the center. Since the base is a square, it's value is half the length of the side:
\footnotesize\text{CM}=\frac{AB}{2}=\frac{230.6 \ \text{m}}{2}=115.3\ \text{m}
We can calculate the angle
\alpha
\begin{align*} \footnotesize\alpha & \footnotesize = \arctan{\left(\frac{\text{OC}}{\text{CM}}\right)} =\\ &\footnotesize=\arctan{\left(\frac{146.7\ \text{m}}{115.3\ \text{m}}\right)} = 51.83\degree \end{align*}
To calculate the angle in the corner of the base, we need the measure of half its diagonal
\beta
\begin{align*} \footnotesize\beta & \footnotesize = \arctan{\left(\frac{\text{OC}}{\text{AC}}\right)} =\\ &\footnotesize=\arctan{\left(\frac{146.7\ \text{m}}{115.3\times\sqrt{2}\ \text{m}}\right)} = 41.98\degree \end{align*}
🙋 Ancient Egyptians measured the slope of a right pyramid using the seked, a unit corresponding to the numbers of horizontal cubits corresponding to a rise of one cubit in height. The base angle of the Great Pyramid of Giza has a seked of
5\tfrac{1}{2}
sekeds.
What about the angle on each face?
We can use the formulas above to find the value of
\gamma
\delta
; however, we need to calculate the length of the slanted side first:
\begin{align*} \footnotesize \text{OB} & \footnotesize =\sqrt{\text{CO}^2+\text{AC}^2} \\ & \footnotesize = \sqrt{146.7^2 + 2\times(115.3)^2}\\ &\footnotesize = 219.3\ \text{m} \end{align*}
Egyptians and mathematics went hand in hand. However, we are sure that they weren't marking angles with Greek letters!
Let's proceed with the calculations.
\begin{align*} \footnotesize\gamma & \footnotesize = \arccos{\left(\frac{\text{PN}}{\text{ON}}\right)} =\\ &\footnotesize=\arccos{\left(\frac{115.3\ \text{m}}{219.3\ \text{m}}\right)}= 58.29\degree \end{align*}
Finally, the angle of a face at the apex of the pyramid,
\delta
\begin{align*} \footnotesize\delta & \footnotesize = 180\degree - 2\times\gamma=\\ &\footnotesize = 180\degree-116.58=63.42\degree \end{align*}
Our pyramid angle calculator can help you with many types of regular right pyramids. Select the type of base you need; we included:
Regular pentagon;
Regular hexagon;
Regular heptagon; and
Regular octagon.
Insert the available measurements in the calculator, and find the results!
🙋 Our tools work in reverse, too. For example, you can insert the value of the base angles and find out the pyramid's height!
Do you want to expand your pyramidal knowledge outside of geometry? Visit our Minecraft's pyramid block calculator!
What is the best angle for pyramid power?
Pyramids are surrounded by mysticism and occultism. The truth is that there's no such thing as pyramid power, hidden purposes, and alien involvement. Pyramids, and in particular ancient pyramids, are neat but not mystic! The math behind them, however, is all true!
What is the angle of a hexagonal pyramid with side 2 and height 3?
To calculate the angle at the base of a hexagonal pyramid, follow these steps:
Calculate the length of the segment MC connecting the side's midpoint to the centroid.
Apply the inverse tangent function to the ratio of the height and the length just calculated: OC/MC.
For a hexagon with side 2 and height 3, it means:
MC = 2 × cos(30°) = 2 × (sqrt(3)/2) = sqrt(3)
α = arctan(3/sqrt(3)) = arctan(sqrt(3)) = 60°
How do I calculate the angles at the base of a square pyramid?
To calculate the angles at the base of a pyramid, you can use the trigonometric formulas of right triangles. Calling the height h and the side L, the value of the base angle is:
α = arctan(h/(L/2))
We calculate the angle in the corner of the base using half the diagonal:
ß = arctan(h/(sqrt(2) × L/2)) = arctan(h/(L/sqrt(2)))
What are the angles of the Great Pyramid of Giza?
The Great Pyramid of Giza has a base angle of 51.83°. We can calculate it using the inverse trigonometric function arctangent, knowing the height and the side of the pyramid:
α = arctan(146.7/115.3) = 51.83°
The angle in the corner of the base is slightly smaller: 41.98°.
Base and measurements
Side (AB)
Segment MC
Height (OC)
Slanted side (BO)
Distance calculator helps you find the distance between two points on a Cartesian coordinate system.
Right rectangular pyramid calculator gives you all the information about the area and volume of a pyramid.
Right Rectangular Pyramid Calc: find A, V, A_l, A_b
|
Explaining Affine Rotation - Nextjournal
If you are working on 2D or 3D graphics you have probably encountered 3x3 or 4x4 matrices used to translate or rotate the points of a geometrical figure.
You could lookup the definition of affine transformations on wikipedia. However unless you already understand the math well it does not explain very well why the affine transformation matrices look the way they do.
Here we are going to focus on explaining the rotation matrix, because that is I believe least obvious to people. But for the sake of completeness here are the scale, translate and rotation matrices.
If you want to scale a vector
u
c_x
along the x-axis and the factor
c_y
along the y-axis to get a new vector
v
you can use the matrix
S
In similar fashion if you want to translate vector
u
d_x
along the x-axis and
d_y
along the y-axis to produce vector
v
you use translation matrix
T
Next comes the rotation. How this works is less obvious. Here we are representing a point
p
u
extending from the origin of our coordinate system to the point
p
. Then we are rotating this vector by an angle
\theta
. This is done with a rotation matrix
R
If you remember the definition of
\cos(\theta)
is equal to the the length of the side adjacent to the angle
\theta
\sin(\theta)
is equal to the equal to the length of the side opposite of the angle
\theta
This is for a unit circle. That is a circle where the radius is 1. We can draw a triangle inside it where the hypothenuse is equal to the radius.
We can also think of the opposite side as representing the y-axis and the adjacent side representing the x-axis. So for a vector
v
r
forming an angle
\theta
with the x-axis you can easily calculate its components:
The coordinates for a vector is always given relative to some basis. The basis are a bunch vectors representing each axis. By default the basis is defined as
[1, 0]
vector for the x-axis and
[0, 1]
vector for the y-axis. Thus a vector with coordinates
x
y
can be seen as the addition of two formed from the unit vectors defining the basis.
Here is an example in coded in the Julia programming language where
x = 3
y = 4
v = 3*[1, 0] + 4*[0, 1]
linear-algebra (Julia)
2-element Array{Int64,1}: 3 4
Which again you can express equally through matrix multiplication:
An example from Julia using matrix multiplication.
v = A*[3, 4]
Basically you can make any vector by combining two other vectors. This allows us to simplify how we look at rotation. You can think of the vector
u
being rotated as being made up of vector representing its x-component and another one representing its y-component. These two vectors can be combined to create the
u
So when calculating a rotation of
u
\theta
degrees, we can handle this as two separate rotations. One rotation of the x-axis basis
[1, 0]
and a rotation of the y-axis basis
[0, 1]
We can study the circle below to see what happens if we try to rotate the x-axis basis by
\theta
degrees. We can imagine that the basis vector starts of as
AB
and then gets rotated to its new orientation
AC
. We can easily see that the x-component of
AC
\cos(\theta)
while the y-component is
\sin(\theta)
This lets us conclude that the new x-basis
b_x
should be defined as the vector:
We can perform a similar analysis for the y-basis defined by
AB
in the circle below. It gets rotated counter-clockwise
\theta
degrees giving us a new y-basis
AC
In this case we can see the x-component of this new basis is defined by
BC
. The length is
\sin(\theta)
. But since this is in the negative half of the circle, the x-coordinate will be
-\sin(\theta)
. In this case the y-coordinate is defined by
AB
which is of length
\cos(\theta)
. We can thus write the new y-basis as:
We can thus write our vector as a combination of these two basis vectors. Which gives us:
To make this 2x2 matrix also usable to express translation, it is often expanded to a 3x3 matrix like this:
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Preferred orientation and elastic anisotropy of illite-rich shalePreferred orientation and elastic anisotropy | Geophysics | GeoScienceWorld
University of California, Department of Earth and Planetary Science, Berkeley, California. E-mail: wenk@berkeley.edu; ivan.lonardelli@ing.unitn.it.
Ivan Lonardelli;
Ivan Lonardelli
Hermann Franz;
HASYLAB, DESY, Hamburg, Germany. E-mail: fhermann@hasylab.edu.
Chevron Energy Technology Company, Seismic Analysis & Property Estimation, San Ramon, California. E-mail: knih@chevron.com.
Lawrence Berkeley National Laboratory, Earth Sciences Division, Berkeley, California. E-mail: snakagawa@lbl.gov.
Hans-Rudolf Wenk, Ivan Lonardelli, Hermann Franz, Kurt Nihei, Seiji Nakagawa; Preferred orientation and elastic anisotropy of illite-rich shale. Geophysics 2007;; 72 (2): E69–E75. doi: https://doi.org/10.1190/1.2432263
Shales display significant seismic anisotropy that is attributed in part to preferred orientation of constituent minerals. This orientation pattern has been difficult to quantify because of the poor crystallinity and small grain size of clay minerals. A new method is introduced that uses high-energy synchrotron X-rays to obtain diffraction images in transmission geometry and applies it to an illite-rich shale. The images are analyzed with the crystallographic Rietveld method to obtain quantitative information about phase proportions, crystal structure, grain size, and preferred orientation (texture) that is the focus of the study. Textures for illite are extremely strong, with a maximum of 10 multiples of a random distribution for (001) pole figures. From the three-dimensional orientation distribution of crystallites, and single-crystal elastic properties, the intrinsic anisotropic elastic constants of the illite aggregate (excluding contribution from aligned micropores) can be calculated by appropriate medium averaging. The illitic shale displays roughly transverse isotropy with
C11
C22
and more than twice as strong as
C33
. This method will lend itself to investigate complex polymineralic shales and quantify the contribution of preferred orientation to macroscopic anisotropy.
Rock physics of organic shales
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Bootstrap options - How training is performed | CatBoost
Bootstrap types
Frequency of resampling and reweighting
The bootstrap_type parameter affects the following important aspects of choosing a split for a tree when building the tree structure:
To prevent overfitting, the weight of each training example is varied over steps of choosing different splits (not over scoring different candidates for one split) or different trees.
When building a new tree, CatBoost calculates a score for each of the numerous split candidates. The computation complexity of this procedure is
O(|C|\cdot n)
|C|
is the number of numerical features, each providing many split candidates.
n
is the number of examples.
Usually, this computation dominates over all other steps of each CatBoost iteration (see Table 1 in the CatBoost: unbiased boosting with categorical features). Hence, it seems appealing to speed up this procedure by using only a part of examples for scoring all the split candidates.
Depending on the value of the bootstrap_type parameter, these ideas are implemented as described in the list below:
MVS (supported only on CPU)
The weight of an example is set to the following value:
w=a^{t} {, where:}
t
is defined by the bootstrap_type parameter
a=-log(\psi)
\psi
is independently generated from Uniform[0,1] . This is equivalent to generating values
a
for all the examples according to the Bayesian bootstrap procedure (see D. Rubin “The Bayesian Bootstrap”, 1981, Section 2).
The Bayesian bootstrap serves only for the regularization, not for speeding up.
Command-line version: --bagging-temperature
[0; \inf)
Command-line version: --sampling-unit
w_{i}
o_{i}
w_{j}
g_{j}
o_{i_{j}}
g_{j}
Corresponds to Stochastic Gradient Boosting (SGB, refer to the Stochastic gradient boosting for details). Each example is independently sampled for choosing the current split with the probability defined by the subsample parameter. All the sampled examples have equal weights. Though SGB was originally proposed for regularization, it speeds up calculations almost
\left(\frac{1}{subsample}\right)
w_{i}
o_{i}
w_{j}
g_{j}
o_{i_{j}}
g_{j}
Supported only on CPU.
Implements the importance sampling algorithm called Minimum Variance Sampling (MVS).
Scoring of a split candidate is based on estimating of the expected gradient in each leaf (provided by this candidate), where the gradient
g_{i}
for the example
is calculated as follows:
g_{i} = \frac{\partial L(y_{i}, z)}{\partial z}|_ {z=M(i)}
L
is the loss function
y_{i}
is the target of the example
i
M(i)
is the current model prediction for the example
i
(see the Algorithm 2 in the CatBoost: unbiased boosting with categorical features).
For this estimation, MVS samples the subsample examples
i
such that the largest values of
|g_i|
are taken with probability
p_{i}=1
and each other example
is sampled with probability
\displaystyle\frac{|g_i|}{\mu}
\mu
is the threshold for considering the gradient to be large if the value is exceeded.
Then, the estimate of the expected gradient is calculated as follows:
\hat{E\, g} = \frac{\sum_{i:\ sampled\ examples} \frac{g_i}{p_i}}{\sum_{i:\ sampled\ examples} \frac{1}{p_i}} { , where}
The numerator is the unbiased estimator of the sum of gradients.
The denominator is the unbiased estimator of the number of training samples.
This algorithm provides the minimum variance estimation of the L2 split score for a given expected number of sampled examples:
s=\sum_{i:\ all\ training\ examples} p_{i}
Since the score is a fractional function, it is important to reduce the variance of both the numerator and the denominator. The mvs_reg (--mvs-reg) hyperparameter affects the weight of the denominator and can be used for balancing between the importance and Bernoulli sampling (setting it to 0 implies importance sampling and to
\infty
- Bernoulli).. If it is not set manually, the value is set based on the gradient distribution on the current iteration.
MVS can be considered as an improved version of the Gradient-based One-Side Sampling (GOSS, see details in the
paper) implemented in LightGBM, which samples a given number of top examples by values
|g_i|
with the probability 1 and samples other examples with the same fixed probability. Due to the theoretical basis, MVS provides a lower variance of the estimate
\hat{E\, g}
than GOSS.
MVS may not be the best choice for regularization, since sampled examples and their weights are similar for close iterations.
Command-line version: --mvs-reg
\infty
w_{i}
o_{i}
w_{j}
g_{j}
o_{i_{j}}
g_{j}
Refer to the paper for details; supported only on GPU)
The weights of examples are i.i.d. sampled from the Poisson distribution with the parameter
-log(1-subsample)
providing the expected number of examples with positive weights equal to the subsample parameter . If subsample is equal to 0.66, this approximates the classical bootstrap (sampling
n
examples with repetitions).
w_{i}
o_{i}
w_{j}
g_{j}
o_{i_{j}}
g_{j}
All training examples are used with equal weights.
w_{i}
o_{i}
w_{j}
g_{j}
o_{i_{j}}
g_{j}
The frequency of resampling and reweighting is defined by the sampling_frequency parameter:
It is recommended to use MVS when speeding up is an important issue and regularization is not. It is usually the case when operating large data. For regularization, other options might be more appropriate.
Estimating Uncertainty for Massive Data Streams
N. Chamandy, O. Muralidharan, A. Najmi, and S. Naid, 2012
Computational Statistics & Data Analysis, 38(4):367–378, 2002
T. B. Johnson and C. Guestrin
G. Ke, Q. Meng, T. Finley, T. Wang, W. Chen, W. Ma, Q. Ye, and T.-Y. Liu..
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Effect of Seeds Size on Germination of Faba Bean Plant
Noha Fahad Alngiemshy1, Jana Saleh Alkharafi1, Norah Saud Alharbi1, Noorah Saleh Al-Sowayan2*
1 Department of Biology, College of Science, Qassim University, Buraydah, Saudi Arabia.
2 Department of Biology, Faculty of Science, Qassim University, Buraydah, Saudi Arabia.
Abstract: The study was conducted in Al-Qassim region in February 2020 and the study aimed to know the effect of seed size on germination and some morphological parameters in Faba bean (Vicia faba L.) plant, by dividing the seeds into three sizes large 2.5 - 2.3 cm, medium 2.0 - 2.2 cm, and small 1.7 - 1.9 cm and planting them. The results showed that the germination percentage in the large seeds was the highest (100%), and the size of the seeds affected the length of the root and shoot as the large seeds gave the highest length for root (8.37 cm) and shoot (14.2 cm) compared to the medium and small seeds. So, for these results, the seed size is closely related to root and shoot length, number of leaves.
Keywords: Faba Bean, Seed Size, Germination
Faba bean (Vicia faba L.) is also known as broad bean one of the most important crops in the world, and fourth on the level of legumes in importance [1] [2], and it is being grown around the world as a food source [3]. The central habitat of faba bean plant is the Middle East, where legumes are generally a staple food there. Their high protein content can be replacement meat for reduced income people [3] [4]. They also are cover crops that improve soil fertility and play an important role in nitrogen fixation, because of their symbiosis between faba bean and Rhizobium bacteria. Furthermore, the other plants can benefit from this [5] [6].
The seeds defined as a young embryonic plant in the state of dormancy, and when factors are available, they sprout. Some environmental conditions such as oxygen, temperature, light, and water required for seed germination [7], but there is a factor for the size of the seed that affects germination rates and speed, which is one of the most important factors affecting seed growth [8], germination speed, crop strength, and seed vigor [9], seed classification according to their size are popular methods in agriculture, as many scientists agree that the seed size is associated with increased growth and the strength of the crop. Therefore, it is essential to know the effect of seed size on germination.
The seeds affect the germination percentage at the time of germination; the seed size in legumes affects cereal production [10]. Usually, small seeds are germination faster, but larger seeds have a higher germination rate and higher growth and greater yield [1] due to the availability of more food, and it is known that the amount of food is related to the size of the seeds [11]. There is a positive relationship between seed size and seedling growth [12] and also in crop improvement [10]. The amount of starch varies due to the different seed sizes, and the amount of starch may be associated with germination [13].
It has observed that there is an effect on the difference in the size of Afzelia africana seeds on germination [14]. This study aims to investigate the effect of faba bean seed size on the germination percentage and some growth vigor of the plant.
The experiment conducted in the Qassim region located in the north-central of the Kingdom of Saudi Arabia between February and March 2020. The total precipitation rain was 0 mm, and the average temperature was 20˚C.
The soil used in the experiment was sandy. Seeds Vicia faba L. obtained from the seed market in the Qassim region. Seeds were divided manually into three groups by using the ruler based on length and width; the large seeds ranged between 2.5 - 2.3 cm, the medium seeds ranged between 2.0 - 2.2 cm, and the small seeds ranged between 1.7 - 1.9 cm was 20 large Seeds, 20 medium seeds and 20 small Seeds. The groups were soaked in water from the tap for 24 hours separately. After that, they germinated in sterile Petri dishes on filter paper moistened with water. Petri dishes were put in the dark at a temperature of 25˚C for seven days and during these days get wet with water.
The sand soil filled in the 30 plastic packages (12 cm in diameter and 9 cm in height) that have perforated holes in the base to drain the excess water by 15 g per package, then the germinated seeds were planted at a depth of 3.0 cm and watering once a day after two-week was calculated germination percentage according to the following equation
\text{Germination percentage}\text{\hspace{0.17em}}\text{\%}=\frac{\text{number}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}\text{germinated}\text{\hspace{0.17em}}\text{seeds}}{\text{number}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}\text{total}\text{\hspace{0.17em}}\text{seeds}\text{\hspace{0.17em}}\text{planted}}\times 100
[9]
Moreover, it took root length, shoot length, and the number of leaves.
The analysis of variance was carried out for the results using the least significant difference (LSD) at 0.05 probability levels.
Seed germination is mainly affected by the seed size. A study has carried out on three different seed sizes: small, medium, and large. 100% of the large size seeds appeared at the end of the experiment, while the medium and the small size seeds had a lower appearance percentage of 85% and 70%, respectively. The seeds size also affects the apparition of seedlings. In the study, the large and medium size seeds took four days to appear, while the small size seeds took only two days to appear.
3.2. Root and Shoot Length
The study shows that the root length was affected by the size of the seed. The root length of the large seeds is longer than the other seeds, and the root of the medium seeds was longer than the small seeds’ roots, as presented in Figure 1 and Figure 2. While Table 1 shows that there are significant differences (P ≤ 0.05) between the root lengths of the large seeds and the medium seeds and between the large seeds and the small seeds and between the medium seeds and the small seeds.
Similarly, for the shoot length, as shown in Figure 2 and Figure 3, the shoot lengths of the large seeds were longer than the others, and the shoots of the medium seeds were in between them. The significant differences (P ≤ 0.05), as shown in Table 1, were found between all seed sizes.
3.3. Number of Leaves
The number of leaves varied according to the size of the seed. There is a positive relationship between seed size and the number of leaves, as shown in Figure 4.
Figure 1. The root length (cm) of faba bean plant after two weeks from planting.
Figure 2. 1: Seedlings of large seed; 2: Seedlings of medium seed; 3: Seedling the smallseed. After two weeks from planting.
Figure 3. The shoot length (cm) of faba bean plant after two weeks from planting.
Figure 4. The number of leaves of faba bean plant after two weeks from planting.
Table 1. Effect of seed size on root length (cm), shoot length (cm) and number of leaves.
The large seeds have more leaves than the other seeds, while the medium seeds have a lower number of leaves than the large seeds and more than the small ones. The significant effect (P ≤ 0.05) on the number of leaves is similar to the root length and shoot length, as shown in Table 1, found significant differences between all seed sizes.
Morphological parameters of faba bean plants, such as root length, shoot length, and the number of leaves, appear higher in plants grown from large seeds compared to plants with other seed sizes. Small seeds germinate faster than medium and large seeds, so there is an inverse relationship between germination speed and seed size. The smaller the size of the seeds, the faster the germination. The result is consistent with this research [7] on Prosopis africana seeds. The germination percentage in large seeds was higher than medium and small seeds, there consistent with [15], but there was no effect of the difference in the size of wheat seeds in the germination percentage [9].
This study showed that the length of roots is affected by the size of the seeds, similar to the research [8], where the roots in large seeds were much longer than small seeds and medium seeds. The result indicates that the seed size affects the root length. Larger seed size may have resulted in higher germination due to increased endosperm.
There is a difference between the plant length; it is clear that the shoot length is also the effect by seed sizes, a decrease in the shoot length with a decrease in the seed size similar to the research [8] [16], so the shoot length was shorter with small seeds than the shoot length in large and medium seeds, maybe that large seeds because they have more energy stored to support the seedling growth.
Large seeds produce more leaves than medium and small seeds. Hence, there is a correlation between faba bean seed size and the number of leaves, and the results of the study are in accordance with [16] in seeds peanuts and Sunflower in [17].
Results of the current work revealed that the germination percentage in the case of small seeds was faster than that of medium and large seeds, but the large seeds give seedlings longer than medium and small seeds. The results indicate that the size of seeds affects the germination percentage and the length of seedlings. Thus, seed size has a role in improving seed germination.
Cite this paper: Alngiemshy, N. , Alkharafi, J. , Alharbi, N. and Al-Sowayan, N. (2020) Effect of Seeds Size on Germination of Faba Bean Plant. Agricultural Sciences, 11, 465-471. doi: 10.4236/as.2020.115028.
[1] Al-Rifaee, M.O.H.D., Turk, M.A. and Tawaha, A.R.M. (2004) Effect of Seed Size and Plant Population Density on Yield and Yield Components of Local Faba Bean (Vicia faba L. Major). International Journal of Agriculture and Biology, 6, 294-299.
[2] López-Bellido, F.J., López-Bellido, L. and López-Bellido, R.J. (2005) Competition, Growth and Yield of Faba Bean (Vicia faba L.). European Journal of Agronomy, 23, 359-378.
[3] Etemadi, F., Hashemi, M., Zandvakili, O. and Mangan, F.X. (2018) Phenology, Yield and Growth Pattern of Faba Bean Varieties. International Journal of Plant Production, 12, 243-250.
[4] Qabil, N., Helal, A.A., El-Khalek, A. and Rasha, Y.S. (2018) Evaluation of Some New and Old Faba Bean Cultivars (Vicia faba L.) for Earliness, Yield, Yield Attributes and Quality Characters. Zagazig Journal of Agricultural Research, 45, 821-833.
https://doi.org/10.21608/zjar.2018.49119
[5] Pereira, S., Mucha, Â., Goncalves, B., Bacelar, E., Látr, A., Ferreira, H., Oliveira, I., Rosa, E. and Marques, G. (2019) Improvement of Some Growth and Yield Parameters of Faba Bean (Vicia faba) by Inoculation with Rhizobium laguerreae and Arbuscular Mycorrhizal Fungi. Crop & Pasture Science, 70, 595-605.
[6] Kharrat, M., Ben Salah, H. and Halila, H.M. (1991) Faba Bean Status and Prospects in Tunisia. In: Cubero, J.I. and Saxena, M.C., Eds., Present Status and Future Prospects of Faba Bean Production and Improvement in the Mediterranean Countries, CIHEAM, Zaragoza, 169-172.
[7] Dera, B.A., Agera, S.I.N. and Ezugwu, E.U. (2019) Effect of Seed Size and Acid Scarification on Germination and Early Growth of Prosopis africana. Journal of Global Biosciences, 8, 5774-5788.
[8] Ali, S.A. and Idris, A.Y. (2015) Effect of Seed Size and Sowing Depth on Germination and Some Growth Parameters of Faba Bean (Vicia faba L.). Agricultural and Biological Sciences Journal, 1, 1-5.
[9] Baysah, N.S., Olympio, N.S. and Asibuo, J.Y. (2018) Influence of Seed Size on the Germination of Four Cowpea (Vigna unguiculata (L) Walp) Varieties. ISABB Journal of Food and Agricultural Sciences, 8, 25-29.
[10] Adebisi, M.A., Kehinde, T.O., Salau, A.W., Okesola, L.A., Porbeni, J.B.O., Esuruoso, A.O. and Oyekale, K.O. (2013) Influence of Different Seed Size Fractions on Seed Germination, Seedling Emergence and Seed Yield Characters in Tropical Soybean (Glycine max L. Merrill). International Journal of Agricultural Research, 8, 26-33.
[11] Souza, M.L. and Fagundes, M. (2014) Seed Size as Key Factor in Germination and Seedling Development of Copaifera langsdorffii (Fabaceae). American Journal of Plant Sciences, 5, 2566-2573.
[12] Neugschwandtner, R.W., Papst, S., Kemetter, J., Wagentristl, H., Sedlár, O. and Kaul, H.P. (2019) Effect of Seed Size on Soil Cover, Yield, Yield Components and Nitrogen Uptake of Two-Row Malting Barley. Die Bodenkultur: Journal of Land Management, Food and Environment, 70, 89-98.
[13] Ahirwar, J.R. (2012) Effect of Seed Size and Weight on Seed Germination of Alangium lamarckii, Akola, India. Research Journal of Recent Sciences, 1, 320-322.
[14] Folake, A.B. and Olusola, O.A. (2020) Effect of Seed Size on Afzelia africana (Smith) Germination. International Journal of Agriculture, Forestry and Fisheries, 8, 1.
[15] Umeoka, N. and Ogbonnaya, C.I. (2016) Effects of Seed Size and Sowing Depth on Seed Germination and Seedling Growth of Telfairia occidentalis (Hook F.). International Journal of Advances in Chemical Engineering & Biological Sciences, 3, 201-207.
https://doi.org/10.15242/IJACEBS.AE0916207
[16] Olayinka, U.B., Owodeyi, S.O. and Etejere, E.O. (2016) Biological Productivity and Composition of Groundnut in Relation to Seed Size. Environmental and Experimental Biology, 14, 9-14.
https://doi.org/10.22364/eeb.14.02
[17] Ahmed, T.A.M., Mutwali, E.M. and Salih, E.A. (2019) The Effect of Seed Size and Burial Depth on the Germination, Growth and Yield of Sunflower (Helianthus annus L.). American Scientific Research Journal for Engineering, Technology, and Sciences, 53, 75-82.
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Tacit_collusion Knowpia
Tacit collusion is a collusion between competitors, which do not explicitly exchange information and achieving an agreement about coordination of conduct.[1] There are two types of tacit collusion - concerted action and conscious parallelism.[2][3] In a concerted action also known as concerted activity,[4] competitors exchange some information without reaching any explicit agreement, while conscious parallelism implies no communication.[1][5] In both types of tacit collusion, competitors agree to play a certain strategy without explicitly saying so. It is also referred to as oligopolistic price coordination[6] or tacit parallelism.[7]
A dataset of gasoline prices of BP, Caltex, Woolworths, Coles, and Gull from Perth gathered in the years 2001 to 2015 was used to show by statistical analysis the tacit collusion between these retailers.[8] BP emerged as a price leader and influenced the behavior of the competitors. As result, the timing of price jumps became coordinated and the margins started to grow in 2010.
Conscious parallelismEdit
In competition law, some sources use conscious parallelism as a synonym to tacit collusion in order to describe pricing strategies among competitors in an oligopoly that occurs without an actual agreement[9] or at least without any evidence of an actual agreement between the players.[10] In result, one competitor will take the lead in raising or lowering prices. The others will then follow suit, raising or lowering their prices by the same amount, with the understanding that greater profits result.
This practice can be harmful to consumers who, if the market power of the firm is used, can be forced to pay monopoly prices for goods that should be selling for only a little more than the cost of production. Nevertheless, it is very hard to prosecute because it may occur without any collusion between the competitors. Courts have held that no violation of the antitrust laws occurs where firms independently raise or lower prices, but that a violation can be shown when plus factors occur, such as firms being motivated to collude and taking actions against their own economic self-interests.[11][12] This procedure of the courts is sometimes called as setting of a conspiracy theory.[13]
Price leadershipEdit
Oligopolists usually try not to engage in price cutting, excessive advertising or other forms of competition. Thus, there may be unwritten rules of collusive behavior such as price leadership. Price leadership is the form of a tacit collusion, whereby firms orient at the price set by a leader.[14] A price leader will then emerge and set the general industry price, with other firms following suit. For example, see the case of British Salt Limited and New Cheshire Salt Works Limited.[15]
Classical economic theory holds that Pareto efficiency is attained at a price equal to the incremental cost of producing additional units. Monopolies are able to extract optimum revenue by offering fewer units at a higher cost. An oligopoly where each firm acts independently tends toward equilibrium at the ideal, but such covert cooperation as price leadership tends toward higher profitability for all, though it is an unstable arrangement.
There exist two types of price leadership.[14] In dominant firm price leadership, the price leader is the biggest firm. In barometric firm price leadership, the most reliable firm emerges as the best barometer of market conditions, or the firm could be the one with the lowest costs of production, leading other firms to follow suit. Although this firm might not be dominating the industry, its prices are believed to reflect market conditions which are the most satisfactory, as the firm would most likely be a good forecaster of economic changes.
Tacit collusion in auctionsEdit
In repeated auctions, bidders might participate in a tacit collusion to keep bids low.[16] A profitable collusion is possible, if the number of bidders is finite and the identity of the winner is publicly observable. It can be very difficult or even impossible for the seller to detect such collusion from the distribution of bids only. In case of spectrum auctions, some sources claim that a tacit collusion is easily upset:[17]
"It requires that all the bidders reach an implicit agreement about who should get what. With thirty diverse bidders unable to communicate about strategy except through their bids, forming such unanimous agreement is difficult at best."
Nevertheless, Federal Communications Commission (FCC) experimented with precautions for spectrum auctions like restricting visibility of bids, limiting the number of bids and anonymous bidding.[18] So called click-box bidding used by governmental agencies in spectrum auctions restricts the number of valid bids and offers them as a list to a bidder to choose from.[19] Click-box bidding was invented in 1997 by FCC to prevent bidders from signalling bidding information by embedding it into digits of the bids.[20] Economic theory predicts a higher difficulty for tacit collusions due to those precautions.[18] In general, transparency in auctions always increases the risk of a tacit collusion.[21]
Algorithmic tacit collusionEdit
Once the competitors are able to use algorithms to determine prices, a tacit collusion between them imposes a much higher danger.[22] E-commerce is one of the major premises for algorithmic tacit collusion.[23] Complex pricing algorithms are essential for the development of e-commerce.[23] European Commissioner Margrethe Vestager mentioned an early example of algorithmic tacit collusion in her speech on "Algorithms and Collusion" on March 16, 2017, described as follow:[24]
"A few years ago, two companies were selling a textbook called The Making of a Fly. One of those sellers used an algorithm which essentially matched its rival’s price. That rival had an algorithm which always set a price 27% higher than the first. The result was that prices kept spiralling upwards, until finally someone noticed what was going on, and adjusted the price manually. By that time, the book was selling – or rather, not selling – for 23 million dollars a copy."
The book "The Making of a Fly" by Peter Anthony Lawrence written in 1992 achieved indeed a price of $23,698,655.93 on Amazon in 2011.[25] An OECD Competition Committee Roundtable "Algorithms and Collusion" took place in June 2017 in order to address the risk of possible anti-competitive behaviour by algorithms.[26]
It is important to distinguish between simple algorithms intentionally programmed to raise price according to the competitors and more sophisticated self-learning AI algorithms with more general goals. Self-learning AI algorithms might form a tacit collusion without the knowledge of their human programmers as result of the task to determine optimal prices in any market situation.[22][27]
Duopoly exampleEdit
Tacit collusion is best understood in the context of a duopoly and the concept of game theory (namely, Nash equilibrium). Let's take an example of two firms A and B, who both play an advertising game over an indefinite number of periods (effectively saying 'infinitely many'). Both of the firms' payoffs are contingent upon their own action, but more importantly the action of their competitor. They can choose to stay at the current level of advertising or choose a more aggressive advertising strategy. If either firm chooses low advertising while the other chooses high, then the low-advertising firm will suffer a great loss in market share while the other experiences a boost. However, if they both choose high advertising, then neither firms' market share will increase but their advertising costs will increase, thus lowering their profits. If they both choose to stay at the normal level of advertising, then sales will remain constant without the added advertising expense. Thus, both firms will experience a greater payoff if they both choose normal advertising (however this set of actions is unstable, as both are tempted to defect to higher advertising to increase payoffs). A payoff matrix is presented with numbers given:
Firm B normal advertising
Firm B aggressive advertising
Firm A normal advertising
Each earns $50 profit Firm A: $0 profit
Firm B: $80 profit
Firm A aggressive advertising
Firm A: $80 profit
Firm B: $0 profit Each earns $15 profit
Notice that Nash's equilibrium is set at both firms choosing an aggressive advertising strategy. This is to protect themselves against lost sales. This game is an example of a prisoner's dilemma.
In general, if the payoffs for colluding (normal, normal) are greater than the payoffs for cheating (aggressive, aggressive), then the two firms will want to collude (tacitly). Although this collusive arrangement is not an equilibrium in the one-shot game above, repeating the game allows the firms to sustain collusion over long time periods. This can be achieved, for example if each firm's strategy is to undertake normal advertising so long as its rival does likewise, and to pursue aggressive advertising forever as soon as its rival has used an aggressive advertising campaign at least once (see: grim trigger) (this threat is credible since symmetric use of aggressive advertising is a Nash equilibrium of each stage of the game). Each firm must then weigh the short term gain of $30 from 'cheating' against the long term loss of $35 in all future periods that comes as part of its punishment. Provided that firms care enough about the future, collusion is an equilibrium of this repeated game.
To be more precise, suppose that firms have a discount factor
{\displaystyle \delta }
. The discounted value of the cost to cheating and being punished indefinitely are
{\displaystyle \sum _{t=1}^{\infty }\delta ^{t}35={\frac {\delta }{1-\delta }}35}
The firms therefore prefer not to cheat (so that collusion is an equilibrium) if
{\displaystyle 30<{\frac {\delta }{1-\delta }}35\Leftrightarrow \delta >{\frac {6}{13}}}
^ a b Harrington, Joseph E. (2012). "A theory of tacit collusion" (PDF). Retrieved 24 March 2021. {{cite journal}}: Cite journal requires |journal= (help)
^ Vaska, Michael K. (1985). "Conscious Parallelism and Price Fixing: Defining the Boundary". University of Chicago Law Review. 52 (2): 508–535. doi:10.2307/1599667. JSTOR 1599667. Retrieved 27 March 2021.
^ Shulman, Daniel R. (2006–2007). "Matsushita and the Role of Economists with Regard to Proof of Conspiracy". Loyola University Chicago Law Journal. 38: 497. Retrieved 27 March 2021.
^ Buccirossi, Paolo (2008). "Facilitating practices" (PDF). Handbook of Antitrust Economics. 1: 305–351. Retrieved 27 March 2021.
^ Page, William H. (2007). "Communication and Concerted Action". Loyola University Chicago Law Journal. 38: 405. Retrieved 24 March 2021.
^ Fzrachi, Ariel; Stucke, Maurice E. (2019). "Sustainable and Unchallenged Algorithmic Tacit Collusion". Northwestern Journal of Technology and Intellectual Property. 17: 217.
^ Markham, Jesse W. (1951). "The Nature and Significance of Price Leadership". The American Economic Review. 41 (5): 891–905. ISSN 0002-8282. JSTOR 1809090. Retrieved 25 April 2021.
^ "Oligopolies, Conscious Parallelism and Concertatio" (PDF). Archived from the original (PDF) on 2007-07-08. Retrieved 2012-02-26.
^ Ezrachi, Ariel; Stucke, Maurice E. (2017). "Emerging Antitrust Threats and Enforcement Actions in the Online World". Competition Law International. 13: 125.
^ Marks, Randall David (1986). "Can Conspiracy Theory Solve the Oligopoly Problem". Maryland Law Review. 45: 387. Retrieved 16 March 2021.
^ Kevin Scott Marshall, Stephen H. Kalos, The Economics of Antitrust Injury and Firm-specific Damages (2008), p. 228.
^ Rogers III, C. P. (1978). "Summary Judgements in Antitrust Conspiracy Litigation". Retrieved 16 March 2021. {{cite journal}}: Cite journal requires |journal= (help)
^ a b Sloman, John (2006). Economics. Financial Times Prentice Hall. ISBN 978-0-273-70512-3. Retrieved 2 May 2021.
^ Commission, Great Britain: Competition (30 November 2005). British Salt Limited and New Cheshire Salt Works Limited: A Report on the Acquisition by British Salt Limited of New Cheshire Salt Works Limited. The Stationery Office. ISBN 978-0-11-703625-3. Retrieved 2 May 2021.
^ Skrzypacz, Andrzej; Hopenhayn, Hugo (2001). "Tacit Collusion in Repeated Auctions" (PDF). Retrieved 16 April 2021. {{cite journal}}: Cite journal requires |journal= (help)
^ Compte, Olivier (1998). "Communication in Repeated Games with Imperfect Private Monitoring". Econometrica. 66 (3): 597–626. doi:10.2307/2998576. ISSN 0012-9682. JSTOR 2998576. Retrieved 16 April 2021.
^ a b Bajari, Patrick; Yeo, Jungwon (1 June 2009). "Auction design and tacit collusion in FCC spectrum auctions". Information Economics and Policy. 21 (2): 90–100. doi:10.1016/j.infoecopol.2009.04.001. ISSN 0167-6245.
^ "Bundesnetzagentur stärkt Deutschland als Leitmarkt für 5G" (PDF). Bundesnetzagentur. Retrieved 17 April 2021.
^ Commission, United States Federal Communications (June 2008). FCC Record: A Comprehensive Compilation of Decisions, Reports, Public Notices, and Other Documents of the Federal Communications Commission of the United States. Federal Communications Commission. Retrieved 17 April 2021.
^ Bichler, Martin; Gretschko, Vitali; Janssen, Maarten (1 June 2017). "Bargaining in spectrum auctions: A review of the German auction in 2015". Telecommunications Policy. 41 (5–6): 325–340. doi:10.1016/j.telpol.2017.01.005. hdl:10419/145809. ISSN 0308-5961. Retrieved 17 April 2021.
^ a b Ezrachi, A.; Stucke, M. E. (13 March 2020). "Sustainable and unchallenged algorithmic tacit collusion". Northwestern Journal of Technology & Intellectual Property. 17 (2). ISSN 1549-8271.
^ a b "Horizontal Restraint Regulations in the EU and the US in the Era of Algorithmic Tacit Collusion". Journal of Law and Jurisprudence. 13 June 2018. doi:10.14324/111.2052-1871.098.
^ VESTAGER, Margrethe (2017). "Algorithms and competition". European Commission. Archived from the original (Bundeskartellamt 18th Conference on Competition) on 2019-11-29. Retrieved 1 May 2021.
