text
stringlengths 256
16.4k
|
|---|
Instant Runoff Normalized Ratings - Electowiki
Instant Runoff Normalized Ratings, or IRNR is a method devised by Brian Olson.[1] Based on a ratings ballot, the method seeks to give every voter equal power and encourage honest ratings.
The first step is normalizing, which can happen in two ways:
Divide each rating by the sum of the absolute values of the ratings. The sum of absolute values of the ratings will then be 1.
This shall be called IRNR[1] since the normalization factor is the L1 norm.
Divide each rating by the square root of the sum of the squared ratings. The vote will then be a vector with magnitude 1.
One could more generally consider IRNR[p], based on the Lp norm, for any fixed real p with
{\displaystyle 1 \le p \le \infty}
. (To avoid difficulties with dividing by 0, we agree to ignore votes that rank all candidates 0.)
Formula for IRNR[n] normalization:
{\displaystyle \begin{equation}
{C_{new}} =\frac{C_{old}}{\sqrt[n]{\sum \left(\bigl| C_{i}\bigr|^{n}\right)}}
\end{equation}}
{\displaystyle \begin{equation}{C_{old}}\end{equation}}
= rating of candidate C in the vote, before the normalization.
{\displaystyle \begin{equation}{C_{new}}\end{equation}}
= rating of C, after the normalization.
{\displaystyle \begin{equation}{C_{i}}\end{equation}}
= ratings of each candidate in the vote, before the normalization.
The second step consists of summing up the normalized ratings for each candidate. If there are two choices, the highest rated is the winner. If there are more than two choices, the lowest rated choice is disqualified.
The process repeats with a normalization step that ignores disqualified choices. A voter's voting power is thus redistributed among the remaining choices.
It is possible to normalize by first observing the highest score the voter gave to any candidate, and pretending that is the maximum allowed score when interacting with that voter's ballot. In other words, a voter who gave their favorite a 3 out of 5 could have their ballot normalized such that the highest score they give to any candidate in any round of IRNR would be a max of 3 out of 5.
Distributed Voting (specific variant, based on L1 norm)
Java code that implements IRNR
Instant Runoff Normalized Ratings: an Election Method by Brian Olson
↑ "Election Methods Defined". bolson.org. Retrieved 2021-12-18.
Retrieved from "https://electowiki.org/w/index.php?title=Instant_Runoff_Normalized_Ratings&oldid=14742"
|
A Guide to Price Elasticity of Demand | Outlier
Zones of Elasticity
What Makes a Good Elastic or Inelastic?
E_D
measures the change in the quantity demanded of a good relative to a change in its price.
We know from the Law of Demand that there is an inverse relationship between price and quantity demanded for most goods. As the price of a good increases, demand falls; as the price of a good decreases, demand rises. Price elasticity of demand tells us how big or small the resulting percentage change in the quantity demanded is relative to the percentage change in price.
For some goods, the percentage change in the quantity demanded is large relative to the price change. Such goods have a price elasticity of demand that is greater than 1. They are considered elastic.
For other goods, the percentage change in demand is small relative to the price change. These goods have an elasticity that is less than 1. They are considered inelastic.
To calculate the price elasticity of demand, we need to find the ratio of the percentage change in the quantity demanded to the percentage change in price. (Percentage change can be calculated by subtracting the original value from the new value, and then dividing this by the original value.) This ratio uses the absolute value of each percentage change because we only compare the magnitude of the changes and not the direction.
\text{Price Elasticity of Demand ($E_D$)} = \dfrac{|\text{\% Change in the Quantity Demanded}|}{|\text{\% Change in the Price}|}
Notice that the elasticity will be greater than 1 when the numerator is greater than the denominator. In other words, elasticity will be greater than 1 when the percentage change in quantity is greater than the percentage change in price. Likewise, the elasticity will be less than 1 if the percentage change in quantity is less than the percentage change in price. This is how we determine whether a good is elastic (E_D>1) or inelastic (E_D<1).
Say a health food chain increases the price of their smoothies from $10 to $12 and as a result, the number of smoothies demanded (i.e., the quantity demanded) falls from 5,000 to 3,500. You can see these changes represented as the change from Point A to Point B on the demand curve below.
We can calculate the price elasticity of demand in three easy steps.
Step 1. Find the percentage change in price First, we find the percentage change in price, the denominator in our price elasticity of demand equation. To do this, we subtract the original price from the new price and divide the difference by the original price. Therefore, an increase in price from $10 to $12 is equal to 0.20 or a 20% increase.
\%Change\;in\;Price=\frac{New\;Price\;-\;Original\;Price}{Original\;Price}=\frac{\$12-\$10}{\$10}=0.2=20\%
Step 2. Find the percentage change in quantity
Next, we find the percentage change in the quantity demanded using the exact same method we used for the percentage change in price. In this case, a decrease from 5,000 smoothies to 3,500 smoothies equals 0.3, or a 30% decrease.
\%Change\;in\;Quantity\;Demanded=\frac{New\;Quantity\;-\;Original\;Quantity}{Original\;Quantity}=\frac{3500-5000}{5000}=-0.3=-30\%
Step 3. Find the price elasticity of demand using the absolute values of the changes found in Steps 1 and 2
To find the price elasticity of demand, we take the absolute value of the percentage changes we found in Steps 1 and 2. Then, we divide the percentage change in quantity by the percentage change in price.
E_D\;for\;Smoothies\;at\;a\;Health\;Foods\;Chain=\frac{0.3}{0.2}=1.5
In this example, we find an elasticity of 1.5, which is greater than 1. We know from earlier that this indicates that the demand for the smoothies is elastic: the percentage change in the quantity demanded is greater than the percentage change in price. In this case, a 20% increase in price resulted in a 30% decrease in demand. The percentage change in demand was 1.5 times greater than the change in price. Hence, elasticity(
E_D
) = 1.5.
We’ve already established that goods with an elasticity greater than 1 are elastic and that goods with an elasticity less than 1 are inelastic. Now, we can further designate elasticity into five categories or five zones. These are shown in the charts below.
Perfectly Inelastic Goods (
E_D
=0): When the price elasticity of demand is equal to zero, the demand is perfectly inelastic. This means that price changes will have no effect on the quantity demanded. The demand curve for such goods is a vertical line indicating that the quantity demanded will remain unchanged no matter what price is charged. Perfectly inelastic demand is rare, but it can occur when a good is an absolute necessity. Demand for a life-saving drug, for example, could be perfectly inelastic: people who need it may be willing to pay any price to get it.
Inelastic Goods (E_D < 1): When demand for a good is inelastic, the demand is relatively resistant to changes in price, and thus, the percentage change in quantity demanded will be less than the percentage change in price. Inelastic goods have an elasticity of less than 1 and have steep demand curves (but not vertical).
Unit Elastic Goods (
E_D
= 1): A unit elastic good has a price elasticity of demand equal to 1. In such cases, the percentage change in quantity demanded is equal to the percentage change in price. So if the price increases by 15%, demand will fall by 15%, or if the price decreases by 5%, demand will rise by 5%.
Elastic Goods (
E_D
> 1): When demand for a good is elastic, the demand is relatively sensitive to changes in the price, as we saw in our smoothie example. Any change in the product’s price will lead to a larger percentage change in the quantity demanded. Elastic goods have an elasticity of less than 1 and have flatter (but not quite horizontal) demand curves.
Perfectly Elastic Goods (
E_D
= ∞): Finally, perfectly elastic goods are the extreme opposite of perfectly inelastic goods. In this case, even a slight change in price will cause demand to drop to zero. This can happen when a good has plenty of close substitutes. For example, a can of non-organic black beans costs about $1. If a single company tries to raise the price of their canned black beans, consumers can quickly abandon the company’s product and buy beans from a different provider. The demand curve for perfectly elastic goods is horizontal, indicating that any deviation from the existing price will cause demand to drop to zero.
The price elasticity of demand is not equal to the slope of the demand curve.
Because the elasticity of demand is related to the slope of the demand curve, it’s easy to confuse the two, but be careful. The slope of the demand curve and the price elasticity of demand are not the same things.
The slope of the demand curve is approximated by the change in price divided by the change in quantity. The price elasticity of demand is calculated as the percentage change in quantity divided by the percentage change in price!
Slope\;of\;the\;Demand\;Curve=\frac{Change\;in\;Price}{Change\;in\;Quantity}=\frac{\Delta P}{\Delta Q}
Price\;Elasticity\;of\;Demand=\frac{Percentage\;Change\;in\;Quantity}{Percentage\;Change\;in\;Price}=\frac{\%\Delta Q}{\%\Delta P}
What makes the demand for a good elastic or inelastic? The chart below lists some common goods with their estimated price elasticities of demand. We can use these to help us understand factors that influence elasticity.
The main factors that influence the elasticity of demand are the following:
When a good has plenty of close substitutes, consumers can easily reduce their demand for the good and switch over to the substitutes (demand will be elastic). We saw this with the example of canned black beans earlier. This is also why restaurant meals are elastic. Consumers can switch to cheaper restaurants or eat at home more frequently if restaurants raise their prices.
On the other hand, when there are few or no available substitutes, consumers have a harder time adjusting their consumption in response to price changes (demand will be inelastic). This explains why goods such as coffee or cigarettes are inelastic. For people who rely on these goods, there are no close substitutes.
When the price of a luxury good increases, consumers can easily decide to reduce consumption or forgo buying the good altogether, and so demand will be elastic. This explains the elasticity of things such as foreign travel and fancy automobiles. These goods are not necessary, so they can more easily be given up.
On the other hand, if a good is a necessary item, consumers have a harder time adjusting their consumption in response to a price change, and so demand is inelastic. Gasoline, for example, is inelastic because people rely on it to get to work every day.
The portion of income spent on the good
When a good makes up a significant portion of a consumer’s overall budget or income, the consumer is likely to be more sensitive to changes in price (i.e., demand will be elastic). This is true for goods and services such as vacation homes, additional automobiles, or cosmetic surgeries.
On the other hand, when a good makes up an insignificant amount of a consumer’s overall budget or income, the consumer may be more willing to continue purchasing the product despite changes in the price (i.e., demand will be inelastic). This is why salt and matches fall into the inelastic category. Even if a salt grinder doubled in price (a 100% increase), you may only be paying a dollar or two extra out of your entire income.
The time consumers have to adjust to price changes
In many instances, goods are more elastic in the long term, because consumers have more time to adjust their behavior to changes in price. For example, someone who smokes cigarettes or is addicted to drugs may change their behavior and reduce consumption over longer periods. Even for necessary goods such as gasoline, people can adjust their behavior by organizing carpools or finding alternative modes of transportation. Still, changing habits and finding alternatives takes time.
On the other hand, goods tend to be more inelastic in the short term. If the price of gasoline goes up overnight, you still need to get to work the next morning. If you have a strong enough need or if you are addicted to a good, changing your habits in an instant can be difficult.
In this brief, you learned about the price elasticity of demand, but you should know that there are other types of elasticity as well. These include:
This measures the change in the quantity demanded of one good relative to a price change in another good. This comes in handy when you are trying to estimate the relationship between demand for one good and the price of another related good, such as a compliment or a substitute.
This measures the change in demand for a good relative to a change in consumer income.
This measures the change in the quantity of a good that a producer is willing to supply relative to changes in the price they can charge.
The calculations for these different types of elasticities are slightly different from one another. However, the basic intuition behind all of them is the same. In each case, the elasticity measures the responsiveness (or the resulting change) in one factor—typically the quantity demanded or supplied of a good—relative to some other factor such as price or income.
|
Modular Multi-level Converter (MMC) with Induction Machine
This note demonstrates use of a MMC in one of its main applications: electrical drive systems.
C-HIL: Field-oriented control of PMSM using Texas Instruments TMS320F2808 card
Demonstration on how to use a Texas Instruments card as a controller for a Typhoon HIL power system model.
The main motivation for the application note is to present the possibility of modelling Modular Multilevel Converters (MMCs) in a Typhoon HIL environment. Included is a demonstration of the MMC model performance in electrical drive systems, one of its main application areas.
MMCs are an emerging and scalable topology of power converters. MMCs make it possible to produce voltages with low harmonic influence even in high voltage and power applications. The main advantages of MMCs are:
Low harmonic influence on output voltage
No need for output filter
Low switch rating in relation to output voltage etc.
The typical structure of a three phase MMC is shown in Figure 1 . Each phase leg of the converter has two arms, an upper and a lower. Each one is constituted by n number of sub-modules (SMs). In each arm, there is also an inductor
{L}_{d}
{R}_{d}
to compensate for the voltage difference between the upper and lower arms produced when a SM is switched in or out.
Figure 1. Three phase n – level MMC topology [1]
The configuration of a sub–module (SM) is given in Figure 2. Each SM is a simple chopper cell composed of two IGBT switches S1 and S2, two antiparallel diodes D1 and D2, and a capacitor C.
Figure 2. MMC sub – module [1]
With reference to the SM shown in Figure 2, the output voltage
{V}_{SM}
has two values;
{V}_{SM}={V}_{c}
when S1 is on and S2 is off or
{V}_{SM}=0
when S1 is off and S2 is on.
In Figure 3 it is shownshows the implementation of athe MMC using Switching Function.
Figure 3. Graphical representation of generic MMC switching function model
Proper operation of the MMC requires the average voltage stress across each cell capacitor and switching subcomponent must be maintained at
\frac{{V}_{dc}}{N-1}
{V}_{dc}
is voltage on the DC link and N is number of levels. For phase ‘a’, the output phase voltage
{v}_{a0}
can either be expressed in terms of the upper arm cell capacitor voltages and positive pole-to-ground, or in terms of the lower arm cell capacitor voltages and negative pole-to-ground. These are described by the equations:
{v}_{a0}=\frac{1}{2}{V}_{dc}-\sum _{J=1}^{N}{S}_{uj}{V}_{cj}- {R}_{d} {i}_{a1}- {L}_{d} \frac{d{i}_{a1}}{dt}
{v}_{a0}=-\frac{1}{2}{V}_{dc}+\sum _{J=1}^{N}{S}_{lj}{V}_{cj}+ {R}_{d}{i}_{a2}+ {L}_{d} \frac{d{i}_{a2}}{dt}
The voltage across each arbitrary cell capacitor (from the upper or lower arms of phase ‘a’, for example) can be expressed using the switching function as:
{v}_{cj}=\frac{1}{{C}_{m}}\int {S}_{uj}{i}_{a1}\left(t\right)dt
{v}_{cj}=\frac{1}{{C}_{m}}\int {S}_{lj}{i}_{a2}\left(t\right)dt
The model consists of a DC link, three MMC Leg - Switching Function with Nearest Level Control (NLC), and an Induction Machine. The control for the converter and mechanical model is implemented using Signal Processing Components. The control for the drive system is implemented as an open-loop. By changing the parameter for the NLC block, there is a possibility to change number of levels.
Figure 4. MMC with Induction Machine
This application comes with a pre-built SCADA panel Figure 5. The panel offers most essential user interface elements (widgets) to monitor and interact with the simulation in runtime. You can customize it freely to fit your needs.
The SCADA panel made for this MMC converter model, allows you to have a wide insight into the operation of this converter through its various widgets and scopes. On the right corner, you can change the induction machine load per unit. In the Trace graph, you can see the mechanical and electrical torque.
Figure 5. SCADA panel during simulation
In Capture, you can see the currents of all three phases together with all three phase voltages. The third viewport shows capacitor voltage, as well as how the controller balances these voltages across the capacitors. The next three figures show the different behaviors of the converter at different voltage levels. The maximum number of voltage levels depends on the size of the model and the signal processing power of each HIL device.
Figure 6. MMC using 7 levels (HIL 402)
Figure 8. MMC using 17 levels (HIL 404)
examples\models\electrical drives\mmc with indm.tse
HIL device model HIL402, HIL604, HIL404
[1] Prafullachandra M. Meshram, and Vijay B. Borghate, “A Simplified Nearest level control (NLC) Voltage Balancing Method for Modular Multilevel Converter (MMC)” DOI 10.1109/TPEL.2014.2317705, IEEE Transactions on Power Electronics
|
Let's use the antidots we've just created to examine subtraction.
Consider this subtraction problem.
To me, this is really
536
plus the opposite of
123
That is, we're adding one anti-hundred, two anti-tens, and three anti-ones to five hundreds, three tens and six ones.
Put the numbers
536
-123
1\leftarrow 10
Make the annihilations required to show the value of
413
Try this problem on the machine!
Unexplode some dots to clarify the result of the subtraction.
Here’s a very long subtraction problem. Compute it the dots and antidots way and then fix up your answer for society’s sake.
Thinking Question: As you fix up your answer for society, does it seem easier to unexplode from left to right or from right to left?
All correct approaches to mathematics are correct – even this strange approach to subtraction. Actually, if you click on the station below about the Traditional Algorithm, you will see my approach here isn’t actually that new or different!
KThe Traditional Subtraction Algorithm
|
Metropolis-Hastings - Monte-Carlo
Independent MH
ARMS MH
A sample produced by the above algorithm differs from an iid sample. For one thing, such a sample may involve repeated occurrences of the same value, since rejection of
Y_t
leads to repetition of
X^{(t)}
t+1
(an impossible occurrence in absolutely continuous iid settings)
Minimal regularity conditions on both
f
and the conditional distribution
q
f
to be the limiting distribution of the chain
X^{(t)}
\cup_{x\in \mathrm{supp}\, f}\mathrm{supp}\, q(y\mid x)\supset \mathrm{supp}\,,f
f
is the stationary distribution of the Metropolis chain: it satisfies the detailed balance property.
K(x,y) = \rho(x,y)q(y\mid x)+(1-r(x))\delta_x(y)\,.
The MH Markov chain has, by construction, an invariant probability distribution
f
, if it is also an aperiodic Harris chain, then we can apply ergodic theorem to establish a convergence result.
A sufficient condition to be aperiodic: allow events such as
\{X^{(t+1)}=X^{(t)}\}
Property of irreducibility: sufficient conditions such as positivity of the conditional density
q
If the MH chain is
f
-irreducible, it is Harris recurrent.
A somewhat less restrictive condition for irreducibility and aperiodicity.
In the following sections, let's introduce other versions of Metropolis-Hastings.
|
Effect of Squealer Geometry on Tip Flow and Heat Transfer for a Turbine Blade in a Low Speed Cascade | J. Heat Transfer | ASME Digital Collection
Vikrant Saxena,
Mechanical Engineering Department, Louisiana State University, Baton Rouge, LA 70803
e-mail: ekkad@me.lsu.edu
Contributed by the Heat Transfer Division for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received by the Heat Transfer Division June 2, 2003; revision received May 4, 2004. Associate Editor: H. S. Lee.
Saxena, V., and Ekkad, S. V. (May 4, 2004). "Effect of Squealer Geometry on Tip Flow and Heat Transfer for a Turbine Blade in a Low Speed Cascade ." ASME. J. Heat Transfer. August 2004; 126(4): 546–553. https://doi.org/10.1115/1.1777580
A detailed investigation on the effect of squealer geometries on the blade tip leakage flow and associated heat transfer is presented for a scaled up high pressure turbine blade in a low-speed wind tunnel facility. The linear cascade is made of four blades with the two corner blades acting as guides. The tip profile of a first stage rotor blade is used to fabricate the two-dimensional blade. The wind tunnel accommodates an
116°
turn for the blade cascade. The mainstream Reynolds number based on the axial chord length based on cascade exit velocity is
4.83×105.
An upstream wake effect is simulated with a spoked wheel wake generator placed upstream of the cascade. A turbulence grid placed even farther upstream generates a free-stream turbulence of 4.8%. The center blade has a tip clearance gap of 1.56% with respect to the blade span. Static pressure measurements are obtained on the blade surface and the shroud. Results show that the presence of the squealer alters the tip gap flow field significantly and produces lower overall heat transfer coefficients. The effects of different squealer arrangements are basically to study the effect of squealer rim placement on tip leakage flow and associated heat transfer. Detailed heat transfer measurements are obtained using a steady state liquid crystal technique. The effect of periodic unsteady wake effect is also investigated by varying the wake Strouhal number from 0–0.4. Results show that suction side squealers may be favorable in terms of overall reduction in heat transfer coefficients over the tip surface. However, the presence of a full squealer is most beneficial in terms of reducing overall heat load on the tip surface. There is reasonable effect of wake induced periodicity on tip heat transfer.
heat transfer, confined flow, wind tunnels, wakes, turbulence, flow instability
Blades, Cascades (Fluid dynamics), Flow (Dynamics), Heat transfer, Heat transfer coefficients, Leakage flows, Pressure, Wakes, Suction, Liquid crystals, Geometry, Turbine blades, Turbulence
Morphis, G., and Bindon, J. P., 1988, “The Effect of Relative Motion, Blade Edge Radius and Gap Size on the Blade Tip Pressure Distribution in an Annular Turbine Cascade with Clearance,” AMSE Paper 88-GT-256.
Flow Field in the Tip Gap of a Planar Cascade of Turbine Blades
Kaiser, I., and Bindon, J. P., 1997, “The Effect of Tip Clearance on the Development of Loss Behind a Rotor and a Subsequent Nozzle,” ASME Paper 97-GT-53.
Mayle, R. E., and Metzger, D. E., 1982, “Heat Transfer at the Tip of an Unshrouded Turbine Blade,” Proceedings of the 7th International Heat Transfer Conference, 3, pp. 87–92.
Metzger, D. E., Dunn, M. G., and Hah, C., 1990, “Turbine Tip and Shroud Heat Transfer,” Proceeding of International Gas Turbine and Aeroengine Congress and Exposition, Brussels, Belgium, September 1990, Paper No. 90-GT-333.
Yang, T. T., and Diller, T. E., 1995, “Heat Transfer and Flow for a Grooved Turbine Blade Tip in a Transonic Cascade,” Proceedings of International Mechanical Engineering Congress and Exposition, San Francisco, ASME Paper No. 95-WA/HT-29.
Heat Transfer and Flow on the First Stage Blade Tip of a Power Generation Gas Turbine: Part 1—Experimental Results
Heat Transfer and Flow on the First Stage Blade Tip of a Power Generation Gas Turbine: Part 2—Simulation Results
Azad, Gm. S., Han, J. C., and Teng, S., 2000, “Heat Transfer and Pressure Distributions on a Gas Turbine Blade Tip,” Proceedings of International Gas Turbine and Aeroengine Congress and Exposition, ASME paper 2000-GT-194.
Azad, G. S., Han, J. C., and Boyle, R. J., 2000, “Heat Transfer and Flow on the Squealer Tip of a Gas Turbine Blade,” Proceedings of International Gas Turbine and Aeroengine Congress and Exposition, Munich, Germany, 2000-GT-195.
Bunker, R. S., and Bailey, J. C., 2001, “Effect of Squealer Cavity Depth and Oxidation on Turbine Blade Tip Heat Transfer,” International Gas Turbine and Aeroengine Congress and Exposition, New Orleans, June 2001.
Azad, G. S., Han, J. C., Bunker, R. S., and Lee, C. P., 2001, “Effect of Squealer Geometry Arrangement on Gas Turbine Blade Tip Heat Transfer,” Proceedings of International Mechanical Engineering Congress and Exposition, New York, New York, November 2001, IMECE2001/HTD-2431.
Dunn, M. G., and Haldeman, C. W., 2000, “Time-Averaged Heat Flux for a Recessed Tip, Lip and Platform of a Transonic Turbine Blade,” ASME Paper 2000-GT-0197.
Nasir, H., Ekkad, S. V., Kontrovitz, D. M., Bunker, R. S., and Prakash, C., 2003, “Effect of Tip Gap and Squealer Geometry on Detailed Heat Transfer Measurements over a HPT Rotor Blade Tip,” IMECE 2003-41294, ASME IMECE 2003, Washington, D.C.
Wittig, S., Schulz, A., Dullenkopf, K., and Fairbank, J., 1988, “Effects of Free-Stream Turbulence and Wake Characteristics on the Heat Transfer Along a Cooled gas Turbine Blade,” ASME Paper No. 88-GT-179.
Gm. S.
Detailed Heat Transfer Coefficient Distributions on a Large Scale Gas Turbine Blade Tip
Saxena, V., Nasir, H., and Ekkad, S. V., 2003, “Effect of Blade Tip Geometry on Tip Flow and Heat Transfer for a Blade in a Low Speed Cascade,” ASME GT2003-38176, ASME IGTI Conference, Atlanta, GA, June 2003.
Camci, C., Kim, K., and Hippensteele, S. A., 1991, “A New Hue Capturing Technique for Quantitative Interpretation of Liquid Crystal Images Used in Convective Heat Transfer Studies,” ASME Paper 91-GT-277.
Effect of Blade Tip Geometry on Tip Flow and Heat Transfer for a Blade in a Low-Speed Cascade
|
How to Make a Box Plot | Outlier
How to Construct a Box Plot in 7 Steps
A box plot, sometimes called a box and whisker plot, is a graph of the five-number summary of a data set. This graph has two components. The first is a box marking off the 1st quartile (25th percentile), 2nd quartile (the median), and 3rd quartile (75th percentile) of the data. The second component consists of two lines extending outward from the box to mark the minimum and maximum values.
As you can see in the figure below, the five-number summary divides the data into four buckets (or quartiles), each containing 25 percent of the data. You can draw a box plot vertically, as shown in the figure below, or horizontally along a number line.
Benefits of a Box Plot
Box plots are a great visual tool for quickly conveying the center, spread, and skewness of data. They’re often used to provide a high-level comparison of the distribution of data across multiple samples or data sets that share the same units of measurement.
Because a box plot is only anchored by five points (the five-number summary), it does not convey more detailed information about the shape of the distribution. To visualize the entire distribution, you should use a histogram or a probability density graph.
Box Plot Variations
A variation of the box plot is one in which potential outliers are separated from the rest of the data. The outliers are marked by a circle or a dot on the graph. If your data contains extreme outliers, this is a better version of the box plot to use.
Let’s say you have the following data consisting of 18 data points (n=18).
You can construct a box plot in 7 easy steps.
Step 1. Arrange the data from smallest to largest.
Step 2. Find the minimum and maximum of the data.
The minimum and the maximum are simply the smallest and largest values in your data. These will eventually be the endpoints of your whiskers.
If you have outliers and want to exclude them, simply ignore the outliers and select the next smallest or largest values as your maximum and minimum.
Step 3. Find the median.
To find the median, you look for the middle value of your data. If the total number of data points is odd, this is straightforward, as there will only be one number that sits in the middle. If the total number of data points is even, as is the case here, we need to find the middle two data points and then average them.
Step 4. Find Q1.
Now that you have the minimum, maximum, and median, you need to find the first quartile (Q1) and the third quartile (Q3) of the data. Once you have these, you’ll have your five-number summary.
The first quartile is the same thing as the 25th percentile of the data. To find it, we use the locator formula below. This formula lets us find the value that divides the bottom quarter of the data from the upper three-quarters (or 75 percent) of the data.
L=\left(\frac k{100}\right)n=\left(\frac{25}{100}\right)\cdot18=4.5
When L is a whole number, you can calculate Q1 by taking the average of the Lth value in your data and the (Lth +1) value.
When L is not a whole number, you round L up to the next whole number to locate Q1.
Since our value for L is not a whole number, we round up from 4.5 and look for the 5th value in the data set.
Note that there are different ways to calculate the percentiles of data. Depending on the method you use, your calculation for Q1 and Q3 might be slightly different.
The third quartile is the same thing as the 75th percentile. Using the same method we used in Step 4, we can find Q3.
L=\left(\frac k{100}\right)n=\left(\frac{75}{100}\right)\cdot18=13.5
Because L is once again not a whole number, we round up and look for the 14th value in the data set.
Step 6. Construct the box using Q1, Q2, and Q3.
We now have a five-number summary of the data and can start constructing the box plot.
Five-Number Summary:
Min. = 2
Q1 marks the lower-end of the box, and Q3 marks the upper-end. We add a horizontal bar within the box to show the data’s median, or 50th percentile.
Step 7. Add the whiskers!
The last step is to add the whiskers. Simply draw one line connecting the bottom of the box to the minimum value of the data and another line connecting the top-end of the box to the maximum value.
Voila! You have your box plot.
|
Tetradecagon - Simple English Wikipedia, the free encyclopedia
(Redirected from Tetradecagram)
A tetradecagon or 14-gon is a shape with 14 sides and 14 corners.
1 Regular tetradecagon
Regular tetradecagon[change | change source]
The amount of space a regular tetradecagon takes up is
{\displaystyle {\begin{aligned}A&={\frac {14}{4}}a^{2}\cot {\frac {\pi }{14}}={\frac {14}{4}}a^{2}\left({\frac {{\sqrt {7}}+4{\sqrt {7}}\cos \left({{\frac {2}{3}}\arctan {\frac {\sqrt {3}}{9}}}\right)}{3}}\right)\\&\simeq 15.3345a^{2}\end{aligned}}}
Dissection[change | change source]
Coxeter states that every parallel-sided 2m-gon can be divided into m(m-1)/2 rhombs. For the regular tetradecagon, m=7, and it can be divided into 21: 3 sets of 7 rhombs. This decomposition is based on a Petrie polygon projection of a 7-cube, with 21 of 672 faces. [1] The list A006245 defines the number of solutions as 24698, including up to 14-fold rotations and chiral forms in reflection.
Dissection into 21 rhombs
Eric W. Weisstein, Tetradecagon at MathWorld.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Tetradecagon&oldid=6195522"
|
26.8: Appendix - Statistics LibreTexts
https://stats.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fstats.libretexts.org%2FBookshelves%2FIntroductory_Statistics%2FBook%253A_Statistical_Thinking_for_the_21st_Century_(Poldrack)%2F26%253A_The_General_Linear_Model%2F26.08%253A_Appendix
Y = X*\beta + E
This looks very much like the earlier equation that we used, except that the letters are all capitalized, which is meant to express the fact that they are vectors.
We know that the grade data go into the Y matrix, but what goes into the
The rules of matrix multiplication tell us that the dimensions of the matrices have to match with one another; in this case, the design matrix has dimensions of 8 (rows) X 2 (columns) and the Y variable has dimensions of 8 X 1. Therefore, the
\beta
matrix needs to have dimensions 2 X 1, since an 8 X 2 matrix multiplied by a 2 X 1 matrix results in an 8 X 1 matrix (as the matching middle dimensions drop out). The interpretation of the two values in the
\beta
matrix is that they are the values to be multipled by study time and 1 respectively to obtain the estimated grade for each individual. We can also view the linear model as a set of individual equations for each individual:
\hat{y}_1 = studyTime_1*\beta_1 + 1*\beta_2
\hat{y}_2 = studyTime_2*\beta_1 + 1*\beta_2
\hat{y}_8 = studyTime_8*\beta_1 + 1*\beta_2
Remember that our goal is to determine the best fitting values of
\beta
given the known values of
X
Y
. A naive way to do this would be to solve for
\beta
using simple algebra – here we drop the error term
E
because it’s out of our control:
\hat{\beta} = \frac{Y}{X}
The challenge here is that
X
\beta
are now matrices, not single numbers – but the rules of linear algebra tell us how to divide by a matrix, which is the same as multiplying by the inverse of the matrix (referred to as
X^{-1}
). We can do this in R:
26.8: Appendix is shared under a not declared license and was authored, remixed, and/or curated by Russell A. Poldrack via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.
|
Reconfigurable and Adaptive Radar Amplifiers for Spectrum Sharing in Cognitive Radar - IEEE Conference Publication
Conferences > 2019 IEEE Radar Conference (R...
Reconfigurable and Adaptive Radar Amplifiers for Spectrum Sharing in Cognitive Radar
Charles Baylis ; Austin Egbert ; Jose Alcala-Medel ; Angelique Dockendorf ; Caleb Calabrese ; Ellie Langley ; Anthony Martone ; Kyle Gallagher ; Ed Viveiros ; Dimitrios Peroulis ; Abbas Semnani ; Robert J. Marks
Fast Tuning Algorithm
Abstract: The increased use of the wireless spectrum for broadband applications has necessitated a new approach to radar transmission. Future radar systems will be required to adju... View more
The increased use of the wireless spectrum for broadband applications has necessitated a new approach to radar transmission. Future radar systems will be required to adjust operating frequency in real time to avoid interference from other wireless devices operating in the same band. This paper describes the use of a 90 W tunable power-amplifier matching circuit to reconfigure a power amplifier for high power-added efficiency and spectral compliance at multiple frequencies within the S-band. To reconfigure the amplifier in real-time spectrum sharing scenarios, a modified gradient search algorithm is employed to tune the amplifier based on measured data. A look-up table allows previous optimum tuner settings to be stored after the first reconfiguration at each operating frequency, allowing faster optimizations upon re-visits to each frequency. Next research steps are discussed; including the streamlining of system timing, operating the search algorithm under control of spectral prediction and decision-making processes and analyzing the effect of impedance tuning on range and Doppler detection.
Published in: 2019 IEEE Radar Conference (RadarConf)
Conference Location: Boston, MA, USA, USA
The astounding increase of wireless broadband applications continues to encroach upon traditional radar frequency bands. While 3.55 GHz to 3.7 GHz has already been repurposed for radar sharing with the Citizens Broadband Radio Service (CBRS), the National Telecommunications and Information Administration (NTIA) in the United States announced in 2018 that the 3.45 to 3.55 GHz band will also be repurposed for sharing between radar and wireless communication, potentially for the fifth-generation (5G) wireless systems [1]. The ongoing trajectory that includes the loss of 250 MHz of contiguous radar spectrum over the last decade requires that radar systems change their operating methods. Future radar transmitters must be adaptive and reconfigurable; capable of actively sharing spectrum with wireless communication devices.
A typical spectrum sharing scenario facing a radar transmitter could be depicted as shown in Fig. 1 [2]. Future radar systems must predict, detect, and avoid potential RF interferers (RFI). Significant development is needed to facilitate this approach, including the ability to perform spectrum sensing, predict spectral usage, and re-tune transmitter circuitry quickly. In the planned approach, the radar system will attempt to avoid the interferer, maximize signal-to-interference-plus-noise ratio (SINR) while also maximizing the number of subbands in which it can transmit to enhance range resolution' as described by Selvi [3].
Reconfigurable radar transmitter amplifiers provide a means by which the amplifier can be reconfigured to operate at different frequencies with low-loss matching networks. While broadband matching networks can be designed, the low quality factor of the resonators in these networks results in significant loss, which reduces detection range and power efficiency. An evanescent-mode resonant cavity (EVA) matching network has been demonstrated in two generations. The first-generation tuner has been demonstrated by Semnani [4] and can be tuned across much of the radar S-band allocation. Figure 2 shows this tuner, designed and constructed at Purdue University. The resonant cavity positions are adjusted in real-time to re-optimize the tuner for power-added efficiency (PAE) and spectral performance when the circuit is changed to a new operating frequency [5].
Cognitive radar spectrum sharing scenario with an RF interferer (RFI). Radar systems must detect and avoid collisions with other wireless interferers in crowded frequency bands, reprinted from [2].
90 W EVA tuner constructed at purdue university [4]
The scope of this paper summarizes initial experiments on the use of the cavity tuner for fast tuning. The lessons learned from these experiments will be integrated into an approach that allows fast tuning in the cognitive radar spectrum sharing environment described in Fig. 1.
Upon a reconfiguration of center frequency or bandwidth, the tuner must adjust its circuitry quickly to re-optimize PAE while meeting spectral requirements. The first research issue to be addressed is the ability to quickly optimize the tuner based on measurement results. A modified gradient search algorithm is used to perform a constrained optimization of PAE, constrained by compliance with the spectral mask [5], [6]. Spectral mask compliance is described by the metric
Sm=max(s−m),(1)
View Source \begin{equation*} S_{m}= \max(s-m),\tag{1} \end{equation*} which gives the maximum difference between the spectrum and the mask over all frequencies measured [2], [6]. If a value of Sm≤0
S_{m}\leq 0
is obtained, then the spectrum is in compliance with the mask, while Sm>0
S_{m} > 0
indicates that the spectrum is out of mask compliance.
For future visits to a specific proposed operating frequency, a look-up table can be used to speed the search. This has been examined recently [2], where data is shown the for returning to a frequency. The experiment involved re-optimization at random seeded frequencies. The band used for experimentation was the S-band range of 3250 MHz to 3350 MHz. In the experiment, one of ten possible operating frequencies in the band was chosen. The first time the load reflection coefficient, ΓL
\Gamma_{L}
, provided by the matching circuit is optimized at a given frequency, a look-up table was used to store the best performing ΓL
\Gamma_{L}
. During any return to this frequency, the look-up table value was used as the starting point for the modified gradient-based search. The use of the look-up table provided a decrease in optimization measurements in many cases [2].
Figure 3 shows the measurement setup to assess this algorithm for the first-generation tuner of Semnani [4]. A Microwave Technologies MWT-173 field-effect transistor (FET) was used as the device under test with the EVA tuner. We have shown that this tuner can be reconfigured to three different frequencies in the S-band in the following order: 3.3 GHz, 3.1 GHz, and 3.5 GHz; and that measurement results are highly consistent with full load-pull measurements at each frequency [5].
Figure 4 shows an example of the optimization search performed at each frequency in terms of the resonant cavity position numbers n1
n_{1}
n_{2}
. The search progresses through the (n1,n2
n_{1}, n_{2}
) plane in effort to fined the constrained optimum which is defined asthe (n1,n2
n_{1}, n_{2}
) combination providing the maximum efficiency while remaining in the region of spectral mask compliance. In the Fig. 4 comparison with the PAE contours taken from a full load-pull measurement of the (n1,n2
n_{1}, n_{2}
) plane, the endpoint of the fast search is reasonably close to the constrained optimum. Typically the constrained optimum is not the global optimum if maximization of PAE causes the spectrum to violate the mask.
Fast load optimization search results in the (n1,n2
n_{1}, n_{2}
) plane at 3.3 GHz along with traditional load-pull PAE contours. The colored scale on the right shows the PAE in percent.
The implementation of a fast tuning algorithm on the 90 W impedance tuner has demonstrated the ability of a high-power matching network to re-tune over a large part of the S-band. The algorithms have demonstrated that the optimum can be obtained with a small number of measurements. This solves a significant problem on the path toward a reconfigurable, frequency-agile radar. However, other challenges still exist. Some of the present areas of research include the following: (1) streamlining of system timing, (2) operating the circuit search algorithm under control of a spectral prediction and decision-making algorithm, and (3) analyzing the effect of impedance tuning on radar range and Doppler detection.
Streamlining of system timing can be accomplished in a software-defined radio (SDR) controlled setup by analyzing communications and measurement times by each of the equipment pieces. Centralizing the control of the system to the SDR platform eliminates complications from the General Purpose-Interface Bus connections used in the measurement system. Also inclusion of the next-generation tuner in the measurements is expected to decrease the tuning time.
Using a spectral prediction and decision-making algorithm, such as that developed by Selvi [3], to control the tuning of the circuit may allow for advance planning of the impedance tuning. If prior knowledge of previous impedance tuning at different frequency and bandwidth settings is available, then an advance schedule of the impedance tuning frequencies can be created, and impedance tuning can be planned to provide best overall performance in a given part of the schedule. This will be useful because the time to search and optimize an impedance using the mechanical EVA tuners is expected to be significantly longer than the coherent processing interval of the radar. As such, it may be necessary to accept performance that is less than optimal in some situations to ensure the efficiency and output power are acceptable and reasonable at as many of the frequency and bandwidth settings as possible.
Fast impedance tuning is critical for real-time frequency agility, allowing radar transmitters to adaptively avoid RFI from other wireless spectrum users. This work has overviewed fast impedance tuning within the S-band, including the use of a 90 W EVA tuner for fast optimization of PAE of the transmitter amplifier under spectral constraints. These algorithms will be useful in next-generation radar transmitters. A next step, using an SDR platform to control the tuner, is expected to result in faster performance of measurements and communication of measurement results, reducing the “overhead” of the system required for the use of multiple pieces of measurement equipment. Further improvements are expected from using second-generation tuner technology, which takes advantage of faster actuators and miniaturized control. Future development is also needed to integrate the impedance tuning into a decision-making process based on spectral prediction algorithms. In such scenarios, the timing of the tuning will become a limiting factor of the system reconfiguration, and it will be important to determine (1) the effect of the tuner on accurate detection of range and Doppler and (2) potential spectrum occupancy decision influence based on impedance tuning timing and limitations.
The authors wish to thank John Clark of the Army Research Laboratory for his helpful suggestions and comments in the development of this paper.
< Previous | Back to Results | Next >
Optimization and decision-making in electrical distribution networks
Fast Frequency-Agile Real-Time Optimization of High-Power Tuning Network for Cognitive Radar Applications
<",c,' onload="var d=',n,";d.getElementsByTagName('head')[0].",d,"(d.",g,"('script')).",i,"='",a.l,"'\">"].join("")}var c="body",e=h[c];if(!e)return setTimeout(q,100);a.P(1);var d="appendChild",g="createElement",i="src",k=h[g]("div"),l=k[d](h[g]("div")),f=h[g]("iframe"),n="document",p;k.style.display="none";e.insertBefore(k,e.firstChild).id=o+"-"+j;f.frameBorder="0";f.id=o+"-frame-"+j;/MSIE[ ]+6/.test(navigator.userAgent)&&(f[i]="javascript:false");f.allowTransparency="true";l[d](f);try{f.contentWindow[n].open()}catch(s){a.domain=h.domain,p="javascript:var d="+n+".open();d.domain='"+h.domain+"';",f[i]=p+"void(0);"}try{var r=f.contentWindow[n];r.write(b());r.close()}catch(t){f[i]=p+'d.write("'+b().replace(/"/g,String.fromCharCode(92)+'"')+'");d.close();'}a.P(2)};a.l&&setTimeout(q,0)})()}();c[b].lv="1";return c[b]}var o="lightningjs",k=window[o]=g(o);k.require=g;k.modules=c}({}); window.usabilla_live = lightningjs.require("usabilla_live", "//w.usabilla.com/8d05fb3d7436.js"); window.usabilla_live("setButtonZIndex", "99999"); /*]]>{/literal}*/
|
24 | Garbage Collection | Peter Murphy
24 | Garbage Collection
This post walks through an implementation of a mark and sweep garbage collector using python. This post is mainly for my own edification, but when I encountered the problem in one of my classes, I found it a challenging and satisfying graph problem.
For this project, I'm just going to use Python.
On a heap, there are
n
objects numbered
0, ..., n - 1
with sizes
s_0, ..., s_{n-1}
Also given are
roots and
m
pointers, or references stored in objects that refer to other objects.
Write a program that performs a mark-and-sweep collection and outputs the total size of the live heap as well as the total amount of memory which the collector would sweep if the heap were garbage collected.
Report the retained heap size (the additional number of bytes that could be freed if this root were removed from the graph and the collection were repeated) for each of the roots.
The first line of the input contains 3 non-negative integers
n, m, r
r \leq n
n
s_i
that denote the size
s_i
of object
i
Following that are
m
lines with tuples
(i,j)
0 \leq i,j \leq n
, each of which denotes a pair of object indices. A tuple
(i.j)
means that object
i
stores a reference to object
j
, keeping it "alive" privded that
s_i
is reachable from a root.
The decription of the references is followed by a single line with
r
integers denoting the roots of the reachability graph
R_0, R_1, ..., R_{r-1}
which do not have incoming edges that point to them.
R+1
lines, the first of which should output two numbers
l,s
l
represents the total size of the live heap, and
s
the amount of garbage the would be swept if the heap were collected. Following that, for each root, output how much additional memory could be freed if this root were removed, in the order in which the roots appear in the input.
Consider the following input and accompanying representation:
The program should produce the following output
The strategy we're going to employ uses a Breadth First Seach beginning at each of the roots, marking nodes as reachable from the root, then perform one sweep to determine how many bytes would be collected (objects not referenced by any nodes referenced by a root), and then a secondary sweep to determine how many additional bytes would be collected by removing the roots.
The first step is to parse the input and build our graph.
lines = sys.stdin.buffer.read().decode('utf-8').split('\n')
# get n, m, r
args = lines[0].split(' ')
n, m, r = int(args[0]), int(args[1]), int(args[2])
roots = [int(r) for r in lines[-2].split(' ')]
# get the n sizes s_i
sizes = [int(s) for s in lines[1].split(' ')]
We'll represent out graph as a perverse adjacency matrix: a list of dictionaries representing objects on the heap with fields for the object's size, the other objects it references, and whether or not is has been marked by our BFS:
graph = [{'size': s, 'refs': [], 'marked': False} for s in sizes]
obj = lines[i].split(' ')
i, j = int(obj[0]), int(obj[1])
graph[j]['marked'] = False
graph[i]['refs'].append(j)
Next, we'll need to implement a simple BFS algorithm which marks nodes that have been visible:
def mark(node, graph):
# if the object is already referenced (it is bc we're calling it from somewhere)
if not node['marked']:
node['marked'] = True # mark it
# mark all its references too
for ref in node['refs']:
mark(graph[ref], graph)
Next we ma out the reachability graph from each of the roots:
# mark all the root connections
mark(graph[r], graph)
Now, we perform the first sweep to see count and ouput how many bytes would be collected:
# sweep 1
for i, node in enumerate(graph):
if node['marked']:
node['marked'] = False # reset the reachability graph inline
s += node['size']
l = sum(sizes) - s
print(l, s)
Right after that, we can perform the second sweep to account for how many bytes would be reclaimed if each root were removed.
# mark all the other nodes
for other in roots:
if other != r:
mark(graph[other], graph)
# tally the marked sizes, as they would be unreachable
# non roots w/ no incoming edges still get added here
s_i += node['size']
print(s_i - s) # diff of garbage collected
though the graph representation is a bit ugly, it's easier to comprehend and satisfactorily achieves the desired output!
|
Jamilla collected data comparing the weight and cost of pieces of sterling silver jewelry. Her data is listed as (weight in ounces, cost in dollars):
(5,44.00),(8.5,78.50),(12,112.00),(10,93.00),(7,63.50),(9,83.20)
Plot the data on a set of axes.
Use a ruler to draw a line that best approximates the data.
Determine the equation of the line of best fit drawn in (b).
Use your equation to predict the cost of a 50-ounce silver bracelet.
Use eTool below to solve the parts of this problem.
|
Some Thoughts upon Long-Range Interaction and Entangled States
10.4236/jemaa.2018.1012015
Some Thoughts upon Long-Range Interaction and Entangled States
The problems of long-range interaction and associated questions on entangled states are reconsidered in terms of a recently developed revised quantum electrodynamic theory by the author, as being applied to subatomic systems. There are indications that the theories of relativity and quantum mechanics do not necessarily have to be in conflict. But more investigations are required for a full understanding to be obtained on these problems.
Revised Quantum Electrodynamics, Standard Model and Beyond, Zero Point Energy, Long-Range Interaction, Entangled States
One of the remaining problems of fundamental theoretical physics is the apparent contradiction between the limited characteristic velocity of special relativity and the instantaneous behaviour of entangled states through long-range interaction in quantum mechanics. In the present investigation, an attempt is made to dissolve this contradiction, in terms of an earlier revised quantum electrodynamic theory (RQED) by the author [1] [2] , being applied to subatomic systems, such as the interior of elementary particles. Thereby, the vacuum does not become a state of empty space, but is populated by Zero Point Energy modes which also can become electrically polarized to generate electric charges and currents.
In Section 2, a summary is given on resulting modes of the revised theory. This is followed in Section 3 by considerations on long-range interaction between spatially limited particle configurations which may take place in terms of these modes.
2. Modes of the Revised Theory
Depending on the concepts
\text{div}E
\text{curl}\text{ }\text{}E
of the electric field strength
E
, there are three types of modes in RQED theory [1] [2] .
2.1. The EM Mode
\text{div}E=0
\text{curl}\text{ }\text{}E\ne 0
a conventional electromagnetic EM mode arises which satisfies special relativity. It concerns a vacuum state of empty space, with no sources of electric charges and currents. This mode is transverse and does not possess any angular momentum (spin), as shown in Chapter 9 of Reference [1] . There is no steady particle state.
2.2. The EMS Mode
\text{div}E\ne 0
due to arguments based on Zero Point Energy in the vacuum state, and when
\text{curl}\text{ }\text{}E\ne 0
simultaneously, there is an EMS mode. It is of nontransverse and electromagnetic space-charge character, and also becomes consistent with special relativity. In axisymmetric geometry the wave velocity has two components, one along the direction of propagation and one circulating in the transverse direction. The latter gives rise to a nonzero spin, also in a steady state, as determined by the electrically polarized Zero Point Energy populating the vacuum state. Both as a wave and as a steady particle model the EMS mode includes a longitudinal electric field component, i.e. along the axis of symmetry.
The equations of the relativistic EM and EMS modes include the dielectric constant
{\epsilon }_{0}
, the magnetic permeability
{\mu }_{0}
, and the limiting velocity
c={\left(1/{\epsilon }_{0}{\mu }_{0}\right)}^{1/2}
. The two velocity components of the EMS mode are then defined by a velocity vector
C=c\left(0,\mathrm{cos}\theta ,\mathrm{sin}\theta \right)
in a cylindrical frame. For the electric field we have
\left(\frac{{\partial }^{2}}{\partial {t}^{2}}-{c}^{2}{\nabla }^{2}\right)E+\left({c}^{2}\nabla +C\frac{\partial }{\partial t}\right)\left(\text{div}E\right)=0
where the first term is related to
\text{curl}\text{ }\text{}E
2.3. The S Mode
\text{div}E\ne 0
\text{div}E=0
there remains an electric longitudinal space-charge mode. This mode is produced by the Zero Point Energy in vacuo, and has no relation to the theory of relativity. In the field equations all terms with the magnetic field
B
{\mu }_{0}
vanish. Only an electric field
E=-\nabla \varphi
remains. Equation (1) then reduces to
\left({c}^{2}\nabla +C\frac{\partial }{\partial t}\right)\left(\text{div}E\right)=0
C=\left({C}_{r},0,{C}_{z}\right)
in a cylindrical frame. For this mode the components
{C}_{r}
{C}_{z}
can be arbitrarily chosen. This has nothing to do with special relativity and the velocity
c
. In other words, the EM and EMS modes can remain relativistic, independently of the S mode.
With a normal mode representation where
\partial /\partial t=i\omega
\partial /\partial z=ik
Equation (2) yields
\left(-\frac{{\partial }^{2}\varphi }{\partial {r}^{2}}-\frac{1}{r}\frac{\partial \varphi }{\partial r}+{k}^{2}\varphi \right)\left({C}_{r},{C}_{z}\right)=\omega \left(i\frac{\partial \varphi }{\partial r},-k\varphi \right)
Here, it is seen that a given potential
\varphi
cannot satisfy both these equations simultaneously. An S mode does therefore not exist in this general case. But for a flat geometry where
\varphi
r
, a solution may still exist, as given by
{C}_{r}=0
k{C}_{z}=-\omega
We are then free to choose
{C}_{z}<0
|{C}_{z}|/c\gg 1
in a superluminal range. For a finite frequency
\omega
, the wave number
k
then becomes very small and the corresponding axial length quite large.
3. Long-Range Interaction
Almost hundred years ago a discussion started on the concept of local realism, long-range interaction, entangled states, hidden variables and Bell’s inequality, in relation to the theories of relativity and quantum mechanics. It included the long-distance interaction between two particle systems and the influence on their intrinsic properties, such as the spin. The associated problems will here be reconsidered in terms of the modes defined in Section 2.
3.1. Advantageous Points
Two spatially limited particle models based on the EMS mode can in principle become linked at a long mutual distance by an S mode which is not relativistically limited in its speed of transmission:
- The EMS mode remains relativistic also when the S mode satisfies the quantum mechanical conditions of superluminal long-range interaction.
- The EMS mode possesses a nontransverse (longitudinal) electric field component which could make possible a matching to the S mode.
- All field strengths
E
and the magnetic one
B
of the EMS mode are linear functions of a common generating function and its derivatives [1] . A change of sign of the amplitude of the same function therefore changes the sign of all field components, and from symmetry arguments also that of the spin. This would become consistent with the behaviour of entangled states during long-range interaction.
3.2. Points Requiring Clarification
There are also points which require further clarification to justify a formally rigorous model:
- The geometrical matching between the spatially limited particle geometries with a long-range linking mechanism of large transverse extension is a difficult formal task. This also concerns earlier discussions by other scientists.
- The existence of a flat S mode configuration may also be questioned, in particular concerning its large transverse spatial extension as compared to the cross-sections of the EMS particle models.
A recently developed revised quantum electrodynamic theory has added some new ideas to the commonly discussed problems of long-range interaction and entangled states. These ideas indicate that the theories of relativity and quantum mechanics do not necessarily have to get into conflict with each other. The same ideas are consistent with causality, and are not related to events taking place backwards in time. But more investigations are required for a rigorous and detailed analysis to be completed. The present investigation has only to be taken as a proposal on this matter.
Lehnert, B. (2018) Some Thoughts upon Long-Range Interaction and Entangled States. Journal of Electromagnetic Analysis and Applications, 10, 193-196. https://doi.org/10.4236/jemaa.2018.1012015
1. Lehnert, B. (2014) Revised Quantum Electrodynamics. In: Dvoeglazov, V.V., Ed., Contemporary Fundamental Physics, Chs 5, 6, 9, 11.1, 11.2, Nova Science Publishers, New York.
2. Lehnert, B. (2018) Impacts of Revised Quantum Electrodynamics on Fundamental Physics. Journal of Electromagnetic Analysis and Applications, 10, 106-118. https://doi.org/10.4236/jemaa.2018.105008
|
Hermann_Minkowski Knowpia
Hermann Minkowski (/mɪŋˈkɔːfski, -ˈkɒf-/;[2] German: [mɪŋˈkɔfski]; 22 June 1864 – 12 January 1909) was a German mathematician and professor at Königsberg, Zürich and Göttingen. In different sources Minkowski's nationality is variously given as German,[3][4][5] Polish,[6][7][8] or Lithuanian-German,[9] or Russian.[1] He created and developed the geometry of numbers and used geometrical methods to solve problems in number theory, mathematical physics, and the theory of relativity.
Aleksotas, Suwałki Governorate, Kingdom of Poland (now in Kaunas, Lithuania)
Russian Empire[1] or Germany
Lily (1898–1983),
Minkowski is perhaps best known for his work in relativity, in which he showed in 1907 that his former student Albert Einstein's special theory of relativity (1905) could be understood geometrically as a theory of four-dimensional space–time, since known as the "Minkowski spacetime".
Hermann Minkowski was born in the town of Aleksota, the Suwałki Governorate, the Kingdom of Poland, part of the Russian Empire, to Lewin Boruch Minkowski, a merchant who subsidized the building of the choral synagogue in Kovno,[10][11][12] and Rachel Taubmann, both of Jewish descent.[13] Hermann was a younger brother of the medical researcher Oskar (born 1858).[14]
To escape persecution in the Russian Empire the family moved to Königsberg in 1872,[15] where the father became involved in rag export and later in manufacture of mechanical clockwork tin toys (he operated his firm Lewin Minkowski & Son with his eldest son Max).[16]
Minkowski studied in Königsberg and taught in Bonn (1887–1894), Königsberg (1894–1896) and Zurich (1896–1902), and finally in Göttingen from 1902 until his death in 1909. He married Auguste Adler in 1897 with whom he had two daughters; the electrical engineer and inventor Reinhold Rudenberg was his son-in-law.
Minkowski died suddenly of appendicitis in Göttingen on 12 January 1909. David Hilbert's obituary of Minkowski illustrates the deep friendship between the two mathematicians (translated):
Since my student years Minkowski was my best, most dependable friend who supported me with all the depth and loyalty that was so characteristic of him. Our science, which we loved above all else, brought us together; it seemed to us a garden full of flowers. In it, we enjoyed looking for hidden pathways and discovered many a new perspective that appealed to our sense of beauty, and when one of us showed it to the other and we marveled over it together, our joy was complete. He was for me a rare gift from heaven and I must be grateful to have possessed that gift for so long. Now death has suddenly torn him from our midst. However, what death cannot take away is his noble image in our hearts and the knowledge that his spirit continues to be active in us.
Max Born delivered the obituary on behalf of the mathematics students at Göttingen.[17]
The main-belt asteroid 12493 Minkowski and M-matrices are named in Minkowski's honor.[18]
Minkowski in 1883, at the time of being awarded the Mathematics Prize of the French Academy of Sciences
Minkowski was educated in East Prussia at the Albertina University of Königsberg, where he earned his doctorate in 1885 under the direction of Ferdinand von Lindemann. In 1883, while still a student at Königsberg, he was awarded the Mathematics Prize of the French Academy of Sciences for his manuscript on the theory of quadratic forms. He also became a friend of another renowned mathematician, David Hilbert. His brother, Oskar Minkowski (1858–1931), was a well-known physician and researcher.[15]
Minkowski taught at the universities of Bonn, Königsberg, Zürich, and Göttingen. At the Eidgenössische Polytechnikum, today the ETH Zurich, he was one of Einstein's teachers.
Minkowski explored the arithmetic of quadratic forms, especially concerning n variables, and his research into that topic led him to consider certain geometric properties in a space of n dimensions. In 1896, he presented his geometry of numbers, a geometrical method that solved problems in number theory. He is also the creator of the Minkowski Sausage and the Minkowski cover of a curve.[19][failed verification]
In 1902, he joined the Mathematics Department of Göttingen and became a close colleague of David Hilbert, whom he first met at university in Königsberg. Constantin Carathéodory was one of his students there.
Work on relativityEdit
By 1908 Minkowski realized that the special theory of relativity, introduced by his former student Albert Einstein in 1905 and based on the previous work of Lorentz and Poincaré, could best be understood in a four-dimensional space, since known as the "Minkowski spacetime", in which time and space are not separated entities but intermingled in a four-dimensional space–time, and in which the Lorentz geometry of special relativity can be effectively represented using the invariant interval
{\displaystyle x^{2}+y^{2}+z^{2}-c^{2}t^{2}}
(see History of special relativity).
The mathematical basis of Minkowski space can also be found in the hyperboloid model of hyperbolic space already known in the 19th century, because isometries (or motions) in hyperbolic space can be related to Lorentz transformations, which included contributions of Wilhelm Killing (1880, 1885), Henri Poincaré (1881), Homersham Cox (1881), Alexander Macfarlane (1894) and others (see History of Lorentz transformations).
The beginning part of his address called "Space and Time" delivered at the 80th Assembly of German Natural Scientists and Physicians (21 September 1908) is now famous:
Minkowski, Hermann (1915) [1907]. "Das Relativitätsprinzip" . Annalen der Physik. 352 (15): 927–938. Bibcode:1915AnP...352..927M. doi:10.1002/andp.19153521505.
Minkowski, Hermann (1908). "Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern" . Nachrichten der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse: 53–111.
Minkowski, Hermann (1909). "Raum und Zeit" . Jahresbericht der Deutschen Mathematiker-Vereinigung. 18: 75–88. Bibcode:1909JDMaV..18...75M.
Space and Time – Minkowski's Papers on Relativity, Minkowski Institute Press, 2012 ISBN 978-0-9879871-3-6 (free ebook).
Minkowski, Hermann (1907). Diophantische Approximationen: Eine Einführung in die Zahlentheorie. Leipzig-Berlin: R. G. Teubner. Retrieved 28 February 2016. [20]
Mathematical (posthumous)
Minkowski, Hermann (1910). "Geometrie der Zahlen". Leipzig-Berlin: B. G. Teubner Verlag. MR 0249269. Retrieved 28 February 2016. {{cite journal}}: Cite journal requires |journal= (help)[21]
Minkowski, Hermann (1911). Gesammelte Abhandlungen 2 vols. Leipzig-Berlin: R. G. Teubner. Retrieved 28 February 2016. [22] Reprinted in one volume New York, Chelsea 1967.
List of things named after Hermann Minkowski
Brunn–Minkowski theorem
Hermite–Minkowski theorem
Minkowski (crater)
Minkowski problem for polytopes
Minkowski's bound
Minkowski's theorem in geometry of numbers
Minkowski–Hlawka theorem
^ a b Encyclopedia of Earth and Physical Sciences. New York: Marshall Cavendish. 1998. p. 1203. ISBN 9780761405511.
^ "Minkowski". Random House Webster's Unabridged Dictionary.
^ "Hermann Minkowski German mathematician". Encyclopædia Britannica. Retrieved 6 January 2021.
^ Gregersen, Erik, ed. (2010). The Britannica Guide to Relativity and Quantum Mechanics (1st ed.). New York: Britannica Educational Pub. Association with Rosen Educational Services. p. 201. ISBN 978-1-61530-383-0.
^ Bracher, Katherine; et al., eds. (2007). Biographical Encyclopedia of Astronomers (Online ed.). New York: Springer. p. 787. ISBN 978-0-387-30400-7.
^ Hayles, N. Katherine (1984). The Cosmic Web: Scientific Field Models and Literary Strategies in the Twentieth Century. Cornell University Press. p. 46. ISBN 978-0-8014-1742-9.
^ Falconer, K. J. (2013). Fractals: A Very Short Introduction. Oxford University Press. p. 119. ISBN 978-0-19-967598-2.
^ Bardon, Adrian (2013). A Brief History of the Philosophy of Time. Oxford University Press. p. 68. ISBN 978-0-19-930108-9.
^ Safra, Jacob E.; Yeshua, Ilan (2003). Encyclopædia Britannica (New ed.). Chicago, Ill.: Encyclopædia Britannica. p. 665. ISBN 978-0-85229-961-6.
^ А. И. Хаеш (1873). "Коробочное делопроизводство как источник сведений о жизни еврейских обществ и их персональном составе" (in Russian). ...купец Левин Минковский подарил молитвенному обществу при Ковенском казённом еврейском училище начатую им... постройкой молитвенную школу вместе с плацем, с тем, чтобы общество это озаботилась окончанием таковой постройки. Общество, располагая средствами добровольных пожертвований, возвело уже это здание под крышу, но затем средства сии истощились...
^ "Kaunas: dates and facts. Electronic directory".
^ "Box-Tax Paperwork Records". Archived from the original on 8 January 2015. Kovno. In 1873 the merchant (kupez), Levin Minkovsky, gave (as a gift) to the prayer association of the Kovno state Jewish school a lot with an ongoing construction of a prayer school that (the construction) he had started so that the association would take care of completing the construction. The association, having some funds from voluntary contributions, had built the structure up to the roof, but then, ran out of money {{cite web}}: External link in |quote= (help)
^ "Minkowski biography".
^ Oskar Minkowski (1858–1931). Archived 29 December 2013 at the Wayback Machine. The Jewish genealogy site JewishGen.org (Lithuania database, registration required) contains the birth record in the Kovno rabbinical books of Hermann's younger brother Tuvia in 1868 to Boruch Yakovlevich Minkovsky and his wife Rakhil Isaakovna Taubman.
^ a b "Historical note: Oskar Minkowski (1858–1931). An outstanding master of diabetes research". 2006.
^ Report of the Federal Security Agency (p. 183); Tyra lithographed tin toy dog; Rudolph Leo Bernhard Minkowski: A Biographical Memoir.
^ Greenspan, Nancy Thorndike (2005). The End of the Certain World. The Life and Science of Max Born: The Nobel Physicist Who Ignited the Quantum Revolution. Basic Books. pp. 42–43. ISBN 9780738206936.
^ Schmadel, Lutz D. (2007). "(12493) Minkowski". Dictionary of Minor Planet Names – (12493) Minkowski. Springer Berlin Heidelberg. p. 783. doi:10.1007/978-3-540-29925-7_8614. ISBN 978-3-540-00238-3.
^ "Minkowski Sausage", WolframAlpha
^ Dickson, L. E. (1909). "Review: Diophantische Approximationen. Eine Einführung in die Zahlentheorie von Hermann Minkowski" (PDF). Bull. Amer. Math. Soc. 15 (5): 251–252. doi:10.1090/s0002-9904-1909-01753-7.
^ Dickson, L. E. (1914). "Review: Geometrie der Zahlen von Hermann Minkowski". Bull. Amer. Math. Soc. 21 (3): 131–132. doi:10.1090/s0002-9904-1914-02597-2.
^ Wilson, E. B. (1915). "Review: Gesammelte Abhandlungen von Hermann Minkowski". Bull. Amer. Math. Soc. 21 (8): 409–412. doi:10.1090/s0002-9904-1915-02658-3.
Wikiquote has quotations related to Hermann Minkowski.
Wikimedia Commons has media related to Hermann Minkowski.
|
The K-factor and the bending process
Using the K-factor calculator
This K-factor calculator will assist you in finding the K-factor for sheet metal. Sheet metal is the building block of structures, from forming the body of automobiles to the skin of aircraft wings. The roofs on the house or the geyser that supplies water for your warm showers and baths all use sheet metal for the fabrication process. Other applications of sheet metal include piping, medical equipment, machine components, and transformers used in power transmission.
First, the metal is worked into thin sheets for the uses mentioned above and more. It could be any metal as per the demand and loads, but the most commonly manufactured sheets are aluminum, brass, copper, and steel sheets. These sheets undergo different fabrication processes like bending, forming, and punching and are also joined together using welding.
The focus of this article is the K-factor associated with the sheet metal bending process. K-factor deals with the position of the neutral axis, and in this article, you'll learn how to calculate the K-factor.
Before we get into the K-factor, let's look at the bending process for a sheet. You can perform bending using a press brake machine or a bending machine. A press brake consists of a punch and die: the punch presses the sheet down into the die to bend the sheet. The punch and die must be compatible to ensure accuracy and safety.
Let's consider the cross-section of a sheet: the sheet's neutral axis is the line that passes through the points where the stresses and strains are exactly zero. As the punch presses the sheet onto the die, the sheet begins to bend, and the neutral axis shifts.
But why are we concerned about the neutral axis? The neutral axis is the line that divides the cross-section into two regions. The cross-sectional area above the neutral axis (between the inner surface of bending and the neutral axis) experiences compression stresses. In contrast, the region below the neutral axis (between the outer surface and the neutral axis) undergoes tension. The length of the neutral axis does not change upon bending. Instead, it shifts along the thickness direction of the material.
A piece of sheet metal being bent. Above the neutral axis (the line where the total stress is 0), the metal is compressed. Below the axis, the metal experiences tension.