^ "How A Book About Flies Came To Be Priced $24 Million On Amazon". Wired. Retrieved 1 May 2021.
^ "Algorithms and Collusion: Competition Policy in the Digital Age" (PDF). OECD. Retrieved 1 May 2021.
^ Hutchinson, Christophe Samuel; Ruchkina, Gulnara Fliurovna; Pavlikov, Sergei Guerasimovich (2021). "Tacit Collusion on Steroids: The Potential Risks for Competition Resulting from the Use of Algorithm Technology by Companies". Sustainability. 13 (2): 951. doi:10.3390/su13020951.
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How to use the length of a rectangle calculator?
How do I calculate the length of a rectangle?
Welcome to the length of a rectangle calculator, where we'll explain the formula(s) for the length of a rectangle and how to find the length of a rectangle.
Using the length of a rectangle calculator is easy — there's only two steps!
Enter the dimensions that you know of the rectangle.
Find your rectangle length in the bottom box.
The length of a rectangle calculator works both ways — try changing the rectangle's length and see how its other dimensions are affected.
Not enough to just know how to use the length of a rectangle calculator? Read on to learn how to calculate the length of a rectangle.
Depending on what information you have available, there are many ways to calculate the length of a rectangle.
If you have the area A and width w, its length w is determined as h = A/w.
If you have the perimeter P and width w, its length can be found with h = P/2−w.
If you have the diagonal d and width w, it's length is h = √(d²−w²).
If the rectangle's width is not known, you'd need to simultaneously solve the equations above to get the length h.
That's a lot of different formulas for the length of a rectangle! These are all derived from the many formulas that govern a rectangle's dimensions. Those formulas are:
\begin{split} A &= w \times \textcolor{red}h \\ P &= 2(w+\textcolor{red}h) \\ d &= \sqrt{w^2 + \textcolor{red}h^2} \end{split}
w
is the rectangle's width;
h
is the rectangle's length;
A
is the rectangle's area;
P
is the rectangle's perimeter; and
is the length of the rectangle's diagonal, as described by the Pythagorean theorem.
Here's a neat visual:
A rectangle with its length
w
, diagonal
d
, perimeter
P
, and area
A
labelled.
If the length of a rectangle calculator isn't quite what you want, try out our other rectangular calculators:
A rectangle has four sides. Its sides are paired, so really there are only two unique dimensions. Conventionally, the rectangle's length is the longest of these two measurements, but when the rectangle is shown to be standing on the floor, the vertical side is usually called the length.
What is the length of a rectangle with diagonal 5 m and width 3 m?
4 m. Because the connected sides of a rectangle are perpendicular, we can use the Pythagoras theorem to work this one out.
Rearrange the Pythagoras theorem to make the rectangle's length h the subject: h = √(d²−w²).
Plug in your values: h = √(5²−3²) = √(25-9) = √16 = 4.
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The elements of a cone
How to find the slant height of a cone
How to calculate the slant height of cones in other ways
How to use our slant height of a cone calculator
An all-around shape
Our slant height of a cone calculator will answer one of the most pressing questions you may ever have: how long is the side of an ice cream cone?
Here you will learn everything you need about cones and their oblique sides, from the base to the apex:
Which are the elements of a cone?
How to find the slanted height of a cone?
A cone is a solid figure originated by a complete rotation of a right triangle around one of its catheti.
The resulting solid has a circular base and a continuous slant side that connects the edge of the base to the topmost point of the cone, the apex.
When we talk of cones, we usually mean a half-cone: the noun cone refers to a figure obtained by rotating for 180° a tilted line, creating two open "conical" shapes. However, in this tool, we will talk only about the former kind.
A cone is identified by:
A circular base with radius
r
A height
h
perpendicular to the base.
We can identify another element, though:
A slant side called slant height of the cone,
l
A cone with all its elements marked.
In the figure, you can easily identify all of the elements mentioned above.
Here we will teach you how to calculate the slant height of a cone. It is pretty easy, you'll see!
The slant height of a cone is nothing but the hypotenuse of the right triangle that generates the cone. We can apply the Pythagorean theorem to calculate the slant height of any cone, knowing the radius and the height.
The formula for the slant height of a cone is:
l = \sqrt{r^2+h^2}
What if you don't know radius and height, but only one of them, plus the angle at the base of the cone? We need to apply some basic trigonometry!
Calling the angle at the base of the cone
\alpha
, and the angle at the apex
\beta
, two formulas tell you how to find the slant height of a cone:
\begin{align*} &l = \frac{h}{\sin{\alpha}}\\ \\ &l = \frac{r}{\sin{\beta}} \end{align*}
Using our tool will help you apply the formula for the slant height of a cone whenever you need it!
To use our tool, simply insert the values you have at hand, and find out the value of the slanted height.
What about some examples now? Let's talk ice cream: imagine (or go get!) an ice cream cone with radius
2.5\ \text{cm}
15\ \text{cm}
. Insert these values in the appropriate fields of the slant height of a cone calculator. It will apply the formula:
\begin{align*} l &= \sqrt{r^2+h^2}\\ &=\sqrt{2.5^2+15^2}\ \text{cm}\\ &= 15.2\ \text{cm} \end{align*}
Pretty similar to the height? The ice cream cone is a pretty slender one!
What about our other favorite cones, the traffic cones? Take a
28\ \text{in}
height cone (the best type). Measure its diameter at the base. It can be, let's say,
10.5\ \text{in}
. Change the units in our tool, and find out the slant height:
\begin{align*} l &= \sqrt{r^2+h^2}\\ &=\sqrt{10.5^2+28^2}\ \text{in} \\ &= 28.49\ \text{in} \end{align*}
We didn't stop at the slant height of a cone: check out our other tools dedicated to that pointy shape!
What is the slant height of a cone?
The slant height of a cone is the measure of the segment connecting the apex of a cone to the outer rim of its base. It corresponds to the length of the hypotenuse of the right triangle that generates the cone itself.
How do I calculate the slant height of a cone?
If you know the height and the radius of a cone, you must apply the Pythagorean theorem to find the length of the slant height of a cone.
Measure the height and radius of your cone;
Apply the formula `l = sqrt(r² + h²), where:
l is the slant height of the cone;
r is the radius of the base; and
What is the slant height of a cone with radius 10 cm and height 20 cm?
The slant height of such a cone is 22.36 cm. Apply the slant height of a cone formula to find the length of a cone with base radius r = 10 cm and height h = 20 cm:
l = sqrt(r² + h²) = sqrt(10² + 20²) = sqrt(500) = 22.36 cm
What is the slant height of Mount Fuji?
The slant height of Mount Fuji, assuming it to be a geometric cone, is 22.82 km. The volcano has an average base radius r = 22.5 km and we can approximate its height to h = 3.8 km. To climb up Mount Fuji on a straight line, you have to walk:
l = sqrt(r² + h²) = sqrt(22.5² +3.8²) = 22.82 km
Surprisingly, this is pretty close to the length of the actual trail!
Apex angle (ß)
The adjoint matrix calculator will help you determine the adjugate of a square matrix.
The scalene triangle area calculator helps find the area of your scalene triangle.
Scalene Triangle Area Calculator
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Tune Phase-Locked Loop Using Loop-Shaping Design - MATLAB & Simulink - MathWorks 日本
A PLL is a closed-loop system that produces an output signal whose phase depends on the phase of its input signal. The following diagram shows a simple model with a PLL reference architecture block (Integer N PLL with Single Modulus Prescaler) and a PLL Testbench block.
The Mixed-Signal Blockset library provides multiple reference architecture blocks to design and simulate PLL systems in Simulink®. You can tune the components of the Loop Filter block, which is a passive filter, to get the desired open-loop bandwidth and phase margin.
Using the Control System Toolbox software, you can specify the shape of the desired loop response and tune the parameters of a fixed-structure controller to approximate that loop shape. For more information on specifying a desired loop shape, see Loop Shape and Stability Margin Specifications (Control System Toolbox). In the preceding PLL architecture model, the loop filter is defined as a fixed-order, fixed-structure controller. To achieve the target loop shape, the values of the resistances and capacitances of the loop filter are tuned. Doing so improves the open-loop bandwidth of the system and, as a result, reduces the measured lock time.
The PLL block uses the configuration specified in Design and Evaluate Simple PLL Model for the PFD, Charge pump, VCO, and Prescalar tabs in the block parameters. The Loop Filter tab specifies the type as a fourth-order filter, and sets the loop bandwidth to 100 kHz and phase margin to 60 degrees. The values for the resistances and capacitances are automatically computed.
To model the loop filter as a tunable element, first create tunable scalar real parameters (see realp (Control System Toolbox)) to represent each filter component. For each parameter, define the initial value and bounds. Also, specify whether the parameter is free to be tuned.
Using these tunable parameters, create a custom tunable model based on the loop filter transfer function equation specified in the More About section of the Loop Filter block reference page. loopFilterSys is a genss (Control System Toolbox) model parameterized by R2, R3, R4, C1, C2, C3, and C4.
\begin{array}{l}Z\left(s\right)=\frac{{R}_{2}{C}_{2}s+1}{s\left({A}_{4}{s}^{3}+{A}_{3}{s}^{2}+{A}_{2}s+{A}_{1}\right)}\\ \\ {A}_{4}={C}_{1}{C}_{2}{C}_{3}{C}_{4}{R}_{2}{R}_{3}{R}_{4}\\ {A}_{3}={C}_{1}{C}_{2}{R}_{2}{R}_{3}\left({C}_{3}+{C}_{4}\right)+{C}_{4}{R}_{4}\left({C}_{2}{C}_{3}{R}_{3}+{C}_{1}{C}_{3}{R}_{3}+{C}_{1}{C}_{2}{R}_{2}+{C}_{2}{C}_{3}{R}_{2}\right)\\ {A}_{2}={C}_{2}{R}_{2}\left({C}_{1}+{C}_{3}+{C}_{4}\right)+{R}_{3}\left({C}_{1}+{C}_{2}\right)\left({C}_{3}+{C}_{4}\right)+{C}_{4}{R}_{4}\left({C}_{1}+{C}_{2}+{C}_{3}\right)\\ {A}_{1}={C}_{1}+{C}_{2}+{C}_{3}+{C}_{4}\end{array}
Define input and output names for each block. Connect the elements based on signal names (see connect (Control System Toolbox)) to create a tunable closed-loop system (see genss) representing the PLL architecture as shown.
Observe the current open-loop shape of the PLL system with reference to the target loop shape. S represents the inverse sensitivity function and T represents the complementary sensitivity function. By default, Control System Toolbox plots use rad/s as the frequency unit. For more information on how to change the frequency unit to Hz, see Toolbox Preferences Editor (Control System Toolbox).
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Tensor Product Calculator (Kronecker Product)
Tensor Product Calculator
What is the tensor product of matrices?
How to calculate the Kronecker product?
Tensor product of 2x2 matrices
What is the formula for the Kronecker matrix product?
How to use this tensor product calculator?
If you have just stumbled upon this bizarre matrix operation called matrix tensor product or Kronecker product of matrices, look for help no further — Omni's tensor product calculator is here to teach you all you need to know about:
What the Kronecker product is;
What the main properties of Kronecker product are;
How to calculate tensor product of 2x2 matrices by hand; and
What the most general Kronecker product formula looks like.
As a bonus, we'll explain the relationship between the abstract tensor product vs the Kronecker product of two matrices!
⚠️ The Kronecker product is not the same as the usual matrix multiplication! If you're interested in the latter, visit Omni's matrix multiplication calculator.
Matrix tensor product, also known as Kronecker product or matrix direct product, is an operation that takes two matrices of arbitrary size and outputs another matrix, which is most often much bigger than either of the input matrices.
Let's say the input matrices are:
A
r_A
c_A
columns, and
B
r_B
c_B
The resulting matrix then has
r_A \cdot r_B
c_A \cdot c_B
🔎 In particular, we can take matrices with one row or one column, i.e., vectors (whether they are a column or a row in shape). In this case, we call this operation the vector tensor product.
Once we have a rough idea of what the tensor product of matrices is, let's discuss in more detail how to compute it. The Kronecker product is defined as the following block matrix:
\footnotesize A \otimes B = \! \begin{bmatrix} a_{11} {B} & \cdots & a_{1c_A} {B} \\ \vdots &\ddots &\vdots \\a_{r_A1}{B} &\cdots &a_{r_Ac_A}{B} \end{bmatrix}
Hence, calculating the Kronecker product of two matrices boils down to performing a number-by-matrix multiplication many times. As you surely remember, the idea is to multiply each term of the matrix by this number while keeping the matrix shape intact:
\footnotesize a_{ij} B =\! \begin{bmatrix} a_{ij} b_{11} & \cdots & a_{ij} b_{1c_B} \\ \vdots &\ddots &\vdots \\a_{ij}b_{r_B1} &\cdots &a_{ij} b_{r_Bc_B} \end{bmatrix}
Let's discuss what the Kronecker product is in the case of 2x2 matrices to make sure we really understand everything perfectly. Suppose that
\footnotesize A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}\!,\ B = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix}
As we saw above, we have:
\footnotesize A \otimes B = \begin{bmatrix} a_{11} {B} & a_{12} {B} \\a_{21}{B} &a_{22}{B} \end{bmatrix}
Writing the terms of
B
explicitly, we obtain:
\footnotesize A \otimes B = \\ \begin{bmatrix} a_{11} \begin{bmatrix} b_{11} & b_{12} \\b_{21} &b_{22} \end{bmatrix} & a_{12}\begin{bmatrix} b_{11} & b_{12} \\b_{21} &b_{22} \end{bmatrix} \\a_{21}\begin{bmatrix} b_{11} & b_{12} \\b_{21} &b_{22} \end{bmatrix} &a_{22}\begin{bmatrix} b_{11} & b_{12} \\b_{21} &b_{22} \end{bmatrix} \end{bmatrix}
Performing the number-by-matrix multiplication, we arrive at the final result:
\footnotesize A \otimes B = \\ \begin{bmatrix} a_{11} b_{11} & a_{11}b_{12} & a_{12} b_{11} & a_{12} b_{12} \\ a_{11}b_{21} & a_{11}b_{22} & a_{12}b_{21} & a_{12}b_{22} \\ a_{21} b_{11} & a_{21}b_{12} & a_{22} b_{11} & a_{22} b_{12} \\ a_{21}b_{21} & a_{21}b_{22} & a_{22} b_{21} & a_{22} b_{22} \end{bmatrix}
Hence, the tensor product of 2x2 matrices is a 4x4 matrix. It is not hard at all, is it? But you can surely imagine how messy it'd be to explicitly write down the tensor product of much bigger matrices! Fortunately, there's a concise formula for the matrix tensor product — let's discuss it!
We can compute the element
(A\otimes B)_{ij}
of the Kronecker product as:
\footnotesize a_{\lceil i/r_B\rceil ,\lceil j/c_B\rceil } \cdot b_{\left((i-1)\% r_B+1\right),\left((j-1)\% c_B+1\right)}
\lceil x \rceil
is the ceiling function (i.e., it's the smallest integer that is greater than
x
\%
denotes the modulo operation. Recall also that
r_B
c_B
stand for the number of rows and columns of
B
We have discussed two methods of computing tensor matrix product. There's a third method, and it is our favorite one — just use Omni's tensor product calculator!
To compute the Kronecker product of two matrices with the help of our tool, just pick the sizes of your matrices and enter the coefficients in the respective fields.
🙋 Oops, you've messed up the order of matrices? No worries — our tensor product calculator allows you to choose whether you want to multiply
A \otimes B
B \otimes A
. Enjoy!
Tensor matrix product is associative, i.e., for every
A, B, C
\footnotesize ({A} \otimes {B} )\otimes {C} ={A} \otimes ({B} \otimes {C})
Tensor matrix product is also bilinear, i.e., it is linear in each argument separately:
\footnotesize (A + B)\otimes C =A \otimes C +B \otimes C, \\[0.5em] (x{A}) \otimes {B} = x({A} \otimes {B} )
\footnotesize {A} \otimes ({B} +{C} ) ={A} \otimes {B} +{A} \otimes {C}, \\[0.5em] {A} \otimes (x{B} )= x({A} \otimes {B} )
A,B,C
are matrices and
x
(Conjugate) transposition
The transposition of the Kronecker product coincides with the Kronecker products of transposed matrices:
\footnotesize (A\otimes B)^{T}=A^{T}\otimes B^{T}.
The same is true for the conjugate transposition (i.e., adjoint matrices):
\footnotesize (A\otimes B)^{*}=A^{*}\otimes B^{*}.
Singular values and rank
\sigma_1, \ldots, \sigma_{p_A}
are non-zero singular values of
A
s_1, \ldots, s_{p_B}
B
, then the non-zero singular values of
A \otimes B
\sigma_{i}s_j
i=1, \ldots, p_{A}
j=1, \ldots, p_{B}
Recall that the number of non-zero singular values of a matrix is equal to the rank of this matrix. In consequence, we obtain the rank formula:
\footnotesize \operatorname{rank}(A \otimes B) = \operatorname{rank}(A) \cdot \operatorname{rank}(B)
Inverse of tensor product
For the rest of this section, we assume that
A
B
are square matrices of size
m
n
A
B
are both invertible, then
A\otimes B
is invertible as well and
\footnotesize (A\otimes B)^{-1}=A^{-1}\otimes B^{-1}.
Eigenvalues, trace, determinant
\alpha_1, \ldots, \alpha_m
\beta_1, \ldots, \beta_n
A
B
(listed with multiplicities) respectively, then the eigenvalues of
A \otimes B
\alpha_{i}\beta_{j}
i=1,\ldots ,m
j=1,\ldots ,n
Since the determinant corresponds to the product of eigenvalues and the trace to their sum, we have just derived the following relationships:
\footnotesize \det(A \otimes B) = \det(A)^n \det (B)^m
\footnotesize \operatorname{trace}(A \otimes B) = \operatorname{trace}(A) \operatorname{trace}(B)
Is the Kronecker product associative?
Yes, the Kronecker matrix product is associative: (A ⊗ B) ⊗ C = A ⊗ (B ⊗ C) for all matrices A, B, C.
Is the Kronecker product commutative?
No, the Kronecker matrix product is not commutative: A ⊗ B ≠ B ⊗ A for some matrices A, B.
Is tensor product the same as Kronecker product?
The tensor product is a more general notion, but if we deal with finite-dimensional linear spaces, the matrix of the tensor product of two linear operators (with respect to the basis which is the tensor product of the initial bases) is given exactly by the Kronecker product of the matrices of these operators with respect to the initial bases.
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Origins of the Fed Model
Using the Fed Model
Fed Model FAQs
Tim Keefe has 15+ years of experience in many facets of financial services and has held key roles at top-tier investment banks.
The Fed model emerged at the beginning of the 21st century as a stock valuation methodology used by Wall Street gurus and the financial press. The Fed model compares stock yield to bond yield. Proponents almost always cite the following three attributes as the reasons for its popularity:
It is backed by empirical evidence.
It is backed by financial theory.
This article examines the basic concepts behind the Fed model: How it works, and how it was developed, and the article will also outline the challenges to its success and theoretical soundness.
The Fed model is a valuation tool that is used to evaluate the bullishness of the stock market.
It was originally named the "Fed's Stock Valuation Model" by Edward Yardeni, who researched the relationship between bonds and equities in the late 1990s.
The Fed model works by comparing earnings yields with the yield of the 10-year Treasury bond.
Some economists have argued against the Fed model, both on empirical evidence and on theoretical grounds.
The Fed model is a valuation methodology that recognizes a relationship between the forward earnings yield of the stock market (typically the S&P 500 Index,) and the 10-year Treasury bond yield to maturity (YTM).
The yield on a stock is the expected earnings over the next 12 months divided by the current stock price and is symbolized in this article as (E1/PS). This equation is the inverse of the familiar forward P/E ratio but, when shown in the same yield form, it highlights the same concept as the bond yield (YB)—that is, the concept of a return on investment.
Some advocates of the Fed model think the yield relationship varies over time, so they use an average of each period's yield comparison. The more popular method is where the relationship is fixed at the particular value of zero. This technique is referred to as the strict form of the Fed model because it implies that the relationship is strictly based on equality.
In the strict form, the relationship is such that the forward stock yield equals the bond yield:
\begin{aligned} &Y_B = \frac{E_1}{P_S}\\ &\textbf{where:}\\ &Y_B=\text{bond yield}\\ &\frac{E_1}{P_S}=\text{forward stock yield}\\ \end{aligned}
YB=PSE1where:YB=bond yieldPSE1=forward stock yield
1. The difference in the forward stock yield equals 0.
\frac{E_1}{P_S} - Y_B = 0
PSE1−YB=0
2. Alternatively, the ratio of the forward stock yield divided by the bond yield equals 1:
(\frac{E_1}{P_S}) \div Y_B = 1
(PSE1)÷YB=1
The premise behind the model is that bonds and stocks are competing investment products. An investor is constantly making choices between investment products as the relative prices between these products change in the marketplace.
Despite the name, the Fed model is not endorsed or associated with the Federal Reserve.
The name Fed model was manufactured by Wall Street professionals in the late 1990s, but this system is not officially endorsed by the Federal Reserve Board. On July 22, 1997, the Fed's Humphrey-Hawkins Report introduced a graph of the close relationship between long-term Treasury yields and the forward earnings yield of the S&P 500 from 1982 to 1997.
Note: Earnings-price ratio is based on the I/B/E/S International Inc. consensus estimate of earnings over the coming 12 months. All observations reflect prices at mid-month. Source: Federal Reserve
Shortly thereafter, in 1997 and 1999, Edward Yardeni, then at Deutsche Morgan Grenfell, published several research reports further analyzing this bond yield/stock yield relationship. He named the relationship the Fed's Stock Valuation Model, and the name stuck.
The original use of this type of analysis is not known, but a bond yield versus equity yield comparison has been used in practice long before the Fed graphed it out, and Yardini began marketing the idea. In their March 2005 paper titled "The Market P/E Ratio: Stock Returns, Earnings, and Mean Reversion," Robert Weigand and Robert Irons commented that empirical evidence suggests that investors began using the Fed model in the 1960s soon after Myron Gordon described the dividend discount model in the seminal paper "Dividends, Earnings, and Stock Prices" in 1959.
The Fed model evaluates whether the price paid for the riskier cash flows earned from stocks is appropriate by comparing expected return measures for each asset: YTM for bonds and E1/PS for stocks.
This analysis is typically done by looking at the difference between the two expected returns. The value of the spread between (E1/PS) - YB indicates the magnitude of mispricing between the two assets. In general, the bigger the spread, the cheaper the stocks relative to bonds and vice versa. This valuation suggests that a falling bond yield dictates a falling earnings yield, which will ultimately result in higher stock prices. That is PS should rise for any given E1 when bond yields are below the stock yield.
Sometimes, financial market pundits carelessly (or ignorantly) claim that stocks are undervalued according to the Fed model (or interest rates). Although this is a true statement, it is careless because it implies that stock prices will go higher.
The correct interpretation of a comparison between the stock yield and the bond yield is not that stocks are cheap or expensive but that stocks are cheap or expensive relative to bonds. It may be that stocks are expensive and priced to deliver returns below their average long-run returns, but bonds are even more expensive and priced to deliver returns far below their average long-run returns.
It could be possible that stocks could continuously be undervalued according to the Fed model while stock prices fall from their current levels.
Opposition to the Fed model has been based on both observational evidence, and theoretical shortcomings. To begin, although stock and long-term bond yields appear to be correlated from the 1960s forward, they appear to have been less correlated prior to the 1960s.
Also, there may be statistical issues in the way the Fed model has been calculated. Originally, statistical analysis was conducted using ordinary least-squares regression, but bond and stock yields may seem co-integrated, which would require a different method of statistical analysis.
Javier Estrada wrote a paper in 2006 entitled "The Fed Model: The Bad, The Worse, And The Ugly," in which he looked into the empirical evidence using the more appropriate co-integration methodology. His conclusions suggest that the Fed model may not be as good of a tool as originally thought.
In a 2006 paper, Javier Estrada found the Fed model to be "a failure" as an equity pricing model.
Opponents of the Fed model also pose interesting and valid challenges to its theoretical soundness. Concerns arise over comparing stock yields and bond yields because YB is the internal rate of return (IRR) of a bond and accurately represents the expected return on bonds. Remember that IRR assumes that all coupons paid over the life of the bond are reinvested at YB, whereas, E1/PS is not necessarily the IRR of a stock and does not always represent the expected return on stocks.
Furthermore, E1/PS is a real (inflation-adjusted) expected return while YB is a nominal (unadjusted) rate of return. This difference causes a breakdown in the expected return comparison.
Opponents argue that inflation does not affect stocks in the same way it affects bonds. Inflation is typically assumed to pass to stockholders via earnings, but coupons to bondholders are fixed. So, when the bond yield rises due to inflation, PS is not affected because earnings rise by an amount that offsets this increase in the discount rate. In short, E1/PS is a real expected return and YB is a nominal expected return.
Thus, in periods of high inflation, the Fed model will incorrectly argue for a high stock yield and depress stock prices and, in periods of low inflation, it will incorrectly argue for low stock yields and increase stock prices.
The above circumstance is called the illusion of inflation, which Franco Modigliani and Richard A. Cohn presented in their 1979 paper "Inflation, Rational Valuation, and the Market." Unfortunately, the inflation illusion is not as easy to demonstrate as it seems when dealing with corporate earnings. Some studies have shown that a great deal of inflation does pass through to earnings while others have shown the opposite.
The Fed model may or may not be an effective investment tool. However, one thing is certain: If an investor considers stocks real assets that pass inflation through to earnings, they cannot logically invest their capital based on the Fed model.
What Is the Earnings Yield of a Stock?
Earnings yield is calculated by dividing the earnings per share of a stock over a 12-month period by the current share price. This is the inverse of the P/E ratio and is used to determine if a share is overpriced or underpriced.
What Is the First Step in the Fed Model?
The first step in the Fed model is to calculate the forward earnings yield of the stock market, typically using a benchmark index like the S&P 500. This is then compared to the yield of the 10-year Treasury bond to gauge whether the market is bullish or bearish overall.
Average P/E ratios will vary from industry to industry, so there is no fixed bar for what makes a "good" P/E ratio. The median P/E ratio of the S&P 500 companies is around 22, meaning that anything lower than twenty is comparatively inexpensive.
What Is Forward Earnings Yield?
Forward earnings yield is calculated by taking the projected earnings of a given stock over the next twelve months, divided by the current price at the time of calculation. This is the inverse of the forward P/E ratio.
The Federal Reserve Board. "Humphrey-Hawkins Report, July 22, 1997 -- Section 2: Economic and Financial Developments in 1997." Accessed Jan. 23, 2022.
Deutsche Morgan Grenfell. "Topical Study #38: Fed's Stock Market Model Finds Overvaluation," Page 3. Accessed Jan. 23, 2022.
The Journal of Portfolio Management. "The Market P/E Ratio, Earnings Trading, and Stock Return Forecasts." Accessed Jan. 23, 2022.
SSRN. "The Fed Model: The Bad, the Worse, and the Ugly." Accessed Jan. 23, 2022.
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Graph Width Parameters: from Structure to Algorithms (GWP 2022)
Satellite Workshop of ICALP 2022
This is the second edition of GWP, following GWP 2021.
Most optimization problems defined on graphs are computationally hard. For such problems it is natural to restrict the input and ask:
For which graph classes does the problem become efficiently solvable and for which graph classes does it stay hard?
Knowing that a graph has small "width" (for example, treewidth, clique-width, mim-width) has proven to be highly useful for designing efficient algorithms for many well-known problems, such as Feedback Vertex Set, Graph Colouring and Independent Set. That is, boundedness of width enables the application of a problem-specific dynamic programming algorithm or a meta-theorem to solve a certain problem. However, for many graph classes it is not known if the class has small width for some appropriate width parameter. This has resulted in ad-hoc efficient algorithms for special graph classes that may unknowingly make use of the fact that the graph classes under consideration have small width. More generally speaking, rather than solving problems one by one and graph-class by graph-class, the focus of our satellite workshop is:
discovering general properties of graph classes from which we can determine the tractability or hardness of graph problems, and
discovering which graph problems can be solved efficiently on graph classes of bounded width.
For this purpose we aim to bring together researchers from Discrete Mathematics (structure) and Theoretical Computer Science (algorithms).
This will be a hybrid workshop: participants can attend either in person, or remotely via Zoom.
It will be a one-day workshop, consisting of 6 invited talks, and finishing with a session for further discussion and open problems.
For the final session of the workshop, we invite short presentations that highlight an open problem or potential area for future research. If you wish to have a 10-minute slot in our workshop, please send an email with a title and short description to a.munaro@qub.ac.uk by Monday 27 June 2022. Please indicate if you plan to present your open problem in-person or online. Note that the review of contributions may close earlier if the session is filled.
To attend in-person, register for the workshops day of ICALP 2022, as detailed here.
For free online participation, please fill in this form. The deadline for online registration is Friday 1 July.
Benjamin Bergougnoux, Department of Informatics, University of Bergen, Norway.
Édouard Bonnet, LIP, ENS Lyon, France.
Clément Dallard, FAMNIT, University of Primorska, Koper, Slovenia.
Zdeněk Dvořák, Computer Science Institute, Charles University, Prague, Czech Republic.
Paloma T. Lima, Computer Science Department, IT University of Copenhagen, Denmark.
Sophie Spirkl, Department of Combinatorics and Optimization, University of Waterloo, Canada.
All times are in the Central European Summer Timezone (CEST), UTC+2, GMT+2.
9:30 - 10:30: intro/welcome and invited talk
16:45 - 18:00: open problem session
Benjamin Bergougnoux – "A Logic-Based Algorithmic Meta-Theorem for Mim-Width"
Joint work with Jan Dreier and Lars Jaffke.
We introduce a logic called distance neighborhood logic with acyclicity and connectivity constraints (AC DN for short) which extends existential MSO1 with predicates for querying neighborhoods of vertex sets and for verifying connectivity and acyclicity of vertex sets in various powers of a graph. Building upon [Bergougnoux and Kanté, ESA 2019; SIDMA 2021], we show that the model checking problem for every fixed AC DN formula is solvable in nO(w) time when the input graph is given together with a branch decomposition of mim-width w. Nearly all problems that are known to be solvable in polynomial time given a branch decomposition of constant mim-width can be expressed in this framework. We add several natural problems to this list, including problems asking for diverse sets of solutions.
Our model checking algorithm is efficient whenever the given branch decomposition of the input graph has small index in terms of the d-neighborhood equivalence [Bui-Xuan, Telle, and Vatshelle, TCS 2013]. We therefore unify and extend known algorithms for tree-width, clique-width and rank-width. Our algorithm has a single-exponential dependence on these three width measures and asymptotically matches run times of the fastest known algorithms for several problems. This results in algorithms with tight run times under the Exponential Time Hypothesis (ETH) for tree-width and clique-width; the above mentioned run time for mim-width is nearly tight under the ETH for several problems as well.
Our results are also tight in terms of the expressive power of the logic: we show that already slight extensions of our logic make the model checking problem para-NP-hard when parameterized by mim-width plus formula length.
Édouard Bonnet – "Twin-width delineation and win-wins"
A graph class
\mathcal{C}
is said to be delineated if for every hereditary closure
\mathcal{D}
of a subclass of
\mathcal{C}
\mathcal{D}
has bounded twin-width if and only if
\mathcal{D}
is monadically dependent (i.e., cannot express every graph by means of a first-order transduction). An effective strengthening of delineation for a class
\mathcal{C}
implies that tractable FO model checking on
\mathcal{C}
is perfectly understood: On hereditary closures of subclasses
\mathcal{D}
\mathcal{C}
, FO model checking on
\mathcal{D}
is fixed-parameter tractable (FPT) exactly when
\mathcal{D}
has bounded twin-width. We explore which classes are delineated and which are not. Along the same lines, we present FPT algorithms for some W[1]-hard problems in general graphs, on classes of unbounded twin-width, via win-win arguments.
Clément Dallard – TBC
Zdeněk Dvořák – "On fractional treewidth-fragility"
A graph G is fractionally f-treewidth-fragile if for every k and every assignment w of weights to vertices, there exists a subset X of vertices of G such that w(X) ≤ w(G) / k and G − X has treewidth at most f(k); i.e., the treewidth of G can be reduced to constant by removing a small fraction of the weight. A class of graphs is fractionally treewidth-fragile if there exists a function f such that all graphs from the class are fractionally f-treewidth-fragile. We survey the graph classes that are fractionally treewidth-fragile and describe applications of this notion in design of approximation algorithms.
Paloma T. Lima – TBC
Sophie Spirkl – "Induced subgraphs and treewidth"
The results of Robertson and Seymour tell us which subgraphs of large treewidth are always present in graphs of large treewidth: subdivisions of walls. The analogous question for induced subgraphs is still open, and constructions of Sintiari and Trotignon, and of Davies, show that the answer for induced subgraphs is more complicated. I will talk about some recent results in this area. Based on joint work with Tara Abrishami, Bogdan Alecu, Maria Chudnovsky, Sepehr Hajebi, and Kristina Vušković.
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Square feet of a rectangle
How to calculate the square feet of a rectangle?
How to use this square feet of a rectangle calculator?
How do I calculate the square feet of carpet required?
Our square feet of a rectangle calculator is the perfect tool for when you want to calculate the area of a rectangle in square feet. In this article, we shall discuss how to calculate the square feet of a rectangle, how to use this calculator, and some frequently asked questions.
Square feet of a rectangle is the area of that rectangle measured in the imperial units
\text{sq} \cdot \text{ft}
\text{ft}^2
. Apart from this particularity, square feet of a rectangle is computed in the same way as its area:
\qquad \text{SA} = l \times w
\text{SA}
- The surface area of the rectangle measured in
\text{sq} \cdot \text{ft}
\text{ft}^2
l
- The length of the rectangle measured in
\text{ft}
. Usually, it refers to the longest side of the rectangle; and
w
- The width of the rectangle measured in
\text{ft}
. Usually, it refers to the shortest side of the rectangle.
Ensure that the length and width are in feet before the multiplication to avoid further conversions.
To find the square feet of a rectangle, you require its length and width:
Convert the length and width into feet.
Multiply this length and width to obtain the area in square feet.
Double-check your answer using the square feet of a rectangle calculator.
This square feet of a rectangle calculator is simple to use:
Enter the length of the rectangle in its appropriate field. It's alright if its units are not in feet. Just ensure you're getting the units right.
Enter the width (or breadth) of the rectangle in its appropriate field. Again ensure that you've picked the correct units.
Our square feet of a rectangle calculator will automatically calculate the area of the rectangle in square feet and display the result in the appropriate field.
If you're in need for calculating other rectangle related parameters, use our collection below:
What is the area of an American football field?
48,000 sq.ft excluding the end zones. Including the end zones, it is 57,600 sq. ft. The regular play area is a rectangular field 300 ft long and 160 ft wide, whereas the entire field is 360 ft long and 160 ft wide.
To calculate the square feet of carpet required for your floor, follow these simple steps:
Measure the length and width of the floor where you want to lay your carpet. It is convenient to take this measure in feet.
Multiply this length and width to acquire the area of the carpet in square feet.
Celebrate your successful calculation of the carpet area.
Area of the rectangle in square feet
The ceiling function calculator explains all you need to know about this simple yet so useful math function!
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networks(deprecated)/ends - Maple Help
Home : Support : Online Help : networks(deprecated)/ends
finds the ends of an edge in a graph
ends(G)
ends(e, G)
edge or a set or list of edges of G
This routine is used to recover the names of the vertices at the ends of a specified edge.
In the special case where only the graph G is specified then the ends of all of the edges of G are returned as a set.
If an edge e is directed then the ends are returned as a list of length 2 where the second element is the head. For undirected edges the vertex names are returned as a set. Undirected loops will appear as a set containing one vertex name.
Specific edges of interest may be specified either individually or as a set or list of edge names. The result will be a vertex pair, a set of vertex pairs, or a list of vertex pairs as appropriate.
This routine is normally loaded via the command with(networks) but may also be referenced using the full name networks[ends](...).
\mathrm{with}\left(\mathrm{networks}\right):
G≔\mathrm{cycle}\left(5\right):
\mathrm{Eset}≔{\mathrm{addedge}\left({[1,3],[2,2]},G\right)}
\textcolor[rgb]{0,0,1}{\mathrm{Eset}}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{\mathrm{e6}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e7}}}
\mathrm{ends}\left(\mathrm{Eset},G\right)
{[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]}
\mathrm{ends}\left(\mathrm{convert}\left(\mathrm{Eset},\mathrm{list}\right),G\right)
[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]]
\mathrm{ends}\left(G\right)
{[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}}}
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What is the upper and lower fence?
The use of fences in statistics
How to calculate the upper and lower fence
An example of how to calculate the upper and lower fence
How to find the upper and lower fence with our calculator
Welcome to the upper and lower fence calculator, where we'll discuss the use of fences in statistics and show you how to find the lower fence and the upper fence of a dataset. These fences are vital to finding outliers in your dataset.
Before we can make sense of our upper and lower fence calculator, we must define what fences in statistics mean. The upper and lower fences of a dataset are the thresholds, outside of which values can be considered outliers. Outliers, therefore, are any values that fall below the lower fence or above the upper fence.
Besides helping us find outliers, fences can be a suitable replacement for the minimum and maximum in descriptive statistics. In most cases, box plots (an effective way to visualize a 5 number summary) use the minimum and maximum as the box's whiskers. It's much more insightful, though, to use the upper and lower fences for the whiskers and then indicate outlying values with distinct points. Take a look at the figures below:
Two box plots. The box plot on the left uses the minimum and maximum for its whiskers, while the box plot on the right uses the fences for its whiskers. The box plot on the right is more descriptive.
Now that we know what the fences are, we'd love to know exactly how to find the upper and lower fence. There are a few steps involved, so let's get started!
Before we can get to the upper and lower fence formulas, we need two basic but important values: the dataset's quartiles. We denote the first quartile with
Q_1
and the third quartile with
Q_3
. We can then calculate the interquartile range as
\text{IQR} = Q_3 - Q_1
Now that we have the building blocks, how do we find the lower fence, and how do we find the upper fence?
The formula for the upper fence is
\text{Upper fence} = Q_3 + 1.5\times\text{IQR}
The formula for the lower fence is
\text{Lower fence} = Q_1 - 1.5\times\text{IQR}
Some select sources replace the
1.5
in our formulas with other values, like
2
or sometimes even
3
. The choice of the multiplier is problem-specific and depends on the dataset distribution. An outlier in a dataset of children's heights would still be close to the rest of the dataset, so we'd choose a smaller multiplier like
1.5
. A millionaire's salary in a dataset of salaries would be much more outlying, so we'd pick a bigger multiplier, like
3
or more. If you'd like to use a different multiplier for your dataset, you can enter the Advanced mode and change it from the default of
1.5
So, there you have it! We've found the formula for the upper fence and the lower fence, and now we can find outliers in any dataset we want.
Let's use our upper and lower fence formula in a practical example. Suppose we have a dataset of each year's rainfall volumes for January from 2010 to 2021 in New York, and it looks like this:
1.33
1.96
3.12
2.20
1.58
2.04
1.80
6.32
1.90
3.84
2.93
2.34
Can you already guess which value is going to be an outlier?
Let's back up that guess with some math.
We first have to sort the dataset in increasing order:
1.33
1.58
1.80
1.90
1.96
2.04
2.20
2.34
2.93
3.12
3.84
6.32
We can determine that the dataset's first and third quartiles are
Q_1= 1.85
Q_3 = 3.025
Q_1
Q_3
, we can calculate the interquartile range with
\text{IQR} = Q_3 - Q_1 = 3.025 - 1.85 = 1.175
Finally, we can use the upper fence formula to get
\text{Upper fence} = Q_3 + 1.5 \times \text{IQR} = 3.025 + 1.5 \times 1.175 = 4.7875
…and we can use the lower fence formula to get
\text{Lower fence} = Q_1 - 1.5 \times \text{IQR} = 1.85 - 1.5 \times 1.175 = 0.0875
We can look at our data with these upper and lower fences and see that 2017's rainfall of
6.32
is an outlier. Was your initial guess correct?
Our upper and lower fence calculator takes all these steps for you and gives you the fences in the blink of an eye so that you can get to find outliers in your dataset.
Enter your dataset's individual values in the rows. You can input up to 50 values.