The K-factor is the parameter that tells us about the location of the neutral axis. In other words, the K-factor is the ratio of the location of the neutral axis and the material thickness. Mathematically,
\footnotesize K = \frac{180^\circ \cdot \text{BA} }{\pi \cdot \theta \cdot T} - \frac{R_i}{T}
K
is the K-factor;
\theta
is the bend angle;
\text{BA}
is the bend allowance;
T
is the material thickness; and
R_i
is the inner radius.
The neutral axis then lies from the inner surface a distance of the K-factor times the sheet thickness. We denote this distance as
t = K\cdot T
💡 If you want to know more about bend allowance, you can head out to our bend allowance calculator.
Let's calculate the K-factor for a metal sheet having a thickness of
1 \text{ mm}
and bent to an angle of
90^\circ
. Take the bend allowance as
2.1 \text{ mm}
and bend radius as
1 \text{ mm}
To calculate K-factor:
Enter the material thickness,
T = 1 \text{ mm}
Insert the inner radius,
R_i =1 \text{ mm}
Fill in the angle of bending,
\theta = 90^\circ
Punch in the bend allowance,
\text{BA} = 2.1 \text{ mm}
Using the K-factor calculator:
\footnotesize \qquad \begin{align*} K &= \frac{180 \cdot \text{BA} }{\pi \cdot \theta \cdot T} - \frac{R_i}{T} \\[1em] &= \frac{180 \times 2.1 }{\pi \times 90 \times 1} - \frac{1}{1} \\[1em] &= 0.337 \end{align*}
And so we've found that the neutral axis lies at a distance
0.337
times the material thickness from the inner surface, i.e. at a distance of
\footnotesize \begin{align*} t &= K\cdot T \\ &= 0.337\cdot 1\text{ mm} \\ &= 0.337\text{ mm} \end{align*}
What do you mean by K-factor?
The K-factor is the ratio of the location of the neutral axis to the material thickness for sheet metal. The neutral axis divides the cross-section into two regions — compressive and tension. Its position shifts when the sheet undergoes the bending process.
How do I calculate K-factor?
To calculate the K-factor:
Multiply the bending allowance by 180.
Divide the resultant by pi and the bend angle.
Subtract the inner radius from the resulting numeral.
Divide the result by the material thickness to obtain the K-factor.
Mathematically, K = (180×BA) / (π×θ×T) − (Rᵢ/T).
What are the variables that affect K-factor?
The variables that affect K-factor are material type and thickness, type of bending, bending angle, and inner radius. It is also affected by the type of tool used to bend the sheet metal and material properties like yield and tensile strength. The K-factor commonly lies between 0.3 and 0.5.
What is the K-factor for a sheetmetal with bend allowance of 15 mm bent at 60° having thickness and bend radius as 10 mm?
The K-factor for this configuration is 0.432. Consider a bend radius of 10 mm when you bend a 10 mm thick sheet to a 60° bend. Mathematically,
K = (180×BA) / (π×θ×T) − (Rᵢ/T)
K = (180×15) / (π×60×10) − (10/10)
Inside radius (Rᵢ)
Bend angle (θ)
Location of neutral axis (t)
|
Calculation of normal probability for sampling distributions
How this sampling distribution calculator works: an example
How to find the mean of the sampling distribution?
This normal probability calculator for sampling distributions finds the probability that your sample mean lies within a specific range.
It calculates the normal distribution probability with the sample size (n), a mean values range (defined by X₁ and X₂), the population mean (μ), and the standard deviation (σ).
What is the sampling distribution of the mean?
How to find the standard deviation of the sampling distribution.
How to calculate probabilities for sampling distributions.
How to use our normal probability calculator for sampling distributions.
🔎 If you need a calculator that makes the same, but for sample proportions, check our sampling distribution of the sample proportion calculator. If you're interested in the opposite problem: finding a range of possible population values given a probability level, take a look at our sampling error calculator.
Many real-life phenomena follow a normal distribution. For example, the American men's height follows that distribution with a mean of approximately 176.3 cm and a standard deviation of about 7.6 cm. In the following plot, you can see the distribution graph of those heights.
Distribution of American men's heights. You can note that Barack Obama's height is slightly above the mean. At the same time, Danny Devito and Shaquille O'neal are at the very extremes of the percentile, as they're extremely short and high, respectively. Source: Introductory Statistics Explained Edition 1.10CC, by Jeremy Balka
Usually, we use samples to estimate population parameters like a population mean height. The most common example is using the sample mean to estimate the population mean.
If you take different samples from a population, you'll probably get different mean values each time. Therefore, the sample mean is also a random variable that we can describe with some distribution. This distribution is called the sampling distribution of the sample mean, which we will name the sampling distribution for simplicity.
If the original population follows a normal distribution, the sampling distribution will do the same, and if not, the sampling distribution will approximate a normal distribution. The central limit theorem describes the degree to which it occurs.
A common task is to find the probability that the mean of a sample falls within a specific range. We can do it using the same tools for calculating normal distributions (using the z-score). The only difference is that the standard deviation of the sampling distribution is equal to the population standard deviation divided by the square root of the sample size:
\footnotesize σ_{\bar X}=\frac{σ}{\sqrt{n}}
Then, the formula to calculate the z-score is:
\footnotesize z= \frac{X-μ}{σ/ \sqrt{n}}
With the z-score value, you can calculate the probability using available tables or, even better and faster, using our p-value calculator. Read on to look at an example of how to do it.
🙋 If you're interested in the
σ_{\bar X}
term, you can learn more about it in our standard deviation of the sample mean calculator.
The average height of the American women (including all race and Hispanic-origin groups) aged 20 and over is approximately 161.3 cm, with a standard deviation of about 7.1 cm. Let's suppose you randomly sample 7 American women. What is the probability that the average height falls below 160 cm?
To know the answer, follow these steps:
Input the population parameters in the sampling distribution calculator (μ = 161.3, σ = 7.1)
Select left-tailed, in this case.
Input the sample data (n = 7, X = 160).
Your result is ready. It should be 0.314039. Therefore, the probability that the average height of those women falls below 160 cm is about 31.4%.
Alternatively, we can calculate this probability using the z-score formula:
\footnotesize \begin{align*} z_{score}&=\frac{X-μ}{σ/\sqrt{n}}\\\\&= \frac{160-161.3}{ 7.1/ \sqrt{7}}=-0.484433 \end{align*}
\footnotesize \begin{align*} P(\bar X<170)&=P(z_{score}<−0.484433)\\&=0.314039 \end{align*}
If you know the population mean, you know the mean of the sampling distribution, as they're both the same. If you don't, you can assume your sample mean as the mean of the sampling distribution.
The sampling distribution of the mean describes the distribution of possible means you could obtain from infinitely sampling from a given population.
How to calculate probability in sampling distribution?
Define your population mean (μ), standard deviation (σ), sample size, and range of possible sample means.
Input those values in the z-score formula zscore = (X̄ - μ)/(σ/√n).
Considering if your probability is left, right, or two-tailed, use the z-score value to find your probability.
Alternatively, you can use our normal probability calculator for sampling distributions.
How to find the standard deviation of the sampling distribution?
Depending on the information you possess, there are two ways:
If you know the population standard deviation (σ), divide it by the square root of the sample size: σX̄ = σ/√n.
If you don't have σ, estimate it with the sample standard deviation (s): σX̄ = s/√n.
What is the probability of getting a sample mean greater than the population mean?
The probability of getting a sample mean greater than μ (population mean) is 50%, as long as your sampling distribution follows a normal distribution (this occurs if the population distribution is normal or the sample size is large).
X₁ < X̄ < X₂
Normal probability:
P(X₁ < X̄ < X₂)
Z-score of X₁
Z-score of X₂
Use our possible combinations calculator to know how many combinations are possible, given a total number of objects and a sample size.
|
Interface between moist air and mechanical rotational networks - MATLAB - MathWorks Australia
Rotational Mechanical Converter (MA)
Interface between moist air and mechanical rotational networks
The Rotational Mechanical Converter (MA) block models an interface between a moist air network and a mechanical rotational network. The block converts moist air pressure into mechanical torque and vice versa. You can use it as a building block for rotary actuators.
The converter contains a variable volume of moist air. The pressure and temperature evolve based on the compressibility and thermal capacity of this moist air volume. Liquid water condenses out of the moist air volume when it reaches saturation. The Mechanical orientation parameter lets you specify whether an increase in the moist air volume inside the converter results in a positive or negative rotation of port R relative to port C.
The block equations use these symbols. Subscripts a, w, and g indicate the properties of dry air, water vapor, and trace gas, respectively. Subscript ws indicates water vapor at saturation. Subscripts A, H, and S indicate the appropriate port. Subscript I indicates the properties of the internal moist air volume.
\stackrel{˙}{m}
V Volume of moist air inside the converter
The net flow rates into the moist air volume inside the converter are
\begin{array}{l}{\stackrel{˙}{m}}_{net}={\stackrel{˙}{m}}_{A}-{\stackrel{˙}{m}}_{condense}+{\stackrel{˙}{m}}_{wS}+{\stackrel{˙}{m}}_{gS}\\ {\Phi }_{net}={\Phi }_{A}+{Q}_{H}-{\Phi }_{condense}+{\Phi }_{S}\\ {\stackrel{˙}{m}}_{w,net}={\stackrel{˙}{m}}_{wA}-{\stackrel{˙}{m}}_{condense}+{\stackrel{˙}{m}}_{wS}\\ {\stackrel{˙}{m}}_{g,net}={\stackrel{˙}{m}}_{gA}+{\stackrel{˙}{m}}_{gS}\end{array}
\stackrel{˙}{m}
{\stackrel{˙}{m}}_{wS}
{\stackrel{˙}{m}}_{gS}
{\stackrel{˙}{m}}_{wS}
{\stackrel{˙}{m}}_{gS}
, and ΦS are determined by the moisture and trace gas sources connected to port S of the converter.
\frac{d{x}_{wI}}{dt}{\rho }_{I}V+{x}_{wI}{\stackrel{˙}{m}}_{net}={\stackrel{˙}{m}}_{w,net}
\frac{d{x}_{gI}}{dt}{\rho }_{I}V+{x}_{gI}{\stackrel{˙}{m}}_{net}={\stackrel{˙}{m}}_{g,net}
\left(\frac{1}{{p}_{I}}\frac{d{p}_{I}}{dt}-\frac{1}{{T}_{I}}\frac{d{T}_{I}}{dt}\right){\rho }_{I}V+\frac{{R}_{a}-{R}_{w}}{{R}_{I}}\left({\stackrel{˙}{m}}_{w,net}-{x}_{w}{\stackrel{˙}{m}}_{net}\right)+\frac{{R}_{a}-{R}_{g}}{{R}_{I}}\left({\stackrel{˙}{m}}_{g,net}-{x}_{g}{\stackrel{˙}{m}}_{net}\right)+{\rho }_{I}\stackrel{˙}{V}={\stackrel{˙}{m}}_{net}
\stackrel{˙}{V}
is the rate of change of the converter volume.
{\rho }_{I}{c}_{vI}V\frac{d{T}_{I}}{dt}+\left({u}_{wI}-{u}_{aI}\right)\left({\stackrel{˙}{m}}_{w,net}-{x}_{w}{\stackrel{˙}{m}}_{net}\right)+\left({u}_{gI}-{u}_{aI}\right)\left({\stackrel{˙}{m}}_{g,net}-{x}_{g}{\stackrel{˙}{m}}_{net}\right)+{u}_{I}{\stackrel{˙}{m}}_{net}={\Phi }_{net}-{p}_{I}\stackrel{˙}{V}
{p}_{I}={\rho }_{I}{R}_{I}{T}_{I}
{R}_{I}={x}_{aI}{R}_{a}+{x}_{wI}{R}_{w}+{x}_{gI}{R}_{g}
The converter volume depends on the rotation of the moving interface:
V={V}_{dead}+{D}_{\mathrm{int}}{\theta }_{\mathrm{int}}{\epsilon }_{\mathrm{int}}
If you connect the converter to a Multibody joint, use the physical signal input port q to specify the rotation of port R relative to port C. Otherwise, the block calculates the interface rotation from relative port angular velocities. The interface rotation is zero when the moist air volume inside the converter is equal to the dead volume. Then, depending on the Mechanical orientation parameter value:
If Pressure at A causes positive rotation of R relative to C, the interface rotation increases when the moist air volume increases from dead volume.
If Pressure at A causes negative rotation of R relative to C, the interface rotation decreases when the moist air volume increases from dead volume.
The torque balance on the mechanical interface is
{\tau }_{\mathrm{int}}=\left({p}_{env}-{p}_{I}\right){D}_{\mathrm{int}}{\epsilon }_{\mathrm{int}}
Flow resistance and thermal resistance are not modeled in the converter:
\begin{array}{l}{p}_{A}={p}_{I}\\ {T}_{H}={T}_{I}\end{array}
{x}_{wsI}={\phi }_{ws}\frac{{R}_{I}}{{R}_{w}}\frac{{p}_{wsI}}{{p}_{I}}
{\stackrel{˙}{m}}_{condense}=\left\{\begin{array}{ll}0,\hfill & \text{if }{x}_{wI}\le {x}_{wsI}\hfill \\ \frac{{x}_{wI}-{x}_{wsI}}{{\tau }_{condense}}{\rho }_{I}V,\hfill & \text{if }{x}_{wI}>{x}_{wsI}\hfill \end{array}
{\Phi }_{condense}={\stackrel{˙}{m}}_{condense}\left({h}_{wI}-\Delta {h}_{vapI}\right)
\begin{array}{l}{\phi }_{wI}=\frac{{y}_{wI}{p}_{I}}{{p}_{wsI}}\\ {y}_{wI}=\frac{{x}_{wI}{R}_{w}}{{R}_{I}}\\ {r}_{wI}=\frac{{x}_{wI}}{1-{x}_{wI}}\\ {y}_{gI}=\frac{{x}_{gI}{R}_{g}}{{R}_{I}}\\ {x}_{aI}+{x}_{wI}+{x}_{gI}=1\end{array}
Flow resistance between the converter inlet and the moist air volume is not modeled. Connect a Local Restriction (MA) block or a Flow Resistance (MA) block to port A to model pressure losses associated with the inlet.
Thermal resistance between port H and the moist air volume is not modeled. Use Thermal library blocks to model thermal resistances between the moist air mixture and the environment, including any thermal effects of a chamber wall.
The block does not model the mechanical effects of the moving interface, such as hard stops, friction, and inertia.
Physical signal output port that measures the rate of condensation in the converter.
Moist air conserving port associated with the converter inlet.
Thermal conserving port associated with the temperature of the moist air mixture inside the converter.
Select the alignment of moving interface with respect to the volume of moist air inside the converter:
Pressure at A causes positive rotation of R relative to C — Increase in the moist air volume results in a positive rotation of port R relative to port C.
Pressure at A causes negative rotation of R relative to C — Increase in the moist air volume results in a negative rotation of port R relative to port C.
Rotational offset of port R relative to port C at the start of simulation. A value of 0 corresponds to an initial moist air volume equal to Dead volume.
Interface volume displacement — Displaced moist air volume per unit rotation
Displaced moist air volume per unit rotation of the moving interface.
Dead volume — Volume of moist air when the interface rotation is 0
Volume of moist air when the interface rotation is 0.
Cross-sectional area of the converter inlet, in the direction normal to the moist air flow path.
Pressure outside the converter acting against the pressure of the converter moist air volume. A value of 0 indicates that the converter expands into vacuum.
Translational Mechanical Converter (MA) | Rotational Multibody Interface
|
Engineering Acoustics/Acoustic wave solution in Cartesian coordinates - Wikibooks, open books for an open world
Engineering Acoustics/Acoustic wave solution in Cartesian coordinates
The one dimensional acoustic wave equation is described by the following second order partial differential equation.
{\displaystyle {\frac {\partial ^{2}P}{\partial x^{2}}}={\frac {1}{C_{o}^{2}}}{\frac {\partial ^{2}P}{\partial t^{2}}}}
It can be solved using separation of variables also known as the Fourier method. Suppose the pressure is the product of one function only dependent on space and another function only dependent on time.
{\displaystyle P(x,t)=X(x)T(t)}
Substituting back into the wave equation
{\displaystyle X''T={\frac {1}{C_{o}^{2}}}XT''}
{\displaystyle {\frac {X''}{X}}={\frac {1}{C_{o}^{2}}}{\frac {T''}{T}}=-\lambda ^{2}}
This substitution leads to two homogeneous second order ordinary differential equations, one in time and one in space.
{\displaystyle T''+C_{o}^{2}\lambda ^{2}T=0}
{\displaystyle X''+\lambda ^{2}X=0}
The time function is expected to be dependent on the angular frequency of the wave.
{\displaystyle T=Ce^{j\omega t}}
Substituting and solving for the constant which is define as the wave number, K.
{\displaystyle -\omega ^{2}+C_{o}^{2}\lambda ^{2}=0}
{\displaystyle K^{2}=\lambda ^{2}={\frac {\omega ^{2}}{C_{o}^{2}}}}
The wave number is an important quantity relating the angular velocity of the wave to its propagation speed in the medium. It can be expressed in different forms.
{\displaystyle K={\frac {\omega }{C_{o}}}={\frac {2\pi f}{C_{o}}}={\frac {2\pi }{\lambda }}}
{\displaystyle f}
is the frequency in hertz and
{\displaystyle \lambda }
is the wavelength.
The second differential equation can be solved using the wave number. The spacial function is given a general form.
{\displaystyle X=Ce^{jrx}}
Substituting and solving fo{\displaystyle r}
{\displaystyle -r^{2}+K^{2}=0}
{\displaystyle r=\pm K}
The solution of the 1-D acoustic wave equation is obtained.
{\displaystyle P(x,t)=(C_{1}e^{jKx}+C_{2}e^{-jKx})e^{j\omega t}}
The real and imaginary parts of the solution are also solutions to the 1-D wave equation.
{\displaystyle P(x,t)=C_{1}cos(\omega t+Kx)+C_{2}cos(\omega t-Kx)}
{\displaystyle P(x,t)=C_{1}sin(\omega t+Kx)+C_{2}sin(\omega t-Kx)}
Using phasor notation, the solution is written in more compact form.
{\displaystyle \mathbf {P(x,t)} =\mathbf {P} e^{j(\omega t\pm Kx)}}
The actual solution is recovered by taking the real part of the above complex form. The value of the constants above is determined by applying initial and boundary conditions. In general, any function of the following form is a solution for periodic waves.
{\displaystyle P(x,t)=f_{1}(\omega t+Kx)+f_{2}(\omega t-Kx)}
and similarly, for progressive waves,
{\displaystyle P(x,t)=f(ct+x)+g(ct-x)}
{\displaystyle P(x,t)=f(\xi )+g(\eta )}
{\displaystyle f}
{\displaystyle g}
are arbitrary functions, that represent two waves traveling in opposing directions. These are known as the d'Alembert solutions. The form of these two functions can be found by applying initial and boundary conditions.
Retrieved from "https://en.wikibooks.org/w/index.php?title=Engineering_Acoustics/Acoustic_wave_solution_in_Cartesian_coordinates&oldid=3414735"
|
An EM wave has amplitude of electric field Eo and amplitude of magnetic field is Bo The electric field - Physics - Electromagnetic Waves - 8841089 | Meritnation.com
Zafar answered this
The electric and magnetic field in a plane electromagnetic wave are given by .
\mathbit{E}\mathbf{ }\mathbf{=}\mathbf{ }{\mathbit{E}}_{\mathbf{o}}\mathbf{ }\mathbf{ }\mathbit{s}\mathbit{i}\mathbit{n}\mathbf{ }\mathbit{w}\mathbf{ }\mathbf{ }\mathbf{\left(}\mathbit{t}\mathbf{ }\mathbf{-}\mathbf{ }\frac{\mathbf{x}}{\mathbf{c}}\mathbf{ }\mathbf{\right)}\mathbf{ }\mathbf{ }\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\phantom{\rule{0ex}{0ex}}\mathbit{a}\mathbit{n}\mathbit{d}\mathbf{ }\mathbf{ }\mathbit{B}\mathbf{ }\mathbf{=}\mathbf{ }{\mathbit{B}}_{\mathbf{o}}\mathbf{ }\mathbf{ }\mathbf{ }\mathbit{s}\mathbit{i}\mathbit{n}\mathbf{ }\mathbit{w}\mathbf{ }\mathbf{ }\mathbf{\left(}\mathbit{t}\mathbf{ }\mathbf{-}\mathbf{ }\frac{\mathbf{x}}{\mathbf{c}}\mathbf{ }\mathbf{\right)}\mathbf{ }\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\left(}\mathbf{2}\mathbf{\right)}
According to given problem
\mathbit{E}\mathbf{=}\mathbf{ }\frac{\mathbf{3}}{\mathbf{4}}\mathbf{ }{\mathbit{E}}_{\mathbf{o}}
\mathbf{⇒}\mathbf{ }\mathbf{ }\frac{\mathbf{3}}{\mathbf{4}}\mathbf{ }{\mathbit{E}}_{\mathbf{o}}\mathbf{ }\mathbf{ }\mathbf{=}\mathbf{ }{\mathbit{E}}_{\mathbf{o}}\mathbf{ }\mathbf{ }\mathbit{s}\mathbit{i}\mathbit{n}\mathbf{ }\mathbit{w}\mathbf{ }\mathbf{ }\mathbf{\left(}\mathbit{t}\mathbf{ }\mathbf{-}\frac{\mathbf{x}}{\mathbf{c}}\mathbf{ }\mathbf{\right)}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{\left[}\mathbf{ }\mathbf{ }\mathbf{ }\mathbit{u}\mathbit{s}\mathbit{i}\mathbit{n}\mathbit{g}\mathbf{ }\mathbit{e}\mathbit{q}\mathbf{ }\mathbf{ }\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{\right]}\phantom{\rule{0ex}{0ex}}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{⇒}\mathbf{ }\mathbf{ }\mathbit{s}\mathbit{i}\mathbit{n}\mathbf{ }\mathbit{w}\mathbf{ }\mathbf{ }\mathbf{\left(}\mathbit{t}\mathbf{ }\mathbf{-}\frac{\mathbf{x}}{\mathbf{c}}\mathbf{ }\mathbf{\right)}\mathbf{ }\mathbf{=}\mathbf{ }\frac{\mathbf{3}}{\mathbf{4}}\phantom{\rule{0ex}{0ex}}\mathbf{ }\mathbit{B}\mathbf{ }\mathbf{=}\mathbf{ }{\mathbit{B}}_{\mathbf{o}}\mathbf{ }\mathbf{×}\mathbf{ }\frac{\mathbf{3}}{\mathbf{4}}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{⇒}\mathbf{ }\mathbf{ }\mathbit{B}\mathbf{ }\mathbf{=}\mathbf{ }\frac{\mathbf{3}\mathbf{ }\mathbf{ }{\mathbf{B}}_{\mathbf{o}}\mathbf{ }}{\mathbf{4}}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{\left[}\mathbf{ }\mathbf{ }\mathbf{ }\mathbit{u}\mathbit{s}\mathbit{i}\mathbit{n}\mathbit{g}\mathbf{ }\mathbit{e}\mathbit{q}\mathbf{ }\mathbf{ }\mathbf{\left(}\mathbf{2}\mathbf{\right)}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{\right]}
\mathbf{⇒}\mathbf{ }\mathbf{ }\frac{\mathbf{3}}{\mathbf{4}}\mathbf{ }{\mathbit{E}}_{\mathbf{o}}\mathbf{ }\mathbf{ }\mathbf{=}\mathbf{ }{\mathbit{E}}_{\mathbf{o}}\mathbf{ }\mathbf{ }\mathbit{s}\mathbit{i}\mathbit{n}\mathbf{ }\mathbit{w}\mathbf{ }\mathbf{ }\mathbf{\left(}\mathbit{t}\mathbf{ }\mathbf{-}\frac{\mathbf{x}}{\mathbf{c}}\mathbf{ }\mathbf{\right)}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{\left[}\mathbf{ }\mathbf{ }\mathbf{ }\mathbit{u}\mathbit{s}\mathbit{i}\mathbit{n}\mathbit{g}\mathbf{ }\mathbit{e}\mathbit{q}\mathbf{ }\mathbf{ }\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{\right]}\phantom{\rule{0ex}{0ex}}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{⇒}\mathbf{ }\mathbf{ }\mathbit{s}\mathbit{i}\mathbit{n}\mathbf{ }\mathbit{w}\mathbf{ }\mathbf{ }\mathbf{\left(}\mathbit{t}\mathbf{ }\mathbf{-}\frac{\mathbf{x}}{\mathbf{c}}\mathbf{ }\mathbf{\right)}\mathbf{ }\mathbf{=}\mathbf{ }\frac{\mathbf{3}}{\mathbf{4}}\phantom{\rule{0ex}{0ex}}\mathbf{ }\mathbit{B}\mathbf{ }\mathbf{=}\mathbf{ }{\mathbit{B}}_{\mathbf{o}}\mathbf{ }\mathbf{×}\mathbf{ }\frac{\mathbf{3}}{\mathbf{4}}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{⇒}\mathbf{ }\mathbf{ }\mathbit{B}\mathbf{ }\mathbf{=}\mathbf{ }\frac{\mathbf{3}\mathbf{ }\mathbf{ }{\mathbf{B}}_{\mathbf{o}}\mathbf{ }}{\mathbf{4}}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{\left[}\mathbf{ }\mathbf{ }\mathbf{ }\mathbit{u}\mathbit{s}\mathbit{i}\mathbit{n}\mathbit{g}\mathbf{ }\mathbit{e}\mathbit{q}\mathbf{ }\mathbf{ }\mathbf{\left(}\mathbf{2}\mathbf{\right)}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{\right]}
Bo= Eo/c where c is spd of light .
now substituting Eo as 3/4Eo and c as 3 10^8, we solve for Bo.
hence Bo= 3Eo/4c
since c= 310^8
we get, Bo = 3Eo /410^-8
|
Section 60.10 (07IN): Sheaves on the crystalline site—The Stacks project
Section 60.10: Sheaves on the crystalline site (cite)
60.10 Sheaves on the crystalline site
Notation and assumptions as in Situation 60.7.5. In order to discuss the small and big crystalline sites of $X/S$ simultaneously in this section we let
\[ \mathcal{C} = \text{CRIS}(X/S) \quad \text{or}\quad \mathcal{C} = \text{Cris}(X/S). \]
A sheaf $\mathcal{F}$ on $\mathcal{C}$ gives rise to a restriction $\mathcal{F}_ T$ for every object $(U, T, \delta )$ of $\mathcal{C}$. Namely, $\mathcal{F}_ T$ is the Zariski sheaf on the scheme $T$ defined by the rule
\[ \mathcal{F}_ T(W) = \mathcal{F}(U \cap W, W, \delta |_ W) \]
for $W \subset T$ is open. Moreover, if $f : T \to T'$ is a morphism between objects $(U, T, \delta )$ and $(U', T', \delta ')$ of $\mathcal{C}$, then there is a canonical comparison map
\begin{equation} \label{crystalline-equation-comparison} c_ f : f^{-1}\mathcal{F}_{T'} \longrightarrow \mathcal{F}_ T. \end{equation}
Namely, if $W' \subset T'$ is open then $f$ induces a morphism
\[ f|_{f^{-1}W'} : (U \cap f^{-1}(W'), f^{-1}W', \delta |_{f^{-1}W'}) \longrightarrow (U' \cap W', W', \delta |_{W'}) \]
of $\mathcal{C}$, hence we can use the restriction mapping $(f|_{f^{-1}W'})^*$ of $\mathcal{F}$ to define a map $\mathcal{F}_{T'}(W') \to \mathcal{F}_ T(f^{-1}W')$. These maps are clearly compatible with further restriction, hence define an $f$-map from $\mathcal{F}_{T'}$ to $\mathcal{F}_ T$ (see Sheaves, Section 6.21 and especially Sheaves, Definition 6.21.7). Thus a map $c_ f$ as in (60.10.0.1). Note that if $f$ is an open immersion, then $c_ f$ is an isomorphism, because in that case $\mathcal{F}_ T$ is just the restriction of $\mathcal{F}_{T'}$ to $T$.
Conversely, given Zariski sheaves $\mathcal{F}_ T$ for every object $(U, T, \delta )$ of $\mathcal{C}$ and comparison maps $c_ f$ as above which (a) are isomorphisms for open immersions, and (b) satisfy a suitable cocycle condition, we obtain a sheaf on $\mathcal{C}$. This is proved exactly as in Topologies, Lemma 34.3.19.
The structure sheaf on $\mathcal{C}$ is the sheaf $\mathcal{O}_{X/S}$ defined by the rule
\[ \mathcal{O}_{X/S} : (U, T, \delta ) \longmapsto \Gamma (T, \mathcal{O}_ T) \]
This is a sheaf by the definition of coverings in $\mathcal{C}$. Suppose that $\mathcal{F}$ is a sheaf of $\mathcal{O}_{X/S}$-modules. In this case the comparison mappings (60.10.0.1) define a comparison map
\begin{equation} \label{crystalline-equation-comparison-modules} c_ f : f^*\mathcal{F}_{T'} \longrightarrow \mathcal{F}_ T \end{equation}
of $\mathcal{O}_ T$-modules.
Another type of example comes by starting with a sheaf $\mathcal{G}$ on $(\mathit{Sch}/X)_{Zar}$ or $X_{Zar}$ (depending on whether $\mathcal{C} = \text{CRIS}(X/S)$ or $\mathcal{C} = \text{Cris}(X/S)$). Then $\underline{\mathcal{G}}$ defined by the rule
\[ \underline{\mathcal{G}} : (U, T, \delta ) \longmapsto \mathcal{G}(U) \]
is a sheaf on $\mathcal{C}$. In particular, if we take $\mathcal{G} = \mathbf{G}_ a = \mathcal{O}_ X$, then we obtain
\[ \underline{\mathbf{G}_ a} : (U, T, \delta ) \longmapsto \Gamma (U, \mathcal{O}_ U) \]
There is a surjective map of sheaves $\mathcal{O}_{X/S} \to \underline{\mathbf{G}_ a}$ defined by the canonical maps $\Gamma (T, \mathcal{O}_ T) \to \Gamma (U, \mathcal{O}_ U)$ for objects $(U, T, \delta )$. The kernel of this map is denoted $\mathcal{J}_{X/S}$, hence a short exact sequence
\[ 0 \to \mathcal{J}_{X/S} \to \mathcal{O}_{X/S} \to \underline{\mathbf{G}_ a} \to 0 \]
Note that $\mathcal{J}_{X/S}$ comes equipped with a canonical divided power structure. After all, for each object $(U, T, \delta )$ the third component $\delta $ is a divided power structure on the kernel of $\mathcal{O}_ T \to \mathcal{O}_ U$. Hence the (big) crystalline topos is a divided power topos.
Comment #4066 by Dan Dore on March 19, 2019 at 19:50
Maybe it's a good idea to remind the reader here of Definition 7.45.1, defining global sections for a site without a terminal object, then spelling out what this means for the cristalline site?
This whole chapter needs a thorough revision. For the moment I'll only fix mathematical errors. But yes, I do agree it is a good idea and hopefully your comment will help others who visit this page. Thank you.
Comment #5434 by Hao on August 02, 2020 at 11:22
A small typo: In 07IQ we should interchange
T
T'
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 07IN. 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 07IN, in case you are confused.
|
Spindle Spacing Calculator
What are deck and stair spindles?
How to use this spindle spacing calculator
How to calculate spindle spacing
Spindle spacing formula for evenly-spaced spindles
Slanted spindle spacing
This spindle spacing calculator will help you determine the recommended spacing between your railing spindles when constructing stair or deck railings. With this spindle calculator, you will get even uniform spindle spacings, not just between each spindle but also between its supporting walls or posts.
In this stair and deck spindle calculator, you will learn:
What deck and stair spindles are, and why we need them;
How to use our spindle spacing calculator;
How to calculate spindle spacing yourself using two methods; and
How to calculate slanted spindle spacing after finding its corresponding horizontal measurement.
Ready to start learning? Then keep on reading 🙂.
🙋 Please note that when using this tool as a stair spindle calculator, we assume you will install the spindles on a base or shoe railing and not directly on the stair treads.
Spindles, in construction and carpentry, are slender support members of a structure used to create a see-through barrier, like a fence, for staircases and bridges or an open parapet for elevated surfaces like decks, terraces, and balconies. We usually install spindles vertically from a bottom base (i.e., the floor or a base or shoe rail) to the underside of a handrail or a guard rail.
These railings, at adequate heights, offer guidance and protection for people from accidentally falling over and off the aforementioned elevated surfaces. However, we also want to protect people, especially children, from passing through the spaces between the spindles.
Because of that, we want to space our spindles properly and only allow a maximum spindle spacing of 10 cm or 4 inches, as mandated by the U.S. Building Code. That is a safe measurement to prevent a small child's head from passing through or getting stuck between spindles.
This calculator will help you decide how to space your spindles to pass your local code and have beautiful and evenly spaced spindles when done correctly.
🔎 Balusters, which are similar objects to spindles, have the same functions as spindles. However, spindles are usually more slender versions of balusters. We also usually see spindles made from wood or metal materials. While most of the time, balusters are made from wood, stone, or cement.
Our spindle spacing calculator is an easy and intuitive tool to use. Nevertheless, here are the steps you can follow while you use it:
Choose what you want to calculate, whether you wish to calculate spindle spacing for stairs or a deck.
Select the spacing option you want to display. Pick centered spindles if you're going to space your spindles equally and find their balanced end spacings. Choose evenly-spaced spindles if you want to have an even spindle spacing throughout the railing. If you want to compare the results from both options, select both spacing options.
Enter the inside railing distance or the distance between two walls or posts where you want to install a railing.
Input the width of the spindle you wish to use. If you plan to use spindles with an intricate design with varying widths, enter the minimum width of your spindle. The point at which you measure the spindle width shall coincide with the imaginary guiding line where you will be measuring the spindle spacing during the actual construction.
Enter your desired maximum allowable spacing for your spindles. However, please keep in mind the mandated building code in your area to keep your railings safe for kids.
If you selected to calculate stair spindle spacing, you can enter your stair's riser rise and effective tread run or the stair pitch to calculate the slanted spacing of your spindles. You can skip this step if you're using our tool as a deck spindle calculator.
Upon doing these steps, our calculator will immediately display the results for your selected spindle spacing option.
In the next section of this text, you'll understand how to figure out the spindle spacing yourself by considering an example. Ready? Then let's go! 🙂
The best way to understand how to calculate spindle spacing is by considering an example. Let's say we want to use 2-cm wide spindles for a stair railing with a total horizontal railing length of 85 cm from an existing post to the outer side of a 10-cm square end post.
Our first step is to determine the inside railing distance or the "actual space" we'll consider for the calculation. We do that by subtracting the 10-cm width of the end post from the 85-cm railing length to get 75 cm, as shown in the illustration below:
In other cases, we just measure the distance between existing walls or posts to get the actual space.
Then, we determine what we call the "unit length." Unit length is the sum of one space between two spindles and the width of a spindle. Say we only allow a maximum of 10 cm for the spindle spacing. Our unit length will then be:
\scriptsize \begin{align*} \text{unit length} &=w_\text{s} + s_\text{max}\\ &= 2\ \text{cm} - 10\ \text{cm}\\ &= 12\ \text{cm} \end{align*}
w_\text{s}
– Spindle width; and
s_\text{max}
– Maximum allowable or desired spindle spacing.
Next, we divide the inside railing distance,
d
, by the unit length to find how many unit lengths can fit in our actual space. By considering only the whole number of the quotient we will get, we also determine the number of spindles,
n
for our railing. In equation form, we have:
\scriptsize \begin{align*} n &= \frac{d}{\text{unit length}}\\\\ &= \frac{75\ \text{cm}}{12\ \text{cm}}\\\\ &= 6\tfrac{1}{4} \text{unit lengths}\ \text{or}\ 6\ \text{spindles} \end{align*}
That means we can fit 6 of the 12-cm unit length plus an end spacing of 1/4 of a unit length, or 3 cm, within our actual space. We can also interpret that to say that we will need six spindles for this area, as we can see in this illustration:
However, it looks better to center our six spindles so that the distance between the first spindle and the existing post is equal to the distance between the last spindle and the end post. For that, we have to take the average of these end spacings to get balanced end spacings. To do that:
We first multiply our unit length of 12 cm by 6 to get 72 cm.
Then we subtract that 72 cm from 75 cm to obtain the extra space of 3 cm.
Finally, we add 3 cm to 10 cm and divide their sum by 2 to get 6.5 cm for the balanced end spacing.
We can also express that in equation form as shown below:
\scriptsize \begin{align*} s_\text{end} &= \frac{d - n\ \times \text{unit length} + s_\text{max}}{2}\\\\ &= \frac{75\ \text{cm} - 6\ \times 12\ \text{cm} + 10\ \text{cm}}{2}\\\\ &= \frac{75\ \text{cm} - 72\ \text{cm} + 10\ \text{cm}}{2}\\\\ &= \frac{3\ \text{cm} + 10\ \text{cm}}{2}\\\\ &= \frac{13\ \text{cm}}{2}\\\\ &= 6.5\ \text{cm}\\ \end{align*}
s_\text{end}
– Balanced end spacing between the end spindles to the wall or post closest to them.
Now that we know our balanced end spacing, we now have the option to install our first spindle 6.5 cm from the previously installed post and then space the following five spindles 10 cm from each other. Following this procedure will leave us with our last spindle 6.5 cm away from our newly installed post, as we can see in the illustration below:
Now that looks better! 🙂
On the other hand, if we want to have even spindle spacing throughout the entire railing, we need to calculate the spacing between the spindles differently. Let's learn how to do that in the next section of this text.
After knowing the number of unit lengths,
n
, that would fit our actual space, here is how to calculate the spacing for evenly-spaced spindles:
We first determine how much space all our spindles take by multiplying the width of our spindles by
n
to get: 2 cm per spindle × 6 spindles = 12 cm.
Then, we subtract that from our actual space to obtain 75 cm - 12 cm = 63 cm. That is all the spaces between our spindles combined into one.
Since we have 6 unit lengths and one extra space at the end, we have to divide 63 cm by 7 spaces to get the value for an even spindle spacing of 9.0 cm.
We can also use the more general spindle spacing formula that does the same calculation but faster. Here we use the general spindle spacing formula to calculate the same result as explained earlier:
\scriptsize \begin{align*} s_\text{even} &= \frac{d - (n\ \times w_\text{s})}{n + 1}\\\\ &= \frac{75\ \text{cm} - (6\ \times 2\ \text{cm})}{6 + 1}\\\\ &= \frac{75\ \text{cm} - (12\ \text{cm})}{7}\\\\ &= \frac{63\ \text{cm}}{7}\\\\ &= 9.0\ \text{cm} \end{align*}
s_\text{even}
– Even spindle spacing throughout the railing.
With this option, we can install the first spindle 9.0 cm from the existing post, install the next spindle 9.0 cm away from the first spindle, and so on. After installing the sixth spindle, the spacing between the sixth spindle and the newly installed post should also be around 9.0 cm, as shown below:
Learning how to figure out the spindle spacing along a slanted surface like a base rail also comes in handy in some situations (e.g., when using a spindle design with varying diameters or widths for a stair railing, when determining the length of fillet strips). Here is an illustration showing an example of a slanted spindle spacing:
After calculating the horizontal spacing of your spindles by following the instructions described above, we can determine the spacing between spindles along a slanted base rail or a diagonal guideline using this formula:
\footnotesize{S_\text{slanted} = \frac{S_\text{horizontal}}{\cos(\alpha)}}
S_\text{slanted}
– Slanted spacing along the base or shoe rail;
S_\text{horizontal}
– Calculated spacing along the horizontal; and
\alpha
– Base rail angle of inclination equivalent to the stair pitch.
If you have a protractor, you can use that to determine the base rail angle of inclination. We can also calculate it if you don't have any angle-measuring tool right now. Since it would be prettier to make the base rail parallel to the stair's angle of inclination, we can use this stair pitch formula if we need to calculate
\alpha
\footnotesize{\alpha = \tan^{-1}\left(\frac{\text{riser rise}}{\text{effective tread run}}\right)}
\small{\text{riser rise}}
– Vertical distance between two adjacent steps; and
\small{\text{effective tread run}}
– Horizontal distance from two same points on two adjacent steps (e.g., the horizontal distance from the two adjacent stair nosing tips).
Though our spindle calculator can do these steps very quickly, wasn't it fun to learn how to calculate the spacing between spindles ourselves? 🙂 Now that we have all the measurements we need, we can now proceed to the actual construction of our railing 👷♂️👷♀️.
💡 Aside from calculating spindle spacing for stairs and deck railings, you can also use this tool and these methods for determining spindle spacings for your other woodworking projects like, let's say, a chair's backseat or a baby crib 👶.
What is the maximum spacing for spindles?
The maximum spacing for spindles is around 10 cm or 4 inches. We consider this maximum allowed spindle spacing to keep our railings safe by preventing a child's head from getting stuck between spindles. A home inspector may bring a 4-in ball to check if your spindles are up to code.
The number of spindles you need depends on your:
Spindle width;
Inside railing distance; and
Maximum allowed spindle spacing.
Let's say you need to install 3-cm wide spindles between two posts 200 cm apart, and you want to space them 10 cm max.
Add the spindle width and max allowed spacing: 3 cm + 10 cm = 13 cm.
Divide the inside railing distance of 200 cm by 13 cm to get: 200 cm / 13 cm = 15.38.
Round down this result to a whole number. In this case 15 spindles.
How do I evenly space spindles?
To evenly space spindles quickly:
Multiply your spindle's width by the number of spindles you need to find the total width of all your spindles. Say, 15 spindles × 3 cm = 45 cm.
Subtract your spindles' total width from the railing length you will install them to get the total width of all the combined spaces. Say, 200 cm - 45 cm = 155 cm.
Divide this difference by the number of spaces you need, usually the number of spindles plus one. Therefore, we have: 155 cm / (15 + 1) ≈ 9.68 cm.
Can spindles be horizontal?
Spindles can be horizontal. However, horizontal spindles can pose a danger since people can use them as a ladder to climb the railings. Please check your local building code to see if they allow homeowners in your area to install horizontal spindles.
Calculate spindle spacing for...
Evenly-spaced spindles
Inside railing distance
Max. allowable spacing
Estimate the location of holes in a circular pattern using the bolt circle calculator.
Use this square yards calculator to calculate the area in square yards and the cost of the material.
|
Combination vs. permutation
How do I calculate combinations without repetition?
Example: How many combinations of 4 numbers without repetition are possible
Our combinations without repetition calculator finds the number of combinations without repetitions that are possible given the total number of objects and the sample size.
In the next section, we present the formula for combinations without repetition, which is the one this calculator uses. We also show an example of how to calculate combinations without repetition.
Our calculator doesn't only calculate the number of combinations without repetition, but it also generates those combinations without repetition in a list.
This calculator deals only with combinations. Please don't confuse it with permutations ⚠️
We also have a permutation calculator 😉
Combinations and permutations are usually confused. Before using our calculator, it's essential to know the difference:
Combination: The number of ways you can choose r elements out of a set containing n distinct objects, no matter the order.
Permutation: The number of ways you can choose r elements out of a set containing n distinct objects, taking all the possible orders into account. Each possible ordering of the same group of elements accounts for an additional permutation.
As a picture is worth a thousand words, we created the following graphic, which describes perfectly these concepts and their differences.
An example to better understand the combination and permutation concepts.
The formula to calculate combinations without repetition is:
C(n,r) = n! / r!(n-r)!,
C(n,r) — The number of combinations without repetition;
n — The total number of objects (how many distinct objects you have); and
r — Sample size (how many objects we want to choose).
Suppose you're interested in calculating the number of combinations without repetition from 4 numbers out of the decimal systems (numbers from 0 to 9) that are possible. Let's use the combinations without repetition formula to know the answer. Follow these steps:
Identify the relevant values.
We want combinations of 4 numbers; therefore, our samples are of size r = 4.
The total number of objects comes from the decimal system (numbers from 0 to 9). Therefore, n = 10.
In the formula of combinations without repetition, input the values:
C(n,r) = \frac{n!}{r!(n-r)!} = \frac{10!}{4!(10-4)!} = 210
That's it! That's how you calculate the number of combinations without repetition. You can also check the answer with our combinations without repetition generator/calculator.
Now you know how to calculate and generate combinations without repetition. The next step is to look at these other tools created by us:
Combination calculator;
Possible combinations; and
In the following FAQ section, we solve some interesting problems like how many combinations without repetition with 16 numbers or five digits are possible.
How many combinations with 16 numbers without repetition are possible?
There's one (1) possible combination without repetitions and 300,540,195 combinations with repetitions of arranging a group of sixteen numbers (i.e., the 1-16 number list).
The possible combinations without repetitions C(n,r) is one because the total number of objects n (sixteen digits) equals our sample size r (the sixteen digits we'll arrange). In other words, C(n,r) = 1 because n = r.
How many combinations with 5 numbers without repetition are possible?
There's one (1) possible combination without repetitions C(n,r) and 126 combinations with repetitions C'(n,r) of arranging a group of five numbers (i.e., the 1-5 number list).
The number of possible combinations without repetitions equals one because the total number of objects n (five numbers) equals our sample size r (the five digits we'll arrange). In other words, if n = r, then C(n,r) = 1.
This cubic regression calculator will help you determine the polynomial of degree 3 that best fits your two-dimensional dataset.
Cubic Regression Calculator
|
Section 5-6 - DEs - Maple Help
Home : Support : Online Help : Study Guides : Calculus : Chapter 5 - Applications of Integration : Section 5-6 - DEs
An algebraic equation is an open statement that is true when the "openings" in the statement are filled with the appropriate algebraic expressions. Thus,
2 x+3=5
becomes true when
x
, the "opening," is replaced with the number 1.
A differential equation is an open statement in which the openings are are a function and at least one of its derivatives. Table 5.6.1 lists several examples of differential equations.
y\prime \left(x\right)=1
Immediately integrable to the general solution
y\left(x\right)=x+c
y\prime \left(x\right)=y\left(x\right)
Knowledge of the exponential function suggests the general solution
y\left(x\right)=c {e}^{x}
, but this equation is actually separable.
y\prime \left(x\right)+2 y\left(x\right)=x
The equation is first-order, linear, and yields to the "recipe"
y={e}^{∫2 \mathit{ⅆ}x}\left(∫x {e}^{∫2 \mathit{ⅆ}x} \mathit{ⅆ}x+c\right)
The general first-order, linear, equation is
p\left(x\right) y\prime +q\left(x\right) y=r\left(x\right)
\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{{x}^{2}-x y+{y}^{2}}{{x}^{2}-{y}^{2}}
The equation is homogeneous, and becomes first-order, linear, under the change of variables
v\left(x\right)=y\left(x\right)/x
y\prime \left(x\right)+{y}^{2}\left(x\right)=x
The equation is nonlinear. If functions of either
y
y\prime
appear in the equation, it is no longer linear.
y\prime \left(x\right)+2 y\left(x\right)={y}^{3}\left(x\right)
This is a Bernoulli equation, and becomes first-order, linear, under the change of variables
y\left(x\right)={z}^{k}\left(x\right)
for some special value of
k
y\prime \left(x\right) f\left(y\right)=g\left(x\right)
This is the general separable equation whose solution is given implicitly by direct integration:
∫f\left(y\right) \mathit{ⅆ}y=∫g\left(x\right) \mathit{ⅆ}x+c
Table 5.6.1 Examples of differential equations
The separable differential equation affords great opportunity to practice evaluating both indefinite and definite integrals. It will be the main focus of this chapter.
Since the general solution of the first-order differential equation
f\left(y,y\prime ,x\right)=0
contains one arbitrary constant of integration, the solution of such an equation represents a family of curves. One unique member of this family can be distinguished by imposing one algebraic condition of the form
y\left({x}_{0}\right)={y}_{0}
, called an initial condition. A differential equation and its associated initial condition(s) is called an initial value problem (IVP).
An implicit solution of the IVP
y\prime \left(x\right) f\left(y\right)=g\left(x\right),y\left({x}_{0}\right)={y}_{0}
{∫}_{{y}_{0}}^{y}f\left(s\right) ⅆs={∫}_{{x}_{0}}^{x}g\left(s\right) ⅆs
. The differential equation is separable; and the variable of integration on each side is immaterial, as long as it does not duplicate one of the limits.
Obtain the general solution of the differential equation
y\prime \left(x\right)=x y\left(x\right)
Obtain the general solution to the differential equation
x \mathrm{dx}-2 y \sqrt{1+{x}^{2}} \mathrm{dy}=0
Solve the initial-value problem consisting of the differential equation
x-2 y \sqrt{1+{x}^{2}}\cdot y\prime =0
y\left(1\right)=2
. Graph the solution.
Graph the solution of the initial-value problem consisting of the differential equation
y\prime =4x{y}^{2}+8{y}^{2}+x+2
, and the initial condition
y\left(1\right)=0
A tank contains 50 lbs of salt dissolved in 1000 gallons of water. Brine containing
1/100
lb of salt per gal of water enters the tank at a rate of 40 gal per minute, mixes instantaneously, and drains at the same rate. Determine the tank's salt content 15 minutes later.
y>0
, solve the initial-value problem consisting of the differential equation
y\prime \left(t\right)=k y\left(t\right)
y\left(0\right)={y}_{0}
Solve the initial-value problem consisting of the logistic differential equation
\stackrel{.}{y}/y=k \left(c-y\right)
y\left(0\right)={y}_{0}
A species undergoes logistic growth, governed by the formula developed in Example 5.6.7. Observation yields the following three data points.
[\begin{array}{cc}\mathrm{Time in years}& \mathrm{Population Size}\\ 1& 1300\\ 3& 1870\\ 4& 2070\end{array}]
Determine the carrying capacity
c
, the initial population
{y}_{0}
, and the rate constant
k
, if it is known that
k>0
\stackrel{.}{u}=k \left(u-{u}_{s}\right)
u\left(t\right)
represents the temperature of a body in thermal contact with its surroundings at a fixed temperature
{u}_{s}
. This equation, sometimes called Newton's law of cooling, simply states that the rate of change of temperature of the body is proportional to the difference in temperature between the body and its surroundings.
A medical examiner (M.E.) notes the temperature of the body of a deceased person is
85
°F, and the environment in which the body has been located is 65°F. Being careful not to alter the surrounding temperature, the M.E. waits 15 minutes and again checks the body's temperature, finding it to be
80
°F. Using Newton's law of cooling, what estimate can the M.E. make for the time of death of the deceased?
|
Effective Nuclear Charge Calculator | Slater's Rule
A quick review of the nuclear structure
How to calculate the effective nuclear charge: What are Slater's rules?
Example of how to calculate the effective nuclear charge
How to use our effective nuclear charge calculator?
A final review!
The farther an electron moves away from the nucleus of an atom, the weaker their attraction is: discover why with our effective nuclear charge calculator.
Here you will learn what the effective nuclear charge is and how to calculate it using Slater's rules. You will see some examples and get a quick review of the quantum theory behind atoms — and finally, you will learn how to use our Slater's rules calculator. Ready?
We need to take a quick look at the nuclear structure to understand what electron shielding is and how to calculate the effective nuclear charge.
Here are the basics of the atomic orbital model first! An atom is composed by:
A positively charged nucleus made of protons and neutrons. The number of protons defines the atomic number, uniquely identifying a chemical element.
A negatively charged electronic cloud. Each electron can be found in a set of defined regions of space around the nucleus called orbitals.
🙋 Quantum physicist here: orbitals are solutions to the Schrodinger equation, which describes the position of an electron in space and time — but remember, electrons are neither particles nor waves! Wavefunctions are complex quantities (in mathematical language) and bear no physical meaning: we need to take their squared modulus that, according to the rules of quantum mechanics, is proportional to the probability of finding an electron in a given set of coordinates.
An orbital is described by a set of discrete integer numbers called the quantum numbers. That's why we speak of quantized — "quanta" is a Latin word for "discrete quantity". Let's discover them:
n
, which gives an indication on the distance of the electron from the nucleus. The smaller the number, the closer the electron. The value for
n
can be any integer, positive value:
n = 0,1,2,3,\ldots
The azimuthal quantum number,
l
, which describes the shape of the region where it is possible to find the electron. Its values are related to the value of
n
, being the integer numbers from
0
n-1
l=0,1,\ldots,n-1
m
, which is associated with the orientation of the orbitals in space. It varies according to the value of
l
m= -l,\ -l+1,\ \ldots,\ l-1,\ l
Electrons with equal
n
l
but different values of
m
have identical energy: we call the respective orbitals degenerate.
Each orbital is defined by a unique set of quantum numbers and can host at most two electrons, one for each value of spin — another quantum property that can assume one of two values, namely spin up or spin down.
We need to take a closer look at the various orbitals to understand how to calculate the effective nuclear charge. Let's proceed in order with the quantum numbers, starting with the electronic shell closest to the nucleus.
n=1
, the other quantum numbers are
l=0
m=0
. There is a single orbital, called
1s
, with a spherical shape.
n=2
there are two possible values for
l
l=0
l=1
l=0
m=0
too, and we get another spherical orbital,
2s
l=1
there are three possible values of
m
m=-1,0,+1
. We call these orbitals
2p
, and to distinguish the three orientations we add a coordinate to them:
2p_x
2p_y
2p_z
. Instead of spherical, they have a dumbbell shape.
n=3
we have the same orbitals we've just met, with the addition of the ones associated to
l=2
m
can assume five different values:
m=-2,-1,0,1,2
3d
orbitals, and they assume various shapes. Our favorite of these shapes is a donut with a pear on each side.
n=4
we add another set of orbitals, the
4f
. They are described by the set of quantum numbers
n=4
l=3
m=-3,-2,-1,0,1,2,3
— for a total of seven orbitals. Their shapes are even more complex than the one of the
d
Going up with the value of
n
, we would meet even more complex orbitals, like
g
, but they don't appear in the elements we know at this time: talking of them would be meaningless!
The shape of some of the atomic orbitals. In the first line, the
1s
orbital and its spherical shape. Notice also the small size compared to the others: it is in the vicinity of the nucleus. In the second row, there are the
2p
orbitals, with their three orientations. Finally, in the bottom row, you can see the five shapes of the
3d
It is possible to identify each element using its electron configuration. This is a way to specify the occupation of the orbitals, progressively filling the periodic table.
Writing the electron configuration for an element is relatively simple — it gets challenging only for heavier elements that start to misbehave. Let's take a look at the configuration for hydrogen:
\text{H}=1s^1
Hydrogen above has a single electron in the first shell. On the other hand, helium has a full first shell with two electrons. Its configuration is:
\text{He}=1 s^{2}
It goes on like this, following the progression of the orbitals dictated by the following scheme:
Visual representation of the occupation order of orbitals.
Follow the blue arrow from top to bottom, writing down the orbitals in the orders they are crossed. Remember the number of electrons hosted in each shell:
s\rightarrow 2
p\rightarrow 6
d \rightarrow 10
f \rightarrow 14
Let's try with a heavier element, tellurium. Its atomic number is 52, which is also the number of electrons we need to fit in the configuration.
\begin{gather*} \footnotesize 1s^2,\ 2s^2,\ 2p^6,\ 3s^2,\ 3p^6,\ 4s^2, \\ \footnotesize 3d^{10},\ 4p^6,\ 5s^2,\ 4d^10,\ 5p^4 \end{gather*}
After you filled the electron configuration following the graphic rule, you can rewrite it in the most intuitive order:
\begin{gather*}\footnotesize 1s^2,\ 2s^2,\ 2p^6,\ 3s^2,\ 3p^6,\ 3d^{10},\\ \footnotesize 4s^2,\ 4p^6,\ 4d^{10},\ 5s^2,\ 5p^4 \end{gather*}
That's how they are commonly found in textbooks, online, and also on our effective charge calculator.
🔎 You can see that the electron configuration of heavy elements gets a little cumbersome. Chemists found a way to make it easier to write and remember! You can use as reference the electron configuration of noble gases and start writing the configuration from the last one in the periodic table, indicating it with the symbol of the element in square brackets. The electron configuration of the Tellurium would be
\text{Te}=\left[\text{Kr}\right],\ 4d^{10},\ 5s^2,\ 5p^4
. Easier, isn't it?
Electrons feel the attraction of the nucleus since they have opposite charges. However, only a single electron would experience the attractive force in its entirety. For every added electron sharing the same orbital or occupying lower energy orbitals, the negative charge of those particles adds a repulsive component, which contributes to the shielding of the nucleus' electrostatic interaction.
🔎 The methods explained here are approximations that don't take into account the position of the electrons and other factors. However, they fit the observed data.
We need to understand first what is the nuclear charge. It is, straightforwardly, the charge of the nucleus in units of elementary charge, the charge of electrons and protons (with opposite sign). The nuclear charge
Z
thus coincides with the atomic number. Without electrons, that would be the potential energy centered in the nucleus.
When we consider the repulsive interaction of other electrons, however, we see that the farther we get from the nucleus, the lower the charge felt by an electron. We need to talk of effective nuclear charge. We denote it by
Z_\text{eff}
For the first electron around the nucleus, the effective nuclear charge equals the nuclear charge:
Z_\text{eff} = Z
Z_\text{eff}
then decreases approaching
1
for an infinite distance from the nucleus. This is the value of the potential energy experienced by the last electron added to the shell.
The effective nuclear charge has some distinctive trends across the periodic table. It increases following the groups from left to right, decreasing descending into the periods. The ratio
{Z_\text{eff}}/{Z}
is smaller for the elements in the first group, decreasing for heavier elements (where the larger amount of electrons has a more substantial shielding effect).
Now that you know what the effective nuclear charge is, it's time to learn how to calculate it. To do this, allow us to introduce Slater's rules. The concept at the core is that to calculate the effective nuclear charge we need to compute the overall contribution of the shielding electrons.
Slater's rules need the complete electron configuration of an element to be applied. We then choose an electron belonging to a specific orbital. At this point, we have a few different paths that tell us the shielding contribution of each electron in the configuration.
The electrons in orbitals to the right of the chosen one give a zero contribution to the shielding.
If you chose an electron from a
p
s
orbital, with principal quantum number
n=N
Electrons from orbitals with the same principal quantum number have a shielding factor of
0.35
apart from the electrons in
1s
, which shield
0.30
Electrons from orbitals with
n=N-1
0.85
Electrons coming from orbitals with
n=N-2
or less shield
1.00
as they are close to the nucleus.
If you chose an electron from an orbital with
l
d
f
, and again
n=N
n=N
, and equal or bigger
l
(following
s<p<d<f
) shield
0.35
n+N
but smaller
l
than the one of the chosen electron shield
1.00
All other electrons from orbitals with
n<N
1.00
💡 When choosing the electron, it's not essential to specify which position in the orbital we are considering: the shielding effect is not affected by degeneracy.
How do we calculate the effective nuclear charge, then? Each shielding factor is multiplied by the number of electrons in the related orbitals, remembering to subtract one when it comes to the orbital to which the chosen electron belongs. The resulting contributions are summed.
The resulting shielding is called
\sigma
and is used to calculate
Z_\text{eff}
Z_\text{eff}=Z-\sigma
🙋 There is an exception! Hydrogen, having a single electron, has effective nuclear charge equal to the nuclear charge — that poor electron can't shield itself!
Let's choose an element and an orbital. Selenium and
3p
, you say? That's what we were thinking too!
This is the electron configuration of selenium, with the chosen orbital highlighted:
\footnotesize 1s^{2}, 2s^{2}, 2p^{6}, 3s^{2}, \textbf{3p}^\textbf{6}, 3d^{10}, 4s^{2}, 4p^{4}
Ignore the orbitals to the right!
\footnotesize 1s^2, 2s^2, 2p^6, 3s^2, \textbf{3p}^{\textbf{6}}, \sout{3d^{10}},\sout{4s^2}, \sout{4p^4}
We chose a
p
electron, and it's time to apply the appropriate Slater's rules. There are
7
other electrons in the same group, with
n=3
; each contributes with
0.35
. In the orbitals with
n=2
2s
2p
8
electrons, which contribute individually with
0.85
. Lastly, with
n=3-2=1
, we have the two electrons in
1s
, which contribute with a full
1.00
Let's sum up the various contributions then:
\begin{align*} \small \sigma & =7\cdot 0.35+8\cdot 0.85+2\cdot 1.00 \\ & =11.25 \end{align*}
Selenium has a nuclear charge of
Z=34
. The effective nuclear charge is obtained by subtracting from
Z
the value of shielding
\sigma
\small Z_\text{eff} = 34-11.25=23.75
First thing, choose an element from the list. They are in order of increasing atomic number. Its electron configuration will appear just below. Choose the desired electron from the configuration and input the appropriate quantum numbers.
✅ You can't get things wrong with this calculator. If the chosen principal quantum number or the azimuthal quantum number is not available in the electron configuration, we will stop you until you insert a correct set!
At the bottom of the calculator, you will find the values of the total shielding and of the effective nuclear charge
Z_\text{eff}
This is all we needed to say about this topic. Now you should know how to calculate
Z_\text{eff}
without any difficulty, but here is a quick refresher:
Choose an electron from the electron configuration.
Apply Slater's rules to calculate the total shielding.
Remember to ignore electrons from higher orbitals.
To calculate the effective nuclear charge, subtract the shielding from the nuclear charge.
That's it! If you want to learn more about atomic properties, check out our electronegativity calculator, the radioactive decay calculator, or visit the chemistry page at omnicalculator.com!
What are Slater's rules?
Slater's rules are a set of rules used in physical chemistry to calculate the effective nuclear charge experienced by an electron around a nucleus. The rules assign a specific value of shielding to each electron according to its orbital.
Check out the Slater's rule calculator on omnicalculator.com to discover more about it!
How do I calculate the effective nuclear charge?
To calculate the effective nuclear charge:
First, compute the overall shielding effect of the electrons orbiting the nucleus.
Subtract this value from the nuclear charge (equal to the number of protons of the element).
Remember that the value of the effective nuclear charge depends on the orbital you are using in the calculations, since outer electrons don't contribute to the shielding!
What is the trend of the effective nuclear charge?
The effective nuclear charge experienced by the last electron in the negatively charged shell decreases as the period (the rows of the periodic table) increases. This is because a complete shell has been added below it (and according to Slater's rules, the contribution to the shielding is higher), increasing the distance from the nucleus. The effective charge increases within the same period because the contribution to the shielding for orbitals with equal n is less significant.
What is the effective nuclear charge for neon?
Considering the last electron in the electron cloud of neon, in the orbital 2p, we have seven electrons contributing 0.35 to the shielding. We then add the contribution of the two electrons in the orbital 1s with factor 0.85.
The total shielding is 7×0.35 + 2×0.85 = 4.15, and the effective nuclear charge is 10 − 4.15 = 5.85.
Choose an atomic species
Select an electron from the electron configuration:
Electron Selection
You chose an electron in the orbital 1s.
Value of the shielding: 0.00
Effective charge: 1.00
This calculator implements Slater's rules. Other methods may return slightly different results.
The ionic strength calculator is a convenient tool to help you calculate the ionic strength of a solution based on the ions present in it and their charge.
|
Geophysical Networks in Colombia | Seismological Research Letters | GeoScienceWorld
Carlos A. Vargas;
Department of Geosciences, Universidad Nacional de Colombia at Bogotá, Bogotá, Colombia
Alexander Caneva;
Research Center, Universidad Antonio Nariño at Bogotá, Bogotá, Colombia
Hugo Monsalve;
Engineering Faculty, Universidad del Quindío, Armenia, Colombia
Elkin Salcedo;
Department of Geography, Universidad del Valle, Cali, Colombia
Colombian Geological Survey, Bogotá, Colombia
Carlos A. Vargas, Alexander Caneva, Hugo Monsalve, Elkin Salcedo, Héctor Mora; Geophysical Networks in Colombia. Seismological Research Letters 2018;; 89 (2A): 440–445. doi: https://doi.org/10.1785/0220170168
We present a brief description of the Colombian geophysical monitoring networks, mainly from the seismic and volcanic points of view. We present also a description of two significant events that defined the development of the instrumental geophysical infrastructure in Colombia: (1) the 31 March 1983 Popayán earthquake (
Mw
H=4 km
deep) and (2) the eruption of the 13 November 1985 Nevado del Ruiz Volcano (VEI‐3). Seven geophysical networks (seismological, strong motion, volcanological, and Global Positioning System/Global Navigation Satellite System [GPS/GNSS] networks) that are currently monitoring the Colombian territory are described in terms of the operational coverage, technical parameters, and scientific purposes. Networks described correspond to (1) the National Seismological Network of Colombia (RSNC, with 65 stations); (2) the National Strong Motion Network (RNAC, with 177 stations); (3) the Volcanological observatories, located in the Manizales, Pasto, and Popayan cities and comprised of permanent stations that monitor seismicity, deformation, thermal changes in waters and rocks,
SO2
and water emission, as well as gravity, magnetic, and electrical changes in the 12 most active volcanoes of Colombia; (4) the GPS/GNSS Network for Geodynamics (GeoRED, with 108 Continuously Operating Reference Stations, and 382 field stations); (5) the “Sabana de Bogotá” Seismological Network (RSSB), with one short‐period (
T=1 s
) and eight broadband (
T=120 s
) stations, of which three are monitoring seismic, electrical, and magnetic anomalies; (6) the South Western Colombian Seismological Network (OSSO, with 11 stations); and (7) the Quindío Seismological Observatory (OSQ, with 9 stations). Deployment of these networks has been possible due to the active participation of state‐owned and private institutions. Their continuous operation has allowed the growth of the scientific infrastructure countrywide, and has increased knowledge about the geodynamical behavior of this region.
Popayan earthquake 1983
Finite‐Fault Simulation of Broadband Strong Ground Motion from the 2010 Mw 7.0 Haiti Earthquake
The Argentinean National Network of Seismic and Strong‐Motion Stations
The 2011 Mw 9.0 Tohoku Earthquake: Comparison of GPS and Strong‐Motion Data
|
Design and Test of Carbon Nanotube Biwick Structure for High-Heat-Flux Phase Change Heat Transfer | J. Heat Transfer | ASME Digital Collection
e-mail: qcai@teledyne.com
Cai, Q., and Chen, C. (March 9, 2010). "Design and Test of Carbon Nanotube Biwick Structure for High-Heat-Flux Phase Change Heat Transfer." ASME. J. Heat Transfer. May 2010; 132(5): 052403. https://doi.org/10.1115/1.4000469
With the increase in power consumption in compact electronic devices, passive heat transfer cooling technologies with high-heat-flux characteristics are highly desired in microelectronic industries. Carbon nanotube (CNT) clusters have high thermal conductivity, nanopore size, and large porosity and can be used as wick structure in a heat pipe heatspreader to provide high capillary force for high-heat-flux thermal management. This paper reports investigations of high-heat-flux cooling of the CNT biwick structure, associated with the development of a reliable thermometer and high performance heater. The thermometer/heater is a 100-nm-thick and
600 μm
wide Z-shaped platinum wire resistor, fabricated on a thermally oxidized silicon substrate of a CNT sample to heat a
2×2 mm2
wick area. As a heater, it provides a direct heating effect without a thermal interface and is capable of high-temperature operation over
800°C
. As a thermometer, reliable temperature measurement is achieved by calibrating the resistance variation versus temperature after the annealing process is applied. The thermally oxidized layer on the silicon substrate is around
1-μm
-thick and pinhole-free, which ensures the platinum thermometer/heater from the severe CNT growth environments without any electrical leakage. For high-heat-flux cooling, the CNT biwick structure is composed of
250 μm
100 μm
wide stripelike CNT clusters with
50 μm
stripe-spacers. Using
1×1 cm2
CNT biwick samples, experiments are completed in both open and saturated environments. Experimental results demonstrate
600 W/cm2
heat transfer capacity and good thermal and mass transport characteristics in the nanolevel porous media.
annealing, capillarity, carbon nanotubes, cooling, micromechanical devices, temperature measurement, thermal management (packaging), thermometers, thermometer, carbon nanotube, wick structure
Carbon nanotubes, Heat, Heat transfer, Temperature, Thermometers, Platinum, Annealing, Silicon, Resistors, Cooling, Temperature measurement
Heat Pipes With Bidispersed Capillary Structures
Proceedings of the Fifth International Heat Pipe Conference
Liquid Film Evaporation From Bi-Dispersed Capillary Wicks in Heat Pipe Evaporators
Proceedings of the Ninth International Heat Pipe Conference
, Albuquerque, NM, May 1–5, pp.
High Heat Flux Loop Heat Pipes
Vaporization Heat Transfer in Biporous Wicks of Heat Pipe Evaporators
Experimental Study of Evaporative Heat Transfer in Sintered Copper Bidispersed Wick Structures
Experimental Investigation of Boiling Heat Transfer in Bidispersed Media
Evaporation/Boiling on Thin Capillary Wick (I): Thickness Effects
Biporous Sintered Copper for a Closed Loop Heat Pipe Evaporator
Carbon Nanotubes Synthesis, Structure, Properties and Applications
Thermal Transport Measurement of Individual Multiwalled Nanotubes
Exploration of Carbon Nanotube Wick Structure for High Heat Flux Cooling
A Micro-Channel Heat Sink With Integrated Temperature Sensors for Phase Transition Study
12th IEEE International Conference on Micro ElectroMechanical Systems
Heater Integrated Sensor System
Thick Film Heater Elements and Temperature Sensors in Modern Domestic Appliances
Fabrication of a Complicated Heat Transfer Microchannel System for CPU Cooling Study
Proceedings of the Second IEEE International Conference on Nano/Micro Engineered and Molecular Systems
Traceable Temperature
|
US and UK gallons
Gallons and cubic feet
How do I convert gallons to cubic feet?
Other useful converters
With the gallons to cubic feet converter, you can quickly convert cubic feet to gallons. You can easily switch between both volume units with just a simple multiplication or division.
This short text will cover how to convert gallons to cubic feet and give some practical examples. Let's dive right in!
Gallon (gal) is a volume unit, and it can be either a US gallon or UK gallon. A US gallon is exactly 231 cubic inches or 3.785 L, while a UK gallon is 4.546 L. As you can see, UK gallons have a slightly larger capacity than their US counterpart.
Let's take a look at how they compare to cubic feet.
A cubic foot (cu ft or ft³) is another volume unit used in the US and Imperial unit systems. It's defined as the amount of space a 1 foot sided square box occupies. To convert between gallons and cubic feet we would first need to express a single gallon in terms of cubic feet:
1\ \text{US gal} = 0.133681\ \text{ft³}\\ 1\ \text{UK gal} = 0.160544\ \text{ft³}
That means we would need to multiply this number (0.133681 or 160544) by the number of gallons to switch between both units.
To convert gallons to cubic feet:
Write down the volume in gallons (US or UK).
Multiply this number by 0.133681 if it's a US gallon.
Otherwise, multiply the volume in gallons by 0.160544 if it's a UK gallon.
Here's a list of some other helpful volume converters:
How many cubic feet are in 5 gallons?
There are 0.668403 ft³ in 5 US gallons and 0.802718 ft³ in 5 UK gallons. To obtain this result, we multiply the number of gallons by 0.133681 or 0.160544 if it's a US or UK gallon, respectively.
What do you call half a gallon?
2 quarts. A quart is equal to one-fourth of a gallon. Therefore, half a gallon is the same as 2 quarts.
The binary to hexadecimal converter gives you an insight into the relationship between binary and hexadecimal number systems and how to convert from one to the other.
|
He died on 1 September 1648 of complications arising from a lung abscess.
{\displaystyle f={\frac {1}{2L}}{\sqrt {\frac {F}{\mu }}},}
{\displaystyle {\sqrt[{4}]{\frac {2}{3-{\sqrt {2}}}}}}
as the ratio for an equally-tempered semitone (
{\displaystyle {\sqrt[{12}]{2}}}
). It was more accurate (0.44 cents sharp) than Vincenzo Galilei's 18/17 (1.05 cents flat), and could be constructed using straightedge and compass. Mersenne's description in the 1636 Harmonie universelle of the first absolute determination of the frequency of an audible tone (at 84 Hz) implies that he had already demonstrated that the absolute-frequency ratio of two vibrating strings, radiating a musical tone and its octave, is 1 : 2. The perceived harmony (consonance) of two such notes would be explained if the ratio of the air oscillation frequencies is also 1 : 2, which in turn is consistent with the source-air-motion-frequency-equivalence hypothesis.
Battles with occult and mystical thinkers[edit]
Tractatus mechanicus theoricus et practicus, 1644
Tractatus mechanicus theoricus et practicus (in Latin). Paris: Antoine Bertier. 1644.
^ Simmons, George F. (1992/2007). Calculus Gems: Brief Lives and Memorable Mathematics, p. 94. MAA. ISBN 9780883855614.
^ Hauréau, Barthélemy (1852). A. Lanier (ed.). Histoire littéraire du Maine (in French). Vol. 1. p. 321.
^ Murr, Sylvia, ed. (1997). Gassendi et l'Europe (in French). Paris: Vrin. ISBN 978-2-7116-1306-9.
^ Heilbron, J. L. (1979). Electricity in the 17th and 18th Centuries: A Study of Early Modern Physics. University of California Press. ISBN 9780520034785.
Philippe Hamou (2018). Marin Mersenne. Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Marin_Mersenne&oldid=1068206395"
|
Design Compensator Using Automated PID Tuning and Graphical Bode Design - MATLAB & Simulink - MathWorks Italia
Specify Blocks to Tune
Plot Closed-Loop Step Response
Tune Compensator Using Automated PID Tuning
Tune Compensator Using Bode Graphical Tuning
Fine Tune Controller Using Compensator Editor
Simulate Closed-Loop System in Simulink
This example shows how to design a compensator for a Simulink® model using automated PID tuning in the Control System Designer app. It then shows how to fine tune the compensator design using the open-loop Bode editor.
This example uses the watertank_comp_design Simulink model. To open the model, at the MATLAB® command line, enter:
open_system('watertank_comp_design')
This model contains a Water-Tank System plant model and a PID controller in a single-loop feedback system.
Water enters the tank from the top at a rate proportional to the voltage applied to the pump. The water leaves through an opening in the tank base at a rate that is proportional to the square root of the water height in the tank. The presence of the square root in the water flow rate results in a nonlinear plant. Based on these flow rates, the rate of change of the tank volume is:
\frac{d}{dt}Vol=A\frac{dH}{dt}=bV-a\sqrt{H}
Tune the PID controller to meet the following closed-loop step response design requirements:
Rise time less than five seconds
Control System Designer opens and automatically opens the Edit Architecture dialog box.
To specify the compensator to tune, in the Edit Architecture dialog box, click Add Blocks.
In the Select Blocks to Tune dialog box, in the left pane, click the Controller subsystem. In the Tune column, check the box for the PID Controller.
In the Edit Architecture dialog box, the app adds the selected controller block to the list of blocks to tune on the Blocks tab. On the Signals tab, the app also adds the output of the PID Controller block to the list of analysis point Locations.
When Control System Designer opens, it adds any analysis points previously defined in the Simulink model to the Locations list. For the watertank_comp_design, there are two such signals.
Desired Water Level block output — Reference signal for the closed-loop step response
Water-Tank System block output — Output signal for the closed-loop step response
To linearize the Simulink model and set the control architecture, click OK.
By default, Control System Designer linearizes the plant model at the model initial conditions.
The app adds the PID controller to the Data Browser, in the Controllers and Fixed Blocks section. The app also computes the open-loop transfer function at the output of the PID Controller block and adds this response to the Data Browser.
To analyze the controller design, create a closed-loop transfer function of the system and plot its step response.
On the Control System tab, click New Plot, and select New Step.
In the New Step to plot dialog box, in the Select Response to Plot drop-down list, select New Input-Output Transfer Response.
To add an input signal, in the Specify input signals area, click +. In the drop-down list, select the output of the Desired Water Level block.
To add an output signal, in the Specify output signals area, click +. In the drop-down list, select the output of the Water-Tank System block.
To create the closed-loop transfer function and plot the step response, click Plot.
To view the maximum overshoot on the response plot, right-click the plot area, and select Characteristics > Peak Response.
To view the rise time on the response plot, right-click the plot area, and select Characteristics > Rise Time.
Mouse-over the characteristic indicators to view their values. The current design has a:
Maximum overshoot of 47.9%.
Rise time of 2.13 seconds.
This response does not satisfy the 5% overshoot design requirement.
To tune the compensator using automated PID tuning, click Tuning Methods, and select PID Tuning.
In the PID Tuning dialog box, in the Specifications section, select the following options:
Tuning method — Robust response time
Controller Type — PI
Click Update Compensator. The app updates the closed-loop response for the new compensator settings and updates the step response plot.
To check the system performance, mouse over the response characteristic markers. The system response with the tuned compensator has a:
Rise time of 51.2 seconds.
This response exceeds the maximum allowed overshoot of 5%. The rise time is much slower than the required rise time of five seconds.
To decrease the rise time, interactively increase the compensator gain using graphical Bode Tuning.
To open the open-loop Bode editor, click Tuning Methods, and select Bode Editor.
In the Select Response to Edit dialog box, the open-loop response at the output of the PID Controller block is already selected. To open the Bode editor for this response, click Plot.
To view the Bode Editor and Step Response plots side-by-side, on the View tab, click Left/Right.
In the Bode Editor plot, drag the magnitude response up to increase the compensator gain. By increasing the gain, you increase the bandwidth and speed up the response.
As you drag the Bode response upward, the app automatically updates the compensator and the associated response plots. Also, when you release the plot, in the status bar, on the right side, the app displays the updated gain value.
Increase the compensator gain until the step response meets the design requirements. One potential solution is to set the gain to 1.7.
At this gain value, the closed loop response has a:
Maximum overshoot of 4.74%.
To tune the parameters of your compensator directly, use the compensator editor. In the Bode Editor, right-click the plot area, and select Edit Compensator.
In the Compensator Editor dialog box, on the Parameter tab, tune the PID controller gains. For more information on editing compensator parameters, see Tune Simulink Blocks Using Compensator Editor.
While the tuned compensator meets the design requirements, the settling time is over 30 seconds. To improve the settling time, adjust the P and I parameters of the controller manually.
For example, set the compensator parameters to:
This compensator produces a closed-loop response with a:
Maximum overshoot of 0.206%.
Settling time of around three seconds.
Validate your compensator design by simulating the nonlinear Simulink model with the tuned controller parameters.
To write the tuned compensator parameters to the PID Controller block, in Control System Designer, on the Control System tab, click Update Blocks.
In the Simulink model window, run the simulation.
To view the closed-loop simulation output, double-click the Scope block.
The closed-loop response of the nonlinear system satisfies the design requirements with a rise time of less than five seconds and minimal overshoot.
|
<!-- IQ_complex_signals.mediawiki -->
If you are searching for more detailed information please refer to corresponding literature such as references [[#ancre1|[1]]],[[#ancre2|[2]]],[[#ancre3|[3]]].
This section summarize complex numbers properties used in this tutorial. More information can be found on
[[wikipedia: Complex number|complex number Wikipedia page||200px]].
Complex numbers can be represented in the complex plane as vectors. The modulus or magnitude ''r'' of a complex number ''z'' = ''a'' + ''jb'' is
: <math>r=|z|=\sqrt{a^2+b^2}</math>
The phase φ of ''z'' mathematically referred to as the argument is the angle of the radius Oz with the positive real axis.
: <math>\phi=\arg(z)=\arctan(b/a) </math> (for a≠0)
Together, r and φ give another way of representing complex numbers, the polar form and the exponential form.
: <math>z=r \left(cos(\phi) + j sin(\phi) \right) = r e^{j\phi}</math>
[[File:IQ_complex_tutorial_polar1.png|frame|The complex plane]]
Following complex number have a unit magnitude ''r''=1 :
: <math>+1=e^{j0}</math>
: <math>+j=e^{j\pi/2}</math>
: <math>-1=e^{j\pi}</math>
: <math>-j=e^{j3\pi/2} </math>
A complex signal ''c(t)'' can be seen as two real signal ''i(t), q(t)'' combined to create a complex signal. It can also be represented by its time varying amplitude ''a(t)'' and its phase ''φ(t)''
: <math>c(t) = i(t) + j q(t) = a(t) e^{j\phi (t)} </math>
GNURadio software is mainly used to design and study radio communications. Making high frequency transmission requires modulating a high frequency carrier at frequency ''F<sub>0</sub>''. The most common modulation for analog transmissions are Amplitude modulation (AM) Phase modulation (PM) and Frequency modulation (FM).
For analog AM, the modulated signal ''m(t)'' is simply the mathematical product of the carrier ''c(t)'' and the baseband signal to transmit ''a(t)''. The corresponding hardware is a mixer whose scheme and mathematical representation is a multiplier.
: <math>m(t) = a(t) c(t) = a(t) \cos(2\pi f_0t)</math>
We call ''a(t)'' a baseband signal since its spectrum is in a low frequency range starting near 0 Hz (For example [0-20kHz] for an HIFI audio signal).
N.B. Negative frequencies are often omitted in spectrum representation since, for real signal (''a(t)'', ''m(t)'' are real) the power spectrum are symmetric around zero as will be detailed later.
Up to now we have been dealing with real signal. The need for complex signal appears in the next step. Simulation requires sampled signal. Sampling is the operation of observing a continuous signal and taking a finite number of sample at a given sampling rate ''f<sub>s</sub>'' (i.e; one sample each 1/''f<sub>s</sub>'' second). simulator can only make calculations on a finite number of samples, they require sampled signal. Nyquist Sampling theorem states that the sampling rate must be greater than twice the maximum frequency ''F<sub>Max</sub>'' to be able to reconstruct the original signal from the sampled signal.
: <math>f_s > F_{Max}</math>
* study the modulator part which simply multiply the baseband signal and the sine carrier
* look at the influence of the carrier frequency on the modulated signal spectrum (carrier frequency must stay lower than half the sampling rate)
* look at the spectrum shape for sawtooth input and random bit sequence (QT Gui chooser and Selector)
* When transmitting random bits, you can desactivate the interpolating FIR Filter and replace it by a root raised cosine filter
== Spectrum properties of signals==
: <math>X(f) =\int{x(t)e^{-2j\pi ft}dt</math>
: <math>|X(-f)|=|X(f)|</math>
: <math>\text{arg}\{X(-f)\}=-\text{arg}\{X(f)\}</math>
* any complex signal having non null imaginary part exhibits a non-symmetric spectrum.
* as a consequence, every non-symmetric spectrum correspond to a complex signal
=== sampled signals ===
We consider a signal ''x(t))'' and we note its sampled version ''x<sub>s</sub>(t))'' sampled at frequency ''F<sub>s</sub>''.
The spectrum ''X<sub>s</sub>(f)'' of the sampled signal is a periodic function of period ''F<sub>s</sub>''.
: <math>X_s(f)=\sum_k X(f-kF_s)</math>
: <math>m^{bb}(t)=a(t) e^{j \phi(t)} =i(t) + j q(t)</math>
: <math>m(t)=\text{Re} \left[ \big(i(t)+jq(t)\big) e^{j2\pi F_0t} \right]</math>
: <math>m(t)=i(t) \cos(2\pi F_0t) -q(t)) \sin(2\pi F_0t)</math>
The phase ''φ(t)'' of the modulated signal ''m(t)'', is identical to the phase of the complex signal ''c(t)=i(t)+jq(t)''. The equivalent baseband signal ''c(t)'' is represented in a complex plane also refeered to as the IQ plane. The resulting ''m(t)'' can be any modulated in AM, PM or even FM signal.
* <math>i(t)=a(t) \cos(\phi(t))</math>
* <math>q(t)=a(t) \sin(\phi(t))</math>
: <math>\hat{i}(t)=i(t) </math>
: <math>\hat{q}(t)=q(t) </math>
First, we suppose our emitter carrier is <math> \cos (2\pi F_0t)</math> so that every equivalent baseband signal will be defined according to this reference.
=== complex envelope of a pure sine wave ===
We will consider a pure sine wave, close to the carrier having a ''Δf'' frequency shift and ''φ'' phase shift as compared to the carrier. After some math we get its complex equivalent signal.
: <math>m(t) = A \cos (2\pi (F_0+\Delta f)t+ \phi)</math>
: <math>\tilde{m}(t) = A e^{(j(2\pi (F_0+\Delta f)t + \phi))} </math>
: <math>m^{bb}(t) = A e^{2j(\pi (\Delta ft+\phi)} = A e^{j \Delta ft} e^{j\phi}</math>
: <math>m^{bb}(t) = A e^{2j\pi \Delta ft}</math>
* What do you observe when ''Δf''=-1/12=-0.0833 ?
* Slowly increase ''Δf'' to reach ''f<sub>s</sub>/2'' and observe the spectrum really has a single peak at ''Δf''. Explain your observation when ''Δf>f<sub>s</sub>/2'.
This example will consider signal baseband signal ''a(t)'' modulating a carrier at F<sub>0</sub> in AM, and its demodulation. As no phase modulation in used, ''φ(t)''=0 and consequently ''q(t)''=0.
: <math>m(t) = a(t) \cos (2\pi (F_0t)=i(t) \cos (2\pi F_0t) -q(t) \sin (2\pi F_0t)= </math>
: <math>i(t) = a(t) </math>
: <math>q(t) = 0</math>
AM modulation is a special case for which the equivalent baseband complex signal has a null imaginary part and is real. Considering the schematic diagram of an IQ modulator demodulator, when ''q(t))'' is null the diagram is simplified (imaginary path is not used) yielding the well known AM modulation/demodulation scheme.
* orange connections correspond to real signals (float numbers)
* blue connections correspond to complex signals (complex numbers)
Flowgraph [[media:IQ_tutorial_AM_TX_complex.grc|IQ_tutorial_AM_TX_complex.grc]] contains two equivalent diagram for an AM modulation with a sawtooth input:
[[File:AM TX complex.png|thumb|600px|AM modulator flowgraph]]
* the lower one uses an equivalent baseband representation.
** sampling frequency is 25 kHz, 5 times the input rate which equal 5 kHz.
** the sawtoooth correspond to identical generator in both modulator
** each blue input or output is the baseband equivalent of the corresponding signal in the upper AM modulator.
** the carrier frequency can be any value compatible with connected Hardware
** the carrier equivalent signal equal 1 (as stated in the previous section) so it has been disable and replaced by a complex constant
** the Hardware input is the complex equivalent baseband signal
** the carrier frequency is not needed nor used in complex blocks; excepted in the QT GUI spectrum to label the center frequency which is 0 but correspond to the ''F>sub>0</sub>''.
* save the current flowgraph as IQ_tutorial_AM_TX_complex_2.grc.
* remove the complex multiplier, the carrier equivalent baseband (complex constant=1) and the carrier complex source (the disabled one). Reconnect complex to float and throttle blocks to obtain the flowgraph sketched on the right.
==== Further work : construct the AM demodulator flowgraph ====
Use the flowgraph IQ_tutorial_AM_TX_complex_2.grc that you created, add blocks to perform AM demodulation and to recover the input signal ''a(t)'' from the modulated signal. You have to use the equivalent baseband representation of the demodulator.
| 1st indication
| Remind and use the relation between ''a(t)'' and ''c(t)=i(t)+jq(t)'' given above.
| 2nd indication
| An AM demodulator extract the amplitude (magnitude) of the modulated signal...
| Solution including filter
| [[media:IQ_tutorial_AM_TX_complex_3.grc|IQ_tutorial_AM_TX_complex_3.grc]]
=== QPSK example ===
QPSK digital signal exhibit four phase state <math>\phi \in \{\pi/4, 3\pi/4, -3\pi/4, -\pi/4 \}</math>.
: <math>m^{bb}(t) \in \{1+j , -1+j , -1-j, 1-j \}</math>
Normally, we would use a GNURadio "constellation modulator" to simulate QPSK as is done in the excellent [[Guided_Tutorial_PSK_Demodulation|Guided Tutorial on PSK Demodulation]].
[[File:IQ_complex_tutorial_QPSK.png|thumb|600px|QPSK modulator]]
For the present tutorial we will simulate a QPSK without Nyquist filter in order to get phase states which can be simply displayed on a constellation sink. This is not possible with constellation modulator. Our QPSK modulator (complex representation) is build taking into account that the complex signal exhibit 4 different values, its obvious that both ''i(t)'' and ''q(t)'' have only 2 states so they are binary symmetric NRZ line codes:
: <math>i(t), q(t) \in \{+1, -1\}</math>
Flowgraph [[IQ_tutorial_QPSK.grc]] generates 2 sequences of bits, interpolates them to get 2 binary symmetric NRZ line codes. The NRZ signals are combined to create the complex equivalent baseband signal of the QPSK which can be transmitted to any SDR emitter.
* stop the QT GUI spectrum to observe that the complex baseband signal spectrum is no longer symmetric as expected for complex signals.
* use spectrum averaging to see that despite of the previous observation, the power spectral density (average of the magnitude spectrum) is symmetric around 0 (which correspond to ''F<sub>0</sub>'' for the modulated signal.
* Disable the interpolating filters and enable both root raised cosine filter (the filter used in every QPSK emitter). This yields the real spectrum of a QPSK.
== Equivalent baseband scheme ==
This section is not yet ready... Work in progress !!
* give the equivalent baseband representation of a bandpass filter
* explain what is a complex filter and when a filter is complex
=== Unsynchronized demodulator ===
* give the equivalent baseband representation of an unsynchronized receiver (using results obtained above)
* give the equivalent baseband representation of noise source
* explain what is a complex noise.
== Tx/Rx QPSK : the carrier asynchronism problem ==
=== GNURadio Channel model ===
* explain how carrier asynchronism is simulated with the channel model
* give a flowgraph illustrating the problem (with the QPSK used above)
* give a basic solution to compensate carrier asynchronism by muttiplying by exp(2 j pi df t)
* explain the Xlating mecanism, spectrum cycling-shifted
* give a basic solution to compensate carrier asynchronism with a Xlating filter
* note that this solution is basic and not sufficient in real hardware
=== Asynchronism in real hardware ===
* give a recorded example of unmodulated carrier emitted and received by a SDR Dongle
* try to compensate asynchronism
* result : we need more robust block to synchronize: see PSK guided tutorial ...
* <span id="ancre1"[1]> [1] Proakis J., ''Digital Communication'', McGraw Hill Series in Electrical and Computer Engineering, Singapore, 1989</span>
* <span id="ancre2"[2]> [2] Gallager R., ''Principles of digital communication'', Cambridge University Press Cambridge, UK, 2008</span>
* <span id="ancre3"[3]> [3] Benedetto S. and Biglieri E., ''Principles of digital transmission : with wireless applications'', Kluwer Academic/Plenum Publishers, NY, 1999</span>
{\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(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\}}
|
Linear Kernel: Why is it recommended for text classification ?
The Support Vector Machine can be viewed as a kernel machine. As a result, you can change its behavior by using a different kernel function.
The most popular kernel functions are :
the RBF (Gaussian) kernel
the string kernel
The linear kernel is often recommended for text classification
It is interesting to note that :
The original optimal hyperplane algorithm proposed by Vapnik in 1963 was a linear classifier [1]
That's only 30 years later that the kernel trick was introduced.
If it is the simpler algorithm, why is the linear kernel recommended for text classification?
Text is often linearly separable
Most of text classification problems are linearly separable [2]
Linear kernel works well with linearly separable data
Text has a lot of features
The linear kernel is good when there is a lot of features. That's because mapping the data to a higher dimensional space does not really improve the performance. [3] In text classification, both the numbers of instances (document) and features (words) are large.
As we can see in the image above, the decision boundary produced by a RBF kernel when the data is linearly separable is almost the same as the decision boundary produced by a linear kernel. Mapping data to a higher dimensional space using an RBF kernel was not useful.
Linear kernel is faster
Training a SVM with a linear kernel is faster than with another kernel. Particularly when using a dedicated library such as LibLinear [3]
Less parameters to optimize
When you train a SVM with a linear kernel, you only need to optimize the C regularization parameter. When training with other kernels, you also need to optimize the
\gamma
parameter which means that performing a grid search will usually take more time.
Linear kernel is indeed very well suited for text-categorization.
Keep in mind however that it is not the only solution and in some case using another kernel might be better.
The recommended approach for text classification is to try a linear kernel first, because of its advantages.
If however you search to get the best possible classification performance, it might be interesting to try the other kernels to see if they help.
[1] Support Vector Machines Article
[2] Text Categorization with Support Vector Machines: Learning with Many Relevant Features
[3] A Practical Guide to Support Vector Classification
Categories SVM in Practice, Text classification Tags Linear kernel, Text classification Post navigation
|
Home : Support : Online Help : Connectivity : Database Package : Statement : Execute
execute an arbitrary SQL string
statement:-Execute( sql )
string; one or more SQL statements to execute
Execute passes a string of SQL statements to the database to be executed. The string can contain multiple statements and generate multiple return values.
Execute returns the return value of the first statement. The return values of subsequent statements are accessible through the NextResult command.
If only a single SQL statement is being executed, using ExecuteUpdate or ExecuteQuery may be more convenient. However, if a large number of such statements are executed, Execute is more efficient.
\mathrm{driver}≔\mathrm{Database}[\mathrm{LoadDriver}]\left(\right):
\mathrm{conn}≔\mathrm{driver}:-\mathrm{OpenConnection}\left(\mathrm{url},\mathrm{name},\mathrm{pass}\right):
\mathrm{stat}≔\mathrm{conn}:-\mathrm{CreateStatement}\left(\right):
\mathrm{res}≔\mathrm{stat}:-\mathrm{Execute}\left("SELECT name FROM animals WHERE id = 1; SELECT name FROM animals WHERE id = 2; SELECT name FROM animals WHERE id = 3;"\right):
\mathrm{res}:-\mathrm{Next}\left(\right);
\mathrm{res}:-\mathrm{GetData}\left("name"\right)
\textcolor[rgb]{0,0,1}{"fish"}
\mathrm{res}≔\mathrm{stat}:-\mathrm{NextResult}\left(\right):
\mathrm{res}:-\mathrm{Next}\left(\right):
\mathrm{res}:-\mathrm{GetData}\left("name"\right)
\textcolor[rgb]{0,0,1}{"dog"}
\mathrm{res}≔\mathrm{stat}:-\mathrm{NextResult}\left(\right):
\mathrm{res}:-\mathrm{Next}\left(\right):
\mathrm{res}:-\mathrm{GetData}\left("name"\right)
\textcolor[rgb]{0,0,1}{"cat"}
|
Small_but_significant_and_non-transitory_increase_in_price Knowpia
In 1982 the U.S. Department of Justice Merger Guidelines introduced the SSNIP test as a new method for defining markets and for measuring market power directly. In the EU it was used for the first time in the Nestlé/Perrier case in 1992 and has been officially recognized by the European Commission in its "Commission's Notice for the Definition of the Relevant Market" in 1997.[1]
The original concept is believed to have been proposed first in 1959 by economist David Morris Adelman of the Aston University.[2] Several other individuals formulated, apparently independently, similar conceptual approaches during the 1970s.[3] The SSNIP approach was implemented by F. M. Scherer in three antitrust cases: in a 1972 Justice Department attempt to enjoin the merger of Associated Brewing Co. and G. W. Heileman Co., in 1975 during hearings on the U.S. government's monopolization case against IBM, and in a 1981 proceeding precipitated by Marathon Oil Company's effort to avert takeover by Mobil Oil Corporation.[4] Scherer also proposed the basic concept underlying SSNIP along with limitations posed by what has come to be known as "the cellophane fallacy" in the second (1980) edition of his industrial organization textbook.[5] Historical retrospectives suggest that early proponents were unaware of other individuals' conceptual proposals.
The SSNIP test seeks to identify the smallest relevant market within which a hypothetical monopolist or cartel could impose a profitable significant increase in price. The relevant market consists of a "catalogue" of goods and/or services which are considered substitutes by the customer. Such a catalogue is considered "worth monopolising" if should only one single supplier provided it, that supplier could profitably increase its price without its customers turning away and choosing other goods and services from other suppliers.
An alternative method for applying the SSNIP test where demand elasticities cannot be estimated, involves estimating the "critical loss." The critical loss is defined as the maximum sales loss that could be sustained as a result of the price increase without making the price increase unprofitable. Where the likely loss of sales to the hypothetical monopolist (cartel) is less than the Critical Loss, then a 5% price increase would be profitable and the market is defined.[6]
The test consists of observing whether a small increase in price (in the range of 5 to 10 percent) would provoke a significant number of consumers to switch to another product (in fact, substitute product). In other words, it is designed to analyse whether that increase in price would be profitable or if, instead, it would just induce substitution, making it unprofitable.
In general, one uses databases from the firms which may include data on variables such as costs, prices, revenue or sales and over a sufficiently long period (generally over at least two years).
In economic terms, what the SSNIP test does is to calculate the residual elasticity of demand of the firm. That is, how a change in prices by the firm affects its own demand.
As an example, let's suppose the following situation for a firm:
Variable cost per unit = 5
In this case, the firm would make profits equal to 5000:
{\displaystyle \mathrm {Price} \times \mathrm {Sales} -\mathrm {Variable\ cost} \times \mathrm {Sales} }
Now suppose the firm decides to increase its price by a 10 percent, which would imply that the new price would be 11 (10 percent increase). Suppose that the new situation facing the firm is therefore:
{\displaystyle \mathrm {Price} \times \mathrm {Sales} -\mathrm {Variable\ cost} \times \mathrm {Sales} }
As can be seen, such an increase in prices would induce a certain substitution for our hypothetical firm, in fact, 200 units less will be sold. This may be so because some consumers have started to buy a substitute product, the same consumers have bought a smaller quantity of the product given its price increase or maybe because they have stopped buying that type of product.
If we want to know whether such price increase has been profitable, we should solve the following equation:
{\displaystyle \mathrm {Profits} =\mathrm {Price} \times \mathrm {Sales} -\mathrm {Variable\ cost\ per\ unit} \times \mathrm {Sales} =4800.}
In our example, the increase in price produces too much consumer substitution which is not compensated by the increase in price nor the reduction in costs. Overall, the firm would make less profits (4800 compared to 5000). In other words, there are other substitute products that should be included in the relevant market and the product of the firm does not constitute by itself a separate relevant market. The "market" formed by this only product is not "worth monopolising" as an increase in prices would not be profitable. The investigation should continue by including new products which we may guess are substitutes of the one under investigation.
We already know that the previous product is not by itself a relevant market because there do exist other substitute products. Let’s suppose that the previous firm (A) tells us that it considers as competitors the products of B and C. In this case, in the second phase we should include these products to our analysis and repeat the exercise.
Price A = 10; Sales A = 1000; Variable cost per unit A = 5
Price B = 13; Sales B = 800; Variable cost per unit B = 4
Price C = 9; Sales C = 1100; Variable cost per unit C = 4
Given that we want to know if products A, B and C constitute a relevant market, the exercise would consist in supposing that an hypothetical monopolist X would control all three products. In that case, the monopolist would make profits of:
{\displaystyle 10\times 1000-5\times 1000+13\times 800-4\times 800+9\times 1100-4\times 1100=17700}
Now suppose that monopolist X decides to increase the price of product A, maintaining the price of B and C constant. Suppose that a 10 percent increase in the price of A provokes the following situation:
Price A = 11; Sales A = 800; Variable cost per unit A = 5
This means that the price increase of A would provoke that 200 units less of A be sold and instead, 100 more units of B and C will be sold respectively. Given that our hypothetical monopolist controls all three products, its profits will be:
{\displaystyle 11\times 800-5\times 800+13\times 900-4\times 900+9\times 1200-4\times 1200=18900}
As can be seen, the monopolist controlling A, B and C would profitably increase the price of A by 10 percent, in other words, these three products do constitute a market "worth monopolising" and therefore constitutes a relevant market. This result is because X controls all three products which are the only substitutes of A. Thus, X knows that even if its increase in price of A will generate some substitution, a significant share of these consumers will end up buying other products which he controls, therefore overall, his profits will not be reduced but rather increased.
If we had found that such an increase would not have been profitable, we should further include new products which we may imagine are substitutes in a third phase until we arrive at a situation in which such an increase in price would have been profitable, indicating that those products do constitute a relevant market.
Despite its widespread usage, the SSNIP test is not without problems. Specifically:
In evaluating a merger of A and B, performing the SSNIP test on A's products will not necessarily yield the same relevant market as applying the SSNIP test on B's products. (This presented a legal issue in the 2000 Bayer/Aventis Crop Science merger.) So a competition authority investigating A should only consider competitive pressure (or lack thereof) that B puts on A - reverse pressure from A to B is irrelevant.
The SSNIP test relies on total losses in sales after a 5% price increase, not just substitutions to a particular competitor's product. Thus it includes sales losses due to outpricing by competitor 1, a more attractive deal by competitor 2, or to customers saving their money instead of spending it on any of those competitors' products. Mathematically speaking, what is important is the own-price elasticity of the good in question, not its cross-price elasticity relative to any other product. Cross-price elasticities can help determine what products are substitutes (high, positive cross-price elasticities) in succeeding iterations of the SSNIP test, but the attractiveness of controlling a market can only be evaluated with an own-price elasticity.
In succeeding iterations of larger market control, the hypothetical price increase still only applies to the first product. The gains to the hypothetical owner of substitute products come from increasing the price of one base product and thus getting higher revenue from it and spillover from its competitors - not from increasing the price of the base product and the price for its competitors.
The SSNIP test only measures competition based on price and thus cannot be considered a catch-all or fully sufficient tool for defining markets.[7]
Furthermore, many economists have noted an important pitfall in the use of demand elasticities when inferring both the market power and the relevant market. The problem arises from the fact that economic theory predicts that any profit-maximizing firm will set its prices at a level where demand for its product is elastic. Therefore, when a monopolist sets its prices at a monopoly level it may happen that two products appear to be close substitutes whereas at competitive prices they are not. In other words, it may happen that using the SSNIP test one defines the relevant market too broadly, including products which are not substitutes.
This problem is known in the literature as the cellophane paradox after the celebrated Du Pont case. In this case, Du Pont (a cellophane producer) argued that cellophane was not a separate relevant market since it competed with flexible packaging materials such as aluminum foil, wax paper and polyethylene. The problem was that Du Pont, being the sole producer of cellophane, had set prices at the monopoly level, and it was at this level that consumers viewed those other products as substitutes. Instead, at the competitive level, consumers viewed cellophane as a unique relevant market (a small but significant increase in prices would not have them switching to goods like wax or the others). In the case, the US Supreme Court failed to recognise that a high own-price elasticity may mean that a firm is already exercising monopoly power.[8]
^ See Crai International discussion on the issue "Archived copy" (PDF). Archived from the original (PDF) on 2007-09-28. Retrieved 2015-02-06. {{cite web}}: CS1 maint: archived copy as title (link).
^ See M. A. Adelman, "Economic Aspects of the Bethlehem Opinion," Virginia Law Review, vol. 45 (1959), p. 686.
^ See Gregory Werden, "The 1982 Merger Guidelines and the Ascent of the Hypothetical Monopolist Paradigm," Antitrust Law Review, vol. 71 (2003), pp. 253-269.
^ See F. M. Scherer, "On the Paternity of a Market Delineation Approach," working paper, American Antitrust Institute web site, January 2009.
^ F. M. Scherer, Industrial Market Structure and Economic Performance (1980), p. 517.
^ The SSNIP Test.
^ The SSNIP test: some common misconceptions
^ Market Definition and Market Power in Competition Analysis. Archived 2007-06-13 at the Wayback Machine
|
Pfam: Family: SKI (PF01202)
Family: SKI (PF01202)
Summary: Shikimate kinase
Wikipedia: Shikimate kinase
This is the Wikipedia entry entitled "Shikimate kinase". More...
Shikimate kinase Edit Wikipedia article
In enzymology, a shikimate kinase (EC 2.7.1.71) is an enzyme that catalyzes the chemical reaction
ATP + shikimate
{\displaystyle \rightleftharpoons }
ADP + shikimate 3-phosphate
Thus, the two substrates of this enzyme are ATP and shikimate, whereas its two products are ADP and shikimate 3-phosphate.
This enzyme belongs to the family of transferases, specifically those transferring phosphorus-containing groups (phosphotransferases) with an alcohol group as acceptor. The systematic name of this enzyme class is ATP:shikimate 3-phosphotransferase. Other names in common use include shikimate kinase (phosphorylating), and shikimate kinase II. This enzyme participates in phenylalanine, tyrosine and tryptophan biosynthesis.
As of late 2007, 26 structures have been solved for this class of enzymes, with PDB accession codes 1E6C, 1KAG, 1L4U, 1L4Y, 1SHK, 1U8A, 1VIA, 1WE2, 1ZUH, 1ZUI, 1ZYU, 2DFN, 2DFT, 2G1J, 2G1K, 2IYQ, 2IYR, 2IYS, 2IYT, 2IYU, 2IYV, 2IYW, 2IYX, 2IYY, 2IYZ, and 2SHK.
Morell H, Sprinson DB (1968). "Shikimate kinase isoenzymes in Salmonella typhimurium". J. Biol. Chem. 243: 676–7. PMID 4866525.
Shikimate kinase Provide feedback
SCOOP: AAA AAA_14 AAA_16 AAA_17 AAA_18 AAA_22 AAA_24 AAA_28 AAA_33 AAA_5 ABC_tran ADK APS_kinase ATP_bind_2 CoaE CPT Cytidylate_kin Cytidylate_kin2 dNK KTI12 Mg_chelatase NACHT Rad17 RuvB_N tRNA_lig_kinase TsaE Zeta_toxin
Similarity to PfamA using HHSearch: AAA CoaE APS_kinase dNK Thymidylate_kin Cytidylate_kin Cytidylate_kin2 AAA_17 AAA_18 AAA_33
SCOP: 2shk
Shikimate kinase ( EC ) catalyses the fifth step in the biosynthesis of aromatic amino acids from chorismate (the so-called shikimate pathway) [ PUBMED:7612934 ]. The enzyme catalyses the following reaction:
ATP + shikimate = ADP + shikimate-3-phosphate
The protein is found in bacteria (gene aroK or aroL), plants and fungi (where it is part of a multifunctional enzyme that catalyses five consecutive steps in this pathway). In 1994, the 3D structure of shikimate kinase was predicted to be very close to that of adenylate kinase, suggesting a functional similarity as well as an evolutionary relationship [ PUBMED:7703851 ]. This prediction has since been confirmed experimentally. The protein is reported to possess an alpha/beta fold, consisting of a central sheet of five parallel beta-strands flanked by alpha-helices. Such a topology is very similar to that of adenylate kinase [ PUBMED:9600856 ].
The N terminus of threonine synthase-like 1 from metazoan shares protein sequence similarity with shikimate kinase and is included in this entry. However, their functions may be different.
For those sequences which have a structure in the Protein DataBank, we use the mapping between UniProt, PDB and Pfam coordinate systems from the PDBe group, to allow us to map Pfam domains onto UniProt sequences and three-dimensional protein structures. The table below shows the structures on which the SKI domain has been found. There are 76 instances of this domain found in the PDB. Note that there may be multiple copies of the domain in a single PDB structure, since many structures contain multiple copies of the same protein sequence.
A0A0D2FBH4 View 3D Structure Click here
A0A0H3GYI1 View 3D Structure Click here
A0A0H3GZE8 View 3D Structure Click here
A0A0K0EHC9 View 3D Structure Click here
A0A0P0WRJ9 View 3D Structure Click here
A0A0R0F8S5 View 3D Structure Click here
A0A0R0IZ75 View 3D Structure Click here
A0A0R0JH37 View 3D Structure Click here
A0A175VQK6 View 3D Structure Click here
A0A175WJI8 View 3D Structure Click here
A0A1C1CEP6 View 3D Structure Click here
A0A1C1CXW8 View 3D Structure Click here
A0A1D6DWX6 View 3D Structure Click here
A0A1D6HKK5 View 3D Structure Click here
A0A1D6K9Y9 View 3D Structure Click here
A0A1D6KDZ4 View 3D Structure Click here
A0KN29 View 3D Structure Click here
A0LUH0 View 3D Structure Click here
A1BER1 View 3D Structure Click here
A1CP85 View 3D Structure Click here
A1SB14 View 3D Structure Click here
A1SRB6 View 3D Structure Click here
A1TKW2 View 3D Structure Click here
A1WZB0 View 3D Structure Click here
A3LSZ2 View 3D Structure Click here
A3MYR6 View 3D Structure Click here
A3Q9D7 View 3D Structure Click here
A4FBE7 View 3D Structure Click here
A4RD09 View 3D Structure Click here
A4SFH8 View 3D Structure Click here
A4VTX6 View 3D Structure Click here
|
Cubic feet and gallons
Equivalence between cubic feet and gallons
How do I convert cubic feet to gallons?
With our cubic feet to gallons converter, you can easily switch between both volume units. With this tool, you will be able to convert cubic feet to gallons just by typing the volume in cubic feet in the converter.
In this short text, you will learn:
How to convert cubic feet to gallons;
Cubic foot and gallon definitions; and
Difference between US and UK gallons.
Cubic feet (ft³ or cu ft) and gallons (gal) are both imperial/US volume units. What is volume exactly? In simple words, volume is the space an object occupies. We use volume units to measure, for example, the amount of water in a pool, a car's trunk size, amount of liquid inside a bottle, and several other things.
A foot is a unit of length equal to 12 inches or 30.48 cm. A cubic foot is the amount of space enclosed by a box with a 1 ft length of its sides.
Similarly, a US gallon is defined as 231 cubic inches or 231 times the amount of space a box with 1-inch sides occupies.
On the other hand a UK gallon is defined as 4.54609 cubic decimeters.
🙋 Are you having trouble understanding all these different definitions and units? Check Omni's volume conversion tool! It has all the explanations and examples you need to comprehend the subject thoroughly.
Since both are volume units, we can switch between them using a conversion factor which is just a number to indicate the proportion between one unit and another. In the case of cubic feet and gallons:
1\ \text{ft³} = 7.48052\ \text{US gal}\\ 1\ \text{ft³} = 6.22884\ \text{UK gal}
We will need to multiply the volume by this number to convert from cubic feet to gallons or divide it by the same number to switch from gallons to cubic feet.
To convert cubic feet to gallons:
Write down the volume expressed in cubic feet.
Multiply this number by 7.48052 to convert to US gallons.
Otherwise, multiply the volume in cubic feet by 6.22884 to convert to UK gallons.
Feel free to check these other useful converters/tools related to volume calculations:
How many gallons are 1.5 cubic feet?
1.5 cubic feet are 11.2208 US gallons or 9.34325 UK gallons. To convert 1.5 cubic feet to gallons, you need to multiply 1.5 by 7.48052 or 6.22884 if you want the result expressed in US or UK gallons respectively.
Is cubic foot a metric unit?
No, a cubic foot is not a metric unit. It is a US and Imperial system unit used to measure volume.
|
Variogram - Wikipedia
Spatial statistics function
In spatial statistics the theoretical variogram
{\displaystyle 2\gamma (\mathbf {s} _{1},\mathbf {s} _{2})}
is a function describing the degree of spatial dependence of a spatial random field or stochastic process
{\displaystyle Z(\mathbf {s} )}
. The semivariogram
{\displaystyle \gamma (\mathbf {s} _{1},\mathbf {s} _{2})}
is half the variogram.
In the case of a concrete example from the field of gold mining, a variogram will give a measure of how much two samples taken from the mining area will vary in gold percentage depending on the distance between those samples. Samples taken far apart will vary more than samples taken close to each other.
"Semivariance" redirects here. For the measure of downside risk, see Variance § Semivariance.
The semivariogram
{\displaystyle \gamma (h)}
was first defined by Matheron (1963) as half the average squared difference between the values at points (
{\displaystyle \mathbf {s} _{1}}
{\displaystyle \mathbf {s} _{2}}
) separated at distance
{\displaystyle h}
.[1][2] Formally
{\displaystyle \gamma (h)={\frac {1}{2V}}\iiint _{V}\left[f(M+h)-f(M)\right]^{2}dV,}
{\displaystyle M}
is a point in the geometric field
{\displaystyle V}
{\displaystyle f(M)}
is the value at that point. The triple integral is over 3 dimensions.
{\displaystyle h}
is the separation distance (e.g., in meters or km) of interest. For example, the value
{\displaystyle f(M)}
could represent the iron content in soil, at some location
{\displaystyle M}
(with geographic coordinates of latitude, longitude, and elevation) over some region
{\displaystyle V}
with element of volume
{\displaystyle dV}
. To obtain the semivariogram for a given
{\displaystyle \gamma (h)}
, all pairs of points at that exact distance would be sampled. In practice it is impossible to sample everywhere, so the empirical variogram is used instead.
The variogram is twice the semivariogram and can be defined, equivalently, as the variance of the difference between field values at two locations (
{\displaystyle \mathbf {s} _{1}}
{\displaystyle \mathbf {s} _{2}}
, note change of notation from
{\displaystyle M}
{\displaystyle \mathbf {s} }
{\displaystyle f}
{\displaystyle Z}
) across realizations of the field (Cressie 1993):
{\displaystyle 2\gamma (\mathbf {s} _{1},\mathbf {s} _{2})={\text{var}}\left(Z(\mathbf {s} _{1})-Z(\mathbf {s} _{2})\right)=E\left[((Z(\mathbf {s} _{1})-\mu (\mathbf {s} _{1}))-(Z(\mathbf {s} _{2})-\mu (\mathbf {s} _{2})))^{2}\right].}
If the spatial random field has constant mean
{\displaystyle \mu }
, this is equivalent to the expectation for the squared increment of the values between locations
{\displaystyle \mathbf {s} _{1}}
{\displaystyle s_{2}}
(Wackernagel 2003) (where
{\displaystyle \mathbf {s} _{1}}
{\displaystyle \mathbf {s} _{2}}
are points in space and possibly time):
{\displaystyle 2\gamma (\mathbf {s} _{1},\mathbf {s} _{2})=E\left[\left(Z(\mathbf {s} _{1})-Z(\mathbf {s} _{2})\right)^{2}\right].}
In the case of a stationary process, the variogram and semivariogram can be represented as a function
{\displaystyle \gamma _{s}(h)=\gamma (0,0+h)}
of the difference
{\displaystyle h=\mathbf {s} _{2}-\mathbf {s} _{1}}
between locations only, by the following relation (Cressie 1993):
{\displaystyle \gamma (\mathbf {s} _{1},\mathbf {s} _{2})=\gamma _{s}(\mathbf {s} _{2}-\mathbf {s} _{1}).}
If the process is furthermore isotropic, then the variogram and semivariogram can be represented by a function
{\displaystyle \gamma _{i}(h):=\gamma _{s}(he_{1})}
of the distance
{\displaystyle h=\|\mathbf {s} _{2}-\mathbf {s} _{1}\|}
only (Cressie 1993):
{\displaystyle \gamma (\mathbf {s} _{1},\mathbf {s} _{2})=\gamma _{i}(h).}
The indexes
{\displaystyle i}
{\displaystyle s}
are typically not written. The terms are used for all three forms of the function. Moreover, the term "variogram" is sometimes used to denote the semivariogram, and the symbol
{\displaystyle \gamma }
is sometimes used for the variogram, which brings some confusion.[3]
The semivariogram is nonnegative
{\displaystyle \gamma (\mathbf {s} _{1},\mathbf {s} _{2})\geq 0}
, since it is the expectation of a square.
{\displaystyle \gamma (\mathbf {s} _{1},\mathbf {s} _{1})=\gamma _{i}(0)=E\left((Z(\mathbf {s} _{1})-Z(\mathbf {s} _{1}))^{2}\right)=0}
at distance 0 is always 0, since
{\displaystyle Z(\mathbf {s} _{1})-Z(\mathbf {s} _{1})=0}
A function is a semivariogram if and only if it is a conditionally negative definite function, i.e. for all weights
{\displaystyle w_{1},\ldots ,w_{N}}
{\displaystyle \sum _{i=1}^{N}w_{i}=0}
{\displaystyle s_{1},\ldots ,s_{N}}
{\displaystyle \sum _{i=1}^{N}\sum _{j=1}^{N}w_{i}\gamma (\mathbf {s} _{i},\mathbf {s} _{j})w_{j}\leq 0}
which corresponds to the fact that the variance
{\displaystyle var(X)}
{\displaystyle X=\sum _{i=1}^{N}w_{i}Z(x_{i})}
is given by the negative of this double sum and must be nonnegative.[disputed – discuss]
If the covariance function of a stationary process exists it is related to variogram by
{\displaystyle 2\gamma (\mathbf {s} _{1},\mathbf {s} _{2})=C(\mathbf {s} _{1},\mathbf {s} _{1})+C(\mathbf {s} _{2},\mathbf {s} _{2})-2C(\mathbf {s} _{1},\mathbf {s} _{2})}
If a stationary random field has no spatial dependence (i.e.
{\displaystyle C(h)=0}
{\displaystyle h\not =0}
), the semivariogram is the constant
{\displaystyle var(Z(\mathbf {s} ))}
everywhere except at the origin, where it is zero.
{\displaystyle \gamma (\mathbf {s} _{1},\mathbf {s} _{2})=E\left[|Z(\mathbf {s} _{1})-Z(\mathbf {s} _{2})|^{2}\right]=\gamma (\mathbf {s} _{2},\mathbf {s} _{1})}
{\displaystyle \gamma _{s}(h)=\gamma _{s}(-h)}
If the random field is stationary and ergodic, the
{\displaystyle \lim _{h\to \infty }\gamma _{s}(h)=var(Z(\mathbf {s} ))}
corresponds to the variance of the field. The limit of the semivariogram is also called its sill.
In summary, the following parameters are often used to describe variograms:
{\displaystyle n}
: The height of the jump of the semivariogram at the discontinuity at the origin.
{\displaystyle s}
: Limit of the variogram tending to infinity lag distances.
{\displaystyle r}
: The distance in which the difference of the variogram from the sill becomes negligible. In models with a fixed sill, it is the distance at which this is first reached; for models with an asymptotic sill, it is conventionally taken to be the distance when the semivariance first reaches 95% of the sill.
Empirical variogram[edit]
Generally, an empirical variogram is needed for measured data, because sample information
{\displaystyle Z}
is not available for every location. The sample information for example could be concentration of iron in soil samples, or pixel intensity on a camera. Each piece of sample information has coordinates
{\displaystyle \mathbf {s} =(x,y)}
for a 2D sample space where
{\displaystyle x}
{\displaystyle y}
are geographical coordinates. In the case of the iron in soil, the sample space could be 3 dimensional. If there is temporal variability as well (e.g., phosphorus content in a lake) then
{\displaystyle \mathbf {s} }
could be a 4 dimensional vector
{\displaystyle (x,y,z,t)}
. For the case where dimensions have different units (e.g., distance and time) then a scaling factor
{\displaystyle B}
can be applied to each to obtain a modified Euclidean distance.[4]
Sample observations are denoted
{\displaystyle Z(\mathbf {s} _{i})=z_{i}}
. Samples may be taken at
{\displaystyle k}
total different locations. This would provide as set of samples
{\displaystyle z_{1},\ldots ,z_{k}}
at locations
{\displaystyle \mathbf {s} _{1},\ldots ,\mathbf {s} _{k}}
. Generally, plots show the semivariogram values as a function of sample point separation
{\displaystyle h}
. In the case of empirical semivariogram, separation distance bins
{\displaystyle h\pm \delta }
are used rather than exact distances, and usually isotropic conditions are assumed (i.e., that
{\displaystyle \gamma }
is only a function of
{\displaystyle h}
and does not depend on other variables such as center position). Then, the empirical semivariogram
{\displaystyle {\hat {\gamma }}(h\pm \delta )}
can be calculated for each bin:
{\displaystyle {\hat {\gamma }}(h\pm \delta ):={\frac {1}{2|N(h\pm \delta )|}}\sum _{(i,j)\in N(h\pm \delta )}|z_{i}-z_{j}|^{2}}
Or in other words, each pair of points separated by
{\displaystyle h}
(plus or minus some bin width tolerance range
{\displaystyle \delta }
) are found. These form the set of points
{\displaystyle N(h\pm \delta )\equiv \{(\mathbf {s} _{i},\mathbf {s} _{j}):|\mathbf {s} _{i},\mathbf {s} _{j}|=h\pm \delta ;i,j=1,\ldots ,N\}}
. The number of these points in this bin is
{\displaystyle |N(h\pm \delta )|}
. Then for each pair of points
{\displaystyle i,j}
, the square of the difference in the observation (e.g., soil sample content or pixel intensity) is found (
{\displaystyle |z_{i}-z_{j}|^{2}}
). These squared differences are added together and normalized by the natural number
{\displaystyle |N(h\pm \delta )|}
. By definition the result is divided by 2 for the semivariogram at this separation.
For computational speed, only the unique pairs of points are needed. For example, for 2 observations pairs [
{\displaystyle (z_{a},z_{b}),(z_{c},z_{d})}
] taken from locations with separation
{\displaystyle h\pm \delta }
only [
{\displaystyle (z_{a},z_{b}),(z_{c},z_{d})}
] need to be considered, as the pairs [
{\displaystyle (z_{b},z_{a}),(z_{d},z_{c})}
] do not provide any additional information.
Variogram models[edit]
The empirical variogram cannot be computed at every lag distance
{\displaystyle h}
and due to variation in the estimation it is not ensured that it is a valid variogram, as defined above. However some Geostatistical methods such as kriging need valid semivariograms. In applied geostatistics the empirical variograms are thus often approximated by model function ensuring validity (Chiles&Delfiner 1999). Some important models are (Chiles&Delfiner 1999, Cressie 1993):
{\displaystyle \gamma (h)=(s-n)(1-\exp(-h/(ra)))+n1_{(0,\infty )}(h).}
{\displaystyle \gamma (h)=(s-n)\left(\left({\frac {3h}{2r}}-{\frac {h^{3}}{2r^{3}}}\right)1_{(0,r)}(h)+1_{[r,\infty )}(h)\right)+n1_{(0,\infty )}(h).}
{\displaystyle \gamma (h)=(s-n)\left(1-\exp \left(-{\frac {h^{2}}{r^{2}a}}\right)\right)+n1_{(0,\infty )}(h).}
{\displaystyle a}
has different values in different references, due to the ambiguity in the definition of the range. E.g.
{\displaystyle a=1/3}
is the value used in (Chiles&Delfiner 1999). The
{\displaystyle 1_{A}(h)}
function is 1 if
{\displaystyle h\in A}
Three functions are used in geostatistics for describing the spatial or the temporal correlation of observations: these are the correlogram, the covariance and the semivariogram. The last is also more simply called variogram.
The empirical variogram is used in geostatistics as a first estimate of the variogram model needed for spatial interpolation by kriging.
Empirical variograms for the spatiotemporal variability of column-averaged carbon dioxide was used to determine coincidence criteria for satellite and ground-based measurements.[4]
Empirical variograms were calculated for the density of a heterogeneous material (Gilsocarbon).[5]
Empirical variograms are calculated from observations of strong ground motion from earthquakes.[6] These models are used for seismic risk and loss assessments of spatially-distributed infrastructure.[7]
The squared term in the variogram, for instance
{\displaystyle (Z(\mathbf {s} _{1})-Z(\mathbf {s} _{2}))^{2}}
, can be replaced with different powers: A madogram is defined with the absolute difference,
{\displaystyle |Z(\mathbf {s} _{1})-Z(\mathbf {s} _{2})|}
, and a rodogram is defined with the square root of the absolute difference,
{\displaystyle |Z(\mathbf {s} _{1})-Z(\mathbf {s} _{2})|^{0.5}}
. Estimators based on these lower powers are said to be more resistant to outliers. They can be generalized as a "variogram of order α",
{\displaystyle 2\gamma (\mathbf {s} _{1},\mathbf {s} _{2})=E\left[\left|Z(\mathbf {s} _{1})-Z(\mathbf {s} _{2})\right|^{\alpha }\right]}
in which a variogram is of order 2, a madogram is a variogram of order 1, and a rodogram is a variogram of order 0.5.[8]
^ Matheron, Georges (1963). "Principles of geostatistics". Economic Geology. 58 (8): 1246–1266. doi:10.2113/gsecongeo.58.8.1246. ISSN 1554-0774.
^ Ford, David. "The Empirical Variogram" (PDF). faculty.washington.edu/edford. Retrieved 31 October 2017.
^ Bachmaier, Martin; Backes, Matthias (2008-02-24). "Variogram or semivariogram? Understanding the variances in a variogram". Precision Agriculture. Springer Science and Business Media LLC. 9 (3): 173–175. doi:10.1007/s11119-008-9056-2. ISSN 1385-2256.
^ a b Nguyen, H.; Osterman, G.; Wunch, D.; O'Dell, C.; Mandrake, L.; Wennberg, P.; Fisher, B.; Castano, R. (2014). "A method for colocating satellite XCO2 data to ground-based data and its application to ACOS-GOSAT and TCCON". Atmospheric Measurement Techniques. 7 (8): 2631–2644. Bibcode:2014AMT.....7.2631N. doi:10.5194/amt-7-2631-2014. ISSN 1867-8548.
^ Schiappapietra, Erika; Douglas, John (April 2020). "Modelling the spatial correlation of earthquake ground motion: Insights from the literature, data from the 2016–2017 Central Italy earthquake sequence and ground-motion simulations". Earth-Science Reviews. 203: 103139. Bibcode:2020ESRv..20303139S. doi:10.1016/j.earscirev.2020.103139.
^ Sokolov, Vladimir; Wenzel, Friedemann (2011-07-25). "Influence of spatial correlation of strong ground motion on uncertainty in earthquake loss estimation". Earthquake Engineering & Structural Dynamics. 40 (9): 993–1009. doi:10.1002/eqe.1074.
^ Olea, Ricardo A. (1991). Geostatistical Glossary and Multilingual Dictionary. Oxford University Press. pp. 47, 67, 81. ISBN 9780195066890.
Cressie, N., 1993, Statistics for spatial data, Wiley Interscience.
Chiles, J. P., P. Delfiner, 1999, Geostatistics, Modelling Spatial Uncertainty, Wiley-Interscience.
Wackernagel, H., 2003, Multivariate Geostatistics, Springer.
Burrough, P. A. and McDonnell, R. A., 1998, Principles of Geographical Information Systems.
Isobel Clark, 1979, Practical Geostatistics, Applied Science Publishers.
Clark, I., 1979, Practical Geostatistics, Applied Science Publishers.
David, M., 1978, Geostatistical Ore Reserve Estimation, Elsevier Publishing.
Hald, A., 1952, Statistical Theory with Engineering Applications, John Wiley & Sons, New York.
Journel, A. G. and Huijbregts, Ch. J., 1978 Mining Geostatistics, Academic Press.
Wikimedia Commons has media related to Variogram.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Variogram&oldid=1068268501"
|
An Investigation of the Effects of Layer Thickness on the Fracture Behavior of Layered NiAl/V Composites | J. Eng. Mater. Technol. | ASME Digital Collection
An Investigation of the Effects of Layer Thickness on the Fracture Behavior of Layered NiAl/V Composites
Department of Materials Science and Engineering, The Ohio State University, 2041 College Road, Columbus, OH 43210-1179
e-mail: li.285@osu.edu
Applied Mechanics Section, Department of Aerospace Engineering, Applied Mechanics and Aviation, The Ohio State University, 2041 College Road, Columbus, OH 43210-1179
Li, M., Wang, R., Katsube, N., and Soboyejo, W. O. (October 1, 1999). "An Investigation of the Effects of Layer Thickness on the Fracture Behavior of Layered NiAl/V Composites." ASME. J. Eng. Mater. Technol. October 1999; 121(4): 453–459. https://doi.org/10.1115/1.2812401
The effects of vanadium layer thickness (100, 200 and 400 μm) on the resistance-curve behavior of NiAl/V, microlaminates are examined in this paper. The fracture resistance of the NiAl microlaminates reinforced with 20 vol.% of vanadium layers is shown to increase with increasing vanadium layer thickness. The improved fracture toughness (from an NiAl matrix toughness of
6˜.6MPam
to a steady-state toughness of
1˜5MPam
obtained from finite element analysis) is associated with crack bridging and the interactions of cracks with vanadium layers. The reinitiation of cracks in adjacent NiAl layers is modeled using finite element methods and the reinitiation is shown to occur as a result of strain concentrations at the interface between the adjacent NiAl layers and vanadium layers. The deviation of the reinitiated cracks from the pure mode I direction is shown to occur in the direction of maximum shear strain. Toughening due to crack bridging is also modeled using large-scale bridging models. The intrinsic toughness levels of the microlaminates are also inferred by extrapolating the large scale bridging models to arbitrarily large specimen widths. The extrapolations also show that the small-scale bridging intrinsic toughness increases with increasing vanadium layer thickness.
Composite materials, Fracture (Materials), Fracture toughness, Finite element analysis, Finite element methods, Shear (Mechanics), Steady state
Amazigo
Fett, T., and Munz, D., 1994, “Stress Intensity Factors and Weight Functions for One-Dimensional Cracks,” Institut fur Materialforschung, Kernforschungszentrum, Karlsryhe, Germany.
Noebe, R. D., Ritzert, F. J., Misra, A., and Gibala, R., 1991, “Prospects for Ductility and Toughness Enhancement of NiAl by Ductile Phase Reinforcement,” NASA Technical Memorandum 103796, NASA Lewis Research Center, Cleveland, OH.
Sheckherd
Tada, H., Paris, P. C., and Irwin, G. R., 1985, “The Stress Analysis of Cracks Handbook,” Del Research Corporation, St. Louis, MO.
Ye, F., Li, M., and Soboyejo, W. O., 1999, in press, Journal of the American Ceramic Society.
|
AFR Calculator | Air-Fuel Ratio Calculator
What is the air-fuel ratio (AFR)?
Air-fuel ratio of common fuels
How to use the AFR calculator?
The AFR calculator (air-fuel ratio) will give you the rate of air to fuel and the mass of air needed for its complete combustion.
Combustion is a process found in different technologies such as heating devices, internal combustion engines, gas turbines, rocket engines, etc., where AFR is an important parameter.
Keep reading to learn about what is the air-fuel ratio, the AFR of some fuels and how to calculate the stoichiometric air-fuel ratio for fossil fuels.
As Lavoisier found out, oxygen is a key substance in any combustion process, and it turns out that for different types of fuels, the amount of oxygen required is also different.
This specific oxygen requirement is expressed by the parameter known as the air-fuel ratio (AFR), indicating the amount of air needed to achieve the complete combustion of a given quantity of a fuel 🔥.
The AFR is often expressed on a mass basis, as mass of air divided by mass of fuel:
\quad AFR= \dfrac{mass_{air}}{mass_{fuel}}
This air-fuel ratio calculation formula can also be expressed in terms of the molar ratio and the molar mass of each substance as:
\quad AFR= \dfrac{moles_{air}\times M_{air}}{moles_{fuel}\times M_{fuel}}
\quad AFR= \overline{AFR}\times \dfrac{M_{air}}{M_{fuel}}
AFR
– Air-fuel ratio on mass basis;
\overline{AFR}
– Air-fuel ratio on molar basis; and
M_{air}
M_{fuel}
– Respective molar masses.
From the stove in our homes to rockets leaving Earth, we’re surrounded by different appliances and equipment that include the combustion process as part of their functioning system, all of which use a specific fuel. For example, natural gas used on stoves, heating systems and power generation plants 🏭,; gasoline and diesel, mainly used for cars, buses and other means of transportation 🚗,; aviation turbine fuel (ATF) for airplanes ✈ or liquid hydrogen as rocket fuel 🚀.
Here are some air-fuel ratios for most common hydrocarbon fuels:
AFR (mass)
AFR (molar)
💡 Did you know that natural gas is mainly composed of methane (CH4), around 80-90%?
The minimum amount of air needed for complete combustion is known as theoretical or stoichiometric air. That is the quantity of air used when calculating the stoichiometric air-fuel ratio.
The general formula for the complete combustion of a hydrocarbon fuel with theoretical air is:
\text C_\alpha \text H_\beta + a(\text O_2 + 3.76 \text N_2) \longrightarrow b\text C\text O_2 + c\text H_2\text O + d\text N_2
On the left side of the reaction we have:
The reactants of the combustion, which are a hydrocarbon fuel and oxygen.