Optionally, change the multiplier used in the fence formulas in the Advanced mode.
The calculator will determine the fences and display them at the bottom of the list of values, along with your dataset's outliers and the steps taken to calculate them. Happy outlier hunting!
Outliers are values in a dataset that differ significantly from other values. The presence of outliers can be a problem, although it depends on what task you're using the data for. Outliers can be legitimate data, like a CEO's salary in a salary dataset. Outliers can also be invalid or due to mistakes; this could be a poorly calibrated sensor, or a typing error made when copying handwritten data over to a spreadsheet.
How do I find outliers?
To find outliers in your dataset, you need to calculate the upper and lower fences of the dataset. You'd then see which of the dataset's values fall outside of the fences — those values are all outliers.
How do I calculate the upper and lower fences?
Multiply your dataset's interquartile range with 1.5, then add and subtract that from your dataset's first and third quartiles, respectively. Those are your upper and lower fences.
What is the upper fence formula?
You can calculate the upper fence with Q3 + 1.5 × IQR, where Q3 is your third quartile and IQR is your interquartile range. Any value in your dataset above the upper fence is an outlier.
What is the lower fence formula?
You can calculate the lower fence with Q1 − 1.5 × IQR, where Q1 is your first quartile and IQR is your interquartile range. Any value in your dataset below the lower fence is an outlier.
Too few values ‒ try adding some!
Our geometric distribution calculator will help you determine the probability of a certain number of trials needed for success.
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Exhaust-Stream and In-Cylinder Measurements and Analysis of the Soot Emissions From a Common Rail Diesel Engine Using Two Fuels | J. Eng. Gas Turbines Power | ASME Digital Collection
Patrick Kirchen,
, Sonneggstrasse 3, CH-8092 Zurich, Switzlerand
e-mail: pkirchen@mit.edu
Peter Obrecht,
e-mail: obrecht@lav.mavt.ethz.ch
Konstantinos Boulouchos,
e-mail: boulouchos@lav.mavt.ethz.ch
, CH-8408 Winterthur, Switzerland
e-mail: andrea.bertola@kistler.com
Kirchen, P., Obrecht, P., Boulouchos, K., and Bertola, A. (August 16, 2010). "Exhaust-Stream and In-Cylinder Measurements and Analysis of the Soot Emissions From a Common Rail Diesel Engine Using Two Fuels." ASME. J. Eng. Gas Turbines Power. November 2010; 132(11): 112804. https://doi.org/10.1115/1.4001083
The operation and emissions of a four cylinder, passenger car common-rail diesel engine operating with two different fuels was investigated on the basis of exhaust-stream and in-cylinder soot measurements, as well as a thermodynamic analysis of the combustion process. The two fuels considered were a standard diesel fuel and a synthetic diesel (fuel two) with a lower aromatic content, evaporation temperature, and cetane number than the standard diesel. The exhaust-stream soot emissions, measured using a filter smoke number system, as well as a photo-acoustic soot sensor (AVL Micro Soot Sensor), were lower with the second fuel throughout the entire engine operating map. To elucidate the cause of the reduced exhaust-stream soot emissions, the in-cylinder soot temperature and the
KL
factor (proportional to concentration) were measured using miniature, three-color pyrometers mounted in the glow plug bores. Using the maximum
KL
factor value to quantify the soot formation process, it was seen that for all operating points, less soot was formed in the combustion chamber using the second fuel. The oxidation of the soot, however, was not strongly influenced by the fuel, as the relative oxidized soot fraction was not significantly different for the two fuels. The reduced soot formation of fuel two was attributed to the lower aromatic content of the fuel. The soot cloud temperatures for operation with the two fuels were not seen differ significantly. Similar correlations between the cylinder-out soot emissions, characterized using the pyrometers, and the exhaust-stream soot emissions were seen for both fuels. The combustion process itself was only seen to differ between the two fuels to a much lesser degree than the soot formation process. The predominant differences were seen as higher maximum fuel conversion rates during premixed combustion at several operating points, when fuel two was used. This was attributed to the lower evaporation temperatures and longer ignition delays (characterized by the lower cetane number) leading to larger premixed combustion fractions.
air pollution measurement, combustion, diesel engines, evaporation, exhaust systems, petroleum, soot
Cylinders, Emissions, Exhaust systems, Fuels, Soot, Diesel engines, Oxidation, Combustion, Engines, Common rail fuel injectors, Pyrometers
Semmler-Behnke
Health Implications of Nanoparticles
An Association Between Air Pollution and Mortality in 6 United States Cities
Increased Particulate Air Pollution and the Triggering of Myocardial Infarction
Reduction in Fine Particulate Air Pollution and Mortality—Extended Follow-Up of the Harvard Six Cities Study
Diesel Particulates; What They Are and Why
Detailed Modeling of Soot Particle Nucleation and Growth
23rd Symposium (International) on Combustion
The Effect of Aromatic Hydrocarbons on Soot Formation in Laminar Diffusion Flames and in a Diesel Engine
J. Inst. Energy
The Potential for Particle Reductions in Modern Combustion Engines
Minimierung von Partikelemissionen
Soot Formation in Combustion: Mechanisms and Models
Fundamentals of Soot Formation in Flames With Application to Diesel Engine Particulate-Emissions
Das ccs-brennverfahren von volkswagen
New Combustion System Based on a New Fuel Specification
, Vienna.
A Photoacoustic Sensor System for Time Resolved Quantification of Diesel Soot Emissions
Measurement Apparatus for Smoke From Engines Operating Under Steadystate Conditions
,” Standard No. 10054.
Determination of True Temperature and Total Radiation From Luminous Gas Flames
Ind. Eng. Chem. Anal. Ed.
Analysis and In-Cylinder Measurement of Particulate Radiant Emissions and Temperature in a Direct Injection Diesel Engine
A Study on the Application of the Two-Color Method to the Measurement of Flame Temperature and Soot Concentration in Diesel Engines
Correlation and Analysis of In-Cylinder and Engine-Out Soot Emission Measurements From a Multi-Cylinder Diesel Engine
8th Internationales Stuttgarter Symposium
, Stuttgart, Germany, Vol.
Temporal Soot Evolution and Diesel Engine Combustion: Influence of Fuel Composition, Injection Parameters, and Exhaust Gas Recirculation
, 2008, Weg—verbrennungsanalyse: Berechnung des wärmeentwicklungsgesetzes. Version 15.
Development and Validation of a Phenomenological Mean Value Soot Model for Common-Rail Diesel Engines
Eine neue messmethodik der bosch-zahl mit erhöhter empfindlichkeit
Smoke Value Measurement With the Filter-Paper-Method: Application Notes
,” Technical Report No. AT1007E, Rev. 02.
Optical Diagnostics for Soot and Temperature Measurement in Diesel Engines
Ein phänomenologisches Modell zur kombinierten Stickoxid-und Russberechnung bei direkteinspritzenden Dieselmotoren
,” Ph.D. thesis, Universität Stuttgart.
Soot Particle Sizing During High-Pressure Diesel Spray Combustion via Time-Resolved Laser-Induced Incandescence
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Carmichael number - Simple English Wikipedia, the free encyclopedia
A composite number in number theory
In number theory a Carmichael number is a composite positive integer
{\displaystyle n}
, which satisfies the congruence
{\displaystyle b^{n-1}\equiv 1{\pmod {n}}}
{\displaystyle b}
{\displaystyle n}
. Being relatively prime means that they do not have common divisors, other than 1. Such numbers are named after Robert Carmichael.
All prime numbers
{\displaystyle p}
{\displaystyle b^{p-1}\equiv 1{\pmod {p}}}
{\displaystyle b}
{\displaystyle p}
. This has been proven by the famous mathematician Pierre de Fermat. In most cases if a number
{\displaystyle n}
is composite, it does not satisfy this congruence equation. So, there exist not so many Carmichael numbers. We can say that Carmichael numbers are composite numbers that behave a little bit like they would be a prime number.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Carmichael_number&oldid=8027709"
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Schlegel_diagram Knowpia
In geometry, a Schlegel diagram is a projection of a polytope from
{\textstyle \mathbb {R} ^{d}}
{\textstyle \mathbb {R} ^{d-1}}
through a point just outside one of its facets. The resulting entity is a polytopal subdivision of the facet in
{\textstyle \mathbb {R} ^{d-1}}
that, together with the original facet, is combinatorially equivalent to the original polytope. The diagram is named for Victor Schlegel, who in 1886 introduced this tool for studying combinatorial and topological properties of polytopes. In dimension 3, a Schlegel diagram is a projection of a polyhedron into a plane figure; in dimension 4, it is a projection of a 4-polytope to 3-space. As such, Schlegel diagrams are commonly used as a means of visualizing four-dimensional polytopes.
Examples colored by the number of sides on each face. Yellow triangles, red squares, and green pentagons.
A tesseract projected into 3-space as a Schlegel diagram. There are 8 cubic cells visible: the outer cell into which the others are projected, one below each of the six exterior faces, and one in the center.
Various visualizations of the icosahedron
The most elementary Schlegel diagram, that of a polyhedron, was described by Duncan Sommerville as follows:[1]
A very useful method of representing a convex polyhedron is by plane projection. If it is projected from any external point, since each ray cuts it twice, it will be represented by a polygonal area divided twice over into polygons. It is always possible by suitable choice of the centre of projection to make the projection of one face completely contain the projections of all the other faces. This is called a Schlegel diagram of the polyhedron. The Schlegel diagram completely represents the morphology of the polyhedron. It is sometimes convenient to project the polyhedron from a vertex; this vertex is projected to infinity and does not appear in the diagram, the edges through it are represented by lines drawn outwards.
Sommerville also considers the case of a simplex in four dimensions:[2] "The Schlegel diagram of simplex in S4 is a tetrahedron divided into four tetrahedra." More generally, a polytope in n-dimensions has a Schegel diagram constructed by a perspective projection viewed from a point outside of the polytope, above the center of a facet. All vertices and edges of the polytope are projected onto a hyperplane of that facet. If the polytope is convex, a point near the facet will exist which maps the facet outside, and all other facets inside, so no edges need to cross in the projection.
Net (polyhedron) – A different approach for visualization by lowering the dimension of a polytope is to build a net, disconnecting facets, and unfolding until the facets can exist on a single hyperplane. This maintains the geometric scale and shape, but makes the topological connections harder to see.
^ Duncan Sommerville (1929). Introduction to the Geometry of N Dimensions, p.100. E. P. Dutton. Reprint 1958 by Dover Books.
^ Sommerville (1929), p.101.
Victor Schlegel (1883) Theorie der homogen zusammengesetzten Raumgebilde, Nova Acta, Ksl. Leop.-Carol. Deutsche Akademie der Naturforscher, Band XLIV, Nr. 4, Druck von E. Blochmann & Sohn in Dresden. [1]
Coxeter, H.S.M.; Regular Polytopes, (Methuen and Co., 1948). (p. 242)
Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
Wikimedia Commons has media related to Schlegel diagrams.
Weisstein, Eric W. "Schlegel graph". MathWorld.
George W. Hart: 4D Polytope Projection Models by 3D Printing
Nrich maths – for the teenager. Also useful for teachers.
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I know nothing about internet security, so naturally I'm taking xkcd pretty seriously.
The Incidental Economist has left me thinking about password security, in a world where just about everything we do requires passwords of escalating complexity, usually requiring some combination of lowercase letters, uppercase letters, and numerals.
Basically, xkcd.com's author Randall Munroe proposes that we toss out existing password requirements, and instead assign users a fully randomized sequence of three words. I didn't really understand Munroe's math, so here's my own.
So to start, we have to get a sense of how hard it is to crack a three-randomized-word password. As it happens, English is one of the largest languages ever to exist on planet earth, clocking in with at least 140,000 different words. But, it's hard to find all those words in one place, so here's a random word generator that contains a subset of 90,000 words. That's still a lot more words than you or I know. So here's the first three words our word generator produced for me:
Reeumlnthrone Elicitation Pace
If you are curious, "reeumlnthrone" means "to enthrone again." Purely by accident, I was at the Queen's recoronation at Westminster Abbey last summer, but I suppose that a recorronation isn't quite the same thing as reeumlnthroning.
We want to know the expected number of attempts a hacker would have to make to guess our password. So our random variable
X
is the number of attempts until the correct password is guessed. Assuming these attempts are independently distributed, then
X
follows a geometric distribution, where the probability of the
{k}^{th}
attempt being a success is given by
Pr\left(X=k\right)={\left(1-p\right)}^{k-1}p,
p
is the probability of guessing the password in each (independent and identically distributed) attempt. So with 90,000 possible words, picking three at random, the probability that any particular attempt at guessing our password is successful is
p=\frac{1}{{\left(90,000\right)}^{3}}.
I won't do the math here, but it turns out that
X
has an expected value given by
E\left[X\right]=\frac{1}{p}=\frac{1}{7.29×{10}^{14}}.
which means that on average we'd expect a hacker to have to make 729 trillion guesses to get our password. Again, I know nothing about cybersecurity. But that sounds like a lot of guesses.
A useful way to put this into perspective is to ask how long the password would have to be to achieve at least as much security as Munroe's recommendation, assuming that we fully randomize each separate characters chosen from the 26 lowercase letters, 26 uppercase letters, and 10 numerals--which are usually the only characters passwords are allowed. So, that's a total of 62 characters we are picking at random from. Let
n
be the length of our password. Assuming a hacker attempting a sequence of independent and identically distributed guesses at our password, the probability of hacking this password is also geometrically distributed, with expected value of
{\left(62\right)}^{n}
. Thus, we are looking for
n
that solves
{\left(62\right)}^{n}=7.29×{10}^{14}.
Equations like these are the reason god gave us logarithms. The answer is 8.29... but since we can't have fractions of a character in a password, that means we need nine totally random digits to match the password strength of Munroe's method. For reference, most applications impose an 8-character minimum password length.
This is where memorability comes in. There are also random password generators that do exactly what I described above, but the password looks like this:
rsYAA8UHH
I've already memorized Reeumlnthrone Elicit Pace, having typed it only once. I doubt I could possibly ever remember this second password, no matter how many times I type it.
There are, of course, a lot of mathematical caveats here. For one thing, hackers don't generally make independent and identically distributed attempts to guess your password. At the very least, they'd be silly to guess the same password twice. I don't care to recalculate the probabilities assuming the hacker never guesses the same password twice--"without replacement," in probability theory lingo--but needless to say this would reduce the expected number of guesses somewhat. Second, expected value may not be the best metric of security. Another metric that comes to mind is how many attempts it would take to be guaranteed to have found the right password, which is just equal to the total number of password combinations possible. I did not do that because it is an upperbound--the absolute best case scenario--whereas what matters for security is really the lower bound--that is, whether there's a high probability that the hacker can guess the password well before then. Third, and perhaps most importantly, my analysis above was actually fairly skewed against Munroe's method. That's because I implicitly assumed that hackers have information about the password--the password consists of exactly three randomized words chosen at random from a particular database of words--that hackers in real life probably wouldn't have. In addition, I assumed that the alternative to Munroe's method is fully randomized characters, whereas in reality people tend towards relatively predictable passwords based on names, words, dates, and locations that are personally familiar to them. Thus, in real life how long it takes to guess a password depends a lot on the efficiency of the hacker's guessing algorithm.
But, I think a good rule of thumb is that a password consisting of three randomized English words is roughly equal to a fully randomized 9-character password. Of course, nothing is stopping you from picking a fourth word for your password...
Hackers don't guess passwords anyway. It's much, much more important to use a different password with each service than to use strong passwords.
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Effects of Coating Thickness, Test Temperature, and Coating Hardness on the Erosion Resistance of Steam Turbine Blades | J. Eng. Gas Turbines Power | ASME Digital Collection
Shun-sen Wang,
Shun-sen Wang
, Xi’an 710049, P. R. China
Guan-wei Liu,
Guan-wei Liu
Jing-ru Mao,
Jing-ru Mao
e-mail: jrmao@mail.xjtu.edu.cn
Qun-gong He,
Qun-gong He
Zhen-ping Feng
Wang, S., Liu, G., Mao, J., He, Q., and Feng, Z. (November 4, 2009). "Effects of Coating Thickness, Test Temperature, and Coating Hardness on the Erosion Resistance of Steam Turbine Blades." ASME. J. Eng. Gas Turbines Power. February 2010; 132(2): 022102. https://doi.org/10.1115/1.3155796
This paper experimentally examines the influence of coating thickness, test temperature, coating hardness, and defects on the erosion resistance of boride coatings, ion plating CrN coatings, and thermal spraying coatings. The results demonstrate that the erosion rate of coating can be reduced effectively by improving coating hardness and thickness with the absence of the cracks of coating during the coating process. In comparison with thermal spraying coatings, boride coatings and ion plating CrN coatings are more suitable for protecting steam turbine blades from solid particle erosion due to higher erosion resistance. However, blades cannot be protected effectively when coating is thinner than a critical value
θcrit
. Based on our results, it is recommended that the protective coating for the steam turbine blade should be thicker than 0.02 mm. In addition, the effect of temperature on erosion resistance of the coating is strongly dependent on the properties of transition layer between coating and substrate material. For the coating without pinholes or pores in the transition layer, the variation in erosion rate with temperature is consistent with that of uncoated substrate material. However, the erosion rate of coating descends with the elevation of test temperature when a lot of pinholes or pores are produced in the transition layer.
blades, chromium compounds, hardness, ion plating, protective coatings, steam turbines, thermal spraying, wear resistance, erosion, experiment, high temperature, boride coating, steam turbine
Coating processes, Coatings, Erosion, Particulate matter, Steam turbines, Temperature, Blades, Plating
Solid-Particle Erosion and Protective Layers for Steam Turbine Blading
Evaluation of Erosion and Fatigue Resistance of Ion Plated Chromium Nitride Applied to Turbine Blades
Nozzle Passage Aerodynamic Design to Reduce Solid Particle Erosion of a Supercritical Steam Turbine Control Stage
Erosion Rate Testing at High Temperature for Turbomachinery Use
Protective Design and Boride Coating Against Solid Particle Erosion of First-Stage Turbine Nozzles
Advances in Steam Turbine Technology for Power Generation
Influence of Coating Processes and Process Parameters on Surface Erosion Resistance and Substrate Fatigue Strength
Resisting Steam Turbine Abrasion Damage by Using Surface Improvement Systems
Improvement of Steam Turbine Hard Particle Eroded Nozzles Using Metallurgical Coatings
IEEE Power Engineering
Effects of Temperature on the Behavior of Metals Under Erosion by Particulate Matter
CVD Diamond Coating for Erosion Protection at Elevated Temperatures
Resistance of Paint Coatings to Multiple Solid Particle Impact: Effect of Coating Thickness and Substrate Material
Ottmuller
Solid Particle Erosion Behaviour of Thermal Sprayed Ceramic, Metallic and Polymer Coatings
High Velocity Sand Impact Damage on CVD Diamond
Protection of Coated Superalloys From Erosion in Turbomachinery and Other Systems Exposed to Particulate Flows
Balazinski
Klemberg-Sapieha
Predictive Tools for the Design of Erosion Resistant Coatings
Fe Modelling of Surface Stresses in Erosion-Resistant Coatings Under Single Particle Impact
Design of Hard Coating Architecture for the Optimization of Erosion Resistance
Impact Damage in Brittle Materials in the Elastic-Plastic Response Regime
Wear Mechanisms in Ceramics
,” Fundamentals of Friction and Wear of Materials, papers presented at the 1980 ASM Materials Science Seminar, pp. 439–452.
High-Temperature Coatings for Protection Against Turbine Deterioration
Coatings for the Protection of Turbine Blades From Erosion
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Transforming categorical features to numerical features - How training is performed | CatBoost
Before each split is selected in the tree (see Choosing the tree structure), categorical features are transformed to numerical. This is done using various statistics on combinations of categorical features and combinations of categorical and numerical features.
The method of transforming categorical features to numerical generally includes the following stages:
Permutating the set of input objects in a random order.
Converting the label value from a floating point to an integer.
The method depends on the machine learning problem being solved (which is determined by the selected loss function).
How transformation is performed
Regression Quantization is performed on the label value. The mode and number of buckets (
k+1
) are set in the starting parameters. All values located inside a single bucket are assigned a label value class – an integer in the range
[0;k]
defined by the formula: <bucket ID – 1>.
Classification Possible values for label value are 0 (doesn't belong to the specified target class) and "1" (belongs to the specified target class).
Multiclassification The label values are integer identifiers of target classes (starting from "0").
Transforming categorical features to numerical features.
The method is determined by the starting parameters.
Type : Borders
Calculating ctr for the i-th bucket (
i\in[0; k-1]
ctr_{i} = \frac{countInClass + prior}{totalCount + 1} { , where}
countInClass is how many times the label value exceeded
i
for objects with the current categorical feature value. It only counts objects that already have this value calculated (calculations are made in the order of the objects after shuffling).
totalCount is the total number of objects (up to the current one) that have a feature value matching the current one.
prior is a number (constant) defined by the starting parameters.
i\in[0; k]
, creates
k+1
buckets):
ctr_{i} = \frac{countInClass + prior}{totalCount + 1} { , where}
countInClass is how many times the label value was equal to
i
Type : BinarizedTargetMeanValue
ctr = \frac{countInClass + prior}{totalCount + 1} { , where}
countInClass is the ratio of the sum of the label value integers for this categorical feature to the maximum label value integer (
k
totalCount is the total number of objects that have a feature value matching the current one.
How ctr is calculated for the training dataset:
ctr = \frac{curCount + prior}{maxCount + 1} { , where}
curCount is the total number of objects in the training dataset with the current categorical feature value.
maxCount the number of objects in the training dataset with the most frequent feature value.
How ctr is calculated for the validation dataset:
ctr = \frac{curCount + prior}{maxCount + 1} { , where}
curCount computing principles depend on the chosen calculation method:
Full — The sum of the total number of objects in the training dataset with the current categorical feature value and the number of objects in the validation dataset with the current categorical feature value.
SkipTest — The total number of objects in the training dataset with the current categorical feature value
maxCount is the number of objects with the most frequent feature value in one of the combinations of the following sets depending on the chosen calculation method:
Full — The training and the validation datasets.
SkipTest — The training dataset.
This ctr does not depend on the label value.
As a result, each categorical feature values or feature combination value is assigned a numerical feature.
Example of aggregating multiple features
Assume that the objects in the training set belong to two categorical features: the musical genre ( rock , indie ) and the musical style ( dance , classical ). These features can occur in different combinations. CatBoost can create a new feature that is a combination of those listed ( dance rock , classic rock , dance indie , or indie classical ). Any number of features can be combined.
Transforming categorical features to numerical features in classification
CatBoost accepts a set of object properties and model values as input.
The table below shows what the results of this stage look like.
f_{1}
f_{2}
f_{n}
1 2 40 ... rock 1
2 3 55 ... indie 0
3 5 34 ... pop 1
The rows in the input file are randomly shuffled several times. Multiple random permutations are generated.
f_{1}
f_{2}
f_{n}
All categorical feature values are transformed to numerical using the following formula:
avg\_target = \frac{countInClass + prior}{totalCount + 1}
countInClass
is how many times the label value was equal to 1 for objects with the current categorical feature value.
prior
is the preliminary value for the numerator. It is determined by the starting parameters.
totalCount
is the total number of objects (up to the current one) that have a categorical feature value matching the current one.
These values are calculated individually for each object using data from previous objects.
In the example with musical genres,
j \in [1;3]
accepts the values rock , pop , and indie , and prior is set to 0.05.
f_{1}
f_{2}
f_{n}
1 4 53 ... 0,05 0
3 2 40 ... 0,025 1
7 2 45 ... 0,5125 0
One-hot encoding is also supported. Use one of the following training parameters to enable it.
Command-line version parameter
--one-hot-max-size one_hot_max_size one_hot_max_size Use one-hot encoding for all categorical features with a number of different values less than or equal to the given parameter value. Ctrs are not calculated for such features.
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On Perturbation Solutions for Nearly Circular Inclusion Problems in Plane Thermoelasticity | J. Appl. Mech. | ASME Digital Collection
C.-H. Wang, Graduate Student,
C.-H. Wang, Graduate Student
Department of Mechanical Engineering, National Taiwan University of Science and Technology, 43 Keelung Road, Section 4, Taipei, Taiwan 106, R.O.C.
C.-K. Chao, Professor
Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF APPLIED MECHANICS. Manuscript received by the ASME Applied Mechanics Division, October 2, 2000; final revision, June 5, 2001. Associate Editor: J. R. Barber. Discussion on the paper should be addressed to the Editor, Professor Lewis T. Wheeler, Department of Mechanical Engineering, University of Houston, Houston, TX 77204-4792, and will be accepted until four months after final publication of the paper itself in the ASME JOURNAL OF APPLIED MECHANICS.
Wang, C., and Chao, C. (June 5, 2001). "On Perturbation Solutions for Nearly Circular Inclusion Problems in Plane Thermoelasticity ." ASME. J. Appl. Mech. January 2002; 69(1): 36–44. https://doi.org/10.1115/1.1410367
An approximate analytical solution to the nearly circular inclusion problems of arbitrary shape in plane thermoelasticity is provided. The shape of the inclusion boundary considered in the present study is assumed to have the form
r=a0[1+Aθ],
a0
is the radius of the unperturbed circle and
Aθ
is the radius perturbation magnitude that is represented by a Fourier series expansion. The proposed method in this study is based on the complex variable theory, analytical continuation theorem, and the boundary perturbation technique. Originating from the principle of superposition, the solution of the present problem is composed of the reference and the perturbation terms that the reference term is the known exact solution pertaining to the case with circular inclusion. First-order perturbation solutions of both temperature and stress fields are obtained explicitly for elastic inclusions of arbitrary shape. To demonstrate the derived general solutions, two typical examples including elliptical and smooth polygonal inclusions are discussed in detail. Compared to other existing approaches for elastic inclusion problems, our methodology presented here is remarked by its efficiency and applicability to inclusions of arbitrary shape in a plane under thermal load.
thermoelasticity, perturbation theory, inclusions, Fourier series, temperature distribution, stress analysis, thermal stresses
Shapes, Stress, Temperature, Thermoelasticity, Fourier series, Thermal stresses, Heat flux
Thermal Stress due to Disturbance of Uniform Heat Flow by an Insulated Ovaloid Hole
Plane Thermal Stress at an Insulated Hole Under Uniform Heat Flow in an Orthotropic Medium
Green, A. E., and Zerna, W., 1954, Theoretical Elasticity, Oxford University Press, London.
Thermal Stresses in an Anisotropic Plate Disturbed by an Insulated Elliptic Hole or Crack
Mequid
Thermal Stresses in a Generally Anisotropic Body With an Elliptic Inclusion Subject to Uniform Heat Flow
Two-Dimensional Elastic Inclusion Problem
Sendeckyi
Elastic Inclusion Problems in Plane Elastostatics
Analytic Solution for Eshelby’s Problem of an Inclusion of Arbitrary Shape in a Plane or Half-Plane
Interface Design of Neutral Elastic Inclusions
Neutral Inhomogeneities in Conduction Phenomena
A Boundary Perturbation Analysis for Elastic Inclusions and Interfaces
Thermoelastic Plane Problems With Curvilinear Boundaries
Note on Thermal Stress
On Bonded Circular Inclusions in Plane Thermoelasticity
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networks(deprecated)/degreeseq - Maple Help
Home : Support : Online Help : networks(deprecated)/degreeseq
find the degree sequence of a graph
degreeseq(G)
Important: The networks package has been deprecated. Use the superseding command GraphTheory[DegreeSequence] instead.
This routine returns a list of vertex degrees sorted into ascending order.
This routine is normally loaded via the command with(networks) but may also be referenced using the full name networks[degreeseq](...).
\mathrm{with}\left(\mathrm{networks}\right):
G≔\mathrm{complete}\left(4\right):
\mathrm{degreeseq}\left(G\right)
[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]
G≔\mathrm{random}\left(5\right):
\mathrm{ends}\left(G\right)
{{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}}}
\mathrm{degreeseq}\left(G\right)
[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]
GraphTheory[DegreeSequence]
networks(deprecated)[vdegree]
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February - Volume 108, Number 1 April - Volume 108, Number 2 June - Volume 108, Number 3A July - Volume 108, Number 3B August - Volume 108, Number 4 October - Volume 108, Number 5A November - Volume 108, Number 5B December - Volume 108, Number 6
Introduction to the Special Issue “Fifty Years after the 1967 Koyna Earthquake—Lessons Learned about Reservoir Seismicity (RTS)”
N. Purnachandra Rao; Kusumita Arora; Alexander Ponomarev; Aderson Farias do Nascimento
Bulletin of the Seismological Society of America October 02, 2018, Vol.108, 2903-2906. doi:https://doi.org/10.1785/0120180240
Review: Reservoir Triggered Seismicity (RTS) at Koyna, India, over the Past 50 Yrs
Bulletin of the Seismological Society of America July 24, 2018, Vol.108, 2907-2918. doi:https://doi.org/10.1785/0120180019
Lineaments in Deccan Basalts: The Basement Connection in the Koyna–Warna RTS Region
Kusumita Arora; Y. Srinu; D. Gopinadh; R. K. Chadha; Haris Raza; Valentin Mikhailov; Alexander Ponomarev; Elena Kiseleva; Vladimir Smirnov
Bulletin of the Seismological Society of America September 11, 2018, Vol.108, 2919-2932. doi:https://doi.org/10.1785/0120180011
Geodetic Constraints on Tectonic and Anthropogenic Deformation and Seismogenesis of Koyna–Warna Region, India
Vineet K. Gahalaut; Kalpna Gahalaut; Joshi K. Catherine; K. M. Sreejith; Ritesh Agrawal; Rajeev Kumar Yadav; Ch. Mohanalakshmi; M. S. Naidu; V. Rajeshwar Rao
Rate of Change in Lake Level and Its Impact on Reservoir Triggered Seismicity
David W. Simpson; Josh C. Stachnik; Sobit Kh. Negmatoullaev
Bulletin of the Seismological Society of America August 21, 2018, Vol.108, 2943-2954. doi:https://doi.org/10.1785/0120180026
GPS Measurements of Deformation Caused by Seasonal Filling and Emptying Cycles of Four Hydroelectric Reservoirs in India
Rakesh Dumka; Pallabee Choudhury; V. K. Gahalaut; Kalpna Gahalaut; Rajeev Kumar Yadav
Patterns of Reservoir‐Triggered Seismicity in a Low‐Seismicity Region of France
J.‐R. Grasso; A. Karimov; D. Amorese; C. Sue; C. Voisin
The Effects of Weak Dynamic Pulses on the Slip Dynamics of a Laboratory Fault
Gevorg G. Kocharyan; Alexey A. Ostapchuk; Dmitry V. Pavlov; Vadim K. Markov
M. Kinali; S. Pytharouli; R. J. Lunn; Z. K. Shipton; M. Stillings; R. Lord; S. Thompson
Interpretations of Reservoir‐Induced Seismicity May Not Always Be Valid: The Case of Seismicity during the Impoundment of the Kremasta Dam (Greece, 1965–1966)
S. C. Stiros; S. Pytharouli
A Possible Mechanism of Reservoir‐Induced Earthquakes in the Three Gorges Reservoir, Central China
Lifen Zhang; Jinggang Li; Xiaodan Sun; Wulin Liao; Yannan Zhao; Guichun Wei; Chaofeng He
A Detailed Insight into Fluid Infiltration in the Three Gorges Reservoir Area, China, from 3D
VP
VP/VS
QP
QS
Lianqing Zhou; Cuiping Zhao; Jun Luo; Zhangli Chen
Reservoir‐Triggered Seismicity in Brazil: Statistical Characteristics in a Midplate Environment
Lucas Vieira Barros; Marcelo Assumpção; Luis Carlos Ribotta; Vinicius M. Ferreira; Juraci M. de Carvalho; Brigida M. D. Bowen; Diogo F. Albuquerque
A Review of Reservoir Monitoring and Reservoir‐Triggered Seismicity in Canada
Maurice Lamontagne; Garry Rogers; John Cassidy; Jean‐Pierre Tournier; Martin S. Lawrence
A New Case of Triggered Seismicity Associated with the Itezhi‐Tezhi Reservoir, Zambia
I. D. Gupta
El Cuchillo Seismic Sequence of October 2013–July 2014 in the Burgos Basin, Northeastern Mexico: Hydraulic Fracturing or Reservoir‐Induced Seismicity?
Juan C. Montalvo‐Arrieta; Xyoli Pérez‐Campos; Luis G. Ramos‐Zuñiga; Edgar G. Paz‐Martínez; Jorge A. Salinas‐Jasso; Ignacio Navarro de León; Juan A. Ramírez‐Fernández
Seismicity Migration Induced by the Açu Reservoir, Northeast Brazil, and Implications for Fault Hydraulic Variability
Pedro A. R. Ferreira; Joaquim M. Ferreira; Aderson F. do Nascimento; Francisco H. R. Bezerra; Heleno C. Lima Neto; Eduardo A. S. Menezes
Influence of Tehri Reservoir Impoundment on Local Seismicity of Northwest Himalaya
Kalpna Gahalaut; Sandeep Gupta; Vineet Kumar Gahalaut; P. Mahesh
Cellular Seismology Analysis of Reservoir‐Triggered Seismicity Associated with Armenian Dams
Lilit Sargsyan; Natasha E. Toghramadjian; Alan L. Kafka
Luciano Telesca; Tamaz Chelidze
Mw
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Memorylessness - Wikipédia Sunda, énsiklopédi bébas
Dina tiori probabiliti, memorylessness nyaéta sipat penting sebaran probabiliti: sebaran eksponensial sarta sebaran geometrik.
1 Discrete memorylessness
2 Example and motivation for the name memorylessness
2.1 A frequent misunderstanding
3 Continuous memorylessness
Discrete memorylessness[édit | édit sumber]
Suppose X is a discrete random variable whose values lie in the set { 0, 1, 2, ... } or in the set { 1, 2, 3, ... }. The probability distribution of X is memoryless precisely if for any x, y in { 0, 1, 2, ... } or in { 1, 2, 3, ... }, (as the case may be), we have
{\displaystyle P(X>x+y\mid X>x)=P(X>y).}
It can réadily be shown that the only probability distributions that enjoy this discrete memorylessness are geometric distributions. These are the distributions of the number of independent Bernoulli trials needed to get one "success", with a fixed probability p of "success" on éach trial.
Example and motivation for the name memorylessness[édit | édit sumber]
For example, suppose a die is thrown as many times as it takes to get a "1", so that the probability of "success" on éach trial is 1/6, and the random variable X is the number of times the die must be thrown. Then X has a géometric distribution, and the conditional probability that the die must be thrown at léast four more times to get a "1", given that it has alréady been thrown 10 times without a "1" appéaring, is no different from the original probability that the die would be thrown at léast four times. In effect, the random process does not "remember" how many failures have occurred so far.
A frequent misunderstanding[édit | édit sumber]
Memorylessness is often misunderstood by students taking courses on probability: the fact that P(X > 16 | X > 12) = P(X > 4) does not méan that the events X > 16 and X > 12 are independent; i.e., it does not méan that P(X > 16 | X > 12) = P(X > 16). To summarize: "memorlessness" of the probability distribution of the number of trials X until the first success méans
{\displaystyle \mathrm {(Right)} \ P(X>16\mid X>12)=P(X>4).}
It does not méan
{\displaystyle \mathrm {(Wrong)} \ P(X>16\mid X>12)=P(X>16).}
(That would be independence. These two events are not independent.)
Continuous memorylessness[édit | édit sumber]
Suppose that rather than considering the discrete number of trials until the first "success", we consider continuous waiting time T until the arrival of the first phone call at a switchboard. To say that the probability distribtuion of T is memoryless méans that for any positive real numbers s and t, we have
{\displaystyle P(T>t+s\mid T>t)=P(T>s).}
The only difference between this and the discrete version is that instéad of requiring s and t to be positive (or, in some cases, nonnegative) integers, thus achieving discreteness, we allow them to be réal numbers that are not necessarily integers.
It can be shown that the only probability distributions that enjoy this continuous memorylessness are the sebaran eksponensial.
Dicomot ti "https://su.wikipedia.org/w/index.php?title=Memorylessness&oldid=485175"
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Standard Error of Measurement - SAGE Research Methods
Standard Error of Measurement | The SAGE Encyclopedia of Educational Research, Measurement, and Evaluation
The term standard error of measurement indicates the spread of measurement errors when estimating an examinee’s true score from the observed score. Standard error of measurement is most frequently useful in test reliability. An observed score is an examinee’s obtained score, or raw score, on a particular test. A true score would be determined if this particular test was then given to a group of examinees 1,000 times, under identical conditions. The average of those observed scores would yield the best estimate of the examinees’ true abilities. Standard deviation is applied to the average of those scores across persons and administrations to determine the standard error of measurement. Observed score and true score can be used together to determine the amount of error:
{\text{Score}}_{\text{true}}={\text{ Score}}_{\text{observed}}+{\text{ Score}}_{\text{error}}.
However, this true ...
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Solve nonstiff differential equations — high order method - MATLAB ode89 - MathWorks Deutschland
Solve nonstiff differential equations — high order method
y\text{'}=f\left(t,y\right)
y\text{'}=f\left(t,y\right)
M\left(t,y\right)y\text{'}=f\left(t,y\right)
. The solvers use similar syntaxes. The ode23s solver can solve problems with a mass matrix only if the mass matrix is constant. ode15s and ode23t can solve problems with a mass matrix that is singular, known as differential-algebraic equations (DAEs). Specify the mass matrix using the Mass option of odeset.
[t,y] = ode89(odefun,tspan,y0,options) also uses the integration settings defined by options, which is an argument created using the odeset function. For example, set the AbsTol and RelTol options to specify absolute and relative error tolerances, or set the Mass option to provide a mass matrix.
For each event function, specify whether the integration is to terminate at a zero and whether the direction of the zero crossing is significant. Do this by setting the 'Events' option of odeset to a function, such as myEventFcn or @myEventFcn, and create a corresponding function: [value,isterminal,direction] = myEventFcn(t,y). For more information, see ODE Event Location.
{y}^{\prime }=2t.
{y}_{1}^{\prime \prime }-\mu \left(1-{y}_{1}^{2}\right){y}_{1}^{\prime }+{y}_{1}=0,
\mu >0
{y}_{1}^{\prime }={y}_{2}
\begin{array}{cl}{y}_{1}^{\prime }& ={y}_{2}\\ {y}_{2}^{\prime }& =\mu \left(1-{y}_{1}^{2}\right){y}_{2}-{y}_{1}.\end{array}
\mu =1
{y}_{1}
{y}_{2}
{y}_{1}
{y}_{2}
{y}_{1}
{y}_{2}
{y}^{\prime \prime }=\frac{A}{B}ty.
\begin{array}{cl}{y}_{1}^{\prime }& ={y}_{2}\\ {y}_{2}^{\prime }& =\frac{A}{B}t{y}_{1}.\end{array}
odefcn, a local function at the end of this example, represents this system of equations as a function that accepts four input arguments: t, y, A, and B.
Compared to ode45, the ode113, ode78, and ode89 solvers are better at solving problems with stringent error tolerances. A common situation where these solvers excel is in orbital dynamics problems, where the solution curve is smooth and requires high accuracy in each step from the solver.
The two-body problem considers two interacting masses m1 and m2 orbiting a common plane. In this example, one of the masses is significantly larger than the other. With the heavy body at the origin, the equations of motion are
\begin{array}{l}{{\mathit{x}}^{\prime }}^{\prime }=-\mathit{x}/{\mathit{r}}^{3}\\ {{\mathit{y}}^{\prime }}^{\prime }=-\mathit{y}/{\mathit{r}}^{3},\end{array}
\mathit{r}=\sqrt{{\mathit{x}}^{2}+{\mathit{y}}^{2}}.
\begin{array}{l}{\mathit{y}}_{1}=\mathit{x}\\ {\mathit{y}}_{2}={\mathit{x}}^{\prime }\\ {\mathit{y}}_{3}=\mathit{y}\\ {\mathit{y}}_{4}={\mathit{y}}^{\prime }.\end{array}
\begin{array}{l}{{\mathit{y}}_{1}}^{\prime }={\mathit{y}}_{2}\\ {{\mathit{y}}_{2}}^{\prime }=-{\mathit{y}}_{1}/{\mathit{r}}^{3}\\ {{\mathit{y}}_{3}}^{\prime }={\mathit{y}}_{4}\\ {{\mathit{y}}_{4}}^{\prime }=-{\mathit{y}}_{3}/{\mathit{r}}^{3}.\end{array}
twobodyode, a local function included at the end of this example, codes the system of equations for the two-body problem.
% Two-body problem with one mass much larger than the other.
Solve the system of ODEs using ode89. Specify stringent error tolerances of 1e-13 for RelTol and 1e-14 for AbsTol.
[t,y] = ode89(@twobodyode, tspan, y0, opts);
Compared to ode45, the ode89 solver is able to obtain the solution faster and with fewer steps and function evaluations.
f\left(t,y\right)
y\text{'}=5y-3
\begin{array}{l}y{\text{'}}_{1}={y}_{1}+2{y}_{2}\\ y{\text{'}}_{2}=3{y}_{1}+2{y}_{2}\end{array}
ode89 is an implementation of Verner's "most robust" Runge-Kutta 9(8) pair with an 8th-order continuous extension. The solution is advanced with the 9th-order result. The 8th-order continuous extension requires five additional evaluations of odefun, but only on steps that require interpolation.
[1] Verner, J. H. “Numerically Optimal Runge–Kutta Pairs with Interpolants.” Numerical Algorithms 53, no. 2–3 (March 2010): 383–396. https://doi.org/10.1007/s11075-009-9290-3.
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Property (T) in julia - Nextjournal
Marek Kaluba / Mar 18 2019
Property (T) in julia
The code below is designed to run on julia 1.1.0.
Before running any computations we need to set-up the environment and download the pre-computed data.
1.1. Julia environment
git clone https://git.wmi.amu.edu.pl/kalmar/1712.07167.git
Bash in JuliaJulia 1.1
Pkg.activate("1712.07167")
JuliaJulia 1.1
1.2. Running tests (optional)
Now everything seems to be installed, but let's check that things run as they should:
Pkg.test("PropertyT")
1.3. Getting the pre-computed data
We need to download and unpack the data from Zenodo
wget -O oSAutF5_r2.tar.xz "https://zenodo.org/record/1913734/files/oSAutF5_r2.tar.xz?download=1"
tar xvf oSAutF5_r2.tar.xz
The above commands need to be typically run only once.
2. Replicating computations for ArXiv:1712.07167
This section shows how one should be able to replicate the computations presented in has property (T) by M. Kaluba, P.W. Nowak and N. Ozawa.
To speed-up certain computations you may wish to set the environmental variable JULIA_NUM_THREADS to the number of (the physical) cores of cpu.
using GroupRings
using PropertyT
BLAS.set_num_threads(Threads.nthreads())
In the cell below we define the group ring of the special automorphism group of the free group (by reading the division table from pm.jld file) and construct the Laplacian
\Delta = |S_5| - \sum_{s\in S_5} s
remembering that in the ordered basis of the identity comes first with generators following directly after. Note that the generating set of consists of exactlytransvections (due to technical issue with the transition from julia-0.6 to julia-1.0 we can not load the supplied oSAutF5_r2/delta.jld file).
G = SAut(FreeGroup(5))
pm = load("oSAutF5_r2/pm.jld", "pm")
RG = GroupRing(G, pm)
@show RG
S_size = 80
Δ_coeff = SparseVector(maximum(pm), collect(1:(1+S_size)), [S_size; -ones(S_size)])
Δ = GroupRingElem(Δ_coeff, RG);
Δ² = Δ^2;
2.1. Recomputing from scratch group ring structure
The computations above could be re-done from scratch (i.e. without relying on the provided pm) by executing
S = AbstractAlgebra.gens(G);
@time E₄, sizes = Groups.generate_balls(S, Id, radius=2radius) # takes lots of time and space
E₄_rdict = GroupRings.reverse_dict(E₄)
@time pm = GroupRings.create_pm(E₄, E₄_rdict, sizes[radius]; twisted=true) # takes lots of time and space
RG = GroupRing(G, E₄, E₄_rdict, pm)
Δ = PropertyT.spLaplacian(RG, S)
Δ² = Δ^2
2.2. Loading the solution
Next we load the solution :
λ₀ = load("oSAutF5_r2/1.3/lambda.jld", "λ")
lambda_0JuliaJulia 1.1
As can be seen, we will be comparing the accuracy ofbelow against the numerical value of .
P₀ = load("oSAutF5_r2/1.3/SDPmatrix.jld", "P")
@time Q = real(sqrt(P₀))
Now we project the columns of to the augmentation ideal, in interval arithmetic. The returned check_columns_augmentation is a boolean flag to detect if the projection was successful, i.e. if we can guarantee that each column of Q_aug can be represented by an element from the augmentation ideal. (If it were not successful, one may project Q = PropertyT.augIdproj(Q) in the floating point arithmetic prior to the cell below).
Q_aug, check_columns_augmentation = PropertyT.augIdproj(Interval, Q);
@show check_columns_augmentation
if !check_columns_augmentation
@warn "Columns of Q are not guaranteed to represent elements of the augmentation ideal!"
Finally we compute the actual sum of squares decomposition represented by Q_aug:
@time sos = PropertyT.compute_SOS(RG, Q_aug);
The residual of the solution and is
residual = Δ² - @interval(λ₀)*Δ - sos;
norm(residual, 1)
residual_normJuliaJulia 1.1
[8.35381e-06, 8.42859e-06]
thus we can certify that the spectral gapis at least the lower end of the interval
certified = @interval(λ₀) - 2^2*norm(residual, 1)
print(certified.lo)
This, via estimateleads to the lower bound on the Kazhdan constant of
κ = (sqrt(2certified/S_size)).lo
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Are171887 (talk | contribs)
(Added Category: Orbiter Math.)
(Added articles category.)
[[Category: Orbiter Math]]
{{HasPrecis}}
{\displaystyle \left\{Q_{0},P_{0}\right\}}
{\displaystyle \left\{Q_{2},P_{2}\right\}}
{\displaystyle \delta t}
{\displaystyle \left\{Q_{0},P_{0}\right\}}
{\displaystyle t=t_{0}}
{\displaystyle \left\{Q_{2},P_{2}\right\}}
{\displaystyle t=t_{0}+\delta t}
{\displaystyle Q_{1}\leftarrow Q_{0}+{\frac {1}{2}}\,\delta t\,P_{0}}
{\displaystyle P_{2}\leftarrow P_{0}+\delta t\,F\left(Q_{1}\right)}
{\displaystyle Q_{2}\leftarrow Q_{1}+{\frac {1}{2}}\,\delta t\,P_{2}}
{\displaystyle \left\{Q_{0},P_{0}\right\}}
{\displaystyle \delta t}
{\displaystyle F\left(Q_{1}\right)}
{\displaystyle Q_{1}}
{\displaystyle F(Q)}
{\displaystyle F(Q)=-{\frac {\mu \,m}{(Q-Q^{*})^{2}}}}
{\displaystyle Q>Q^{*}}
{\displaystyle \mu }
{\displaystyle Q^{*}}
{\displaystyle \mu =0.00029591220828559115}
{\displaystyle m=1}
{\displaystyle m=1/354710}
{\displaystyle m=1}
{\displaystyle Q^{*}}
{\displaystyle \left\{Q_{x,0},Q_{y,0},Q_{z,0},P_{x,0},P_{y,0},P_{z,0}\right\}}
{\displaystyle \left\{Q_{x,2},Q_{y,2},Q_{z,2},P_{x,2},P_{y,2},P_{z,2}\right\}}
{\displaystyle Q_{x,1}\leftarrow Q_{x,0}+{\frac {1}{2}}\,\delta t\,P_{x,0}}
{\displaystyle Q_{y,1}\leftarrow Q_{y,0}+{\frac {1}{2}}\,\delta t\,P_{y,0}}
{\displaystyle Q_{z,1}\leftarrow Q_{z,0}+{\frac {1}{2}}\,\delta t\,P_{z,0}}
{\displaystyle P_{x,2}\leftarrow P_{x,0}+\delta t\,F_{x}\left(Q_{x,1},Q_{y,1},Q_{z,1}\right)}
{\displaystyle P_{y,2}\leftarrow P_{y,0}+\delta t\,F_{y}\left(Q_{x,1},Q_{y,1},Q_{z,1}\right)}
{\displaystyle P_{z,2}\leftarrow P_{z,0}+\delta t\,F_{z}\left(Q_{x,1},Q_{y,1},Q_{z,1}\right)}
{\displaystyle Q_{x,2}\leftarrow Q_{x,1}+{\frac {1}{2}}\,\delta t\,P_{x,2}}
{\displaystyle Q_{y,2}\leftarrow Q_{y,1}+{\frac {1}{2}}\,\delta t\,P_{y,2}}
{\displaystyle Q_{z,2}\leftarrow Q_{z,1}+{\frac {1}{2}}\,\delta t\,P_{z,2}}
{\displaystyle F_{x}\left(Q_{x},Q_{y},Q_{z}\right)}
{\displaystyle {Q_{x},Q_{y},Q_{z}}}
{\displaystyle F_{y}\left(Q_{x},Q_{y},Q_{z}\right)}
{\displaystyle {Q_{x},Q_{y},Q_{z}}}
{\displaystyle F_{z}\left(Q_{x},Q_{y},Q_{z}\right)}
{\displaystyle {Q_{x},Q_{y},Q_{z}}}
{\displaystyle F_{x}\left(Q_{x},Q_{y},Q_{z}\right)=-{\frac {\mu \,m\,\left(Q_{x}-Q_{x}^{*}\right)}{\left(\left(Q_{x}-Q_{x}^{*}\right)^{2}+\left(Q_{y}-Q_{y}^{*}\right)^{2}+\left(Q_{z}-Q_{z}^{*}\right)^{2}\right)^{3/2}}}}
{\displaystyle F_{y}\left(Q_{x},Q_{y},Q_{z}\right)=-{\frac {\mu \,m\,\left(Q_{y}-Q_{y}^{*}\right)}{\left(\left(Q_{x}-Q_{x}^{*}\right)^{2}+\left(Q_{y}-Q_{y}^{*}\right)^{2}+\left(Q_{z}-Q_{z}^{*}\right)^{2}\right)^{3/2}}}}
{\displaystyle F_{z}\left(Q_{x},Q_{y},Q_{z}\right)=-{\frac {\mu \,m\,\left(Q_{z}-Q_{z}^{*}\right)}{\left(\left(Q_{x}-Q_{x}^{*}\right)^{2}+\left(Q_{y}-Q_{y}^{*}\right)^{2}+\left(Q_{z}-Q_{z}^{*}\right)^{2}\right)^{3/2}}}}
{\displaystyle \left(Q_{x}^{*},Q_{y}^{*},Q_{z}^{*}\right)}
{\displaystyle \left(Q_{x},Q_{y},Q_{z}\right)}
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Surface Temperature in Oscillating Sliding Interfaces | J. Tribol. | ASME Digital Collection
M. Mansouri, Graduate Student,,
M. Mansouri, Graduate Student,
Manuscript received November 5, 2004; revision received June 18, 2004. Review conducted by: C. H. Venner.
Mansouri, M., and Khonsari, M. M. (February 7, 2005). "Surface Temperature in Oscillating Sliding Interfaces ." ASME. J. Tribol. January 2005; 127(1): 1–9. https://doi.org/10.1115/1.1828065
A model is developed to predict the behavior of two sliding bodies undergoing oscillatory motion. A set of four dimensionless groups is introduced to characterize the transient dimensionless surface temperature rise. They are: the Peclet number
Pe,
the Biot number
Bi,
the amplitude of oscillation A, and the Hertzian semi-contact width α. Also considered in the analysis is the effect of the ratio
β=A/α
of the amplitude to the semi-contact width. The results of a series of simulations, covering a range of these independent parameters, are presented, and examples are provided to illuminate the utility of the model.
oscillations, mechanical contact, surface phenomena, sliding friction, Hertzian Line Contact, Fretting, Oscillatory, Transient Temperature
Oscillations, Temperature, Steady state, Heat flux
Theoretical Study of Temperature Rise at Surfaces of Actual Contact Under Oiliness Lubricating Conditions
Proc. Royal Society. N. S. W.
Calculation of the Temperature Development in a Contact Heated in the Contact Surface, and Application to the Problem of the Temperature Rise in a Sliding Contact
On Temperature Transients at Sliding Surfaces
Interfacial Temperature Distribution Within a Sliding Hertzian Contact
American Inst. Chemical Eng.
Prediction and Measurement of Surface Temperature Rise at the Contact Interface for Oscillatory Sliding
Contact Surface Temperature Models for Finite Bodies in Dry and Boundary Lubricated Sliding Systems
Surface Temperatures in Fretting Contact
O¨zisik, M. N., 1993, Heat Conduction, John Wiley & Sons, New York.
Dorfman, L. A., 1963, Hydrodynamic Res´ı`stance and the Heat Loss of Rotating Solids, Oliver & Boyd, Ltd., Edinburgh, Scotland.
Khonsari, M. M., and Hua, D. Y., 1997, Tribology Data Handbook, STLE, Vol. 2, pp. 611–637.
Patankar, S. V., 1980, Numerical Heat Transfer and Fluid Flow, McGraw-Hill Book Company, Hemisphere Pub. Corp., New York.
Numerical Simulations on Supercritical Water Flow Instability
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Introduction to Robotics/Electrical Components/Lecture/Teachers - Wikiversity
Introduction to Robotics/Electrical Components/Lecture/Teachers
< Introduction to Robotics | Electrical Components/Lecture
1 Electricity Flowing in a Circuit
2 Resistors: Resisting current
4 Resistor Color Codes
5 Diodes: 1-way Gates
Electricity Flowing in a Circuit[edit | edit source]
Electricity flows through circuits in the same way as water flows through pipes. Water only flows where the pipes are properly connected. Likewise, electricity will only flow where the components and wires are properly connected.
Notice also that electricity will only flow in a "complete circuit". Electricity must come from a source, and must go to a ground.
There are two quantities that we need to measure when dealing with electricity: Voltage and Current.
Voltage is a measure of the "pressure" of the electricity. More voltage means more potential for the electricity to flow.
Current is a measure of the "amount" of the electricity. More current means more electrons are flowing.
Resistors: Resisting current[edit | edit source]
Resistors are said to "resist current". At the same pressure (voltage), resistors cause less current to flow. Resistors are measured in units called "Ohms", and are labled with a Ω (greek capital omega).
For the same pressure, more ohms means less current, and fewer ohms means more current.
Ohm's Law[edit | edit source]
The relationship between current, voltage, and resistance is shown by Ohm's Law:
{\displaystyle V=IR}
Resistor Color Codes[edit | edit source]
Resistor values are shown using a special color code. A resistor will have 4 colored bands on it, one of the bands will be gold or silver. To read a resistor, put the gold/silver band on the right. The first two bands on the left are the digits, and the third band is an exponent.
To calculate the value of a resistor, we take the first two digits to form a 2-digit number, and we multiply it by 10 to the power of the third band. For instance, if our resistor was:
red green yellow gold
Then the value of the resistor would be:
{\displaystyle 25\times 10^{4}=250000=250k\Omega }
The fourth band is a "tolerance" band, or an error code. Silver means the resistor has a 10% error, and gold means the resistor has a 5% error.
{\displaystyle 25\times 10^{4}=250000=250k\Omega }
{\displaystyle 25\times 10^{4}=250000=250k\Omega }
{\displaystyle 25\times 10^{4}=250000=250k\Omega }
{\displaystyle 25\times 10^{4}=250000=250k\Omega }
{\displaystyle 25\times 10^{4}=250000=250k\Omega }
Diodes: 1-way Gates[edit | edit source]
Retrieved from "https://en.wikiversity.org/w/index.php?title=Introduction_to_Robotics/Electrical_Components/Lecture/Teachers&oldid=1608344"
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Add Structure - Maple Help
Home : Support : Online Help : Science and Engineering : Scientific Error Analysis : Commands : Add Structure
AddStructure( struct, val_proc, uncer_proc, ident_proc, opts )
type; type of the new quantity-with-error structure
val_proc
procedure; returns the central value of a quantity-with-error of type struct
uncer_proc
procedure; returns the absolute uncertainty of a quantity-with-error of type struct
ident_proc
procedure; returns an identifier of a quantity-with-error of type struct
(optional) equation of the form check= true or false; allows overwrite of existing interface
The AddStructure( struct, val_proc, uncer_proc, ident_proc ) command adds the interface of a new quantity-with-error structure to the ScientificErrorAnalysis package for the current session. The interface is defined by a maple type and three procedures.
To add a structure to all future Maple sessions, add the AddStructure command to your Maple initialization file. For more information, see Create Maple Initialization File.
The struct argument specifies the maple type of the quantity-with-error structure.
The val_proc argument is a procedure that, when applied to any quantity-with-error of type struct, returns the central value of the quantity-with-error.
The uncer_proc argument is a procedure that, when applied to any quantity-with-error of type struct, returns the absolute uncertainty of the quantity-with-error.
The ident_proc argument is a procedure that, when applied to any quantity-with-error of type struct, returns an identifier that represents the quantity-with-error.
For a quantity-with-error without functional dependence, the identifier must be a simple maple object, distinct from all other identifiers of quantities-with-error of the same structure. ScientificErrorAnalysis uses these identifiers to maintain a table of correlations between quantities-with-error. See SetCorrelation and combine/errors for more information.
For a quantity-with-error with functional dependence, the identifier must be an algebraic expression containing one or more quantities-with-error of the same structure. The identifier defines the object's functional dependence, which ScientificErrorAnalysis uses to calculate the variance or covariances with other quantities-with-error. See Variance, Covariance, and combine/errors for more information.
If check=true, the type of the new structure cannot be the same as that of an existing structure. The default value of check is true. If check=false, the type of the new structure can be the same as that of an existing structure, in which case the existing interface is overwritten.
It is recommended that you do not overwrite the predefined interfaces.
The AddStructure routine was used to define the interface to the Quantity(...) objects of ScientificErrorAnalysis.
The AddStructure routine was also used to define the interface to two other quantity-with-error structures, the Constant(...) and Element(...) objects of the ScientificConstants package. In the case of the Constant(...) objects, the interface communicates the functional dependence of any derived Constants to ScientificErrorAnalysis.
Toy example.
\mathrm{with}\left(\mathrm{ScientificErrorAnalysis}\right):
\mathrm{AddStructure}\left('\mathrm{specfunc}'\left('\mathrm{anything}',F\right),x↦\mathrm{op}\left(1,x\right),x↦\mathrm{op}\left(2,x\right),x↦\mathrm{op}\left(3,x\right)\right)
\mathrm{o1}≔F\left(10.0,1.0,1\right)
\textcolor[rgb]{0,0,1}{\mathrm{o1}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{F}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{10.0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}\right)
\left(\mathrm{evalf},\mathrm{GetError}\right)\left(\mathrm{o1}\right)
\textcolor[rgb]{0,0,1}{F}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{10.0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.0}
\mathrm{o2}≔F\left(20.0,3.0,2\right)
\textcolor[rgb]{0,0,1}{\mathrm{o2}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{F}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{20.0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3.0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\right)
\mathrm{combine}\left(\mathrm{o1}\mathrm{o2},'\mathrm{errors}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{Quantity}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{200.00}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{36.05551275}\right)
\mathrm{SetCorrelation}\left(\mathrm{o1},\mathrm{o2},0.1\right)
\mathrm{combine}\left(\mathrm{o1}\mathrm{o2},'\mathrm{errors}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{Quantity}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{200.00}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{37.68288736}\right)
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Feature importance - Model analysis | CatBoost
InternalFeatureImportance
PredictionDiff
The individual importance values for each of the input features (the default feature importances calculation method for non-ranking metrics).
For each feature, PredictionValuesChange shows how much on average the prediction changes if the feature value changes. The bigger the value of the importance the bigger on average is the change to the prediction value, if this feature is changed.
See the Regular feature importance file format.
Leaf pairs that are compared have different split values in the node on the path to these leaves. If the split condition is met (this condition depends on the feature F), the object goes to the left subtree; otherwise it goes to the right one.
feature\_importance_{F} = \displaystyle\sum\limits_{trees, leafs_{F}} \left(v_{1} - avr \right)^{2} \cdot c_{1} + \left( v_{2} - avr \right)^{2} \cdot c_{2} { , }
avr = \displaystyle\frac{v_{1} \cdot c_{1} + v_{2} \cdot c_{2}}{c_{1} + c_{2}} { , where}
c_1, c_2
represent the total weight of objects in the left and right leaves respectively. This weight is equal to the number of objects in each leaf if weights are not specified for the dataset.
v_1, v_2
represent the formula value in the left and right leaves respectively.
If the model uses a combination of some of the input features instead of using them individually, an average feature importance for these features is calculated and output. For example, the model uses a combination of features f54, c56 and f77. First, the feature importance is calculated for the combination of these features. Then the resulting value is divided by three and is assigned to each of the features.
If the model uses a feature both individually and in a combination with other features, the total importance value of this feature is defined using the following formula:
feature\_total\_importance_{j} = feature\_importance + \sum\limits_{i=1}^{N}average\_feature\_importance_{i} { , where}
feature\_importance_{j}
is the individual feature importance of the j-th feature.
average\_feature\_importance_{i}
is the average feature importance of the j-th feature in the i-th combinational feature.
O(trees\_count \cdot depth \cdot 2 ^ {depth} \cdot dimension)
Feature importance values are normalized so that the sum of importances of all features is equal to 100. This is possible because the values of these importances are always non-negative.
Formula values inside different groups may vary significantly in ranking modes. This might lead to high importance values for some groupwise features, even though these features don't have a large impact on the resulting metric value.
The individual importance values for each of the input features (the default feature importances calculation method for ranking metrics). This type of feature importance can be used for any model, but is particularly useful for ranking models, where other feature importance types might give misleading results.
For each feature the value represents the difference between the loss value of the model with this feature and without it. The model without this feature is equivalent to the one that would have been trained if this feature was excluded from the dataset. Since it is computationally expensive to retrain the model without one of the features, this model is built approximately using the original model with this feature removed from all the trees in the ensemble. The calculation of this feature importance requires a dataset and, therefore, the calculated value is dataset-dependent.
Depend on the conditions for achieving the best metric value to ensure that the more important is a feature, the higher is the corresponding importance value.
Minimum/maximum best value metric:
feature\_importance_{i} = \pm (metric E_{i}v) - metric(v))
Exact best value metric:
feature\_importance_{i} = abs(metric(E_{i}v) - best\_value) - abs(metric(v) - best\_value)
In general, the value of LossFunctionChange can be negative.
E_{i}v
is the mathematical expectation of the formula value without the
i
-th feature. If the feature
is on the path to a leaf, the new leaf value is set to the weighted average of values of leaves that have different paths by feature value. Weights represent the total weight of objects in the corresponding leaf. This weight is equal to the number of objects in each leaf if weights are not specified for the dataset.
For feature combinations (
F = (f_{1}, ..., f_{n})
) the average value on a leaf is calculated as follows:
E_fv = \displaystyle\left(\frac{(n - 1) v + E_{f}v}{n}\right)
v
is the vector with formula values for the dataset. The values of the training dataset are used if both training and validation datasets are provided.
metric is the loss function specified in the training parameters.
The pool random subset size used for calculation is determined as follows:
subset\_size = min(documentCount, max(2e5, \frac{2e9}{featureCount}))
O(trees\_count \cdot (depth + sub\_samples\_count) \cdot 2 ^ {depth} +
+ Eval\_metric\_complexity(model, sub\_samples\_count) \cdot features\_count) { ,}
sub\_samples\_count = Min(samples\_count, Max(10^5, 10^9 / features\_count))
This feature importance approximates the difference between metric values calculated on the following models:
The model with the
i
-th feature excluded.
The original model with all features.
The importance values both for each of the input features and for their combinations (if any).
See the InternalFeatureImportance file format.
feature\_importance_{F} = \displaystyle\sum\limits_{trees, leafs_{F}} \left(v_{1} - avr \right)^{2} \cdot c_{1} + \left( v_{2} - avr \right)^{2} \cdot c_{2} { , }
avr = \displaystyle\frac{v_{1} \cdot c_{1} + v_{2} \cdot c_{2}}{c_{1} + c_{2}} { , where}
c_{1}, c_{2}
v_{1}, v_{2}
feature\_total\_importance_{j} = feature\_importance + \sum\limits_{i=1}^{N}average\_feature\_importance_{i} { , where}
feature\_importance_{j}
average\_feature\_importance_{i}
O(trees\_count \cdot depth \cdot 2 ^ {depth} \cdot dimension)
The impact of a feature on the prediction results for a pair of objects. This type of feature importance is designed for analyzing the reasons for wrong ranking in a pair of documents, but it also can be used for any one-dimensional model.
For each feature PredictionDiff reflects the maximum possible change in the predictions difference if the value of the feature is changed for both objects. The change is considered only if there is an improvement in the direction of changing the order of documents.
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Revision as of 15:50, 1 February 2021 by CSeguinot (talk | contribs) (→Tx/Rx PSK equivalent baseband simulation)
This tutorial is also intended for non specialist, it involves as less maths as possible and present most results with GNURadio flowgraph. Some examples involving simple modulation scheme used in HAM radio are presented. Whilst introducing complex signal can be seen as increasing complexity, we will see that it drastically simplify studying impairment such as synchronization.
{\displaystyle z=a+jb}
{\displaystyle {\text{Re}}\{z\}=a}
{\displaystyle {\text{Im}}\{z\}=b}
{\displaystyle r=|z|={\sqrt {a^{2}+b^{2}}}}
{\displaystyle \phi =\arg(z)=\arctan(b/a)}
{\displaystyle z=r\left(cos(\phi )+jsin(\phi )\right)=re^{j\phi }}
{\displaystyle z_{1}=r_{1}e^{j\phi _{1}}}
{\displaystyle z_{1}=r_{2}e^{j\phi _{2}}}
{\displaystyle z=z_{1}z_{2}=r_{1}r_{2}e^{j(\phi _{1}+\phi _{2})}}
{\displaystyle +1=e^{j0}}
{\displaystyle +j=e^{j\pi /2}}
{\displaystyle -1=e^{j\pi }}
{\displaystyle -j=e^{j3\pi /2}}
{\displaystyle c(t)=i(t)+jq(t)=a(t)e^{j\phi (t)}}
{\displaystyle m(t)=a(t)c(t)=a(t)\cos(2\pi f_{0}t)}
{\displaystyle M(f)={\frac {1}{2}}{\big (}A(f-f_{0})+A(f+f_{0}){\big )}}
{\displaystyle f_{s}>F_{Max}}
{\displaystyle F_{MaxAudio}}
{\displaystyle F_{Max}=F_{0}+F_{MaxAudio}}
{\displaystyle X(f)=\int {x(t)e^{-2j\pi ft}}dt}
{\displaystyle x(t)\in \mathbb {R} }
{\displaystyle X(-f)=X^{*}(f)}
{\displaystyle |X(-f)|=|X(f)|}
{\displaystyle {\text{arg}}\{X(-f)\}=-{\text{arg}}\{X(f)\}}
{\displaystyle X_{s}(f)=\sum _{k}X(f-kF_{s})}
{\displaystyle X_{s}(f+kF_{s})=X_{s}(f)}
{\displaystyle {\tilde {m}}(t)}
{\displaystyle m(t)=a(t)\cos(2\pi F_{0}t+\phi (t))}
{\displaystyle {\tilde {m}}(t)=a(t)e^{(j(2\pi F_{0}t+\phi (t)))}=a(t)e^{j\phi (t)}e^{j2\pi F_{0}t}=m^{bb}(t)e^{j2\pi F_{0}t}}
{\displaystyle m(t)={\text{Re}}({\tilde {m}}(t))}
{\displaystyle {\tilde {m}}(t)}
{\displaystyle {\tilde {M}}(f)}
{\displaystyle {\tilde {M}}(f)=M^{+}(f)}
{\displaystyle =M^{+}(f)}
{\displaystyle =M(f)}
{\displaystyle m^{bb}(t)}
{\displaystyle m(t)}
{\displaystyle m^{bb}(t)={\tilde {m}}(t)e^{+j2\pi F_{0}t}=a(t)e^{j\phi (t)}}
{\displaystyle e^{+j2\pi F_{0}t}}
{\displaystyle M^{bb}(f)=M+(f+F_{0})}
{\displaystyle =M+(f)}
{\displaystyle {\tilde {m}}(t)}
{\displaystyle M^{bb}(f)}
{\displaystyle m(t)}
{\displaystyle M(f)}
{\displaystyle m^{bb}(t)=a(t)e^{j\phi (t)}=i(t)+jq(t)}
{\displaystyle m(t)={\text{Re}}\left[{\big (}i(t)+jq(t){\big )}e^{j2\pi F_{0}t}\right]}
{\displaystyle m(t)=i(t)\cos(2\pi F_{0}t)-q(t))\sin(2\pi F_{0}t)}
{\displaystyle i(t)=a(t)\cos(\phi (t))}
{\displaystyle q(t)=a(t)\sin(\phi (t))}
{\displaystyle i(t)\cos(2\pi F_{0}t)}
{\displaystyle q(t)\sin(-2\pi F_{0}t)}
{\displaystyle {\hat {i}}(t)=i(t)}
{\displaystyle {\hat {q}}(t)=q(t)}
{\displaystyle \cos(2\pi F_{0}t)}
{\displaystyle m(t)=A\cos(2\pi (F_{0}+\Delta f)t+\phi )}
{\displaystyle {\tilde {m}}(t)=Ae^{(j(2\pi (F_{0}+\Delta f)t+\phi ))}}
{\displaystyle m^{bb}(t)=Ae^{2j(\pi (\Delta ft+\phi )}=Ae^{j\Delta ft}e^{j\phi }}
{\displaystyle m^{bb}(t)=A}
{\displaystyle M^{bb}(f)}
{\displaystyle M^{bb}(f)=\delta (f))}
{\displaystyle +F_{0}}
{\displaystyle +F_{0}}
{\displaystyle M^{+}(f)=\delta (f-F_{0}))}
{\displaystyle -F_{0}}
{\displaystyle F_{0}}
{\displaystyle m^{bb}(t)=Ae^{2j\pi \Delta ft}}
{\displaystyle m(t)=a(t)\cos(2\pi (F_{0}t)=i(t)\cos(2\pi F_{0}t)-q(t)\sin(2\pi F_{0}t)=}
{\displaystyle i(t)=a(t)}
{\displaystyle q(t)=0}
{\displaystyle \phi \in \{\pi /4,3\pi /4,-3\pi /4,-\pi /4\}}
{\displaystyle m^{bb}(t)=a(t)e^{j\phi (t)}=i(t)+jq(t)}
{\displaystyle m^{bb}(t)\in \{1+j,-1+j,-1-j,1-j\}}
{\displaystyle i(t),q(t)\in \{+1,-1\}}
{\displaystyle {\hat {i}}(t)}
{\displaystyle i(t)}
{\displaystyle be^{j(2\pi \Delta ft+\phi )}}
{\displaystyle H(f)}
{\displaystyle H^{bb}(f)}
{\displaystyle H^{bb}(f)=H^{+}(f+F_{0})}
{\displaystyle H(f)}
{\displaystyle H^{bb}(f}
{\displaystyle m(t)=2b\cos(2\pi (F_{0}+\Delta f)t+\phi )}
{\displaystyle m^{bb}(t)=be^{2j(\pi (\Delta ft+\phi )}}
{\displaystyle N(t)}
{\displaystyle N^{bb}(t)}
{\displaystyle N^{bb}(f)=N^{+}(f+F_{0})}
{\displaystyle e^{-2j(\pi (\Delta ft)}}
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Application of reinforcement learning to control traffic signals - The SAS Data Science Blog
Application of reinforcement learning to control traffic signals
By Afshin Oroojlooy on The SAS Data Science Blog December 16, 2020 Topics | Machine Learning
In this article, we summarize our SAS research paper on the application of reinforcement learning to monitor traffic control signals which was recently accepted to the 34th Conference on Neural Information Processing Systems (NeurIPS 2020), Vancouver, Canada. This annual conference is hosted by the Neural Information Processing Systems Foundation, a non-profit corporation that promotes the exchange of ideas in neural information processing systems across multiple disciplines.
Traffic Signal Control Problem (TSCP)
With the emergence of urbanization and the increase in household car ownership, traffic congestion has been one of the major challenges in many highly-populated cities. Traffic congestion can be mitigated by road expansion/correction, sophisticated road allowance rules, or improved traffic signal controlling. Although either of these solutions could decrease travel times and fuel costs, optimizing the traffic signals is more convenient due to limited funding resources and the opportunity of finding more effective strategies. Here we introduce a new framework for learning a general traffic control policy that can be deployed in an intersection of interest and ease its traffic flow.
Let’s first define the TSCP. Consider the intersection in the following figure. There are some lanes entering and some leaving the intersection, shown with \(l_1^{in}, \dots, l_6^{out}\)
l_1^{in}, \dots, l_6^{out}
and \(l_1^{out}, \dots, l_6^{out}\)
l_1^{out}, \dots, l_6^{out}
, respectively. Also, six sets v1 ... v6 with each showing the involved traffic movements in each lane. A phase is defined as a set of non-conflicting traffic movements, which become red or green together. The decision is which phase becomes green at what time, and the objective is to minimize the average travel time (ATT) of all vehicles in the long-term.
Figure 1: A sample of an intersection
There are two main approaches for controlling signalized intersections, namely conventional and adaptive methods. In the former, customarily rule-based fixed cycles and phase times are determined a priori and offline based on historical measurements as well as some assumptions about the underlying problem structure. However, since traffic behavior is dynamically changing, that makes most conventional methods highly inefficient. In adaptive methods, decisions are made based on the current state of the intersection. In this category, methods like Self-organizing Traffic Light Control (SOTL) and MaxPressure brought considerable improvements in traffic signal control; nonetheless, they are short-sighted and do not consider the long-term effects of the decisions on the traffic. Besides, these methods do not use the feedback from previous actions toward making more efficient decisions.
Consider an environment and an agent, interacting with each other in several time-steps. At each time-step t, the agent observes the state of the system, st, takes an action, at, and passes it to the environment, and in response receives reward rt and the new state of the system, s(t+1). The goal is to maximize the sum of rewards in a long time, i.e., \(\sum_{t=0}^T \gamma^t r_t\)
\sum_{t=0}^T \gamma^t r_t
where T is an unknown value and 0<γ<1 is a discounting factor. The agent chooses the action based on a policy π which is a mapping function from state to actions. This iterative process is a general definition for Markov Decision Process (MDP). Reinforcement learning (RL) is an area of deep learning that deals with sequential decision-making problems which can be modeled as an MDP, and its goal is to train the agent to achieve the optimal policy.
RL for TSCP
Several reinforcement learning (RL) models are proposed to address these shortcomings. Although, they need to train a new policy for any new intersection or new traffic pattern. For example, if a policy π is trained for an intersection with 12 lanes, it cannot be used in an intersection with 13 lanes. Similarly, if the number of phases is different between two intersections, even if the number of lanes is the same, the policy of one does not work for the other one.
Similarly, the policy which is trained for the noon traffic-peek does not work for other times during the day.
Why existing RL methods are not universal?
The main reason is that there are a different number of inputs and outputs among different intersections. So, a trained model for one intersection does not work for another one.
AttendLight obtains such functionality through the following framework: Given state-input, which represents the traffic of lane l at time t, the embedding function g(.), and Lp showing the set of participating lanes at phase p, weights
\(w^t_l= \texttt{state-attention} \left(g(s_l^t), \sum_{i \in \mathcal{L}_p} \frac{g(s^t_i)}{|\mathcal{L}_p|} \right)\)
w^t_l= \texttt{state-attention} \left(g(s_l^t), \sum_{i \in \mathcal{L}_p} \frac{g(s^t_i)}{|\mathcal{L}_p|} \right)
are used to obtain the phase-feature \(z_t^p = \sum_{l \in \mathcal{L}_p} w_l^t \times g(s^t_l)\)
z_t^p = \sum_{l \in \mathcal{L}_p} w_l^t \times g(s^t_l)
The policy is also obtained by:
\(\pi^t = \texttt{action-attention} \left( LSTM(z_{p-green}^t), \{ z_p^t \in \text{all red phases}\} \right)\)
\pi^t = \texttt{action-attention} \left( LSTM(z_{p-green}^t), \{ z_p^t \in \text{all red phases}\} \right)
AttendLight
We propose AttendLight to train a single universal model to use it for any intersection with any number of roads, lanes, phases, and traffic flow. To achieve such functionality, we use two attention models: (i) State-Attention, which handles different numbers of roads/lanes by extracting meaningful phase representations \(z_p^t\)
z_p^t
for every phase p. (ii) Action-Attention, which decides for the next phase in an intersection with any number of phases. So, AttendLight does not need to be trained for new intersection and traffic data.
We explored 11 intersection topologies, with real-world traffic data from Atlanta and Hangzhou, and synthetic traffic-data with different congestion rates. This results in 112 intersection instances. We followed two training regimes: (i) Single-env regime in which we train and test on single intersections, and the goal is to compare the performance of AttendLight vs the current state of art algorithms. (ii) Multi-env regime, where the goal is to train a single universal policy that works for any new intersection and traffic data with no re-training. For the multi-env regime, we train on 42 training instances and test on 70 unseen instances.
Single-env regime results
AttendLight achieves the best result on 107 cases out of 112 (96% of cases). Also, on average of 112 cases, AttendLight yields an improvement of 46%, 39%, 34%, 16%, 9% over FixedTime, MaxPressure, SOTL, DQTSC-M, and FRAP, respectively. The following figure shows the comparison of results on four intersections.
Multi-env regime results
There is no RL algorithm in the literature with the same capability, so we compare AttendLight multi-env regime with single-env policies. In average of 112 cases, AttendLight yields improvement of 39%, 32%, 26%, 5%, and -3% over FixedTime, MaxPressure, SOTL, DQTSC-M, and FRAP, respectively. Note that here we compare the single policies obtained by AttendLight model which is trained on 42 intersection instances and tested on 70 testing intersection instances, though in SOTL, DQTSC-M, and FRAP there are 112 (were applicable) optimized policy, one for each intersection. \(\rho_m = \frac{a_m - b_m}{\max(a_m, b_m)}\)
\rho_m = \frac{a_m - b_m}{\max(a_m, b_m)}
Also, tam where am and bm are the ATT of AttendLight and the baseline method. As you can see, in most baselines, the distribution is leaned toward the negative side which shows the superiority of the AttendLight.
With AttendLight, we train a single policy to use for any new intersection with any new configuration and traffic-data. In addition, we can use this framework for Assemble-to-Order Systems, Dynamic Matching Problem, and Wireless Resource Allocation with no or small modifications. See more details on the paper!
Free trial: SAS Visual Data Mining and Machine Learning
Product: SAS Visual Data Mining and Machine Learning
Tags data scientist deep learning Markov Decision Process python Reinforcement Learning
Afshin Oroojloooy, Ph.D., is a Machine Learning Developer in the Machine Learning department within SAS R&D's Advanced Analytics division. He is focused on designing new Reinforcement Learning algorithms for real-world problems, e.g. inventory optimization on multi-echelon networks, traveling salesman problems, vehicle routing problem, customer journey optimization, traffic signal processing, HVAC, treatment planning, just a few to mention.
Analytics | Artificial Intelligence | Cloud | Data for Good | Machine Learning May 05, 2022
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Tor Onion Farming in 2021 | Quantum
The most important change since 2017 is probably proposal 224, which added v3 onions that are 56 characters long, instead of 16. When v3 onions first came out, I actually created one for my website, although, given the excellent write-up by Tudor, I didn’t bother writing my own blog post.
While the 16-character-long v2 onions are now deprecated, I still decided to generate one for compatibility and also out of interest. For this, I used the same tool as before: scallion. Having access to a GTX 1060 6GB compared to the mobile GeForce 940MX that I had in 2017 allowed me to be far more ambitious with my plans, however. Instead of spending a few hours generating an onion matching the seven-character prefix followed by a number (quantum2l7xnxwtb.onion) as I had in 2017, I decided to generate a nine-character prefix followed by a number (correctpw[234567]). This took me about 24 hours in total at 2.7 GH/s, and I finally obtained correctpw3wmw7mw.onion.