The generic formula of a hydrocarbon fuel is represented as
\text C_\alpha \text H_\beta
\alpha
\beta
subscripts indicate the respective number of atoms of carbon and hydrogen.
The combustion air
a(\text O_2 + 3.76\text N_2)
, assuming that air is composed of 21% oxygen and 79% nitrogen.
Here, the coefficient
a
represents the required moles of air to balance the combustion reaction.
And on the right side of the reaction:
Here we find the products of a complete combustion: carbon dioxide (CO2), water (H2O) and nitrogen (N2). Note that for theoretical air there's non free oxygen (O2).
b
c
and
of combustion's products balance the equation.
The coefficients that balance the chemical reaction can be obtained in terms of the number of atoms of carbon (
\alpha
) and hydrogen (
\beta
) of a particular fuel, assigning the following values:
b=\alpha
c=\dfrac{{\beta}}{2}
a=\alpha+\dfrac{{\beta}}{4}
d=3.76 \times \bigg(\alpha+\dfrac{{\beta}}{4}\bigg)
Once these are known, we can calculate the number of moles of combustion air and the moles of fuel and relate them using the AFR equation showed before.
On the given occasions when the molecular formula of the fuel is unknown, this one can be obtained through combustion analysis calculation. Once the molecular formula is known, the procedure to balance the combustion reaction is the same as mentioned above.
To use the AFR calculator, follow these steps:
Select one of the substances from the list of fuels.
Once selected, the calculator will show the Air-fuel ratio (AFR) for that substance.
As an example, if choosing methane (CH4), the calculator will show 17.19:1, indicating that for the combustion of every unit mass of methane (i.e., 1 kg), 17.19 unit mass of air (i.e., 17.19 kg) will be required for its complete combustion.
On the Mass of air and fuel section, enter either the mass of fuel or air, and the calculator will give the mass of the other substance.
💡 The air-fuel ratio calculator also works the other way around. You can enter both the mass of air and fuel, and the calculator will show you the Air-fuel ratio (AFR) as a result.
Mass of air and fuel
Mass of fuel ⛽
Mass of air 🌬
Arrhenius equationAtomic massBeer-Lambert law… 31 more
|
Journal of Biomedical Science and Engineering > Vol.11 No.11, November 2018
2Department of Mechanical Engineering, Delta State Polytechnic, Ogwashi-Uku, Nigeria.
Achebe, C. , Iweriolor, S. and Chukwuneke, J. (2018) Surface Energetics Study and Determination of the Combined Negative Hamaker Coefficient for Hepatitis C Virus Infected Human Blood Cells. Journal of Biomedical Science and Engineering, 11, 307-319. doi: 10.4236/jbise.2018.1111025.
\Delta F\left(d\right)=-\frac{{A}_{132}}{12\pi {d}^{2}}
{A}_{132}
{A}_{132}=-12\pi {d}^{2}\Delta {F}^{adh}\left(d\right)
\Delta {F}_{123}^{adh}={\gamma }_{12}-{\gamma }_{13}
{\gamma }_{12}
{\gamma }_{13}
{A}_{132}
\mp
{A}_{132}
{A}_{132}<0
\sqrt{{A}_{11}}>\sqrt{{A}_{33}}
\sqrt{{A}_{22}}<\sqrt{{A}_{33}}
\sqrt{{A}_{11}}<\sqrt{{A}_{33}}<\sqrt{{A}_{22}}
\sqrt{{A}_{11}}<\sqrt{{A}_{33}}<\sqrt{{A}_{22}}
\sqrt{{A}_{11}}>\sqrt{{A}_{33}}>\sqrt{{A}_{22}}
{A}_{132}=\left(\sqrt{{A}_{11}}-\sqrt{{A}_{33}}\right)\left(\sqrt{{A}_{22}}-\sqrt{{A}_{33}}\right)
{A}_{11},{A}_{22}
{A}_{33}
{\gamma }_{lv}\mathrm{cos}\theta ={\gamma }_{sv}-{\gamma }_{sl}
{\gamma }_{lv}
{\gamma }_{sv}
{\gamma }_{sl}
{\gamma }_{l}
{\gamma }_{s}
{\gamma }_{l}<{\gamma }_{s}
S={\gamma }_{s}-\left({\gamma }_{sl}+{\gamma }_{l}\right)
{\gamma }_{sv}
H={\theta }_{a}-{\theta }_{r}
{\gamma }_{lv}
{\gamma }_{lv}\mathrm{cos}\theta
{\gamma }_{sv}
{\gamma }_{lv}\mathrm{cos}\theta =f\left({\gamma }_{lv},{\gamma }_{sv}\right)
{\gamma }_{sl}={\gamma }_{sv}-f\left({\gamma }_{lv},{\gamma }_{sv}\right)=f\left({\gamma }_{lv},{\gamma }_{sv}\right)
\mp
\begin{array}{c}{A}_{132}=2.05-0.021A+0.12B+0.12AB+0.20{A}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+0.033{B}^{2}+0.20{A}^{2}+0.033{B}^{2}\end{array}
[1] Lanvanchy, D. (2009) The Global of Hepatitis C. WHO, Geneve, Switzerland. Liver Int. Medical Research.
[2] (2013) Albert Lasker Awards for Clinical Medical Research; Year 2000 Winners. World Health Organization: Hepatitis C Facts Sheet. No: 164. Retreieved 24th Sept, 2016.
http://www.laskerfoundation.org/awards
[3] Ray, S.C. and Thomas, D.L. (2010) Principles and Practices of Infectious Disease. Churchill Livingstone, Elsevier Publishers, Philadephia, 2157-2185.
[4] Abdelwahab, K.S. and Ahmed, Z.N. (2018) Status of Hepatitis C Virus Vaccination. World Journal of Gastroenterology, 22, 862-873.
[5] Shiffman, M.I., Salvatore, J. and Hubbard, S. (2007) Treatment of Chronic Hepatitis C Virus with Peg-Interferon, Ribavirin and Epotein Alpha. Hepatology, 46, 371-379.
[6] Achebe, C.H., Omenyi, S.N., Manafa, O.P. and Okoli, D. (2012) HIV-Blood Interactions; Surface Thermodynamics Approach. Proceedings of the International Conference of Engineers and Computer Scientist, Hong Kong, 136-141.
[7] Chukwuneke, J.L. (2015) Surface Energetic Study of MTB. Ph.D. Thesis, UNIZIK, Awka.
[8] Hamaker, H.C. (1936) The London/Van der Wall Attraction between Spherical Particles. Physics, 4, 1058.
[9] Omenyi, S.N., Newmann, A.W., Van Oss, C.J., Absolom, D.R. and Viser, J. (1980) Separation and Purification Methods. Advance Colloid Interface Sci., 18, 133.
[10] Ani, O.I. (2016) Surface Energetics Study of the Interaction between HIV-Blood Blood Cells Treated with Antiretroviral Drugs. Ph.D. Thesis, UNIZIK, Nigeria.
[11] Yuan, Y. and Lee, R. (2013) Contact Angle and Wetting Properties. University of Houston, Texas.
[12] Rullison, C. (2008) Comparison of Different Methods to Measure Contact Angles of Soil Colloids. Journal of Colloids and Interface Science, 328, 299-307.
[13] Krumpfer, J.W., McCarthy, T.J. and Gao, L. (2010) Wetting Properties of Fluids. Physics of Fluids, Faraday Discussions, 146, 103.
[14] Neumann, A.W., Good, R.J., Hope, C.J. and Segpal, M. (1974) Colloids and Interfaces; Equation of State Approach to Determine the Surface Tension of Low Energy Fluids from Contact Angle. Journal of Colloid and Interface Science, 49, 291-304.
[15] Kwok, D.Y. and Neumann, A.W. (1999) Contact Angle Measurements and Contact Angle Interpretation. Advances in Colloid and Interface Science, 81, 167-249.
[16] Ozoihu E.M. (2014) Human Immunodeficiency Virus; Contact Angle Approach. Ph.D. Thesis, Nnamdi Azikiwe University, Awka.
[17] Achebe, C.H. and Omenyi, S.N. (2013) Mathematical Determination of the Critical Absolute Hamaker Constant of the Serum (as Intervening Medium) Which Favors Repulsion in HIV-Blood Interactions Mechanism. Proceedings of World Congress on Engineering, London, 3-5 July 2013, 1380-1384.
|
Section 2-3 - DerivRules - Maple Help
Home : Support : Online Help : Study Guides : Calculus : Chapter 2 - Differentiation : Section 2-3 - DerivRules
Evaluation of derivatives using the definition hinges on the ability to evaluate the limit of the difference quotient. This is tedious and time consuming. Even worse, the limits can be extremely difficult to evaluate. The differentiation rules are a collection of general rules for computing derivatives formed from polynomials, powers, roots, and constant multiples, sums, differences, products, quotients of these types of functions. Compositions of functions are handled separately in the next section, on the Chain rule.
Be careful not to confuse the limit laws and differentiation rules. While many are analogous, others are completely different.
Like the tutor, the tutor is an excellent tool for learning the names and function of the differentiation rules and mastering how they can be applied to evaluate a derivative. The three examples provided in this section illustrate the use of this tool.
In general, derivatives can be evaluated by applying the definition, as in Examples 2.2.1 - 3 in the previous section. The first few rules listed in Table 2.3.1 are fairly simple, almost obvious. But, starting with the Product rule, the rules are not so obvious and are quite different from the corresponding limit laws.
\frac{d}{\mathrm{dx}}\left(k\right)= 0
\frac{\mathrm{dx}}{\mathrm{dx}}=1
\frac{d}{\mathrm{dx}}\left({x}^{n}\right)= n {x}^{n-1}
n≠0
\left(kf\left(x\right)\right)\prime = k f\prime \left(x\right)
\left(f\left(x\right)+g\left(x\right)\right)\prime = f\prime \left(x\right)+g\prime \left(x\right)
\left(f\left(x\right)-g\left(x\right)\right)\prime = f\prime \left(x\right)- g\prime \left(x\right)
\left(f\left(x\right)g\left(x\right)\right)\prime = f\left(x\right)g\prime \left(x\right)+g\left(x\right)f\prime \left(x\right)
{\left(\frac{f\left(x\right)}{g\left(x\right)}\right)}^{\prime }=\frac{g\left(x\right)f\prime \left(x\right)-f\left(x\right)g\prime \left(x\right)}{{g}^{2}\left(x\right)}
g\left(x\right)\ne 0
Table 2.3.1 Differentiation rules: the operators
\frac{d}{\mathrm{dx}}
and prime (#) are used interchangeably,
k
denotes a constant, and
f
g
are functions differentiable at
x
Note the conditions listed in the third column. If the conditions for a rule are not satisfied, the rule cannot be used to evaluate a derivative.
The Constant Multiple rule simply says that when differentiating the product of a function times a constant multiple, the derivative of the function is computed and multiplied by the constant.
The Sum and Difference rules simply say that the derivative of a sum or difference is the sum or difference of the derivatives.
The Product rule may best be learned as "the first times the derivative of the second plus the second times the derivative of the first."
In actuality, the pattern being followed is best seen if the derivative of a product of three factors is written. The derivative of a product of three factors is given by
\left(fgh\right)\prime = f\prime g h + f g\prime h + f g h\prime
The product of three factors is written three times, in each of these three terms just one of the functions is differentiated, and the terms are added. With this pattern in mind, the Product rule for two factors is seen to obey exactly this schematic, but the vocalization in terms of "first" and "second" makes articulating and applying the rule much simpler.
Finally, the Quotient rule can be stated as "denominator times derivative of the numerator, minus numerator times derivative of the denominator, all over the denominator squared." (With this version of the rule, the letters DN lead off the vocalization. Remembering that D stands for denominator and that DNA is the biological key to life, no student should ever reverse the roles of numerator and denominator in the Quotient rule.)
In some classrooms, students learn the Quotient rule with "top" and "bottom" replacing "numerator" and "denominator," respectively. In either event, articulating a pattern in language is far easier than trying to memorize it as a string of symbols.
Proofs of the Differentiation Rules
f\left(x\right)=k
k
is a constant. Apply Definition 2.2.1.
f\prime \left(x\right)
=\underset{h→0}{lim}\frac{f\left(x+h\right)-f\left(x\right)}{h}
=\underset{h→0}{lim}\frac{k-k}{h}
=\underset{h→0}{lim}\frac{0}{h}
=\underset{h→0}{lim}0
=0
This result is eminently reasonable because the graph of a constant function is a horizontal line, and the slope of such a line is zero.
f\left(x\right)=x
and apply Definition 2.2.1.
f\prime \left(x\right)
=\underset{h→0}{lim}\frac{f\left(x+h\right)-f\left(x\right)}{h}
=\underset{h→0}{lim}\frac{\left(x+h\right)-x}{h}
=\underset{h→0}{lim}\frac{h}{h}
=\underset{h→0}{lim}1
=1
This result is eminently reasonable because the graph of
f\left(x\right)=x
is a straight line with slope 1.
Apply the Binomial theorem to
{\left(x+h\right)}^{n}
n
{\left(x+h\right)}^{n}
=\underset{k=0}{\overset{n}{∑}}\left(\genfrac{}{}{0}{}{n}{k}\right){x}^{n} {h}^{n-k}
={x}^{n}+n h {x}^{n-1}+\left(\genfrac{}{}{0}{}{n}{2}\right) {h}^{2} {x}^{n-2}+\left(\genfrac{}{}{0}{}{n}{3}\right) {h}^{3} {x}^{n-3}+⋯
={x}^{n}+n h {x}^{n-1}+{h}^{2}\left({\left(\genfrac{}{}{0}{}{n}{2}\right)x}^{n-2}+\left(\genfrac{}{}{0}{}{n}{3}\right)h {x}^{n-3}+⋯\right)
={x}^{n}+n h {x}^{n-1}+\mathrm{O}\left({h}^{2}\right)
The Landau order symbol
\mathrm{O}\left({h}^{p}\right)
denotes a quantity that is a multiple of
{h}^{2}
g=\mathrm{O}\left({h}^{2}\right)
g/h=\mathrm{O}\left(h\right)
g/h=\mathrm{O}\left(h\right)→0
h→0
f\left(x\right)={x}^{n}
n
is a positive integer. Apply Definition 2.2.1.
f\prime \left(x\right)
=\underset{h→0}{lim}\frac{f\left(x+h\right)-f\left(x\right)}{h}
=\underset{h→0}{lim}\frac{{\left(x+h\right)}^{n}-{x}^{n}}{h}
=\underset{h→0}{lim}\frac{\left({x}^{n}+n h {x}^{n-1}+\mathrm{O}\left({h}^{2}\right)\right)-{x}^{n}}{h}
=\underset{h→0}{lim}\frac{n h {x}^{n-1}+\mathrm{O}\left({h}^{2}\right)}{h}
=n {x}^{n-1}+\underset{h→0}{lim}\mathrm{O}\left(h\right)
=n {x}^{n-1}+0
=n {x}^{n-1}
To extend the Power rule to integers
n<0
, apply the Quotient rule to
{x}^{n}=1/{x}^{-n}
, remembering that
-n
\frac{d}{\mathrm{dx}}\left({x}^{n}\right)
=\frac{d}{\mathrm{dx}}\left(\frac{1}{{x}^{-n}}\right)
=\frac{{x}^{-n}\frac{d}{\mathrm{dx}}\left(1\right) -1\cdot \frac{d}{\mathrm{dx}}\left({x}^{-n}\right)}{{\left({x}^{-n}\right)}^{2}}
=\frac{0-\left(-n\right){x}^{\left(-n\right)-1}}{{x}^{-2 n}}
=n{x}^{-n-1+2 n}
=n {x}^{n-1}
To extend the Power rule to rational numbers
a=p/q
y={x}^{a}
{y}^{q}={x}^{p}
and differentiate both sides, using the Chain rule (see Section 2.4) on the left, remembering that
y=y\left(x\right)
Differentiate via the Chain rule
\frac{d}{\mathrm{dx}}\left({y}^{q}\right)
=\frac{d}{\mathrm{dx}}\left({x}^{p}\right)
q {y}^{q-1}y\prime \left(x\right)
=p {x}^{p-1}
y\prime \left(x\right)
y\prime \left(x\right)
=\frac{p {x}^{p-1}}{q {y}^{q-1}}
=\frac{p {x}^{p-1}}{q {\left({x}^{p/q}\right)}^{q-1}}
=\frac{p {x}^{p-1}}{q {x}^{p-p/q}}
=a {x}^{p-1-\left(p-p/q\right)}
=a {x}^{p/q-1}
=a {x}^{a-1}
Note: The typical calculus text will state the Power rule and declare it applicable for all real exponents, but postpone the full proof until additional differentiation techniques have first been established. Proofs based on implicit differentiation (Section 2.5) or logarithmic differentiation (Table 2.6.2) amount to a proof based on the Chain rule; hence, the choice made above.
F\left(x\right)=k f\left(x\right)
and apply Definition 2.2.1 to
F
F\prime \left(x\right)
=\underset{h→0}{lim}\frac{F\left(x+h\right)-F\left(x\right)}{h}
=\underset{h→0}{lim}\frac{k f\left(x+h\right)-k f\left(x\right)}{h}
=k \underset{h→0}{lim}\frac{f\left(x+h\right)-f\left(x\right)}{h}
=k f\prime \left(x\right)
The Sum rule for derivatives follows from the Sum rule for limits.
\frac{d}{\mathrm{dx}}\left(f\left(x\right)+g\left(x\right)\right)
=\underset{h→0}{lim}\frac{\left(f\left(x+h\right)+g\left(x+h\right)\right)-\left(f\left(x\right)+g\left(x\right)\right)}{h}
=\underset{h→0}{lim}\frac{\left(f\left(x+h\right)-f\left(x\right)\right)+\left(g\left(x+h\right)-g\left(x\right)\right)}{h}
=\underset{h→0}{lim}\frac{f\left(x+h\right)-f\left(x\right)}{h}+\underset{h→0}{lim}\frac{g\left(x+h\right)-g\left(x\right)}{h}
=f\prime \left(x\right)+g\prime \left(x\right)
The Difference rule for derivatives follows from the Difference rule for limits.
\frac{d}{\mathrm{dx}}\left(f\left(x\right)-g\left(x\right)\right)
=\underset{h→0}{lim}\frac{\left(f\left(x+h\right)-g\left(x+h\right)\right)-\left(f\left(x\right)-g\left(x\right)\right)}{h}
=\underset{h→0}{lim}\frac{\left(f\left(x+h\right)-f\left(x\right)\right)-\left(g\left(x+h\right)-g\left(x\right)\right)}{h}
=\underset{h→0}{lim}\frac{f\left(x+h\right)-f\left(x\right)}{h}-\underset{h→0}{lim}\frac{g\left(x+h\right)-g\left(x\right)}{h}
=f\prime \left(x\right)-g\prime \left(x\right)
The derivative of a product is not the product of the derivatives!
\frac{d}{\mathrm{dx}}\left(f\left(x\right)g\left(x\right)\right)
=\underset{h→0}{lim}\frac{f\left(x+h\right)g\left(x+h\right)-f\left(x\right)g\left(x\right)}{h}
=\underset{h→0}{lim}\frac{f\left(x+h\right)g\left(x+h\right)-f\left(x+h\right)g\left(x\right)+f\left(x+h\right)g\left(x\right)-f\left(x\right)g\left(x\right)}{h}
=\underset{h→0}{lim}\frac{f\left(x+h\right)\left(g\left(x+h\right)-g\left(x\right)\right)+\left(f\left(x+h\right)-f\left(x\right)\right)g\left(x\right)}{h}
=\underset{h→0}{lim}\frac{f\left(x+h\right)\left(g\left(x+h\right)-g\left(x\right)\right)}{h}+g\left(x\right)\underset{h→0}{lim}\frac{f\left(x+h\right)-f\left(x\right)}{h}
=\underset{h→0}{lim}f\left(x+h\right) \underset{h→0}{lim}\frac{g\left(x+h\right)-g\left(x\right)}{h}+g\left(x\right)\underset{h→0}{lim}\frac{f\left(x+h\right)-f\left(x\right)}{h}
=f\left(x\right) g\prime \left(x\right)+g\left(x\right) f\prime \left(x\right)
The device of adding and subtracting the same term is used often in analysis. Here, the term added and subtracted to the numerator in the second line is
f\left(x+h\right)g\left(x\right)
To go from the fourth line to the fifth, the Product rule for limits is applied. Since
f
is differentiable, it is continuous, so its limit exists, and
f\left(x+h\right)→f\left(x\right)
h→0
The derivative of a quotient is not the quotient of the derivatives!
\frac{d}{\mathrm{dx}}\left(\frac{f\left(x\right)}{g\left(x\right)}\right)
=\underset{h→0}{lim}\frac{\frac{f\left(x+h\right)}{g\left(x+h\right)}-\frac{f\left(x\right)}{g\left(x\right)}}{h}
=\underset{h→0}{lim}\frac{f\left(x+h\right)g\left(x\right)-f\left(x\right)g\left(x+h\right)}{h g\left(x\right)g\left(x+h\right)}
=\underset{h\to 0}{lim}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\frac{f\left(x+h\right)g\left(x\right)-f\left(x\right)g\left(x\right)+f\left(x\right)g\left(x\right)-f\left(x\right)g\left(x+h\right)}{hg\left(x\right)g\left(x+h\right)}
=\underset{h→0}{lim}\frac{\left(f\left(x+h\right)-f\left(x\right)\right)g\left(x\right)-f\left(x\right)\left(g\left(x+h\right)-g\left(x\right)\right)}{h g\left(x\right)g\left(x+h\right)}
=\underset{h\to 0}{lim}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\frac{\left(f\left(x+h\right)-f\left(x\right)\right)g\left(x\right)}{hg\left(x\right)g\left(x+h\right)}-\underset{h\to 0}{lim}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\frac{f\left(x\right)\left(g\left(x+h\right)- g\left(x\right)\right)}{hg\left(x\right)g\left(x+h\right)}
=\underset{h\to 0}{lim}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\frac{f\left(x+h\right)-f\left(x\right)}{hg\left(x+h\right)}-\frac{f\left(x\right)}{g\left(x\right)}\underset{h→0}{lim}\frac{g\left(x+h\right)-g\left(x\right)}{h g\left(x+h\right)}
=\frac{\underset{h→0}{lim}\frac{f\left(x+h\right)-f\left(x\right)}{h}}{\underset{h→0}{lim}g\left(x+h\right)}-\frac{f\left(x\right)}{g\left(x\right)}\frac{\underset{h→0}{lim}\frac{g\left(x+h\right)-g\left(x\right)}{h}}{\underset{h→0}{lim}g\left(x+h\right)}
=\frac{f\prime \left(x\right)}{g\left(x\right)}-\frac{f\left(x\right)}{g\left(x\right)} \frac{g\prime \left(x\right)}{g\left(x\right)}
=\frac{g\left(x\right)f\prime \left(x\right)-f\left(x\right)g\prime \left(x\right)}{{g}^{2}\left(x\right)}
In the third line, the term
f\left(x\right)g\left(x\right)
is added and subtracted. In going from the sixth line to the seventh, the Quotient rule for limits is applied. The very last line is the result of adding fractions over a common denominator.
Table 2.3.2 summarizes the key points in Section 2.3.
The Differentiation rules in Table 2.3.1 provide the tools necessary for differentiating a function without explicitly evaluating a limit. With these rules it is possible to differentiate any polynomial or rational function.
Note the Product and Quotient rules since they do not parallel the corresponding Limit laws.
The derivative of a product is not the product of the derivatives.
The derivative of a quotient is not the quotient of the derivatives.
The Power rule applies to any integer.
Table 2.3.2 Key points in Section 2.3
Apply the rules in Table 2.3.1 to obtain the derivative of
f\left(x\right)={x}^{3}-3{x}^{2}+x+3
f\left(t\right)=\frac{t-1}{{t}^{2}+2}
f\left(x\right)=\left(5 {x}^{2}-7 x+12\right) \left({x}^{3}-3{x}^{2}+x+3\right)
|
Single-linkage clustering - Wikipedia
Agglomerative hierarchical clustering method
A drawback of this method is that it tends to produce long thin clusters in which nearby elements of the same cluster have small distances, but elements at opposite ends of a cluster may be much farther from each other than two elements of other clusters. This may lead to difficulties in defining classes that could usefully subdivide the data.[1]
1 Overview of agglomerative clustering methods
2 Naive algorithm
3.4 The single-linkage dendrogram
4 Other linkages
5 Faster algorithms
Overview of agglomerative clustering methods[edit]
In the beginning of the agglomerative clustering process, each element is in a cluster of its own. The clusters are then sequentially combined into larger clusters, until all elements end up being in the same cluster. At each step, the two clusters separated by the shortest distance are combined. The function used to determine the distance between two clusters, known as the linkage function, is what differentiates the agglomerative clustering methods.
In single-linkage clustering, the distance between two clusters is determined by a single pair of elements: those two elements (one in each cluster) that are closest to each other. The shortest of these pairwise distances that remain at any step causes the two clusters whose elements are involved to be merged. The method is also known as nearest neighbour clustering. The result of the clustering can be visualized as a dendrogram, which shows the sequence in which clusters were merged and the distance at which each merge took place.[2]
Mathematically, the linkage function – the distance D(X,Y) between clusters X and Y – is described by the expression
{\displaystyle D(X,Y)=\min _{x\in X,y\in Y}d(x,y),}
where X and Y are any two sets of elements considered as clusters, and d(x,y) denotes the distance between the two elements x and y.
Naive algorithm[edit]
The following algorithm is an agglomerative scheme that erases rows and columns in a proximity matrix as old clusters are merged into new ones. The
{\displaystyle N\times N}
proximity matrix
{\displaystyle D}
contains all distances
{\displaystyle d(i,j)}
. The clusterings are assigned sequence numbers
{\displaystyle 0,1,\ldots ,n-1}
{\displaystyle L(k)}
is the level of the
{\displaystyle k}
-th clustering. A cluster with sequence number m is denoted (m) and the proximity between clusters
{\displaystyle (r)}
{\displaystyle (s)}
{\displaystyle d[(r),(s)]}
The single linkage algorithm is composed of the following steps:
Begin with the disjoint clustering having level
{\displaystyle L(0)=0}
and sequence number
{\displaystyle m=0}
Find the most similar pair of clusters in the current clustering, say pair
{\displaystyle (r),(s)}
{\displaystyle d[(r),(s)]=\min d[(i),(j)]}
where the minimum is over all pairs of clusters in the current clustering.
Increment the sequence number:
{\displaystyle m=m+1}
. Merge clusters
{\displaystyle (r)}
{\displaystyle (s)}
into a single cluster to form the next clustering
{\displaystyle m}
. Set the level of this clustering to
{\displaystyle L(m)=d[(r),(s)]}
Update the proximity matrix,
{\displaystyle D}
, by deleting the rows and columns corresponding to clusters
{\displaystyle (r)}
{\displaystyle (s)}
and adding a row and column corresponding to the newly formed cluster. The proximity between the new cluster, denoted
{\displaystyle (r,s)}
and old cluster
{\displaystyle (k)}
{\displaystyle d[(r,s),(k)]=\min\{d[(k),(r)],d[(k),(s)]\}}
If all objects are in one cluster, stop. Else, go to step 2.
Working example[edit]
This working example is based on a JC69 genetic distance matrix computed from the 5S ribosomal RNA sequence alignment of five bacteria: Bacillus subtilis (
{\displaystyle a}
), Bacillus stearothermophilus (
{\displaystyle b}
), Lactobacillus viridescens (
{\displaystyle c}
), Acholeplasma modicum (
{\displaystyle d}
), and Micrococcus luteus (
{\displaystyle e}
First clustering
Let us assume that we have five elements
{\displaystyle (a,b,c,d,e)}
and the following matrix
{\displaystyle D_{1}}
of pairwise distances between them:
{\displaystyle D_{1}(a,b)=17}
is the lowest value of
{\displaystyle D_{1}}
, so we cluster elements
{\displaystyle a}nd
{\displaystyle b}
First branch length estimation
{\displaystyle u}
denote the node to which
{\displaystyle a}nd
{\displaystyle b}
are now connected. Setting
{\displaystyle \delta (a,u)=\delta (b,u)=D_{1}(a,b)/2}
ensures that elements
{\displaystyle a}nd
{\displaystyle b}
{\displaystyle u}
. This corresponds to the expectation of the ultrametricity hypothesis. The branches joining
{\displaystyle a}nd
{\displaystyle b}
{\displaystyle u}
then have lengths
{\displaystyle \delta (a,u)=\delta (b,u)=17/2=8.5}
(see the final dendrogram)
First distance matrix update
We then proceed to update the initial proximity matrix
{\displaystyle D_{1}}
into a new proximity matrix
{\displaystyle D_{2}}
(see below), reduced in size by one row and one column because of the clustering of
{\displaystyle a}
{\displaystyle b}
. Bold values in
{\displaystyle D_{2}}
correspond to the new distances, calculated by retaining the minimum distance between each element of the first cluster
{\displaystyle (a,b)}
and each of the remaining elements:
{\displaystyle D_{2}((a,b),c)=min(D_{1}(a,c),D_{1}(b,c))=min(21,30)=21}
{\displaystyle D_{2}((a,b),d)=min(D_{1}(a,d),D_{1}(b,d))=min(31,34)=31}
{\displaystyle D_{2}((a,b),e)=min(D_{1}(a,e),D_{1}(b,e))=min(23,21)=21}
Italicized values in
{\displaystyle D_{2}}
are not affected by the matrix update as they correspond to distances between elements not involved in the first cluster.
Second clustering
We now reiterate the three previous actions, starting from the new distance matrix
{\displaystyle D_{2}}
{\displaystyle D_{2}((a,b),c)=21}
{\displaystyle D_{2}((a,b),e)=21}
are the lowest values of
{\displaystyle D_{2}}
, so we join cluster
{\displaystyle (a,b)}
{\displaystyle c}
and with element
{\displaystyle e}
Second branch length estimation
{\displaystyle v}
{\displaystyle (a,b)}
{\displaystyle c}
{\displaystyle e}
are now connected. Because of the ultrametricity constraint, the branches joining
{\displaystyle a}
{\displaystyle b}
{\displaystyle v}
{\displaystyle c}
{\displaystyle v}
{\displaystyle e}
{\displaystyle v}
are equal and have the following total length:
{\displaystyle \delta (a,v)=\delta (b,v)=\delta (c,v)=\delta (e,v)=21/2=10.5}
We deduce the missing branch length:
{\displaystyle \delta (u,v)=\delta (c,v)-\delta (a,u)=\delta (c,v)-\delta (b,u)=10.5-8.5=2}
Second distance matrix update
We then proceed to update the
{\displaystyle D_{2}}
matrix into a new distance matrix
{\displaystyle D_{3}}
(see below), reduced in size by two rows and two columns because of the clustering of
{\displaystyle (a,b)}
{\displaystyle c}
{\displaystyle e}
{\displaystyle D_{3}(((a,b),c,e),d)=min(D_{2}((a,b),d),D_{2}(c,d),D_{2}(e,d))=min(31,28,43)=28}
{\displaystyle D_{3}}
matrix is:
((a,b),c,e)
So we join clusters
{\displaystyle ((a,b),c,e)}
an{\displaystyle d}
{\displaystyle r}
denote the (root) node to which
{\displaystyle ((a,b),c,e)}
an{\displaystyle d}
are now connected. The branches joining
{\displaystyle ((a,b),c,e)}
an{\displaystyle d}
{\displaystyle r}
then have lengths:
{\displaystyle \delta (((a,b),c,e),r)=\delta (d,r)=28/2=14}
We deduce the remaining branch length:
{\displaystyle \delta (v,r)=\delta (a,r)-\delta (a,v)=\delta (b,r)-\delta (b,v)=\delta (c,r)-\delta (c,v)=\delta (e,r)-\delta (e,v)=14-10.5=3.5}
The single-linkage dendrogram[edit]
The dendrogram is now complete. It is ultrametric because all tips (
{\displaystyle a}
{\displaystyle b}
{\displaystyle c}
{\displaystyle e}
, an{\displaystyle d}
) are equidistant from
{\displaystyle r}
{\displaystyle \delta (a,r)=\delta (b,r)=\delta (c,r)=\delta (e,r)=\delta (d,r)=14}
The dendrogram is therefore rooted by
{\displaystyle r}
, its deepest node.
Other linkages[edit]
The naive algorithm for single linkage clustering is essentially the same as Kruskal's algorithm for minimum spanning trees. However, in single linkage clustering, the order in which clusters are formed is important, while for minimum spanning trees what matters is the set of pairs of points that form distances chosen by the algorithm.
Alternative linkage schemes include complete linkage clustering, average linkage clustering (UPGMA and WPGMA), and Ward's method. In the naive algorithm for agglomerative clustering, implementing a different linkage scheme may be accomplished simply by using a different formula to calculate inter-cluster distances in the algorithm. The formula that should be adjusted has been highlighted using bold text in the above algorithm description. However, more efficient algorithms such as the one described below do not generalize to all linkage schemes in the same way.
Comparison of dendrograms obtained under different clustering methods from the same distance matrix.
Single-linkage clustering Complete-linkage clustering Average linkage clustering: WPGMA Average linkage clustering: UPGMA
Faster algorithms[edit]
The naive algorithm for single-linkage clustering is easy to understand but slow, with time complexity
{\displaystyle O(n^{3})}
.[5] In 1973, R. Sibson proposed an algorithm with time complexity
{\displaystyle O(n^{2})}
and space complexity
{\displaystyle O(n)}
(both optimal) known as SLINK. The slink algorithm represents a clustering on a set of
{\displaystyle n}umbered items by two functions. These functions are both determined by finding the smallest cluster
{\displaystyle C}
that contains both item
{\displaystyle i}
and at least one larger-numbered item. The first function,
{\displaystyle \pi }
, maps item
{\displaystyle i}
to the largest-numbered item in cluster
{\displaystyle C}
. The second function,
{\displaystyle \lambda }
{\displaystyle i}
to the distance associated with the creation of cluster
{\displaystyle C}
. Storing these functions in two arrays that map each item number to its function value takes space
{\displaystyle O(n)}
, and this information is sufficient to determine the clustering itself. As Sibson shows, when a new item is added to the set of items, the updated functions representing the new single-linkage clustering for the augmented set, represented in the same way, can be constructed from the old clustering in time
{\displaystyle O(n)}
. The SLINK algorithm then loops over the items, one by one, adding them to the representation of the clustering.[6][7]
An alternative algorithm, running in the same optimal time and space bounds, is based on the equivalence between the naive algorithm and Kruskal's algorithm for minimum spanning trees. Instead of using Kruskal's algorithm, one can use Prim's algorithm, in a variation without binary heaps that takes time
{\displaystyle O(n^{2})}
{\displaystyle O(n)}
to construct the minimum spanning tree (but not the clustering) of the given items and distances. Then, applying Kruskal's algorithm to the sparse graph formed by the edges of the minimum spanning tree produces the clustering itself in an additional time
{\displaystyle O(n\log n)}
{\displaystyle O(n)}
^ Everitt B (2011). Cluster analysis. Chichester, West Sussex, U.K: Wiley. ISBN 9780470749913.
^ Legendre P, Legendre L (1998). Numerical Ecology. Developments in Environmental Modelling. Vol. 20 (Second English ed.). Amsterdam: Elsevier.
^ Erdmann VA, Wolters J (1986). "Collection of published 5S, 5.8S and 4.5S ribosomal RNA sequences". Nucleic Acids Research. 14 Suppl (Suppl): r1-59. doi:10.1093/nar/14.suppl.r1. PMC 341310. PMID 2422630.
^ Olsen GJ (1988). "Phylogenetic analysis using ribosomal RNA". Methods in Enzymology. 164: 793–812. doi:10.1016/s0076-6879(88)64084-5. PMID 3241556.
^ Murtagh F, Contreras P (2012). "Algorithms for hierarchical clustering: an overview". Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery. Wiley Online Library. 2 (1): 86–97. doi:10.1002/widm.53.
^ Sibson R (1973). "SLINK: an optimally efficient algorithm for the single-link cluster method" (PDF). The Computer Journal. British Computer Society. 16 (1): 30–34. doi:10.1093/comjnl/16.1.30.
^ Gan G (2007). Data clustering : theory, algorithms, and applications. Philadelphia, Pa. Alexandria, Va: SIAM, Society for Industrial and Applied Mathematics American Statistical Association. ISBN 9780898716238.
^ Gower JC, Ross GJ (1969). "Minimum spanning trees and single linkage cluster analysis". Journal of the Royal Statistical Society, Series C. 18 (1): 54–64. doi:10.2307/2346439. JSTOR 2346439. MR 0242315. .
Linkages used in Matlab
Retrieved from "https://en.wikipedia.org/w/index.php?title=Single-linkage_clustering&oldid=1025838177"
|
Control of Circumferential Wall Stress and Luminal Shear Stress Within Intact Vascular Segments Perfused Ex Vivo | J. Biomech Eng. | ASME Digital Collection
Department of Surgery, Department of Bioengineering, McGowan Institute for Regenerative Medicine, The Center for Vascular Remodeling and Regeneration,
Department of Mechanical Engineering and Material Science, Department of Bioengineering,
e-mail: vorpda@upmc.edu
El-Kurdi, M. S., Vipperman, J. S., and Vorp, D. A. (July 10, 2008). "Control of Circumferential Wall Stress and Luminal Shear Stress Within Intact Vascular Segments Perfused Ex Vivo." ASME. J Biomech Eng. October 2008; 130(5): 051003. https://doi.org/10.1115/1.2948419
Proportional, integral, and derivative (PID) controllers have proven to be robust in controlling many applications, and remain the most widely used control system architecture. The purpose of this work was to use this architecture for designing and tuning two PID controllers. The first was used to control the physiologic arterial circumferential wall stress (CWS) and the second to control the physiologic arterial shear stress (SS) imposed on intact vascular segments that were implanted into an ex vivo vascular perfusion system (EVPS). In order to most accurately control the stresses imposed onto vascular segments perfused ex vivo, analytical models were derived to calculate the CWS and SS. The mid-vein-wall CWS was calculated using the classical Lamé solution for thick-walled cylinders in combination with the intraluminal pressure and outer diameter measurements. Similarly, the SS was calculated using the Hagen–Poiseuille equation in combination with the flow rate and outer diameter measurements. Performance of each controller was assessed by calculating the root mean square of the error (RMSE) between the desired and measured process variables. The performance experiments were repeated ten times
(N=10)
and an average RMSE was reported for each controller. RMSE standard deviations were calculated to demonstrate the reproducibility of the results. Sterile methods were utilized for making blood gas and temperature measurements in order to maintain physiologic levels within the EVPS. Physiologic blood gases (pH,
pO2
pCO2
) and temperature within the EVPS were very stable and controlled manually. Blood gas and temperature levels were recorded hourly for several
(N=9)
24h
perfusion experiments. RMSE values for CWS control
(0.427±0.027KPa)
indicated that the system was able to generate a physiologic CWS wave form within 0.5% error of the peak desired CWS over each cardiac cycle. RMSE values for SS control
(0.005±0.0007dynes∕cm2)
indicated that the system was able to generate a physiologic SS wave form within 0.3% error of the peak desired SS over each cardiac cycle. Physiologic pH,
pO2
pCO2
, and temperature levels were precisely maintained within the EVPS. The built-in capabilities and overall performance of the EVPS described in this study provide us with a novel tool for measuring molecular responses of intact vascular segments exposed to precisely simulated arterial biomechanical conditions.
biochemistry, biocontrol, biomechanics, blood vessels, carbon compounds, haemorheology, hydrogen, medical computing, medical control systems, molecular biophysics, oxygen, vascular, perfusion system, PID control
Biological tissues, Biomechanics, Blood, Blood vessels, Control algorithms, Control equipment, Control systems, Cylinders, Design, Errors, Flow (Dynamics), Gases, Maintenance, Physiology, Pressure, Safety, Shear stress, Signals, Stress, Temperature, Waves, Pumps, Pistons, Temperature measurement, Rollers, Biomedicine, Modeling
Simulation In Vitro of Pulsatile Vascular Hemodynamics Using a Cad/Cam-Designed Cam Disc and Roller Follower
Measurement In Vitro of Pulsatile Arterial Diameter Using a Helium-Neon Laser
Biomechanics of the Arterial Wall Under Simulated Flow Conditions
Effect of Variations in Pressure and Flow on the Geometry of Isolated Canine Carotid Arteries
Pulsatile Perfusion System for Ex Vivo Investigation of Biochemical Pathways in Intact Vascular Tissue
The Influence of Hemodynamics and Wall Biomechanics on the Thrombogenicity of Vein Segments Perfused In Vitro
Gene Expression is Altered in Perfused Arterial Segments Exposed to Cyclic Flexure Ex Vivo
A Device for the Application of Cyclic Twist and Extension on Perfused Vascular Segments
El-Kurdi
Evaluation of Endothelium-Derived Nitric Oxide Mediated Vasodilation Utilizing Ex Vivo Perfusion of an Intact Vessel
Enhancement of Tissue Factor Expression by Vein Segments Exposed to Coronary Arterial Hemodynamics
The PID Control Algorithm: How It Works, How to Tune It, and How to Use It
,” http://www.jashaw.com/pidbookhttp://www.jashaw.com/pidbook.
Subspace State Space System Identification for Industrial Processes
Proceedings of the 5th IFAC Symposium on Dynamics and Control of Process Systems
, Jun. 8–10, pp.
Numerical Algorithms for Subspace State Space System Identification (N4sid)
,” ASME Design Engineering Technical Conferences.
Risk Stratification of Individual Coronary Lesions Using Local Endothelial Shear Stress: A New Paradigm for Managing Coronary Artery Disease
Early Remodeling of Saphenous Vein Grafts: Proliferation, Migration and Apoptosis of Adventitial and Medial Cells Occur Simultaneously With Changes in Graft Diameter and Blood Flow
A New Biomechanical Perfusion System for Ex Vivo Study of Small Biological Intact Vessels
|
Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other systems by adding unary operators
{\displaystyle \Diamond }
{\displaystyle \Box }
, representing possibility and necessity respectively. For instance the modal formula
{\displaystyle \Diamond P}
can be read as "possibly
{\displaystyle P}
" while
{\displaystyle \Box P}
can be read as "necessarily
{\displaystyle P}
". Modal logics can be used to represent different phenomena depending on what kind of necessity and possibility is under consideration. When
{\displaystyle \Box }
is used to represent epistemic necessity,
{\displaystyle \Box P}
{\displaystyle P}
is epistemically necessary, or in other words that it is known. When
{\displaystyle \Box }
is used to represent deontic necessity,
{\displaystyle \Box P}
{\displaystyle P}
is a moral or legal obligation.[1]
In the standard relational semantics for modal logic, formulas are assigned truth values relative to a possible world. A formula's truth value at one possible world can depend on the truth values of other formulas at other accessible possible worlds. In particular,
{\displaystyle \Diamond P}
is true at a world if
{\displaystyle P}
is true at some accessible possible world, while
{\displaystyle \Box P}
{\displaystyle P}
is true at every accessible possible world. A variety of proof systems exist which are sound and complete with respect to the semantics one gets by restricting the accessibility relation. For instance, the deontic modal logic D is sound and complete if one requires the accessibility relation to be serial.
1 Syntax for modal operators
2.1.1 Basic notions
2.1.2 Frames and completeness
2.2 Topological semantics
3.2 Decision methods
4 Modal logics in philosophy
4.1 Alethic logic
4.1.1 Physical possibility
4.1.2 Metaphysical possibility
4.4 Deontic logic
4.4.1 Intuitive problems with deontic logic
4.5 Doxastic logic
5 Metaphysical questions
Syntax for modal operators[edit]
The syntax rules for modal operators
{\displaystyle \Box }
{\displaystyle \Diamond }
are very similar to those for universal and existential quantifiers; In fact, any formula with modal operators
{\displaystyle \Box }
{\displaystyle \Diamond }
, and the usual logical connectives in propositional calculus (
{\displaystyle \land ,\lor ,\neg ,\rightarrow ,\leftrightarrow }
) can be rewritten to a de dicto normal form, similar to prenex normal form. One major caveat: Whereas the universal and existential quantifiers only binds to the propositional variables or the predicate variables following the quantifiers, since the modal operators
{\displaystyle \Box }
{\displaystyle \Diamond }
quantifies over accessible possible worlds, they will bind to any formula in their scope. For example,
{\displaystyle (\exists x(x^{2}=1))\land (0=y)}
{\displaystyle \exists x(x^{2}=1\land 0=y)}
{\displaystyle (\Diamond (x^{2}=1))\land (0=y)}
is not logically equivalent to
{\displaystyle \Diamond (x^{2}=1\land 0=y)}
; Instead,
{\displaystyle \Diamond (x^{2}=1\land 0=y)}
{\displaystyle (\Diamond (x^{2}=1))\land \Diamond (0=y)}
Relational semantics[edit]
A relational model is a tuple
{\displaystyle {\mathfrak {M}}=\langle W,R,V\rangle }
{\displaystyle W}
is a set of possible worlds
{\displaystyle R}
is a binary relation on
{\displaystyle W}
{\displaystyle V}
is a valuation function which assigns a truth value to each pair of an atomic formula and a world, (i.e.
{\displaystyle V:W\times F\to \{0,1\}}
{\displaystyle F}
is the set of atomic formulae)
{\displaystyle W}
is often called the universe. The binary relation
{\displaystyle R}
is called an accessibility relation, and it controls which worlds can "see" each other for the sake of determining what is true. For example,
{\displaystyle wRu}
means that the world
{\displaystyle u}
is accessible from world
{\displaystyle w}
. That is to say, the state of affairs known as
{\displaystyle u}
is a live possibility for
{\displaystyle w}
. Finally, the function
{\displaystyle V}
is known as a valuation function. It determines which atomic formulas are true at which worlds.
Then we recursively define the truth of a formula at a world
{\displaystyle w}
in a model
{\displaystyle {\mathfrak {M}}}
{\displaystyle {\mathfrak {M}},w\models P}
{\displaystyle V(w,P)=1}
{\displaystyle {\mathfrak {M}},w\models \neg P}
{\displaystyle w\not \models P}
{\displaystyle {\mathfrak {M}},w\models (P\wedge Q)}
{\displaystyle w\models P}
{\displaystyle w\models Q}
{\displaystyle {\mathfrak {M}},w\models \Box P}
iff for every element
{\displaystyle u}
{\displaystyle W}
{\displaystyle wRu}
{\displaystyle u\models P}
{\displaystyle {\mathfrak {M}},w\models \Diamond P}
iff for some element
{\displaystyle u}
{\displaystyle W}
{\displaystyle wRu}
{\displaystyle u\models P}
According to this semantics, a formula is necessary with respect to a world
{\displaystyle w}
if it holds at every world that is accessible from
{\displaystyle w}
. It is possible if it holds at some world that is accessible from
{\displaystyle w}
. Possibility thereby depends upon the accessibility relation
{\displaystyle R}
, which allows us to express the relative nature of possibility. For example, we might say that given our laws of physics it is not possible for humans to travel faster than the speed of light, but that given other circumstances it could have been possible to do so. Using the accessibility relation we can translate this scenario as follows: At all of the worlds accessible to our own world, it is not the case that humans can travel faster than the speed of light, but at one of these accessible worlds there is another world accessible from those worlds but not accessible from our own at which humans can travel faster than the speed of light.
Frames and completeness[edit]
The choice of accessibility relation alone can sometimes be sufficient to guarantee the truth or falsity of a formula. For instance, consider a model
{\displaystyle {\mathfrak {M}}}
whose accessibility relation is reflexive. Because the relation is reflexive, we will have that
{\displaystyle {\mathfrak {M}},w\models P\rightarrow \Diamond P}
{\displaystyle w\in G}
regardless of which valuation function is used. For this reason, modal logicians sometimes talk about frames, which are the portion of a relational model excluding the valuation function.
A relational frame is a pair
{\displaystyle {\mathfrak {M}}=\langle G,R\rangle }
{\displaystyle G}
is a set of possible worlds,
{\displaystyle R}
{\displaystyle G}
{\displaystyle w\models \Diamond P}
if and only if for some element u of G, it holds that
{\displaystyle u\models P}
and w R u.
{\displaystyle w\models \Diamond P}
{\displaystyle u\models P}
All of these logical systems can also be defined axiomatically, as is shown in the next section. For example, in S5, the axioms
{\displaystyle P\implies \Box \Diamond P}
{\displaystyle \Box P\implies \Box \Box P}
{\displaystyle \Box P\implies P}
(corresponding to symmetry, transitivity and reflexivity, respectively) hold, whereas at least one of these axioms does not hold in each of the other, weaker logics.
Topological semantics[edit]
A topological model is a tuple
{\displaystyle \mathrm {X} =\langle X,\tau ,V\rangle }
{\displaystyle \langle X,\tau \rangle }
{\displaystyle V}
is a valuation function which maps each atomic formula to some subset of
{\displaystyle X}
. The basic interior semantics interprets formulas of modal logic as follows:
{\displaystyle \mathrm {X} ,x\models P}
{\displaystyle x\in V(P)}
{\displaystyle \mathrm {X} ,x\models \neg \phi }
{\displaystyle \mathrm {X} ,x\not \models \phi }
{\displaystyle \mathrm {X} ,x\models \phi \land \chi }
{\displaystyle \mathrm {X} ,x\models \phi }
{\displaystyle \mathrm {X} ,x\models \chi }
{\displaystyle \mathrm {X} ,x\models \Box \phi }
iff for some
{\displaystyle U\in \tau }
we have both that
{\displaystyle x\in U}
{\displaystyle \mathrm {X} ,y\models \phi }
{\displaystyle y\in U}
Axiomatic systems[edit]
Modern treatments of modal logic begin by augmenting the propositional calculus with two unary operations, one denoting "necessity" and the other "possibility". The notation of C. I. Lewis, much employed since, denotes "necessarily p" by a prefixed "box" (□p) whose scope is established by parentheses. Likewise, a prefixed "diamond" (◇p) denotes "possibly p". Similar to the quantifiers in first-order logic, "necessarily p" (□p) does not assume the range of quantification (the set of accessible possible worlds in Kripke semantics) to be non-empty, whereas "possibly p" (◇p) often implicitly assumes
{\displaystyle \Diamond \top }
(viz. the set of accessible possible worlds is non-empty). Regardless of notation, each of these operators is definable in terms of the other in classical modal logic:
N, Necessitation Rule: If p is a theorem/tautology (of any system/model invoking N), then □p is likewise a theorem (i.e.
{\displaystyle (\models p)\implies (\models \Box p)}
{\displaystyle \Box p\to \Box \Box p}
{\displaystyle p\to \Box \Diamond p}
{\displaystyle \Box p\to \Diamond p}
{\displaystyle \Diamond p\to \Box \Diamond p}
K through S5 form a nested hierarchy of systems, making up the core of normal modal logic. But specific rules or sets of rules may be appropriate for specific systems. For example, in deontic logic,
{\displaystyle \Box p\to \Diamond p}
(If it ought to be that p, then it is permitted that p) seems appropriate, but we should probably not include that
{\displaystyle p\to \Box \Diamond p}
. In fact, to do so is to commit the appeal to nature fallacy (i.e. to state that what is natural is also good, by saying that if p is the case, p ought to be permitted).
Decision methods[edit]
Modal logics in philosophy[edit]
Alethic logic[edit]
Physical possibility[edit]
Metaphysical possibility[edit]
Epistemic logic[edit]
K, Distribution Axiom:
{\displaystyle \Box (p\to q)\to (\Box p\to \Box q)}
{\displaystyle \Diamond P}
= P is the case at some time t
{\displaystyle \Box P}
= P is the case at every time t
{\displaystyle \Diamond _{1}P}
= FP
{\displaystyle \Box _{1}P}
= GP
{\displaystyle \Diamond _{2}P}
= P and/or FP
{\displaystyle \Box _{2}P}
= P and GP
Deontic logic[edit]
Deontic logics commonly lack the axiom T semantically corresponding to the reflexivity of the accessibility relation in Kripke semantics: in symbols,
{\displaystyle \Box \phi \to \phi }
. Interpreting □ as "it is obligatory that", T informally says that every obligation is true. For example, if it is obligatory not to kill others (i.e. killing is morally forbidden), then T implies that people actually do not kill others. The consequent is obviously false.
One other principle that is often (at least traditionally) accepted as a deontic principle is D,
{\displaystyle \Box \phi \to \Diamond \phi }
, which corresponds to the seriality (or extendability or unboundedness) of the accessibility relation. It is an embodiment of the Kantian idea that "ought implies can". (Clearly the "can" can be interpreted in various senses, e.g. in a moral or alethic sense.)
Intuitive problems with deontic logic[edit]
{\displaystyle (K\to \Box Q)}
{\displaystyle \Box (K\to Q)}
But (1) and K together entail □Q, which says that it ought to be the case that you have stolen a small amount of money. This surely isn't right, because you ought not to have stolen anything at all. And (2) doesn't work either: If the right representation of "if you have stolen some money it ought to be a small amount" is (2), then the right representation of (3) "if you have stolen some money then it ought to be a large amount" is
{\displaystyle \Box (K\to (K\land \lnot Q))}
. Now suppose (as seems reasonable) that you ought not to steal anything, or
{\displaystyle \Box \lnot K}
. But then we can deduce
{\displaystyle \Box (K\to (K\land \lnot Q))}
{\displaystyle \Box (\lnot K)\to \Box (K\to K\land \lnot K)}
{\displaystyle \Box (K\land \lnot K\to (K\land \lnot Q))}
(the contrapositive of
{\displaystyle Q\to K}
); so sentence (3) follows from our hypothesis (of course the same logic shows sentence (2)). But that can't be right, and is not right when we use natural language. Telling someone they should not steal certainly does not imply that they should steal large amounts of money if they do engage in theft.[17]
Doxastic logic[edit]
Metaphysical questions[edit]
Further applications[edit]
Retrieved from "https://en.wikipedia.org/w/index.php?title=Modal_logic&oldid=1080806127"
|
Three-axle vehicle tractor body with translational and rotational motion - Simulink - MathWorks India
\begin{array}{l}{\overline{F}}_{b}=\left[\begin{array}{c}{F}_{x}\\ {F}_{y}\\ {F}_{z}\end{array}\right]=m\left({\stackrel{˙}{\overline{V}}}_{b}+\overline{\omega }×{\overline{V}}_{b}\right)\\ \\ {\overline{M}}_{b}=\left[\begin{array}{c}L\\ M\\ N\end{array}\right]=I\stackrel{˙}{\overline{\omega }}+\overline{\omega }×\left(I\overline{\omega }\right)\\ \\ I=\left[\begin{array}{ccc}{I}_{xx}& -{I}_{xy}& -{I}_{xz}\\ -{I}_{yx}& {I}_{yy}& -{I}_{yz}\\ -{I}_{zx}& -{I}_{zy}& {I}_{zz}\end{array}\right]\end{array}
\left[\begin{array}{ccc}\stackrel{˙}{\varphi }\text{ }\text{\hspace{0.17em}}& \stackrel{˙}{\theta }\text{\hspace{0.17em}}\text{ }\text{ }& \stackrel{˙}{\psi }\end{array}{\right]}^{T}
\left[\begin{array}{c}p\\ q\\ r\end{array}\right]=\left[\begin{array}{c}\stackrel{˙}{\varphi }\\ 0\\ 0\end{array}\right]+\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\varphi & \mathrm{sin}\varphi \\ 0& -\mathrm{sin}\varphi & \mathrm{cos}\varphi \end{array}\right]\left[\begin{array}{c}0\\ \stackrel{˙}{\theta }\\ 0\end{array}\right]+\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\varphi & \mathrm{sin}\varphi \\ 0& -\mathrm{sin}\varphi & \mathrm{cos}\varphi \end{array}\right]\left[\begin{array}{ccc}\mathrm{cos}\theta & 0& -\mathrm{sin}\theta \\ 0& 1& 0\\ \mathrm{sin}\theta & 0& \mathrm{cos}\theta \end{array}\right]\left[\begin{array}{c}0\\ 0\\ \stackrel{˙}{\psi }\end{array}\right]\equiv {J}^{-1}\left[\begin{array}{c}\stackrel{˙}{\varphi }\\ \stackrel{˙}{\theta }\\ \stackrel{˙}{\psi }\end{array}\right]
\left[\begin{array}{c}\stackrel{˙}{\varphi }\\ \stackrel{˙}{\theta }\\ \stackrel{˙}{\psi }\end{array}\right]=J\left[\begin{array}{c}p\\ q\\ r\end{array}\right]\text{\hspace{0.17em}}=\left[\begin{array}{ccc}1& \left(\mathrm{sin}\varphi \mathrm{tan}\theta \right)& \left(\mathrm{cos}\varphi \mathrm{tan}\theta \right)\\ 0& \mathrm{cos}\varphi & -\mathrm{sin}\varphi \\ 0& \frac{\mathrm{sin}\varphi }{\mathrm{cos}\theta }& \frac{\mathrm{cos}\varphi }{\mathrm{cos}\theta }\end{array}\right]\left[\begin{array}{c}p\\ q\\ r\end{array}\right]
\begin{array}{l}{\overline{F}}_{b}=\left[\begin{array}{c}{F}_{x}\\ {F}_{y}\\ {F}_{z}\end{array}\right]=\left[\begin{array}{c}{F}_{d}{}_{{}_{x}}\\ {F}_{d}{}_{{}_{y}}\\ {F}_{d}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{F}_{g}{}_{{}_{x}}\\ {F}_{g}{}_{{}_{y}}\\ {F}_{g}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{F}_{ext}{}_{{}_{x}}\\ {F}_{ext}{}_{{}_{y}}\\ {F}_{ext}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{F}_{FL}{}_{{}_{x}}\\ {F}_{FL}{}_{{}_{y}}\\ {F}_{FL}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{F}_{FR}{}_{{}_{x}}\\ {F}_{FR}{}_{{}_{y}}\\ {F}_{FR}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{F}_{ML}{}_{{}_{x}}\\ {F}_{ML}{}_{{}_{y}}\\ {F}_{ML}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{F}_{MR}{}_{{}_{x}}\\ {F}_{MR}{}_{{}_{y}}\\ {F}_{MR}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{F}_{RL}{}_{{}_{x}}\\ {F}_{RL}{}_{{}_{y}}\\ {F}_{RL}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{F}_{RR}{}_{{}_{x}}\\ {F}_{RR}{}_{{}_{y}}\\ {F}_{RR}{}_{{}_{z}}\end{array}\right]\\ \\ {\overline{M}}_{b}=\left[\begin{array}{c}{M}_{x}\\ {M}_{y}\\ {M}_{z}\end{array}\right]=\left[\begin{array}{c}{M}_{d}{}_{{}_{x}}\\ {M}_{d}{}_{{}_{y}}\\ {M}_{d}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{M}_{ext}{}_{{}_{x}}\\ {M}_{ext}{}_{{}_{y}}\\ {M}_{ext}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{M}_{FL}{}_{{}_{x}}\\ {M}_{FL}{}_{{}_{y}}\\ {M}_{FL}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{M}_{FR}{}_{{}_{x}}\\ {M}_{FR}{}_{{}_{y}}\\ {M}_{FR}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{M}_{ML}{}_{{}_{x}}\\ {M}_{ML}{}_{{}_{y}}\\ {M}_{ML}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{M}_{MR}{}_{{}_{x}}\\ {M}_{MR}{}_{{}_{y}}\\ {M}_{MR}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{M}_{RL}{}_{{}_{x}}\\ {M}_{RL}{}_{{}_{y}}\\ {M}_{RL}{}_{{}_{z}}\end{array}\right]+\left[\begin{array}{c}{M}_{RR}{}_{{}_{x}}\\ {M}_{RR}{}_{{}_{y}}\\ {M}_{RR}{}_{{}_{z}}\end{array}\right]+{\overline{M}}_{F}\end{array}
{J}_{ij}={I}_{ij}+m\left({|R|}^{2}{\delta }_{ij}-{R}_{i}{R}_{j}\right)
\begin{array}{l}\overline{w}=\sqrt{{\left(\stackrel{˙}{x}-{w}_{x}\right)}^{2}+{\left(\stackrel{˙}{x}-{w}_{x}\right)}^{2}+{\left({w}_{z}\right)}^{2}}\\ \\ {F}_{dx}=-\frac{1}{2TR}{C}_{d}{A}_{f}{P}_{abs}{\left(}^{\overline{w}}\\ {F}_{dy}=-\frac{1}{2TR}{C}_{s}{A}_{f}{P}_{abs}{\left(}^{\overline{w}}\\ {F}_{dz}=-\frac{1}{2TR}{C}_{l}{A}_{f}{P}_{abs}{\left(}^{\overline{w}}\end{array}
\begin{array}{l}{M}_{dr}=-\frac{1}{2TR}{C}_{rm}{A}_{f}{P}_{abs}{\left(}^{\overline{w}}\left(a+c\right)\\ {M}_{dp}=-\frac{1}{2TR}{C}_{pm}{A}_{f}{P}_{abs}{\left(}^{\overline{w}}\left(a+c\right)\\ {M}_{dy}=-\frac{1}{2TR}{C}_{ym}{A}_{f}{P}_{abs}{\left(}^{\overline{w}}\left(a+c\right)\end{array}
x,\stackrel{˙}{x},\stackrel{¨}{x}
y,\stackrel{˙}{y},\stackrel{¨}{y}
z,\stackrel{˙}{z},\stackrel{¨}{z}
FSusp=\left[\begin{array}{cccccc}{F}_{FLx}& {F}_{FRx}& {F}_{MLx}& {F}_{MRx}& {F}_{RLx}& {F}_{RRx}\\ {F}_{FLy}& {F}_{FRy}& {F}_{MLy}& {F}_{MRy}& {F}_{RLy}& {F}_{RRy}\\ {F}_{FLz}& {F}_{FRz}& {F}_{MLz}& {F}_{MRz}& {F}_{RLz}& {F}_{RRz}\end{array}\right]
MSusp=\left[\begin{array}{cccccc}{M}_{FLx}& {M}_{FRx}& {M}_{MLx}& {M}_{MRx}& {M}_{RLx}& {M}_{RRx}\\ {M}_{FLy}& {M}_{FRz}& {M}_{MLy}& {M}_{MRy}& {M}_{RLy}& {M}_{RRy}\\ {M}_{FLz}& {M}_{FRz}& {M}_{MLz}& {M}_{MRz}& {M}_{RLz}& {M}_{RRz}\end{array}\right]
\text{FExt}={F}_{ext}=\left[\begin{array}{ccc}{F}_{ex{t}_{x}}& {F}_{ex{t}_{y}}& {F}_{ex{t}_{z}}\end{array}\right]or\left[\begin{array}{c}{F}_{ex{t}_{x}}\\ {F}_{ex{t}_{y}}\\ {F}_{ex{t}_{z}}\end{array}\right]
\text{MExt}={M}_{ext}=\left[\begin{array}{ccc}{M}_{ex{t}_{x}}& {M}_{ex{t}_{y}}& {M}_{ex{t}_{z}}\end{array}\right]or\left[\begin{array}{c}{M}_{ex{t}_{x}}\\ {M}_{ex{t}_{y}}\\ {M}_{ex{t}_{z}}\end{array}\right]
\beta =\frac{{V}_{y}}{{V}_{x}}
z1I=\left[\begin{array}{ccc}{I}_{xx}& {I}_{xy}& {I}_{xz}\\ {I}_{yx}& {I}_{yy}& {I}_{yz}\\ {I}_{zx}& {I}_{zy}& {I}_{zz}\end{array}\right]
z2I=\left[\begin{array}{ccc}{I}_{xx}& {I}_{xy}& {I}_{xz}\\ {I}_{yx}& {I}_{yy}& {I}_{yz}\\ {I}_{zx}& {I}_{zy}& {I}_{zz}\end{array}\right]
z3I=\left[\begin{array}{ccc}{I}_{xx}& {I}_{xy}& {I}_{xz}\\ {I}_{yx}& {I}_{yy}& {I}_{yz}\\ {I}_{zx}& {I}_{zy}& {I}_{zz}\end{array}\right]
z4I=\left[\begin{array}{ccc}{I}_{xx}& {I}_{xy}& {I}_{xz}\\ {I}_{yx}& {I}_{yy}& {I}_{yz}\\ {I}_{zx}& {I}_{zy}& {I}_{zz}\end{array}\right]
z5I=\left[\begin{array}{ccc}{I}_{xx}& {I}_{xy}& {I}_{xz}\\ {I}_{yx}& {I}_{yy}& {I}_{yz}\\ {I}_{zx}& {I}_{zy}& {I}_{zz}\end{array}\right]
z6I=\left[\begin{array}{ccc}{I}_{xx}& {I}_{xy}& {I}_{xz}\\ {I}_{yx}& {I}_{yy}& {I}_{yz}\\ {I}_{zx}& {I}_{zy}& {I}_{zz}\end{array}\right]
z7I=\left[\begin{array}{ccc}{I}_{xx}& {I}_{xy}& {I}_{xz}\\ {I}_{yx}& {I}_{yy}& {I}_{yz}\\ {I}_{zx}& {I}_{zy}& {I}_{zz}\end{array}\right]
|
Formula for conversion from Fahrenheit to Celsius
Other temperature scale converters
Omni's Fahrenheit to Celsius converter allows you to convert the temperature values from Fahrenheit to Celsius temperature scales.
Continue reading to learn more about the different temperature scales, i.e., Fahrenheit and Celsius, and the formula for conversion between the two. You will also see how to convert Fahrenheit to Celsius using our converter.
Both degree Celsius (°C) and degree Fahrenheit (°F) are units of temperature.
On the Celsius temperature scale, the freezing point of water at 1 atm pressure is 0 °C, and the boiling point of water is 100 °C. The difference between these two points is divided into 100 equal parts, each part representing 1 °C.
On the Fahrenheit temperature scale the freezing point of water is 32 °F, and the boiling point is 212 °F. The difference between these two fixed points is further divided into 180 equal parts, each division representing 1 °F.
Using the information mentioned in the previous section, we will write the formula to convert between Fahrenheit and Celsius temperature scales as:
\small \begin{align*} \frac{T_c - 0}{100} &= \frac{T_F - 32}{180} \\ \\ \frac{T_c}{5} &= \frac{T_F - 32}{9} \\ \end{align*}
T_c
- Temperature on Celsius scale; and
T_F
- Temperature on Fahrenheit scale.