For the v3 onion, I used the tried-and-true tool mkp224o, just as Tudor had. This is a CPU-based tool, and could in fact be run simultaneously with the scallion.
However, things have changed quite a bit since 2018. In 2019, a batched mode was introduced to mkp224o, making it more than 10× faster than it had been. On my 12-core Ryzen 9 3900X, I was able to hit 75 MH/s, compared to the paltry 4.8 MH/s Tudor managed with his grand fleet of cloud servers. However, despite the much-increased capacity, I still only attempted a 7 character prefix followed by a number (correct[234567]), as 75 MH/s was two orders of magnitude slower than the gigahashes per second I managed with my GPU. Still, it took only around 10 hours for me to get a good collection of onions matching the desired prefix, and from this list, I picked correct2qlofpg4tjz5m7zh73lxtl7xrt2eqj27m6vzoyoqyw4d4pgyd.onion.
In short, v2 onions hadn’t changed at all, while v3 vanity onions could be generated an order of magnitude faster thanks to significant improvements to mkp224o.
When generating onions, it is always helpful to be able to estimate how long the process should take before you commit to it. This estimate can not be very precise though, as the process is probabilistic and memoryless. This is a bit sad: if you have a 50% chance of getting the onion you want after 1 trillion hashes, and you have already done 1 trillion hashes, you still only have a 50% chance of getting the onion after another trillion hashes. In this situation, there is no progress.
In Tudor’s post, he modelled this as a Poisson process and used the exponential distribution to calculate the probability of finding a match after a certain time. This is, strictly speaking, not correct, as the hashing process is discrete: each hash either produces a match, or it doesn’t. Thus, each hash is best modelled as a Bernoulli trial, and the geometric distribution models this exact situation: the number of trials (hashes) it takes before we get one success (produces a match). The exponential distribution, on the other hand, is the continuous analogue of the geometric distribution and is best used to model continuous processes, although, in this situation, it is a good approximation.
In any case, let us move onto the derivation.
Probability of Single Hash
First, we need to look at the probability of a single hash matching our desired prefix.
For simple prefixes, such as quantum, this is simple. Onion domain names are base32-encoded, and so there are 32 different possibilities for each character. Therefore, there is a
1/32
chance that any given character will be what we want. For a seven character prefix, seven characters need to match simultaneously, and so the total probability is
1/32^7
. In general, for an
x
character long prefix, the probability of a match is
1/32^x
However, simple prefixes can be hard to read, and it’s generally better to end the prefix with a number so that it is easy to tell where the prefix ends. This is the form that I used for the onions described in this post. In base32, there are six digits used: 234567. Therefore, the probability that any given character is a digit is
6/32
3/16
. Therefore, for an
x
character long prefix followed by a number, the probability of a match is
6/32^{x+1}
For the prefix correctpw[234567], there are 9 characters followed by a number, so the probability is
p = \frac{6}{32^{x+1}} = \frac{6}{32^{10}} = 5.329 \times 10^{-15}.
Hashes Required
p
be the probability that an individual hash will match our desired pattern, which we computed in the previous step. As described before, the number of hashes required is best modelled with the geometric distribution. Therefore, we define the random variable
X \sim \text{Geo}(p)
X
is the number of hashes required.
The expected value, i.e. the number of hashes after which there is a 50% chance that the desired onion will be generated, is
1/p
To know the probability that after
x
hashes, the desired onion is generated, i.e.
\Pr(X \le x)
, we can use the cumulative distribution function (CDF), which describes this exact quantity. For the geometric distribution, the CDF is
1-(1-p)^x
Continuing with the example of correctpw[234567], the expected value would be
E[X] = \frac{1}{p} = \frac{32^{10}}{6} = 1.876 \times 10^{14}.
This is around 188 trillion hashes (terahashes). Here is the plot for the CDF:
Conversion to Time
Now, we can just divide the number of hashes by the hash rate. Let
T
be the time it takes for
X
hashes and
H
be the hash rate. Using my GTX 1060, which was able to do 2.7 GH/s, as an example:
E[T] = \frac{E[X]}{H} = \frac{1.876 \times 10^{14}~\text{hashes}}{2.7 \times 10^9~\text{hashes/s}} \allowbreak = 69\,481~\text{s} = 19.3~\text{h}
From this we can see that it should have taken me around 19 hours.
For convenience, here are the expressions for the time required directly, using the convention that
T
is the random variable for the time taken, and
H
is the hash rate:
\begin{align*} E[T] &= \frac{E[X]}{H} = \frac{1}{pH} \\ E[T \le t] &= 1-(1-p)^{tH} \end{align*}
For simple prefixes of length
x
, these reduce to:
\begin{align*} E[T] &= \frac{32^x}{H} \\ E[T \le t] &= 1-(1-32^x)^{tH} \end{align*}
For prefixes of length
x
followed by a number, these reduce to:
\begin{align*} E[T] &= \frac{32^{x+1}}{6H} \\ E[T \le t] &= 1-\left(1-\frac{32^{x+1}}{6}\right)^{tH} \end{align*}
Have you wondered how a sandbox for Linux and/or FreeBSD works? This post describes how such a sandbox works on a real-world online judge — DMOJ.
DMOJ is now accessible as dmojsites2fpbeve.onion and my blog is accessible as quantum2l7xnxwtb.onion. Learn to make your site accessible as an onion too!
I decided to take it as a challenge to get a full perfect score on the industry standard Qualys SSL Labs Server Test. I achieved my goal, but at what cost?
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FrattiniSubgroup - Maple Help
Home : Support : Online Help : Mathematics : Group Theory : FrattiniSubgroup
construct the Frattini subgroup of a group
FrattiniSubgroup( G )
The Frattini subgroup of a finite group
G
is the set of "non-generators" of
G
g
G
is a non-generator if, whenever
G
is generated by a set
S
containing
, it is also generated by
S∖{g}
G
is also equal to the intersection of the maximal subgroups of
G
. The Frattini subgroup of a finite group is nilpotent.
The FrattiniSubgroup( G ) command returns the Frattini subgroup of a group G. The group G must be an instance of a permutation group.
\mathrm{with}\left(\mathrm{GroupTheory}\right):
G≔\mathrm{SmallGroup}\left(32,5\right):
F≔\mathrm{FrattiniSubgroup}\left(G\right)
\textcolor[rgb]{0,0,1}{F}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\Phi }\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{< a permutation group on 32 letters with 5 generators >}}\right)
\mathrm{GroupOrder}\left(F\right)
\textcolor[rgb]{0,0,1}{8}
\mathrm{IsNilpotent}\left(F\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
F≔\mathrm{FrattiniSubgroup}\left(\mathrm{DihedralGroup}\left(12\right)\right)
\textcolor[rgb]{0,0,1}{F}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\Phi }\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{\mathbf{D}}}_{\textcolor[rgb]{0,0,1}{12}}\right)
\mathrm{GroupOrder}\left(F\right)
\textcolor[rgb]{0,0,1}{2}
\mathrm{GroupOrder}\left(\mathrm{FrattiniSubgroup}\left(\mathrm{Alt}\left(4\right)\right)\right)
\textcolor[rgb]{0,0,1}{1}
The GroupTheory[FrattiniSubgroup] command was introduced in Maple 17.
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Introduction to Robotics/Electrical Components/Assignment/Teachers - Wikiversity
Introduction to Robotics/Electrical Components/Assignment/Teachers
The instructor may wish to assign the reading portion of the assignment to be completed before the lecture, and the questions to be assigned after the lecture. Students will likely need to be able to read resistor color codes and employ Ohm's Law in the lab.
Calculate using ohm's law:
{\displaystyle v=ir}
{\displaystyle 5=(2\times 10^{-3})r}
{\displaystyle r={\frac {5}{(2\times 10^{-3})}}=2.5k\Omega }
Ohm's law is linear. Compare v = ir with y = mx, which is the equation for a line. If r is a constant like m, then y is v and i is x.
10 × 101 (5%)
Retrieved from "https://en.wikiversity.org/w/index.php?title=Introduction_to_Robotics/Electrical_Components/Assignment/Teachers&oldid=224529"
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The Design and Experimental Validation of an Ultrafast Shape Memory Alloy ResetTable (SMART) Latch | J. Mech. Des. | ASME Digital Collection
John A. Redmond,
, 2350 Hayward Street, 2250 G.G. Brown, Ann Arbor, MI 48109-2125
e-mail: jredmond@umich.edu
Diann Brei,
Jonathan Luntz,
, MC 480 106 256, 30500 Mound Road, Warren, MI 48090-9055
e-mail: alan.l.browne@gm.com
e-mail: nancy.l.johnson@gm.com
Kenneth A. Strom
e-mail: kenneth.a.strom@gm.com
Redmond, J. A., Brei, D., Luntz, J., Browne, A. L., Johnson, N. L., and Strom, K. A. (May 25, 2010). "The Design and Experimental Validation of an Ultrafast Shape Memory Alloy ResetTable (SMART) Latch." ASME. J. Mech. Des. June 2010; 132(6): 061007. https://doi.org/10.1115/1.4001393
Latches are essential machine elements utilized by all sectors (military, automotive, consumer, manufacturing, etc.) with a growing need for active capabilities such as automatic release and reset, which require actuation. Shape memory alloy (SMA) actuation is an attractive alternative technology to conventional actuation (electrical, hydraulic, etc.) because SMA, particularly in the wire form, is simple, inexpensive, lightweight, and compact. This paper introduces a fundamental latch technology, referred to as the T-latch, which is driven by an ultrafast SMA wire actuator that employs a novel spool-packaged architecture to produce the necessary rotary release motion within a compact footprint. The T-latch technology can engage passively, maintain a strong structural connection in multiple degrees of freedom with zero power consumption, actively release within a very short timeframe (
<20 ms
, utilizing the SMA spooled actuator), and then repeat operation with automatic reset. The generic architecture of the T-latch and governing operational behavioral models discussed within this paper provide the background for synthesizing basic active latches across a broad range of applications. To illustrate the utility and general operation of the T-latch, a proof-of-concept prototype was designed, built, and experimentally characterized regarding the basic functions of engagement, retention, release, and reset for a common case study of automotive panel lockdown. Based on the successful demonstration and model validation presented in this study, the T-latch demonstrates its promise as an attractive alternative technology to conventional technologies with the potential to enable simple, low-cost, lightweight, and compact active latches across a broad range of industrial applications.
design engineering, flip-flops, intelligent actuators, intelligent materials, machinery, packaging, shape memory effects, shape memory alloy, T-latch, active latch, rotary actuator, high-speed latch, SMA spooling, SMA packaging
Actuators, Design, Friction, Gates (Closures), Separation (Technology), Shape memory alloys, Springs, Stress, Wire, Engineering prototypes, Manufacturing, Packaging
A Remotely Activated SMA Actuated Naval Tie Down System
Development of a Non-Explosive Release Device for Aerospace Applications
Proceedings of the 26th Aerospace Mechanisms Symposium
Non-Pyrotechnic Release System
No-Shock Separation Mechanism
Shape Memory Actuated Release Devices
Flywheel Nut Separable Connector and Method
Development of a New, No-Shock Separation Mechanism for Spacecraft Release Applications
Fast Acting Non-Pyrotechnic 10 kN Separation Nut
Eighth European Space Mechanisms and Tribology Symposium
Development and Transition of Low-Shock Spacecraft Release Devices
Aerospace Conference Proceedings
Deployment and Release Devices Efforts at the Air Force Research Laboratory Space Vehicles Directorate
High Stiffness Latching of Deployable Space Structure Joints Using Melting Metal Technology
Design and Demonstration of Bolt Retractor Separation System for x-38 Deorbit Propulsion State
Proceedings of the Tenth European Space Mechanisms and Tribology Symposium
Multiple Payload Adapters; Opening the Doors to Space
Aerospace Conference Proceedings 2003
Proof-of-concept investigation of Active Velcro for Smart Attachment Mechanisms
42nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit
Defense Advanced Research Projects Agency—Smart Materials and Structures Demonstration Program Overview
Aersopace Applications of Shape Memory Alloys
Proc. IMechE, Part G: J. Aerospace Engineering
The Design and Experimental Validation of an Ultrafast Smart Latch
Fiat Designs New Door Lock
,” http://www.magnaclosures.comhttp://www.magnaclosures.com
Barvosa-Carter
Active Material Reversible Attachments: Shape Memory Polymer Based
Enhancing the Force Capability of Permanent Magnet Latching Actuators for Electromechanical Valve Systems
Proceedings of the Shape Memory Technology Conference
Application of SMA Technology to Auxiliary Functions in Appliances
Osvatic
Washing Machine Lid Lock With Memory Wire Actuator
,” http://dynalloy.com/Links.htmlhttp://dynalloy.com/Links.html
ITW Global Appliance Group
,” http://www.itwappliance.com/products/door_locks.htmhttp://www.itwappliance.com/products/door_locks.htm
Albrecth
Load/Unload Technology for Disk Drives
Monajemy
Pawl Latch Mechanism Design and Control for Load/Unload Technology
Experimental Investigation of Disc Drive Seek Control When Subject to a Nonlinear Magnetic Bias
Inertial Magnetic Latch Design Considering Actuator Load Unload
Towards Self-Disassembling Vehicles
J. Sustainable Product Design
Shape Memory Alloy Actuators for Active Disassembly Using ‘Smart’ Materials of Consumer Electronics Products
Design for Active Dissassembly (DfAD)
Proceedings of the IEEE International Symposium on Electronics and the Environment
Design for Disassembly With High-Stiffness Heat-Reversible Locator-Snap Systems
High-Stiffness, Lock-and-Key Heat-Reversible Locator-Snap Systems for the Design for Disassembly
Proceedings of the 2007 IEEE/RSJ International Conference on Robots and Systems, Self-Reconfigurable Robotics Workshop
Kokagi
An Active Micro Joining Mechanism for 3D Assembly
Dynamic Braille Display Using SMA Coil Actuator and Magnetic Latch
Medical and Welfare Applications of Shape Memory Alloy Microcoil Actuators
Mechanism Design of the Flapper Actuator in Chinese Braille Display
Development of Advanced Actuators Using Shape Memory Alloys and Electrorheological Fluids
Applications of Smart Structures to Aircraft for Performance Enhancement
Can. Aeronautics Space J.
Namuduri
Mechamatronics: An Automotive Perspective
Adaptronics and Smart Structures: Basics, Materials, Design, and Applications
, Chap. 6.4, pp.
Automotive Materials: Technology Trends and Challenges in the 21st Century
Behavioral Model and Experimental Validation for Spool-Packaged Shape Memory Alloy Linear Actuators
Proceedings of the 19th International Conference on Adaptive Structures and Technologies
Effect of Bending on the Performance of Spool-Packaged Shape Memory Alloy Actuators
Federal Motor Vehicle Safety Standard 206, U.S. Department of Transportation, National Highway Traffic Safety Administration.
A Thermodynamical Constitutive Model for Shape Memory Materials, Part I: The Monolithic Shape Memory Alloy
A Reduced-Order Thermomechanical Model and Analytical Solution for Uniaxial Shape Memory Alloy Actuators
The General Case of Friction of a String Round a Cylinder
Product Datasheet for E91 AA Alkaline Battery
,” Energizer Technical Information, http://data.energizer.com/PDFs/E91.pdfhttp://data.energizer.com/PDFs/E91.pdf
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Единицы Измерения - FreeCAD Documentation
This page is a translated version of the page Units and the translation is 5% complete.
Некоторые сведения об единицах измерения:
Единицы измерения реализованные в OCC
3 Purpose and principles: proposal of an extension of the unit management system
5.1 Context 1: opening a data file
5.2 Context 2: switching the unit system at runtime
6.1 Logic of unit scaling
6.1.1 Unit coherence throughout the FreeCAD running instance
6.1.3 Base and derived units
6.1.4 Base and derived unit symbols
6.2.2 Unit dictionary
6.2.4 Unit management API
6.2.4.1 Checking the unit dictionary
6.2.4.2 Единицы измерения масштабирования
6.2.5 Motivations for such a management: example of application
A complete list of all supported units can be found here.
Purpose and principles: proposal of an extension of the unit management system
An extension unit management system is proposed in the following sections, developping the concept of unit system, activated during a running FreeCAD instance. The interest in defining such a new concept is to work more easily with as many type of physical units as one wants (even user-created ones), without increasing the complexity of unit management for the user, nor for FreeCAD developpers.
In short, event of unit scaling are localized precisely, and carried out in a generic fashion.
Achieving such a flexibility is most notably required when one starts to deal with material properties that can have very different units, difficult to manage one by one manually.
The reasoning proposed allows handling the units such as described in the Guide for the Use of the International System of Units (SI) and The International System of Units (SI) both from NIST.
In this proposal, one first recall in Brainstorming section what are the possible contexts for which unit management is required.
In Organizing section, we present the data model retained to achieve unit management, based on 3 objects, the unit, the unit dictionary, and the unit system. Finally, a short API of a 4th object called the unit manager is presented as well.
Thanks to this extension, one aims to ease unit scaling that can occurs between different business tasks. For instance, technical drawings can be done in standard unit system, while FE modelling can be managed in an unit system more suited for it.
The exchange of data between these two kind of activities become easier with this extension.
In this section are highlighted the contexts of use of such an unit management system. From these contexts, we are then able to defined its technical specifications.
Essentially 2 contexts are given as example.
Context 1: opening a data file
This case is probably the most frequent case. You receive a file containing for instance a geometrical model, or describing a material with quite a lot of properties. The geometrical model is expressed in meters, or the material properties according the international unit system.
You are an expert FE modelling, and you usually work with millimeter for length, MegaPascal for stress, tonne for mass...
In this context, unit management is required to scale data from an initial unit system defined in the input file to a user-defined target unit system.
Context 2: switching the unit system at runtime
In this case, you can be at the same time the guy that carries out a drawing, and the guy that will manage the FE modelling. Similarly to the previous case, the unit systems for these 2 tasks are not the same, and you need to switch the initial unit system at runtime to your favorite one.
Logic of unit scaling
In the Brainstorming section have been presented 2 contexts when using unit scaling. Some items should be highlighted from these two contexts.
Unit coherence throughout the FreeCAD running instance
The system proposed is based on a primary assumption: the user is working in a coherent unit system. For instance, this means that if the user expresses length in millimeters, necessarily areas will be expressed in terms of squared millimeters, not squared meters. This is hypothesis one.
Because of hypothesis one, it is possible and relevant to define an unit system. An unit system applies to:
a running FreeCAD instance into which you are working
or it may also apply globally to the content of an input file
According Guide for the Use of the International System of Units (SI) from NIST, they are 7 physical base units. We chose to express a unit system in terms of these 7 base units.
When working within an instance of FreeCAD, the user has thus to define first the unit system according to which she/he is working before she/he decides to switch to another unit system, or before importing data from an input file.
This unit system will apply till the user decides to change it. If she/he does, all data with dimensions will be scaled.
Considering hypothesis one, all data that the user will input manually in FreeCAD are assumed to be coherent with the chosen unit system.
The benefit to work with a unit system defined at a FreeCAD running instance level, or at data file level (instead of unit which are defined at the data level) is then that unit management is considerably simplified.
Here are some examples of unit systems.
meter, kilogram, second, ampere, Kelvin, mole, candela
millimeter, tonne, millisecond, ampere, Kelvin, mole, candela
millimeter, kilogramme, millisecond, ampere, Kelvin, mole, candela
Derived units are created by combination of base units. For instance, an acceleration (m/s) combines at the same time length and time. An interesting picture presenting the relationships between base and derived units can be seen here also from NIST.
Thanks to the definition of unit system, it is possible for the user to work with any kind of derived units, without the need for FreeCAD developpers to foresee them in advance.
Base and derived unit symbols
According to The International System of Units (SI), the symbols to specify a units are officially approved. Two consequences can be highlighted from this.
it is not easy for a computer program to work with unit symbols because some are Greek letters for instance. Hence they can be a bit difficult to process by a program
while some units and their symbols can be used widely, they may be not approved officially, like for instance tonne unit (see p55 of Guide for the Use of the International System of Units (SI))
To overcome these limitations and remain flexible, the proposed system favors the use of unit magnitudes instead of unit symbols, which remain nonetheless available for an ergonomy reason.
The three core objects of the unit management system are presented, namely the unit, the unit dictionary and the unit system.
As a foreword, it is important to highlight that a unit object in itself only indicates a dimension like length, mass, time... It doesn't specify a magnitude like meter, millimeter, kilometer... This last information is specified through the unit system.
Compulsory string indicating the dimension of the unit. The dimension of the 7 base units are indicated below (from Guide for the Use of the International System of Units (SI)).
Dimension attribute allows identifying the unit. Two units cannot share the same dimension.
Compulsory integer array of size 7 (number of base units) that defines what the unit is. The signature of the 7 base units are:
From these 7 units, we are then able to express all derived units defined in Guide for the Use of the International System of Units (SI) and create new ones as needed such as for instance:
Signature is the attribute thanks to which unit scaling can be achieved in a generic way.
Array of [real, string] (meaning [magnitude, symbol]) that lists all symbols known by FreeCAD. Thanks to this array, the unit scaling API becomes more ergonomic because symbols and related magnitudes are linked.
This array can be extended as required.
For instance, the list of symbols of the LENGTH unit, and their related magnitudes is:
Standard symbols can be found on NIST website and p23 to 26 and p32 (metric ton or tonne) of The International System of Units (SI).
All the units available in FreeCAD, and new ones created by the user, should be stored in unit dictionary, which is an XML file (FreeCAD configuration file), so as to be retrieved when needed, i.e. when achieving unit scaling.
Array of units, contained in the unit dictionary.
A unit system is the object that allows the user defining the current unit magnitude of each base units with which she/he is working. For instance, knowing that the user is working with millimeter, tonne, and second, thanks to the use of a unit system, FreeCAD can know that energy is expressed in terms of milliJoule, force in terms of Newton, and stress in terms of MegaPascal. Hence a unit system is only defined by a name (for instance Standard unit system) and a magnitude table specifying for each of the 7 base units, what is its corresponding magnitude.
String allowing to the user identifying what is the unit system.
By specifying the magnitude of the 7 base units, a unit system is defined.
For instance [1e-03, 1e+03, 1, 1, 1, 1, 1], meaning millimeter, tonne, second, ampere, Kelvin, mole, candela
Unit management API
Only the logic of some methods is presented, in order to highlight some features. These methods could belong to an object called Unit manager.
Checking the unit dictionary
The unit dictionary can be an XML file (FreeCAD configuration file). It contains a list of defined units. Such a dictionary is required for the proposed unit management system to work.
It must fulfills some conditions that should be checked before activating the unit management system. These conditions are:
check that all base units are defined
check that a dimension is not defined twice through the units
check that a symbol is not defined twice in all the existing symbols
check that the signatures of all units have all the same size
chacke that a standard symbol (for which magnitude is 1) is defined for all units
A unit dictionary defines a set of units and their known magnitudes. When managing a unit, it is relevant to check that its signature is compatible with the set of units registered in the unit dictionary, so as to process it. This check includes:
check that the input signature length is of the same size than the unit dictionary unit signatures
Единицы измерения масштабирования
Knowing a value, an initial unit by its symbol, the target unit by its symbol, scale the value.
Knowing a value, an initial unit by its symbol, the target unit system, scale the value.
Knowing a value, an initial unit system, the target unit by its symbol, scale the value.
Motivations for such a management: example of application
Let's assume that we are going to setup a finite element model. To build our model, we need the mesh, material properties, and to define numerical parameters. Considering that they can be tens of material properties to manage, expressed with different units, sometimes not always very common, it is interesting for the user to only have to specify a global unit system, without caring much.
FreeCAD would then just do the job.
As FreeCAD developpers and FreeCAD users do not necessarily know all units that can be defined in the material property files, it is interesting to rely on a generic system.
Let's assume that in such a file we have a fair number of exotic material properties expressed with exotic units, and that we want to work in a specific unit system.
It is easy with the proposed extension to scale any of these properties by knowing their signatures, magnitudes, and the target unit system.
For each of the properties, the scaling is obtained by multiplying the initial property value with the factor
{\displaystyle {\frac {initialMagnitude}{targetMagnitude}}}
The targetMagnitude is then simply obtained with the operation
{\displaystyle \prod _{bu}targetMagnitude_{bu}^{signature_{bu}}}
, bu standing for base unit.
It becomes thus very easy to manage any number of properties with any kind of units with very few lines of Python.
The Expressions page for a list of all known units.
The documentation of Quantity.
The Std UnitsCalculator tool.
Retrieved from "http://wiki.freecadweb.org/index.php?title=Units/ru&oldid=1116520"
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Cyclomatic Complexity Calculator | Software Metrics
Reviewed by Anna Szczepanek, PhD and Adena Benn
Understanding programming complexity
Representing complexity: the control-flow graph
Calculating complexity: the cyclomatic complexity
How to calculate the cyclomatic complexity
An example of cyclomatic complexity calculation
How to use our cyclomatic complexity calculator?
Loop after loop, your code may get hard to read and understand: our cyclomatic complexity calculator can tell you if adding that nested if would be too much!
Here you will learn an essential part of theoretical computer science, and we hope that it will help you write better code. Keep reading to learn:
What is programming complexity, and why does it matter;
How to measure programming complexity: the cyclomatic complexity;
How to calculate the cyclomatic complexity: the cyclomatic complexity formula; and
A neat example.
We are here to tell you how to reduce your cyclomatic complexity; what are you waiting for?
Studying complex system physics, professors tend to repeat a short mantra: complicated and complex are two different things. The distinction is pretty easy to make when talking of problems.
Complicated means a messy problem, full of variables but theoretically solvable with enough computational power. Think of the exact trajectory of a golf ball. We can calculate the effect of every single air molecule on it: it would take a lot of time, but eventually, we would have an exact result.
Complex means a problem where even a small number of elements develop iterations which causes the emergence (scientists love this word) of non-deterministic behaviors or deep connections that may make it hard to understand and modify the problem.
Your code, any program, can be analyzed in terms of complexity. Each block of instructions that interacts with others can repeat; there may be bifurcations only to return to the main path.
An increase in complexity reflects a growing set of interactions that, after a while, may reach a point where it would be impossible to modify the program or even understand it!
Frances Allen, the first woman to win a Turing award (the Nobel prize of computer science), introduced a way to represent a program using a graph that made it easy to understand the underlying complexity: the control-flow graph.
To build a control-flow graph, follow these steps:
Each instruction corresponds to a node;
Each jump in the code (from one instruction to the other) corresponds to an edge;
It is possible to reduce the number of nodes by grouping subsequent nodes if the edge connecting them:
Departs from a node with a single outbound edge; and
Ends in a node with a single entry.
This process allows us to reduce to minimize the control-flow graph.
Consider this simple program to check if a number is prime or not, written in pseudocode.
DECLARE n, i, flag
PRINT "Enter a positive integer: "
FOR i = 2; i <= n/2; INCREMENT i)
IF n MOD i == 0
PRINT "Prime number."
PRINT "Not a prime number."
We can draw the control-flow graph by assigning a node to each instruction:
The control-flow graph for a program that checks if a number is prime or not,
There are a lot of instructions, though. Let's reduce them. The next representation will be the one we will use to teach you the cyclomatic complexity of a program.
The same program can be reduced to hide instructions that follow simple connections.
As you can see, this representation highlights loops and bifurcations. In fact, complexity is all about them. In a program, we can find many types of instruction increasing the complexity:
Conditional statements: if, if...else, if...else if;
Loops: for, while, do...while;
Break statements;
And many more. You can easily understand the representation of such elements when you think of their function.
To measure the complexity of a program, scientists introduced metrics that allow quantifying how many interactions it contains (and how bad they are). There are many types of complexity metrics, but one is particularly straightforward: the cyclomatic complexity metric
M
The concept of cyclomatic complexity heavily borrows terms from graph theory in math. It uses nodes, edges, and components to measure the number of linearly independent paths in the code.
Edges and nodes are the basic elements of a graph. A set of independent edges and nodes is called a connected component. Each program has at least a connected component, but that's the only sure thing!
The cyclomatic complexity formula is simple. Make sure to know the elements we introduced before: nodes, edges, and connected components. Then, apply the formula:
M = E - N + 2\cdot C
N
is the number of nodes;
E
is the number of edges; and
C
is the number of connected components.
🔎 Why do we use
M
to indicate the cyclomatic complexity? It comes from the name of its creator, Thomas J. McCabe, Sr.
It is easier to understand the cyclomatic complexity with a few examples:
A simple code without loops and conditions has cyclomatic complexity
M=1
. We can reduce it to a single node without edges; hence the cyclomatic complexity formula reduces to:
M = 0-2+2\cdot 1 = 0
A code with a single if...else condition has a control-flow graph where a bifurcation departs from a node; there are two possible sets of instructions (two nodes) that reconnect in a final node. The total number of nodes is
4
, and there are analogously
4
edges. The cyclomatic complexity is
M=2
Look at the diagram:
The control-flow graph of an if...else statement.
M=4-4+2\cdot 1=2
In a while loop, we can identify the starting node (a condition), a set of instructions to be executed in the loop, and an exit node containing the instructions executed after the loop. The total number of nodes is
3
, which equals the number of edges. The cyclomatic complexity formula tells us that there are:
M=3-3+2\cdot 1=2
Two independent paths in the program.
The control-flow graph for a while loop.
As you can see, the cyclomatic complexity is easy to calculate. Remember to count the correct number of nodes, edges, and components.
We can try to compute the cyclomatic complexity of a bigger program. We chose a code that computes the Collatz conjecture (you can learn more at our Collatz conjecture calculator).
Take a look at the pseudocode:
DECLARE n, i
PRINT "Insert the starting number."
PRINT "Insert a non negative number"
IF n MOD 2 == 0
SET n = n/2
SET n = 3 × n + 1
The code prints the Collatz sequence and repeats three times the final loop
4,2,1
. We drew the control-flow graph for you:
The control-flow graph for the Colllatz conjecture. That's a lot of for loops!
Let's count the nodes and the edges. Note that there is again a single connected component.
14
nodes; and
18
We calculate the cyclomatic complexity:
M = 18-14 + 2\cdot 1 = 6
Our cyclomatic complexity calculator is easy to use: insert the parameters of your code, and find out its complexity.
Even if using it is easy, the answer you get can help you write better, more agile code. If you follow the guidelines of the very creator of the cyclomatic complexity, remember to split modules when they reach a certain complexity, originally
M=10
🙋 Check our other computer science tools at Omni Calculator like the password entropy calculator or the IP subnet calculator!
Cyclomatic complexity is a metric that measures the complexity of a program by computing the number of linearly independent paths in the code. The higher the number of independent paths, the higher the difficulty in reading and modifying the code.
How do I calculate the cyclomatic complexity?
To calculate the cyclomatic complexity, follow the next steps:
Count the nodes N and edges E in the control-flow graph;
Count the number of independent components P (disjoined groups of nodes and edges); and
Apply the cyclomatic complexity formula:
M = E - N + 2 × P
Remember that the number of components in a code without functions and methods is 1.
What is the cyclomatic complexity of nested if...else statements?
Nested if...else statements have a cyclomatic complexity of 4. Draw the control-flow graph: it includes 10 nodes (an entry and exit node for each statement, for a total of 6, plus 4 instructions) and 12 edges. There is a single component. Apply the cyclomatic complex formula:
M = 12 - 10 + 2 × 1 = 4
How to reduce the cyclomatic complexity?
To reduce cyclomatic complexity, split your program into smaller modules. This will make your code easier both to read and manipulate. You can also try to reduce the number of if statements: remember that:
Number of components (C)
Number of edges (E)
Cyclomatic complexity (M)
The matrix rank calculator is an easy-to-use tool to calculate the rank of any matrix with up to four rows or columns.
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GruenbergKegelGraph - Maple Help
Home : Support : Online Help : Mathematics : Group Theory : GruenbergKegelGraph
construct the Gruenberg-Kegel graph of a group
GruenbergKegelGraph( G )
G
, the Gruenberg-Kegel graph (also known as the prime graph) of
G
is the graph with vertices the prime divisors of the order of
G
, and for which two vertices
p
q
are adjacent if
G
has an element of order
\mathrm{pq}
The GruenbergKegelGraph( 'G' ) command returns the Gruenberg-Kegel graph of the finite group G.
Commands in the GraphTheory package can be used to visualize the graph returned by this command, as well as to analyze its properties.
\mathrm{with}\left(\mathrm{GroupTheory}\right):
The vertices of the Gruenberg-Kegel graph of the Monster sporadic finite simple group are the so-called supersingular primes.
\mathrm{GKG}≔\mathrm{GruenbergKegelGraph}\left(\mathrm{Monster}\left(\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{GKG}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 1: an undirected unweighted graph with 15 vertices, 23 edge\left(s\right), and 3 self-loop\left(s\right)}}
\mathbf{use}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{GraphTheory}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{HighlightVertex}\left(\mathrm{GKG},\mathrm{SelfLoops}\left(\mathrm{GKG}\right),'\mathrm{stylesheet}'=\left['\mathrm{shape}'="pentagon",'\mathrm{color}'="red"\right]\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end use}:
\mathbf{use}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{GraphTheory}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{HighlightVertex}\left(\mathrm{GKG},\mathrm{map}\left(\mathrm{op},\mathrm{select}\left(c→\mathrm{nops}\left(c\right)=1,\mathrm{ConnectedComponents}\left(\mathrm{GKG}\right)\right)\right),'\mathrm{stylesheet}'=\left['\mathrm{shape}'="7gon",'\mathrm{color}'="green"\right]\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end use}:
The self-loops indicate those supersingular primes
p
for which the Monster has an element of order
{p}^{2}
\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{GKG}\right)
The Gruenberg-Kegel graph of a Frobenius group is never connected.
G≔\mathrm{FrobeniusGroup}\left(2238,1\right):
\mathrm{GKG}≔\mathrm{GruenbergKegelGraph}\left(G\right)
\textcolor[rgb]{0,0,1}{\mathrm{GKG}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 2: an undirected unweighted graph with 3 vertices and 1 edge\left(s\right)}}
\mathrm{GraphTheory}:-\mathrm{ConnectedComponents}\left(\mathrm{GKG}\right)
[[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{373}]]
The GroupTheory[GruenbergKegelGraph] command was introduced in Maple 2020.
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Response curves to autoinducer induction in the population-average model. lux01 (A) and lux02 (B) operons. The normalized GFP concentration is plotted as a function of the exogenous autoinducer concentration
{c}_{A}^{\ast }
: steady-state response for increasing (arrow-free upper blue curve) and decreasing (arrow-free red curve) autoinducer concentration, response under 10 h induction time for increasing (blue curve with arrow) autoinducer concentration, transient response after 2 hours of induction (lower blue curve) from initially non-induced cells, decreasing-concentration trajectories (green curves) for cells weakly induced (2 hours) at
{c}_{A}^{\ast }=100\text{nM}
, 75 nM and 50 nM, and decreasing-concentration trajectories (red curve with arrow) for cells fully induced (10 hours) at
{c}_{A}^{\ast }=100\text{nM}
. The decreasing-concentration trajectories reduce the value of
{c}_{A}^{\ast }
hourly by 25% (similar to the experiments in [10]). The gray-shaded region between the increasing and decreasing steady-state curves reveals bistability in the range
2\text{nM}<{c}_{A}^{\ast }<15\text{nM}
(lux01) and
0\text{nM}<{c}_{A}^{\ast }<15\text{nM}
(lux02).
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Slope, intercepts, and the general line equation
How do you find the y-intercept of a line?
How do you find the x-intercept of a line?
How do you find the line equation from its intercepts?
How to calculate x- and y-intercepts using this y-intercept calculator
This y-intercept calculator is the perfect tool to calculate the x- and y-intercept of any given line. Additionally, you can use it to find the line equation from its slope and the x- or y-intercept.
Finding intercepts of straight lines is a simple process, but it is pretty common to get the basics mixed up. Let's discuss the following basics in this article so that you're always ready:
How do you find the y-intercept of any line?
How do you find the x-intercept of any line?
If you're interested in finding the line equation in different forms, we recommend our popular slope intercept form calculator and point slope form calculator.
We can express the most general form of a straight line in 2-dimensional space as:
ax + by + c = 0
a
x
term;
b
y
c
is the constant term; and
x
y
are the variables representing the two dimensions.
You can plot this line on a graph sheet if you know at least two points that lie on this line. We define the y-intercept of this line as the point at which it crosses (or intersects) the y-axis. Specifically, it refers to the y-coordinate of this point, although it is also common to call the point itself the y-intercept.
Similarly, the line's x-intercept would be the point (or the x-coordinate) where it intersects the x-axis.
The slope (or gradient) of a line is the amount of change in
y
x
. You can learn more about the slope of a line using our slope calculator.
We can express the slope, y-intercept and x-intercept of any line
ax + by + c = 0
using these equations:
\begin{align*} y_c &= - c/b\\ x_c &= - c/a \\ m &= - a/b \end{align*}
y_c
is the y-intercept of the line;
x_c
is the x-intercept of the line; and
m
is the slope of the line.
In the following sections, we'll prove these equations with an example — but first, let's discuss another form of a line equation.
We can also express a line equation in terms of its slope and y-intercept:
y = mx + c
m
is the line's slope; and
c
is the line's y-intercept, i.e.
c = y_c
We could rewrite it to include the y-intercept from the start:
y = mx + y_c
You'll find this form very useful when formulating most line equations if you can calculate the slope and y-intercept beforehand.
To find the y-intercept of a line given by ax + by + c = 0, follow these simple steps:
Substitute the value x = 0 into the line equation to get by + c = 0.
Rearrange this equation to find the y-intercept y꜀, as y꜀ = −c/b.
Verify your results using our y-intercept calculator.
Or, if the line equation is in the slope-intercept form y = mx + c, you can directly extract the term c as the line's y-intercept y꜀.
For example, consider a line given by the equation
2x + 3y -2 = 0
. The y-intercept lies on the intersection of the y-axis (the line defined by
x=0
) and our line
2x + 3y -2 = 0
. So, we insert
x=0
2x + 3y -2 = 0
\begin{align*} 2\cdot 0+ 3y - 2 &= 0\\ 3y - 2 &=0\\ 3y &= 2\\ \therefore y_c &= \frac{2}{3} \end{align*}
To find the x-intercept of a line given by ax + by + c = 0, follow these simple steps:
Substitute the value x = 0 into the line equation to get ax + c =0.
Rearrange this equation to find the y-intercept x꜀, as x꜀ = −c/a.
These steps are applicable even if the line equation is in slope-intercept form: y = mx + c, giving you x꜀ = −c/m.
Again, consider the line
2x + 3y -2 = 0
. Its x-intercept lies on the intersection point of the x-axis (
y=0
2x + 3y -2 = 0
y=0
2x + 3y -2 = 0
\begin{align*} 2x + 3 \cdot 0 - 2 &= 0\\ 2x - 2 &=0\\ 2x &= 2\\ \therefore x_c &= 1 \end{align*}
The line 2x + 3y - 2 = 0 with its slope and intercepts.
To find the line equation from its x-intercept (x꜀, 0) and y-intercept (0, y꜀), follow these steps:
Determine the slope m of the line using m = (0 − y꜀)/(x꜀ − 0) to get m = −y꜀/x꜀.
Formulate the line equation in the slope-intercept form y = mx + c, keeping in mind that c = y꜀.
Simplify and rearrange as required, or use the equation as it is.
Once again, let's consider the line
2x + 3y -2 = 0
(1,0)
x-intercept and
(0,\frac{2}{3})
y-intercept. Can we find the line equation with just these intercepts? Let's find out.
We can determine the slope
m
of this line using these two intercept points
(0,\frac{2}{3})
(1,0)
\qquad\begin{align*} m &= \frac{0-\frac{2}{3}}{1-0}\\ \therefore m &= -\frac{2}{3} \end{align*}
Formulate the line equation in the slope-intercept form
y = mx + c
\qquad y = -\frac{2}{3}x + \frac{2}{3}
Simplify this equation and rearrange it to get
2x + 3y - 2 = 0
You can use this y-intercept calculator in three modes:
To calculate the x- and y-intercepts along with the line's slope from its general equation:
Choose the mode "Line equation is ax + by + c = 0".
Enter the values for a, b, and c, and the calculator will provide you with all the answers!
To calculate the slope, y-intercept, and x-intercept of a line from its slope-intercept form:
Choose the mode "Line equation is y = mx + c".
Enter the values for m and c.