For example, using the formula for conversion from Fahrenheit to Celsius, we can show that 212 °F is equivalent to 100 °C:
\small \begin{align*} T_c&= \frac{212 - 32}{9} \times 5 \\ & = \frac{180}{9} \times 5 \\ & = 100 \degree \text{C} \end{align*}
Now let us see how we can convert Fahrenheit to Celsius using our converter:
Enter the temperature value on the Fahrenheit scale.
The converter will display the temperature on the Celsius scale.
You can also use this tool to convert from Celsius to Fahrenheit.
If you want to convert to other temperature scales, make sure to check out our other temperature converters:
To convert from Fahrenheit to Celsius temperature scale, follow the given instructions:
Subtract 32 from the temperature value on the Fahrenheit scale.
Divide the result by 9 and multiply by 5.
Congrats! You have converted from Fahrenheit to Celsius temperature scale.
Is there a point where Celsius and Fahrenheit are equal?
Yes, at -40, both Celsius and Fahrenheit are equal, as shown below:
Use the formula for conversion from Celsius and Fahrenheit, i.e., T(°C) = (T(°F) - 32)/9 × 5.
Substitute x in the above equation for both temperature scales, x(°C) = (x (°F) - 32)/9 × 5.
Solve for x, and you will get x = -40.
Hence both Celsius and Fahrenheit are equal at -40 °C and -40 °F.
💡 You can also convert from Celsius to Fahrenheit!
You can use the ft to m converter to convert feet to meters.
|
What is the standard deviation of the distribution of the sample mean?
How to calculate the standard deviation of the sample mean?
Example of how to use this calculator
Additional tools to complement your calculations of the standard deviation of the distribution of sample mean
Calculating the standard deviation of the sample mean (aka standard deviation of the sampling distribution of the mean) is an excellent way to make sense of how the sample size influences the error of our estimates.
When the standard deviation of the mean is multiplied by a critical value, like the z-score or t-statistic, we obtain a margin of error that allows stating a confidence interval of our prediction. Therefore, calculating the standard deviation of the sampling distribution of the mean indicates where the population mean could be.
First of all, it's important to clarify that this term is known by various names such as:
Standard deviation of the mean;
Standard deviation of the sample mean;
Standard deviation of the distribution of sample means; and
Standard deviation of the sampling distribution of the sample mean.
It's also essential to know some definitions and concepts:
Statistic: a point estimate or numeric characteristic of a sample (i.e., sample mean). It's different from a parameter, such as the population mean.
Sampling distribution: the probability distribution of a randomly sampled statistic. In other words, it is the distribution of all the possible values that a statistic could take using the same sample size.
Sampling distribution of the mean: this is an extension of the previous concept. If you have a population, take infinite samples of n size, and plot their means in a histogram, you get a probability distribution. That probability distribution is what we call the sampling distribution of the mean, and like any other distribution, it has its own mean and standard deviation.
The following diagram shows how to generate a sampling distribution of the mean. In reality, we don't use only six samples but an almost infinite number (i.e., 100,000).
Sampling distribution of the sample mean generation process.
That said, we can define the standard deviation mean as the standard deviation of a distribution of means (like the one shown in the last diagram). Additionally, the mean of this sampling distribution is equal to the population mean.
Now that you know the concept let's see how to calculate the standard deviation of the sample mean.
The formula to find the standard deviation of the sample mean is:
σX̄ = σ/√n
σX̄ – Standard deviation of the sample mean;
σ – Population standard deviation; and
n - Sample size.
We previously said that if we know the mean of the sampling distribution (μX̄), we also know the population mean (μ), as they're equal (μX̄ = μ). In practice, we never know μX̄, but we can estimate it using the sample mean (X̄).
σX̄ indicates how X̄ approximates to μ. The smaller σX̄, the nearest μ can be from our estimate. As σ is constant, the only way to diminish σX̄ is to increase the sample size. Therefore, increasing n is a way to reduce our estimations' sampling error.
We know that the mean and standard deviation of height of the adult American female population is about μ = 161.3 cm and σ = 7.1 cm. Now, suppose you randomly take samples of 100 women and take their mean height each time. What is the standard deviation of the sample mean?
Input 7.1 in the population standard deviation box.
Input 100 in the sample size box.
That's it. The answer should be 0.71. Therefore, the standard deviation of the sampling distributions of means n = 100 is 0.71.
You can check the results using the formula:
σ_{\bar X} = \frac{σ}{\sqrt{n}} = \frac{7.1}{\sqrt{100}} = 0.71
Now that you know how to calculate the standard deviation of the distribution of sample means, you can go one step beyond and use these other tools:
What is the difference between sample distribution and sampling distribution?
The difference between a sample and sampling distribution is:
Sampling distribution: it's the term we usually hear. It refers to the probability distribution of a randomly sampled statistic. The sampling distribution of the mean is an example.
Sample distribution: accounts for the distribution of the observations within only one sample. Each sample distribution possesses a mean, which helps form the sampling distribution.
How to find the mean and standard deviation of the sampling distribution?
To find the standard deviation of the sample mean (σX̄), divide the population standard deviation (σ) by the square root of the sample size (n): σX̄ = σ/√n.
Contrary to the standard deviation, to calculate the mean of the sampling distribution of the mean (μX̄), you only need the population mean (μ), as they are both the same (μX̄ = μ).
What is the standard deviation of the sample means called?
The standard deviation of the sample means is called by different names:
Standard deviation of the distribution of sample means.
Mathematically, you calculate the standard deviation of the sample mean with the formula σX̄ = σ/√n.
The standard error of the mean (SE(X̄)) is a different statistic that uses the sample standard deviation (s) instead of σ. Its formula is SE(X̄) = s/√n.
How to create a sampling distribution of the mean?
Follow these steps to create a sampling distribution of the mean:
Define a sample size.
Take a random sample of that size and calculate its mean.
Plot that mean in a histogram.
Repeat this process an almost infinite number of times (i.e., 100,000 times) until the distribution converges.
If we're in front of a normal distribution, a standard deviation of 1 (σ = 1) means that 68.27% of the distribution values lie within one standard deviation from the mean. In mathematical notation: Pr( μ-1σ ≤ X ≤ μ+1σ) ≈ 68.27%.
Standard deviation of the sample mean (σₓ)
|
How to use our acres to hectares converter
Other surface conversion tools!
Whether you need to convert the surface of your plot or are studying geography, our acres to hectares converter will be helpful!
What is an acre;
What is a hectare;
How to convert from acres to hectares; and
Some neat examples.
An acre is a surface measurement unit of the US customary system. It corresponds to
1/640
square miles. The traditional acre is defined as the area with sides equal to one chain and one furlong, two traditional — and today obsolete — length measurement units.
A chain is
66
feet long; and
A furlong is
660
feet long;
We can think of the acre as
\footnotesize 1\ \text{acre} = 66 × 660\ \text{ft} = 43560\ \text{ft}
🙋 You can indicate the acres in your measurements both with the extended form
\text{acre}
\text{ac}
The acre is both an old measurement unit and a unit used in agriculture: these features give it a colorful history and origin!
Many countries used the acre in the past specifically to measure the size of arable land and of properties. Its definition was often dependent on the work done in a certain time. In particular, the traditional acre corresponds to the surface that a single man can work in a day with the help of a single team of oxen.
Nowadays, thanks to technology, a single man can work way more than an acre a day!
A hectare, symbol
\text{ha}
, is a unit similar to the acre in scope but different when it comes to definition.
First of all, the hectare originates within the metric system: it is nothing but a square with sides equal to
100\ \text{m}
. This gives us the conversion:
\footnotesize 1\ \text{ha} = 100 × 100\ \text{m} = 10,000\ \text{m}^2
Hectares are commonly used across the world to measure fields, land, and plots. Do you know that an official rugby field has a surface of almost exactly one hectare? Luckily, we don't measure surfaces in rugby fields!
The conversion between acres and hectares is a useful one, for sure. To convert acres to hectares, use the following formula:
1\ \text{acre} = 0.404686\ \text{ha}
As you can see, the acre is slightly smaller than half a hectare. If you are more familiar with fractions, you can use an alternative approximate formula:
1\ \text{acre} = \frac{1}{2.5}×1\ \text{ha}
To know how many hectares are to an acre or to any value you need, use our acres to hectares converter. You only have to insert the measurement in the appropriate field, and we will do all the math!
Do you want to know how many hectares is the Yellowstone National Park? Here we are. The park has a staggering surface of
2,219,791\ \text{acres}
. To put this number in perspective, four US States are smaller than the park: Rhode Island, Delaware, Connecticut, and New Jersey. To calculate how many hectares correspond to that number, apply the conversion:
\footnotesize \begin{align*} &2,219,791\ \text{acres} = \\ &=2,219,791 \cdot 0.404686\ \text{ha} =\\ &= 898,317\ \text{ha} \end{align*}
This incredible amount of land is worth protecting: a speckle of Nature we need to preserve.
We covered almost all of the possible conversions between measurement units for the area! Here is a comprehensive list:
How many hectares is the Hundred Acres Wood?
40.4 ha. The forest where Winnie-the-Pooh and his friends reside has a surface — at least in the stories by Milne — of 100 acres. Apply the conversion between acres and hectares to find the value of its surface in the metric unit:
100 acres = 100 × 0.404 ha = 40.4 ha
How do I convert from acres to hectares?
To convert from acres to hectares:
Measure the surface using the US surface measurement unit;
Multiply that quantity by 0.404686;
About 0.4. An acre covers a smaller surface than a hectare. To calculate the hectares from the acres, you can either:
Multiply the measurement in acres by 0.404686; or
Divide the measurement in acres by 2.471.
What are 5 acres in hectares?
5 acres is about 2 hectares. Use the following conversion to find out how many hectares are in 2 acres:
1 acre = 1/2.5 ha
Then simply substitute the known values:
5 acres = 5/2.5 ha = 2 ha
This formula is most helpful when you deal with multiples of 5!
AcreageAcres to square feet converterAngle conversion… 137 more
|
k=A\mathrm{exp}\left(-{E}_{a}/RT\right)
\text{d}x/\text{d}t=Af\left(x\right)\mathrm{exp}\left(-\frac{{E}_{a}}{RT}\right)
x=\frac{\left({w}_{0}-w\right)}{\left({w}_{0}-{w}_{\infty }\right)}
\beta =\frac{\text{d}T}{\text{d}t}
\frac{{\text{d}}^{2}x}{\text{d}{t}^{2}}=\left\{\frac{{E}_{a}\beta }{R{T}^{2}}+A{f}^{\prime }\left(x\right)\mathrm{exp}\left(-{E}_{a}/RT\right)\right\}\frac{\text{d}x}{\text{d}t}
\mathrm{ln}\left(\frac{\beta }{{T}_{\text{peak}}^{2}}\right)=\mathrm{ln}\left(\frac{AR}{{E}_{a}}\right)-\left(\frac{{E}_{a}}{R{T}_{\text{peak}}}\right)
Equation (6) is a straight line graph, of
\mathrm{ln}\left(\beta /{T}_{\text{peak}}^{2}\right)
1/{T}_{\text{Peak}}
, The line slope is
{E}_{a}/R
and the intercept on the vertical axis is an
\mathrm{ln}\left(AR/{E}_{a}\right)
, which are used to determined Ea and A.
x=\frac{\left({w}_{0}-w\right)}{\left({w}_{0}-{w}_{\infty }\right)}
\Delta {S}^{\ne }=R\left(\mathrm{ln}A-\mathrm{ln}\frac{\sigma e\chi {T}_{\text{peak}}}{h}\right)
\Delta {H}^{\ne }={E}_{a}-R{T}_{\text{peak}}
\Delta {G}^{\ne }=\Delta {H}^{\ne }-{T}_{\text{peak}}\Delta {S}^{\ne }
The chemical analysis (ultimate analysis) was used to estimate the chemical composition of pine sawdust. Sulphur is neglected in this study since its amount is small in comparison with those of other elements. When this is taken into consideration, the chemical formula for the pine sawdust material is
{\text{C}}_{\text{23}\text{.74}}{\text{H}}_{\text{32}\text{.03}}{\text{O}}_{\text{16}\text{.16}}\text{N}
|
Nuclear Reactions | Boundless Chemistry | Course Hero
To balance a nuclear equation, the mass number and atomic numbers of all particles on either side of the arrow must be equal.
Produce a balanced nuclear equation
A balanced nuclear equation is one where the sum of the mass numbers (the top number in notation) and the sum of the atomic numbers balance on either side of an equation.
Nuclear equation problems will often be given such that one particle is missing.
Instead of using the full equations, in many situations a compact notation is used to describe nuclear reactions.
baryon: A heavy subatomic particle created by the binding of quarks by gluons; a hadron containing three quarks. They have half-odd integral spin and are thus fermions.
Nuclear reactions may be shown in a form similar to chemical equations, for which invariant mass, which is the mass not considering the mass defect, must balance for each side of the equation. The transformations of particles must follow certain conservation laws, such as conservation of charge and baryon number, which is the total atomic mass number. An example of this notation follows:
^6_3\text{Li}\ +\ \ ^2_1\text{H}\rightarrow \ ^4_2\text{He}\ + \ ?
To balance the equation above for mass, charge, and mass number, the second nucleus on the right side must have atomic number 2 and mass number 4; it is therefore also helium-4. The complete equation therefore reads:
^6_3\text{Li}\ +\ \ ^2_1\text{H}\rightarrow \ ^4_2\text{He}\ \ +\ \ ^4_2\text{He}
^6_3\text{Li}\ +\ \ ^2_1\text{H}\rightarrow 2\ ^4_2\text{He}
Lithium-6 plus deuterium gives two helium-4s.: The visual representation of the equation we used as an example.
Compact Notation of Radioactive Decay
Instead of using the full equations in the style above, in many situations a compact notation is used to describe nuclear reactions. This style is of the form A(b,c)D, which is equivalent to A + b gives c + D. Common light particles are often abbreviated in this shorthand, typically p for proton, n for neutron, d for deuteron, α representing an alpha particle or helium-4, β for beta particle or electron, γ for gamma photon, etc. The reaction in our example above would be written as Li-6(d,α)α.
Balancing a Radioactive Decay Equation
In balancing a nuclear equation, it is important to remember that the sum of all the mass numbers and atomic numbers, given on the upper left and lower left side of the element symbol, respectively, must be equal for both sides of the equation. In addition, problems will also often be given as word problems, so it is useful to know the various names of radioactively emitted particles.
^{ 235 }_{ 92 }\text{U} \rightarrow \ \ ^{ 231 }_{ 90 }\text{Th}\ +\ \ ?
This could be written out as uranium-235 gives thorium-231 plus what? In order to solve, we find the difference between the atomic masses and atomic numbers in the reactant and product. The result is an atomic mass difference of 4 and an atomic number difference of 2. This fits the description of an alpha particle. Thus, we arrive at our answer:
^{ 235 }_{ 92 }\text{U} \rightarrow \ \ ^{ 231 }_{ 90}\text{Th} + \ \ ^{ 4 }_{ 2 }\text{He}
^{ 214 }_{ 84 }\text{Po}\ +\ \ 2\ ^{ 4 }_{ 2 }\text{He}\ +\ 2^0_{-1}\text{e}\rightarrow \ \ \ ?
This could also be written out as polonium-214, plus two alpha particles, plus two electrons, give what? In order to solve this equation, we simply add the mass numbers, 214 for polonium, plus 8 (two times four) for helium (two alpha particles), plus zero for the electrons, to give a mass number of 222. For the atomic number, we take 84 for polonium, add 4 (two times two) for helium, then subtract two (two times -1) for two electrons lost through beta emission, to give 86; this is the atomic number for radon (Rn). Therefore, the equation should read:
^{ 214 }_{ 84 }\text{Po}+2^{ 4 }_{ 2 }\text{He}+2^0_{-1}\text{e}\rightarrow\ \ ^{222}_{86}\text{Rn}
Writing nuclear equations: Describes how to write the nuclear equations for alpha and beta decay.
A nucleus weighs less than its sum of nucleons, a quantity known as the mass defect, caused by release of energy when the nucleus formed.
nucleon: One of the subatomic particles of the atomic nucleus, i.e. a proton or a neutron.
strong force: The nuclear force, a residual force responsible for the interactions between nucleons, deriving from the color force.
mass defect: The difference between the calculated mass of the unbound system and the experimentally measured mass of the nucleus.
Nuclear binding energy is the energy required to split a nucleus of an atom into its component parts: protons and neutrons, or, collectively, the nucleons. The binding energy of nuclei is always a positive number, since all nuclei require net energy to separate them into individual protons and neutrons.
Nuclear binding energy accounts for a noticeable difference between the actual mass of an atom's nucleus and its expected mass based on the sum of the masses of its non-bound components.
Recall that energy (E) and mass (m) are related by the equation:
\text{E}=\text{mc}^2
Here, c is the speed of light. In the case of nuclei, the binding energy is so great that it accounts for a significant amount of mass.
The actual mass is always less than the sum of the individual masses of the constituent protons and neutrons because energy is removed when when the nucleus is formed. This energy has mass, which is removed from the total mass of the original particles. This mass, known as the mass defect, is missing in the resulting nucleus and represents the energy released when the nucleus is formed.
Mass defect (Md) can be calculated as the difference between observed atomic mass (mo) and that expected from the combined masses of its protons (mp, each proton having a mass of 1.00728 amu) and neutrons (mn, 1.00867 amu):
\text{M}_\text{d}=(\text{m}_\text{n}+\text{m}_\text{p})-\text{m}_\text{o}
Once mass defect is known, nuclear binding energy can be calculated by converting that mass to energy by using E=mc2. Mass must be in units of kg.
Once this energy, which is a quantity of joules for one nucleus, is known, it can be scaled into per-nucleon and per- mole quantities. To convert to joules/mole, simply multiply by Avogadro's number. To convert to joules per nucleon, simply divide by the number of nucleons.
Nuclear binding energy can also apply to situations when the nucleus splits into fragments composed of more than one nucleon; in these cases, the binding energies for the fragments, as compared to the whole, may be either positive or negative, depending on where the parent nucleus and the daughter fragments fall on the nuclear binding energy curve. If new binding energy is available when light nuclei fuse, or when heavy nuclei split, either of these processes result in the release of the binding energy. This energy—available as nuclear energy—can be used to produce nuclear power or build nuclear weapons. When a large nucleus splits into pieces, excess energy is emitted as photons, or gamma rays, and as kinetic energy, as a number of different particles are ejected.
Nuclear binding energy is also used to determine whether fission or fusion will be a favorable process. For elements lighter than iron-56, fusion will release energy because the nuclear binding energy increases with increasing mass. Elements heavier than iron-56 will generally release energy upon fission, as the lighter elements produced contain greater nuclear binding energy. As such, there is a peak at iron-56 on the nuclear binding energy curve.
Nuclear binding energy curve: This graph shows the nuclear binding energy (in MeV) per nucleon as a function of the number of nucleons in the nucleus. Notice that iron-56 has the most binding energy per nucleon, making it the most stable nucleus.
The rationale for this peak in binding energy is the interplay between the coulombic repulsion of the protons in the nucleus, because like charges repel each other, and the strong nuclear force, or strong force. The strong force is what holds protons and neutrons together at short distances. As the size of the nucleus increases, the strong nuclear force is only felt between nucleons that are close together, while the coulombic repulsion continues to be felt throughout the nucleus; this leads to instability and hence the radioactivity and fissile nature of the heavier elements.
Calculate the average binding energy per mole of a U-235 isotope. Show your answer in kJ/mole.
First, you must calculate the mass defect. U-235 has 92 protons, 143 neutrons, and has an observed mass of 235.04393 amu.
\text{M}_\text{d}=(\text{m}_\text{n}+\text{m}_\text{p})-\text{m}_\text{o}
Md = (92(1.00728 amu)+143(1.00867 amu)) - 235.04393 amu
Md = 1.86564 amu
Calculate the mass in kg:
1.86564 amu x
\frac{1\ \text{kg}}{6.02214\times10^{26}\ \text{amu}}
= 3.09797 x 10-27 kg
Now calculate the energy:
E = 3.09797 x 10-27 kg x (2.99792458 x 108
\frac{\text{m}}{\text{s}}
E =2.7843 x 10-10 J
Now convert to kJ per mole:
2.7843\times10^{-10}\frac{\text{Joules}}{\text{atom}}\ \times \frac {6.02\times10^{23}\ \text{atoms}}{\text{mole}}\times \frac{1\ \text{kJ}}{1000\ \text{joules}} =
\frac{\text{kJ}}{\text{mole}}
Nuclear reaction. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/Nuclear_reaction. License: CC BY-SA: Attribution-ShareAlike
baryon. Provided by: Wiktionary. Located at: http://en.wiktionary.org/wiki/baryon. License: CC BY-SA: Attribution-ShareAlike
File:Li6-D%20Reaction.svg%20-%20Wikipedia,%20the%20free%20encyclopedia. Provided by: Wikipedia. Located at: http://en.wikipedia.org/w/index.php?title=File:Li6-D_Reaction.svg&page=1. License: CC BY-SA: Attribution-ShareAlike
Writing nuclear equations. License: Public Domain: No Known Copyright. License terms: Standard YouTube license
Nuclear binding energy. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/Nuclear_binding_energy. License: CC BY-SA: Attribution-ShareAlike
mass defect. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/mass%20defect. License: CC BY-SA: Attribution-ShareAlike
nucleon. Provided by: Wiktionary. Located at: http://en.wiktionary.org/wiki/nucleon. License: CC BY-SA: Attribution-ShareAlike
strong force. Provided by: Wiktionary. Located at: http://en.wiktionary.org/wiki/strong_force. License: CC BY-SA: Attribution-ShareAlike
File:Binding energy curve - common isotopes.svg - Wikipedia, the free encyclopedia. Provided by: Wikipedia. Located at: http://en.wikipedia.org/w/index.php?title=File:Binding_energy_curve_-_common_isotopes.svg&page=1. License: Public Domain: No Known Copyright
The Origins of the Elements.pptx
CHE MISC • Technological University of Peru
6ae23b11-Nuclear_Chemistry_ch12.doc
SPA 3178 • New London High School, New London
Stellar And fusion star.pptx
CHEM 141 • Embry-Riddle Aeronautical University
2019_04_03_Nuclear_Fusion.ppt
Nuclear Chemistry.pdf
2016_03_30_Nuclear_Fusion(1)
2015_10_20_Atoms_and_Nuclear_Energy
2016_03_23_Atoms_and_Nuclear_Energy (1)
NUCLEAR REACTION1.docx
5.2.3 Part 1_ Introduction to Nuclear Reactions-Honors (1).pdf
CHEM 100 • Irvine Valley College
Ch 4 Nuclear Reactions
balancing-nuclear-reactions-converted.docx
CHEMISTRY 121 • Pattonville High School
Nuclear Reactions.pdf
CHEM 1700 • Saint Paul College
RaKirah__Nuclear_Reactions_Practice.docx
CHEM 101 • Phillip O Berry Academy Of Tec
Nuclear-Reactions.docx
CHEM PHYSICAL C • Centro Escolar University
Nuclear Reactions Worksheet (1).pdf
CHEMISTRY 2003350 • Freedom High School
11.02 Chemistry Quiz- Nuclear Reactions.docx
CHEMISTRY 101 • Keystone Academy
5.2.3 Part 1_ Introduction to Nuclear Reactions-Honors.pdf
Ch 24 Nuclear Reactions Quiz A.docx
CHEM 101 • Ehove Career Center
Ch 24 Nuclear Reactions Quiz B KEY.docx
Ch 24 Nuclear Reactions Quiz B.docx
Nuclear_Reactions_Worksheet_Fission_and_Fusion.docx
CHEMISTRY 2005 • Brashear Hs
|
Note on the Formation of Supermassive Black Holes
Abstract: Supermassive black holes formed during the lepton epoch of the Big Bang. What follows is a description of how this may have happened.
Keywords: Supermassive Black Holes, Electron-Positron Model
The electron-positron model of supermassive black holes is given in two recent papers [1] [2]. The equilibrium conditions from [1] are reproduced below. The intermediate masses (Table 1) range from 103 to
8\times {10}^{6}\text{\hspace{0.05em}}{M}_{\odot }
. They are supported against gravity by electron degeneracy pressure and are characterized by the central Fermi energy
{ϵ}_{F0}
. They are in a quantum ground state and do not radiate. Larger masses (Table 2) are supported by ideal gas and radiation pressure, with a central thermal energy
k{T}_{0}
. They are equal in size to the Schwarzschild radius, so that the gas and radiation are confined.
Clusters of black holes can explain the dark matter in elliptical galaxies, dwarf galaxies, star clusters and galactic clusters [3]. Spiral galaxies evolved around a single black hole, and the missing mass must be treated separately [4]. In these
and other cases, it is the arrival of black holes in the early Universe that helps to explain recent discoveries in astronomy.
2. Formation of Black Holes
A great profusion of electrons and positrons occurred during the lepton epoch of the Big Bang. It is estimated that for every nucleon there were 109 leptons, so that the mass of leptons exceeded that of nucleons by a factor of 106 [5]. It may be said that this period began at time
t={10}^{-4}\text{s}
T={10}^{12}\text{K}
\rho ={10}^{14}\text{g}\cdot {\text{cm}}^{-3}
, when the final creation of muons took place. This was followed by annihilation and by the decay of muons into electrons and neutrinos (but no photons). The latter reaction depleted the photon density in favor of electrons and positrons. The lepton energy density
{u}_{\text{lep}}=3{\rho }_{\text{lep}}kT/m
will be greater than that of radiation
{u}_{\text{rad}}=a{T}^{4}
{\rho }_{\text{lep}}>\frac{am}{3k}\text{ }{T}^{3}
The corresponding Jeans mass is [6]
{M}_{J}={\left(\frac{750}{\pi }\right)}^{1/2}{\left(\frac{k}{Gm}\right)}^{3/2}{\left(\frac{{T}^{3}}{{\rho }_{\text{lep}}}\right)}^{1/2}<2\times {10}^{8}{M}_{\odot }
Any mass greater than
{M}_{J}
will be unstable toward gravitational collapse. In response to the surge in density, the lepton mass shattered into billions of spherical masses, each with a decreasing rate of expansion. They became supermassive black holes.
The time required for expansion from the highly compressed state to equilibrium may be estimated from R/c in the tables. This ranges from 102 to 106 seconds. During this length of time, a great deal of annihilation occurred. In the present-day Universe, the mass ratio of dark (leptonic) matter to normal (baryonic) matter is five or six to one. Comparison with the above-cited factor of 106 shows that a minute portion of leptons achieved equilibrium as a black hole. The vast majority annihilated and replenished the expanding radiation field.
The conventional treatment of black holes is mathematical. It posits a solution to the field equations of general relativity, which yields a singular metric at the Schwarzschild radius [7]. This implies that light cannot escape from a black hole. However, this phenomenon was already known to Laplace and others [8]. Since that time, the difficulty has been to find a suitable concentration of mass. The electron-positron model solves that problem, and it provides a physical understanding of black holes.
Cite this paper: Dalton, K. (2020) Note on the Formation of Supermassive Black Holes. Journal of High Energy Physics, Gravitation and Cosmology, 6, 321-323. doi: 10.4236/jhepgc.2020.63025.
[2] Dalton, K. (2014) The Galactic Black Hole. Hadronic J., 37, 241-245.
[3] Dalton, K. (2019) Black Hole Clusters: The Dark Matter. JHEPGC, 5, 989-991.
[4] Dalton, K. (2013) The Missing Mass of the Milky Way Galaxy. Hadronic J., 36, 499.
[5] Harrison, E. (2000) Cosmology. 2nd Edition, Cambridge University Press, Cambridge, Chap. 20.
[6] Carroll, B. and Ostlie, D. (2007) An Introduction to Modern Astrophysics. 2nd Edition, Addison-Wesley, San Francisco, Sect. 12.2.
[7] Adler, R., Bazin, M. and Schiffer, M. (1975) Introduction to General Relativity. 2nd Edition, McGraw-Hill, Tokyo, Sect. 6.8.
|
torque - Maple Help
Home : Support : Online Help : Science and Engineering : Units : Known Units : torque
Torque has the dimension force length(radius). The SI composite unit of torque is the newton meter(radius).
A torque is the product of a radial distance, for example, meter(radius), and a tangential force, that is, the component of the force vector tangent to the radial vector. Multiplying a unit of torque by a unit of planar angle results in a unit of energy (the energy required to rotate the body through the given angle).
\mathrm{convert}\left('N''m\left(\mathrm{radius}\right)','\mathrm{dimensions}','\mathrm{base}'=\mathrm{true}\right)
\frac{\textcolor[rgb]{0,0,1}{\mathrm{length}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{length}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{radius}}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{mass}}}{{\textcolor[rgb]{0,0,1}{\mathrm{time}}}^{\textcolor[rgb]{0,0,1}{2}}}
\mathrm{convert}\left(1,'\mathrm{units}','N''m\left(\mathrm{radius}\right)','\mathrm{lbf}''\mathrm{ft}\left(\mathrm{radius}\right)'\right)
\frac{\textcolor[rgb]{0,0,1}{2500000000000000}}{\textcolor[rgb]{0,0,1}{3389544870828501}}
When the standard or natural modes for combining units are selected, Maple by default requires the correct annotation to the length unit, as in the previous example. This also prevents conversions from units of torque to units of energy; they differ only in the annotations. In the default simple mode, this is not required.
\mathrm{convert}\left(1,'\mathrm{units}','N''m\left(\mathrm{radius}\right)','\mathrm{lbf}''\mathrm{ft}'\right)
\frac{\textcolor[rgb]{0,0,1}{2500000000000000}}{\textcolor[rgb]{0,0,1}{3389544870828501}}
\mathrm{convert}\left(1,'\mathrm{units}','N''m\left(\mathrm{radius}\right)','J'\right)
\textcolor[rgb]{0,0,1}{1}
\mathrm{Units}[\mathrm{UseMode}]\left('\mathrm{standard}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{simple}}
\mathrm{convert}\left(1,'\mathrm{units}','N''m\left(\mathrm{radius}\right)','\mathrm{lbf}''\mathrm{ft}'\right)
Error, (in `convert/units`) unable to convert `N*m(radius)` to `ft*lbf`
\mathrm{convert}\left(1,'\mathrm{units}','N''m\left(\mathrm{radius}\right)','J'\right)
Error, (in `convert/units`) unable to convert `N*m(radius)` to `J`
\mathrm{convert}\left(1,'\mathrm{units}','N''m\left(\mathrm{radius}\right)','J','\mathrm{symbolic}'\right)
\textcolor[rgb]{0,0,1}{1}
\mathrm{with}\left(\mathrm{Units}[\mathrm{Standard}]\right):
\mathrm{unit}
32.523\mathrm{Unit}\left('N''m\left(\mathrm{radius}\right)'\right)
\textcolor[rgb]{0,0,1}{32.523}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{N}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{radius}}\right)⟧
\cdot 45\mathrm{Unit}\left('\mathrm{degrees}'\right)
\textcolor[rgb]{0,0,1}{25.54350447}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{J}⟧
|
Quadratic function - Wikipedia @ WordDisk
Forms of a univariate quadratic function
Graph of the univariate function
Roots of the univariate function
The square root of a univariate quadratic function
Bivariate (two variable) quadratic function
In algebra, a quadratic function, a quadratic polynomial, a polynomial of degree 2, or simply a quadratic, is a polynomial function with one or more variables in which the highest-degree term is of the second degree.
A quadratic polynomial with two real roots (crossings of the x axis) and hence no complex roots. Some other quadratic polynomials have their minimum above the x axis, in which case there are no real roots and two complex roots.
Polynomial function of degree two
For the zeros of a quadratic function, see Quadratic equation and Quadratic formula.
For example, a univariate (single-variable) quadratic function has the form[1]
{\displaystyle f(x)=ax^{2}+bx+c,\quad a\neq 0}
in the single variable x. The graph of a univariate quadratic function is a parabola whose axis of symmetry is parallel to the y-axis, as shown at right.
If the quadratic function is set equal to zero, then the result is a quadratic equation. The solutions to the univariate equation are called the roots of the univariate function.
{\displaystyle f(x,y)=ax^{2}+by^{2}+cxy+dx+ey+f\,\!}
with at least one of a, b, c not equal to zero, and an equation setting this function equal to zero gives rise to a conic section (a circle or other ellipse, a parabola, or a hyperbola).
A quadratic function in three variables x, y, and z contains exclusively terms x2, y2, z2, xy, xz, yz, x, y, z, and a constant:
{\displaystyle f(x,y,z)=ax^{2}+by^{2}+cz^{2}+dxy+exz+fyz+gx+hy+iz+j,}
with at least one of the coefficients a, b, c, d, e, or f of the second-degree terms being non-zero.
In general there can be an arbitrarily large number of variables, in which case the resulting surface of setting a quadratic function to zero is called a quadric, but the highest degree term must be of degree 2, such as x2, xy, yz, etc.
This article uses material from the Wikipedia article Quadratic function, 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.
|
Lewis Acid and Base Molecules | Introduction to Chemistry | Course Hero
Lewis Acid and Base Molecules
Recognize Lewis acids and bases in chemical reactions.
A Lewis acid is an electron-pair acceptor; a Lewis base is an electron-pair donor.
Some molecules can act as either Lewis acids or Lewis bases; the difference is context-specific and varies based on the reaction.
Lewis acids and bases result in the formation of an adduct rather than a simple displacement reaction, as with classical acids and bases. An example is HCl vs H+: HCl is a classical acid, but not a Lewis acid; H+ is a Lewis acid when it forms an adduct with a Lewis base.
covalent bonda chemical bond in which two atoms are connected to each other by sharing two or more electrons
nucleophileliterally "lover of nuclei," Lewis bases are often referred to as this because they seek to donate their electron pairs to electron-poor species, such as H+
A Lewis acid is defined as an electron-pair acceptor, whereas a Lewis base is an electron-pair donor. Under this definition, we need not define an acid as a compound that is capable of donating a proton, because under the Lewis definition, H+ itself is the Lewis acid; this is because, with no electrons, H+ can accept an electron pair.
A Lewis base, therefore, is any species that donates a pair of electrons to a Lewis acid. The "neutralization" reaction is one in which a covalent bond forms between an electron-rich species (the Lewis base) and an electron-poor species (the Lewis acid). For this reason, Lewis bases are often referred to as nucleophiles (literally, "lovers of nuclei"), and Lewis acids are sometimes called electrophiles ("lovers of electrons"). This definition is useful because it not only covers all the acid-base chemistry with which we are already familiar, but it describes reactions that cannot be modeled by Arrhenius or Bronsted-Lowry acid-base chemistry. For now however, we will consider how the Lewis definition applies to classic acid-base neutralization.
Applying the Lewis Definition to Classical Acid-Base Chemistry
Consider the familiar reaction of NaOH and HCl:
NaOH(aq)+HCl(aq)\rightarrow NaCl(aq)+H_2O(l)
We have previously described this as an acid-base neutralization reaction in which water and a salt are formed. This is still completely correct, but the Lewis definition describes the chemistry from a slightly different perspective. When considering Lewis acids and bases, the only real reaction of interest is the net ionic reaction:
OH^-(aq)+H^+(aq)\rightarrow H_2O(l)
Under the Lewis definition, hydroxide acts as the Lewis base, donating its electron pair to H+. Thus, in this version of the neutralization reaction, what interests us is not the salt that forms, but the covalent bond that forms between OH- and H+ to form water. A significant hallmark for Lewis acid-base reactions is the formation of such a covalent bond between the two reacting species. The reaction's final product is known as an adduct, because it forms from the addition of the Lewis base to the Lewis acid.
Lewis acids and basesLewis acids (BF3, top, and H+, bottom) react with Lewis bases (F-, top, NH3, bottom) to form products known as adducts. Note that the first reaction cannot be described by Arrhenius or Bronsted-Lowry acid-base chemistry.
Beyond Classical Acid-Base Chemistry
By treating acid-base reactions in terms of electron pairs instead of specific substances, the Lewis definition can apply to reactions that do not fall under other definitions of acid-base reactions. For example, a silver cation behaves as a Lewis acid with respect to ammonia, which behaves as a Lewis base, in the following reaction:
Ag^+(aq) + 2\;NH_3 \rightarrow [Ag(NH_3)_2]^+
This reaction results in the formation of diamminesilver(I), a complex ion; it is perfectly described by Lewis acid-base chemistry, but is unclassifiable according to more traditional Arrhenius and Bronsted-Lowry definitions.
Application to Organic Chemistry
In organic chemistry, it is useful to understand that nucleophiles are Lewis bases and electrophiles are Lewis acids. Nearly all reactions in organic chemistry can be considered Lewis acid-base processes.
What are acids and bases?This lesson continues to describe acids and bases according to their definition. We first look at the Bronsted-Lowry theory, and then describe Lewis acids and bases according to the Lewis Theory.
"covalent bond."
http://en.wiktionary.org/wiki/covalent_bond Wiktionary
"adduct."
http://en.wiktionary.org/wiki/adduct Wiktionary
"Lewis acids and bases."
http://en.wikipedia.org/wiki/Lewis_acids_and_bases Wikipedia
"Acid–base reaction."
http://en.wikipedia.org/wiki/Acid%E2%80%93base_reaction%23Lewis_definition Wikipedia
"Lewis Acids and Bases - The WikiPremed MCAT Course."
http://www.wikipremed.com/mcat_course.php?code=0212000103000000 WikiPremed
"LewisAcid."
http://commons.wikimedia.org/wiki/File:LewisAcid.png Wikimedia
Lect-20160401.ppt
CHEM 030.102 #2 • Johns Hopkins University
Lewis Acid and Base Worksheet.docx.pdf
CHEMISTRY 12 • West Vancouver Secondary
#5 lewis acid and base
CHEM 006 • Austin Community College District
WHAT IS A LEWIS ACID AND BASE 2.0.pptx
CHEMESTRY NELI04044 • University of Guanajuato
Introduction to Acid and Base and pH.pptx
3.02__Brnsted_and_Lewis_Acids_and_Bases.pdf
CHEMISTRY 123 • University of the Fraser Valley
Week 2 Lewis Acid Base.pdf
CHEMISTRY 1020 • University of New England
15.2 - Lewis Acids and Bases.pdf
CHEM 1128Q • University Of Connecticut
SLG Chem2 LG 4.3 Definition of Acid and Base (Lewis Concept).pdf
CHEMISTRY Chemistry • Mackinaw Academy
Lewis_Acids_and_Bases.pdf
CHEM 3401 • Valdosta State University
CH2 HW2- Lewis Acid-Base.docx
CHEM 241 • Western Carolina University
p Lewis Acids and Bases_st4.pdf
CHEMISTRY 098461 • University of Malaya
Topic IV Acid and Base(I)--Introduction to Acids and Alkalis.pdf
CHEM 02 • S.K.H. Lui Ming Choi Secondary School
5+Chapter+5+Lewis+acids+and+bases.pdf
BONUS QUESTION 5 Lewis acids and bases Solution.pdf
C. Lewis Acids and Bases.docx
5 Chapter 5 Lewis acids and bases.pdf
Acid Base HW 20-3 Lewis Acids and Bases.pdf
18. Lewis Acids and Bases!.pdf
Lewis Acid and Base prep for 2444.pdf
CHEM 2444 • University Of Connecticut
18.1 LEWIS ACIDS AND BASES.ppt
CHEM CHEMISTRY • Decatur High School, Decatur
Arrhenius, Bronsted-Lowry, and Lewis Acids and Bases.pdf
CHEMISTRY Chemistry • Centennial High, Corona
Unit 9 - Module on Lewis Acids and Bases.pdf
CHEM 112A • Queens University
Acid and Base Homework Answers .pdf
CHEM 10004 • University of Melbourne
Lewis Acid-Base Lab COMPLETE.docx
CHEM N330 • Indiana University, Bloomington
|
Section 3-5 - Curvature - Maple Help
Home : Support : Online Help : Study Guides : Calculus : Chapter 3 - Applications of Differentiation : Section 3-5 - Curvature
Section 3.5: Curvature of a Plane Curve
y=y\left(x\right)
\mathrm{κ}=\frac{|y″|}{{\left(1+y{\prime }^{2}\right)}^{3/2}}
x=x\left(t\right),y=y\left(t\right)
\mathrm{κ}=\frac{\left|\stackrel{.}{x} \stackrel{..}{y}-\stackrel{.}{y} \stackrel{..}{x}\right|}{{\left({\stackrel{.}{x}}^{2}+{\stackrel{.}{y}}^{2}\right)}^{3/2}}
r=r\left(\mathrm{θ}\right)
\mathrm{κ}=\frac{|{r}^{2}+2 r{\prime }^{2}-r r″|}{{\left({r}^{2}+r{\prime }^{2}\right)}^{3/2}}
y=y\left(x\right)
\mathrm{κ}
x
t
r=r\left(\mathrm{θ}\right)
\mathrm{θ}
\mathrm{κ}
\mathrm{κ}=\frac{d\mathrm{θ}}{\mathrm{ds}}
\mathrm{θ}
s=s\left(x\right)
y\prime =\mathrm{tan}\left(\mathrm{θ}\right)
\mathrm{θ}=\mathrm{arctan}\left(y\prime \right)
s
\mathrm{ds}=\sqrt{{\mathrm{dx}}^{2}+{\mathrm{dy}}^{2}}=\mathrm{dx}\sqrt{1+{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^{2}}=\mathrm{dx} \sqrt{1+y{\prime }^{2}}
\frac{\mathrm{ds}}{\mathrm{dx}}=\sqrt{1+y{\prime }^{2}}
\mathrm{κ}
\mathrm{θ}
s
\frac{d\mathrm{θ}}{\mathrm{ds}}
=\frac{d}{\mathrm{ds}} \mathrm{arctan}\left(y\prime \right)
=\frac{d}{\mathrm{dx}}\left(\mathrm{arctan}\left(y\prime \right)\right) \frac{\mathrm{dx}}{\mathrm{ds}}
=\frac{y″}{1+y{\prime }^{2}} \frac{1}{\mathrm{ds}/\mathrm{dx}}
=\frac{y″}{1+y{\prime }^{2}} \frac{1}{\sqrt{1+y{\prime }^{2}}}
=\frac{y″}{{\left(1+y{\prime }^{2}\right)}^{3/2}}
Second-Order Contact
The graphs of two functions
f
g
make second-order contact at
x=a
if the values of
g
, and their first two derivatives, agree at
x=a
. Table 3.5.2 lists these three conditions as equations, and provides amusing interpretations for this degree of contact between two curves.
Analytic Condition
f\left(a\right)=g\left(a\right)
Curves touch
f\prime \left(a\right)=g\prime \left(a\right)
Curves kiss
f″\left(a\right)=g″\left(a\right)
Curves hug
Table 3.5.2 Conditions for second-order contact
The center of curvature for a plane curve that is the graph of
y=f\left(x\right)
is the center of the circle of curvature, the circle that makes second-order contact with the plane curve. The radius of the circle of curvature is the radius of curvature. Because the curvature of a circle of radius
r
\mathrm{κ}=1/r
, the radius of curvature is
R=1/\mathrm{κ}
Table 3.5.3 lists formulas for
\left(h,k\right)
, the coordinates of the center of curvature, and for the radius of curvature.
h=a-\frac{\stackrel{.}{y}\left(a\right)\left(1+{\stackrel{.}{y}}^{2}\left(a\right)\right)}{\stackrel{..}{y}\left(a\right)}
k=y\left(a\right)+\frac{1+{\stackrel{.}{y}}^{2}\left(a\right)}{\stackrel{..}{y}\left(a\right)}
R=\frac{{\left(1+{\stackrel{.}{y}}^{2}\left(a\right)\right)}^{3/2}}{|\stackrel{..}{y}\left(a\right)|}
Table 3.5.3 Center and radius of curvature
The overdots represent differentiation with respect to the independent variable; because some of these derivatives are squared, this notation is used in place of the prime.
The trajectory traced by the center of curvature as the circle of curvature traverses the curve
C
is called the evolute of
C
. The curve
C
is called the involute.
y=m x+b
{\left(x-h\right)}^{2}+{\left(y-k\right)}^{2}={r}^{2}
\mathrm{κ}=1/r
Obtain and graph the curvature
\mathrm{κ}\left(x\right)
y={x}^{2}
x=1
, obtain the equation of the circle of curvature for
y={x}^{2}
x=1
, the first and second derivatives for the curve and the circle of curvature agree.
Obtain the evolute for
C
y\left(x\right)={x}^{2}
, and show that it is the locus of the center of curvature.
y\left(x\right)={x}^{3/2},x≥0
|
Abstract: Airborne particulate matter (PM) filter sample processing is susceptible to error and can present issues associated with organizing samples, tracking data, and maintaining weighing conditions. While filter weighing facilities should implement robust quality assurance and control checks to ensure that data collection is accurate and filter storage is secure, mistakes and accidents can still occur that compromise valuable data. This paper presents a novel approach to PM filter sample processing that allows for data validation or data recovery while ensuring data integrity. The technique approximates the original, unused pre-sampling weight of polytetrafluoroethylene (PTFE) filters after PM collection to determine PM mass-deposition (MD). The method describes the extraction of PM loaded on PTFE filters via sonication in relatively non-toxic solvents, methanol and distilled water. The extraction method is compared to the standard gravimetric PM MD determination method for a set of 265 PTFE filters with mean post-sampling filter mass of 116 ± 3.6 mg, mean estimated PM MD using the standard method of 367 ± 589 μg, and mean estimated PM MD using the extraction method of 371 ± 589 μg. A Deming regression comparison of the two methods yields a slope of 0.9983 and a Pearson’s r of 0.999. A Bland-Altman assessment of the percent and absolute differences between the two methods shows the limits of agreement between 32.5% and 25.5% and -61.9 and 50.1 μg, respectively. The 99% confidence interval of the mean difference in mass deposition between the two methods is -5.8 ± 4.5 μg. These data demonstrate that estimating pre-sampling PTFE filter mass by extracting PM from sampled filters is a viable technique for gravimetric filter analysis. This method is of use in recovering pre-sampling filter weights that have been lost, incorrectly measured, or otherwise compromised.
Keywords: PTFE Filters, Gravimetric Analysis, Particulate Matter, Particle Extraction, Gravimetry
{Μ}_{o}={X}_{post}-{X}_{pre}–{F}_{o}
{Μ}_{ex}={X}_{post}-{X}_{ex}-{F}_{ex}
{F}_{o}=\frac{{\sum }_{n=1}^{N}{}_{\left({f}_{b-post}\right)n}{-}_{\left({f}_{b-pre}\right)n}}{N}
{F}_{ex}=\frac{{\sum }_{n=1}^{N}{}_{\left({f}_{b-post}\right)n}{-}_{\left({f}_{b-ex}\right)n}}{N}
{y}_{LOD}={B}_{ex}+3\ast {\sigma }_{Bex}
Cite this paper: Garland, C. , Delapena, S. and Pennise, D. (2018) An Alternative Technique for Determining Gravimetric Particle Mass Deposition on Filter Substrate: The Particle Extraction Method. Open Journal of Air Pollution, 7, 309-321. doi: 10.4236/ojap.2018.74016.
|
MapleALGEB_Printf - Maple Help
Home : Support : Online Help : Connectivity : Calling External Routines : ExternalCalling : C Application Programming Interface : MapleALGEB_Printf
MaplePrintf(kv, format, hw_arg1, hw_arg2, ..., hw_argN)
MapleALGEB_Printf(kv, format, mpl_arg1, mpl_arg2, ..., mpl_argN)
MapleALGEB_SPrintf(kv, format, mpl_arg1, mpl_arg2, ..., mpl_argN)
hw_arg1, ..., hw_argN
mpl_arg1, ..., mpl_argN
MaplePrintf is the same as the printf function in the C standard library, except that it directs output to the Maple user interface instead of stdout. The extra arguments must all be hardware types (for example, int, float, and char*).
MapleALGEB_Printf is the same as the Maple printf function. The extra arguments must all be Maple objects (of type ALGEB). Of note in the format specification, %a, formats an object of any Maple type, whereas, for example, %d formats only Maple integers.
MapleALGEB_SPrintf is the same as the Maple sprintf function. It returns a Maple String object containing the formatted output.
When an argument to MapleALGEB_Printf or MapleALGEB_SPrintf is an expression sequence, it is formatted as if it were a list.
ALGEB M_DECL MyArgs( MKernelVector kv, ALGEB *args )
M_INT i, n;
MaplePrintf(kv,"External routine called with %ld argumentsn",
(long)(n=MapleNumArgs(kv,(ALGEB)args)));
MapleALGEB_Printf(kv,"arg[%d] = %an",ToMapleInteger(kv,i),args[i]);
return( ToMapleString(kv,"") );
return( MapleALGEB_SPrintf(kv,"%a",args[1]) );
\mathrm{with}\left(\mathrm{ExternalCalling}\right):
\mathrm{dll}≔\mathrm{ExternalLibraryName}\left("HelpExamples"\right):
\mathrm{show}≔\mathrm{DefineExternal}\left("MyArgs",\mathrm{dll}\right):
\mathrm{show}\left(1,2\right)
External routine called with 2 arguments
\textcolor[rgb]{0,0,1}{"1"}
\mathrm{show}\left({x}^{2}+1,[1,2],\mathrm{sqrt}\left(2\right)\right)
arg[1] = x^2+1
arg[2] = [1, 2]
arg[3] = 2^(1/2)
\textcolor[rgb]{0,0,1}{"x^2+1"}
|
What is t-statistic and Student's t-test?
What is the t-statistic formula?
How to use this t-statistic calculator?
A t-statistic example
Use the t-statistic calculator (t-value calculator or t test statistic calculator) to compute the t-value of a given dataset using its sample mean, population mean, standard deviation and sample size.
Read further where we answer to the following questions:
How do I calculate t-statistic?
What is the difference between T-score vs. Z-score?
In statistics, the t-statistic, or t-value, is a measure that describes the relationship between a sample and its population. The t-statistic is central to the Student's t-test, which is a test for evaluating hypotheses about the population mean.
More precisely, the t-statistic is used to determine whether to support or reject the null hypothesis. It is used in conjunction with the p-value, or critical value, which indicates the probability that your results could have happened by chance. It is comparable to the z-statistic, with the difference being that the t-statistic is applied for small sample sizes or unknown population standard deviations.
You need to use the following t-statistic formula to calculate the t-value:
t = \frac{\bar x - \mu}{s/\sqrt n}
\bar x
- Sample mean;
\mu
- Population mean;
n
- Sample size; and
s
- Standard deviation of the sample.
To compute the t-statistic, you need to provide the following four variables:
Sample mean,
\bar x
Population mean,
\mu
Sample size,
s
Sample standard deviation,
s
Alternatively, you can use the tool in reverse; for example, you can recover the sample mean from the t-statistic, provided you input all other values.
Let's say you are a basketball player and your game score is 15 (x̄) on average over 36 (n) games, with a standard deviation of 6 (s). You know that an average basketball player scores 10 (μ). Should your performance be considered above average? Or are your scores due to luck? Finding the t-statistic and the probability value will give you some insight. More specifically, finding the t-statistic with the p-value will let you know if there is a significant difference between your mean and the population mean of everyone else.
Applying the previously stated t-statistic formula, you can obtain the following equation.
t = \dfrac{15 - 10}{6 / \sqrt{36}} = 5
Now, we know that the t-statistic equals 5, but what does it mean? To gain more knowledge, you should compare this value with a particular threshold (or significance level), let's say 5 percent (α = 5%) of a Student-t distribution. Since the sample size is relatively large (n > 30) we can use the critical value of the standard normal distribution. The critical value of a 5% threshold in a standard normal distribution is 1.645. Since our t-statistic is above the critical value, we can say that you play better than the average.
🙋 In fact, we have just performed a Student's t-test! Visit our dedicated t-test calculator to learn more.
Both t-score and Z-score aim to make comparisons and decide on the dissimilarity between the sample and the population mean. The main difference between T-score vs. Z-score comes from the knowledge about the population standard deviation. For Z-score, we assume it is given, while for T-score you need to estimate it. In addition, T-score can be applied when you have a small sample size (less than 30 elements).
To calculate t-statistic:
Determine the sample mean (x̄, x bar), which is the arithmetic mean of your data set.
Find the population mean (μ, mu).
Compute the sample standard deviation (s) by taking the square root of the variance. To find the variance, if it is not given, take each value in the sample, subtract it from the sample mean, square the difference and sum them up.
Calculate the t-statistic as (x̄ - μ) / (s / √n), where n denotes the sample size.
What is the origin of Student's t-distribution?
The student t-test was devised by Gosset, who developed the connected statistical theory in 1908. At the time, Gosset worked at the Guinness Brewery in Dublin, which had an internal policy of forbidding employees from publishing to preclude potential loss of trade secrets. Gosset, however, found a loophole: he was writing under the pseudonym of ‘Student’. As a consequence, the statistical student t distribution became known as student t rather than Gosset's t. So, next time you enjoying a pint of Guinness with your friend, you have a compelling story to share.
The Poisson distribution calculator is a tool for determining the probability of a certain number of independent events happening in a given time.
|
RTableDataBlock - Maple Help
Home : Support : Online Help : Connectivity : Calling External Routines : ExternalCalling : C Application Programming Interface : RTableDataBlock
access the rtable data block in external code
RTableDataBlock(kv, rt)
RTableDataBlock returns a pointer to the data contained in the given rtable, rt. The returned pointer must be cast into the correct hardware datatype to access elements. The data type can be obtained from the data_type field of the RTableSettings structure returned by RTableGetSettings.
Before modifying data in the data block, ensure the rtable rt is not read-only. The read-only flag value can be obtained from the read_only field of the RTableSettings structure returned by RTableGetSettings.
Do not directly modify the data block of an rtable with an indexing function. The data stored in position [1,1] of an rtable with a special indexing function may not correspond to the [1,1] element accessed from the rtable.
An error is raised if an attempt is made to get the data block from a Maple-sparse rtable.
ALGEB M_DECL MySumElems( MKernelVector kv, ALGEB *args )
FLOAT64 val, *data;
MapleRaiseError(kv,"float[8] rtable expected for parameter 1");
n = RTableNumElements(kv,rt);
for( val=0,i=0; i<n; ++i ) {
val += data[i];
return( ToMapleFloat(kv,val) );
\mathrm{with}\left(\mathrm{ExternalCalling}\right):
\mathrm{dll}≔\mathrm{ExternalLibraryName}\left("HelpExamples"\right):
\mathrm{sum_elems}≔\mathrm{DefineExternal}\left("MySumElems",\mathrm{dll}\right):
M≔\mathrm{Matrix}\left(3,\left(i,j\right)↦3\cdot i-3+j,\mathrm{datatype}=\mathrm{float}[8]\right)
\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{2.}& \textcolor[rgb]{0,0,1}{3.}\\ \textcolor[rgb]{0,0,1}{4.}& \textcolor[rgb]{0,0,1}{5.}& \textcolor[rgb]{0,0,1}{6.}\\ \textcolor[rgb]{0,0,1}{7.}& \textcolor[rgb]{0,0,1}{8.}& \textcolor[rgb]{0,0,1}{9.}\end{array}]
\mathrm{sum_elems}\left(M\right)
\textcolor[rgb]{0,0,1}{45.}
|
Energy Analysis of a Residential Combined Heat and Power System Based on a Proton Exchange Membrane Fuel Cell 1 | J. Electrochem. En. Conv. Stor | ASME Digital Collection
Energy Analysis of a Residential Combined Heat and Power System Based on a Proton Exchange Membrane Fuel Cell 1
M. Minutillo,
M. Minutillo
, Via Di Biasio 43, 03043 Cassino, FR, Italy
e-mail: minutillo@unicas.it
e-mail: perna@unicas.it
Minutillo, M., and Perna, A. (November 26, 2008). "Energy Analysis of a Residential Combined Heat and Power System Based on a Proton Exchange Membrane Fuel Cell ." ASME. J. Fuel Cell Sci. Technol. February 2009; 6(1): 014502. https://doi.org/10.1115/1.2971197
In this work the preliminary results of the research activity regarding the development of a microcogeneration unit prototype based on a proton exchange membrane fuel cell for residential application have been presented. The combined heat and power (CHP) system, which has been designed to optimize the integration of commercial and precommercial components, is equipped with two fuel cell stacks, a natural gas steam reforming unit, a heat recovery unit, electrical devices such as batteries, dc/ac converters, and auxiliary components such as compressors and pumps. In order to evaluate the electrical and thermal energy production and to estimate the system efficiency, an energy analysis has been carried out by using a numerical model. The simulation results pointed out that the microcogeneration system is able to provide
2.2 kWel
2.5 kWth
with electrical and CHP efficiencies (refer to the low heating value) of 40% and 88%, respectively. Furthermore, the primary energy savings, achievable by using the cogeneration system in comparison with a separate generation of electricity and heat from a centralized power plant and conventional boilers, have been evaluated.
cogeneration, energy conservation, fuel cell power plants, heat recovery, natural gas technology, proton exchange membrane fuel cells
Cogeneration systems, Combined heat and power, Proton exchange membrane fuel cells, Thermal energy, Heat, Stress, Natural gas, Heat recovery, Fuel cells
Directive 2004/8/EC on the promotion of cogeneration based on an useful heat demand in the internal energy market and amending Directive 92/42/EEC, Official Journal of the European Union, Feb. 21, 2004
Modelling and Testing a PEM Fuel Cell Fueled by a Steam Reforming System
Proceedings of the FUELCELL2006, Fourth International Conference on Fuel Cell Science, Engineering and Technology
, 2000, Aspen Plus 1.2, User Guide.
Reaction and Surface Characterization Studies of Ru/Al2O3 Catalysts for CO Preferential Oxidation in Reformed Gas
, “Behaviour Modeling of a PEMFC Operating on Diluted Hydrogen Feed,” Int. J. Energy Res., to be published.
Performance Modelling of Ballard Mark IV Solid Polymer Electrolyte Fuel Cell
Jalalzadeh-Azar
A Comparison of Electrical and Thermal–Load Following CHP Systems
|
Restriction in flow area in moist air network - MATLAB - MathWorks Australia
Restriction in flow area in moist air network
The Local Restriction (MA) block models the pressure drop due to a localized reduction in flow area, such as a valve or an orifice, in a moist air network. Choking occurs when the restriction reaches the sonic condition.
The restriction consists of a contraction followed by a sudden expansion in flow area. The moist air accelerates during the contraction, causing the pressure to drop. The moist air separates from the wall during the sudden expansion, causing the pressure to recover only partially due to the loss of momentum.
Moist air flow through this block can choke. If a Mass Flow Rate Source (MA) block or a Controlled Mass Flow Rate Source (MA) block connected to the Local Restriction (MA) block specifies a greater mass flow rate than the possible choked mass flow rate, the simulation generates an error. For more information, see Choked Flow.
The block equations use these symbols.
\stackrel{˙}{m}
S Cross-sectional area
Subscripts a, w, and g indicate the properties of dry air, water vapor, and trace gas, respectively. Subscripts lam and tur indicate the laminar and turbulent regime, respectively. Subscripts A and B indicate the appropriate port. Subscript R indicates the restriction.
\begin{array}{l}{\stackrel{˙}{m}}_{A}+{\stackrel{˙}{m}}_{B}=0\\ {\stackrel{˙}{m}}_{wA}+{\stackrel{˙}{m}}_{wB}=0\\ {\stackrel{˙}{m}}_{gA}+{\stackrel{˙}{m}}_{gB}=0\end{array}
{\Phi }_{A}+{\Phi }_{B}=0
When the flow is not choked, the mixture mass flow rate (positive from port A to port B) in the turbulent regime is
\begin{array}{l}{\stackrel{˙}{m}}_{tur}={C}_{d}{S}_{R}\left({p}_{A}-{p}_{B}\right)\sqrt{\frac{2{\rho }_{R}}{|{p}_{A}-{p}_{B}|{K}_{tur}}}\\ {K}_{tur}=\left(1+\frac{{S}_{R}}{S}\right)\left(1-\frac{{\rho }_{R}}{{\rho }_{in}}\frac{{S}_{R}}{S}\right)-2\frac{{S}_{R}}{S}\left(1-\frac{{\rho }_{R}}{{\rho }_{out}}\frac{{S}_{R}}{S}\right)\end{array}
Subscripts in and out indicate the inlet and outlet, respectively. If pA ≥ pB, the inlet is port A and the outlet is port B; otherwise, they are reversed. The cross-sectional area S is assumed to be equal at ports A and B. SR is the area at the restriction.
The mixture mass flow rate equation is derived by combining the equations from two control volume analyses:
Momentum balance for flow area contraction from the inlet to the restriction
Momentum balance for sudden flow area expansion from the restriction to the outlet
In the analysis for the flow area contraction, pressure pin acts on the area at the inlet, S, and pressure pR acts on the area at the restriction, SR. The pressure acting on the area outside the restriction, S − SR, is assumed to be (pinS + pRSR)/(S + SR).
In the analysis for the flow area expansion, the pressure acting on both the area at the restriction, SR, and the area outside the restriction, S − SR, is assumed to be pR, because of flow separation from the restriction. The pressure acting on the area at the outlet , S, is equal to pout.
The mixture mass flow rate (positive from port A to port B) in the laminar regime is linearized with respect to the pressure difference:
{\stackrel{˙}{m}}_{lam}={C}_{d}{S}_{R}\left({p}_{A}-{p}_{B}\right)\sqrt{\frac{2{\rho }_{R}}{\Delta {p}_{threshold}{\left(1-\frac{{S}_{R}}{S}\right)}^{2}}}
where the threshold for transition between the laminar and turbulent regime is defined based on the laminar flow pressure ratio, Blam, as
\Delta {p}_{threshold}=\left(\frac{{p}_{A}+{p}_{B}}{2}\right)\left(1-{B}_{lam}\right)
|{p}_{A}-{p}_{B}|\ge \Delta {p}_{threshold}
, the flow is assumed to be turbulent and therefore
{\stackrel{˙}{m}}_{unchoked}={\stackrel{˙}{m}}_{tur}
|{p}_{A}-{p}_{B}|<\Delta {p}_{threshold}
{\stackrel{˙}{m}}_{unchoked}
smoothly transitions to
{\stackrel{˙}{m}}_{lam}
When the flow is choked, the velocity at the restriction is equal to the speed of sound and cannot increase any further. Assuming the flow is choked, the mixture mass flow rate is
{\stackrel{˙}{m}}_{choked}={C}_{d}{S}_{R}{p}_{R}\sqrt{\frac{{\gamma }_{R}}{R{T}_{R}}}
{\gamma }_{R}={c}_{pR}/\left({c}_{pR}-R\right)
. Therefore, the actual mixture mass flow rate is equal to
{\stackrel{˙}{m}}_{unchoked}
, but is limited in magnitude by
{\stackrel{˙}{m}}_{choked}
{\stackrel{˙}{m}}_{A}=\left\{\begin{array}{ll}-{\stackrel{˙}{m}}_{choked},\hfill & \text{if }{\stackrel{˙}{m}}_{unchoked}\le -{\stackrel{˙}{m}}_{choked}\hfill \\ {\stackrel{˙}{m}}_{unchoked},\hfill & \text{if -}{\stackrel{˙}{m}}_{choked}<{\stackrel{˙}{m}}_{unchoked}<{\stackrel{˙}{m}}_{choked}\hfill \\ {\stackrel{˙}{m}}_{choked},\hfill & \text{if }{\stackrel{˙}{m}}_{unchoked}\ge {\stackrel{˙}{m}}_{choked}\text{ }\hfill \end{array}
The expression for the pressure at the restriction is obtained by considering the momentum balance for flow area contraction from the inlet to the restriction only.
{p}_{R}={p}_{in}-\frac{1}{2{\rho }_{R}}{\left(\frac{{\stackrel{˙}{m}}_{A}}{{C}_{d}{S}_{R}}\right)}^{2}\left(1+\frac{{S}_{R}}{S}\right)\left(1-\frac{{\rho }_{R}}{{\rho }_{in}}\frac{{S}_{R}}{S}\right)
The local restriction is assumed adiabatic, so the mixture specific total enthalpies are equal. Therefore, the changes in mixture specific enthalpies are:
\begin{array}{l}{h}_{A}-{h}_{R}=\left(\frac{1}{{\rho }_{R}^{2}{S}_{R}^{2}}-\frac{1}{{\rho }_{A}^{2}{S}^{2}}\right)\frac{{\stackrel{˙}{m}}_{A}^{2}}{2{C}_{D}^{2}}\\ {h}_{B}-{h}_{R}=\left(\frac{1}{{\rho }_{R}^{2}{S}_{R}^{2}}-\frac{1}{{\rho }_{B}^{2}{S}^{2}}\right)\frac{{\stackrel{˙}{m}}_{B}^{2}}{2{C}_{D}^{2}}\end{array}
Input physical signal that controls the air flow restriction area. The signal saturates when its value is outside the minimum and maximum restriction area limits, specified by the block parameters.
Moist air conserving port associated with the inlet or outlet of the local restriction. This block has no intrinsic directionality.
Lower bound for the restriction cross-sectional area. You can use this parameter to represent the leakage area. The input signal AR saturates at this value to prevent the restriction area from decreasing any further.
Upper bound for the restriction cross-sectional area. The input signal AR saturates at this value to prevent the restriction area from increasing any further.
Discharge coefficient — Ratio of actual mass flow rate to theoretical mass flow rate through the restriction
Ratio of actual mass flow rate to the theoretical mass flow rate through the restriction. The discharge coefficient is an empirical parameter that accounts for nonideal effects.
Laminar flow pressure ratio — Pressure ratio at which air flow transitions between laminar and turbulent regimes
Pressure ratio at which the moist air flow transitions between laminar and turbulent regimes. The pressure loss is linear with respect to mass flow rate in the laminar regime and quadratic with respect to mass flow rate in the turbulent regime.
|
How to use the Kelvin to Celsius converter
Formula for Kelvin to Celsius conversion
Other helpful temperature converters
Welcome to our Kelvin to Celsius converter, where you can easily convert one temperature into the other. It doesn't matter whether you want to know how to convert Kelvin to Celsius or how to convert Celsius to Kelvin. We've covered both ways of conversion in our tool and below article.
We've also discussed the formula of Kelvin to Celsius conversion to help you learn more about the kelvin unit in this article. Wondering how much is 1 kelvin in Celsius or 0 Celsius in kelvin? Well, you're about to find out!
To use the Kelvin to Celsius converter:
Enter your Kelvin temperature in the first field, e.g., 0 K.
The Kelvin to Celsius calculator will convert and display it in Celsius in the second field, i.e., -273.15 °C.
🔎 Kelvin is an absolute temperature scale having 0 K as the coldest temperature possible.
You can also enter your Celsius temperature first to obtain your temperature in Kelvin.
Furthermore, there is a unit switcher in the Celsius to Kelvin calculator in both fields that you can also switch to Fahrenheit if you ever needed it.
Here's the formula to convert Kelvin to Celsius:
\degree C = K - 273.15
\degree C
- the unit temperature in degrees Celsius; and
K
- the absolute unit temperature in kelvin.
The convert Celsius to Kelvin formula is the opposite:
K = \degree C + 273.15
Here are a few examples that use the Celsius to Kelvin equation:
Let's convert 1 Kelvin to Celsius.
°C = 1 - 273.15
Thus, if we convert 1 K into Celsius, we get - 272.15 °C.
Now, let's convert 100 degrees Celsius to Kelvin.
Rearranging the formula, we get:
K = °C + 273.15
K = 100 + 273.15
Thus, if we convert 100 °C into Kelvin, we get 373.15 K.
Of course, using our tool is a more convenient way to use the Celsius to Kelvin equation very quickly!
Here are some other temperature converters that you may find helpful:
Why is the Kelvin scale not measured in degrees?
Degrees is a relative term, whereas Kelvin is an absolute temperature scale with defined endpoints. It has 0 as its smallest value that measures the coldest temperature possible, i.e., where every element's atomic and molecular motion comes to a stop.
How do I convert 100 degrees Celsius to Kelvin?
The temperature of 100 °C is 375.15 K. You can convert Celsius to Kelvin by adding 273.15 to Celsius:
373.15 K = 100 °C + 273.15
What is 0 Celsius in Kelvin?
The temperature of 0 Celsius is 273.15 kelvins.
We can find this by adding 273.15 to the Celsius temperature, e.g., 100 degrees Celsius is 373.15 kelvin, i.e.:
K - the absolute temperature in kelvin; and
°C - the temperature in degrees of Celsius.
|
The IYRU Fifteen Metre class yachts are constructed to the First International rule of 1907. A total of twenty 15mR yachts were built between 1907 and 1917, the four that have survived are still actively raced.
The 15mR Ma'oona in 1908. +
1907 (design rule)
The IYRU International Rule was set up in 1907 to replace the YRA 1901 revised Linear Rating Rule. The IYRU 15mR boats would replace the YRA 52-raters and open competition to foreign nations, replacing local or national systems with a unified rating system across Europe. The rule changed several times, but the 15mR boats only raced in the first rule of 1907. The twenty boats that were built, were raced in Spain, France, Britain and Germany. The rule was proposed for competition in the 1908 Olympics but there were no entries.
1907 RuleEdit
The four restored 15mRs in their first races together
The 15-Metre class is a construction class, meaning that the boats are not identical but are all designed to meet specific measurements in a formula, in this case the In their heyday, Metre classes were the most important group of international yacht racing classes, and they are still actively raced around the world. "Metre" does not refer to the length of the boat, but to her rating; the length overall of 15mR boats measuring almost 30 metres (98 ft).
The 15mR formula used in the First International Rule from 1907 to 1920:
{\displaystyle 15~{\mbox{m}}={\frac {{\textrm {L}}+B+{\frac {1}{2}}G+3d+{\frac {1}{3}}{\sqrt {S}}-F}{2}}}
L = load waterline length in metres
B = beam in metres
G = chain girth in metres
d = difference between girth and chain in metres
S = sail area in square metres
F = freeboard in metres
1907 Ma'oona Alfred Mylne
Robert McAlister & Son J. Talbot Clifton later sold to Almeric Paget
1907 Shimna William Fife III
Alexander Robertson & Sons William Yates later rechristened Slec, and Yildiz in 1938. damaged and broken up in Turkey 1949.
1908 Mariska D1 William Fife III
Fife & Son A. K. Stothert restored by the Charpentiers Réunis de Méditerranée in 2009
1909 Ostara D2 Alfred Mylne
Robert McAlister & Son William P. Burton
1909 Anémone II C. Maurice Chevreux
Chantier Vincent, Cannes Philippe de Vilmorin
1909 Encarnita Joseph Guédon
Karpard de Pasajes Marquis of Cuba
1909 Hispania D5 William Fife III
Karpard de Pasajes King Alfonso XIII restored by the Astilleros de Mallorca in 2012[1]
1909 Tuiga D3 D9 1 William Fife III
Fife & Son 17th Duke of Medinaceli owned in the 1920s by Warwick Brookes.[2] rechristened Betty IV, Dorina, Kismet III. restored by Fairlie Restorations in 1993[3]
1909 Vanity D4 William Fife III
Fife & Son W. & Benn Payne
1910 Paula II D2 D8 Alfred Mylne
Robert McAlister & Son Ludwig Sanders
1910 Tritonia D3 Alfred Mylne
Alexander Robertson & Sons Graham C. Lomer later rechristened Jeano, Gerd II, Rinola, Fortuna II, Cisne Branco and Albatroz. served in the Gremio de Vela da Escola Navala in Brazil until 1986
1910 Sophie-Elisabeth D6 D4 William Fife III
Fife & Son L. Biermann 1913/1914 sold to G.Eyde, Norway, and rechristened Beduin,[4] later Magda X
1911 Senta Max Oertz
Max Oertz Duke of Saxe-Altenburg
1912 Istria D7 Charles E. Nicholson
Camper & Nicholsons Charles C. Allom World's first Marconi topmast. broken up in Norway 1924.
1912 The Lady Anne D10 William Fife III
Fife & Son George Coats restored by fairlie Restorations in 1999 with her 1914 rig configuration
1913 Pamela D1 Charles E. Nicholson
Camper & Nicholsons S. Glen L. Bradley
1913 Paula III D2 D8 Charles E. Nicholson
Camper & Nicholsons Ludwig Sanders
1913 Isabel Alexandra D5 Johan Anker
Anker & Jensen E. Luttrop
1913 Maudrey D3 William Fife III
Fife & Son W. Blatspiel Stamp
1917 Neptune Johan Anker
Mandrup Abel S. Klouman
^ "Hispania" (in Spanish). Fundación Isla Ebusitana. Archived from the original on 2016-05-02. Retrieved 2019-08-21.
^ "Royal Southern Yacht Club". The Times. 1922-08-07. p. 3, column F.
^ "Fairlie Restorations". Archived from the original on 2016-03-04. Retrieved 2015-11-09.
^ "Die Yacht", 1914,Vol.27, p.672
"15mR Class". Archived from the original on 2015-10-11. Retrieved 2015-11-09.
Yacht Club de Monaco, Grupo Panorama (2012-08-22). vintage photographs and current footage of the 15mR class. youtube. Archived from the original on 2021-12-15.
Dr. William Collier (April 1994). "Tuiga et les Quinze Mètres JI". Le Chasse Marée. Abri du Marin (78): 38.
Dr. Daniel Charles (August 2005). Tuiga 1909. Yachting Heritage. ISBN 978-0-9550777-0-8.
François Chevalier (2008). "Tuiga". Classic Yachts. Thomas Reed Publications. ISBN 978-1-4081-0518-4. Archived from the original on 2015-09-08. Retrieved 2015-07-15.
|
Thermodynamics - Wikiquote
branch of physics concerned with heat, work, temperature, and thermal or internal energy
Thermodynamics is a branch of physics that studies the movement of energy and how energy instills movement. More precisely, it studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. 19th century physicists defined three Laws of thermodynamics to sum up the basic principles of the subject; in the 20th century, an unofficial "zeroth law" was added.
1.1 Second Law of Thermodynamics/Entropy
1.3 Unsourced/Anonymous
V.I. Arnold, "Contact geometry: The geometrical method of Gibbs' thermodynamics," in Proceedings of the Gibbs Symposium, D. Caldi and G. Mostow, eds. (American Mathematical Society, 1990), p. 163.
In order to consider in the most general way the principle of the production of motion by heat, it must be considered independently of any mechanism or any particular agent. It is necessary to establish principles applicable not only to steam engines but to all imaginable heat-engines, whatever the working substance and whatever the method by which it is operated.
Nicolas Léonard Sadi Carnot, Reflections on the Motive Power of Heat and on Machines Fitted to Develop Power (1824)
̈Machines which do not receive their motion from heat... can be studied even to their smallest details by the mechanical theory. ...A similar theory is evidently needed for heat-engines. We shall have it only when the laws of Physics shall be extended enough, generalized enough, to make known beforehand all of the effects of heat acting in a determined manner on any body.
The production of heat alone is not sufficient to give birth to the impelling powerː it is necessary that there should also be cold; without it the heat would be useless. And in fact, if we should find about us only bodies as hot as our furnaces... What should we do with it if once produced? We should not presume that we might discharge it into the atmosphere... the atmosphere would not receive it. It does receive it under the actual condition of things only because.. it is at a lower temperature, otherwise it... would be already saturated.
Heat can evidently be a cause of motion only by virtue of the changes of volume or of form which it produces in bodies. These changes are not caused by uniform temperature but rather by alternations of heat and cold.
Albert Einstein (author), Paul Arthur, Schilpp (editor). Autobiographical Notes. A Centennial Edition. Open Court Publishing Company. 1979. p. 31 [As quoted by Don Howard, John Stachel. Einstein: The Formative Years, 1879-1909 (Einstein Studies, vol. 8). Birkhäuser Boston. 2000. p. 1]
Newton and his theories were a step ahead of the technologies that would define his age. Thermodynamics, the grand theoretical vision of the nineteenth century, operated in the other direction with practice leading theory. The sweeping concepts of energy, heat, work and entropy, which thermodynamics (and its later form, statistical mechanics) would embrace, began first on the shop floor. Originally the domain of engineers, thermodynamics emerged from their engagement with machines. Only later did this study of heat and its transformation rise to the heights of abstract physics and, finally, to a new cosmological vision.
James Clerk Maxwell, Theory of Heat "Preface" (1871)
James Clerk Maxwell, Theory of Heat Ch.3 "Calorimetry" (1871)
Isn’t thermodynamics considered a fine intellectual structure, bequeathed by past decades, whose every subtlety only experts in the art of handling Hamiltonians would be able to appreciate?
Arnold Sommerfeld, as quoted in Salvatore Califano's Pathways to Modern Chemical Physics (2012) and by many other science authors
William Thomson, Mathematical and Physical Papers, Vol.2 (1884) "On Mechanical Antecedents of Motion, Heat and Light" (originally published 1854, 1855)
Thermodynamics is more like a mode of reasoning than a body of physical law. ...we can think of thermodynamics as a certain pattern of arrows that occurs again and again in very different physical contexts, but, wherever this pattern of explanation occurs, the arrows can be traced back by the methods of statistical mechanics to deeper laws and ultimately to the principles of elementary particle physics. ...the fact that a scientific theory finds applications to a wide variety of different phenomena does not imply anything about the autonomy of this theory from deeper physical laws.
Steven Weinberg, Dreams of a Final Theory (1992) p. 41.
Second Law of Thermodynamics/EntropyEdit
The Second Law recognizes that there is a fundamental dissymmetry in Nature... All around us are aspects of the dissymmetry: hot objects become cool, but cool objects do not spontaneously become hot; a bouncing ball comes to rest, but a stationary ball does not spontaneously begin to bounce. Here is the feature of Nature that both Kelvin and Clausius disentangled from the conservation of energy: although the total quantity of energy must be conserved in any process (which is their revised version of what Carnot had taken to be the conservation of the quantity of caloric), the distribution of that energy changes in an irreversible manner. The Second Law is concerned with the natural direction of change of the distribution of energy, something that is quite independent of its total quantity.
P. W. Atkins, The Second Law (1984)
The choice (or accident) of initial conditions creates a sense of time directionality in a physical environment. The 'arrow' of entropy increase is a reflection of the improbability of those initial conditions which are entropy-decreasing in a closed physical system. ...
Everywhere... in the Universe, we discern that closed physical systems evolve in the same sense from ordered states towards a state of complete disorder called thermal equilibrium. This cannot be a consequence of known laws of change, since... these laws are time symmetric—they permit... time-reverse... The initial conditions play a decisive role in endowing the world with its sense of temporal direction. ...some prescription for initial conditions is crucial if we are to understand... A Theory of Everything needs to be complemented by some such independent prescription which appeals to simplicity, economy, or some other equally metaphysical notion to underpin its credibility. The only radically different alternative... a belief that the type of mathematical description of Nature... —that of causal equations with starting conditions—is just an artefact of our own preferred categories of thought and merely an approximation... At a deeper level, a sharp divide between those aspects of reality that we habitually call 'laws' and... 'initial conditions' may simply not exist.
John D. Barrow, Theories of Everything: The Quest for Ultimate Explanation (1991) pp. 38-39.
The second law of thermodynamics is, without a doubt, one of the most perfect laws in physics. Any reproducible violation of it, however small, would bring the discoverer great riches as well as a trip to Stockholm. The world’s energy problems would be solved at one stroke. It is not possible to find any other law (except, perhaps, for super selection rules such as charge conservation) for which a proposed violation would bring more skepticism than this one. Not even Maxwell’s laws of electricity or Newton’s law of gravitation are so sacrosanct, for each has measurable corrections coming from quantum effects or general relativity. The law has caught the attention of poets and philosophers and has been called the greatest scientific achievement of the nineteenth century. Engels disliked it, for it supported opposition to Dialectical Materialism, while Pope Pius XII regarded it as proving the existence of a higher being.
Ivan P. Bazarov, "Thermodynamics" (1964)
If one applies this to the universe in total, one reaches a remarkable conclusion. ...Namely, if, in the universe, heat always shows the endeavour to change its distribution in such a way that existing temperature differences are thereby smoothened, then the universe must continually get closer and closer to the state, where the forces cannot produce any new motions, and no further differences exist.
Rudolf Clausius (1864) as quoted by Helge S. Kragh, Entropic Creation: Religious Contexts of Thermodynamics and Cosmology (2013)
Rudolf Clausius (1868) as quoted by Jed Z. Buchwald, Robert Fox, The Oxford Handbook of the History of Physics (2013)
{\displaystyle E=hv}
{\displaystyle E}
{\displaystyle v}
{\displaystyle h}
{\displaystyle S=kA}
{\displaystyle S}
{\displaystyle A}
{\displaystyle k}
Sir Arthur Stanley Eddington, The Nature of the Physical World (1915), chapter 4
Seth Lloyd, writing in Nature 430, 971 (26 August 2004); doi:10.1038/430971a
Life is nature's solution to the problem of preserving information despite the second law of thermodynamics.
Howard L. Resnikoff, The Illusion of Reality (1989), ISBN 0387963987, p. 74
The reactions that break down large molecules into small ones do not require an input of energy, but the reactions that build up large molecules require and input of energy. This is consistent with the laws of thermodynamics, which say that large, orderly molecules tend to break down into small, disorderly molecules.
No one has yet succeeded in deriving the second law from any other law of nature. It stands on its own feet. It is the only law in our everyday world that gives a direction to time, which tells us that the universe is moving toward equilibrium and which gives us a criteria for that state, namely, the point of maximum entropy, of maximum probability. The second law involves no new forces. On the contrary, it says nothing about forces whatsoever.
Brian L Silver, The Ascent of Science (1998)
C. P. Snow, 1959 Rede Lecture entitled "The Two Cultures and the Scientific Revolution".
My own introduction to entropy was as an undergraduate mechanical engineering student. Neither I nor any of the other students knew anything about the molecular theory of heat, and I bet that the professor didn't either. The course... was so confusing that I...couldn't make any sense of it. Worst of all was the concept of entropy. We were told that if you heat something a small amount, the change in thermal energy, divided by the temperature, is the change of its entropy. Everyone copied it down but no one understood what it meant. It was as incomprehensible to me as "The change in the number of sausages divided by the onionization is called floogelweiss."
William Thomson, Mathematical and Physical Papers, Vol.1 p. 179 (1882) "On the Dynamical Theory of Heat with Numerical Results Deduced from Mr Joule's Equivalent of a Thermal Unit and M. Regnault's Observations on Steam" originally from Transactions of the Royal Society of Edinburgh, March, 1851 and Philosophical Magazine iv, 1852
William Thomson, Mathematical and Physical Papers, Vol.1 p. 512 (1882) "On a Universal Tendency in Nature to the Dissipation of Mechanical Energy" originally from the Proceedings of the Royal Society of Edinburgh for April 19, 1852, also Philosophical Magazine, Oct. 1852
In classical physics, most of the fundamental laws of nature were concerned either with the stability of certain configurations of bodies, e.g. the solar system, or else with the conservation of certain properties of matter, e.g. mass, energy, angular momentum or spin. The outstanding exception was the famous Second Law of Thermodynamics, discovered by Clausius in 1850. This law, as usually stated, refers to an abstract concept called entropy, which for any enclosed or thermally isolated system tends to increase continually with lapse of time. In practice, the most familiar example of this law occurs when two bodies are in contact: in general, heat tends to flow from the hotter body to the cooler. Thus, while the First Law of Thermodynamics, viz. the conservation of energy, is concerned only with time as mere duration, the Second Law involves the idea of trend. Milne developed his cosmology by taking this idea of trend to be fundamental, regarding the expansion of the universe as its supreme manifestation.
HumorousEdit
It’s the Second Law of Thermodynamics: Sooner or later everything turns to shit.
Woody Allen, Husbands and wives (1992).
Homer Simpson, after Lisa constructs a perpetual motion machine whose energy increases with time, in the The PTA Disbands episode of The Simpsons.
A common scientific joke expressing the four laws. Often attributed to C. P. Snow without a source, such as in Mahon, Tom (2011). Reconnecting.calm. The first to third laws were expressed like this in Astounding Science Fiction, November 1956[1] by Dr. Wayne Batteau of Harvard University.
You can't win; you can only break even.
You can't reach absolute zero.
Second: You can't break even, except on a very cold day.
Third: It doesn't get that cold.
Third: It doesn't get that cold, even in Wisconsin.
Zeroth: There is a game.
Second: You must lose.
Third: You can't quit.
Unsourced/AnonymousEdit
Murphy's Law about Thermodynamics: Things get worse under pressure.
"Old Chemists never die: they reach thermodynamic equilibrium"
S happens.
Bumper sticker on the second law of thermodynamics.
You can't unscramble eggs.
Look up thermodynamics in Wiktionary, the free dictionary
Thermodynamic Jokes - Joachim Verhagen's science jokes site.
↑ Archive of Astounding Science Fiction November 1956[1]
Retrieved from "https://en.wikiquote.org/w/index.php?title=Thermodynamics&oldid=3105158#Second_Law_of_Thermodynamics/Entropy"
|
What is Fahrenheit scale?
Convert Fahrenheit to Celsius, Kelvin and other temperature units
How to use the Fahrenheit converter
Welcome to our Fahrenheit converter. Here we will tell you what Fahrenheit is and what its origin is. We will also inform you how to convert Fahrenheit to Celsius or kelvins and a few other temperature units.
In 1724 a german physicist of polish origin, Daniel Gabriel Fahrenheit, proposed his temperature scale where he measured and designated 32 °F as the temperature at which pure water freezes. Having a 180 °F separation between two defining points, 212 °F became its boiling point.
Based on the thermodynamic temperature scale, the coldest attainable Fahrenheit temperature is -459.67 °F.
Knowing what the Fahrenheit scale is, let's look into how we can convert Fahrenheit to different temperature units.
We use the following Fahrenheit formulas to convert degrees Fahrenheit to different temperature units:
Fahrenheit °F to degree Celsius °C:
\degree C = (\degree F − 32) \times \frac{5}{9}
where, 212 °F is equal to 100 °C.
Fahrenheit °F to Kelvin K:
K = (\degree F − 32) \times \frac{5}{9} + 273.15
where, 212 °F is equal to 373.15 K.
Fahrenheit °F to degree Rankine °R:
\degree R = \degree F + 459.67
where, 212 °F is equal to 671.67 °R.
Fahrenheit °F to degree Delisle °De:
\degree De = (212 − \degree F) \times \frac{5}{6}
where, 212 °F is equal to 0 °De.
Fahrenheit °F to degree Newton °N:
\degree N = (\degree F − 32) \times \frac{11}{60}
where, 212 °F is equal to 33 °N.
Fahrenheit °F to degree Réaumur °Ré:
\degree Ré = (\degree F − 32) \times \frac{4}{9}
where, 212 °F is equal to 80 °Ré.
Fahrenheit °F to degree Rømer °Rø:
\degree Rø = (\degree F − 32) \times\frac{7}{24} + 7.5
where, 212 °F is equal to 60 °Rø.
Now you know how to convert Fahrenheit to Celsius and other temperature units. Next, we'll guide you on how easy it is to use our Fahrenheit converter.
Here's how you can use our Fahrenheit calculator:
Enter your Fahrenheit temperature in the first field, e.g., 32 °F.
The conversion results will auto-populate in the rest of the temperature fields, i.e.:
32 °F = 0 °C;
32 °F= 273.15 K;
32 °F = 491.67 °R;
32 °F = 150 °De;
32 °F = 0 °N;
32 °F = 0 °Ré; and
32 °F = 7.5 °Rø.
So, if you'll ever need to know how to calculate Fahrenheit from Celsius, or vice-versa, how to calculate Celsius from Fahrenheit, remember about our Fahrenheit calculator!
Here's a list of our other temperature converters that you may find helpful:
Kelvin to Fahrenheit converter;
Celsius converter; and
How do I convert 200 Fahrenheit to Celsius?
Subtract 32 from 200, to obtain 168;
Multiply 168 by 5 to get 840;
Divide 840 by 9 and you get the result, i.e., 93.33 °C.
We use the following formula to convert Fahrenheit into Celsius:
How do I convert 38 Celsius to Fahrenheit?
Follow the below steps to convert Celsius to Fahrenheit:
Use the Fahrenheit formula:
Place the value of 38 °C into the formula:
°F = (38 °C × 9/5) + 32 = 100.4
Thus, 38 °C is equal to 100.4 °F.
What is the freezing and boiling point of water in Fahrenheit?
The freezing point of water is 32 °F, whereas the boiling point of water is 212 °F. This temperature varies ever so slightly, depending on the purity of the water.
Other temperatures
The square meter converter allows you to translate square meters to any other area unit.
|
Neutrophil - Citizendium
2 Polymorphonuclear neutrophil
3 Chemotaxis and migration
5 Phagocytosis, degranulation and lysis
5.1 The respiratory burst
5.2 Degranulation
A neutrophil is a granular, nucleated leukocyte , and cytoplasm containing fine inconspicuous granules and stainable by neutral dyes.[1] They are quick to respond to infection challenge, often within hours. They are ephemeral cells, with a half-life on the order of 6 hours.[2] Neutrophils are more associated with acute inflammatory response than macrophages, which are more involved in chronic inflammation. [3] They normally make up 40% to 60% percent of all white cells; there may be an additional 0 to 3 percent of immature band neutrophils. [4]
While their role has classically been associated with phagocytosis, a means of cell-mediated immune response in which they are attracted to targets via chemokines, more and more information has been accumulated about their role in releasing cytokines, especially interleukin 12 (IL-12). IL-12 encourages the production of interferon gamma (γ-interferon). [5]
In the creation of white cells, the original progenitor is the pluripotent stem cell. Under the influence of interleukins IL-1, IL-3, and IL-6, they form committed stem cells (i.e., progenitors). The cytokine granulocyte-macrophage colony-stimulating factor (GM-CSF) causes differentiation into the neutrophils, with maturation to polymorphonuclear neutrophils via juvenile and band forms.
Immature forms of neutrophil are juvenile and band. Band neutrophils are immature cells, released by the bone marrow in response to demand. They have a single nucleus resembling a band, a sausage, or the letters C or U. [6]
The mature form of polymorphonuclear neutrophil (PMN) has a nucleus with three to five lobes connected by slender threads of chromatin. The average mature neutrophil has a diameter of about 13-15µm and a granulated cytoplasm.
Chemotaxis and migration
The neutrophil is able to chemotactically migrate to infected tissue, moving upstream against a concentration gradient of Interleukin-6, [7] secreted by activated endothelial cells, mast cells and macrophages. When nearing the site of infection, the neutrophil will attach repeatedly to the endothelial blood vessel lining viz. selectins, thus dissipating kinetic energy. When the neutrophil has been slowed down enough, it will migrate from the blood vessel lumen to the infected tissue via an integrin-mediated pathway.
Because of their relative abundance compared to other leukocytes, neutrophil granulocytes are often the first to arrive at sites of new infection. The neutrophil granulocyte is activated through recognition of target molecules, viz the many different surface receptors the neutrophil expresses, some of which bind molecules that are unique to pathogens[8] such as peptidoglycan, a component of bacterial cell walls. Apart from the innate ability to recognize potential pathogens, the neutrophil is also able to be activated by target-bound opsonins (antibodies), thus making the neutrophil a humoral effector cell.
Phagocytosis, degranulation and lysis
The neutrophil granulocyte is a phagocyte, capable of ingesting and lysing many pathogens. It does, however, not present antigens on its surface. Upon ingesting a pathogen, a phagosome is formed, into which oxidants and lytic enzymes are secreted.
The respiratory burst
In order to facilitate the lysis of ingested pathogens, the neutrophil synthesizes strong oxidants in what has been termed the respiratory burst, though it does not involve respiration. In the respiratory burst, the enzyme NADPH oxidase is activated, producing large amounts of superoxide, which spontaneously or catalyzed by superoxide dismutase dismutates to hydrogen peroxide, as per the following reaction:
{\displaystyle 2{\mbox{O}}_{2}^{-}+2{\mbox{H}}_{2}{\mbox{O}}\rightarrow {\mbox{O}}_{2}+{\mbox{H}}_{2}{\mbox{O}}_{2}+2{\mbox{OH}}^{-}}
Hydrogen peroxide and chloride are subsequently converted by the enzyme myeloperoxidase to hypochlorous acid:
{\displaystyle {\mbox{H}}_{2}{\mbox{O}}_{2}+2{\mbox{Cl}}^{-}\rightarrow 2{\mbox{HOCl}}}
Bromide is able to substitute for chloride in the reaction. Both hypochlorous and hypobromous acid are strong oxidants, and are directly able to kill the phagocytosed organisms.
Apart from phagocytic, oxidative lysis, the cytoplasm of the neutrophil contain many granulae, which contain soluble antimicrobial proteins, including lactoferrin, bactericidal/permeability increasing protein and defensins. These proteins generally confer their antimicrobial activity by increasing the permeability of bacterial, viral and fungal membranes, resulting in osmotic lysis. The exocytotic process in which these proteins are secreted to the surroundings is termed degranulation. This term is, however, most closely associated with the eosinophil granulocyte.
↑ National Library of Medicine, Medical Subject Headings (MeSH)
↑ Ganong, William F. (Nineteenth edition, 1999), Review of Medical Physiology, Appleton & Lange ,pp. 494-496
↑ , Phagocytosis, Conjoint 401-403, University of Washington
↑ "Complete Blood Count", MedLine Plus
↑ Denkers EY, Del Rio L, Bennouna S (2003), Neutrophil production of IL-12 and other cytokines during microbial infection, in Cassatella, Marco A., "The Neutrophil: An Emerging Regulator of Inflammatory and Immune Response", Chem Immunol Allergy
↑ American Proficiency Institute (2004), Educational Commentary -- Blood Cell Identification
↑ Dr. Abraham L. Kierszchenbaum (2006), Histology and cell biology
↑ Finn Geneser (2007), Histology - textbook and atlas
Retrieved from "https://citizendium.org/wiki/index.php?title=Neutrophil&oldid=542911"
|
Non-Detection of Dark Matter Particles: A Case for Alternate Theories of Gravity
C. Sivaram1, Kenath Arun2,3*, A. Prasad4, Louise Rebecca3
1 Indian Institute of Astrophysics, Bangalore, India.
2 Department of Physics and Electronics, CHRIST (Deemed to be University), Bangalore, India.
3 Department of Physics, Christ Junior College, Bangalore, India.
4 Center for Space Plasma & Aeronomic Research, The University of Alabama in Huntsville, Huntsville, Alabama, USA.
Abstract: While there is overwhelming evidence for dark matter (DM) in galaxies and galaxy clusters, all searches for DM particles have so far proved negative. It is not even clear whether only one particle is involved or a combination of particles, their masses not precisely predicted. This non-detectability raises the possible relevance of modified gravity theories: MOND, MONG, etc. Here we consider a specific modification of Newtonian gravity (MONG) which involves gravitational self-energy, leading to modified equations whose solutions imply flat rotation curves and limitations of sizes of clusters. The results are consistent with current observations including that involving large spirals. This modification could also explain the current Hubble tension. We also consider the effects of dark energy (DE) in terms of a cosmological constant.
Keywords: Dark Matter, Dark Energy, Modification of Newtonian Gravity, Hubble Tension
Over the past few decades, there has been a plethora of sophisticated experiments involving massive sensitive detectors trying to catch faint traces of the elusive Dark Matter (DM) particles. But so far all of these efforts have been to no avail. Most of these detectors are designed to look for Weakly Interacting Massive Particles (WIMPS) which are much heavier than the proton (several GeV), with there being no definite prediction for the masses of these heavy particles [1]. Another possible candidate is the axion, which is expected to have a much smaller mass (10−3 - 10−5 eV). Here again there is no definite theoretical prediction for the axion mass. A recent work [2] has put limits on very light axion like particle. There have also been several astrophysical searches for DM particles.
So considering these negative results of all experiments so far, can we try to understand why we don’t see them? While evidence for DM as such is overwhelming, otherwise galaxies and galaxy clusters would fly apart but for the additional gravity they provide, the question is what type of particles constitutes them, or can there be alternate ideas to understand the enhanced gravity. Is only one type of DM particle involved or a combination? Theories do not predict what combination of particles or what type of particles will fix the ratio of DM to baryonic matter as about six. We suggest some reasons why we have not seen these elusive particles.
2. Non-Detection of Dark Matter
One such suggestion that has been made [3] [4] is that the DM particles (of favoured mass range) could form degenerate objects of a Neptune mass or less. The first DM clumps to form (as these particles do not couple to radiation) could form primordial planets at large redshifts. So this clumping of DM into objects of different masses would substantially reduce the flux of free DM particles, so that the number of expected events in detectors would be reduced, accounting for negative results so far. Even axions could clump [5].
Another possibility is that the DM particles could have much weaker cross-sections and their masses may not be in the range assumed. This would lead to non-detection. A more drastic conclusion would be that the particles in the predicted mass range, fluxes and coupling may not exist. They may be a different kind of particle, interacting only gravitationally with higher masses, the fluxes would be smaller and number of events less.
In the absence of detection of DM particles so far, it is natural to explore alternate possibilities such as modification of Newtonian gravity that could explain the galaxy rotation curves and motion of clusters. There have been recent approaches in this direction [6]. One such alternative picture, the Modification of Newtonian dynamics (MOND) was initially proposed as an alternative to account for the flat rotation curves of spiral galaxies, without invoking DM in the halo [7] [8].
The theory required an ad hoc introduction of a fundamental acceleration
{a}_{0}\approx {10}^{-8}\text{cm}/{\text{s}}^{\text{2}}
. When the acceleration approaches
{a}_{0}
, the Newtonian law giving the field strength is modified as:
a=\frac{{\left(GM{a}_{0}\right)}^{1/2}}{r}
where a is the acceleration, r is the radial distance, M is the central mass. And this gives a constant velocity, i.e. flat rotation curve for the galaxies, with the constant velocity, (at the galactic outskirts) given by,
{v}_{c}={\left(GM{a}_{0}\right)}^{1/4}
These results can also be arrived at by considering a minimum acceleration given by [9]:
{a}_{\mathrm{min}}=\frac{GM}{{r}_{\mathrm{max}}^{2}}
{r}_{\mathrm{max}}
is the radius of the structure corresponding to the minimum acceleration and it sets the limit for the size of large scale structures, which follows from Equation (2) as,
{r}_{\mathrm{max}}={\left(\frac{GM}{{a}_{\mathrm{min}}}\right)}^{1/2}
, hence we get,
{v}_{c}={\left(GM{a}_{\mathrm{min}}\right)}^{1/4}
This velocity is independent of r beyond
{r}_{\mathrm{max}}
, which is consistent with observation. For Milky Way, this constant velocity (Equation (3)) ~300 km/s, same order as that observed. For clusters of galaxies, such as Virgo (
M\approx 1.25\times {10}^{15}{M}_{\odot }
) and Coma (
M\approx 7\times {10}^{14}{M}_{\odot }
) clusters, the velocity (from Equation (3)) is ~1500 km/s which is again in accordance with what is observed. The constraints on the size of large scale structures such as galaxies, clusters and super clusters, i.e.
{r}_{\mathrm{max}}={\left(\frac{GM}{{a}_{\mathrm{min}}}\right)}^{1/2}
, closely matches with observations (Table 1) [10].
It was also discussed in recent papers that the requirement that the attractive gravitational binding self-energy density of large scale structures (such as galaxies, clusters, superclusters, etc.) should at least be equal to the background repulsive DE (cosmological constant Λ) density implies a mass-radius relation of the type [11] [12] [13]:
\frac{M}{{r}^{2}}=\frac{{c}^{2}}{G}\sqrt{\text{Λ}}\approx 1\text{\hspace{0.17em}}\text{g}/{\text{cm}}^{\text{2}}
for the observed value of
\text{Λ}~{10}^{-56}\text{ }{\text{cm}}^{-2}
. Here M and r correspond to mass and radius of these structures. This relation holds true for primeval galaxies as well as those at present epoch [14]. This relation can also be obtained by rearranging Equation (2), i.e.,
\frac{M}{{r}_{\mathrm{max}}^{2}}=\frac{{a}_{\mathrm{min}}}{G}\approx 1\text{\hspace{0.17em}}\text{g}/{\text{cm}}^{\text{2}}
, where the minimum acceleration is
{a}_{\mathrm{min}}~{10}^{-8}\text{cm}/{\text{s}}^{\text{2}}
Table 1. Observed and calculated sizes of clusters and superclusters.
The flat rotation curves can also be explained by considering Modifications of Newtonian Gravity (MONG). By adding a gravitational self-energy term to the Poisson’s equation we get,
{\nabla }^{2}\varphi +K{\left(\nabla \varphi \right)}^{2}=4\pi G\rho
\varphi \left(~\frac{GM}{r}\right)
is the gravitational potential and the constant
K~{G}^{2}/{c}^{2}
. The gravitational self-energy density is given by
K{\left(\nabla \varphi \right)}^{2}
, and also contributes to the gravitational field along with the matter density
\rho
. For small values of the density
\rho
, (for e.g. at the outskirts of galaxies) we have,
{\nabla }^{2}\varphi +K{\left(\nabla \varphi \right)}^{2}=0
The solution of this equation yields,
\varphi ={K}^{\prime }\mathrm{ln}\frac{r}{{r}_{\mathrm{max}}}
{K}^{\prime }=\frac{GM}{{r}_{\mathrm{max}}}
is a constant. This gives the force of the form,
F=\frac{{K}^{″}}{r}
{K}^{″}={\left(GM{a}_{\mathrm{min}}\right)}^{1/2}
, is again a constant. The balance of centripetal force and gravitational force then gives,
{v}^{2}/r={K}^{″}/r
This then implies the independence of v on r (i.e. flat rotation curve,
{v}^{2}={K}^{″}
, which is a constant) for larger distances from the centre of the galaxy (i.e. for
r>{r}_{\mathrm{max}}
). Including both gravitational self-energy and DE densities, the Poisson’s equation now takes the form,
{\nabla }^{2}\varphi +K{\left(\nabla \varphi \right)}^{2}-\Lambda {c}^{2}=0
The general solution (for the potential
\varphi
) can be written as:
\varphi =\frac{GM}{r}+{K}^{\prime }\mathrm{ln}\frac{r}{{r}_{\mathrm{max}}}+\Lambda {r}^{2}{c}^{2}
We now make use of this general solution for different regimes of interest in the galaxy structure. Where matter density dominates, i.e.
r<{r}_{\mathrm{max}}
\varphi \approx \frac{GM}{r}
(solution of
{\nabla }^{2}\varphi =4\pi G\rho
) which gives a velocity varying linearly with distance. For
r>{r}_{\mathrm{max}}
{\left(\nabla \varphi \right)}^{2}
term dominates, and
\varphi
goes as
{K}^{\prime }\mathrm{ln}\frac{r}{{r}_{\mathrm{max}}}
, (giving a constant velocity) accounting for DM (solution of
{\nabla }^{2}\varphi +K{\left(\nabla \varphi \right)}^{2}=0
r\gg {r}_{\mathrm{max}}
\varphi
\Lambda {r}^{2}{c}^{2}
, DE dominates (i.e. the cosmological constant term).
In the case of the Milky Way, the velocity flattens out beyond ~2 kpc, which is what is obtained from the above results (Figure 1). Other galaxies also show similar typical rotation curves, with MONG matching with observation [15].
Figure 1. Galaxy rotation curve (for Milky Way) from modified Newtonian gravity.
3. Implications for Super-Spiral Galaxies and Hubble Constant
In this connection, the extra term in the Poisson equation given by MONG, i.e. Equation (9) and its solution given by Equation (10) could have interesting consequences for current observations of super-spirals [16], wherein their large extent (450,000 light-years) is associated with large rotation velocities of up to ~450 km/s at their periphery. Conventionally such large velocity would imply a large amount of DM, i.e. about
~{10}^{13}{M}_{\odot }
. However, our extra term would give a velocity given by:
v={\left(GM{a}_{\mathrm{min}}\right)}^{1/4}{\left(\mathrm{ln}\frac{r}{{r}_{\mathrm{max}}}\right)}^{1/2}
{r}_{\mathrm{max}}
corresponds to the radius at which acceleration approaches
{a}_{\mathrm{min}}
{r}_{\mathrm{max}}=20\text{\hspace{0.17em}}\text{kpc}
and super spiral extant
r={r}_{SS}\approx 200\text{\hspace{0.17em}}\text{kpc}
, this would give velocities ~450 km/s. In other words, the logarithmic term makes gravity stronger above
{r}_{\mathrm{max}}
(i.e., potential going as
\mathrm{ln}r
, instead of 1/r), so that we do not need such colossal amounts of DM. This also implies a logarithmic correction to the Tully-Fisher relation.
The usual Friedmann equation now gets modified to:
\frac{\stackrel{̇}{{R}^{2}}}{{R}^{2}}=\frac{8\pi G\rho }{3}+{\left(GM{a}_{0}\right)}^{1/2}\mathrm{ln}\frac{R}{{R}_{\mathrm{max}}}
(R is the scale factor)
The second term can be seen as a modification in potential energy due to gravitational self-energy density in the usual balance between kinetic and potential energy terms,
\frac{8\pi G\rho }{3}
(in the expanding Universe), i.e. the usual Newtonian analogue agreeing with the GR result.
{a}_{0}={10}^{-8}\text{cm}/{\text{s}}^{\text{2}}
, and the Universe having expanded at present to
R>{10}^{28}\text{cm}
the modified term will also contribute. With
R\approx 2\times {10}^{28}\text{cm}
\rho \approx {10}^{-29}\text{g}/\text{cc}
, and the mass of the Universe,
M=2{\pi }^{2}{R}^{3}\rho \approx {10}^{56}\text{g}
, the usual first term is ≈1021, whereas the second term ≈1020. This suggests that this extra term now manifesting itself would cause an increase of the expansion rate, i.e. a change in Hubble constant (
{\stackrel{˙}{R}}^{2}/{R}^{2}
) by ~5%. This could perhaps account for the faster expansion rate seen at the present epoch.
Here we show that the modification of the gravitational field can provide an alternate explanation for the discrepancy in the value of the Hubble constant as implied by Planck observations of the CMBR in the early Universe and that deduced from other distance indicators in the present epoch [17] [18] [19]. The DM effects can be, in principle and at least partially, be explained through the framework of extended gravity which has been considered earlier [20] [21]. Hence we see that the specific modification of Newtonian gravity, involving gravitational self-energy leads to modified equations and the solutions imply flat rotation curves and limit the sizes of clusters. This modification can also account for current observations involving super-spiral galaxies and can account for the Hubble tension [22].
Cite this paper: Sivaram, C. , Arun, K. , Prasad, A. and Rebecca, L. (2021) Non-Detection of Dark Matter Particles: A Case for Alternate Theories of Gravity. Journal of High Energy Physics, Gravitation and Cosmology, 7, 680-686. doi: 10.4236/jhepgc.2021.72039.
[1] Arun, K., Gudennavar, S.B. and Sivaram, C. (2017) Dark Matter, Dark Energy, and Alternate Models: A Review. Advances in Space Research, 60, 166.
[2] Reynolds, C.S., et al. (2020) Astrophysical Limits on Very Light Axion-Like Particles from Chandra Grating Spectroscopy of NGC 1275. The Astrophysical Journal, 890, 59.
[3] Sivaram, C. and Arun, K. (2011) New Class of Dark Matter Objects and Their Detection. The Open Astronomy Journal, 4, 57-63.
[4] Sivaram, C., Arun, K. and Kiren, O.V. (2019) Primordial Planets Predominantly of Dark Matter. Earth, Moon and Planets, 122, 115-119.
[5] Hogan, C.J. and Rees, M.J. (1988) Axion Miniclusters. Physics Letters B, 205, 228.
[6] Sivaram, C., Arun, K. and Rebecca, L. (2020) MOND, MONG, MORG as Alternatives to Dark Matter and Dark Energy, and Consequences for Cosmic Structures. Journal of Astrophysics and Astronomy, 41, Article No. 4.
[8] Milgrom, M. (1983) A Modification of the Newtonian Dynamics—Implications for Galaxies. The Astrophysical Journal, 270, 371-383.
[9] de Sabbata, V. and Sivaram, C. (1993) On Limiting Field Strengths in Gravitation. Foundations of Physics Letters, 6, 561-570.
[10] Rebecca, L., Arun, K. and Sivaram, C. (2018) Dark Energy Constraints on Masses and Sizes of Large Scale Cosmic Structures. Astrophysics and Space Science, 363, Article No. 149.
[11] Sivaram, C. and Arun, K. (2012) Primordial Rotation of the Universe, Hydrodynamics, Vortices and Angular Momenta of Celestial Objects. The Open Astronomy Journal, 5, 7-11.
[12] Sivaram, C. and Arun, K. (2013) Holography, Dark Energy and Entropy of Large Cosmic Structures. Astrophysics and Space Science, 348, 217.
[13] Sivaram, C., Arun, K. and Kiren, O.V. (2013) Some Consequences of a Universal Tension Arising from Dark Energy for Structures from Atomic Nuclei to Galaxy Clusters. The Open Astronomy Journal, 6, 90.
[14] Sivaram, C., Arun, K. and Rebecca, L. (2020) Planckian Pre Big Bang Phase of the Universe. Astrophysics and Space Science, 365, 17.
[15] Rebecca, L., Arun, K. and Sivaram, C. (2020) Dark Matter Density Distributions and Dark Energy Constraints on Structure Formation Including MOND. Indian Journal of Physics, 94, 1491.
[16] Ogle, P.M., et al. (2019) A Break in Spiral Galaxy Scaling Relations at the Upper Limit of Galaxy Mass. The Astrophysical Journal Letters, 884, L11.
[17] Aghanim, N., et al. (2020) Astronomy and Astrophysics, 641, Article No. A6.
[18] Riess, A., et al. (2019) Large Magellanic Cloud Cepheid Standards Provide a 1% Foundation for the Determination of the Hubble Constant and Stronger Evidence for Physics Beyond Lambda CDM. The Astrophysical Journal, 876, 85.
[21] Corda, C. (2018) The Future of Gravitational Theories in the Era of the Gravitational Wave Astronomy. International Journal of Modern Physics D, 27, Article ID: 1850060.
[22] Sivaram, C., Arun, K. and Rebecca, L. (2021) The Hubble Tension: Change in Dark Energy or a Case for Modified Gravity? Indian Journal of Physics.
|
MLVSS Calculator
Reviewed by Purnima Singh, PhD and Steven Wooding
MLVSS definition
MLVSS procedure for industry
MLVSS calculation formula
How to calculate MLVSS in a laboratory
If you have to monitor an activated sludge process, you will surely find this mixed liquor volatile suspended solids (MLVSS) calculator helpful. As well as helping you find the industrial MLVSS for wastewater, it also allows you to quickly find it if you have a sample in a laboratory.
To help you better understand why calculating MLVSS is necessary, we will:
Give you the MLVSS definition in the context of a microbiological suspension in an aeration tank;
Walk you through the procedure of each MLVSS calculation formula so you can calculate MLVSS on your own; and
Explain the difference between MLSS and MLVSS in wastewater treatment.
As civilization increased in size and scope, one issue became more and more pressing: how to deal with waste, namely, human waste? While not the most pleasant of topics, when you have a city with millions of souls, this becomes an existential issue, as dumping it into a river or pit will likely cause the nearby sources of drinking water to become contaminated. This is what causes many of the cholera epidemics found throughout the world.
The issue of waste management got so bad in London that the summer of 1858 was known as 'The Great Stink', and led to the construction of the world's first modern sewer system, designed by Sir Joseph Bazalgette.
We have come a long way since then. Now, instead of dumping sewage straight into bodies of water, much of the developed world attempts to remove waste material from the water before reintroducing it into the water system.
Part of this system is the activated sludge process. Here, raw, untreated wastewater enters an aeration tank. It is then treated with aerobic microorganisms and the air is pumped through the vessel. The microorganisms digest much of the organic matter in the sewage, and the contents of the tank are moved to a clarifier-settler tank, where clean water can be extracted. Some of the sludge that is left behind is reintroduced back into the aerobic tank as activated sludge. This is where the microorganisms originally came from.
That brief description of modern wastewater treatment probably isn't enough for your keen inquisitive mind; you're probably still asking "What is MLVSS?!" As MLVSS stands for mixed liquor volatile suspended solids, it may be useful to go through what each of these terms means:
Mixed liquor – The name for the concoction of raw wastewater and activated sludge found in the aeration tank.
Volatile – Components of the liquor that are destroyed when exposed to temperatures of 550 °C. In this case, all volatile solids are assumed to be the bacteria that break down the sewage. This is the key difference between the MLSS and MLVSS definitions.
Suspended solids – The mass or concentration of solids in the liquor that do not dissolve but are instead dispersed throughout the liquid. The cut-off point between a suspended solid and a dissolved solid is usually taken to be 2 microns.
From this, we can derive the MLVSS definition: The mass or concentration of suspended bacteria within a combination of raw sewage and activated sludge. The more precise industrial definition of MLVSS for wastewater is:
"The concentration or mass of bacteria needed to break down a certain amount of organic material in sewage."
Before we get to the MLVSS calculation formula, it is worth discussing all of the terms that you need to accurately calculate the MLVSS of wastewater in an industrial setting.
Flow – The volume of mixed liquor that enters (or exits) the aeration tank per unit time. While this is usually given in MGD (millions of gallons per day) or as millions of m3 per day, you can input any value of flow into our tool for ease of calculation. It'll then do the conversion into MGD for you!
COD – Chemical oxygen demand. This is the amount of oxygen required to oxidize every atom or molecule in the mixed liquor that can be oxidized. COD is very closely related to BOD (biological oxygen demand), the concentration of oxygen needed to allow the aerobic bacteria to break down the organic matter – their "food". COD is preferred to BOD because BOD usually takes 5 days to determine, while we can find COD in around 2 hours. There are certain terms that you need to know that involve COD:
Average untreated in-fluent COD – The average COD of the mixed liquor entering the aeration tank.
Average primary treated COD – The average COD of the mixed liquor after being treated in the aeration tank.
Primary effluent COD – The COD of the system while it is in the aeration tank.
COD added to aeration – The amount of oxygen needed to be added to the aeration tank to meet the system's oxygen needs, given as mass per unit time.
Aeration volume - This is the volume of the aeration tank, in which part of the activated sludge process takes place. This requires four variables to calculate, or you can input the volume directly:
Sidewall depth; and
Free-board (the distance between the top of the liquor and the top of the sidewall).
F/M ratio – Food to microorganism ratio. This is the amount of digestible organic matter entering the aeration tank to the amount of bacteria in the aeration tank. This ratio is usually between 0.3 and 0.6, and it will vary depending on the needs of the process.
The tool actually contains several different formulas that allow you to calculate the MLVSS of wastewater, and so we will now go through each of the industrial MLVSS procedures one-by-one. The first converts flow in gallons per hour to flow in millions of gallons per day:
\!\scriptsize {\text{Flow}}_{\text{MGD}} = \frac{{\text{Flow}}_{\text{GPH}} \times 24}{1000000}
{\text{Flow}}_{\text{MGD}}
– Flow in millions of gallons per day; and
{\text{Flow}}_{\text{GPH}}
– Flow in gallons per hour.
If you wish, you can easily substitute m3 for gallons.
The next determines the COD of the effluent:
\!\scriptsize {\begin{gather*}\text{Primary}\\ \text{effluent}\\ \text{COD}\end{gather*}} = {\begin{gather*}\text{Average}\\ \text{untreated}\\ \text{in-fluent} \text{COD}\end{gather*}} - {\begin{gather*}\text{Average}\\ \text{primary}\\ \text{treated} \text{COD}\end{gather*}}
What each of these variables means is described in the section above. The units of concentration used for each of them aren't important – all that matters is that they're the same.
The following formula uses the two previous MVLSS calculation formulae to find the mass of oxygen needed to meet the COD each day:
\!\scriptsize \begin{gather*}\text{COD}\\ \text{added to}\\ \text{aeration}\end{gather*} = \text{Flow}_{\text{MGD}} \times {\begin{gather*}\text{Primary}\\ \text{effluent}\\ \text{COD}\end{gather*}} \times 8.34
{\text{Primary effluent COD}}
– Given in mg per liter;
8.34
– Conversion factor between millions of mg per liter and pounds per gallon; and
{\text{COD added to aeration}}
– Given in pounds per day.
Again, you can find the definition of these variables in the section above. Of note, one million mg per liter equals 8.34 pounds per gallon, which is why that conversion factor is used. If you are using other combinations of units for millions of volume per time and concentration, we recommend you use our calculator here and change the units as appropriately or find the correct conversion factor with our dedicated concentration calculator.
To find the value of MLVSS as a concentration, you will need to find the volume of the aeration tank (this is essential if you wish to find the concentration of MLVSS in wastewater treatment):
\!\scriptsize V = l \times w \times (\text{Side wall depth} - \text{Free-board})
V
– Volume of the aeration tank;
l
– Length of the tank;
w
– Width of the tank;
\text{Sidewall depth}
– Total depth of the tank; and
\text{Free-board}
– Distance between the top of the tank and the wastewater level.
You can use whatever units of length you like, as long as they are all the same (this will give you the units of volume in that unit cubed).
If we know the
\text{F/M ratio}
, we can now answer the question "How to find MLVSS?":
\!\scriptsize \text{MLVSS (mass)} = \frac{\text{COD added to aeration}}{\text{F/M ratio}}
The mass unit and time unit of both
\text{COD added to aeration}
\text{F/M ratio}
must be the same for correct calculation of MLVSS. In the U.S.A., the units are usually
\text{lbs}/{\text{day}}
\text{lbs}\cdot\text{day}/\text{lbs}
To calculate the MLVSS value as a concentration:
\!\scriptsize \text{MLVSS (conc.)} = \frac{\text{MLVSS (mass)}}{V \times 8.34 \times 10^{-6}}
It is often easier to use the MLVSS calculation formula for laboratories than take numerous on-site readings. To do this, you must follow two MLVSS procedures: one to calculate the concentration of MLSS (mixed liquor suspended solids) in the liquor, and one to find the concentration of fixed solids. To find the MLSS concentration, you must take a volume of the wastewater, filter it, dry it thoroughly, and finally weight it:
\!\scriptsize \begin{align*} &\text{MLSS (conc.)} =\\\\ &\quad\frac{\text{(Sample + filter weight)} - \text{Filter weight}}{\text{Volume of sample}} \end{align*}
Regarding units, if you keep the weights the same, you won't have to deal with any pesky conversions!
Finding the fixed solids is very similar, but instead of thorough drying, you must heat the sample sufficiently to ensure conflagration:
\!\scriptsize \begin{align*} &\text{Fixed solids (conc.)} =\\\\ &\ \ \frac{\text{(Ash + crucible weight)} - \text{Crucible weight}}{\text{Volume of sample}} \end{align*}
The same rules for units apply as above. The final step is then to subtract fixed solids from MLSS to get MLVSS:
\!\scriptsize \begin{gather*}\text{MLVSS}\\ \text{(conc.)}\end{gather*} = \text{MLSS (conc.)} - \begin{gather*}\text{Fixed}\\ \text{solids (conc.)}\end{gather*}
What is MLSS and MLVSS in wastewater treatment?
MLSS is the amount of solid material suspended in wastewater, while MLVSS is the amount of bacterial material suspended in wastewater. This is because MLSS is short for mixed liquor suspended solids, while MLVSS stands for mixed liquor volatile suspended solids.
How do I calculate MLVSS?
To calculate the MLVSS of industrial wastewater:
Find the flow of the aeration tank in millions of gallons per day.
Multiply that value by the COD of the primary effluent in milligrams per liter and 8.34.
Divide the value from Step 2 by the food to microorganism ratio in pounds. This result is the MLVSS in pounds.
Is MLVSS the same as VSS?
All MLVSS is VSS, while not all VSS is MLVSS. This is because MLVSS are volatile suspended solids found specifically within mixed liquor, also known as wastewater or sewage.
What is the MLVSS when the COD is 3000 lb per day and the F/M ratio is 0.65?
In this instance, the MLVSS in pounds is 4615. This result is found by dividing the COD of the primary effluent by the food to microorganism ratio. If you divide this poundage by the volume of the wastewater in the aeration tank and 8.34, you'll get the MLVSS as a concentration
Mixed liquor suspended solid experiment
Weight of filter paper + residue
Fixed solids experiment
Weight of crucible
Weight of crucible + ash
MLVSS (conc.)
The vegetable yield calculator helps you find the crop yield that you'll get from your garden/farm and also serves as a small farm planner that guides you on the spacing required.
Vegetable Yield Calculator
|
How to use the garbage bag size calculator
Example of using the garbage bag size calculator
How to use the formula to calculate what garbage bag size we need
List of standard garbage bag sizes commonly available
Industrial vs. kitchen garbage bag sizes
Our garbage bag size calculator is here to help you measure garbage bag sizes suitable for your trash cans.
We will also share with you how to do these calculations manually so you can calculate what garbage bag size you need, anywhere, at any time.
Furthermore, we'll also tell you about the standard garbage bag sizes commonly available so that you can make the best choice for your trash cans 🚮.
On another note, you can also calculate how much toilet paper you may need using our toilet paper calculator 🧻.
In just a few easy steps, you can use our garbage bag size calculator to find the right bag size for your trash cans.
Shape of your trash can 🗑️
You can choose from one of the following trash can shapes.
Overhang or tie
This is the extra height of the garbage bag to be hung at the rim of the trash can. Ideally, it should be between 3 to 6 in or 7 to 15 cm, depending on the size of your trash can.
In this group, you can enter the size of your trash can, in the following fields:
Width (for rectangle & square only)
Length (for rectangle only)
Diameter (for circle only)
Suitable garbage bag size
Once you've entered your trash can size, the calculator will present you with the most suitable garbage bag size in the following fields:
Let's take an example by calculating the garbage bag size of a 22-inch trash can, where the height of our trash can is 30 inches, and it's of a circle shape.
We select Circle as the shape of our trash can.
Keeping the default Overhang or tie size of 4 inches.
In the diameter, under the trash can size, enter 22 inches.
Enter the height of our trash can, i.e., 30 inches.
Thus, our suitable garbage bag should have a width of approximately 35 inches with a height of 45 inches.
Here are the formulas that we use to find the width and height of our garbage bags manually:
For rectangle and square trash cans:
\bold G _{w} = T _{l} + T _{w}
\bold G _{h} = T _{h} + G _{o} + G _{w} / 4
For circle trash cans:
\bold G _{w} = T _{d} × \pi / 2
\bold G _{h} = T _{h} + G _{o} + T _{d} / 2
\bold G _{w}
– Suitable garbage bag width;
\bold G _{h}
– Suitable garbage bag height;
T _{l}
– Length of the trash can;
T _{w}
– Width of the trash can;
T _{d}
– Diameter of the trash can;
T _{h}
– Height of the trash can;
G _{o}
– Overhanging bag height; and
\pi
– Math constant with an approximate value of 3.1416.
🙋 We add
\bold G _{w}/4
T _{d} / 2
, i.e., the half of our trash can width to our garbage bag height to ensure that the garbage bag covers the bottom of our trash can adequately.
We will now explore a couple of example calculations.
Let's take an example using the values from the previous example to calculate the garbage bag size of a 22-inch trash can, where the height of our circle trash can is 18 inches while we want the overhanging to be 4 inches.
\bold G _{w} = 22 \times \pi / 2 = 34.56
\bold G _{h} = 30 + 4 + 22 / 2 = 45
Thus, a suitable garbage bag for our trash can should have an approximate width of 35 inches and a height of 45 inches.
Let's take another example to find a kitchen garbage bag size, where the square trash can has a length and width of 20 inches and a height of 36 inches.
Placing the values in our formula, with an overhanging height of 6 inches, we get:
\bold G _{w} = 20 + 20 = 40
\bold G _{h} = 36 + 6 + 40 / 4 = 52
Thus, a suitable garbage bag size for this kitchen's trash can is 40 × 52 inches in width and height, respectively.
Here's a list of some standard garbage bag sizes commonly available:
Small garbage bag sizes (W × H):
Medium garbage bag sizes (W × H):
Large garbage bag sizes (W × H):
Garbage bags come in many colors and sizes. But what sets them apart is their thickness, usually measured in mil.
💡 A mil is a thousandth of an inch, i.e., 0.001 inch.
In the kitchen, we usually use medium, low-density garbage bags, ranging between 0.5 to 0.9 mil. They are economical and sufficient to dispose of every day's junk.
However, when we're renovating or cleaning our house, we require bags with more durability to resist broken glass, nails, wood shavings, and other sharp items. For that, bags between 2.0 to 3.0 mil are more suited.
But on the downside, the majority of these garbage bags end up as single-use plastic. And we already have more than 5 trillion plastic pieces floating in our oceans.
Here's our plastic footprint calculator that you can use to determine how much we have unknowingly contributed to this wreckage.
Let's all switch to biodegradable garbage bags to reduce our plastic bag footprint. 😀
How do I calculate what garbage bag size I need?
Sum the length and width of your rectangular trash can to obtain your garbage bag width.
Then sum the height of your trash can with a quarter of your garbage bag width and add 4 inches of overhang. That's your garbage bag height.
What garbage bag size do I need for an 8" × 12" × 22" bin?
You need a 20" × 31" garbage bag size for an 8" × 12" × 22" bin. Here's how the calculation goes:
Sum the length and width of the trash can, i.e., 8 + 12 to get the garbage bag width of 20".
Sum the 22 height of your trash can with 5, i.e., a quarter of the garbage bag width, and 4" overhang to get your garbage bag height: 31".
What are the largest sizes of garbage bags?
Some of the largest available garbage bag sizes are:
What sizes do garbage bags come in?
Here's a list of common garbage bag sizes:
|
Square inches and square feet - Imperial units of area
More area conversion tools!
Welcome to Omni's square inch to square foot converter, a tool that helps you convert any area from square inches to square feet. Maybe you're wondering how many square feet are 20 inches by 20 inches or are 12 inches by 12 inches a square foot, then you have landed at the right place.
If you're interested in area conversions, we invite you to keep reading to find:
How many feet are in a square inch; and
How to convert square inches to square feet yourself.
We use different units to measure area quantities. In the imperial system of units, the most commonly used to measure area are the square inches (in2) and the square feet (ft2).
There's a direct correlation between these two units, and it's given by:
1 in2 = 1/144 ft2 = 0.00694444 ft2
This way, you can express the same property (area) with another unit of measurement. Even though they'll be numerically different, both represent the same area.
For instance, if a given region has an area of 432 in2, and we apply the conversion factor expressed above, we can say that this is equivalent to 3 ft2.
🙋 Give the square inch to square foot converter a try! How many square feet are in an area of 864 in2?
To calculate any surface area, we use linear or one-dimensional measures and go on from there. In this particular case, we'll be referring to inches and feet, which are imperial units to measure distances.
The appropriate conversion factor between these two is 12 inches equal to 1 foot:
12\text { in} = 1\text{ ft}
By multiplying this linear conversion factor by itself, we can go from one-dimensional quantities to two-dimensional or area values:
{12\text{ in}} \times {12\text{ in}} = 1\text{ ft} \times 1\text{ ft}
144 \text { in}^2 = 1 \text{ ft}^2
The above corresponds to the conversion factor to go from areas in square inches to square feet or the other way around.
Finally, the next step is to apply this factor to swap between square inches and square feet.
For example, to convert an area of
36\text { in}^2
\text{ ft}^2
, we simply divide
36\text { in}^2
144\text { in}^2
36\text { in}^2 \times (1\text { ft}^2 / 144 \text { in}^2) = 0.25 \text { ft}^2
If you enjoyed using the square inch to square foot converter, you might like to take a look at some other of our wonderful area conversion tools:
How many square feet are in a square inch?
There are 0.00694444 ft2 in a square inch.
To convert any area value from square inches to square feet use the conversion factor of 144 in2 = 1 ft2, from here
1 in2 = 1/144 ft2 = 0.00694444 ft2.
There are 2.77778 ft2 in a 20 by 20 inches area.
To get this value you can follow these steps:
Multiply 20 inches by 20 inches to get the value in square inches.
20 in × 20 in = 400 in²
For this conversion, the area conversion factor is:
Utilize this conversion factor and perform the corresponding algebraic operation:
400 in² × (1 ft² / 144 in²) = 2.77778 ft²
20 in × 20 in = 400 in² = 2.77778 ft²
Are 12 inches by 12 inches equal to a square foot?
Yes, 12 inches by 12 inches area equals one square foot. To verify this value:
Keep in mind that 12 inches = 1 ft.
From the above, we can affirm that multiplying 12 inches by itself is the same as multiplying 1 foot by 1 foot:
12 in × 12 in = 1 ft × 1 ft
12 in × 12 in = 144 in² = 1 ft²
|
MyOpenMath/Solutions/Big-O - Wikiversity
MyOpenMath/Solutions/Big-O
< MyOpenMath | Solutions
An excellent introduction to this subject can be found at this document from web.mit.edu:
big_o.pdf.
In this introduction to Big O notation, we solve two problems: one simple and the other so tricky I got a bit lost. The advantage of Big-O notation is that you can quickly "see" an answer without doing elaborate perturbation theory. Instead you just learn a few low order approximations for small
{\displaystyle \epsilon }
. A few examples are
{\displaystyle \sin(\epsilon )\approx \epsilon }
{\displaystyle \cos \epsilon \approx 1-{\tfrac {1}{2}}\epsilon ^{2}}
. All we need for this discussion is the first order approximation for
{\displaystyle (1+\epsilon )^{p}}
1 Ruler misalignment
1.2 Defining the small parameter
2 Two slit diffraction with narrow slits
2.1 When the screen is far from the slits
2.2 When the screen is not far from the slits
Ruler misalignment[edit | edit source]
{\displaystyle h}
but measure
{\displaystyle y}
Have you ever pondered the fact that you can measure your height without carefully verifying that the ruler is perfectly vertical? In the language of Big O notation, if the actual height is
{\displaystyle h}
, the error is second order in the distance,
{\displaystyle x}
, between the actual and proper locations of the bottom of the ruler (see figure). To understand how this all works, begin with the Pythagorean theorem and express the erroneously measured height as:
{\displaystyle y={\sqrt {h^{2}+x^{2}}}=h\;\left(1+{\frac {x^{2}}{h^{2}}}\right)^{1/2}}
Advanced mathematics with numbers that have dimensions is best done by creating dimensionless variables, and this is especially true when analyzing approximations. A handy approximation is that whenever
{\displaystyle \epsilon <<1}
{\displaystyle (1+\epsilon )^{p}=1+p\epsilon +{\mathcal {O}}\epsilon ^{2}+...}
Here the "big-O" informs us that the next term is proportion to
{\displaystyle \epsilon ^{2}}
{\displaystyle {\mathcal {O}}}
-symbol allows us to avoid consideration of this term, while at the same time, preserve the location of these higher order terms, in case the calculation needs to be improved. We define,
{\displaystyle \Delta y=y-h}
, to be the error that arises from ruler misalignment. We presume that this error will be small ... but small compared with what? This problem has two large terms,
{\displaystyle (h,y)}
, and two small ones
{\displaystyle (x,\Delta y)}
. The big-O notation will help us sort things out. From the two equations displayed above:
{\displaystyle {\frac {y}{h}}=1+{\frac {\Delta y}{h}}=1+{\frac {1}{2}}\left({\frac {x}{y}}\right)^{2}+{\mathcal {O}}\left({\frac {x}{y}}\right)^{4}}
{\displaystyle x=0.1y}
{\displaystyle \Delta y/h=.005}
, which implies that:
A horizontal displacement of one end of ruler's length by ten percent will increase the measured height by approximately half of one percent.