Sit back and relax as the calculator takes care of the rest.
To find an equation with the intercepts given, use the mode "Line equation is to be determined."
Enter the values of x-intercept and y-intercept.
Enjoy the fast and accurate results.
Our calculator will also present you with a summary of results and a helpful graph in all these modes!
Pat yourself on your back for learning something new today! We believe you're ready to explain to others how to find the slope, y-intercept, and x-intercept of a line.
What is the y-intercept of the line 2x + 3y = -9?
−3 is the y-intercept of the line 2x + 3y = −9. To find this yourself, follow these steps:
Substitute x = 0 into the line's equation to get 2×0 + 3y = −9, or 3y = −9.
Divide both sides by 3 to get y = −3.
Do all straight lines have a y-intercept?
No. Some lines run parallel to the y-axis, and thus don't have a y-intercept. However, every line in two dimensions has at least one intercept, be it x- or y-intercept.
Line equation is...
General line equation: ax + by + c =0
y-intercept (y꜀)
x-intercept (x꜀)
This mixed number to improper fraction calculator works like a charm - it not only converts one fraction form to the other, but also it shows the step by step solution.
Rectangle scale factor
Calculate linear and area scale factors between two rectangles using this rectangle scale factor calculator.
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Krishnappa H K, N K Srinath and Ramakanth Kumar P “Magic Square Construction Algorithms and Their Applications“See 2 more sources
Grasha Jacob, Dr. A. Murugan “On the Construction of Doubly Even Order Magic Squares“William H. Richardson “Magic Squares of Even Order (4n + 2)“
Properties of magic squares
The elusive 2 by 2 magic square
How to calculate a magic square: from 3x3 magic squares to infinity
Pulling squares out of a hat
How to use our magic square calculator
Sprinkle some magic in your math with our magic square calculator! In this calculator, we will lead you in an exploration of this interesting mathematical puzzle.
The history of magic squares;
The properties of magic squares;
How to calculate a magic square (with a step by step guide on the calculation of some of them);
Where is the 2 by 2 magic square; and
How to shuffle a magic square.
And much more, from the weird types of magic squares to how many 3x3 magic squares are there. Abracadabra!
A magic square is a square grid of integer numbers arranged in such a way that their positions respect three rules:
No numbers repeat;
The summation the numbers in each row and column returns the same value; and
The sum of the values in both diagonals is equal to the values of summed rows and columns.
However, the magic doesn't end here. It is possible to arrange numbers in such a way to satisfy other rules - this generates different types of magic squares. We then have:
Pandiagonal magic squares - Where the sum of the numbers on a broken diagonal is the same as the sums of the rows and columns;
Associative magic squares - Where cells symmetrical with respect to the center sum to the value of
n^2-1
n
is the order of the square; and
Most-perfect magic squares - This square is both pandiagonal and:
Each 2 by 2 subsquare has a total sum equal to a quarter of the value of its rows divided by
n
Cells separated by
\tfrac{n}{2}
on a diagonal sum to
n^2+1
🔎 A broken diagonal is an offset diagonal that "wraps" around the corner of a square. To visualize it, imagine putting two identical magic squares side by side: a broken diagonal would start in one and end in the next.
Why stop at squares, though? Mathematicians have sprinkled magic over circles, triangles, hexagons, and many other shapes.
Generalizing a magic square in higher dimensions gives us the magic hypercubes, where the sums are generalized over the various faces of the solid and the spatial diagonals. But this is definitely too complicated. Back to squares one! 😉
Where there's magic, there's the attention of humanity, and magic squares are no exception. The first traces of magic squares date back to 190 BC, where the first 3 by 3 magic square appears in the record. In the following centuries, many cultures started exploring these mathematical curiosities.
🙋 We don't think it was necessary, but here's a disclaimer anyway: magic squares are not magic! It's only a matter of numbers and math - close to magic, but still real!
Magic squares landed in Europe, where mathematicians quickly removed any esoterism (always the killjoys) and expanded the theoretical knowledge around the topic. Magic squares turned into a mathematical curiosity, with studies on the possible permutations, their properties, and variations.
Magic squares are all about addition. We already saw where to look for sums in the introduction, but here's a diagram to make things clear.
The rows, columns, and diagonals of a "normal" magic square sum to the same value!
The sum result is the magic constant
M
, and its value depends on the order
n
of the magic square.
M=\frac{n\cdot (n^2+1)}{2}
n=3
M=15
n=4
M=34
, and so on. However, a magic square remains magical when each of its numbers is summed to or multiplied by a constant. In that case, the magic constant can be bigger. This article and our magic square calculator will only consider magic squares populated by the numbers
1,2,...,n^2-1,n^2
The magic square of order
1
is trivial: this means that it theoretically satisfies the requirements to be a magic square, but... meh.
If something goes wrong in physics or math, it's likely that a factor two went missing somewhere. That hideous number decided to ruin magic squares for everyone, too.
n=2
is, in fact, the only one for which we can't build a magic square. You can try to understand why by yourself (it's pretty easy), or you can read below!
The impossible 2 by 2 magic square.
Take a generic 2 by 2 magic square, and write the possible sums for the rows and columns:
\begin{cases} a+b=M\\ c+d=M\\ a+c=M\\ b+d=M \end{cases}
Now consider the first and the third equations. They both equal
M
a+b=a+c
b=c
: since there can't be identical number in a magic square, we can't build a 2 by 2 specimen!
Building a magic square is not difficult! There are different algorithms that you can use: here we will explain how to build every possible magic square!
This is probably the easiest magic square to build on a piece of paper. Take a pen and follow the steps! We will start calculating a 3 by 3 magic square:
Chose the cell in the middle of a side (we chose the top one), and place the number
1
Move one square up, and one square left, and place the next number,
2
We are out of the square: imagine to "tile" around it with its copies, and place the number
2
in the corresponding place of the original square.
Proceed in this fashion until the movement brings you to an occupied square. In that case, move down one square.
After nine iterations, you should have filled the square!
How to build a 3 by 3 magic square.
For magic squares with even order the process is slightly more complex. We need to identify two possible behaviors:
Magic squares with singly even order, which means
n\ \text{mod}\ 4=2
Magic squares with doubly even order, that is, multiples of
4
n\ \text{mod}\ 4=0
We will begin with doubly even magic squares.
🔎 We included an animation to show you the sequence of steps you have to take. The animation is for a 4×4 magic square, but the algorithm is valid for any order.
Building a 4 by 4 magic square.
To fill a doubly even magic square, you must follow the diagonals!
Start by placing the number
1
in the upper left corner.
Moving as if you were reading, identify the next cell lying on one of the diagonals.
Fill the cell with the value
(i-1) \cdot n + j
n
is the order of the square,
i
the index of the row, and
j
the index of the column of the cell (in this case,
(1-1)\cdot(4)+4=4
Proceed as if you were "reading" the square, filling the diagonals. The last number will be
16
in the lower right corner.
Return to the topmost row and fill the empty squares with the remaining numbers in descending order (in the case of
n=4
15
14
12
,...).
For any other doubly even magic square, you have to subdivide the square into the adequate number of 4×4 squares, fill all of their diagonals (thus creating a sort of tilted reticulate over the square), finally filling the empty squares, disregarding the subdivision.
This is how an 8 by 8 magic square looks. We highlighted the 4 by 4 squares and their diagonals, respectively in red and blue.
As you can see in the 8×8 magic square, the filling process is not difficult at all.
The last algorithm we use in our magic square calculator allows you to fill singly even magic squares.
We resort again to a subdivision in smaller magic squares, but this time of odd order. Follow our steps:
Divide the magic square with order
N
into four odd order magic squares with order
\tfrac{N}{2}
, name them (in reading order)
S_1
S_3
S_4
S_2
Use the algorithm for odd order magic squares to fill them, then sum the number in each square to
(i-1)\cdot n^2
is the index of the square. For a square with
N=6
S_1
remains unchanged;
S_2=S_1 + 9
S_3=S_1 + 18
S_4=S_1 + 27
Let's take a look at the square now!
The first step to fill a singly even magic square.
It looks neat, doesn't it. Well, let's check the sum of the columns:
8+3+4+35+30+31=111
1+5+9+28+32+36=111
Now check the rows:
8+1+6+26+19+24=84
35+28+33+17+10+15=138
Oh no! We need to shuffle the numbers, but just a little.
Let's introduce
k
, a number that "generates" singly even magic squares of order
N
N=4\cdot k + 2
To correctly fill singly even magic square follow these three final rules.
Swap the elements of the upper and lower squares in the first and last
k-1
columns of the square.
Swap all but the central element of the column
k
of the squares
S_1
S_4
Swap only the central element of the
(k+1)^{\text{th}}
column between the upper and lower square.
Take a look at the rearranged square:
After a bit of reshuffling, our magic square is magic again!
Let's check the sums of the rows that were wrong before.
8+28+33+17+10+15=111
35+1+6+26+19+24=111
You can check the others, but trust us: the magic is back!
Depending on the order of the square, there can be many possible combinations that fulfill the magic requirements. Some of them are "trivial", others a bit more complex.
There is a single way to calculate a 3 by 3 magic square with the algorithm we've already seen. We can rotate and reflect it, but the relative positions of the number wouldn't change. We say that the 3x3 magic square is unique.
The story is different for higher orders. For
n>3
, we can transform a magic square into another by:
Rotation: four possible magic squares.
Reflection: by mirroring, you can create three magic squares.
Combination of rotations and reflections (you may find duplicates)!
Exchange of columns and rows. We can identify:
The exchange of a pair of rows and columns intersecting on a diagonal. We swap the columns
x
n-x+1
followed by the swap of the same index rows. In an even magic square there are
\tfrac{n}{2}
such columns, in an odd one
\tfrac{n-1}{2}
: the resulting combinations are respectively
2^{\tfrac{n}{2}}
2^{\tfrac{n-1}{2}}
The exchange of a pair of rows and columns on the same side of the center with the corresponding pairs on the other side. The squares we can obtain are
2^{\tfrac{n\cdot(n-2)}{8}}
for even orders and
2^{\tfrac{(n-1)\cdot(n-3)}{8}}
for odd orders.
A generalized exchange, where we swap any pair of non-central rows (columns)
x
y
, alongside a swap of the rows (columns)
n-x+1
n-y+1
x
y
are on the same side (and
x \neq y
). This is the second case we've seen. If
x=n-y+1
we have a "diagonal" exchange.
Dividing the square into four sub-squares and swapping them with their opposite across the center.
Our magic square calculator allows you to create a magic square of any (possible) size. You only have to choose the size of the square, say the magic word ("math!"). We will generate the square, and if the size is too big, we will ask you if you really want to visualize it.
In any case, we give you the possibility of downloading the magic square as a text file - if you know how to code and you want to play a little with magic, we can give you the raw materials!
A magic square is a grid of numbers where the sum of the values in every row, column, and diagonal equals a magic constant. Magic squares appeared early in history and then pervaded mathematics with their interesting properties.
How many 3x3 magic squares are there?
Accounting for rotations and reflections, there is a single way to fill a 3x3 magic square: it is unique. For larger magic squares, the number of possibilities grows: for a 4 by 4 magic square, we can find 880 distinct solutions.
How to fill a 3 by 3 magic square?
You can fill a 3×3 magic square using an algorithm or by intuition. The sum of rows, columns, and diagonals is 15.
The center is 5 (it appears in four sums).
Since the number in the center is odd, and 15 is odd too, each pair of numbers on its side has same evenness.
All the pairs of numbers encircling the center sum to 10. There are four combinations: 1 + 9, 2 + 8, 3 + 7, and 4 + 6.
Fill the square by starting with a pair and checking the sums!
What is the magic constant of a 3 by 3 magic square?
The magic constant, that is the sum of the numbers in each row, column, and diagonal of a 3 by 3 magic square, is 15. To find it, sum all of the numbers in the square:
And divide the result by the number of rows (or columns):
You can download the magic square as a text file, CSV ready. Do you want to proceed?
Rectangle diagonal angle
The rectangle diagonal angle calculator helps you find the angles created by the diagonals of a given rectangle.
Rectangle Diagonal Angle Calculator
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Reviewed by Gabriela Diaz
What's the right trapezoid?
How to calculate the slant of right angle trapezoid?
How to calculate the length of right trapezoid?
Other trapezoid-related calculators
The right trapezoid calculator is indeed the right tool for computing all the properties of our favorite trapezoid.
We'll show you what's the difference between a 'typical' and a right trapezoid, and teach you how to calculate the slant of the right angle trapezoid. Follow the text for more!
A right trapezoid is a trapezoid with one of its legs perpendicular to both of the bases. In other words, it means that such a trapezoid must contain two right angles.
It should be our favorite type of trapezoids - it's much easier to calculate! It is because the height of the trapezoid is equal to one of its sides.
Don't you worry - our right trapezoid calculator is here to help you with all the necessary steps. 👌 We'll calculate:
Right trapezoid area;
Right trapezoid slant side;
Right trapezoid height;
Right trapezoid perimeter;
Right trapezoid angles; and
Right trapezoid median - available in the advanced mode.
💡 A rectangle is also a right trapezoid - it does satisfy the condition described above!
We can easily calculate the slant side of a right angle trapezoid using the Pythagorean theorem:
d = \sqrt{(a-b)^2 + (c^2) }
d is the slant (long) side of a trapezoid;
a is the longer base;
b is the shorter base; and
c is the short side (trapezoid's height).
A right trapezoid is indeed special - its height is equal to the length of its shorter side. We can calculate it easily, using a modification of the Pythagorean theorem:
b = \sqrt{ d^2 - (a-c)^2}
b is the shorter base;
a is the longer base; and
Happy with our right trapezoid calculator? Check out all the other amazing tools that'll help you compute all the trapezoid properties you may need:
Does a trapezoid with one right angle exist?
No, it doesn't. A trapezoid may have two right angles or none at all.
Why is that so? 🤔
Two of the trapezoid bases must be parallel to each other. That's why, if one of the sides is at right angle (perpendicular) to one of the bases, it also must be perpendicular to the second one.
c (side/height)
d (slant side)
How to find the height of a triangle? What is the height of a triangle formula? Check out with this triangle height calculator!
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Principal energy levels in atomic physics
In chemistry and atomic physics, an electron shell may be thought of as an orbit followed by electrons around an atom's nucleus. The closest shell to the nucleus is called the "1 shell" (also called the "K shell"), followed by the "2 shell" (or "L shell"), then the "3 shell" (or "M shell"), and so on farther and farther from the nucleus. The shells correspond to the principal quantum numbers (n = 1, 2, 3, 4 ...) or are labeled alphabetically with the letters used in X-ray notation (K, L, M, …).
Each shell can contain only a fixed number of electrons: the first shell can hold up to two electrons, the second shell can hold up to eight (2 + 6) electrons, the third shell can hold up to 18 (2 + 6 + 10) and so on. The general formula is that the nth shell can in principle hold up to 2(n2) electrons.[1] For an explanation of why electrons exist in these shells, see electron configuration.[2]
The 1913 Bohr model of the atom attempted an arrangement of electrons in their sequential orbits, however, at that time Bohr continued to increase the inner orbit of the atom to eight electrons as the atoms got larger. Bohr built his 1913 model of electrons in elements thus:[3]
“From the above we are led to the following possible scheme for the arrangement of the electrons in light atoms:
Periodic table of Bohr in 1913 showing electron configurations in his second paper where he went to the 24th element.[4][5]
The shell terminology comes from Arnold Sommerfeld's modification of the Bohr model. During this period Bohr was working with Walther Kossel, whose papers in 1914 and in 1916 called the orbits “shells.”[6][7] Sommerfeld retained Bohr's planetary model, but added mildly elliptical orbits (characterized by additional quantum numbers ℓ and m) to explain the fine spectroscopic structure of some elements.[8] The multiple electrons with the same principal quantum number (n) had close orbits that formed a "shell" of positive thickness instead of the circular orbit of Bohr's model which orbits called “rings” were described by a plane.[9]
The existence of electron shells was first observed experimentally in Charles Barkla's and Henry Moseley's X-ray absorption studies. Moseley’s work did not directly concern the study of electron shells, because he was trying to prove that the periodic table was not arranged by weight, but by the charge of the protons in the nucleus.[10] However, because in a neutral atom, the number of electrons equals the number of protons, this work was extremely important to Niels Bohr who mentioned Moseley’s work several times in his interview of 1962.[11] Moseley was part of Rutherford’s group, as was Niels Bohr. Moseley measured the frequencies of X-rays emitted by every element between calcium and zinc, and found that the frequencies became greater as the elements got heavier, leading to the theory that electrons were emitting X-rays when they were shifted to lower shells.[12] This led to the conclusion that the electrons were in Kossel’s shells with a definite limit per shell, labeling the shells with the letters K, L, M, N, O, P, and Q.[13][14] The origin of this terminology was alphabetic. Barkla, who worked independently from Moseley as an X-ray spectrometry experimentalist, first noticed two distinct types of scattering from shooting X-rays at elements in 1909 and named them "A" and "B". Barkla described these two types of X-ray diffraction: the first was unconnected with the type of material used in the experiment, and could be polarized. The other second diffraction beam he called "fluorescent" because it depended on the irradiated material.[15] It was not known what these lines meant at the time, but in 1911 Barkla decided there might be scattering lines previous to "A", so he began at "K".[16] However, later experiments indicated that the K absorption lines are produced by the innermost electrons. These letters were later found to correspond to the n values 1, 2, 3, etc. that were used in the Bohr model. They are used in the spectroscopic Siegbahn notation.
The work of assigning electrons to shells was continued from 1913 to 1925 by many chemists and a few physicists. Niels Bohr was one of the few physicists who followed the chemist's work[17] of defining the periodic table, while Arnold Sommerfeld worked more on trying to make a relativistic working model of the atom that would explain the fine structure of the spectra from a classical orbital physics standpoint through the Atombau approach.[18] Einstein and Rutherford, who did not follow chemistry, were unaware of the chemists who were developing electron shell theories of the periodic table from a chemistry point of view, such as Irving Langmuir, Charles Bury, J.J. Thomson, and Gilbert Lewis, who all introduced corrections to Bohr’s model such as a maximum of two electrons in the first shell, eight in the next and so on, and were responsible for explaining valency in the outer electron shells, and the building up of atoms by adding electrons to the outer shells.[19][20] So when Bohr outlined his electron shell atomic theory in 1922, there was no mathematical formula for the theory. So Rutherford said he was hard put "to form an idea of how you arrive at your conclusions".[21][22] Einstein said of Bohr's 1922 paper that his "electron-shells of the atoms together with their significance for chemistry appeared to me like a miracle – and appears to me as a miracle even today".[23] Arnold Sommerfeld, who had followed the Atombau structure of electrons instead of Bohr who was familiar with the chemists' views of electron structure, spoke of Bohr's 1921 lecture and 1922 article on the shell model as "the greatest advance in atomic structure since 1913".[24][25][26] However, the electron shell development of Niels Bohr was basically the same theory as that of the chemist Charles Rugeley Bury in his 1921 paper.[27][28][29]
As work continued on the electron shell structure of the Sommerfeld-Bohr Model, Sommerfeld had introduced three "quantum numbers n, k, and m, that described the size of the orbit, the shape of the orbit, and the direction in which the orbit was pointing."[30] Because we use k for the Boltzmann constant, the azimuthal quantum number was changed to ℓ. When the modern quantum mechanics theory was put forward based on Heisenberg's matrix mechanics and Schrödinger's wave equation, these quantum numbers were kept in the current quantum theory but were changed to n being the principal quantum number, and m being the magnetic quantum number.
However, the final form of the electron shell model still in use today for the number of electrons in shells was discovered in 1923 by Edmund Stoner, who introduced the principle that the nth shell was described by 2(n2). Seeing this in 1925, Wolfgang Pauli added a fourth quantum number, "spin", during the old quantum theory period of the Sommerfeld-Bohr solar system atom to complete the modern electron shell theory.[31]
3D views of some hydrogen-like atomic orbitals showing probability density and phase (g orbitals and higher are not shown).
Each shell is composed of one or more subshells, which are themselves composed of atomic orbitals. For example, the first (K) shell has one subshell, called 1s; the second (L) shell has two subshells, called 2s and 2p; the third shell has 3s, 3p, and 3d; the fourth shell has 4s, 4p, 4d and 4f; the fifth shell has 5s, 5p, 5d, and 5f and can theoretically hold more in the 5g subshell that is not occupied in the ground-state electron configuration of any known element.[2] The various possible subshells are shown in the following table:
g 4 18 5th shell and higher (theoretically) (next in alphabet after f)[32]
The first column is the "subshell label", a lowercase-letter label for the type of subshell. For example, the "4s subshell" is a subshell of the fourth (N) shell, with the type (s) described in the first row.
The second column is the azimuthal quantum number (ℓ) of the subshell. The precise definition involves quantum mechanics, but it is a number that characterizes the subshell.
The third column is the maximum number of electrons that can be put into a subshell of that type. For example, the top row says that each s-type subshell (1s, 2s, etc.) can have at most two electrons in it. In each case the figure is 4 greater than the one above it.
The fourth column says which shells have a subshell of that type. For example, looking at the top two rows, every shell has an s subshell, while only the second shell and higher have a p subshell (i.e., there is no "1p" subshell).
The final column gives the historical origin of the labels s, p, d, and f. They come from early studies of atomic spectral lines. The other labels, namely g, h and i, are an alphabetic continuation following the last historically originated label of f.
Therefore, the K shell, which contains only an s subshell, can hold up to 2 electrons; the L shell, which contains an s and a p, can hold up to 2 + 6 = 8 electrons, and so forth; in general, the nth shell can hold up to 2n2 electrons.[1]
Although that formula gives the maximum in principle, in fact that maximum is only achieved (in known elements) for the first four shells (K, L, M, N). No known element has more than 32 electrons in any one shell.[33][34] This is because the subshells are filled according to the Aufbau principle. The first elements to have more than 32 electrons in one shell would belong to the g-block of period 8 of the periodic table. These elements would have some electrons in their 5g subshell and thus have more than 32 electrons in the O shell (fifth principal shell).
Subshell energies and filling order
Further information: Aufbau principle
For multielectron atoms n is a poor indicator of electron's energy. Energy spectra of some shells interleave.
The states crossed by same red arrow have same
{\displaystyle n+\ell }
value. The direction of the red arrow indicates the order of state filling.
Although it is sometimes stated that all the electrons in a shell have the same energy, this is an approximation. However, the electrons in one subshell do have exactly the same level of energy, with later subshells having more energy per electron than earlier ones. This effect is great enough that the energy ranges associated with shells can overlap.
The filling of the shells and subshells with electrons proceeds from subshells of lower energy to subshells of higher energy. This follows the n + ℓ rule which is also commonly known as the Madelung rule. Subshells with a lower n + ℓ value are filled before those with higher n + ℓ values. In the case of equal n + ℓ values, the subshell with a lower n value is filled first.
Because of this, the later shells are filled over vast sections of the periodic table. The K shell fills in the first period (hydrogen and helium), while the L shell fills in the second (lithium to neon). However, the M shell starts filling at sodium (element 11) but does not finish filling till copper (element 29), and the N shell is even slower: it starts filling at potassium (element 19) but does not finish filling till ytterbium (element 70). The O, P, and Q shells begin filling in the known elements, but they are not complete even at the heaviest known element, oganesson (element 118).
The list below gives the elements arranged by increasing atomic number and shows the number of electrons per shell. At a glance, the subsets of the list show obvious patterns. In particular, every set of five elements (in electric blue) before each noble gas (group 18, in yellow) heavier than helium have successive numbers of electrons in the outermost shell, namely three to seven.
The list below is primarily consistent with the Aufbau principle. However, there are a number of exceptions to the rule; for example palladium (atomic number 46) has no electrons in the fifth shell, unlike other atoms with lower atomic number. The elements past 108 have such short half-lives that their electron configurations have not yet been measured, and so predictions have been inserted instead.
Wikimedia Commons has media related to Electron shell diagrams.
^ a b Re: Why do electron shells have set limits ? madsci.org, 17 March 1999, Dan Berger, Faculty Chemistry/Science, Bluffton College
^ a b Electron Subshells. Corrosion Source.
^ See Wikipedia periodic table.
^ Niels Bohr, “On the Constitution of Atoms and Molecules, Part II Systems containing only a Single Nucleus Philosophical Magazine 26:857--875 (1913)
^ Kragh, Helge. “Niels Bohr’s Second Atomic Theory.” Historical Studies in the Physical Sciences, vol. 10, University of California Press, 1979, pp. 123–86, https://doi.org/10.2307/27757389.
^ W. Kossel, “Über Molekülbildung als Folge des Atombaues,” Ann. Phys., 1916, 49, 229-362 (237).
^ Translated in Helge Kragh, Aarhus, LARS VEGARD, ATOMIC STRUCTURE, AND THE PERIODIC SYSTEM, Bull. Hist. Chem., VOLUME 37, Number 1 (2012), p.43.
^ Donald Sadoway, Introduction to Solid State Chemistry, Lecture 5 Archived 29 June 2011 at the Wayback Machine
^ Bohr, Niels (1913). On the Constitution of Atoms and Molecules, Part I. _Philosophical Magazine_ 26:1--25.
^ Uhler, Horace Scudder. “On Moseley’s Law for X-Ray Spectra.” Proceedings of the National Academy of Sciences of the United States of America, vol. 3, no. 2, National Academy of Sciences, 1917, pp. 88–90, http://www.jstor.org/stable/83748.
^ Niels Bohr interview 1962 Session III https://www.aip.org/history-programs/niels-bohr-library/oral-histories/4517-3
^ Kumar, Manjit. Quantum: Einstein, Bohr, and the great debate about the nature of reality / Manjit Kumar.—1st American ed., 2008. Chap.4.
^ Barkla, Charles G. (1911). "XXXIX.The spectra of the fluorescent Röntgen radiations". Philosophical Magazine. Series 6. 22 (129): 396–412. doi:10.1080/14786440908637137. Previously denoted by letters B and A (...). The letters K and L are, however, preferable, as it is highly probable that series of radiations both more absorbable and more penetrating exist.
^ Charles G. Barkla M.A. D.Sc. (1911) XXXIX. The spectra of the fluorescent Röntgen radiations, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 22:129, 396-412, DOI: 10.1080/14786440908637137
^ T.Hirosigeand S.Nisio,"Formation of Bohr's Theory of Atomic Constitution",Jap. Stud.Hist.Set.,No. 3(1964),6-28.
^ See Periodic Table for full history.
^ Niels Bohr Collected Works, Vol. 4, p. 740. Postcard from Arnold Sommerfeld to Bohr, 7 March 1921.
^ Pais, Abraham (1991), Niels Bohr’s Times, in Physics, Philosophy, and Polity (Oxford: Clarendon Press), quoted p. 205.
^ Schilpp, Paul A. (ed.) (1969), Albert Einstein: Philosopher-Scientist (New York: MJF Books). Collection first published in 1949 as Vol. VII in the series The Library of Living Philosophers by Open Court, La Salle, IL, Einstein, Albert ‘Autobiographical Notes’, pp.45-47.
^ Bury, Charles R. (July 1921). "Langmuir's Theory of the Arrangement of Electrons in Atoms and Molecules". Journal of the American Chemical Society. 43 (7): 1602–1609. doi:10.1021/ja01440a023. ISSN 0002-7863.
^ The Genesis of the Bohr Atom, John L. Heilbron and Thomas S. Kuhn, Historical Studies in the Physical Sciences, Vol. 1 (1969), pp. vi, 211-290 (81 pages), University of California Press,p. 285-286.
^ Jue, T. (2009). "Quantum Mechanic Basic to Biophysical Methods". Fundamental Concepts in Biophysics. Berlin: Springer. p. 33. ISBN 978-1-58829-973-4.
^ Electron & Shell Configuration Archived 28 December 2018 at the Wayback Machine. Chemistry.patent-invent.com. Retrieved on 1 December 2011.
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Compute solution to given system of LMIs - MATLAB feasp - MathWorks América Latina
Solve System of LMIs
Compute solution to given system of LMIs
[tmin,xfeas] = feasp(lmisys,options,target)
[tmin,xfeas] = feasp(lmisys,options,target) computes a solution xfeas (if any) of the system of LMIs described by lmisys. The vector xfeas is a particular value of the decision variables for which all LMIs are satisfied.
Given the LMI system
{N}^{T}LxN\le {M}^{T}R\left(x\right)M,
xfeas is computed by solving the auxiliary convex program:
Minimize t subject to NTL(x)N–MTR(x)M≤tI.
The global minimum of this program is the scalar value tmin returned as first output argument by feasp. The LMI constraints are feasible if tmin ≤ 0 and strictly feasible if tmin < 0. If the problem is feasible but not strictly feasible, tmin is positive and very small. Some post-analysis may then be required to decide whether xfeas is close enough to feasible.
The optional argument target sets a target value for tmin. The optimization code terminates as soon as a value of t below this target is reached. The default value is target = 0.
Note that xfeas is a solution in terms of the decision variables and not in terms of the matrix variables of the problem. Use dec2mat to derive feasible values of the matrix variables from xfeas.
The optional argument options gives access to certain control parameters for the optimization algorithm. This five-entry vector is organized as follows:
options(1) is not used.
options(3) resets the feasibility radius. Setting options(3) to a value R > 0 further constrains the decision vector x = (x1, . . ., xN) to lie within the ball
\sum _{i=1}^{N}{x}_{i}^{2}<{R}^{2}
In other words, the Euclidean norm of xfeas should not exceed R. The feasibility radius is a simple means of controlling the magnitude of solutions. Upon termination, feasp displays the f-radius saturation, that is, the norm of the solution as a percentage of the feasibility radius R.
The default value is R = 109. Setting options(3) to a negative value activates the “flexible bound” mode. In this mode, the feasibility radius is initially set to 108, and increased if necessary during the course of optimization
options(4) helps speed up termination. When set to an integer value J > 0, the code terminates if t did not decrease by more than one percent in relative terms during the last J iterations. The default value is 10. This parameter trades off speed vs. accuracy. If set to a small value (< 10), the code terminates quickly but without guarantee of accuracy. On the contrary, a large value results in natural convergence at the expense of a possibly large number of iterations.
Setting option(i) to zero is equivalent to setting the corresponding control parameter to its default value. Consequently, there is no need to redefine the entire vector when changing just one control parameter. To set the maximum number of iterations to 10, for instance, it suffices to type
options=zeros(1,5) % default value for all parameters
options(2)=10
When the least-squares problem solved at each iteration becomes ill conditioned, the feasp solver switches from Cholesky-based to QR-based linear algebra (see Memory Problems for details). Since the QR mode typically requires much more memory, MATLAB® may run out of memory and display the message
??? Error using ==> feaslv
You should then ask your system manager to increase your swap space or, if no additional swap space is available, set options(4) = 1. This will prevent switching to QR and feasp will terminate when Cholesky fails due to numerical instabilities.
Consider the problem of finding P > I such that:
{A}_{1}^{T}P+P{A}_{1}<0,
{A}_{2}^{T}P+P{A}_{2}<0,
{A}_{3}^{T}P+P{A}_{3}<0,
{A}_{1}=\left(\begin{array}{cc}-1& 2\\ 1& -3\end{array}\right),{A}_{2}=\left(\begin{array}{cc}-0.8& 1.5\\ 1.3& -2.7\end{array}\right),{A}_{3}=\left(\begin{array}{cc}-1.4& 0.9\\ 0.7& -2.0\end{array}\right).
This problem arises when studying the quadratic stability of the polytope of the matrices,
Co\left\{{A}_{1},{A}_{2},{A}_{3}\right\}
To assess feasibility using feasp, first enter the LMIs.
p = lmivar(1,[2 1]);
A1 = [-1 2;1 -3];
A2 = [-0.8 1.5; 1.3 -2.7];
A3 = [-1.4 0.9;0.7 -2.0];
lmiterm([1 1 1 p],1,A1,'s'); % LMI #1
lmiterm([-4 1 1 p],1,1); % LMI #4: P
lmiterm([4 1 1 0],1); % LMI #4: I
Call feasp to a find a feasible decision vector.
Result: best value of t: -3.136305
The result tmin = -3.1363 means that the problem is feasible. Therefore, the dynamical system
\underset{}{\overset{˙}{x}}=A\left(t\right)x
A\left(t\right)\in Co\left\{{A}_{1},{A}_{2},{A}_{3}\right\}.
To obtain a Lyapunov matrix P proving the quadratic stability, use dec2mat.
P = dec2mat(lmis,xfeas,p)
It is possible to add further constraints on this feasibility problem. For instance, the following command bounds the Frobenius norm of P by 10 while asking tmin to be less than or equal to –1.
options = [0,0,10,0,0];
[tmin,xfeas] = feasp(lmis,options,-1);
*** new lower bound: -3.726964
f-radius saturation: 91.385% of R = 1.00e+01
The third entry of options sets the feasibility radius to 10 while the third argument to feasp, -1, sets the target value for tmin. This constraint yields tmin = -1.011 and a matrix P with largest eigenvalue
{\lambda }_{max}\left(P\right)
P = dec2mat(lmis,xfeas,p);
e = eig(P)
The feasibility solver feasp is based on Nesterov and Nemirovski's Projective Method described in:
Nesterov, Y., and A. Nemirovski, Interior Point Polynomial Methods in Convex Programming: Theory and Applications, SIAM, Philadelphia, 1994.
Nemirovski, A., and P. Gahinet, “The Projective Method for Solving Linear Matrix Inequalities,” Proc. Amer. Contr. Conf., 1994, Baltimore, Maryland, p. 840–844.
The optimization is performed by the C-MEX file feaslv.mex.
mincx | gevp | dec2mat
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Volume 61 Issue 3 | Kyoto Journal of Mathematics
Home > Journals > Kyoto J. Math. > Volume 61 > Issue 3
Kyoto J. Math. 61 (3), (September 2021)
Gromov–Witten invariants of local
{\mathbb{P}}^{2}
and modular forms
Tom Coates, Hiroshi Iritani
Kyoto J. Math. 61 (3), 543-706, (September 2021) DOI: 10.1215/21562261-2021-0010
KEYWORDS: Gromov–Witten Invariants, geometric quantization, modular form, mirror symmetry, toric Calabi–Yau 3-fold, 14N35, 14J33, 53D45, 53D50
We construct a sheaf of Fock spaces over the moduli space of elliptic curves
{E}_{y}
{\mathrm{\Gamma }}_{1}\left(3\right)
-level structure, arising from geometric quantization of
{H}^{1}\left({E}_{y}\right)
, and a global section of this Fock sheaf. The global section coincides, near appropriate limit points, with the Gromov–Witten potentials of local
{\mathbb{P}}^{2}
and of the orbifold
\left[{\mathbb{C}}^{3}∕{\mathrm{\mu }}_{3}\right]
. This proves that the Gromov–Witten potentials of local
{\mathbb{P}}^{2}
are quasimodular functions for the group
{\mathrm{\Gamma }}_{1}\left(3\right)
, as predicted by Aganagic, Bouchard, and Klemm, and it proves the crepant resolution conjecture for
\left[{\mathbb{C}}^{3}∕{\mathrm{\mu }}_{3}\right]
in all genera.
Ideal-adic completion of quasi-excellent rings (after Gabber)
Kazuhiko Kurano, Kazuma Shimomoto
KEYWORDS: ideal-adic completion, lifting problem, local uniformization, excellent ring, quasi-excellent ring, 13B35, 13F25, 13F40
We give a detailed proof of a result of Gabber (unpublished) on the lifting problem of quasi-excellent rings, extending the previous work of Nishimura and Nishimura. As a corollary, we establish that an ideal-adic completion of an excellent (resp., quasi-excellent) ring is excellent (resp., quasi-excellent).
Noncommutative homological mirror symmetry of elliptic curves
KEYWORDS: homological mirror symmetry, noncommutative mirror functor, LG/CY correspondence, Elliptic curve, 53D37, 14A22
We prove an equivalence of two
{A}_{\mathrm{\infty }}
-functors, via Orlov’s Landau–Ginzburg/ Calabi–Yau (LG/CY) correspondence. One is the Polishchuk–Zaslow mirror symmetry functor of elliptic curves, and the other is a localized mirror functor from the Fukaya category of
{T}^{2}
to a category of noncommutative matrix factorizations. As a corollary, we prove that the noncommutative mirror functor
\mathcal{L}{\mathcal{M}}_{gr}^{{\mathbb{L}}_{t}}
realizes homological mirror symmetry for any t.
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Three-Dimensional Numerical Simulations of Flows Past Smooth and Rough/Bare and Helically Straked Circular Cylinders Allowed to Undergo Two Degree-of-Freedom Motions | J. Offshore Mech. Arct. Eng. | ASME Digital Collection
Shell Global Solutions (US) Inc.,
Fluid Flow & Flow Assurance
, Westhollow Technology Center, Houston, TX 77082
Raghu G. Menon,
Raghu G. Menon
Pontaza, J. P., Menon, R. G., and Chen, H. (March 30, 2009). "Three-Dimensional Numerical Simulations of Flows Past Smooth and Rough/Bare and Helically Straked Circular Cylinders Allowed to Undergo Two Degree-of-Freedom Motions." ASME. J. Offshore Mech. Arct. Eng. May 2009; 131(2): 021301. https://doi.org/10.1115/1.3058697
We simulate the flow past smooth and rough rigid circular cylinders that are either bare or outfitted with helical strakes. We consider operating conditions that correspond to high Reynolds numbers of
105
106
, and allow for two degree-of-freedom motions such that the structure is allowed to respond to flow-induced cross-flow and in-line forces. The computations are performed using a parallelized Navier–Stokes in-house solver using overset grids. For smooth surface simulations at a Reynolds number of
105
, we use a Smagorinsky large eddy simulation turbulence model and for the Reynolds number cases of
106
we make use of the unsteady Reynolds-averaged Navier–Stokes equations with a two-layer
k
-epsilon turbulence model. The rough surface modifications of the two-layer
k
-epsilon turbulence model due to Durbin et al. (2001, “Rough Wall Modification of Two-Layer k-Epsilon,” ASME J. Fluids Eng., 123, pp. 16–21) are implemented to account for surface roughness effects. In all our computations we aim to resolve the boundary layer directly by using adequate grid spacing in the near-wall region. The predicted global flow parameters under different surface conditions are in good agreement with experimental data, and significant vortex-induced vibration suppression is observed when using helically straked cylinders.
confined flow, flow instability, flow simulation, Navier-Stokes equations, turbulence
Cylinders, Flow (Dynamics), Surface roughness, Reynolds number, Drag (Fluid dynamics), Vortex-induced vibration, Turbulence, Circular cylinders, Degrees of freedom, Simulation, Computer simulation
Rough Wall Modification of Two-Layer k-Epsilon
Prediction of Unsteady Loading on a Circular Cylinder in High Reynolds Number Flows Using Large Eddy Simulation
,” OMAE Paper No. 2005–67044.
Detached-Eddy Simulation Past a Circular Cylinder
An Experimental Study Into Vortex Induced Vibration of a Cylinder at High Reynolds Numbers
,” Oceanic Consulting Corporation Report. Data quoted with permission.
Flow Around Circular Cylinders: Fundamentals
Supplemental VIV Experiments With Cylinders at High Reynolds Numbers
Surface Roughness Effects on the Mean Flow Past Circular Cylinders
, 2006, Shell Global Solutions internal report. Data quoted with permission.
Supercritical Reynolds Number Simulation for Two-Dimensional Flow Over Circular Cylinders
Vortex-Induced Vibration Tests of a Flexible Smooth Cylinder at Supercritical Reynolds Numbers
Proceedings of the Seventh International Offshore and Polar Engineering Conference
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Today I watched quite interesting hangout. It was a quick (30min) discussion. Attenders:
Few days ago @DHH wrote some “controversial” post on his blog about TDD and in general presented his point of view to TDD. As a result of that we got a huge discussion on the Twitter about TDD. Some people says that every professional should use TDD and and so on. As a result of that guys from ThoughtWorks organized this hangout.