This calculation is only an estimate that lacks a proper proof because higher order terms have been neglected. On the other hand, it is likely to be correct, since the next term in the expansion is of order
{\displaystyle (x/y)^{4}\approx 10^{-4}=0.01\%}
Defining the small parameter[edit | edit source]
This section might seem unnecessary, but the next calculation is so weird that it might help to discuss it here: Whether something is "first" or "second" order depends on you choose to define things. Here we have chosen,
{\displaystyle \epsilon ={\frac {x^{2}}{y^{2}}}}
and we are working to "first order" in
{\displaystyle \epsilon }
, even though it is "second" order in
{\displaystyle x/y}
Note the approximation that the two paths to the screen are parallel
Two slit diffraction when screen is close
A better way to visualize
{\displaystyle \Delta r}
as the base of a triangle for nearly parallel paths. Click for explanation
Two slit diffraction with narrow slits[edit | edit source]
When the screen is far from the slits[edit | edit source]
Problem: Two narrow slits are separated by 0.8 mm. The 15-th fringe appears 89 mm from the center of the diffraction pattern, and the screen is 9 m from the slits. What is the wavelength?
The standard textbook solution to this problem uses the formula,
{\displaystyle n\lambda =d\sin \theta }
, where :
{\displaystyle 15\lambda ={\frac {0.8{\text{ mm}}\times 89{\text{ mm}}}{{\sqrt {9000^{2}+89^{2}}}{\text{ mm}}}}}
{\displaystyle \Rightarrow \lambda \approx 527.4{\text{ nm}}}
This solution is only valid when the distance to the screen,
{\displaystyle L}
is much greater than the distance between the slits,
{\displaystyle d}
. In the next section we will derive an exact equation, and then use big-O notation to recover the standard formula in the limit that
{\displaystyle d/L}
To learn about two slit diffraction visit:
OpenStax College Physics Chapter 27-3
wikibooks:Waves/Double_slit_Diffraction
To see a hand written solution on MyOpenMath visit:
https://myopenmaths3.s3.amazonaws.com/ufiles/2556987/interference_question_ID_420941.pdf
When the screen is not far from the slits[edit | edit source]
The wavelength and dimensions of the device in the previous section were chosen so that the simple formula would yield the correct answer. But how we solve the problem when the spacing between the slits is close to the distance to the screen. The geometry is shown in the figure to the right. It helps to define,
{\displaystyle R={\sqrt {L^{2}+y^{2}}},}
{\displaystyle r_{1}={\sqrt {L^{2}+(y-d/2)^{2}}}={\sqrt {R^{2}-yd+d^{2}/4}}}
{\displaystyle r_{2}={\sqrt {L^{2}+(y+d/2)^{2}}}={\sqrt {R^{2}+yd+d^{2}/4}}}
Note from the figure that
{\displaystyle y/R=\sin \theta }
, and that the two paths are effectively parallel when
{\displaystyle d<<R}
. The exact formula for the path difference is:
{\displaystyle \Delta r=r_{2}-r_{1}={\sqrt {T^{2}+yd}}-{\sqrt {T^{2}+yd}},}
{\displaystyle T={\sqrt {L^{2}+y^{2}+{\frac {d^{2}}{4}}}}={\sqrt {R^{2}+{\frac {d^{2}}{4}}}}}
From the formulas stated without proof in the previous section, we are looking to show that:
{\displaystyle \Delta r\approx {\frac {yd}{R}}=d\sin \theta }
We also seek insight into the nature of higher order correction terms in order to estimate when this simple formula is likely to be valid. The standard approach would be to perform a Taylor series expansion of the function
{\displaystyle f(d)=\Delta r}
{\displaystyle d}
as the variable. But in order to highlight big-O notation, we employ the aforementioned expansion:
{\displaystyle (1+\epsilon )^{1/2}=1+{\frac {\epsilon }{2}}+{\mathcal {O}}\epsilon ^{2}}
Wright this expression with
{\displaystyle \epsilon }
{\displaystyle -\epsilon }
, and subtract the two:
{\displaystyle (1+\epsilon )^{1/2}-(1-\epsilon )^{1/2}=}
{\displaystyle \left[1+{\frac {\epsilon }{2}}+{\mathcal {O}}\epsilon ^{2}\right]}
{\displaystyle -\left[1-{\frac {\epsilon }{2}}+{\mathcal {O}}\epsilon ^{2}\right]}
When subtracting in the big-O notation, it is essential to realize that in general,
{\displaystyle {\mathcal {O}}\epsilon ^{2}-{\mathcal {O}}\epsilon ^{2}={\mathcal {O}}\epsilon ^{2}\neq {\mathcal {O}}\epsilon ^{2}}
{\displaystyle {\mathcal {O}}\epsilon ^{2}}
{\displaystyle C\epsilon ^{2}}
{\displaystyle C}
is some unknown constant. The difference between unknown constants is not usually zero. However in this case the exact cancellation of all even terms leaves us with an expression containing only terms that are odd in
{\displaystyle \epsilon }
{\displaystyle (1+\epsilon )^{1/2}-(1-\epsilon )^{1/2}=\epsilon +{\mathcal {O}}\epsilon ^{3}+...}
The absence of a second order term suggests that the first order term is likely to be sufficient for reasonably small values of
{\displaystyle \epsilon }
. The physics of this problem informs us that,
{\displaystyle \Delta r=n\lambda ,}
so that we seek and expression for,
{\displaystyle \Delta r={\sqrt {T^{2}+yd}}-{\sqrt {T^{2}+yd}}=T\cdot \left[\,{\sqrt {1+\epsilon }}+{\sqrt {1-\epsilon }}\;\right]}
We see here that our small parameter is,
{\displaystyle \epsilon ={\frac {yd}{T^{2}}}}
{\displaystyle \Delta r=\epsilon \cdot T+{\mathcal {O}}\epsilon ^{3}\cdot T={\frac {yd}{T}}+T\cdot {\mathcal {O}}\epsilon ^{3}}
One final task remains: Since since
{\displaystyle y/R=\sin \theta ,}
we need to replace
{\displaystyle yd/T}
{\displaystyle yd/R}
(...plus small terms.) It is left as an exercise for the reader to show that:
{\displaystyle {\frac {1}{T}}={\frac {1}{R}}\left\{1-{\mathcal {O}}\left({\frac {d^{2}}{4R^{2}}}\right)\right\}}
{\displaystyle T}
{\displaystyle R}
are very close to each other, differing only at second order in our small parameter. Unless
{\displaystyle \theta {\text{ is very close to }}\pi /2,}
the three lengths
{\displaystyle T,R,{\text{and }}L}
are all of the same order and are not small:
{\displaystyle {\mathcal {O}}T={\mathcal {O}}R={\mathcal {O}}L={\mathcal {O}}\epsilon ^{0}}
{\displaystyle yd/T={\mathcal {O}}\epsilon }
, so that up to but not including third order, the difference in path length is:
{\displaystyle \Delta r={\frac {yd}{R}}\left[1+{\mathcal {O}}\epsilon ^{2}\right]\approx yd\sin \theta +...}
which agrees with the formula found in most physics books.
{\displaystyle n\lambda \approx d\sin \theta }
is a good approximation even with the screen this close to the slits.
If your small parameters have small parameters, you need analysis.
While not exact, the familiar formula for fringes when the screen is far away, the approximate formula,
{\displaystyle n\lambda =d\sin \theta }
, works surprisingly well for the screen close to the slits. Here,
{\displaystyle [d,y,L]=[321,274,404]].}
The fringe number was
{\displaystyle n=1}
(first maximum.)
Yet, the "small parameter" is not very small:
{\displaystyle yd/R^{2}\approx 0.37}
The approximate formula for the first fringe is depicted in the figure as
{\displaystyle \lambda _{1}}
, which equals the length of the line segment,
{\displaystyle {\overline {CD}}}
The actual wavelength is
{\displaystyle \lambda ={\overline {CE}}}
{\displaystyle E}
was by creating the (dotted) arc of length
{\displaystyle r_{1}}
, which intersect with line
{\displaystyle {\overline {CP}}}
{\displaystyle r_{2}}
. The approximation,
{\displaystyle n\lambda _{1}=d\sin \theta }
{\displaystyle \lambda _{1}=180.18...=\lambda \times 1.037...}
{\displaystyle \lambda =173.66...}
is the exact wavelength, calculated from:
{\displaystyle n\lambda ={\sqrt {{\sqrt {L^{2}+y^{2}+{\frac {d^{2}}{4}}}}+yd}}-{\sqrt {{\sqrt {R^{2}L^{2}+y^{2}+{\frac {d^{2}}{4}}}}+yd}},}
Another approach[edit | edit source]
The big-O approach led us initially to a rather awkward small parameter,
{\displaystyle \epsilon ={\frac {yx}{T^{2}}}={\frac {yx}{R^{2}+x^{2}/4}}\approx {\frac {yx}{R^{2}}}+{\mathcal {O}}{\frac {x^{3}y}{R^{4}}}}
In other words, the convenient small parameter differs from the useful small parameter by a small parameter. Weird, huh?
We could also solve this problem with a Taylor series. In anticipation of doing differential calculus, we replace
{\displaystyle d}
{\displaystyle x}
as the variable to represent the distance between the slits. Now define the path length difference
{\displaystyle \Delta r}
by the function
{\displaystyle F}
{\displaystyle F(x)={\sqrt {{\sqrt {R^{2}+{\frac {x^{2}}{4}}}}+yx}}-{\sqrt {{\sqrt {R^{2}+{\frac {x^{2}}{4}}}}-yx}}}
This looks like a lot of trouble, but symbolic software is available that can make this almost effortless. This expression also shows us why the big-O approach got into trouble. There are really two small dimensionless parameters lurking in this problem, and we can distinguish between them with subscripts:
{\displaystyle \epsilon _{R}={\frac {x^{2}}{4R^{2}}}{\text{, and }}\epsilon _{y}={\frac {yx}{R^{2}}}}
{\displaystyle {\frac {\Delta r}{R}}={\sqrt {{\sqrt {1+\epsilon _{R}}}+\epsilon _{y}}}-{\sqrt {{\sqrt {1+\epsilon _{R}}}-\epsilon _{y}}}}
In other words we need to understand the function,
{\displaystyle f(x,y)={\sqrt {{\sqrt {1+x}}+y}}-{\sqrt {{\sqrt {1+x}}-y}}}
{\displaystyle x}
{\displaystyle y}
are small. What I would do here is a two-dimensional expansion:
{\displaystyle f(x,y)=f_{0}+\left.{\frac {\partial f}{\partial x}}\right|_{0}x+\left.{\frac {\partial f}{\partial y}}\right|_{0}y+{\frac {1}{2}}\left.{\frac {\partial ^{2}f}{\partial x^{2}}}\right|_{0}x^{2}+{\frac {1}{2}}\left.{\frac {\partial ^{2}f}{\partial y^{2}}}\right|_{0}y^{2}+\left.{\frac {\partial ^{2}f}{\partial x\partial y}}\right|_{0}xy+...}
Retrieved from "https://en.wikiversity.org/w/index.php?title=MyOpenMath/Solutions/Big-O&oldid=2269957"
MyOpenMath/Electromagnetism
|
Tbsp to ml Converter
Why tablespoons?
What is the conversion between tablespoons and milliliters?
What about teaspoons?
Our tbsp to ml converter will help you avoid that situation where you see the recipe for the best cake ever, but all of the measurements are in kitchen tools and not metric units!
We want you to bake that cake: that's why we made this tbsp to ml converter! a tablespoon of sugar makes the recipe easier! Keep reading to learn:
Why do we use tablespoons?
Where is the tablespoon?
How to convert from tablespoons to milliliters.
We can already answer the third question: check in your drawer! 🥄
A tablespoon is a type of cutlery used to... wait. Sorry, let's try again.
A tablespoon is a measurement unit for volume used most often in cooking. You will see tablespoons all over the places in recipes in English, in place of the more "scientific" metric units. Tablespoons are abbreviated with tbsp (don't mistake them for the teaspoons' smaller counterparts).
Tablespoons, cups, and sticks. If you're used to reading recipes with metric units, like grams and liters, finding them may throw you off.
To be honest, it's a great unit of measurement. Everyone has tablespoons at home (the same thing is true for cups, teaspoons, and all the others). Like for many other "non-metric" units, this quantity is perfectly balanced: when milliliters are too little and liters too much, the tablespoon helps you always call for a reasonable number!
The milliliter is a multiple of the liter, the S.I. measurement unit for the volume.
1
liter equals
1000
milliliters. The symbol of the milliliter is
\text{ml}
The milliliter is a relatively small unit: you need only
20
drops of water to make one. This is why it's not the best unit for cooking measurements; in fact, you will almost always need
20\ \text{ml}
50\ \text{ml}
, or some other bigger multiple in your recipes.
A tablespoon equals
15
milliliters. Let's get this right:
1\ \text{tbsp}=15\ \text{ml}
Try our volume converter if you need other calculators for your cooking endeavors. And if you are in a rush, maybe your oven is already heated, we prepared a list of specific converters for all your needs:
And remember that after all, the spoon doesn't exist!
What are tbsp and tsp?
The abbreviations tbsp and tsp mean respectively:
These are common volume units used in cooking.
Is a tablespoon always the same?
No, and it is not important! A typical quantity used in the conversion between tablespoons and milliliters is 1 tbsp = 15 ml. Still, it's not a fixed value, and it varies from country to country and also from kitchen to kitchen!
How do I convert from tablespoons to milliliters?
To convert tbsp to ml, you have to multiply the number of tablespoons by 15:
Check out various volume converters at omnicalculator.com!
How much are 2 tablespoons in milliliters?
Two tablespoons are the same as 30 ml. This is an average quantity for medications like cough syrup: it fits in a small cup.
|
Revision as of 02:39, 1 November 2016 by Andy1978 (talk | contribs) (→Common problems)
This FAQ is intended to supplement, not replace, the GNU Octave manual (HTML, PDF). Before posting a question to the help@octave.org mailing list, you should first check to see if the topic is covered in the manual.
5.2 Pre-compiled binaries
5.4 How can I install Octave on Android? What is this Octave app in the Google Play store?
6.3 How do install or load all Octave packages?
7.1.11 Do-Until loop structure
8.10 How do I run a Matlab P-file in Octave?
10 Differences between Octave and Matlab
Octave is free software and does not legally bind you to cite it. However, we have invested a lot of time and effort in creating GNU Octave, and we would appreciate if you would cite if you used.
Ideally, you should cite the software itself, not a book or manual, like so:
@software{octave,
author = {John W. Eaton and others},
title = {GNU Octave},
url = {http://www.octave.org},
It is recommended to do so on a first draft submitted. However, some editors may disallow this, in which case you can still make a general reference to Octave in the text, such as:
If a more traditional reference is required, there is also a manual for each Octave release that can be cited:
author = {John W. Eaton, David Bateman, S\oren Hauberg, and Rik Wehbring},
url = {http://www.gnu.org/software/octave/doc/interpreter},
Note that there are two reasons for citing the software used. One is giving recognition to the work done by others which we already addressed. The other is giving details on the system used so that experiments can be replicated. This is just as important, if not more.
For this, you should cite the version of Octave and all packages used, as well as any details of your setup as part of your Methods. In addition, you should make your source available. See How to cite and describe software for more details and an in depth discussion.
Be around. Be social. Participate in the help and maintainers mailing lists. Find things about Octave you don't like, and start thinking about how to fix them. Many people who now contribute to Octave first spent several years helping in the mailing list before they started to delve into the code. A good way to learn Octave is to understand the problems other people are having with it, so being helpful in the mailing lists not only helps Octave as a whole, but it also prepares you to be a better Octave contributor.
If you feel ready to dive right into the code, read here and here for guidance. But do not send an email to the mailing lists listing your skills and offering to help. We won't just suggest things for you to do. We lack volunteers and we do need your help, but because of that, we also lack the time to provide good guidance and mentoring. If there is a specific short-term project you would like to work on, say so, and just do it. Then ask for help or advice when you're doing it. It is a lot more important that you do something that you're actually interested on than something we suggested because it only matches your skills.
Look at our Projects and Summer of Code Project Ideas if you need specific inspiration for coding tasks that we would like to get done. See also the list of short projects.
When one downloads code from File Exchange and use it on non Mathworks software (such as Octave), they are violating the Matlab central Terms of Use. While the BSD licenses does allow one to use such code in Octave, it also allows others to further impose restrictions which Mathworks does through the MATLAB Central Terms of Use of their site:
It should suffice -- although interpretations of this vary -- to contact the author directly and have them send you the code personally, or download the code from the author's own website, if available.
Each new Octave release introduces many new features. The following are a distilled list of the major changes. A complete list of user visible changes can be seen by running news at the Octave prompt, and a full list of changes is on the Template:Filepath distributed with the Octave sources.
See NEWS on the development branch
See also: Build from source
The Octave project does not normally distribute its own binaries, but other projects do. For example, Linux and BSD distributions provide by their respective Octave binaries. Windows is a recent exception, in that binaries are available directly from the Octave project (starting from version 4.0). For an up-to-date listing, see:
As of today, Octave binaries are available at least on Debian, Ubuntu, RedHat, Suse, and Fedora GNU/Linux; Mac OS X; and Windows (versions 98, 2000, XP, Vista, 7, 8, and 10).
Octave currently runs on Unix-like systems, Mac OS X, and Windows. It should be possible to make Octave work on other systems as well. If you are interested in porting Octave to other systems, please contact the maintainers' mailing list .
How can I install Octave on Android? What is this Octave app in the Google Play store?
There is an unofficial Octave app available for Android in the Google Play store. Please see Android for more information.
This version of Octave is no longer freely available and has become 'nagware'. This is a sad turn of events.
How do install or load all Octave packages?
When Octave starts, it runs ~/.octaverc. If you want Octave to automatically load a package, simply add a pkg load pkg-name command to it. If the files does not exist, create it.
Similar to the do-while loop in C and C++, Octave allows a do-until loop which does not exist in Matlab
Borrowed from other languages, octave broadcasting allows easy and readable vectorialization.
This only works with gnuplot as graphics_toolkit, NOT with fltk. See Bug#33180
Alternatively, one may use arbitrary precision arithmetic, which has as much precision as is practical to hold in your computer's memory. The symbolic package has a vpa() function for arbitrary precision arithmetic. Note that arbitrary precision arithmetic must be implemented in software which makes it much slower than hardware floats.
You can't. Matlab P-files (files with a .p file extension), also known as P-code, are obfuscated files than cannot be run outside of Matlab itself. The original source Matlab m-files that were used to generate the P-files should be used in Octave instead.
Sound input from a sound card and output to a sound card is fully supported in Octave 4.0.0 and newer in all operating systems. Older versions of Octave had very limited audio support that was essentially Linux-specific. If you have problems with the audio I/O functions using Octave 4.0 or a newer version, please report them on the bug tracker.
{\displaystyle 1/10}
{\displaystyle 0.0{\overline {0011}}_{b}}
{\displaystyle 1/6=0.1{\overline {6}}_{d}}
This isn't an Octave bug. It happens with any program that uses IEEE 754 floating point arithmetic. To be fair, IEEE 754 also specifies decimal floating point arithmetic, which has never seen wide adoption. The reason why Octave and other programs use IEEE 754 binary floats is that they are fast, because they are implemented in hardware or system libraries. Unless you are using very exotic hardware, Octave will use your computer's processor for basic floating point arithmetic.
Another approach to the problem is interval arithmetic with the Interval package. Then, the exact result would always be enclosed by two binary floats. Again, this is slower since only the most basic interval arithmetic operations can be performed in hardware.
Users with this kind of problem should try to install/update their Intel OpenGL drivers for Windows or consider installing Mesa drivers from http://qt-project.org/wiki/Cross-compiling-Mesa-for-Windows
See also https://www.opengl.org/wiki/FAQ#Why_is_my_GL_version_only_1.4_or_lower.3F
A large number of the Matlab core functions (ie those that are in the core and not a toolbox) are implemented, and certainly all of the commonly used ones. There are a few functions that aren't implemented, usually to do with specific missing Octave functionality (GUI, DLL, Java, ActiveX, DDE, web, and serial functions). Some of the core functions have limitations that aren't in the Matlab version. For example the sprandn function can not force a particular condition number for the matrix like Matlab can. Another example is that testing and the runtests function work differently in Matlab and Octave.
Up to Octave 2.9.9 there was no support for graphic handles in Octave itself. In the 3.2.N versions of Octave and beyond, the support for graphics handles is converging towards full compatibility. The patch function is currently limited to 2-D patches, due to an underlying limitation in gnuplot, but the experimental OpenGL backend is starting to see an implementation of 3-D patches.
Yes! It was officially released with Octave 4.0.0. It was also available since version 3.8.0 has an experimental feature (use the --force-gui option to start Octave).
None of the GUIs for Octave that had been developed thus far were part of Octave and there is a reason for it. All of them failed at a very important point, integration with Octave. They treated Octave as a foreign black box using pipes for communication. This approach is bound to fail with each new version. Any fix made to make them work with a new Octave versions would only be temporary. This included QtOctave (now abandoned and incompatible with newer versions of Octave), Xoctave (which is proprietary and commercial), and GUI Octave (which was proprietary and no longer available).
QtOctave was a great and very useful tool. It looked beautiful and we are thankful to its developers for working on such a nice tool. However, it would have never been stable as it was. But most of all, the developers made it free software so we could reuse large chunks of it which were incorporated in what is now the Octave GUI.
The current default is OpenGL, through the "qt" graphics toolkit. The "ftlk" toolkit also relies on OpenGL.
Alternatively, Gnuplot can be used as a graphics backend by selecting the "gnuplot" graphics toolkit.
The development of Octave is committed to being both compatible with Matlab and adding additional features. Toward those ends, the development community has chosen to introduce a native OpenGL backend that supports Matlab handle graphics and its uicontrols. Starting with the 3.8 release, Octave uses OpenGL graphics by default (with FLTK widgets in Octave 3.8 and Qt widgets in Octave 4.0 and later).
There are no plans to remove the gnuplot backend. Even though the default graphics toolkit is now "qt", which uses OpenGL graphics with Qt widgets, the gnuplot backend will still be available as long as our users find it useful.
|
Screw_turbine Knowpia
Archimedes Screw Generator (ASG),[1] also known as Archimedes/Archimedean Screw Turbine (AST),[2] Archimedean turbine or screw turbine is a hydraulic machine that convert the potential energy of water on an upstream level into work. This hydropower converter is driven by the weight of water, similar to water wheels, and can be considered as a quasi-static pressure machine.[2]
Reverse action of the "Archimedean screw", the principle of the "screw turbine" gaining energy from water flowing down through the screw.
Screw turbines typically have three or four flights (second row)
Two parallel screw turbines capable of producing 75 kW each, in Monmouth, Wales
Video of a 40 kW screw turbine in Munich, Germany
Archimedes screws can be used to generate power if they are driven by flowing fluid instead of lifting fluid.[2] Water transiting the screw from high to low elevation generates a torque on the helical plane surfaces, causing the screw to rotate.[2] The Archimedes screw generator consists of a rotor in the shape of an Archimedean screw which rotates in a semicircular trough. Water flows into the screw and its weight presses down onto the blades of the turbine, which in turn forces the turbine to turn. Water flows freely off the end of the screw into the river. The upper end of the screw is connected to a generator through a gearbox. The Archimedes screw is theoretically a reversible hydraulic machine, and there are examples of single installations where screws can be used alternately as pumps and generators.[3]
The required parameters to design Archimedes screws.[3] The Archimedes screw design parameters are similar in designing Archimedes screw turbines (generators) and Archimedean screw pumps [1].
A screw turbine at a small hydro power plant in Goryn, Poland.
The Archimedean screw is an ancient invention, attributed to Archimedes of Syracuse (287–212 BC.), and commonly used to raise water from a watercourse for irrigation purposes. In 1819 the French engineer Claude Louis Marie Henri Navier (1785–1836) suggested using the Archimedean screw as a type of water wheel. In 1916 William Moerscher applied for a U.S. patent on the hydrodynamic screw turbine.[4]
12 kW screw turbine at the Cragside estate
The Archimedean screw turbine is applied on rivers with a relatively low head (from 0.1 m to 10 m)[2] and on low flows (0.01 m³/s up to around 10 m³/s on one turbine). Due to the construction and slow movement of the blades of the turbine, the turbine is considered to be friendly to aquatic wildlife. It is often labelled as "fish friendly". The Archimedean turbine may be used in situations where there is a stipulation for the preservation and care of the environment and wildlife.
An Archimedes Screw Turbine (AST) hydroelectricity powerplant can be considered as a system with three major components: a reservoir, a weir, and the AST (which is connected to the system by a control gate and trash rack).[2] At most real AST locations, the incoming flow must be divided between the AST and a parallel weir. Typically, a minimum flow over the weir is mandated for the protection of the local environment. Other outlets as well as a fish ladder could be considered as the other components of this system.[2] A comprehensive guide about the principles of designing Archimedes screw turbines and screw hydropower plants is available in "Archimedes Screw Turbines: A Sustainable Development Solution for Green and Renewable Energy Generation—A Review of Potential and Design Procedures".[2]
Flow Rate ModelEdit
To design Archimedes screw turbines and hydropower plants, it is essential to estimate the amount of water is passing through the screw turbine since the amount of power generated by an Archimedes screw turbine is proportional to the volume flow rate of water through it.[2] The volume of water that enters an Archimedes screw turbine depends on the inlet water depth and the screw's rotation speed.[5] To estimates the total flow rate passing through an Archimedes screw turbine for different rotation speeds (ω) and inlet water levels the following equation could be used:[2][5]
{\displaystyle Q=\alpha Q_{Max}(A_{E}/A_{Max})^{\beta }(\omega /\omega _{M})^{\gamma }}
{\displaystyle \alpha }
{\displaystyle \beta }
{\displaystyle \gamma }
are constants related to the screw properties. Preliminary investigations suggest that
{\displaystyle \alpha =1.242}
{\displaystyle \beta =1.311}
{\displaystyle \gamma =0.822}
give reasonable predictions of
{\displaystyle Q}
for a wide range of small to full-scale AST sizes.[2][5]
Design Archimedes screw geometryEdit
By determination of
{\displaystyle D_{O}}
other design parameters of Archimedes screws could be easily calculated using an easy an step by step analytical method.[1][3] Studies shows that the volume of flow passes through Archimedes screws is a function of inlet depth, diameter and rotation speed of the screw.[5][2] Therefore, the desired volumetric flow rate
{\displaystyle Q}
{\displaystyle (m^{3}/s)}
and rotation speed
{\displaystyle \omega }
{\displaystyle (rad/s)}
the following analytical equation could be used to determinate the Archimedes screws overall diameter
{\displaystyle D_{O}}
{\displaystyle (m)}
{\displaystyle D_{O}=(16\pi Q}/{\sigma \omega (2\theta _{O}-sin2\theta _{O})-\delta ^{2}(2\theta _{i}-sin(2\theta _{i}))^{1/3}}
Based on the common standards that the Archimedes screw designers use this analytical equation could be simplified as:[1]
{\displaystyle D_{O}=\eta Q^{3/7}}
The value of η could simply determinate using the
{\displaystyle \eta }
graph[1] or
{\displaystyle \Theta }
graph.[3] By determination of
{\displaystyle D_{O}}
other design parameters of Archimedes screws could be easily calculated using an easy an step by step analytical method.[1][3]
Woolston, Cheshire weir on the River Mersey 486 kW, Under construction[6][7][8]
Devon, Totnes 320 kW , Commissioned December 2015[9]
Romney, Berkshire, 270 kW, Installed to provide a renewable source of energy to Windsor Castle, Commissioned July 2013[10]
Bealey’s Weir, Radcliffe, 100 kW, Commissioned May 2012[11]
Mapledurham, River Thames, UK’s largest flow capacity (8 m³/s) single screw, 99 kW.[12]
Buckfast, River Dart, screw turbine and fishpass, 84 kW[12]
UK’s first community owned hydro scheme, and fishpass, 63 kW at New Mills.[12]
UK’s first grid connected screw turbine, 50 kW at River Dart Country Park.[12]
Bainbridge, community owned screw turbine, 37 kW[12]
Tipton, River Otter, 30 kW[12]
Rochdale, screw turbine and fishpass, 20 kW[12]
Cragside, the birthplace of hydroelectricity, 12 kW[13]
Hanover Pond on the Quinnipiac River in Meriden, Connecticut, 105 kW (or 920,000 kWh/year), grid connected, commissioned April, 2017; the first Screw Turbine installation in the US.[14][15]
The first Archimedes screw turbine was installed in Canada in 2013 near Waterford, Ontario.[2]
YoosefDoost, A, W.-D. Lubitz: Design Guideline for Hydropower Plants Using One or Multiple Archimedes Screws, Processes, 2021. doi:10.3390/pr9122128
YoosefDoost, A, W.-D. Lubitz: Archimedes Screw Design: An Analytical Model for Rapid Estimation of Archimedes Screw Geometry, Energies 2021. doi:10.3390/en14227812
YoosefDoost, A, W.-D. Lubitz: Archimedes Screw Turbines: A Sustainable Development Solution for Green and Renewable Energy Generation—A Review of Potential and Design Procedures, Sustainability, 2020. doi:10.3390/su12187352.
P. J. Kantert: Manual for Archimedean Screw Pump, Hirthammer Verlag 2008, ISBN 978-3-88721-896-6
P. J. Kantert: Praxishandbuch Schneckenpumpe. Hirthammer Verlag 2008, ISBN 978-3-88721-202-5
William Moerscher - Patent US1434138
K. Brada, K.-A. Radlik - Water Screw Motor to Micro Power Plant - First Experiences of Construction and Operation (1998)
K. Brada - Micro Power Plant with Water Screw Motor (1995)
K. Brada, K.-A. Radlik - Water Power Screw - Characteristic and Use (1996)
K. Brada, K.-A. Radlik, (1996). Water screw motor for micropower plant. 6th Intl. Symp. Heat exchange and renewable energy sources, 43–52, W. Nowak, ed. Wydaw Politechniki Szczecińskiej, Szczecin, Poland.
^ a b c d e f g YoosefDoost, Arash; Lubitz, William David (January 2021). "Archimedes Screw Design: An Analytical Model for Rapid Estimation of Archimedes Screw Geometry". Energies. 14 (22): 7812. doi:10.3390/en14227812.
^ a b c d e f g h i j k l m YoosefDoost, Arash; Lubitz, William David (2020). "Archimedes Screw Turbines: A Sustainable Development Solution for Green and Renewable Energy Generation—A Review of Potential and Design Procedures". Sustainability. 12 (18): 7352. doi:10.3390/su12187352.
^ a b c d e YoosefDoost, Arash; Lubitz, William David (December 2021). "Design Guideline for Hydropower Plants Using One or Multiple Archimedes Screws". Processes. 9 (12): 2128. doi:10.3390/pr9122128.
^ William Moerscher, Water-power system, U.S. Patent 1,434,138, granted Oct 31, 1922.
^ a b c d YoosefDoost, Arash; Lubitz, William David (2021), Ting, David S.-K.; Vasel-Be-Hagh, Ahmad (eds.), "Development of an Equation for the Volume of Flow Passing Through an Archimedes Screw Turbine", Sustaining Tomorrow, Cham: Springer International Publishing, pp. 17–37, doi:10.1007/978-3-030-64715-5_2, ISBN 978-3-030-64714-8, S2CID 234121383, retrieved 2021-02-09
^ "Woolston | Planning Application". Warrington Borough Council. Retrieved 2021-03-22.
^ "Woolston | Project Overview". renfin.eu. Retrieved 2021-03-22.
^ "Woolston | Local News Items". Warrington Worldwide. Retrieved 2021-03-22.
^ "Totnes | MannPower Consulting". www.mannpower-hydro.co.uk. Retrieved 2016-08-05.
^ "Romney | MannPower Consulting". www.mannpower-hydro.co.uk. Retrieved 2016-08-05.
^ "Bealeys Weir | MannPower Consulting". www.mannpower-hydro.co.uk. Retrieved 2016-08-05.
^ a b c d e f g "Hydro Power Case Studies, Micro-Hydro Case Studies - Western Renewable Energy". www.westernrenew.co.uk. Retrieved 2016-08-05.
^ "Hydropower returns to Cragside". National Trust. Retrieved 2016-08-09.
^ Andrew Ragall, Ancient technology in Meriden's Hannover Pond dam begins generating electricity, Meriden Record Journal, April 27, 2017.
^ New England Hydropower Energizes First Archimedes Screw Turbine in U.S., PR Newswire, April 27, 2017.
Wikimedia Commons has media related to Archimedes' screw power plants.
spaansbabcock.com/products/screw-turbine Hydropower screw
Information on one of the manufacturers
(in Polish) The first screw turbine in Poland
|
Although the quadratic formula always works as a strategy to solve quadratic equations, for many problems it is not the most efficient method. Sometimes it is faster to factor or complete the square or even just “out-think” the problem. For each equation below, choose the method you think is most efficient to solve the equation and explain your reason. Then solve the problems that can be factored.
x^2+7x−8=0
Note the expression in this equation can be factored.
(x+2)^2=49
49
5x^2−x−7=0
Note the expression in this equation cannot be factored.
x^2+4x=−1
4
to both sides will create a perfect square.
|
Cubic Cell Calculator
The basics of crystallography
The types of unit cells and crystal families
The cubic unit cell and the cubic crystal family
Calculations in a cubic unit cell
How to use our cubic cell calculator to calculate the lattice constant
The cubic cell calculator will introduce you to the world of crystals: learn the mathematics underlying the most regular structure in nature, and find examples where you would never imagine!
Some basics concepts of crystallography;
The crystal structures in two and three dimensions;
The types of cubic unit cells and the related lattices; and
The calculations of the lattice parameter for the cubic lattices.
And much more. This article will make this topic... crystal clear! If you want to dig deeper in crystallography, check out our miller indices calculator and our Bragg's law calculator too!
Crystallography deals with the arrangement of atoms in the ordered structures of crystalline substances.
Crystals are all around us, but more often than not, we dismiss them, not even noticing their tidiness and marvelous order. Go in your kitchen: the coarse salt and the metal of your pan are examples of crystalline materials.
The number of atomic species in a crystal defines the basis of the crystal itself. The basis is the first element required to describe a crystal.
🔎 The word "crystal" comes from the Greek for "frozen drop" or "ice drop". They weren't off at all: ice is a crystal!
Take a crystalline substance, and start breaking it. Piece after piece, you would be able to separate the sample until a final break would give you something that is not the crystal anymore: you just found the unit cell of that substance, the smallest possible element with all the properties of the crystal.
Unit cells are tightly packed: there can't be any empty spaces between them. This highly restricts their possible shapes.
There's one last concept we need to introduce: the Bravais lattice. The lattice is an array of vectors that describes the positions of the elements in a crystalline structure. The lattice is the second (and last) element used to describe a crystalline structure: you could say that crystalline structure = basis + lattice. You can find more about lattices at our lattice energy calculator.
For a two-dimensional crystalline structure, only the following shapes pack tightly:
Rectangle; and
Centered rectangle.
The two-dimensional crystalline structures. The vectors
\vec{a}
\vec{b}
identify the unit cells. 1) Cubic cell; 2) Rectangular cell; 3) Rhomboidal cell; 4) Parallelogrammatic cell. Notice how the rhomboidal cell packs in a hexagonal lattice.
From there, we can identify the corresponding crystal families, the groups to which we can assign crystalline structure according to non-translational (rotation, inversion, and roto-inversion) transformations and translational transformations:
Monoclinic for the parallelogram unit cell;
Hexagonal for the rhomboidal unit cell;
Tetragonal for the square unit cell; and
Orthorhombic for both rectangular unit cells.
The centered rectangular unit cell is a particular two-dimensional cell that contains more than a single atom.
Increasing the dimensions to three gives us a slightly more complex picture: even if all of the unit cells were parallelepipeds, we can identify six crystal families:
Hexagonal; and
Different geometric transformations return a possible set of 14 Bravais lattices among these six families. We will analyze in detail the ones related to the cubic crystal family.
A cubic unit cell generates three Bravais lattices:
The simple cubic lattice;
The body centered cubic (BCC) lattice; and
The face centered cubic (FCC) lattice.
The three possible types of cubic lattices. From left to right: the simple cubic lattice, the body centered lattice, and the face centered lattice.
Let's analyze them in detail!
The simple cubic lattice
In the simple cubic lattice, each atom sits at a corner of a cube. In a single cell, each corner contains an eighth of each atom. Since there are eight corners, the total "amount" of atoms in the simple cubic cell is one (the cell is a proper unit cell).
The body centered cubic lattice
Trace all the diagonals connecting maximally separated pairs of atoms in a simple cubic cell, and place an atom at their intersection: that's how you obtain a body centered cubic lattice. The addition of the atom in the center brings the total contained by the cell to
2
The face centered cubic lattice
On each face of the cube, trace the two diagonals and add an atom at their intersection. This would add
6
halves of atoms, bringing the total to
3
🔎 The fourth possible lattice is the base centered cubic one. However, placing an atom at the center of each base would allow us to define a smaller simple cubic cell. Draw it if you don't believe us!
The geometry of a unit cell is determined by a set of numerical values called lattice parameters or lattice constants.
The maximum number of lattice parameters is
6
: three angles and three spatial directions. However, symmetries reduce that number in many instances. For a cubic cell, in fact, the angles are already defined, all equal to
90\degree
. Similarly, the three spatial lattice parameters are equal to each other: this leaves us with a single lattice constant,
a
We can define the lattice constant with a simple mathematical formula for each of the three types of cubic lattices.
The only variable in the lattice parameter determination is the atomic radius of the species composing the crystal,
r
. We will first show the calculations for monoatomic crystals.
The lattice constant of a simple cubic crystal is:
a = 2\times r
Considering the cell is tightly packed with atoms, the minimum length of the side of the cube would be twice the radius.
What if we place an atom in the middle of the cell? The lattice constant of a BCC cell reflects its presence:
a = 4\times \frac{r}{\sqrt{3}}
This result comes from the formula for the diagonal of a cube,
d=\sqrt{3}\times l
. Pack the cube tightly with an atom in the center, and you'll find that the diagonal has length equal to four times the atomic radius.
The last possible lattice is the face centered cubic one. To find the lattice parameter of such crystal, follow a reasoning similar to the one we applied to the BCC lattice.
Pack each face of the cube tightly with atoms. On each diagonal (
\sqrt{2}\times l
long), you can fit again four atomic radiuses. This gives us:
a = 4\times \frac{r}{\sqrt{2}}
🔎 The face centered cubic is a crystalline structure that manages to minimize empty spaces. Crystallographers call such structure closest packed. There are only two types of them: cubic and hexagonal.
Let's sum up the results:
Lattice constant of a simple cubic cell:
a = 2\times r
Lattice constant of a BCC cell:
a = 4\times \frac{r}{\sqrt{3}}
Lattice constant of an FCC cell:
a = 4\times \frac{r}{\sqrt{2}}
Calculating the lattice constant for diatomic crystals (where the basis is
2
) is a different matter. The values are usually computed experimentally using techniques like X-rays diffraction.
Our cubic cell calculator will give you the value of the lattice constant in a few quick steps.
First, input in the cubic cell calculator the value of the atomic radius of the atoms making up the crystal. You can find those values in tables like the one on this page.
Then, choose the type of cubic cell of interest, and we will return the value of the lattice parameter!
🔎 The default measurement unit of the atomic radius is the ångström, denoted as
\text{\AA}
. It corresponds to
0.1
nanometers, or
1\times10^{-10}
meters. This is the perfect unit to measure atoms!
Take as example aluminum. Its atomic radius is
r= 1.43\ \text{\AA}
, and it crystallizes in a face centered cubic lattice. Its lattice parameter would be:
a = 4 \times \frac{1.43\ \text{\AA}}{\sqrt{2}}=4.045\ \text{\AA}
Probably around half of your kitchenware is made of aluminum — you can find a perfect face centered cubic crystal right in your kitchen! While you're there, check if you have a cast iron pan. Cast iron is an alloy of iron and carbon. Carbon lies in the defects of a body centered cubic iron crystal. What is the lattice constant of the body centered cubic iron? The atomic radius of iron is
r= 1.24\ \text{\AA}
a = 4 \times \frac{1.24\ \text{\AA}}{\sqrt{3}}=2.8637\ \text{\AA}
Which are the three types of cubic cells?
A cubic call can be of three types:
Simple cubic, with one atom in each corner;
Face centered cubic, with an additional atom at the center of each face; and
Body centered cubic, where a single atom is added in the center of the cell.
Only the second structure maximizes the occupation of space in the cell.
What are the lattice parameters of a cubic cell?
The lattice parameters are quantities that indicate the shape of a crystalline cell. There are six lattices parameters, three dimensions, and three angles. Their number can be smaller in the case of symmetries in the unit cell — for example, there is only one spatial lattice parameter in a cubic cell, a.
How do I calculate the lattice constant of a cubic cell?
There are three formulas to calculate a cubic cell constant parameter a, depending on the type of cell itself,
For a simple cubic cell: a = 2 × r;
For a body centered cubic cell: a = 4 × r / √3; and
For a face centered cubic cell: a = 4 × r / √2.
Here, r is the value of the radius.
What is the lattice constant of polonium?
Polonium has a lattice constant of 3.369 Å. The radioactive element is the only known atomic species that packs in the relatively unstable simple cubic structure. The value of the lattice parameter comes from the equation:
a = 2 × r = 2 × 1.67 Å = 3.369 Å
a is the lattice constant; and
r is the atomic radius.
Atomic radius (r)
Our Rate Constant Calculator will compute the rate constant, the rate of the reaction or the concentration of the substance using the rate laws.
|
F-statistic calculator
Verbeek, M. “A Guide to Modern Econometrics“ (2017)See 1 more source
Hayashi, F. “Econometrics“ (2000)
What is F-statistic?
How to calculate the F-statistic using an F-statistic table?
How to calculate the F-statistic in linear regression?
The F-statistic calculator (or F-test calculator) helps you compare the equality of the variances of two populations with normal distributions based on the ratio of the variances of a sample of observations drawn from them.
Read further, and learn the following:
What is an F-statistic;
What is the F-statistic formula; and
How to interpret an F-statistic in regression.
Broadly speaking, an F-statistic is a test procedure that compares variances of two given populations. While an F-test may appear in various statistical or econometric problems, we apply it most frequently to regression analysis containing multiple explanatory variables. In this vein, an F-statistic is comparable to a T-statistic, with the main difference of having a linear combination of multiple regression coefficients (F-test) instead of testing only an individual one (T-test).
In the following article, we introduce the F-test in its most basic form using the F-distribution table for better intuition. Then we show how to calculate F-statistic in linear regressions (see the calculator's Multiple regression mode) and explain how to interpret an F-statistic in regression analysis.
The best way to grasp the essence of F-test statistics is to consider its most basic form. Let's consider two populations, from which we each draw an equal number of observation samples. If we want to test whether the two populations are likely to have the same variance (denoted by
S^2_i
i = 1, 2
), we need to follow these steps:
Specify the null hypothesis
H_0
(which in our simple case is that the two variances are equal) and the alternative hypothesis
H_1
(which supposes that the two variances are different).
\footnotesize \qquad \begin{align*} H_0 : S^2_1 &= S^2_2 \\ H_1 : S^2_1 &\neq S^2_2 \end{align*}
Determine the variance of the samples (here you may find our variance calculator useful).
Calculate the F-test statistic by dividing the two variances.
\footnotesize \qquad F = \frac{S^2_1}{S^2_2}
Determine the degrees of freedom
(\text{df}_i)
of the two samples, with
n
being the number of observations taken from the two populations in each case.
\footnotesize \qquad \text{df}_i = n_i - 1
Choose the significance level of the F-statistic
(\alpha)
— for example,
\alpha = 0.05
corresponds to a 95 percent confidence interval.
Check the critical value of the F-statistic in the F-distribution table as follows:
Look for the appropriate F-statistic table with the given significance level
(\alpha)
Find the right column at the top of the F-table statistics that correspond to the degree of freedom of your first sample (nominator).
Check the row on the side that corresponds to the degree of freedom of your second sample (denominator).
Read the F critical value at the intersection, which represents the shaded area on the F-distribution graph below.
Compare the F-statistic critical value to the previously obtained F-value. If the F-value is larger than the critical value collected from the F-table statistic
(F > F_\text{critical})
, you can reject the null hypothesis. That is, we can state with a high confidence that the variances in the two observation samples are not equal.
Analysts mainly apply F-statistic on multiple regressions models (and so can you, with our F-test statistic calculator in Multiple regression mode). It's therefore a good idea that we step further in this direction from the previous basic analysis.
Let's assume we have the following regression model (full model, or unrestricted model), where we would like to know if it is more significant than its reduced form (restricted model). In other words, we are testing whether the restricted coefficients (or the effects of the restricted variables) are jointly non-significant (equal to zero) in the population:
\footnotesize \text{Full model} \\ y = \beta_0 + \beta_1x_1+ \beta_2x_2+ \beta_3x_3+ \hat{u} \\[1em] \footnotesize \text{Restricted model} \\ y = \beta_0 + \beta_1x_1+ \hat{u}
\beta_0
is the constant or intercept,
y
is the dependent variable (also called the regressand, response variable, explained variable, or output variable);
x_i\ , \ i = 1, 2, 3
is the independent variable (also called the regressor, *explanatory variable, controlled variable, or input variable);
\beta_i\ ,\ i = 1, 2, 3
are the coefficients;
\hat{u}
is the residual (or error term).
To conduct the F-test and obtain the F-statistic (or F-value), we need to take the following steps:
State the hypothesis we want to test.
In our case, the null hypothesis
(H_0)
is that the last two coefficients are jointly equal to zero in the unrestricted model. Or, stating the same differently, the joint effect of the related independent variables is insignificant.
In turn, the alternative hypothesis
(H_1)
is that at least one of these coefficients is not equal to zero.
\footnotesize \qquad \text{Specific case} \\ \qquad H_0: \beta_2= \beta_3 = 0 \\ \qquad \text{General case} \\ \qquad H_0: \beta_{K-J+1} = \cdots = \beta_K = 0
J
is the number of restrictions (in the present case,
J=2
K
is the total number of coefficients (in the present case,
K = 3
Now, to gain information on which model fits better, we need to obtain the sum square of residuals (
\text{SSR}
), where we expect that the sum square of residuals of the restricted model is larger than that of the full model (i.e.
\text{SSR}_R > \text{SSR}_F
\footnotesize \qquad SSR = \sum^N_{i=1} \hat{u}^2_i
However, the real question is to determine whether the sum square of residuals of the restricted model is significantly larger than the one in the full model (i.e.
\text{SSR}_R \gg \text{SSR}_F
). To do so, we need to apply the following F-statistic formula to estimate the F-ratio.
\qquad \footnotesize F = \frac{(\text{SSR}_R-\text{SSR}_F) / J}{(1 - \text{SSR}_F)/(N-K)}
F
is the F-statistic;
\text{SSR}_F
is the sum square of residuals of the full model;
\text{SSR}_R
is the sum square of residuals of the restricted model;
J
is the number of restrictions;
K
is the total number of coefficients; and
N
is the number of observations representing the population.
Naturally, the larger the F-statistic, the more evidence we have to reject the null hypothesis (note that the F-statistic increases when the difference between the two variances gets larger). However, to be more precise, we need to find a critical value of the F-statistic to decide on the rejection. In other words, if
F
is larger than its critical value, we can reject the null hypothesis.
Now, we can proceed in the way we described in the previous section by finding the critical F-value
(F^J_{N-K;\alpha})
in the F distribution table with a specified significance level F-statistic
(\alpha)
and looking for the intercept corresponding to the degrees of freedom, where
\text{df}_1 = J
is at the top and
\text{df}_1 = N-K
is at the side of the table (we can also say that
F
has an F-distribution with
J
N − K
degrees of freedom). If
F
So how to interpret F-statistic in regression?
The F-test can be interpreted as testing whether the increase in variance moving from the restricted model to the more general model is significant. We may write it formally in the following way:
\footnotesize P\{F > F^J_{N-K;\alpha}\} = \alpha
\alpha
is the significance level of the test. For example, if
N − K = 40
J = 4
, the critical value at the 5% level is
F^J_{N-K; \alpha} = 2.606
What is the difference between F-test vs T-test?
There are some differences between the F-test vs a T-test.
The T-test is applied to test the significance of one explanatory variable, but the F-test studies the whole model.
While the T-test is used to compare the means of two populations, F-test is applied for comparing two population variances.
The T-statistic is based on the student t-distribution, while the F-statistic follows the F-distribution under the null hypothesis.
While the T-test is a univariate hypothesis test where the standard deviation is unknown, the F-test is applied to determine the equality of the two normal populations.
Can an F-statistic be negative?
No. Since variances always take a positive value (squared values), both the numerator and the denominator of the F-statistic formula must always be positive, resulting in a positive F-value.
What is a high F-statistic?
While a large F-value tends to indicate that the null hypothesis can be rejected, you can confidently reject the null if the T-value is larger than its critical value.
Is the F-distribution symmetric?
No. The curve of the F-distribution is not symmetrical but skewed to the right (the curve has a long tail on its right side), where the shape of the curve depends on the degrees of freedom.
How to calculate F-statistic?
To calculate F-statistic, in general, you need to follow the below steps.
Determine the F-value by the formula of F = [(SSE₁ – SSE₂) / m] / [SSE₂ / (n−k)], where SSE is the residual sum of squares, m is the number of restrictions and k is the number of independent variables.
Find the critical value for the F-statistic as determined by F-statistic = variance of the group means / mean of the within-group variances.
Find the F-statistic in the F-table.
What is the F-statistic of two populations with variances of 10 and 5?
The F-statistic of two populations with variances of 10 and 5 is 2. To get this result, it suffices to divide the two variances.
Sum square of residuals — full model (S₀)
Sum square of residuals — restricted model (S₁)
Number of excluded coefficients (J)
Total number of coefficients (K)
F-statistic (F)
Correlation Coefficient Calculator (Matthews)
This skewness calculator finds both the skewness and kurtosis of a dataset and interprets these values, telling you how skewed or peaked your distribution is.
|
A rectangular prism has a cylindrical hole removed, as shown at right.
If the length of the radius of the cylindrical hole is
0.5
cm, find the volume of the solid.
\text{Volume of cube}-\text{volume of cylinder}=\text{volume of solid}
\text{Volume of cylinder}=\pi r^2h\\ v=\pi\left(0.5\right)^2\left(6\right)\\ v=1.5\pi\approx4.71\ \text{cm}^3
\text{Volume of cube}=l\cdot w\cdot h\\ v=5\left(2\right)\left(6\right)=60\ \text{cm}^3
\text{Volume of cube}-\text{volume of cylinder}=60\ \text{cm}^3-4.71\ \text{cm}^3
≈ 55.3\text{ cm}³
What could this geometric figure represent? That is, if it were a model for something that exists in the world, what might it be? Also, how might you change it to make it a better model?
|
Section 5.5: Numeric Evaluation of Iterated Integrals
Maple has a number of built-in routines for evaluating iterated integrals numerically. The study of such methods properly belongs in a Numerical Analysis course, and will not be considered here. In fact, numeric methods for iterated integrals do not appear in the typical calculus text.
Just as for the numeric evaluation of single integrals, Maple applies the evalf command to the inert form of the iterated integral to effect a numeric calculation. This can be done either by command or interactively, via the Context Panel. The examples below illustrate the technique.
The Maple help page for numeric evaluation of definite integrals is available here.
{∫}_{0}^{3}{∫}_{{y}^{1/3}}^{2}\frac{{\mathit{ⅇ}}^{x- y}}{{x}^{2}+\mathrm{cosh}\left(y\right)} \mathit{ⅆ}x \mathit{ⅆ}y
{∫}_{0}^{\mathrm{\pi }/2}{∫}_{{x}^{2}}^{10-{x}^{2}}\mathrm{sin}\left({x}^{2}+2 x y-{y}^{2}\right) \mathit{ⅆ}y \mathit{ⅆ}x
{∫}_{-1}^{3}{∫}_{{x}^{2}}^{2 x+3}\mathrm{cosh}\left(x \mathrm{cos}\left(y\right)\right) \mathit{ⅆ}y \mathit{ⅆ}x
|
Fees - Whitepaper
Dtravel is committed to fee transparency, precision and simplicity, as detailed below:
The nominal guest fee (
f_g
) is initially set to 10%.
The nominal host fee (
f_h
) is initially set to 0%.
The nominal total fee (
f
) is calculated according to the following equation:
f = f_g + f_h
The base price (
p_b
) is set by the host, and:
The guest price (
p_g
), paid by the guest, is calculated as
p_g = (1 + f_g) p_b
The host price (
p_h
), received by the host, is calculated as
p_h = (1 - f_h) p_b
The effective guest fee (
f'_g(u)
) for a guest
u
f'_g(u) = f_g - c_g(u)f
c_g(u)
u
's cashback percentage.
The effective host fee (
f'_h(u)
) for a host
u
f'_h(u) = f_h - c_h(u) f
c_h(u)
u
's the cashback percentage.
The effective total fee (
f'(u_g, u_h)
f'(u_g, u_h) = f'_g(u_g) + f'_h(u_h)
When the guest pays for a booking:
(c_g(u_g) f ) p_b
(c_h(u_h) f)p_b
are immediately converted to amounts
C_g(u_g)
C_h(u_h)
of TRVL and temporarily sent to a Cashback Reserve. When the booking is completed
C_g(u_g)
C_h(u_h)
are sent from this Reserve to the guest and the host, respectively.
f'(u_g, u_h) \ p_b
is sent to the Community Treasury. From there, it may be sent further to other locations (e.g. Protection Pool) or used according to the rules established by the Dtravel DAO.
Note that, with an initial nominal total fee of 10% and cashbacks and giveaways of up to 50% of the fee, the effective total fee can be as low as 5%.
Alice is a host and sets the base price for a night in her property to US$100.
Bob is a guest and books Alice's property for a guest price of US$110.
Alice has chosen to receive payments in TRVL and she has a Level 2 Premium Member NFT.
The Level 2 gives her a 2.0 boost factor for her cashback.
Bob has chosen to pay in TRVL and he has a Level 1 Premium Member NFT.
The Level 1 gives him a 1.5 boost factor for his cashback.
Furthermore, the price of 1 TRVL is currently US $1.
When Bob makes the payment of 110 TRVL for the booking:
A guest cashback of 1.5 TRVL is sent to the Cashback Reserve.
A host cashback of 2 TRVL is sent to the Cashback Reserve.
The amount of 6.5 TRVL is sent to the Community Treasury.
When the stay is completed:
Bob receives 1.5 TRVL from the Cashback Reserve.
Alice receives 2 TRVL from the Cashback Reserve.
Thus, the effective total fee was 6.5%.
|
Home : Support : Online Help : Connectivity : Calling External Routines : ExternalCalling : C Application Programming Interface : MapleAlloc
MapleAlloc allocates n bytes of memory and returns a pointer to it. MapleDispose frees this memory so it can be reused for other purposes.
MapleAlloc is used to access temporary memory that will be cleaned up by the Maple garbage collector. The memory returned by MapleAlloc cannot be protected from gc. To allocate memory that persists, use standard non-Maple memory allocation methods in combination with a MaplePointer.
ALGEB M_DECL MyConcat( MKernelVector kv, ALGEB *args )
name = MapleToString(kv,args[1]);
f = MapleToFloat64(kv,args[2]);
tmp = (char*)MapleAlloc(kv,(len+30)*sizeof(char));
sprintf(tmp,"%s%f",name,f);
return( ToMapleName(kv,tmp,FALSE) );
\mathrm{with}\left(\mathrm{ExternalCalling}\right):
\mathrm{dll}≔\mathrm{ExternalLibraryName}\left("HelpExamples"\right):
\mathrm{fcat}≔\mathrm{DefineExternal}\left("MyConcat",\mathrm{dll}\right):
l≔\mathrm{fcat}\left(\mathrm{foo},1.1\right)
\textcolor[rgb]{0,0,1}{l}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{foo1.100000}}
\mathrm{assign}\left(\mathrm{eval}\left(l,1\right),17\right)
l
\textcolor[rgb]{0,0,1}{17}
Note: The name returned is local (it does not match the global name).
\mathrm{`foo1.100000`}
\textcolor[rgb]{0,0,1}{\mathrm{foo1.100000}}
\mathrm{dismantle}[\mathrm{hex}]\left(\mathrm{eval}\left(l,1\right)\right)
NAME(200007F630A080934,5): `foo1.100000`
\mathrm{dismantle}[\mathrm{hex}]\left(\mathrm{`foo1.100000`}\right)
NAME(200007F6306FFCAC4,5): `foo1.100000`
|
The origin of centigrade and degrees Celsius. The celsius to centigrade formula
How to use the centigrade to Celsius converter
Formula to convert centigrade to Celsius with examples
Here's our centigrade to Celsius converter where you can input your temperature in centigrade to get your temperature in Celsius or do the opposite, find conversion of Celsius to centigrade.
The following article will tell you about the origin of degrees Celsius and centigrade. You will also learn how to convert Celsius to centigrade as we share some examples, e.g., what is 0 centigrade to Celsius, and the formula to convert centigrade to Celsius.
In 1742, a Swedish astronomer named Anders Celsius created a temperature scale known as the centigrade scale. The name derives from the Latin word centum, which means 100, and gradus, which means steps.
The initial centigrade scale measured the temperature in reverse order, where 0 degrees centigrade was considered the boiling point and 100 degrees as the freezing point. In 1743, Jean-Pierre Christin proposed reversing this scale.
Thus, in 1948, in honor of Anders Celsius, the centigrade scale was renamed to Celsius scale. In other words, Celsius and centigrade are the same units and that's what our Celsius to centigrade calculator uses.
Here's how you convert centigrade to Celsius using the calculator:
Enter your centigrade temperature in the first field, e.g., 17 degrees centigrade.
Your result in Celsius will display in the second field, i.e., 17 degrees Celsius.
Thus, if we convert 17 degrees centigrade to Celsius, the result is 17 degrees Celsius. Yes, that's it! The conversion of Celsius to centigrade is a simple 1:1 conversion. Still, you can come back to our centigrade to degrees Celsius converter whenever you have doubts about the proper conversion.
To convert centigrade to Celsius, we use the following Celsius to centigrade formula:
\degree \text{centigrade} = \degree \text{Celsius}
From this, we can understand that centigrade is equal to Celsius.
Let's take an example of converting 34 degrees centigrade to Celsius.
Placing values, in the formula, we get:
34 \degree \text{centigrade} = 34 \degree \text{Celsius}
Similarly, if we convert 34 degrees Celsius to centigrade, we get the same result.
Thus, we conclude that the centigrade is equal to Celsius and vice versa.
💡 If we convert 0 centigrade to Celsius, we get 0 degrees Celsius.
Here are some other temperature converters that you may find useful:
Temperature converter;
Fahrenheit converter;
How do I convert 0 centigrade to Celsius?
0 degrees centigrade is equal to 0 degrees Celsius.
Celsius and centigrade refer to the same temperature scale, where the temperature of the centigrade is always equal to the temperature of Celsius and vice versa.
Under the right conditions, the freezing point of pure water is 0 degrees Celsius, or 0 degrees centigrade. However, in certain conditions, the water can remain liquid at temperatures lower than 0 degrees Celsius or frozen at temperatures higher than 0 degrees Celsius.
What is the coldest temperature in Celsius?
The temperature of -273.15 degrees Celsius is the coldest temperature, where every natural particle, i.e., atom or molecule, stops moving.
It is also known as the absolute zero on the Kelvin scale.
° centigrade
° Celsius
Our Celsius to Fahrenheit converter will help you find the temperature from Celsius to Fahrenheit and vice versa.
|
Computer Science – Mininook
Category: Computer Science (page 1 of 2)
November 22, 2017 / Robbie / 0 Comments
Earlier today, I came up with this analogy for the internet and net neutrality:
Let’s say UPS owns I-95, FedEx owns I-64, and Joe’s Shipping owns I-295 and 288. With net neutrality, anyone could drive on any interstate, including UPS trucks on 64, without additional cost. There is some negotiation between companies for the interchanges between 64, 95, and 295.
Without net neutrality, it becomes much more problematic. FedEx could charge a fee for UPS trucks on 64, and vice versa. Joe’s Shipping is such a small company, that they may not be able to afford new charges to make deliveries using 64 or 95; therefore, they end up needing to use back roads which would affect delivery speed. They would lose to the companies who could take the faster routes and ultimately can’t compete with UPS and FedEx speeds. So, they fold and sell 295 and 288 to the other companies.
Perhaps you own a store along 64, but depend on a supplier from the DC area for your products. If the supplier is a small company, you or them would have to add the additional cost of shipping fees from both shipping companies to use both roads to get the delivery to you (i.e. UPS delivery fee plus the FedEx fee that UPS pays to use 64). But, if the supplier is already a big player, like Amazon or Walmart, they will likely have a second warehouse off 64, so they can still offer the lower FedEx-only shipping charges. Therefore, small suppliers can't compete with already established large corporations.
And, what would be even worse: what if UPS and FedEx owned their own supply companies? Then perhaps you buy their products and shipping, because they charge anyone else extra to use either of their roads.
And that’s where we are today. Comcast and Verizon own large swaths of the internet and its interconnection, and they produce content (tv, movies, websites, etc). AT&T, which also owns portions of the internet, are trying to acquire Time Warner, including their production companies.
So, that should be terrifying. Even if they are transparent about how much they charge, it’s still not neutral. There aren’t enough back-channels to help all content get everywhere.
Now, I know you might be thinking "well, I pay for Ting, Google Fiber, [insert your good company here] internet, so they won't play favorites with content." But, it's not just about them; the internet is a very deep and complex network. At its base is a backbone controlled by multiple different companies, some that you may have never heard of. Your web content may pass through a few different companies on top of the one that you actually pay for internet access. Without net neutrality, any one of them along the way has the ability to stop or slow your data or charge a fee.
There are a few things you can try to test out the internet for yourself and see what companies you'll need to deal with to do rather mundane things online. These are: traceroute and whois, and they're freely available in Terminal (MacOS), the Linux, and I believe Windows' Command Prompt.
Example Usage: My website
Let's take a look at getting to my site, robbiehott.com, from my in-law's house. From the terminal, we will execute the command traceroute robbiehott.com which will provide us with the following response:
traceroute to robbiehott.com (208.113.162.147), 100 hops max, 60 byte packets
1 gateway (10.0.0.1) 2.745 ms 3.198 ms 4.602 ms
3 ge-3-1-sr01.palmyra.va.richmond.comcast.net (68.86.127.69) 25.772 ms 25.879 ms 27.830 ms
5 ae-18-ar02.charlvilleco.va.richmond.comcast.net (68.86.173.213) 34.516 ms 34.505 ms 34.451 ms
6 be-21508-cr02.ashburn.va.ibone.comcast.net (68.86.91.53) 36.430 ms 20.432 ms 24.491 ms
7 hu-0-11-0-3-pe04.ashburn.va.ibone.comcast.net (68.86.88.78) 27.858 ms 27.145 ms 27.138 ms
9 207.88.14.164.ptr.us.xo.net (207.88.14.164) 33.410 ms 36.523 ms 38.295 ms
10 207.88.14.181.ptr.us.xo.net (207.88.14.181) 37.674 ms 36.776 ms 39.484 ms
12 ip-208-113-156-4.dreamhost.com (208.113.156.4) 26.594 ms 27.676 ms 28.477 ms
13 ip-208-113-156-14.dreamhost.com (208.113.156.14) 22.865 ms 23.073 ms 23.237 ms
This list shows all the steps between my laptop and my website. You'll notice it's backwards; that is, these are the step to my website. However, the website data will take roughly the same path back to my laptop. Let's unpack this a little:
Step 1 is the gateway, i.e. the router in the house that my laptop connects to on wifi. If your first entry starts with 10 or 192.168, then that is a local network and likely your router.
Steps 2-8 are all routers or computers at Comcast. Steps 3 and 5-7 specifically tell us that they are comcast.net, and we see my request going from Palmyra to Charlottesville to Ashburn.
Steps 9-11 are all routers or computers at MCI Communications (remember them? well, they're actually Verizon now). They don't advertise that fact here, but I'll show you how to get that information in a minute.
Steps 12-13 are computers at DreamHost, where my website resides.
How do I know that step 9 is Verizon? Our second command will give us that information: by typing whois 207.88.14.164 into our terminal, we get a response from a registrar that details the owner of that particular address. In this case, the important part is:
In an age without net neutrality, my site could be slowed down by either Comcast or Verizon, even though my website is hosted at DreamHost. You'll see images of "plans" that speculate paying extra for the "news websites" package or the "streaming video sites" package, but the actual case is more complicated than that. My in-laws could pay Comcast extra for the "personal websites" package, but that won't affect Verizon's handling of my website data.
This is a simple example because it is likely that DreamHost pays Verizon for internet access and my in-laws pay Comcast, but there are cases in which the internet traffic will pass through an intermediary company. I encourage you to go forth and test this out. You'll find companies like Fox News that pay a company called Akamai, which provides those "warehouses" from my analogy--places on your network that may use only your internet provider to deliver faster responses. You'll see companies like Level3 that you may have never heard of.
When you're done, and you're convinced something needs to be done, there are a few things you can do to try to influence what's happening at the FCC:
Call your representatives in Congress and ask them to support net neutrality. (Don't email, call. Someone has to take your call.)
Comment with the FCC. They are supposed to take these into account when making the decision.
March 21, 2014 / Robbie / 0 Comments
So, I always am using some command line shortcuts to do various tasks, and often have to look up the tricks every time I need to do something remotely fancy. Here are some of my most-used helpful hints:
To remove the leading spaces and tabs from each line of text on standard in (so use with a pipe for the input), this sed command will work well:
Reformatting XML/HTML files so that line returns inside tags are removed:
xmllint --format --noblanks infile.xml > outfile.xml
Boots: New Machine Learning Approaches to Modeling Dynamical Systems
February 10, 2014 / Robbie / 0 Comments
Large streams of data, mostly unlabeled.
Machine learning approach to fit models to data. How does it work? Take the raw data, hypothesize a model, use a learning algorithm to get the model parameters to match the data.
What makes a good machine learning algorithm?
Performance guarantees:
\theta \approx \theta^*
(statistical consistency and finite sample bounds)
Real-world sensors, data, resources (high-dimensional, large-scale, ...)
For many types of dynamical systems, learning is provably intractable. You must choose the right class of model, or else all bets are off!