Youtube playlist (Parts from 1 to 6)
% <![CDATA[ \begin{align*} & \phi(x,y) = \phi \left(\sum_{i=1}^n x_ie_i, \sum_{j=1}^n y_je_j \right) = \sum_{i=1}^n \sum_{j=1}^n x_i y_j \phi(e_i, e_j) = \\ & (x_1, \ldots, x_n) \left( \begin{array}{ccc} \phi(e_1, e_1) & \cdots & \phi(e_1, e_n) \\ \vdots & \ddots & \vdots \\ \phi(e_n, e_1) & \cdots & \phi(e_n, e_n) \end{array} \right) \left( \begin{array}{c} y_1 \\ \vdots \\ y_n \end{array} \right) \end{align*} %]]>
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Dynamics of the quorum sensing switch: stochastic and non-stationary effects | BMC Systems Biology | Full Text
From: Dynamics of the quorum sensing switch: stochastic and non-stationary effects
Individual cell trajectories for autoinducer induction in the stochastic model. Individual cell trajectories (blue lines), cell population average (orange line) and deterministic solution (red dashed line) for an induction experiment at
{c}_{A}^{\ast }=25\text{nM}
for the lux01 operon in the stochastic model. Individual cell trajectories show the heterogeneous distribution of cell jumping times. While some cells achieve full induction of the operon before the deterministic case, the global response of the population reaches steady-state at ∼30 hours, slower than the deterministic solution.
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Output settings - Training parameters | CatBoost
R package, Command-line
[0; \inf)
None (Turned on and set to 0.5)
Supported processing units CPU and GPU
The name of the output file to save the ROC curve points to. This parameter can only be set in cross-validation mode if the Logloss loss function is selected. The ROC curve points are calculated for the test fold.
The output file is saved to the catboost_info directory.
None (the file is not saved)
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IsStronglyRegular - Maple Help
Home : Support : Online Help : Mathematics : Discrete Mathematics : Graph Theory : GraphTheory Package : IsStronglyRegular
test if graph is strongly regular
Strongly regular graphs in SpecialGraphs
IsStronglyRegular(G,opts)
(optional) equation of the form parameters=true or parameters=false
parameters : keyword option of the form parameters=true or parameters=false. This specifies whether the parameters [k, lambda, mu] should be returned when the graph is strongly regular. The default is false.
The IsStronglyRegular(G) command returns true if G is a strongly regular graph and false otherwise.
An undirected graph G is strongly regular if there exist integers k, lambda, and mu such that every vertex has k neighbors and for every pair of vertices (u,v), u and v have exactly lambda neighbors in common if they are themselves adjacent, and exactly mu neighbors in common if they are not.
Note that some parts of this definition may be satisfied trivially, in which a complete graph every pair of vertices is adjacent, so the choice of mu could be arbitrary and therefore mu is undefined.
Any strongly regular graph is regular, but the converse is not true.
The following are graphs in the SpecialGraphs subpackage which are strongly regular.
Cameron graph
Suzuki graph
\mathrm{with}\left(\mathrm{GraphTheory}\right):
\mathrm{with}\left(\mathrm{SpecialGraphs}\right):
G≔\mathrm{Graph}\left({{1,2},{1,3},{2,3},{3,4}}\right)
\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 1: an undirected unweighted graph with 4 vertices and 4 edge\left(s\right)}}
\mathrm{DegreeSequence}\left(G\right)
[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]
\mathrm{IsStronglyRegular}\left(G\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
P≔\mathrm{PetersenGraph}\left(\right)
\textcolor[rgb]{0,0,1}{P}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 2: an undirected unweighted graph with 10 vertices and 15 edge\left(s\right)}}
\mathrm{DegreeSequence}\left(P\right)
[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]
\mathrm{IsStronglyRegular}\left(P,'\mathrm{parameters}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]
\mathrm{DrawGraph}\left(P\right)
C≔\mathrm{ClebschGraph}\left(\right)
\textcolor[rgb]{0,0,1}{C}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 3: an undirected unweighted graph with 16 vertices and 40 edge\left(s\right)}}
\mathrm{IsStronglyRegular}\left(C,'\mathrm{parameters}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]
\mathrm{DrawGraph}\left(C\right)
The GraphTheory[IsStronglyRegular] command was introduced in Maple 2019.
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Screening Cowpea (Vigna unguiculata (L.) Walp.) Genotypes for Enhanced N 2Fixation and Water Use Efficiency under Field Conditions in Ghana
4Department of Crop Sciences, Tshwane University of Technology, Pretoria, South Africa
To explore the variations in symbiotic N2 fixation and water use efficiency in cowpea, this study evaluated 25 USDA cowpea genotypes subjected to drought under field conditions at two locations (Kpachi and Woribogu) in the Northern region of Ghana. The 15N and 13C natural abundance techniques were respectively used to assess N2 fixation and water use efficiency. The test genotypes elicited high symbiotic dependence in association with indigenous rhizobia, deriving between 55% and 98% of their N requirements from symbiosis. Consequently, the amounts of N-fixed by the genotypes showed remarkable variations, with values ranging from 37 kg∙N-fixed∙ha−1 to 337 kg∙N-fixed∙ha−1. Most genotypes elicited contrasting symbiotic performance between locations, a finding that highlights the effect of complex host/soil microbiome compatibility on the efficiency of the cowpea-rhizobia symbiosis. The test genotypes showed marked variations in water use efficiency, with most of the genotypes recording higher δ13C values when planted at Kpachi. Despite the high symbiotic dependence, the grain yield of the test cowpeas was low due to the imposed drought, and ranged from 56 kg/ha to 556 kg/ha at Kpachi, and 143 kg/ha to 748 kg/ha at Woribogu. The fact that some genotypes could grow and produce grain yields of 627 - 748 kg/ha under drought imposition is an important trait that could be tapped for further improvement of cowpea. These findings highlight the importance of the cowpea-rhizobia symbiosis and enhanced water relations in the crop’s wider adaptation to adverse edaphoclimatic conditions.
{\delta }^{\text{15}}\text{N}(‰)=\frac{{\left[{}^{\text{15}}\text{N}/{}^{\text{14}}\text{N}\right]}_{\text{sample}}-{\left[{}^{\text{15}}\text{N}/{}^{\text{14}}\text{N}\right]}_{\text{atm}}}{{\left[{}^{\text{15}}\text{N}/{}^{\text{14}}\text{N}\right]}_{\text{atm}}}\times 1000
\%\text{Ndfa}=\frac{{\delta }^{15}{\text{N}}_{\text{ref}}-{\delta }^{15}{\text{N}}_{\text{leg}}}{{\delta }^{15}{\text{N}}_{\text{ref}}-\text{B}}\times 100
{\delta }^{\text{13}}\text{C}=\left[\frac{{\left({}^{\text{13}}\text{C}/{}^{\text{12}}\text{C}\right)}_{\text{sample}}}{{\left({}^{\text{13}}\text{C}/{}^{\text{12}}\text{C}\right)}_{\text{standard}}}-1\right]\times 1000
Yahaya, D., Denwar, N., Mohammed, M. and Blair, M.W. (2019) Screening Cowpea (Vigna unguiculata (L.) Walp.) Genotypes for Enhanced N2 Fixation and Water Use Efficiency under Field Conditions in Ghana. American Journal of Plant Sciences, 10, 640-658. https://doi.org/10.4236/ajps.2019.104047
1. Kermah, M., Franke, A.C., Adjei-Nsiah, S., Ahiabor, B.D.K., Abaidoo, R.C. and Giller, K.E. (2018) N2-Fixation and N Contribution by Grain Legumes under Different Soil Fertility Status and Cropping Systems in the Guinea Savanna of Northern Ghana. Agriculture, Ecosystems and Environment, 261, 201-210. https://doi.org/10.1016/j.agee.2017.08.028
2. Nyemba, R.C. and Dakora, F.D. (2010) Evaluating N2 Fixation by Food Grain Legumes in Farmers’ Fields in Three Agro-Ecological Zones of Zambia, Using 15N Natural Abundance. Biology and Fertility of Soils, 46, 461-470. https://doi.org/10.1007/s00374-010-0451-2
3. Mohale, K.C., Belane, A.K. and Dakora, F.D. (2014) Symbiotic N Nutrition, C Assimilation, and Plant Water Use Efficiency in Bambara Groundnut (Vigna subterranea L. Verdc) Grown in Farmers’ Fields in South Africa, Measured Using 15N and 13C Natural Abundance. Biology and Fertility of Soils, 50, 307-319. https://doi.org/10.1007/s00374-013-0841-3
4. Marandu, E.T., Semu, E., Mrema, J.P. and Nyaki, A.S. (2013) Contribution of Legume Rotations to the Nitrogen Requirements of a Subsequent Maize Crop on a Rhodic Ferrasol in Tanga, Tanzania. Tanzania Journal of Agricultural Sciences, 12, 23-29. https://doi.org/10.4314/eajsci.v2i2.40343
5. Belane, A.K. and Dakora, F.D. (2010) Symbiotic N2 Fixation in 30 Field-Grown Cowpea (Vigna unguiculata L. Walp.) Genotypes in the Upper West Region of Ghana Measured Using 15N Natural Abundance. Biology and Fertility of Soils, 46, 191-198. https://doi.org/10.1007/s00374-009-0415-6
6. Oteng-Frimpong, R. and Dakora, F.D. (2018) Selecting Elite Groundnut (Arachis hypogaea L) Genotypes for Symbiotic N Nutrition, Water-Use Efficiency and Pod Yield at Three Field Sites , Using 15N and 13C Natural Abundance. Symbiosis, 75, 229-243. https://doi.org/10.1007/s13199-017-0524-1
7. Chibeba, M.A., Kyei-boahen, S., Guimaraes, M.D.F., Nogueira, A.M. and Hungria, M. (2017) Isolation, Characterization and Selection of Indigenous Bradyrhizobium Strains with Outstanding Symbiotic Performance to Increase Soybean Yields in Mozambique. Agriculture, Ecosystems and Environment, 246, 291-305. https://doi.org/10.1016/j.agee.2017.06.017
8. Mapope, N. and Dakora, F.D. (2016) N2 Fixation, Carbon Accumulation, and Plant Water Relations in Soybean (Glycine max L. Merrill) Varieties Sampled from Farmers’ Fields in South Africa, Measured Using 15N and 13C Natural Abundance. Agriculture, Ecosystems and Environment, 221, 174-186. https://doi.org/10.1016/j.agee.2016.01.023
9. Samago, T.Y., Anniye, E.W. and Dakora, F.D. (2018) Grain Yield of Common Bean (Phaseolus vulgaris L.) Varieties Is Markedly Increased by Rhizobial Inoculation and Phosphorus Application in Ethiopia. Symbiosis, 75, 245-255. https://doi.org/10.1007/s13199-017-0529-9
10. Mohammed, M., Jaiswal, S.K., Sowley, E.N.K., Ahiabor, B.D.K. and Dakora, F.D. (2018) Symbiotic N2 Fixation and Grain Yield of Endangered Kersting’s Groundnut Landraces in Response to Soil and Plant Associated Bradyrhizobium Inoculation to Promote Ecological Resource-Use Efficiency. Frontiers in Microbiology, 9, 2105. https://doi.org/10.3389/fmicb.2018.02105
11. Unkovich, M., Herridge, D., Peoples, M., Cadisch, G., Boddey, B., Giller, K., Alves, B. and Chalk, P. (2008) Measuring Plant-Associated Nitrogen Fixation in Agricultural Systems. Australian Centre for International Agricultural Research, Canberra, Australia.
12. Mitra, J. (2001) Genetics and Genetic Improvement of Drought Resistance in Crop Plants. Current Science, 80, 758-763.
13. Watanabe, I., Terao, T. and Tomio, T. (1998) Drought Tolerance of Cowpea (Vigna unguiculata (L.) Walp.) 2: Field Trial in the Dry Season of Sudan Savanna and Dry Matter Production of Potted Plants under Water-Stress. JIRCAS Journal for Scientific Papers, 6, 29-37.
14. Picasso, V.D., Casler, M.D. and Undersander, D. (2019) Resilience, Stability, and Productivity of Alfalfa Cultivars in Rainfed Regions of North America. Crop Science, 59, 800-810. https://doi.org/10.2135/cropsci2018.06.0372
15. Belane, A.K. and Dakora, F.D. (2012) Elevated Concentrations of Dietarily-Important Trace Elements and Macronutrients in Edible Leaves and Grain of 27 Cowpea (Vigna unguiculata L. Walp.) Genotypes: Implications for Human Nutrition and Health. Food and Nutrition Sciences, 3, 377-386. https://doi.org/10.4236/fns.2012.33054
16. FAO (2016). http://www.fao.org/faostat/en. Accessed on 2018-05-23
17. Hall, A.E. (2012) Phenotyping Cowpeas for Adaptation to Drought. Frontiers in Physiology, 3, 155-162. https://doi.org/10.3389/fphys.2012.00155
18. Agyeman, K., Osei-Bonsu, I. and Tetteh, E.N. (2014) Growth and Yield Performance of Improved Cowpea (Vigna unguiculata L.) Varieties in Ghana. Agricultural Science, 2, 44-52. https://doi.org/10.12735/as.v2i4p44
19. Cobbinah, F.A., Addo-Quaye, A.A. and Asante, I.K. (2011) Characterization, Evaluation and Selection of Cowpea (Vigna unguiculata (L.) Walp.) Accessions with Desirable Traits from Eight Regions of Ghana. ARPN Journal of Agricultural and Biological Science, 6, 21-32.
20. Van Loon, A.F. (2015) Hydrological Drought Explained. Wiley Interdisciplinary Reviews: Water, 2, 359-392. https://doi.org/10.1002/wat2.1085
21. Hadebe, S.T., Modi, A.T. and Mabhaudhi, T. (2017) Drought Tolerance and Water Use of Cereal Crops: A Focus on Sorghum as a Food Security Crop in Sub-Saharan Africa. Journal of Agronomy and Crop Science, 203, 177-191. https://doi.org/10.1111/jac.12191
22. Medrano, H., Tomás, M., Martorell, S., Flexas, J., Hernández, E., Rosselló, J., Pou, A., Escalona, J. and Bota, J. (2015) From Leaf to Whole-Plant Water Use Efficiency (WUE) in Complex Canopies: Limitations of Leaf WUE as a Selection Target. The Crop Journal, 3, 220-228. https://doi.org/10.1016/j.cj.2015.04.002
23. Farquhar, G.D., Ehleringer, J.R. and Hubick, K.T. (1989) Carbon Isotope Discrimination and Photosynthesis. Annual Review of Plant Physiology and Plant Molecular Biology, 40, 503-537. https://doi.org/10.1146/annurev.pp.40.060189.002443
24. Owusu, K. and Waylen, P.R. (2013) The Changing Rainy Season Climatology of Mid-Ghana. Theoretical and Applied Climatology, 112, 419-430. https://doi.org/10.1007/s00704-012-0736-5
25. Walkley, A. and Black, I.A. (1934) An Examination of the Degtjareff Method for Determining Soil Organic Matter, and a Proposed Modification of the Chromic Acid Titration Method. Soil Science, 37, 29-38. https://doi.org/10.1097/00010694-193401000-00003
26. Bray, R.H. and Kurtz, L.T. (1945) Determination of Total, Organic, and Available Forms of Phosphorus in Soils. Soil Science, 59, 39-46. https://doi.org/10.1097/00010694-194501000-00006
27. Shearer, G. and Kohl, D.H. (1986) N2-Fixation in Field Settings: Estimations Based on Natural 15N Abundance. Australian Journal of Plant Physiology, 13, 699-756. https://doi.org/10.1071/PP9860699
28. Pausch, R.C., Mulchi, C.L., Lee, E.H. and Meisinger, J.J. (1996) Use of 13C and 15N Isotopes to Investigate O3 Effects on C and N Metabolism in Soybeans. Part II. Nitrogen Uptake, Fixation, and Partitioning. Agriculture, Ecosystems and Environment, 60, 61-69. https://doi.org/10.1016/S0167-8809(96)01062-6
29. Maskey, S.L., Bhattarai, S., Peoples, M.B. and Herridge, D.F. (2001) On-Farm Measurements of Nitrogen Fixation by Winter and Summer Legumes in the Hill and Terai Regions of Nepal. Field Crops Research, 70, 209-221. https://doi.org/10.1016/S0378-4290(01)00140-X
30. Craig, H. (1957) Isotopic Standards for Carbon and Oxygen and Correction Factors for Mass-Spectrometric Analysis of Carbon Dioxide. Geochimica et Cosmochimica Acta, 12, 133-149. https://doi.org/10.1016/0016-7037(57)90024-8
31. Statsoft Inc. (2011) Statistica (Data Analysis Software System). Version 10. https://www.Statsoft.Com
32. Belane, A.K. and Dakora, F.D. (2011) Levels of Nutritionally-Important Trace Elements and Macronutrients in Edible Leaves and Grain of 27 Nodulated Cowpea (Vigna unguiculata L. Walp.) Genotypes Grown in the Upper West Region of Ghana. Food Chemistry, 125, 99-105. https://doi.org/10.1016/j.foodchem.2010.08.044
33. Mbah, G.C. and Dakora, F.D. (2017) Nitrate Inhibition of N2 Fixation and Its Effect on Micronutrient Accumulation in Shoots of Soybean (Glycine max L. Merr.), Bambara Groundnut (Vigna subterranea L. Vedc) and Kersting’s Groundnut (Macrotyloma geocarpum Harms.). Symbiosis, 2, 205-216. https://doi.org/10.1007/s13199-017-0531-2
34. Belane, A.K. and Dakora, F.D. (2015) Assessing the Relationship between Photosynthetic C Accumulation and Symbiotic N Nutrition in Leaves of Field-Grown Nodulated Cowpea (Vigna unguiculata L. Walp.) Genotypes. Photosynthetica, 53, 562-571. https://doi.org/10.1007/s11099-015-0144-z
35. Mohammed, M., Jaiswal, S.K. and Dakora, F.D. (2018) Distribution and Correlation between Phylogeny and Functional Traits of Cowpea (Vigna unguiculata L. Walp.)—Nodulating Microsymbionts from Ghana and South Africa. Scientific Reports, 8, Article No. 18006. https://doi.org/10.1038/s41598-018-36324-0
36. Hobbie, E.A., MacKo, S.A. and Shugart, H.H. (1998) Patterns in N Dynamics and N Isotopes during Primary Succession in Glacier Bay, Alaska. Chemical Geology, 152, 3-11. https://doi.org/10.1016/S0009-2541(98)00092-8
37. O’Leary, M.H. (1988) Carbon Isotopes in Photosynthesis: Fractionation Techniques May Reveal New Aspects of Carbon Dynamics in Plants. BioScience, 38, 328-336. https://doi.org/10.2307/1310735
38. Belane, A.K., Asiwe, J. and Dakora, F.D. (2011) Assessment of N2 Fixation in 32 Cowpea (Vigna unguiculata L. Walp.) Genotypes Grown in the Field at Taung in South Africa, Using 15N Natural Abundance. African Journal of Biotechnology, 10, 11450-11458.
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Mutual inductor in electrical systems - MATLAB - MathWorks United Kingdom
Mutual inductor in electrical systems
The Mutual Inductor block models a mutual inductor, described with the following equations:
V1=L1\frac{dI1}{dt}+M\frac{dI2}{dt}
V2=L2\frac{dI2}{dt}+M\frac{dI1}{dt}
M=k\sqrt{L1·L2}
V1 Voltage across winding 1
I1 Current flowing into the + terminal of winding 1
L1, L2 Winding self-inductances
k Coefficient of coupling, 0 < k < 1
This block can be used to represent an AC transformer. If inductance and mutual inductance terms are not important in a model, or are unknown, you can use the Ideal Transformer block instead.
Self-inductance of the first winding. The default value is 10 H.
Self-inductance of the second winding. The default value is 0.1 H.
Coefficient of coupling, which defines the mutual inductance. The parameter value should be greater than zero and less than 1. The default value is 0.9.
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Direction of the Vector Calculator
How to calculate the direction of a vector?
How to find the direction angle of the vector?
How do I calculate a unit vector in the direction of another vector?
How to find a vector of some magnitude in the direction of another?
How to find the magnitude and direction of two vectors?
If you want to calculate the direction of a vector, you're in the right place. This calculator finds the direction angle of a vector and calculates a unit vector in this direction.
Vectors are a powerful tool to represent many physical quantities in our physical world. They represent forces, velocities, and many other quantities derived from them.
Besides direction, finding the magnitude of the vector is also possible if you choose the advanced mode of the calculator. Therefore, with this tool, you can find the magnitude and direction angle of any vector.
You can express or calculate the direction of a vector v⃗ in two ways:
Calculating the direction angle of the vector v⃗. The direction angle is the angle that v⃗ forms with the positive x-axis, counting in the counterclockwise direction.
Calculating a unit vector in the direction of the same vector. This unit vector is called the direction vector.
To calculate the angle
\theta
that a 2D vector
\vec{v} = (x, y)
forms with the horizontal axis, use this equation:
\theta = \arctan\left(\frac{y}{x}\right)
The only problem with this equation is that it doesn't give us the angle about the positive x-axis, but only about the nearest horizontal axis. If your vector lies in the first quadrant of the Cartesian plane, like the vector pointing to
P(3, 5)
in the image, that's not a problem.
But what if the vector lies in any of the other quadrants? Suppose you want to find the direction angle
\theta
Q = (-2, 4)
of the previous image. If we used the previous formula to find the direction angle, we wouldn't obtain the correct angle, as we'd get the angle
\gamma
instead of the direction angle
\theta
How can we deal with this? Well, in this case, you could have noticed that
\theta = 180^\circ - \gamma
. We can extend this reasoning to the other cases and come up with the following equations to calculate the direction of the vector in each quadrant:
In the first quadrant,
\theta_\text{I} = \arctan(\frac{y}{x})
In the second quadrant,
\theta_\text{II} = 180° - \arctan(\frac{y}{x})
In the third quadrant,
\theta_\text{III} = 180° + \arctan(\frac{y}{x})
In the fourth quadrant,
\theta_\text{IV} = 360° - \arctan(\frac{y}{x})
🙋 The term
\arctan(\frac{y}{x})
gives an angle in radians, and you must convert it to degrees before using it in the second, third, or fourth quadrant equations. Visit our angle converter to learn how to do it.
To find a unit vector û in the direction of another vector v⃗ = (x, y, z), follow these steps:
Find the magnitude of the vector v⃗:
|v⃗| = √(x² + y² + z²)
Divide each coefficient of the vector v⃗ by the magnitude of v⃗:
û = v⃗/|v⃗| = (x/|v⃗|, y/|v⃗|, z/|v⃗|).
That's it. û is the unit vector in the direction of v⃗.
To find a vector of a specific magnitude in the direction of another vector v⃗ = (x, y, z):
Find a unit vector û in the direction of v⃗. To do it, divide each coefficient of the vector v⃗ by the vector's magnitude:
û = v⃗/|v⃗| = (x/|v⃗|, y/|v⃗|, z/|v⃗|)
Multiply the magnitude of the desired vector by the unit vector û. That will result in the desired vector.
To find the magnitude and direction of two vectors, you must find the resultant vector (you can use our vector addition calculator to do it) and apply to it the steps described above.
Now that you know how to find the magnitude and direction angle of a vector, let's look at some numerical examples and FAQs.
How to find a vector of magnitude 3 in the direction of v = 12i - 5k?
To find a vector of magnitude 3 in the direction of v⃗ = 12i − 5k:
Find the magnitude of v⃗:
|v⃗| = √(12² + (-5)²) = 13
Find a unit vector û in the direction of v⃗. To do it, divide v⃗ by its magnitude:
û = v⃗/|v⃗| = (12/13)i − (5/13)k
Multiply the desired magnitude 3 by the unit vector û. We obtain the vector w⃗:
w⃗ = 3û = (36/13)i − (15/13)k
which has the desired direction and magnitude.
How to calculate the unit vector in the direction of v = i + j + 2k
To calculate a unit vector in the direction of v⃗ = i + j + 2k:
|v⃗| = √(1² + 1² + 2²) = √6 ≈ 2.4495
Divide the vector v⃗ by its magnitude:
û = v⃗/|v⃗| = (1/√6)i + (1/√6)j + (2/√6)k
Is the dot product of two vectors in the same direction positive or negative?
The dot product of two vectors in the same direction is always positive. That's because the dot product of two vectors in the same direction equals the multiplication of their magnitudes, and their magnitudes are always positive.
How do I find the magnitude and direction of two vectors' sum?
To find the magnitude and direction of two vectors' sum:
Find the resultant of the two vectors.
Sum the square of each of the components of the resultant vector.
Take the square root of the previous result, and this is the magnitude of your two vectors' sum!
To calculate the direction of the vector v⃗ = (x, y), use the formula θ = arctan(y/x), where θ is the smallest angle the vector forms with the horizontal axis, and x and y are the components of the resultant vector.
Find a unit vector with the same direction as your given vector.
Direction angle θ
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SVD Calculator (Singular Value Decomposition)
SVD Calculator
How to use this SVD calculator?
How to calculate SVD of a matrix?
Is singular value decomposition unique?
The singular value decomposition of matrices will never cause you any problems again — with the help of our SVD calculator, you will quickly master this important topic in linear algebra. Scroll down and learn:
How do I find the SVD of a matrix using our SVD calculator?
How to calculate the SVD of a matrix by hand?
Is the singular value decomposition unique?
Singular value decomposition (SVD) is a way of factorizing a matrix: any real matrix
A
m \times n
A = U\Sigma V^T
U
V
are orthogonal matrices of sizes
m\times m
n\times n
\Sigma
is a rectangular matrix of the same size as
A
m \times n
) which has non-negative numbers on its diagonal and zeroes everywhere else. The diagonal elements of
\Sigma
are in fact the singular values of
A
🙋 If
A
is complex, replace the transposition
V^T
with the complex conjugation
V^*
U
V
then become unitary matrices, but
\Sigma
still features real non-negative numbers on its diagonal.
Here's a visualization of singular value decomposition of a
4\times 3
M
Cmglee, CC BY-SA 4.0, via Wikimedia Commons
Once we know what the singular value decomposition of a matrix is, it'd be beneficial to see some examples. Calculating SVD by hand is a time-consuming procedure, as we will see in the section on How to calculate SVD of a matrix. We bet the quickest way to generate examples of SVD is to use Omni's singular value decomposition calculator!
Working with this SVD calculator is simple!
Pick the matrix size: the number of rows and the number of columns in
A
Enter the matrix entries in their dedicated fields.
The components of singular value decomposition
U
\Sigma
V^T
will appear at the bottom of the calculator.
Do you want to verify the results? Just perform the matrix multiplication of the result's three matrices and compare that outcome with your initial matrix. Remember that numerical computations and rounding may cause tiny discrepancies!
Do you want to understand how the SVD calculator got its results? In the next section, we will discuss all the theory that stands behind the singular value decomposition and explain step-by-step how to find the SVD of a matrix. Ready?
Here's how to calculate the singular value decomposition of a
m \times n
A
by hand. We will see that SVD is closely related to the eigenvalues and eigenvectors of
A
As we remember, we can easily find the eigenvalues and eigenvectors for square matrices, yet
A
can be rectangular in SVD. What can we do? Let's consider two square matrices that are closely related to
A
: these matrices are
A^TA
AA^T
V
are eigenvectors of
A^TA
The non-zero elements of
\Sigma
are the non-zero singular values of
A
, i.e., they are the square roots of the non-zero eigenvalues of
A^TA
Once we know
V
\Sigma
, we can recover
U
from the SVD formula (
A = U\Sigma V^T
In more details, to find SVD by hand:
A^TA
Compute the eigenvalues and eigenvectors of
A^TA
Draw a matrix of the same size as
A
and fill in its diagonal entries with the square roots of the eigenvalues you found in Step 2. This is
\Sigma
Write down the matrix whose columns are the eigenvectors you found in Step 2. This is
V
The SVD equation
A = U\Sigma V^T
transforms to
AV = U\Sigma
. We can rewrite this in terms of columns as
A v_i = \sigma_i u_i
. This tells us how to compute
U
, as the columns of
U
u_i = \frac 1 {\sigma_i} A v_i
i
\sigma_i \neq 0
U
needs more columns to fill its size, you can pick arbitrary vectors, but you have to make sure that
U
is an orthogonal matrix. Therefore, you must pick vectors that have unit length and are orthogonal to all the columns in
U
(and the ones you're adding).
No, the SVD is not unique. Even if we agree to have the diagonal elements of Σ in descending order (which makes Σ unique), the matrices U and V are still non-unique.
What does SVD do to a matrix?
SVD decomposes an arbitrary rectangular matrix A into the product of three matrices UΣVᵀ, which is subject to some constraints. These U and V are orthogonal matrices. Σ has the same size as A and contains the singular values of A as its diagonal entries.
What is SVD of a symmetric matrix?
If A is real symmetric, then its singular values (the diagonal elements of Σ) coincide with the absolute values of its eigenvalues. The columns of U and V are the unit eigenvectors of A. In particular, if the eigenvalues of A are all strictly positive (i.e. A is positive definite), then U = V and the SVD of A coincides with the eigendecomposition of A.
What is SVD of a unitary matrix?
For unitary matrices, SVD is trivial. Namely, if A is unitary (i.e., AA = AA* = I), then all of the singular values of A are equal to 1. Hence, in SVD we have U = A , Σ = I, and V = I.
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Magic, maths and money: March 2012
Markets, Morality and Mathematics
Labels: history of financial engineering, philosophy of financial mathematics, science policy
One of the key critics of Aquinas’ argument that a merchant could charge what heliked, providing the future was uncertain, was Pierre Jean Olivi, who was born near Béziers in Languedoc around 1248. Olivi entered the Franciscan order when he was twelve and was sent to Paris to study theology in 1267, and although he spent four years at the University he did not graduate with a masters degree. When he left Paris he appears to have started working on a theological work that took him over twenty years to complete and addressed a a range of questions, including the nature of free will. During this time he travelled widely in southern France and Italy and came into conflict with the Church hierarchy.
Franciscans had an oath of poverty, and within a couple of generations of the founding of the order this oath began to be re-interpreted. Some Franciscans took the view that they kept to the oath if they did not own anything, others believed that this was a loop-hole, the oath required that a Franciscan should limit their use of goods. Olivi became a leader of this ‘rigorist’ or ‘spiritual’ wing of the Order. In 1282 he was accused of heresy and his writings destroyed, though he successfully defended himself in 1287 and was able to carry on teaching until his death in 1298. However, his tomb quickly attracted pilgrims and the Church, faced with a growing cult banned his writings in 1299, destroyed his tomb in 1312, and finally, when the Holy Roman Emperor, Louis the Bavarian, used some of Olivi’s arguments to attack the Papacy, he was condemned, again, as a heretic and all his works were obliterated. Umberto Eco’s book, The Name of the Rose, and the subsequent film, have these events in the background.
Olivi’s fideism, the view that faith and reason are independent of each other and that you cannot rationalise faith, meant that he was sceptical towards Aristotelian empiricism and it was in this context that he made a revolutionary observation with regard to the selling of grain to the starving Rhodeans. He argued that the metaphysical probability of more grain arriving in Rhodes, giving the merchant excessive profits, had a certain reality, which Aquinas was ignoring by focusing on the ‘physical reality’ of the prices being offered in the market. Olivi said
The judgement of the value of a thing in exchange seldom or never can be made except through conjecture or probable opinion, and not so precisely, or as if understood and measured by one invisible point, but rather as a fitting latitude within which the diverse judgements of men will differ in estimation.
This does not mean Olivi, despite his belief in absolute poverty, felt the merchant should charge a lower price for the grain. While Aquinas felt the market price was justifiable, but it was more moral for the merchant to lower the price, Olivi believed that the market mechanism was important. It was to the common good if prices did rise during a famine as it would encourage an increase in the supply of food.
Olivi applied this approach to the question of loans. As the historian Joel Kaye explains
if someone intends to invest his money in trade or profit, and instead, out of charity, lends the money to a friend in need, can he expect back from his friend not only the sum lent but in addition the profit he lost in not investing in trade? Olivi’s answer to this question was an unqualified yes: the borrower was responsible for indemnifying the lender for his loss of “probable profit”. and for restoring a “probable equivalence”.
Olivi introduces the idea that market exchange is about equating expectations.
Olivi, despite his position in the Spiritual Franciscan movement, seems to have been a close observer of markets. As well as developing the ideas of Aquinas and Albert the Great, he commented that the market price depended not just on ‘need’ but on three factors; its scarcity, usefulness and desirability. Since desirability is subjective, different people will value the same good differently and based on these ideas, Olivi was able to explain the ‘value paradox’, why water, essential to life, was less valuable than gold, of no use, because gold is scarcer than water.
As a result of his condemnation for heresy, Olivi’s economic thought was retained only by a few Franciscans who secretly read his works, but, of course, could not acknowledge the influence he had on them. As a result has been only recently realised that the writings of another Franciscan, San Bernardino of Siena, were based on Olivi. Bernardino was born into a noble family in Tuscany in 1380 but was orphaned when he was six. Raised by an religious aunt he spent much of his spare time studying law and nursing the sick, remaining in Siena when it was hit by plague in 1400 and miraculously surviving. Bernardino became a strict Franciscan, abstaining from all pleasures, developed a reputation as an inspiring preacher, and now he is he regarded as the most important Italian missionary of the fifteenth century.
Just as it is surprising that Olivi observed markets, it is also surprising that such an ascetic as Bernardino wrote the first book on entrepreneurship, On Contracts and Usury between 1431 and 1433. Bernardino realised that to be successful a merchant had to be well informed; of prices, the qualities of goods and the market, be diligent in keeping accounts; be hard working; and, importantly, be willing to take on risk and he recognised that there were very few people who had all the these qualities. The book was written at a time when the Catholic Church had condensed morality into three ‘Christian’ virtues, Hope (Spes), Faith (Fides) and Charity (Caritas), and four ‘Pagan’ or ‘Cardinal’ virtues, Courage (Fortitudo), Justice (Iustitia), Temperance (Temperantia), and Prudence(Prudentia). An ethical life was one that exhibited all, not just some, of the virtues and within this context a merchant could, just as much as a knight, be seen as being ethical.
Seven Virtues c. 1460 Pesellino (Francesco di Stefano) workshop, Birmingham Museum of Art, Birmingham, AL
The thirteenth century saw a flowering of European science, driven by a flood of classical texts being translated from Arabic. Aristotle became “the Philosopher” and his works came in for particular attention. In Nicomachean Ethics Aristotle considers the justice of economic exchange and argues that reciprocal, fair, exchange in the market as being fundamental to a well functioning society since it binds individuals together. Exchange is not performed in order to generate a profit, for gain, but to correct for inequalities and to establish a social equilibrium. So, for medieval scholars, like Aquinas and Olivi, the virtue of Justice should be central to the actions of a merchant.
Prudence is the ability to judge between different courses of action, it is at the root of reason and rationality and can be seen as the motivation for all science. This virtue is the one most closely associated, in the modern mind at least, with effective merchants. Temperance is the virtue least associated with modern bankers, its corresponding vice being gluttony. However, the modern understanding of temperance as denial is not only how a medieval friar would have understood the virtue. Temperance is at the root of humility, an acceptance that the human is not all-knowing. A good merchant would exhibit the virtue by allowing for the unforeseen, and consequently, diversifying, or at least not betting the house on a single venture. Prudence and Temperance complement each other.
Courage, the remaining Pagan Virtue, is demonstrated by the merchant in being able to commit to a risky venture. However, Courage untempered by the other virtues is rashness, and should be avoided.
Faith is the ability to believe without seeing, and was central to Olivi’s whole philosophy. The Latin root is fides, which gives federal, and captures the concept of trust, the very essence of finance. While Faith is backward looking, you build trust, Hope is its forward-looking complement. When Christian Huygens was translating his ‘On the Reckoning at Games of Chance’ from the Dutch into the more scholarly Latin he had a lot of trouble translating the word kans (‘chance’), which would normally be translated as sors. Eventually, he, or his editor van Schooten, chose expectatio , giving the English term ‘expectation’ (in the mathematical sense). While the English are left expecting, Huygens had also considered using the term spes, or ‘Hope’ and the French have taken this line, using the word espérance when referring to mathematical expectation. Statistics can be seen as the mathematical expression of Faith, while Probability captures Hope. Again, a merchant would need to express these virtues if they wished to be successful in business.
Charity, along with Temperance, is the virtue least likely to be associated with merchants. While we now think of charity in terms of giving to others, in the past it was associated with a love, or care, for others. When business people talk about being ‘customer focused’ they are talking about exhibiting the virtue of Charity. Shakespeare’s play The Merchant of Venice is not about the moneylender Shylock, but ‘Antonio, a merchant of Venice’ who characterises Christian love or agape demonstrated by his sacrifices for his young friend Bassanio. The view that Antonio and Bassanio were physical lovers is a modern misreading based on an ignorance that the medieval mind was much capable of distinguishing storge (famillial love), philia (friendship), eros (physical love) and agape (spitritual love).
In the U.S. the review of the financial crisis of 2007–2009 was not undertaken by the regulator, which was possibly not inclined to give a thorough and independent review of what crippled the economy, but passed to an independent commission, The National Commission on the Causes of the Financial and Economic Crisis in the United States. They concluded that
We conclude there was a systemic breakdown in accountability and ethics. The integrity of our financial markets and the public’s trust in those markets are essential to the economic well–being of our nation. The soundness and the sustained prosperity of the financial system and our economy rely on the notions of fair dealing, responsibility, and transparency. In our economy, we expect businesses and individuals to pursue profits, at the same time that they produce products and services of quality and conduct themselves well.
Unfortunately–as has been the case in past speculative booms and busts–we witnessed an erosion of standards of responsibility and ethics that exacerbated the financial crisis. This was not universal, but these breaches stretched from the ground level to the corporate suites. They resulted not only in significant financial consequences but also in damage to the trust of investors, businesses, and the public in the financial system.
Olivi and Bernardino would be spinning in their graves.
The status of morality and ethics in finance has changed significantly between the seventeenth century, when Huygens and Bernoulli constructed mathematical probability on the basis of commercial ethics, and today when ethics seems to have been expunged from the science of economics. The change point is often associated with the early Victorian period, associated with Romanticism. The ‘liberal’ philosopher John Stuart Mill argued that (political) economics
is concerned with [man] solely as a being who desires to possess wealth, and who is capable of judging the comparative efficacy of means for obtaining that end.
Around the same time, the future poet-laureate, Alfred, Lord Tennyson, wrote about nature “red in tooth and claw”. In 1859 Darwin published the On the Origin of Species by Means of Natural Selection, or the Preservation of Favoured Races in the Struggle for Life which explained evolution in terms of natural selection. In the popular perception, nature became seen as being driven by a bitter struggle for survival, un-regulated by a divine architect. Modern science was demoting man from being created in God’s image to the status of a higher ape, as Darwin wrote in his 1871 book, The Descent of Man
My object in this chapter is to shew that there is no fundamental difference between man and the higher mammals in their mental faculties.
The Descent of Man goes on to argue that “the civilised races of man will almost certainly exterminate, and replace, the savage races throughout the world”, and Adolf Hitler would be the champion of the Romantic triumph of will over reason in the twentieth century.
The reaction to Romantic Fascism was ‘positive science’, and consequence was the abandonment of six of the virtues from economics, and to focus on what Deirdre McCloskey has described as “Prudence Only”, blind rationality. Prof McCloskey is, perhaps, being too extreme, the widespread use of mathematics suggests residual Faith and Hope, but there has definitely been an associated lack of humility (Temperance) The irony is, ethics have been banished from finance, in the main, by academics teaching students to focus on ‘rational expectations’ (Prudent Hope) and ignore Temperance (the counterbalance to Gluttony) and Charity (the counterbalance to Greed), not because the markets are fundamentally unethical and immoral, but because this is seen to be “good science”.
Ian Hislop, the British commentator also discusses how banking changed around the time that Romanticism dominated culture in When Bankers Were Good.
{}^{1}
Kaye [1998, p 121]
{}^{2}
{}^{3}
{}^{4}
Kaye [1998, p 119], also Franklin [2001, pp 265–267]
{}^{5}
Rothbard [1996, pp 60–61], Kaye [1998, pp 123–124]
{}^{6}
Rothbard [1996, pp 81], Kaye [1998, p 118]
{}^{7}
Rothbard [1996, pp 81–82]
{}^{8}
Kaye [1998, p 51]
{}^{9}
Hacking [1984, p 95]
{}^{10}
FCIC [2011] p xxii
{}^{11}
Sylla [2006], Sylla [2003]
{}^{12}
Friedman [1953]
{}^{13}
Persky [1995, quoting Mill, p 223]
{}^{14}
MacCulloch [2009, pp 861–862]
{}^{15}
Darwin [1871, pp 36]
{}^{16}
Darwin [1871, pp 200–201]
{}^{17}
McCloskey [2007]
C. Darwin. The descent of man, and selection in relation to sex. John Murray, 1871. darwin-online.org.uk/contents.html.
FCIC. The Financial Crisis Inquiry Report. Technical report, The National Commission on the Causes of the Financial and Economic Crisis in the United States, 2011.
J. Franklin. The Science of Conjecture: Evidence and Probability before Pascal. Johns Hopkins University Press, 2001.
I. Hacking. The emergence of probability. Cambridge University Press, 1984.
D. MacCulloch. A History of Christianity. Allen Lane, 2009.
J. Persky. Retrospectives: The ethology of Homo economicus. The Journal of Economic Perspectives, 9(2):221–231, 1995.
M. N. Rothbard. Economic Thought before Adam Smith. Edward Elgar, 1996.
E. D. Sylla. Commercial arithmetic, theology and the intellectual foundations of Jacob Bernoulli’s Art of Conjecturing. In G. Poitras, editor, Pioneers of Financial Economics: contributions prior to Irving Fisher, pages 11–45. Edward Elgar, 2006.
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Created by Krishna Nelaturu
Bessel functions and its differential equation
How do you calculate the Bessel function of the first kind?
How do you calculate the Bessel function of the second kind?
How do you calculate Hankel functions?
Recurrence relation of Bessel functions
How to find Bessel function values using this Bessel function calculator
Are you struggling to calculate or validate Bessel function values? Do you wish you could plot the Bessel function to get that extra information? If your answer is yes, you've come to the right place because our Bessel function calculator does everything for you!
Bessel functions are fairly advanced mathematical topics that can be perplexing to anyone. This article covers the basics, such as the Bessel differential equation, how to calculate Bessel functions of the first and second kinds, and the recurrence relations for Bessel functions, so you're well equipped to solve your problem using Bessel functions.
Bessel differential equation is a second-order differential equation given by:
\small x^2\frac{d^2y}{dx^2}+ x\frac{dy}{dx} + \left(x^2-\nu^2 \right)y = 0
\nu
is an arbitrary complex number.
Since this is a second-order differential equation, there have to be two linearly independent solutions. We call these solutions Bessel functions of the first and second kind. All Bessel functions are also commonly referred to as cylinder functions.
The order of the Bessel function is given by
\nu
, and although it can be an arbitrary complex number, the most critical cases are when
\nu
is an integer or half-integer. In this calculator, we shall exclusively use real-valued
\nu
x
can be complex.
You can refresh your knowledge of complex number operations using our complex number calculator.
We use the following power-series to evaluate the Bessel function of the first kind:
\scriptsize \! J_\nu(x) \!= \!\sum_{k=0}^\infty \frac{(-1)^k}{\Gamma(k\!+\!1)\Gamma(k\!+\!\nu\!+\!1)}\! \left( \frac{x}{2}\right)^{2k+\nu}
J_\nu(x)
– Bessel function of the first kind;
\nu
– Order of the Bessel function;
x
– Arbitrary real or complex number; and
\Gamma(z)
– Gamma function, an extension of the factorial function to non-integer values.
You can learn more about this
\Gamma(z)
using our gamma function calculator.
J_\nu(x)
for the first three orders. Notice how they resemble a decaying sine or cosine wave.
For non-integer
\nu
J_\nu(x)
J_{-\nu}(x)
are linearly independent. However, for integer
\nu
, they're related as follows:
\small J_{-n}(x) = (-1)^n J_{n}(x)
💡 For computation, it is impractical to expand this series to
\infin
. We should get sufficiently precise values after a finite number of iterations.
Computing the Bessel function of the second kind is tricker than
J_\nu(x)
because it has different formulae for different
\nu
To evaluate the Bessel function of the second kind for non-integer
\nu
, we use the formula:
\small Y_\nu(x) = \frac{J_\nu(x)\cos(\nu\pi) - J_{-\nu}(x)}{\sin(\nu\pi)}
Y_\nu(x)
is the Bessel function of the second kind of order
\nu
If the order
\nu
n
, we should take the limit of
Y_\nu(x)
as the order
\nu
approaches the integer
n
\small Y_n(x) = \lim_{\nu \to n} Y_\nu(x)
For non-negative integer
n
, this limit reduces to:
\scriptsize\begin{align*} Y_n(x) = & -\frac{\left(\frac{x}{2}\right)^{-n}}{\pi}\sum_{k=0}^{n-1}\frac{(n-k-1)!}{k!}\!\left(\frac{x^2}{4}\right)^k\\ & +\frac{2}{\pi} \ln\left(\frac{x}{2}\right)J_n(x)\\ &-\frac{\left(\frac{x}{2}\right)^{n}}{\pi}\sum_{k=0}^{\infin}(\psi(k+1)\\ &\qquad\quad+\psi(n+k+1))\frac{\left(\frac{x^2}{4}\right)^k}{k!(n+k)!} \end{align*}
\psi(z)
is the digamma function, the logarithmic derivative of the gamma function
\Gamma(z)
\qquad \small \psi(z) = \frac{\Gamma'(z)}{\Gamma(z)}
If you're groaning at this complexity, we have some good news! Since we're using
\psi(n)
exclusively for non-negative
n
, we can use a simpler formulation for the digamma function, given by:
\qquad \small \psi(n) = H_{n-1} - \gamma
H_{n-1}
(n-1)^{th}
harmonic number; and
\gamma
– Euler–Mascheroni constant.
Y_\nu(x)
for the first three orders.
But what about negative integer
n
? Again, we have some good news! Similar to
J_\nu(x)
Y_\nu(x)
also follows the relation:
\qquad \small Y_{-n}(x) = (-1)^n Y_{n}(x)
which we can utilize to calculate
Y_{-n}(x)
Y_{n}
The Bessel functions of the third kind, also known as the Hankel functions, are two linearly independent solutions to the Bessel differential equation. We express them as linear combinations of the first two kinds of Bessel functions:
\qquad \small \begin{align*} H_\nu^{(1)}(x) = J_\nu(x) + i Y_\nu(x)\\ H_\nu^{(2)}(x) = J_\nu(x) - i Y_\nu(x) \end{align*}
H_\nu^{(1)}(x) , H_\nu^{(2)}(x)
– The Hankel functions; and
i
– The imaginary unit.
The Bessel functions we discussed so far exhibit the following recurrence relations:
\small\begin{align*} C_\nu(z) &= \frac{z}{2\nu}[C_{\nu-1}(z) + C_{\nu+1}(z)] \\\\ C'_\nu(z) &= \frac{1}{2} [C_{\nu-1}(z) - C_{\nu+1}(z)] \end{align*}
C_\nu(z)
– Any cylinder function, i.e.,
J_\nu(z)
Y_\nu(z)
C\rq_\nu(z)
– Derivative of any cylinder function; and
z
– Any arbitrary real or complex number.
\small J\rq_0(z) = -J_1(z)
\small Y\rq_0(z) = -Y_1(z)
You can use these recurrence relations for Bessel functions to calculate the derivatives of the desired Bessel function quickly.
This Bessel function calculator will solve for Bessel functions of the first, second, and third kind simultaneously. All you need to input are the order
\nu
x
, the point at which you desire to evaluate. Keep in mind that:
\nu
must be a real number;
x
can be real or complex; and
This Bessel function calculator will plot the Bessel function of the first two kinds, as long as the number
x
Note that the order
\nu
must be within the range
[-99, 99]
to keep the computational time to a minimum. Any higher order will cause noticeable lag in most computers.
\Re(x)
[-20, 20]
to maintain computational accuracy since there is a different method for calculating Bessel functions for large
x
If you wish to perform calculations beyond these limits, please get in touch with us!
How do I calculate bandwidth with Bessel function table?
To estimate bandwidth using a Bessel function table, you must know the modulating index β and modulating frequency fₘ:
Find the minimum value of Jᵥ(β) above 0.01 (or any value deemed the minimum significant value) by referring to a Bessel function table.
Determine the number of sideband pairs N in the signal, equal to the order v of the Bessel function Jᵥ(β).
Substitute N and fₘ in the formula B = 2fₘN to get the bandwidth B.
Be proud of yourself for solving a not-so-simple problem in a not-so-complex manner!
What is the maximum value of the Bessel function of the first kind?
J₀(0) = 1 is the maximum value of the Bessel function of the first kind. It occurs when the order ν = 0 and x = 0. To calculate this, follow these steps:
Substitute ν = 0 and x = 0 into the integrand cos(ντ - x sin(τ)) to get cos(0) = 1.
Evaluate the integral ∫1 ∙ dτ to get [τ].
Apply the lower limit 0 and the upper limit π to [τ] to get [π-0] = π.
Divide by π to get π/π = 1.
Verify this result using Omni's Bessel function calculator!
Where do the singularities lie for the Bessel functions?
For negative non-integer orders, the Bessel function of the first kind has a singularity at x = 0. For all orders, the Bessel function of the second kind has a singularity at x = 0.
Are Bessel functions periodic?
The Bessel functions are not periodic, although they look like decaying sine or cosine waves on a graph.
Krishna Nelaturu
x is a...
Bessel function parameters
The LCM calculatorcalculates the least common multiple of two to fifteen numbers.
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Multiple Solutions for Kirchhoff Equations under the Partially Sublinear Case
Wenjun Feng, Xiaojing Feng, "Multiple Solutions for Kirchhoff Equations under the Partially Sublinear Case", Journal of Function Spaces, vol. 2015, Article ID 610858, 4 pages, 2015. https://doi.org/10.1155/2015/610858
Wenjun Feng1 and Xiaojing Feng2
2School of Mathematical Sciences, Shanxi University, Taiyuan 030006, China
We prove the infinitely many solutions to a class of sublinear Kirchhoff type equations by using an extension of Clark’s theorem established by Zhaoli Liu and Zhi-Qiang Wang.
In this paper we study the existence and multiplicity of solutions for the following Kirchhoff type equations:where , are positive constants.
When is a smooth bounded domain in , the problemhas been studied in several papers. Perera and Zhang [1] considered the case where is asymptotically linear at 0 and asymptotically 4-linear at infinity. They obtained a nontrivial solution of the problems by using the Yang index and critical group. Then, in [1] they considered the cases where is 4-sublinear, 4-superlinear, and asymptotically 4-linear at infinity. By various assumptions on near 0, they obtained multiple and sign changing solutions. Cheng and Wu [2] and Ma and Rivera [3] studied the existence of positive solutions of (2) and He and Zou [4] obtained the existence of infinitely many positive solutions of (2), respectively; Mao and Luan [5] obtained the existence of signed and sign-changing solutions for problem (2) with asymptotically 4-linear bounded nonlinearity via variational methods and invariant sets of descent flow; Sun and Tang [6] studied the existence and multiplicity results of nontrivial solutions for problem (2) with the weaker monotony and 4-superlinear nonlinearity. For (2), Sun and Liu [7] considered the cases where the nonlinearity is superlinear near zero but asymptotically 4-linear at infinity, and the nonlinearity is asymptotically linear near zero but 4-superlinear at infinity. By computing the relevant critical groups, they obtained nontrivial solutions via Morse theory.
Comparing with (1) and (2), is in place of the bounded domain . This makes the study of problem (1) more difficult and interesting. Wu [8] considered a class of Schrödinger Kirchhoff type problem in and a sequence of high energy solutions are obtained by using a symmetric Mountain Pass Theorem. In [9], Alves and Figueiredo study a periodic Kirchhoff equation in ; they get the nontrivial solution when the nonlinearity is in subcritical case and critical case. Liu and He [10] obtained multiplicity of high energy solutions for superlinear Kirchhoff equations in . Li et al. in [11] proved the existence of a positive solution to a Kirchhoff type problem on by using variational methods and cutoff functional technique.
In [12], Jin and Wu consider the following problem: where constants , , or 3, and . By using the Fountain Theorem, they obtained the following theorem.
Theorem A (see [12]). Assume that the following conditions hold.
If the following assumptions are satisfied,() as uniformly for any ,()there are constants and such that where ()there exists such that() () for each and for each , where is the group of orthogonal transformations on ,() for any ,then problem (3) has a sequence of radial solutions.
Recently, Liu and Wang [13] obtained an extension of Clark’s theorem as follows.
Theorem B (see [13]). Let be a Banach space, . Assume is even and satisfies the (PS) condition, bounded from below, and . If, for any , there exists a -dimensional subspace of and such that , where , then at least one of the following conclusions holds.(i)There exists a sequence of critical points satisfying for all and as .(ii)There exists such that for any there exists a critical point such that and .
Theorem A obtained the existence of infinitely many solutions under the case that is sublinear at infinity in . It is worth noticing that there are few papers concerning the sublinear case up to now. Motivated by the above fact, in this paper our aim is to study the existence of infinitely many solutions for (1) when satisfies sublinear condition in at infinity. Our tool is extension of Clark’s theorem established in [13]. Now, we state our main result.
Theorem 1. Assume that satisfies () and the following conditions:()There exist , , such that and .()Consider uniformly in some ball , where .() is a positive continuous function such that .
Then (1) possesses infinitely many solutions such that as .
Remark 2. Throughout the paper we denote by various positive constants which may vary from line to line and are not essential to the problem.
The paper is organized as follows: in Section 2, some preliminary results are presented. Section 3 is devoted to the proof of Theorem 1.
In this section, we will give some notations that will be used throughout this paper.
Let be the completion of with respect to the inner product and norm Moreover, we denote the completion of with respect to the norm by . To avoid lack of compactness, we need to consider the set of radial functions as follows: Here we note that the continuous embedding is compact for any .
Define a functional by Then we have from () that is well defined on and is of , and It is standard to verify that the weak solutions of (1) correspond to the critical points of functional .
3. Proofs of the Main Result
Proof of Theorem 1. Choose such that is odd in , for and , and for and . In order to obtain solutions of (1) we consider Moreover, (13) is variational and its solutions are the critical points of the functional defined in by From (), it is easy to check that is well defined on and , and Note that is even, and . For , Hence, it follows from (14) that We now use the same ideas to prove the (PS) condition. Let be a sequence in so that is bounded and . We will prove that contains a convergent subsequence. By (17), we claim that is bounded. Assume without loss of generality that converges to weakly in . Observe that Hence, we have It is clear that and as . In the following, we will estimate , by using (), for any , which implies Therefore, converges strongly in and the (PS) condition holds for . By () and (), for any , there exists such that if and then , and it follows from (14) that This implies, for any , if is a -dimensional subspace of and is sufficiently small then , where . Now we apply Theorem B to obtain infinitely many solutions for (13) such that Finally we show that as . Let be a solution of (13) and . Let and set . Multiplying both sides of (13) with implies By using the iterating method in [13], we can get the following estimate: where is a number in and is independent of and . By (23) and Sobolev Imbedding Theorem [14], we derive that as . Therefore, are the solutions of (1) as is sufficiently large. The proof is completed.
The authors would like to express their sincere gratitude to one anonymous referee for his/her constructive comments for improving the quality of this paper. This work was supported by the National Natural Science Foundation of China (Grant nos. 11071149, 11271299, and 11301313), Natural Science Foundation of Shanxi Province (2012011004-2, 2013021001-4, and 2014021009-1), and the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (no. 2015101).
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Copyright © 2015 Wenjun Feng and Xiaojing Feng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Canonical Correlation - SAGE Research Methods
Canonical Correlation | The SAGE Encyclopedia of Educational Research, Measurement, and Evaluation
Canonical correlation is a statistical measure for expressing the relationship between two sets of variables. Formally, given two random vectors x ∈ Rdx and y ∈ Rdy with some joint (unknown) distribution D, the canonical correlation analysis (CCA) seeks vectors u ∈ Rdx and v ∈ Rdy, such that the random vectors when projected along these directions, that is, variables u > x and v > y, are maximally correlated. Equivalently, we can write CCA as the following optimization problem: find u ∈ Rdx, v ∈ Rdy that:
\begin{array}{l}{\text{Maximize}}_{dx\text{\hspace{0.17em}}dy}\text{\hspace{0.17em}}\rho \left(u>x,v>y\right),\hfill \\ \text{u}\in \text{R},v\in \text{R}\hfill \end{array}
where the correlation, ρ(u > x, v > y), between two random variables, is defined as
\rho \left(u>x,v>y\right)=\sqrt{\mathrm{cov}\left(u>\sqrt{x,v>}y\right)}
. Assuming that vectors x and y are 0 mean, we can write CCA as the problem var(u > x) var(u > ...
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May 2022 Oracle lower bounds for stochastic gradient sampling algorithms
Niladri S. Chatterji,1 Peter L. Bartlett,2 Philip M. Long3
1Department of Computer Science, Stanford University, 353 Jane Stanford Way, Stanford, CA 94305, USA
2University of California, Berkeley & Google, 367 Evans Hall #3860, Berkeley, CA 94720-3860, USA
3Google, 1600 Amphitheatre Parkway, Mountain View, CA, 94043, USA
{\mathbb{R}}^{\mathit{d}}
\mathrm{\Omega }\left({\mathit{\sigma }}^{2}\mathit{d}/{\mathit{\epsilon }}^{2}\right)
{\mathit{\sigma }}^{2}\mathit{d}
We gratefully acknowledge the support of the NSF through grants IIS-1619362 and IIS-1909365. Part of this work was completed while NC was interning at Google.
We are grateful to Aditya Guntuboyina for pointing us towards the literature on Le Cam deficiency. We would also like to thank Jelena Diakonikolas, Sébastien Gerchinovitz, Michael Jordan, Aldo Pacchiano, Aaditya Ramdas and Morris Yau for many helpful conversations. We thank Kush Bhatia for helpful comments that improved the presentation of the results.
Niladri S. Chatterji. Peter L. Bartlett. Philip M. Long. "Oracle lower bounds for stochastic gradient sampling algorithms." Bernoulli 28 (2) 1074 - 1092, May 2022. https://doi.org/10.3150/21-BEJ1377
Received: 1 October 2020; Revised: 1 March 2021; Published: May 2022
Keywords: information theoretic lower bounds , Markov chain Monte Carlo , Sampling lower bounds , stochastic gradient Monte Carlo
Niladri S. Chatterji, Peter L. Bartlett, Philip M. Long "Oracle lower bounds for stochastic gradient sampling algorithms," Bernoulli, Bernoulli 28(2), 1074-1092, (May 2022)
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Monolayer - Wikipedia @ WordDisk
A monolayer is a single, closely packed layer of atoms, molecules,[1] or cells. In some cases it is referred to as a self-assembled monolayer. Monolayers of layered crystals like graphene and molybdenum disulfide are generally called 2D materials.
Diagram of ambiphilic molecules floating on a water surface.
A Langmuir monolayer or insoluble monolayer is a one-molecule thick layer of an insoluble organic material spread onto an aqueous subphase in a Langmuir-Blodgett trough. Traditional compounds used to prepare Langmuir monolayers are amphiphilic materials that possess a hydrophilic headgroup and a hydrophobic tail. Since the 1980s a large number of other materials have been employed to produce Langmuir monolayers, some of which are semi-amphiphilic, including polymeric, ceramic or metallic nanoparticles and macromolecules such as polymers. Langmuir monolayers are extensively studied for the fabrication of Langmuir-Blodgett film (LB films), which are formed by transferred monolayers on a solid substrate. A Gibbs monolayer or soluble monolayer is a monolayer formed by a compound that is soluble in one of the phases separated by the interface on which the monolayer is formed.
The monolayer formation time or monolayer time is the length of time required, on average, for a surface to be covered by an adsorbate, such as oxygen sticking to fresh aluminum. If the adsorbate has a unity sticking coefficient, so that every molecule which reaches the surface sticks to it without re-evaporating, then the monolayer time is very roughly:
{\displaystyle t={\frac {3\times 10^{-4}\,\mathrm {Pa} \cdot \mathrm {s} }{P}}}
where t is the time and P is the pressure. It takes about 1 second for a surface to be covered at a pressure of 300 µPa (2×10−6 Torr).
Monolayer phases and equations of state
A Langmuir monolayer can be compressed or expanded by modifying its area with a moving barrier in a Langmuir film balance. If the surface tension of the interface is measured during the compression, a compression isotherm is obtained. This isotherm shows the variation of surface pressure (
{\displaystyle \Pi =\gamma ^{o}-\gamma }
{\displaystyle \gamma ^{o}}
is the surface tension of the interface before the monolayer is formed) with the area (the inverse of surface concentration
{\displaystyle \Gamma ^{-1}}
). It is analogous with a 3D process in which pressure varies with volume.
A variety of bidimensional phases can be detected, each separated by a phase transition. During the phase transition, the surface pressure doesn't change, but the area does, just like during normal phase transitions volume changes but pressure doesn't. The 2D phases, in increasing pressure order:
Bidimensional gas: there are few molecules per area unit, and they have few interactions, therefore, analogous of the equations of state for 3D gases can be used: ideal gas law
{\displaystyle \Pi A=RT}
{\displaystyle A}
is the area per mole. As the surface pressure increases, more complex equations are needed (Van der Waals, virial...)
If the area is further reduced once the solid phase has been reached, collapse occurs, the monolayer breaks and soluble aggregates and multilayers are formed
Gibbs monolayers also follow equations of state, which can be deduced from Gibbs isotherm.
For very dilute solutions
{\displaystyle \gamma =\gamma _{o}-mC}
, through Gibbs isotherm another analogous of ideal gas law is reached
{\displaystyle \Pi =\Gamma RT}
For more concentrated solutions and applying Langmuir isotherm
{\displaystyle \Gamma =\Gamma _{\max }{\frac {C}{a+C}}}
{\displaystyle \Pi =\Gamma _{\max }RT\left(1+{\frac {C}{a}}\right)}
Monolayers have a multitude of applications both at the air-water and at air-solid interphases.
Nanoparticle monolayers can be used to create functional surfaces that have for instance anti-reflective or superhydrophobic properties.[2][3]
Monolayers are frequently encountered in biology. A micelle is a monolayer, and the phospholipid lipid bilayer structure of biological membranes is technically two monolayers. Langmuir monolayers are commonly used to mimic cell membrane to study the effects of pharmaceuticals or toxins.[4]
In cell culture a monolayer refers to a layer of cells in which no cell is growing on top of another, but all are growing side by side and often touching each other on the same growth surface.
Evaporation suppressing monolayers
Ter Minassian-Saraga, L. (1994). "Thin films including layers: terminology in relation to their preparation and characterization (IUPAC Recommendations 1994)" (PDF). Pure and Applied Chemistry. 66 (8): 1667–1738 (1672). doi:10.1351/pac199466081667. S2CID 95035065.
"Functional Nanoscale and Nanoparticle Coatings - Biolin Scientific". Biolin Scientific. Retrieved 2017-08-03.
"Influence of Thermal Separation of Oleic Acid on the Properties of Quantum Dots Solutions and Optoelectronic of Their Langmuir Monolayers - BioNanoScience". BioNanoScience. doi:10.1007/s12668-017-0412-4.
"Interactions of biomolecules in cell membrane models" (PDF). Archived from the original (PDF) on 2017-08-03. Retrieved 2017-08-03.
This article uses material from the Wikipedia article Monolayer, and is written by contributors. Text is available under a CC BY-SA 4.0 International License; additional terms may apply. Images, videos and audio are available under their respective licenses.
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Sensitivity Analysis of a Heat Exchanger Tube Fitted With Cross-Cut Twisted Tape With Alternate Axis | J. Heat Transfer | ASME Digital Collection
M. E. Nakhchi,
M. E. Nakhchi
Mashhad 91775-1111, Iran
e-mail: abolfazl@um.ac.ir
Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received October 12, 2018; final manuscript received January 30, 2019; published online February 27, 2019. Assoc. Editor: Danesh K. Tafti.
Nakhchi, M. E., and Esfahani, J. A. (February 27, 2019). "Sensitivity Analysis of a Heat Exchanger Tube Fitted With Cross-Cut Twisted Tape With Alternate Axis." ASME. J. Heat Transfer. April 2019; 141(4): 041902. https://doi.org/10.1115/1.4042780
Numerical simulations are used to analyze the thermal performance of turbulent flow inside heat exchanger tube fitted with cross-cut twisted tape with alternate axis (CCTA). The design parameters include the Reynolds number
(5000<Re<15,000)
, cross-cut width ratio
(0.7<b/D<0.9)
, cross-cut length ratio
(2<s/D<2.5)
, and twist ratio
(2<y/D<4)
. The objective functions are the Nusselt number ratio
(Nu/Nus)
, the friction factor ratio
(f/fs)
, and the thermal performance
(η)
. Response surface method (RSM) is used to construct second-order polynomial correlations as functions of design parameters. The regression analysis shows that heat transfer ratio decreased with increasing both the Reynolds number and the width to diameter ratio of the twisted tape. This means that the twisted tape has more influence on heat transfer at smaller inlet fluid velocities. Sensitivity analysis reveals that among the effective input parameters, the sensitivity of
Nu/Nus
to the Reynolds number is the highest. The results reveal that thermal performance enhances with increasing the width to diameter ratio of the twisted tape
(b/D)
. The maximum thermal performance factor of 1.531 is obtained for the case of
Re=5000, b/D=0.9, s/D=2.5
y/D=4
cross-cut twisted tape, thermal performance, heat transfer enhancement, response surface method, sensitivity analysis
Design, Friction, Heat exchangers, Heat transfer, Response surface methodology, Reynolds number, Sensitivity analysis, Temperature, Turbulence, Flow (Dynamics), Fluid dynamics, Fluids, Polynomials
Influences of Corrugation Profiles on Entropy Generation, Heat Transfer, Pressure Drop, and Performance in a Wavy Channel
Influences of Wavy Wall and Nanoparticles on Entropy Generation Over Heat Exchanger Plat
Nakhchi
Optimization of the Heat Transfer Coefficient and Pressure Drop of Taylor–Couette–Poiseuille Flows Between an Inner Rotating Cylinder and an Outer Grooved Stationary Cylinder
Heat Transfer Enhancement in Annular Flow With Outer Grooved Cylinder and Rotating Inner Cylinder: Review and Experiments
Experimental Study of Convective Heat Transfer in the Entrance Region of an Annulus With an External Grooved Surface
Friction Factor and Nusselt Number in Annular Flows With Smooth and Slotted Surface
Mirzakhanlari
Vortex Generators Position Effect on Heat Transfer and Nanofluid Homogeneity: A Numerical Investigation and Sensitivity Analysis
Sensitivity Analysis and Multi-objective Optimization of a Heat Exchanger Tube With Conical Strip Vortex Generators
Heat Transfer Enhancement Due to Swirl Effects in Oval Tubes Twisted About Their Longitudinal Axis
Experimental and CFD Studies on Heat Transfer and Friction Factor Characteristics of a Tube Equipped With Modified Twisted Tape Inserts
Performance Assessment in a Heat Exchanger Tube With Alternate Clockwise and Counter-Clockwise Twisted-Tape Inserts
Thermal Characteristics in Round Tube Fitted With Serrated Twisted Tape
Heat Transfer and Friction Factor Characteristics in Turbulent Flow Through a Tube Fitted With Perforated Twisted Tape Inserts
Kiatkittipong
Heat Transfer Enhancement by Multiple Twisted Tape Inserts and TiO2/Water Nanofluid
Nanofluid Turbulent Flow in a Pipe Under the Effect of Twisted Tape With Alternate Axis
Performance Assessment of Turbular Heat Exchanger Tubes Containing Rectangular-Cut Twisted Tapes With Alternate Axes
Thermo-Hydraulic Analysis for a Novel Eccentric Helical Screw Tape Insert in a Three Dimensional Tube
Thermo-Fluid Performance and Entropy Generation Analysis for a New Eccentric Helical Screw Tape Insert in a 3D Tube
Experimental Optimization of Geometrical Parameters on Heat Transfer and Pressure Drop Inside Sinusoidal Wavy Channels
Chetehouna
Sensitivity Analysis of Fluid Properties and Operating Conditions on Flow Distribution in Non-Uniformly Heated Parallel Pipes
Numerical Simulation and Sensitivity Analysis of Heat Transfer Enhancement in a Flat Heat Exchanger Tube With Discrete Inclined Ribs
Rios-Iribe
Cervantes-Gaxiola
González-Llanes
Reyes-Moreno
Hernández-Calderón
Heat Transfer Analysis of a Non-Newtonian Fluid Flowing Through a Circular Tube With Twisted Tape Inserts
Heat Transfer and Pressure Drop Correlations for Twisted-Tape Inserts in Isothermal Tubes—Part II: Transition and Turbulent Flows
Hajmohammad
Multi-Objective Optimization of Cost and Thermal Performance of Double Walled Carbon Nanotubes/Water Nanofluids by NSGA-II Using Response Surface Method
Application of Response Surface Method and Multi-Objective Genetic Algorithm to Configuration Optimization of Shell-and-Tube Heat Exchanger With Fold Helical Baffles
Multi-Objective RSM Optimization of Fin Assisted Latent Heat Thermal Energy Storage System Based on Solidification Process of Phase Change Material in Presence of Copper Nanoparticles
Sensitivity Analysis by the Use of a Surrogate Model During Large Break LOCA on ZION Nuclear Power Plant With CATHARE-2 V2.5 Code
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How do I use the center of a circle calculator?
How do I calculate the center of a circle?
How do I find the center of a physical circle?
Welcome to the center of a circle calculator that finds the center of a circle for you. Here, we'll show you how to calculate the center of a circle from the various circle equations. We'll also cover finding the center of a circle without any math!
The center of a circle calculator is easy to use.
Select the circle equation for which you have the values.
Fill in the known values of the selected equation.
You can find the center of the circle at the bottom.
Read on if you want to learn some formulas for the center of a circle!
Circles can be defined with multiple equations. If you have a mathematical formula for your circle, pick the correct one from the headings below. We'll then explain how to calculate the center of the circle from there.
The standard equation of a circle
\small (x - A)^2 + (y - B)^2 = C
C = r^2
, or the radius squared.
With this equation, we can find the center of the circle at point
(A, B)
. Be careful of the signs!
The parametric equation of a circle is defined as:
\small \begin{split} x &= A + r\!\cdot\!\cos{(\alpha)} \\ y &= B + r\!\cdot\!\sin{(\alpha)} \end{split}
In this form, we can calculate the center of the circle as
(A,B)
A less common circle equation is the general equation of a circle:
\small x^2 + y^2 + D\!\cdot\!x + E\!\cdot\!y + F = 0
In the general equation, we can calculate the center of the circle as
\left(-\frac{D}{2}, -\frac{E}{2}\right)
If you have a circle drawn on paper, there's no center of a circle formula. Instead, follow these steps:
Draw two (or more) chords on the circle.
Find these chords' midpoints.
From the midpoints, draw lines that are perpendicular to the chords.
The point where these lines intersect is the circle's center.
Congrats, you can find the center of the circle!
Need to know more about circles? Try some of our other circle calculators, like:
General to standard form of a circle calculator; and
What is the center of a circle represented by the equation (x+9)² + (y−6)² = 10²?
The center of this circle is (−9, 6), with a radius of 10. The equation (x+9)² + (y−6)² = 10² is in the standard circle equation form (x−A)² + (y−B)² = C, making A = −9 and B = 6.
What is the center of a circle represented by the equation (x−5)² + (y+6)² = 4²?
The center of this circle is (5, −6), with a radius of 4. The equation (x−5)² + (y+6)² = 4² is in the standard circle equation form (x−A)² + (y−B)² = C, making A = 5 and B = −6.
What is the center of a circle given the equation (x−5)² + (y+7)² = 81?
The center of this circle is (5, −7), with a radius of √81 = 9. The equation (x−5)² + (y+7)² = 81 is in the standard circle equation form (x−A)² + (y−B)² = C, making A = 5 and B = −7.
Pick a circle equation and input its parameters. Then find the coordinates of your circle's center below.
Equation choice
(x − A)² + (y − B)² = C
Circle center coordinates
This right circular cone calc finds a (area), v (volume), a_l (lateral area) and a_b (base area) of a right circular cone.
How to find the volume of a cube? Why is the formula for the volume of a cube so simple? Learn all the answers and calculate all the results with Omni.
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Feature interaction - Model analysis | CatBoost
The value of the feature interaction strength for each pair of features.
All splits of features
f1
f2
in all trees of the resulting ensemble are observed when calculating the interaction between these features.
If splits of both features are present in the tree, then we are looking on how much leaf value changes when these splits have the same value and they have opposite values.
See the Interaction file format.
interaction(f_{1}, f_{2}) = \sum_{trees} \left |\sum_{leafs: split(f_1)=split(f_2)} LeafValue { } - \sum_{leafs: split(f_1)\ne split(f_2)}LeafValue \right |
The sum inside the modulus always contains an even number of terms. The first half of terms contains leaf values when splits by
f1
have the same value as splits by
f2
, the second half contains leaf values when two splits have different values, and the second half is in the sum with a different sign.
The larger the difference between sums of leaf values, the bigger the interaction. This process reflects the following idea: let's fix one feature and see if the changes to the other one will result in large formula changes.
The value of the feature interaction strength for each pair of features that are used in the model. Internally the model uses feature combinations as separate features. All feature combinations that are used in the model are listed separately. For example, if the model contains a feature named F1 and a combination of features {F2, F3}, the interaction between F1 and the combination of features {F2, F3} is listed in the output file.
The rows are sorted in descending order of the feature interaction strength value.
See the InternalInteraction file format.
interaction(f_{1}, f_{2}) = \sum_{trees} \left |\sum_{leafs: split(f_1)=split(f_2)} LeafValue { } - \sum_{leafs: split(f_1)\ne split(f_2)}LeafValue \right |
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May 3, 2014 • First Post
To be honest I don’t remember when I actually started thinking about writing down most the solutions I found over the internet or implemented myself. I realized that from time to time I go back to solutions, code snippets or config files I created or modified to fulfill some specific requirements. Good is it I wrote it down somewhere, put on Gist or Pastebin but what if I did not ? Let’s say I just configured Nginx as loadbalancer and for example after a year or so I need to make something exactly the same or at last similar. What’s then ? I need to go through the NGINX’s documentation again or start searching over google. Sure in some case I remember exactly how I solved some specific problem but when it was some time ago then … This is the reason why I decided to create this simple blog. I hope it wont evolve to any unpredictable way. You can thing about this blog like a about my notebook. If somebody else will find my notes useful/helpful - even better. Let’s make this World better.
P.S As you probably noticed Im not a native speaker, so in case you guys will find some mistakes in my post (probably more than one) please let me know ASAP.
Our universe (in SI units):
% <![CDATA[ \begin{align*} & \phi(x,y) = \phi \left(\sum_{i=1}^n x_ie_i, \sum_{j=1}^n y_je_j \right) = \sum_{i=1}^n \sum_{j=1}^n x_i y_j \phi(e_i, e_j) = \\ & (x_1, \ldots, x_n) \left( \begin{array}{ccc} \phi(e_1, e_1) & \cdots & \phi(e_1, e_n) \\ \vdots & \ddots & \vdots \\ \phi(e_n, e_1) & \cdots & \phi(e_n, e_n) \end{array} \right) \left( \begin{array}{c} y_1 \\ \vdots \\ y_n \end{array} \right) \end{align*} %]]>
@widget = Widget(params[:id])
format.json { render json: @widget }
//PipeExample function
func PipeExample() error {
w.Write([]byte("test\n"))
// Copy data from std in to std out
func Copy(in io.ReadSeeker, out io.Writer) error {
w := io.MultiWriter(out, os.Stdout)
if _, err := io.Copy(w, in); err != nil {
in.Seek(0, 0)
if _, err := io.CopyBuffer(w, in, buf); err != nil {
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Square Knowpia
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adjacent sides. It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length. A square with vertices ABCD would be denoted
{\displaystyle \square }
ABCD.[1]
A convex quadrilateral with successive sides a, b, c, d whose area is
{\displaystyle A={\tfrac {1}{2}}(a^{2}+c^{2})={\tfrac {1}{2}}(b^{2}+d^{2}).}
[4]: Corollary 15
All four internal angles of a square are equal (each being 360°/4 = 90°, a right angle).
The central angle of a square is equal to 90° (360°/4).
The external angle of a square is equal to 90°.
The diagonals of a square are equal and bisect each other, meeting at 90°.
The diagonal of a square bisects its internal angle, forming adjacent angles of 45°.
Opposite sides of a square are parallel.
The perimeter of a square whose four sides have length
{\displaystyle \ell }
{\displaystyle P=4\ell }
{\displaystyle A=\ell ^{2}.}
{\displaystyle A={\frac {d^{2}}{2}}.}
{\displaystyle A=2R^{2};}
since the area of the circle is
{\displaystyle \pi R^{2},}
the square fills
{\displaystyle 2/\pi \approx 0.6366}
of its circumscribed circle.
{\displaystyle A=4r^{2};}
hence the area of the inscribed circle is
{\displaystyle \pi /4\approx 0.7854}
of that of the square.
{\displaystyle 16A\leq P^{2}}
The diagonals of a square are
{\displaystyle {\sqrt {2}}}
(about 1.414) times the length of a side of the square. This value, known as the square root of 2 or Pythagoras' constant,[1] was the first number proven to be irrational.
A square can be inscribed inside any regular polygon. The only other polygon with this property is the equilateral triangle.
{\displaystyle 2(PH^{2}-PE^{2})=PD^{2}-PB^{2}.}
{\displaystyle d_{i}}
is the distance from an arbitrary point in the plane to the i-th vertex of a square and
{\displaystyle R}
is the circumradius of the square, then[9]
{\displaystyle {\frac {d_{1}^{4}+d_{2}^{4}+d_{3}^{4}+d_{4}^{4}}{4}}+3R^{4}=\left({\frac {d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}}{4}}+R^{2}\right)^{2}.}
{\displaystyle L}
{\displaystyle d_{i}}
are the distances from an arbitrary point in the plane to the centroid of the square and its four vertices respectively, then [10]
{\displaystyle d_{1}^{2}+d_{3}^{2}=d_{2}^{2}+d_{4}^{2}=2(R^{2}+L^{2})}
{\displaystyle d_{1}^{2}d_{3}^{2}+d_{2}^{2}d_{4}^{2}=2(R^{4}+L^{4}),}
{\displaystyle R}
is the circumradius of the square.
{\displaystyle |x|+|y|=2}
{\displaystyle \max(x^{2},y^{2})=1}
specifies the boundary of this square. This equation means "x2 or y2, whichever is larger, equals 1." The circumradius of this square (the radius of a circle drawn through the square's vertices) is half the square's diagonal, and is equal to
{\displaystyle {\sqrt {2}}.}
Then the circumcircle has the equation
{\displaystyle x^{2}+y^{2}=2.}
{\displaystyle \left|x-a\right|+\left|y-b\right|=r.}
The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars) Cyclic symmetries in the middle column are labeled as g for their central gyration orders. Full symmetry of the square is r8 and no symmetry is labeled a1.
A square is a special case of many lower symmetry quadrilaterals:
A rhombus with equal diagonals
^ a b c Weisstein, Eric W. "Square". mathworld.wolfram.com. Retrieved 2020-09-02.
^ Zalman Usiskin and Jennifer Griffin, "The Classification of Quadrilaterals. A Study of Definition", Information Age Publishing, 2008, p. 59, ISBN 1-59311-695-0.
^ "Problem Set 1.3". jwilson.coe.uga.edu. Retrieved 2017-12-12.
^ Josefsson, Martin, "Properties of equidiagonal quadrilaterals" Forum Geometricorum, 14 (2014), 129-144.
^ "Quadrilaterals - Square, Rectangle, Rhombus, Trapezoid, Parallelogram". www.mathsisfun.com. Retrieved 2020-09-02.
^ 1999, Martin Lundsgaard Hansen, thats IT (c). "Vagn Lundsgaard Hansen". www2.mat.dtu.dk. Retrieved 2017-12-12. {{cite web}}: CS1 maint: numeric names: authors list (link)
^ "Geometry classes, Problem 331. Square, Point on the Inscribed Circle, Tangency Points. Math teacher Master Degree. College, SAT Prep. Elearning, Online math tutor, LMS". gogeometry.com. Retrieved 2017-12-12.
^ Wells, Christopher J. "Quadrilaterals". www.technologyuk.net. Retrieved 2017-12-12.
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