Spectral Learning approaches to machine learning
|
Section 5.7: Double Integration in Polar Coordinates
When Cartesian coordinates are changed to polar coordinates, the points in the Cartesian plane remain where they are, and their "names" simply change. Thus, a Cartesian curve
y=f\left(x\right)
expressed in polar coordinates by
r \mathrm{sin}\left(\mathrm{θ}\right)=f\left(r \mathrm{cos}\left(\mathrm{θ}\right)\right)
does not change shape or location. Just its representation changes.
r=\sqrt{{x}^{2}+{y}^{2}},\mathrm{θ}=\mathrm{arctan}\left(y/x\right)
prescribes the new name for each Cartesian point. This is what Figure 5.7.1 shows. The concentric circles and polar rays of polar coordinates are superimposed on the Cartesian plane, so objects retain their location and shape. Just their descriptions change.
Figure 5.7.1 Polar coordinates
When the change to polar coordinates is viewed as a transformation (or mapping) from the Cartesian plane, objects change both position and shape. An object that was in the Cartesian plane, is moved over to the
r\mathrm{θ}
-plane. Figure 5.7.2 provides one way of visualizing the distortion experienced when the Cartesian plane is mapped onto the polar plane. The red horizontal and green vertical lines in the Cartesian plane become the curved red and green lines, respectively, in the polar plane.
Figure 5.7.3 shows a unit square in the Cartesian plane. Figure 5.7.4 shows the image of this square under the mapping
r=\sqrt{{x}^{2}+{y}^{2}},\mathrm{θ}=\mathrm{arctan}\left(y/x\right)
that takes the Cartesian square over to the
r\mathrm{θ}
Figure 5.7.3 A unit square in the Cartesian plane
p1:=plot([seq([sqrt(b^2+t^2),arctan(b,t),t=1..2],b=1..2)],color=[black,green]):
p2:=plot([seq([sqrt(t^2+a^2),arctan(t,a),t=1..2],a=1..2)],color=[blue,red]):
Figure 5.7.4 Polar-plane image of Cartesian square
Because of the distortion experienced by the square, the element of area in the Cartesian plane, namely,
\mathrm{dA}=\mathrm{dy} \mathrm{dx}
\mathrm{dx} \mathrm{dy}
\mathrm{dA}\prime =\left|\frac{∂\left(x,y\right)}{∂\left(r,\mathrm{θ}\right)}\right| \mathrm{dr} d\mathrm{θ}
\mathrm{dA}\prime =\left|\frac{∂\left(x,y\right)}{∂\left(r,\mathrm{\theta }\right)}\right| d\mathrm{θ} \mathrm{dr}
. The scaling factor is the absolute value of the Jacobian
\frac{∂\left(x,y\right)}{∂\left(r,\mathrm{\theta }\right)}
, which is best obtained from the inverse of the mapping formulas:
x=r \mathrm{cos}\left(\mathrm{θ}\right),y=r \mathrm{sin}\left(\mathrm{θ}\right)
. Thus, the scaling factor for area in polar coordinates is
\left|\frac{∂\left(x,y\right)}{∂\left(r,\mathrm{\theta }\right)}\right|
|\begin{array}{cc}\frac{∂x}{∂r}& \frac{∂x}{∂\mathrm{θ}}\\ \frac{∂y}{∂r}& \frac{∂y}{∂\mathrm{θ}}\end{array}|
\left|\begin{array}{cc}\mathrm{cos}\left(\mathrm{θ}\right)& -r \mathrm{sin}\left(\mathrm{θ}\right)\\ \mathrm{sin}\left(\mathrm{θ}\right)& r \mathrm{cos}\left(\mathrm{θ}\right)\end{array}\right|
r\left({\mathrm{cos}}^{2}\left(\mathrm{θ}\right)+{\mathrm{sin}}^{2}\left(\mathrm{θ}\right)\right)=r
Calculate the area of contained in one loop of the 4-leaf rose
r=\mathrm{cos}\left(2 \mathrm{θ}\right)
Calculate the area that is inside the cardioid
r=1+ \mathrm{cos}\left(\mathrm{θ}\right)
r=1
Calculate the area that is inside the circle
r=3 \mathrm{sin}\left(\mathrm{θ}\right)
r=1+\mathrm{sin}\left(\mathrm{θ}\right)
r=3 \mathrm{cos}\left(\mathrm{θ}\right)
r=1+\mathrm{cos}\left(\mathrm{θ}\right)
r=3 \mathrm{cos}\left(\mathrm{θ}\right)
r=2-\mathrm{cos}\left(\mathrm{θ}\right)
Calculate the area that is inside the large loop, but outside the small inner loop, of the limaçon
r=1/2+\mathrm{cos}\left(\mathrm{θ}\right)
r=4 \mathrm{sin}\left(\mathrm{θ}\right)
r=2
Calculate the area that is common to the circle
r=3 \mathrm{cos}\left(\mathrm{θ}\right)
and the cardioid
r=1+\mathrm{cos}\left(\mathrm{θ}\right)
Calculate the area that is inside the lemniscate
{r}^{2}=4 \mathrm{cos}\left(2 \mathrm{θ}\right)
r=\mathrm{cos}\left(\mathrm{θ}\right)
Calculate the area that is inside both the rose
r=\mathrm{sin}\left(2 \mathrm{θ}\right)
r=\mathrm{sin}\left(\mathrm{θ}\right)
r=1+\mathrm{cos}\left(\mathrm{θ}\right)
r=3 \mathrm{cos}\left(\mathrm{θ}\right)
Give a geometric construction showing that for polar coordinates,
\mathrm{dA}\prime =r \mathrm{dr} d\mathrm{θ}
r d\mathrm{θ} \mathrm{dr}
|
Weighted Variance Goal - MATLAB & Simulink
I/O Transfer Selection
Weighted Variance Goal limits the noise impact on the outputs of the frequency-weighted transfer function WL(s)H(s)WR(s), where H(s) is the transfer function between inputs and outputs you specify. WL(s) and WR(s) are weighting functions you can use to model a noise spectrum or emphasize particular frequency bands. Thus, you can use Weighted Variance Goal to tune the system response to stochastic inputs with a nonuniform spectrum such as colored noise or wind gusts.
Weighted Variance minimizes the response to noise at the inputs by minimizing the H2 norm of the frequency-weighted transfer function. The H2 norm measures:
The total energy of the impulse response, for deterministic inputs to the transfer function.
The square root of the output variance for a unit-variance white-noise input, for stochastic inputs to the transfer function. Equivalently, the H2 norm measures the root-mean-square of the output for such input.
In the Tuning tab of Control System Tuner, select New Goal > Frequency-weighted variance attenuation to create a Weighted Variance Goal.
When tuning control systems at the command line, use TuningGoal.WeightedVariance to specify a weighted gain goal.
Use this section of the dialog box to specify noise input locations and response outputs. Also specify any locations at which to open loops for evaluating the tuning goal.
Specify stochastic inputs
Select one or more signal locations in your model as noise inputs. To constrain a SISO response, select a single-valued input signal. For example, to constrain the gain from a location named 'u' to a location named 'y', click Add signal to list and select 'u'. To constrain the noise amplification of a MIMO response, select multiple signals or a vector-valued signal.
Specify stochastic outputs
Select one or more signal locations in your model as outputs for computing response to the noise inputs. To constrain a SISO response, select a single-valued output signal. For example, to constrain the gain from a location named 'u' to a location named 'y', click Add signal to list and select 'y'. To constrain the noise amplification of a MIMO response, select multiple signals or a vector-valued signal.
Compute output variance with the following loops open
Use the Left weight WL and Right weight WR text boxes to specify the frequency-weighting functions for the tuning goal.
WL provides the weighting for the output channels of H(s), and WR provides the weighting for the input channels.
You can specify scalar weights or frequency-dependent weighting. To specify a frequency-dependent weighting, use a numeric LTI model whose magnitude represents the desired weighting as a function of frequency. For example, enter tf(1,[1 0.01]) to specify a high weight at low frequencies that rolls off above 0.01 rad/s. To limit the response to a nonuniform noise distribution, enter as WR an LTI model whose magnitude represents the noise spectrum.
If the tuning goal constrains a MIMO transfer function, scalar or SISO weighting functions automatically expand to any input or output dimension. You can specify different weights for each channel by specifying MIMO weighting functions. The dimensions H(s) must be commensurate with the dimensions of WL and WR. For example, if the constrained transfer function has two inputs, you can specify diag([1 10]) as WR.
If you are tuning in discrete time, you can specify the weighting functions as discrete-time models with the same sampling time as you use for tuning. If you specify the weighting functions in continuous time, the tuning software discretizes them. Specifying the weighting functions in discrete time gives you more control over the weighting functions near the Nyquist frequency.
Use this section of the dialog box to specify additional characteristics of the weighted variance goal.
This tuning goal also imposes an implicit stability constraint on the weighted closed-loop transfer function between the specified inputs to outputs, evaluated with loops opened at the specified loop-opening locations. The dynamics affected by this implicit constraint are the stabilized dynamics for this tuning goal. The Minimum decay rate and Maximum natural frequency tuning options control the lower and upper bounds on these implicitly constrained dynamics. If the optimization fails to meet the default bounds, or if the default bounds conflict with other requirements, on the Tuning tab, use Tuning Options to change the defaults.
For Weighted Variance Goal, f(x) is given by:
f\left(x\right)={‖WL\text{\hspace{0.17em}}H\left(s,x\right)\text{\hspace{0.17em}}WR‖}_{2}.
H(s,x) is the closed-loop transfer function between the specified inputs and outputs, evaluated with parameter values x.
{‖\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}‖}_{2}
f\left(x\right)=\frac{1}{\sqrt{{T}_{s}}}{‖WL\left(z\right)\text{\hspace{0.17em}}H\left(z,x\right)\text{\hspace{0.17em}}WR\left(z\right)‖}_{2}.
Ts is the sample time of the discrete-time transfer function H(z,x).
|
Calculus1 Visualizations - Maple Help
Home : Support : Online Help : Applications and Example Worksheets : Calculus : Calculus1 Visualizations
Calculus 1 Visualization
The visualization component of the Calculus1 subpackage of the Student package consists of a collection of routines that can be used to both work with and visualize various concepts in an introductory single-variable calculus course. These worksheets demonstrates the basics of this functionality.
\mathrm{restart}
\mathrm{with}\textcolor[rgb]{1,0,0}{}\left(\mathrm{Student}\left[\mathrm{Calculus1}\right]\right)\textcolor[rgb]{1,0,0}{:}
\textcolor[rgb]{1,0,0}{}
The following worksheets show how the various routines work. In some cases, examples show to use these visualization routines in conjunction with the single-stepping Calculus1 routines.
Tangents, Inverses, and Sampling
Various Theorems about Derivatives
|
IsStellated - Maple Help
Home : Support : Online Help : Mathematics : Geometry : 3-D Euclidean : Polyhedra : IsStellated
test if the given polyhedron is of stellated form
IsStellated(ngon)
The routine IsStellated returns true if the given polyhedron is of stellated form; false otherwise.
The command with(geom3d,IsStellated) allows the use of the abbreviated form of this command.
\mathrm{with}\left(\mathrm{geom3d}\right):
\mathrm{stellate}\left(\mathrm{sico},\mathrm{icosahedron}\left(\mathrm{ico},\mathrm{point}\left(o,0,0,0\right),1\right),46\right)
\textcolor[rgb]{0,0,1}{\mathrm{sico}}
\mathrm{IsStellated}\left(\mathrm{sico}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{draw}\left(\mathrm{sico},\mathrm{style}=\mathrm{patch},\mathrm{lightmodel}=\mathrm{light4},\mathrm{shading}=\mathrm{XY},\mathrm{orientation}=[-142,-127],\mathrm{title}=\mathrm{`stellated icosahedron - 46`}\right)
|
Functions for which the remainder term goes to zero for all
x
in some interval about the expansion point are essentially given by an "infinite polynomial" or, in terms of Chapter 8, by an infinite series. Thus, a function with an appropriately behaved remainder has a power series representation, and this series is called a Taylor series.
When the expansion point is
x=0
, the power series representation of
is sometimes called a Maclaurin series, but some authors will simplify the terminology and use just the term "Taylor series" for all convergent power-series.
Thus, if a power series converges to
f\left(x\right)
, then that series is the Taylor series of
. But given an arbitrary function
f
, even one for which all derivatives exist, the expansion (called the formal Taylor expansion)
\underset{n=0}{\overset{\infty }{∑}}\frac{{f}^{\left(n\right)}\left(c\right)}{n!}{\left(x-c\right)}^{n}
which is formed by the "Taylor series recipe" may not converge to
f\left(x\right)
Once again, a convergent power series is the Taylor series for the limit function, but the formal Taylor expansion of
may not be the power-series representation of
f
may not have a power-series representation.
Section 8.5 deals with functions that indeed have a Taylor series representation. Determining which functions actually have a power-series (and hence a Taylor series) representation is no small matter. The most satisfying answers to this question are given for functions of a complex variable, that is, for functions
f\left(z\right)
z=x+i y
. For such functions, if one derivative exists in a neighborhood, all derivatives exist and the Taylor expansion actually represents the function. But for functions of the real variable
x
, the situation is not so sanguine. Real functions can have just a finite number of derivatives and no more. Moreover, even functions with all derivatives may not have a Taylor series that converges back to the function, as is the case with the example function in Example 8.5.1.
{\stackrel{^}{R}}_{n}\left(x\right)=\frac{{f}^{\left(n+1\right)}\left(x\right)}{\left(n+1\right)!}{\left(x-a\right)}^{n+1}
(not just
{R}_{n}\left(x\right)=\frac{{f}^{\left(n+1\right)}\left(c\right)}{\left(n+1\right)!}{\left(x-a\right)}^{n+1}
from Theorem 3.3.1) goes to zero as
n→\infty
, then the formal Taylor series of
f\left(x\right)
does indeed converge to, i.e., represent,
f\left(x\right)
. But it is no easy matter to show this for an arbitrary function, essentially because of the need to have either a representation of
{f}^{\left(n\right)}\left(x\right)
or an estimate of how these derivatives behave as
n→\infty
. In the examples below, this is done for a few of the elementary functions, but in general, determining whether or not a function has a Taylor series representation relies heavily on the theory of complex variables.
f\left(x\right)={\left(1+x\right)}^{c}
has a Maclaurin series for any real number
c
\left|x\right|<1
. The series is a generalization of the Binomial expansion where
c=k
, for positive integers
k
c=k
, a positive integer, the expansion is of the form
{\left(1+x\right)}^{k}=\underset{n=0}{\overset{k}{∑}}\left(\genfrac{}{}{0}{}{n}{k}\right){x}^{n}
\left(\genfrac{}{}{0}{}{n}{k}\right)=\frac{n!}{k! \left(n-k\right)!}
c
, the expansion is then
{\left(1+x\right)}^{c}=\underset{n=0}{\overset{\infty }{∑}}\left(\genfrac{}{}{0}{}{c}{n}\right){x}^{n}
, where now
\left(\genfrac{}{}{0}{}{c}{n}\right)=\frac{c\cdot \left(c-1\right)\cdot ⋯\cdot \left(c-\left(n-1\right)\right)}{n!}
\left(\genfrac{}{}{0}{}{c}{0}\right)=1
Table 8.5.1 lists the first few values of
\left(\genfrac{}{}{0}{}{c}{n}\right)
\left(\genfrac{}{}{0}{}{c}{0}\right)\equiv 1
\left(\genfrac{}{}{0}{}{c}{3}\right)=\frac{c\left(c-1\right)\left(c-2\right)}{3!}
\left(\genfrac{}{}{0}{}{c}{1}\right)=c
\left(\genfrac{}{}{0}{}{c}{4}\right)=\frac{c\left(c-1\right)\left(c-2\right)\left(c-3\right)}{4!}
\left(\genfrac{}{}{0}{}{c}{2}\right)=\frac{c \left(c-1\right)}{2!}
\left(\genfrac{}{}{0}{}{c}{5}\right)=\frac{c\left(c-1\right)\left(c-2\right)\left(c-3\right)\left(c-4\right)}{5!}
Table 8.5.1 First few values of
\left(\genfrac{}{}{0}{}{c}{n}\right)
The binomial coefficient in Maple is generalized to include the case where
c
is not an integer. A template for this function is available in the Expression palette, and this template is equivalent to invoking Maple's binomial command.
Show that the formal Taylor expansion of
f\left(x\right)=\left\{\begin{array}{cc}{\mathrm{e}}^{-1/{x}^{2}}& x≠0\\ 0& x=0\end{array}\right\
is identically zero, so that this expansion does not represent
f\left(x\right)
. Hint: Show that
{f}^{\left(n\right)}\left(0\right)=0
n=1,2,\dots
In Examples 8.5.2-7, show that
{\stackrel{^}{R}}_{n+1}\left(x\right)
for the given function
f\left(x\right)
n→\infty
, establishing that
f
has a Maclaurin series. Find the terms of that series.
f\left(x\right)={e}^{x}
f\left(x\right)=\mathrm{cos}\left(x\right)
f\left(x\right)=\mathrm{cosh}\left(x\right)
f\left(x\right)=\mathrm{sin}\left(x\right)
f\left(x\right)=\mathrm{sinh}\left(x\right)
f\left(x\right)=\mathrm{ln}\left(1+x\right)
\sqrt[3]{8+x}
{x}^{2}/\sqrt{1-{x}^{3}}
{\left(\frac{x}{1-x}\right)}^{5}
Expand the integrand in
{∫}_{0}^{1}\mathrm{sin}\left({x}^{2}\right) \mathit{ⅆ}x
and integrate termwise to obtain an estimate of the integral guaranteed correct to three decimal places. Compare to the value Maple provides.
{∫}_{0}^{1}{J}_{0}\left(x\right) \mathit{ⅆ}x
Obtain the Maclaurin series for
{e}^{3 {x}^{2}}
by manipulating the series for
{e}^{x}
{x}^{2}\mathrm{cos}\left(2 x\right)
\mathrm{cos}\left(x\right)
x/\left(1-{x}^{2}\right)
from an appropriate geometric series.
\frac{1-{x}^{2}}{1+{x}^{2}}
from appropriate geometric series.
{p}_{1}\left(x\right)
{p}_{2}\left(x\right)
, degree 8 Maclaurin polynomials for
f\left(x\right)={e}^{x}
g\left(x\right)=\mathrm{ln}\left(1+x\right)
Form the product
{p}_{1}\cdot {p}_{2}
Obtain the degree 11 Maclaurin polynomial for the product
f\left(x\right)\cdot g\left(x\right)
Obtain the first 10 coefficients
{c}_{n}
formed from the Cauchy product of the coefficients of
{p}_{1}
{p}_{2}
Observe that the coefficients formed from the Cauchy product (Theorem 8.2.4) are always correct, but those formed from the product of the polynomials are correct only to a limited degree in
x
|
Building the PIXELCUBE (Pixelcube Part 1) - coral
Building the PIXELCUBE (Part 1)
I’ve spent the last few weeks on building a cube made out of a bunch of PVC vertices and acrylic edges but with no face, where the edges are filled with over 60 individually addressable LED’s per meter. It’s not done yet, but i felt that sharing an update on why I’m doing this and why it turned out to be harder than i thought would be nice.
Idea & Background
To be honest here, I don’t know how this really started. There is a possibility i was watching something, together with an influx of other ideas that led me to the conclusion that i just had to build a large light fixture. I talked with my friends from Sweden about building something interesting for ANDERSTORPSFESTIVALEN and i guess this is the idea i subconsciously came up with. This is in a way very representative for how my mind works, once it is clear what I want the idea is the result is almost finished inside my head, the end result is clear to me early on but mapping this onto a process of developing is the hard part.
This cube would be a huge physical light fixture that behaved like an LED wall but without the somewhat boring LED wall form factor. If you’ve ever worked with LED walls you know that after a while the 2-dimensional plate tends to become very flat, even if you build intricate shapes with mapping, as light is only projected at a 180 degree angle. There is no volume to it, meaning that perspective is hard to achieve. Since a LED wall maps to a screen, content tend to be 2-dimensional clips, pre-rendered and then applied in a manner to sync to the music. While there is some available software to do generative works, most of this caps out at 60 FPS + has a visible latency due to the processing needed to generate -> feed over HDMI -> go into processor -> get fed to wall.
I wanted to build something that’s different to this, a volumetric cube with the same LED’s that’s in a good LED wall. Having high dynamic range, fast dimming and aggressive driving speed allows for driving speeds over 240 FPS, meaning you’re able to do aggressive PoV (persistence of vision) effects and stroboscopic patterns. All of this while not projecting 2D content onto a 3D fixture, rather building something that’s inherently volumetric
Here’s an early concept around how the cube could look:
Building this monster
Here’s the thing: I had no idea of how to construct this cube when starting. Absolutely zero knowledge about building something physical except for electronics. So in order to prototype how this would even look, I glued together an old Amazon cardboard box and hanged in my kitchen in order to make it easier to visualize what’s needed in order to build it.
I went through a couple of iterations on building materials and eventually decided to buy the acrylic rods needed and just commit to experimenting instead of trying to solve this on paper. After a long internal debate, i ended up deciding the cube would be 50 inches per side (+ some for the fittings), since 50 inches is exactly 127 cm (very pleasing). I ordered the acrylic rods and once they arrived i immediately realized i had a couple of issues.
Turns out that these acrylic tubes are measured in outside diameter, whereas PVC pipe fittings are measured in inside diameter with a standard thickness, meaning my 1/2 rods had no great counterpart. I ended up trying to solve this problem for a fairly long time, trying different approaches with heat shrink tubing, electrical tape and some other experiments in order to adapt them with no good solution.
No idea of how to make the LED strip “float” inside the acrylic tube. I thought i could mount it on a L angle aluminium bar, but how would i fixate the bar with such tight clearances inside the fittings?
No good way of actually fixating the rods inside the fittings. Since the cube is to be suspended at 45/45 degrees rotation, it needs to have a strong rigidity in the edges, something that friction alone can’t provide. I experimented with a bunch of different solutions to this problem with no really good outcome.
With all these problem stacking up i started to get a bit negative about this project. It was hard to solve, at least if i wanted it to be close to what i imagined it and not take shortcuts in regards to the build quality. I did not want to have multiple support wires inside the cube and I could not find a good way of actually mounting it. In times like these it turns out that an outside perspective is key. I had two good friends visiting (Love & Pajlada) and they came over to my house in order to hang out and look at the cube. Something Love immediately suggested is that i should 3D print the parts i needed, both to hold the fittings and to hold the L angle. I knew basically nothing about 3D printing except from a small part that I sent away for 3D print on a previous project. This did not stop me however, i purchased a Monoprice MP Select Mini 3D Printer V2 and started printing some test parts I modelled. Turns out this is exactly what i needed.
This just shows how important it is to have an outside perspective when getting stuck in projects. Having Love vet my ideas and add his perspective basically made this project possible. It’s a possibility that i would have come to this conclusion eventually but never as fast as i did here. The fittings are now tight and I’ve modelled in holders for bolts so i can use it as a mounting plate for the screws. On top of this the parts also hold the L angle which has the LED strip mounted on it.
After experimenting together over the weekend, we put together the cube temporarily, just feeding the strips through the rods and taping it with electrical tape in order to hold it’s form. After the first power-up it was obvious that this was i was striving after from the start:
LED & Processing hardware
The vision puts a couple of different requirements in place, namely a driver that can generate patterns at 240 FPS and do music analysis and a LED strip that’s powerful enough to deliver on this. With these restrictions in place, i had to start looking into what could be used to build this piece. For LED’s it turned out to be rather obvious, the chip / strip that has what i need is the APA102, also referred to as the “superled”. The APA102 has a lot of features in contrast to the WS2812 with the primary one being a separate Data / Clock line, meaning the timing of the data is less sensitive. You also get a much higher dimming frequency + a global intensity allowing for LUT lookup to get a higher dynamic range. There’s really no negative part about this LED except for the price.
In order to drive these LED’s i need a platform that’s fast enough to do super granular FFT analysis, has audio input, networking on-board and preferably some groundwork already done. There’s a bunch of directions that could be taken here:
Since i wanted this to be a no hassle setup, the TouchDesigner setup goes away immediately. On top of this, TouchDesigner has a node based layout and for some reason this never plays well with my way of thinking. I’ve tried these node based UI’s multiple times and never felt that they’ve been powerful compared to just writing it.
An Arduino is just too slow.
Raspberry Pi is what i ended up prototyping all this on. It’s a great device but sadly it just doesn’t hold up when you start trying to push multiple volumetric patterns at 200 FPS.
PixelBlaze (ESP8266 & ESP32)
PixelBlaze is a great alternative. Ben Hecke has built a very neat small controller on the ESP8266 platform (with a ESP32 upgrade in the works) that’s the easiest to setup and get started with. It basically runs itself however i felt a tad limited by the CPU speed and general pattern blending. It was hard to re-use components of patterns or mix multiple patterns.
SnickerDoodle (ARM/FPGA)
The SnickerDoodle is a good alternative but coding FPGA is a nightmare. On top of this the actual ARM core isn’t that fast, it runs a 866 MHz Dual-Core ARM Cortex-A9 which is slower than the Raspberry Pi so a lot of functionality would have to be shifted to the FPGA, even more headache. It’s also not really rapid prototyping working with FPGA’s. I have one of these sitting around but this is a “in case of emergency” solution.
With all this considered, this is the actual solution:
Intel NUC with a FT232H chip attached
This is the route that I will end up taking with the finished cube. It gives me enough CPU speed in a small package, less of a headache in terms of setup and can be mounted inside enclosures. The FT232H chip speaks SPI at up to 30MHz, which is perfect since i only need 20. However, this requires me to write most of the software to drive it.
How does this cube even get powered? After looking at the datasheet of the LED’s it turns out they can pull up to 50mA per LED. With 75 LED’s per rod and 12 rods, that’s about 900 LED’s to drive at 5V, not exactly something you can do with a regular USB port.
(75: pixels: x: 12: rods): x: 50: mA: ==: <span class="small-caps">45 000</span>: mA
The PSU to drive this needs to be at minimum 45A at 5V in order to drive the LED’s at peak load. Mean Well has exactly what i need. These PSU’s are commonly sighted within LED walls so I felt safe with this option. Still, pushing 45A at 5V requires a thicker cable, since the drop over distance will be substantial. Now this is a question of how the cube gets wired, since the APA102 LED’s also needs to be connected in sequence. In order to figure this out, i drew it out in Sketchup in order to visualize the connection order. Once i drew it i realized this is common graph theory issue, how can you walk all the edges in the shortest amount of trips. (entry -> a - ab - bc - cd - da - ae - ef - fg - gh - he - ef (through skip) - fb - bc (through skip) - cg - gh (through skip) - hd )
When building the rods I add an extra signal wire and 5V carrier on the back side of the L angle, that way i can create a common voltage rail for the 5V stuff and have backfeed for signal in order to wire it up as seen in the Sketchup.
In order to connect all this, i’ll use regular Molex connectors in the ends with a custom split one for 5V and serial ones for data in order to wire it all.
Software & Design philosophy
So as mentioned in the beginning of this post, in order to break free from the 2D video mapped onto 3D textures, the cube has to support volumetric patterns. Think of volumetric patterns as resolving pixels inside an XYZ space, you could for example use a 3D model have the intersecting LED’s light up on start. The software should be able to seamlessly mix between patterns and needs input from multiple sources, such as Websocket, OSC, Audio and Time. After discussing the structure with Love & Pajlada, the processing pipeline ended up looking like this:
This is the layout of the intended processing pipeline. The pipeline is almost like a node based editor, in the sense that it uses multiple generators producing floats for every single pixel between 0.0 and 1.0, the operators you chain together with various operations is what creates the effect. You could for example use a stroboscope effect, chain it to a colorlookup that changes the RGB output to red and in that way create a red strobe, a hue shift or colorizer becomes powerful especially when using pre-defined palettes. The generator & color lookup can also make use of the FFT data and the beat in order to generate dynamic, music-driven effects with high resolution.
It helps to start with the layout of the flow, since it makes it easy to figure out what software has to be written and what tools to use. In this case i decided on Go for a couple of different reasons.
I know Go. Use what you know, the intent here is not to build distributed software, but to make this cube look good.
Go has great SPI support. Go supports the SPI chip on the Raspberry Pi, the FT232H chip and the APA102 led through periph.io. This makes it easy to prototype and move between different hardware stacks. I can do the prototyping on the Raspberry Pi, developing on my Mac and effortlessly move this to a Intel based Linux box without having to deal with compiling for different archs.
Go is fast enough. Compared to Node.JS and the other languages i evaluated, Go hits the sweet spot between rapid iteration and execution speed, which is needed for achieving a solid 200 FPS.
There are bindings for aubio and PortAudio. Getting audio in and performing analysis on it turns out to be very easy with good bindings available for these two great libraries. Aubio itself provides you with tools to analyze and make sense out of a PCM audio stream, detecting interesting data like tempo, onset, beat and loudness. On top of that you also get a good FFT implementation. PortAudio makes it easy to acquire audio across all platforms, lending into the goal of portability between platforms.
I’ve been working on this since late October and intend to finish this before June 30th 2019 in order to bring it to ANDERSTORPSFESTIVALEN. As you can see, at the time of writing this post the cube is not yet done. I will post updates here on the blog once more progress has been made. I will end this post with a video that I shot when we tested the cube:
Building a weird lamp (Pixelcube Part 2) I needed a way to test my setup and have something small visible when i work on the Pixelcube in order to see what sort of mounting will work for
|
Miller Indices Calculator - Cubic crystals
Miller Indices Calculator
How to calculate Miller indices for planes
Example: Using the Miller indices calculator
The Miller indices calculator determines the interplanar distance for cubic crystal systems. It was introduced in 1839 by a British mineralogist named Prof. William Hallowes Miller. Miller indices are part of X-ray crystallography, which are useful to study the optical properties of atoms and molecules using their crystal structures.
In addition to the optical properties, it is also helpful to observe dislocations during the plastic deformations in an atomic structure, i.e., the strain at the atomic level. The knowledge is crucial in nanofabrication or machining nano wafers. There are different kinds of crystal lattices; this calculator focuses on cubic structures in particular.
The topic is a fundamental concept in material sciences that has implications in various fields. This article will help you understand the definition of Miller indices and steps on how to calculate Miller indices.
Miller indices are the lynchpin in a notation system that denotes the family of lattice planes for a given Bravais lattice. The notation is written as
hkl
but has several variations based on what it represents. For instance:
(h,k,l)
[hkl]
\text{<}hkl\text{>}
(hkl)
{hkl}
Family of planes
The Miller indices represent the reciprocal of the intercepts a plane makes with each axis. In a 3-dimensional plane, consider a crystal face parallel to the XY plane. The intercepts for the crystal face with the axes are
(\infin, \infin, 1)
. Therefore, the Miller indices for the face are
(1/\infin, 1/\infin, 1) = (100)
If the intercepts are fractions, the indices are converted to integers by multiplying by their common denominator. The negative intercepts are represented with a bar on the index.
Using the Miller indices and the lattice constant, you can determine the interplanar distance for the unit cell. Mathematically, this is:
\quad d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}
h,k,l
– Miller indices;
a
– Lattice constant; and
d_{hkl}
– Interplanar distance.
To calculate Miller indices using this calculator:
Locate the x, y, and z axes intercept for the crystal face.
Find the reciprocal of the intercept to obtain the Miller indices.
Enter the lattice constant for the cubic cell.
Fill in the Miller indices,
(hkl)
The Miller indices calculator will return the interplanar distance for the cubic cell.
The list of compounds and elements contains the lattice constant you can directly use upon selecting.
Find the interplanar distance for a cubic unit cell having Miller indices (201) and lattice constant
2\r{\text{ A}}
Enter the lattice constant for the cubic cell as
2\r{\text{ A}}
Fill in the Miller indices as 2, 0 and 1, respectively.
The interplanar distance,
d_{hkl}
\qquad \scriptsize d_{hkl} = \frac{2}{\sqrt{2^2 + 0^2 + 1^2}} = 0.8944\r{\text{ A}}
The Miller indices are the notation to depict a plane or a family of planes in a two or three-dimensional cartesian coordinate space. You can also represent a point using the Miller indices. The parenthesis enclosure explains what the indices denote. For instance, if the Miller indices are enclosed in ( ), e.g. (1,0,0) separated by commas, it represents a point. The index inside ( ) without commas represent a plane such as (100) or (001).
To determine the Miller indices:
Find the x, y, and z intercepts for the face.
Calculate the reciprocal of the intercepts to obtain the Miller indices.
In the case of the Miller indices being fractions, multiply the index with the common denominator to convert them into integers.
What are different types of crystal systems?
There are seven different types of crystal systems which are triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. Each system has a different set of symmetry properties, such as the cubic system having four threefold axes of rotation. Some of the cubic crystal system elements are nickel, copper, sodium chloride, silver, and gold.
What are some applications of Miller indices?
The applications of Miller indices include the field of X-ray crystallography, the study of material deformation in the plastic region considering the strains at the atomic level otherwise known as dislocations. Other areas include diffraction, surface tension, nanofabrication, and machining.
Miller index, h
Miller index, k
Miller index, l
Interplanar distance, d
The molality calculator helps you find the molal concentration, given number of moles of solute and mass of the solvent.
With this vapor pressure of water calculator, you can find the vapor pressure at a particular temperature according to five different formulas. This calculator does not only work for standard 0-100 °C range but also for temperatures above 100 °C and below the freezing point. Awesome!
|
Lemma 37.14.5 (08KU)—The Stacks project
Comment #3176 by Matthieu Romagny on February 05, 2018 at 11:58
In Lemma Tag 08KU, item (2)(b), isn't
\mathcal{F}'
flat over
X'
rather than over
X
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 08KU. 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 08KU, in case you are confused.
|
Two-Photon Absorption Calculator
What is two-photon absorption?
How to calculate two-photon excitation rate – Two-photon absorption equation
How to use the two-photon absorption calculator – TPA calculation example
Omni's two-photon absorption calculator allows you to determine the number of two-photon excitations per molecule for a given laser source.
Continue reading this article to know what two-photon absorption (TPA) is and how to calculate the excitation rate using the two-photon absorption equation. You will also find an example of TPA calculation.
Two-photon absorption (TPA) is a process where an atom or a molecule absorbs two photons simultaneously. The energy of the two photons can be identical or different. This absorption results in the excitation of the atom or molecule from one energy state (usually the ground state) to a higher-energy state via an induced virtual state (shown by the dashed horizontal line in figure 1).
As shown in figure 1, the difference in the energy of the two states is equal to the sum of the energy of the two absorbed photons.
\quad E_n - E_0 = 2\frac{hc}{\lambda}
The phenomenon was first predicted by Maria Goppert-Mayer in 1931, and Kaiser and Garret experimentally verified it in 1963.
Fig 1: Energy level diagram of a two-photon absorption process.
To calculate the number of two-photon excitations per molecule
N
, we will use the formula:
\quad N = \frac{1}{2} \cdot \delta \cdot \phi^2 \cdot \tau
\delta
– Two-photon absorption (TPA) cross-section measured in GM. One Gm is
10^{-50}\ \rm cm^4 \cdot s \cdot ph^{-1}
\tau
– Exposure time; and
\phi
– Photon flux at the center of a Gaussian beam.
💡 Do you know the two-photon absorption cross-section is expressed in units of GM to honor Maria Goppert-Mayer?
Since the energy carried by a photon is
h\nu
, the number of photons crossing a unit area per unit time (i.e., the photon flux) is related to the intensity of the beam
I
by the expression:
\quad \phi = \frac{I}{h\nu} = \frac{I \lambda}{hc}
The intensity of the laser beam with power
P
and the beam radius
w
\quad I = \frac{2P}{\pi w^2}
Figure 2: Laser power two-photon absorption cross-section.
where the beam radius is related to the full width at half maximum (FWHM) as (see figure 2):
\quad w = \frac{\rm FWHM}{\sqrt{2\ \ln 2}}
In the next section, we will use the two-photon absorption calculator to calculate the number of excitations per molecule for a given laser pulse.
Let's calculate the photon flux and number of excitations per molecule when a sample is irradiated for
1\ \rm s
10\ \rm W
laser source of wavelength
840\ \rm nm
. The two-photon absorption cross-section is
210\ \rm GM
, and the FWHM of the focussed laser beam is
20 \ \rm\mu m
Enter the two-photon absorption cross-section in GM (
210\ \rm GM
Type the power of the laser source (
10\ \rm W
), the wavelength of the beam (
840\ \rm nm
), and the FWHM of the focussed beam (
20 \ \mu \rm m
Enter the exposure time (
1\ \rm s
The two-photon absorption calculator will give you the photon flux at the center of the beam (
9.33 \times 10^{24}\ \rm ph/(cm^2.s)
) and the number of excitations per molecule (
91.4
What does photon absorption mean?
Photon absorption is a process in which an atomic electron absorbs the energy of an incident photon. If the photon's energy is higher than the binding energy of the electron, the electron is ejected from the atom. Otherwise, the electron gets excited to a higher energy state within the atom.
Can a free electron absorb a photon?
No, a free electron cannot absorb or emit a photon. The conditions for conservation of energy and momentum are not satisfied in the process if the electron is free. Hence, only electrons that are bound to atoms can absorb photons.
How do you measure two-photon absorption cross-section?
Some of the techniques used to measure two-photon absorption cross-section are:
Two-photon excited fluorescence (TPEF) spectroscopy;
Z-scan approach;
Mass-sedimentation approach; and
Nonlinear transmission method.
What are the applications of two-photon absorption?
The two-photon absorption technique has several applications, including:
The study of novel materials and investigate the relationship between their molecular structure and electronic and optical properties.
Performing high-resolution imaging of live cell and tissue samples.
Cross-section (δ)
Focus size FWHM
Exposure time (τ)
Photon flux (ϕ)
x10²⁴
ph/(cm²•s)
Excitations per molecule (N)
|
What is noise figure and noise factor?
How to calculate the noise figure: the noise figure formula
Applications of noise figure
Noise figure is a measure of any device's contribution to the overall noise of the system in which the device is present, and our noise figure calculator calculates that value. By using this information, you can identify how much noise is being produced in that system.
In the below article we explain the noise factor and noise figure definitions, which are similar terms with a slight difference in the way they are calculated. We use the latter, for example, in the noise figure of cascaded amplifier formula to estimate the degradation of signal-to-noise ratio (SNR) in such a system.
Continue reading to not only learn more about the formula of noise figure, but also the practical applications of noise figures in various fields of life.
The noise figure is the common logarithm of the ratio of input SNR to output SNR, where SNR refers to the signal-to-noise ratio. This ratio is a measure of the strength of the desired signal in comparison to the acceptable level of background noise. The noise factor is a similar idea but does not use logarithms.
We can categorize any unwanted disturbance that impacts the quality of the signal as noise, and it can disrupt data transfer of texts, graphics, audio, and video. So, studying the noise factor of a circuit or system is significant if you wish to improving the system's performance.
Noise factor and noise figure definitions
The noise factor and figure measure the degradation of the signal-to-noise ratio.
When we calculate the value using a linear equation, it is noise factor, but it is a noise figure when we use a common logarithm.
When we connect multiple devices in a consecutive or cascaded manner, the total noise figure of such a system is called a cascaded noise figure.
The formula for noise figure of cascaded amplifier is:
Noise_{total} = 10\log_{10} \lparen n_1 + \sum_M^{i=2}\frac{n_i-1}{\prod_{i-1}^{j=1} g_j}\rparen
Gain_{total} = \sum_M^{i=1} g_i
Noise_{total}
- Noise figure of the cascade;
Gain_{total}
- Total gain of the cascade;
n_i
- Noise figure of the
i
component;
g_i
- Gain of the
i
component; and
M
- Number of components.
The noise figure calculator allows you to determine the value of the noise figure in multiple ways, depending on the scenario in front of you. There are 4 types of calculations provided in the calculator, each with its own formula that we need to apply to determine how to calculate the noise figure based on your inputs.
The methods of calculations are:
The signal-to-noise ratios;
The signal-to-noise ratios in decibels (dB);
Convert from noise factor to noise figure; and
Cascaded noise figure.
Once you have selected how you want to determine the noise figure value, you may input the required variables.
For Methods 1 and 2, you should enter:
The signal-to-noise ratio at the input (
SNRi
The signal-to-noise ratio at the output (
SNRo
The result will be noise figure in dB.
Method 3 is useful when you do not have the SNR and still need to determine the noise figure of a device. You may use the conversion of noise factor to noise figure. For this, you need to input the value of the noise factor, and the result will be the noise figure in decibels.
Method 4 is for when you have multiple devices connected and have each device's noise and gain value. You have the choice to enter values for up to 10 devices.
The result will be in terms of total noise and total gain.
For instance, let's consider Method 1. If your
SNRi
SNRo
is 7, then the noise figure
NF
NF = 0.5799 \text{ dB}
The noise figure gives a value that specifies the noise performance of any device.
A simple circuit which we can calculate noise figure of.
The noise figure values are essential for wireless networking and RF circuits. RF circuits are electronic circuits that transfer or receive radio signals and operate at a particular radio frequency. Our noise figure calculator calculates the noise figure and noise factor values of a system based on the signal-to-noise ratio of that system.
The noise figure formula in dB(decibels) is:
NF = 10 \times \log_{10}(SNRi/SNRo)
NF
- Noise figure in decibels (dB);
SNRi
- Signal-to-noise ratio at the input; and
SNRo
- Signal-to-noise ratio at the output.
So, let's suppose you have
SNRi
SNRo
SNRi
SNRo
Then, you determine the
\log
at base 10 of the ratio obtained in the first step.
Next, you multiply it by 10.
The result is a noise figure value of 3.9794 dB.
You can find some of the applications of noise figures below. They are primarily used to study the noise and performance of a device:
The noise figure gives an estimated difference in the noise output of a receiver in comparison to the noise output of a model receiver, keeping the overall gain, bandwidth, and temperature constant.
We usually use the noise figure in the circuit design of radio receivers to understand the noise performance.
The noise figure of a device tells us about the noise contribution of that device.
How can I calculate the noise figure?
You can follow the given steps to calculate the noise figure of a device:
Calculate the signal-to-noise ratio at the input terminal (SNRi) of the device.
Calculate the signal-to-noise ratio at the output terminal (SNRo) of the device.
Divide the SNRi by the SNRo.
Calculate the base 10 logarithm of the ratio obtained in Step 3.
Finally, multiply it by 10.
The resultant value is the noise figure. The noise figure formula, NF, in dB is the following:
NF = 10 × log₁₀(SNRi/SNRo)
What is a good noise figure?
Ideally, the noise figure should be 0 dB, but that is not achievable. Even the highest performance systems have some noise figure. Still, the smaller the noise figure of a system, the better it is.
The noise figure measures the signal-to-noise degradation of the system under consideration.
What is the noise figure if the SNRs are 40 and 35 dB?
The noise figure is 5 dB if the signal-to-noise ratio at input is 40 dB and signal-to-noise ratio at output is 35 dB.
We achieve the above result only if the SNR values are expressed in decibels. Instead, if the values were expressed in unitless ratios, then the resultant noise figure would be 0.5799.
Can noise figure be negative?
No, the noise figure value cannot be negative.
The signal-to-noise ratio at the input is always greater than the signal-to-noise ratio at the output, which means that the noise factor should always be greater than one.
When you take the logarithm of the noise factor, it determines the noise figure, and the log of a value greater than one is a positive number.
Find noise figure using
Noise (stage 1)
Gain (stage 1)
|
p\left(\textcolor[rgb]{0,0.8,1}{ }\left[\left[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\right]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{ }\left[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\right]\right] \right)
p(\textcolor[rgb]{0.501960784313725,0,0}{ }\left[\begin{array}{cc}\textcolor[rgb]{0.501960784313725,0,0}{1}& \textcolor[rgb]{0.501960784313725,0,0}{2}\\ \textcolor[rgb]{0.501960784313725,0,0}{3}& \textcolor[rgb]{0.501960784313725,0,0}{4}\end{array}\right])
\mathrm{LinearAlgebra}:-\mathrm{Rank}\left( \mathrm{Array}\left( \left[ \left[ 1,2 \right], \left[3,4\right] \right] \right) \right);
\textcolor[rgb]{0,0,1}{2}
\mathrm{LinearAlgebra}:-\mathrm{Determinant}\left( \left[ \left[ 6, 7 \right],\left[8,9\right] \right]\right);
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}
p\left( \mathrm{Array}\left( \left[ \left[ 1, 2 \right], \left[3, 4\right] \right] \right) \right);
\textcolor[rgb]{0,0,1}{\mathrm{Matrix}}
p\left( \mathrm{Vector}\left[\mathrm{row}\right]\left( \left[ 5, 6,7 \right] \right) \right);
\textcolor[rgb]{0,0,1}{\mathrm{Matrix}}
p\left( \left[ \left[ 1,2, 3\right], \left[ 4, 5, 6 \right] \right] \right);
\textcolor[rgb]{0,0,1}{\mathrm{Matrix}}
Adding a ~ prefix to the m::~Matrix parameter specification in the example above tells Maple it will accept something similar to a Matrix. You can now pass in a Vector. Behind the scenes an
n
A ≔ \mathrm{Array}\left(2..3,6..7\right):
\mathrm{ArrayDims}\left(A\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}{6}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{7}
p\left(A\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}{2}
p\left( 〈1,2;3,4〉 \right);
\textcolor[rgb]{0,0,1}{1}
\mathrm{Frac}\left(1.5\right);
\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}
\mathrm{Frac}\left(\frac{3}{2}\right);
\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}
p\left("a string"\right);
\textcolor[rgb]{0,0,1}{"a string"}
p\left(\mathrm{`a name`}\right);
\textcolor[rgb]{0,0,1}{"a name"}
p\left(\mathrm{expect}+\mathrm{an}+\mathrm{Error}\right);
|
Section 5-9 - Hydrostatic Force - Maple Help
Home : Support : Online Help : Study Guides : Calculus : Chapter 5 - Applications of Integration : Section 5-9 - Hydrostatic Force
Section 5.9: Hydrostatic Force
Pressure is defined as force per unit area. Given a pressure and an area over which is applies, the total force exerted is then the product of the pressure and the area. Such a force is called hydrostatic when the pressure is applied by a fluid such as water.
A
is the area of a plane surface parallel to the surface of a fluid weighing
\mathrm{δ}
lbs/
{\mathrm{ft}}^{3}
, and at depth
d
in the fluid, the total weigh of the fluid "sitting" on the submerged surface is
F=A d \mathrm{δ}
. The pressure on this surface is
P=F/A=d \mathrm{δ}
{\mathrm{f}\mathrm{t}}^{2}
It has been verified empirically that at depth
d
, the pressure is the same for any direction. This observation makes it possible to calculate the total force on submerged surfaces that are not parallel to the surface of the fluid. In particular, the total hydrostatic force
F
on a submerged surface that is perpendicular to the surface of a fluid is
F=∫P \mathit{ⅆ}A
\mathrm{dA}
, the element of area, is taken as a narrow horizontal strip along which the pressure is constant.
It can be shown that the total hydrostatic force on such a vertical surface is the product of the area of the surface with the pressure at the centroid of the surface.
Under heavy seas, the top of a cruise ship's circular porthole one foot in diameter is submerged under 20 ft of water. If sea water weighs
62.5 \mathrm{lbs}
{\mathrm{ft}}^{3}
, find the total force on the porthole.
The water behind a trapezoidal dam (dimensions 25 × 15 × 15 × 15 in feet, longest edge uppermost), is to the top edge. Find the total hydrostatic force on this structure if water weighs
62.5 \mathrm{lbs}
{\mathrm{ft}}^{3}
|
In computer science, the longest palindromic substring or longest symmetric factor problem is the problem of finding a maximum-length contiguous substring of a given string that is also a palindrome. For example, the longest palindromic substring of "bananas" is "anana". The longest palindromic substring is not guaranteed to be unique; for example, in the string "abracadabra", there is no palindromic substring with length greater than three, but there are two palindromic substrings with length three, namely, "aca" and "ada". In some applications it may be necessary to return all maximal palindromic substrings (that is, all substrings that are themselves palindromes and cannot be extended to larger palindromic substrings) rather than returning only one substring or returning the maximum length of a palindromic substring.
Manacher (1975) invented a linear time algorithm for listing all the palindromes that appear at the start of a given string. However, as observed e.g., by Apostolico, Breslauer & Galil (1995), the same algorithm can also be used to find all maximal palindromic substrings anywhere within the input string, again in linear time. Therefore, it provides a linear time solution to the longest palindromic substring problem. Alternative linear time solutions were provided by Jeuring (1994), and by Gusfield (1997), who described a solution based on suffix trees. Efficient parallel algorithms are also known for the problem.[1]
The longest palindromic substring problem should not be confused with the different problem of finding the longest palindromic subsequence.
1 Slow algorithm
2 Manacher's algorithm
Slow algorithm[edit]
This algorithm is slower than Manacher's algorithm, but is a good stepping stone for understanding Manacher's algorithm. It looks at each character as the center of a palindrome and loops to determine the largest palindrome with that center.
The loop at the center of the function only works for palindromes where the length is an odd number. The function works for even-length palindromes by modifying the input string. The character '|' is inserted between every character in the inputs string, and at both ends. So the input "book" becomes "|b|o|o|k|". The even-length palindrome "oo" in "book" becomes the odd-length palindrome "|o|o|".
Longest_Palindrome_SLOW(string S) {
string S' = S with a bogus character (eg. '|') inserted between each character (including outer boundaries)
array PalindromeRadii = [0,...,0] // The radius of the longest palindrome centered on each place in S'
// note: length(S') = length(PalindromeRadii) = 2 × length(S) + 1
while Center < length(S') {
// Determine the longest palindrome starting at Center-Radius and going to Center+Radius
while Center-(Radius+1) >= 0 and Center+(Radius+1) < length(S') and S'[Center-(Radius+1)] = S'[Center+(Radius+1)] {
Radius = Radius+1
// Save the radius of the longest palindrome in the array
PalindromeRadii[Center] = Radius
Center = Center+1
longest_palindrome_in_S' = 2*max(PalindromeRadii)+1
longest_palindrome_in_S = (longest_palindrome_in_S'-1)/2
return longest_palindrome_in_S
The runtime of this algorithm is
{\displaystyle O(n^{2})}
. The outer loop runs
{\displaystyle n}
times and the inner loop can run up to
{\displaystyle n/2}
Manacher's algorithm[edit]
Below is the pseudocode for Manacher's algorithm. The algorithm is faster than the previous algorithm because it exploits when a palindrome happens inside another palindrome.
For example, consider the input string "abacaba". By the time it gets to the "c", Manacher's algorithm will have identified the length of every palindrome centered on the letters before the "c". At the "c", it runs a loop to identify the largest palindrome centered on the "c": "abacaba". With that knowledge, everything after the "c" looks like the reflection of everything before the "c". The "a" after the "c" has the same longest palindrome as the "a" before the "c". Similarly, the "b" after the "c" has a longest palindrome that is at least the length of the longest palindrome centered on the "b" before the "c". There are some special cases to consider, but that trick speeds up the computation dramatically.[citation needed]
Longest_Palindrome(string S) {
// At the start of the loop, Radius is already set to a lower-bound for the longest radius.
// In the first iteration, Radius is 0, but it can be higher.
// Below, Center is incremented.
// If any precomputed values can be reused, they are.
// Also, Radius may be set to a value greater than 0
OldCenter = Center
OldRadius = Radius
// Radius' default value will be 0, if we reach the end of the following loop.
while Center <= OldCenter + OldRadius {
// Because Center lies inside the old palindrome and every character inside
// a palindrome has a "mirrored" character reflected across its center, we
// can use the data that was precomputed for the Center's mirrored point.
MirroredCenter = OldCenter - (Center - OldCenter)
MaxMirroredRadius = OldCenter + OldRadius - Center
if PalindromeRadii[MirroredCenter] < MaxMirroredRadius {
PalindromeRadii[Center] = PalindromeRadii[MirroredCenter]
else if PalindromeRadii[MirroredCenter] > MaxMirroredRadius {
PalindromeRadii[Center] = MaxMirroredRadius
else { // PalindromeRadii[MirroredCenter] = MaxMirroredRadius
Radius = MaxMirroredRadius
break // exit while loop early
Manacher's algorithm is faster because it reuses precomputed data when a palindrome exists inside another palindrome. There are 3 cases of this. They are represented by the "if / else if / else" statement in the pseudocode.
The first case is when the palindrome at MirroredCenter lies completely inside the "Old" palindrome. In this situation, the palindrome at Center will have the same length as the one at MirroredCenter. For example, if the "Old" palindrome is "abcbpbcba", we can see that the palindrome centered on "c" after the "p" must have the same length as the palindrome centered on the "c" before the "p".
The second case is when the palindrome at MirroredCenter extends outside the "Old" palindrome. That is, it extends "to the left" (or, contains characters with a lower index than any inside the "Old" palindrome). Because the "Old" palindrome is the largest possible palindrome centered on OldCenter, we know the characters before and after it are different. Thus, the palindrome at Center will run exactly up to the border of the "Old" palindrome, because the next character will be different than the one inside the palindrome at MirroredCenter. For example, if the string was "ababc", the "Old" palindrome could be "bab" with the Center being the second "b" and the MirroredCenter being the first "b". Since the palindrome at the MirroredCenter is "aba" and extends beyond the boundaries of the "Old" palindrome, we know the longest palindrome at the second "b" can only extend up to the border of the "Old" palindrome. We know this because if the character after the "Old" palindrome had been an "a" instead of a "c", the "Old" palindrome would have been longer.
The third and last case is when the palindrome at MirroredCenter extends exactly up to the border of the "Old" palindrome. In this case, we don't know if the character after the "Old" palindrome might make the palindrome at Center longer than the one at MirroredCenter. But we do know that the palindrome at Center is at least as long as the one at MirroredCenter. In this case, Radius is initialized to the radius of the palindrome at MirroredCenter and the search starts from there. An example string would be "abcbpbcbp" where the "Old" palindrome is "bcbpbcb" and the Center is on the second "c". The MirroredCenter is the first "c" and it has a longest palindrome of "bcb". The longest palindrome at the Center on the second "c" has to be at least that long and, in this case, is longer.
The algorithm runs in linear time. This can be seen by noting that Center strictly increases after each outer loop and the sum Center + Radius is non-decreasing. Moreover, the number of operations in the first inner loop is linear in the increase of the sum Center + Radius while the number of operations in the second inner loop is linear in the increase of Center. Since Center ≤ 2n+1 and Radius ≤ n, the total number of operations in the first and second inner loops is
{\displaystyle O(n)}
and the total number of operations in the outer loop, other than those in the inner loops, is also
{\displaystyle O(n)}
. The overall running time is therefore
{\displaystyle O(n)}
^ Crochemore & Rytter (1991), Apostolico, Breslauer & Galil (1995).
Apostolico, Alberto; Breslauer, Dany; Galil, Zvi (1995), "Parallel detection of all palindromes in a string", Theoretical Computer Science, 141 (1–2): 163–173, doi:10.1016/0304-3975(94)00083-U .
Crochemore, Maxime; Rytter, Wojciech (1991), "Usefulness of the Karp–Miller–Rosenberg algorithm in parallel computations on strings and arrays", Theoretical Computer Science, 88 (1): 59–82, doi:10.1016/0304-3975(91)90073-B, MR 1130372 .
Crochemore, Maxime; Rytter, Wojciech (2003), "8.1 Searching for symmetric words", Jewels of Stringology: Text Algorithms, World Scientific, pp. 111–114, ISBN 978-981-02-4897-0 .
Gusfield, Dan (1997), "9.2 Finding all maximal palindromes in linear time", Algorithms on Strings, Trees, and Sequences, Cambridge: Cambridge University Press, pp. 197–199, doi:10.1017/CBO9780511574931, ISBN 0-521-58519-8, MR 1460730 .
Jeuring, Johan (1994), "The derivation of on-line algorithms, with an application to finding palindromes", Algorithmica, 11 (2): 146–184, doi:10.1007/BF01182773, hdl:1874/20926, MR 1272521, S2CID 7032332 .
Manacher, Glenn (1975), "A new linear-time "on-line" algorithm for finding the smallest initial palindrome of a string", Journal of the ACM, 22 (3): 346–351, doi:10.1145/321892.321896, S2CID 10615419 .
Longest Palindromic Substring Part II., 2011-11-20, archived from the original on 2018-12-08 . A description of Manacher’s algorithm for finding the longest palindromic substring in linear time.
Akalin, Fred (2007-11-28), Finding the longest palindromic substring in linear time, retrieved 2016-10-01 . An explanation and Python implementation of Manacher's linear-time algorithm.
Jeuring, Johan (2007–2010), Palindromes, retrieved 2011-11-22 . Haskell implementation of Jeuring's linear-time algorithm.
Palindromes (deadlink). Java implementation of Manacher's linear-time algorithm.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Longest_palindromic_substring&oldid=1089432932"
|
Last night, while on patrol, Agent 008 came upon a spaceship! He hid behind a tree and watched a group of little space creatures carry all sorts of equipment out of the ship. But suddenly, he sneezed. The creatures jumped back into their ship and sped off into the night. 008 noticed that they had dropped something, so he went to pick it up. It was a calculator! What a great find. He noticed that it had a
\boxed{\text{LOG}}
button, but he noticed something interesting:
\log10
did not equal
1
! With this calculator,
\log10\approx0.926628408
. He tried some more:
\log100\approx1.853256816
\log1000\approx2.779885224
What base do the space creatures work in? Explain how you can tell.
You are looking for the base of the log; let’s call it
b
\log_b(10)\approx0.926628408
Rewrite that equation in exponential form. How can you solve it?
Use Guess and Check if you can’t think of another method.
Does the same base work for
\log100
\log1000
How many fingers do you think the space creatures have?
How many fingers do you have? What is the base of the log on your calculator?
|
Home : Support : Online Help : Science and Engineering : Units : Environments : Standard : Differentiation
differentiation and partial differentiation in the Standard Units environment
In the Standard Units environment, the diff function differentiates an expression with respect to a name that can have a unit. The result is the derivative of the expression, with respect to the variable of differentiation, with a unit, the input expression unit divided by the variable of differentiation unit if any.
\mathrm{unit}
\mathrm{with}\left(\mathrm{Units}[\mathrm{Standard}]\right):
-3.532{x}^{2}\mathrm{Unit}\left('J'\right)
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{3.532}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{J}⟧
\mathrm{diff}\left(,x\mathrm{Unit}\left('s'\right)\right)
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{7.064}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{W}⟧
32{x}^{2}\mathrm{Unit}\left('\mathrm{ft}'\right)+7x\mathrm{Unit}\left('\mathrm{inch}'\right)+45\mathrm{Unit}\left('m'\right)
\left(\frac{\textcolor[rgb]{0,0,1}{6096}}{\textcolor[rgb]{0,0,1}{625}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{889}}{\textcolor[rgb]{0,0,1}{5000}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{45}\right)\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{m}⟧
\mathrm{diff}\left(,x\mathrm{Unit}\left('s'\right),x\mathrm{Unit}\left('s'\right)\right)
\frac{\textcolor[rgb]{0,0,1}{12192}}{\textcolor[rgb]{0,0,1}{625}}\textcolor[rgb]{0,0,1}{}⟦\frac{\textcolor[rgb]{0,0,1}{m}}{{\textcolor[rgb]{0,0,1}{s}}^{\textcolor[rgb]{0,0,1}{2}}}⟧
4{x}^{4}-3x+2
\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}
\mathrm{diff}\left(,x\mathrm{Unit}\left('s'\right)\right)
\left(\textcolor[rgb]{0,0,1}{16}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{3}\right)\textcolor[rgb]{0,0,1}{}⟦\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{s}}⟧
|
Luhn algorithm - Wikipedia
(Redirected from Luhn)
Simple checksum formula
The Luhn algorithm or Luhn formula, also known as the "modulus 10" or "mod 10" algorithm, named after its creator, IBM scientist Hans Peter Luhn, is a simple checksum formula used to validate a variety of identification numbers, such as credit card numbers, IMEI numbers, National Provider Identifier numbers in the United States, Canadian Social Insurance Numbers, Israeli ID Numbers, South African ID Numbers, Swedish National identification numbers, Swedish Corporate Identity Numbers (OrgNr), Greek Social Security Numbers (ΑΜΚΑ), SIM card numbers and survey codes appearing on McDonald's, Taco Bell, and Tractor Supply Co. receipts. It is described in U.S. Patent No. 2,950,048, granted on August 23, 1960.[1]
The algorithm is in the public domain and is in wide use today. It is specified in ISO/IEC 7812-1.[2] It is not intended to be a cryptographically secure hash function; it was designed to protect against accidental errors, not malicious attacks. Most credit cards and many government identification numbers use the algorithm as a simple method of distinguishing valid numbers from mistyped or otherwise incorrect numbers.
1.1 Example for computing check digit
1.2 Example for validating check digit
3 Pseudocode implementation
The check digit is computed as follows:
If the number already contains the check digit, drop that digit to form the "payload." The check digit is most often the last digit.
With the payload, start from the rightmost digit. Moving left, double the value of every second digit (including the rightmost digit).
Sum the digits of the resulting value in each position.
Sum the resulting values from all positions repeatedly until a single digit,
{\displaystyle s}
remains (i.e. do Casting out nines on it).
The check digit is calculated by
{\displaystyle 10-(s\operatorname {mod} 10)}
Example for computing check digit[edit]
Assume an example of an account number "7992739871" (just the "payload", check digit not yet included):
7 9 (1+8) 9 4 7 6 9 7 (1+6) 7 2
The sum of the resulting digits is 67.
The check digit is equal to
{\displaystyle 10-(67\operatorname {mod} 10)=3}
Example for validating check digit[edit]
Drop the check digit (last digit) of the number to validate. (e.g. 79927398713 -> 7992739871)
Calculate the check digit (see above)
Compare your result with the original check digit. If both numbers match, the result is valid. (e.g.
{\displaystyle (givenCheckDigit=calculatedCheckDigit)\Leftrightarrow (isValidCheckDigit)}
The Luhn algorithm will detect any single-digit error, as well as almost all transpositions of adjacent digits. It will not, however, detect transposition of the two-digit sequence 09 to 90 (or vice versa). It will detect most of the possible twin errors (it will not detect 22 ↔ 55, 33 ↔ 66 or 44 ↔ 77).
Other, more complex check-digit algorithms (such as the Verhoeff algorithm and the Damm algorithm) can detect more transcription errors. The Luhn mod N algorithm is an extension that supports non-numerical strings.
Because the algorithm operates on the digits in a right-to-left manner and zero digits affect the result only if they cause shift in position, zero-padding the beginning of a string of numbers does not affect the calculation. Therefore, systems that pad to a specific number of digits (by converting 1234 to 0001234 for instance) can perform Luhn validation before or after the padding and achieve the same result.
The algorithm appeared in a United States Patent[1] for a hand-held, mechanical device for computing the checksum. Therefore, it was required to be rather simple. The device took the mod 10 sum by mechanical means. The substitution digits, that is, the results of the double and reduce procedure, were not produced mechanically. Rather, the digits were marked in their permuted order on the body of the machine.
Pseudocode implementation[edit]
int sum := integer(purportedCC[nDigits-1])
int parity := (nDigits-2)modulus 2
^ a b US patent 2950048A, Luhn, Hans P., "Computer for verifying numbers", published 1960-08-23
^ "Annex B: Luhn formula for computing modulus-10 "double-add-double" check digits". Identification cards — Identification of issuers — Part 1: Numbering system (Standard). International Organization for Standardization, International Electrotechnical Commission. January 2017. ISO/IEC 7812-1:2017.
Implementation in 88 languages on the Rosetta Code project
Retrieved from "https://en.wikipedia.org/w/index.php?title=Luhn_algorithm&oldid=1084595037"
|
Many programming languages require garbage collection, either as part of the language specification (for example, RPL, Java, C#, D,[3] Go and most scripting languages) or effectively for practical implementation (for example, formal languages like lambda calculus); these are said to be garbage collected languages. Other languages were designed for use with manual memory management, but have garbage-collected implementations available (for example, C and C++). Some languages, like Ada, Modula-3, and C++/CLI, allow both garbage collection and manual memory management to co-exist in the same application by using separate heaps for collected and manually managed objects; others, like D, are garbage-collected but allow the user to manually delete objects and also entirely disable garbage collection when speed is required.
While integrating garbage collection into the language's compiler and runtime system enables a much wider choice of methods,[citation needed] post-hoc GC systems exist, such as Automatic Reference Counting (ARC), including some that do not require recompilation. (Post-hoc GC is sometimes distinguished as litter collection.) The garbage collector will almost always be closely integrated with the memory allocator.
Garbage collection consumes computing resources in deciding which memory to free, even though the programmer may have already known this information. The penalty for the convenience of not annotating object lifetime manually in the source code is overhead, which can lead to decreased or uneven performance.[4] A peer-reviewed paper from 2005 came to the conclusion that GC needs five times the memory to compensate for this overhead and to perform as fast as the same program using idealised explicit memory management, by inserting annotations using an oracle implemented by collecting traces from programs run under a profiler.[5] Interaction with memory hierarchy effects can make this overhead intolerable in circumstances that are hard to predict or to detect in routine testing. The impact on performance was also given by Apple as a reason for not adopting garbage collection in iOS despite it being the most desired feature.[6]
Reference counting[edit]
Escape analysis[edit]
Other dynamic languages, such as Ruby and Julia (but not Perl 5 or PHP before version 5.3,[16] which both use reference counting), JavaScript and ECMAScript also tend to use GC. Object-oriented programming languages such as Smalltalk, RPL and Java usually provide integrated garbage collection. Notable exceptions are C++ and Delphi, which have destructors.
BASIC and Logo have often used garbage collection for variable-length data types, such as strings and lists, so as not to burden programmers with memory management details. On the Altair 8800, programs with many string variables and little string space could cause long pauses due to garbage collection.[17] Similarly the Applesoft BASIC interpreter's garbage collection algorithm repeatedly scans the string descriptors for the string having the highest address in order to compact it toward high memory, resulting in
{\displaystyle O(n^{2})}
performance[18] and pauses anywhere from a few seconds to a few minutes.[19] A replacement garbage collector for Applesoft BASIC by Randy Wigginton identifies a group of strings in every pass over the heap, reducing collection time dramatically.[20] BASIC.System, released with ProDOS in 1983, provides a windowing garbage collector for BASIC that is many times faster.[21]
Limited environments[edit]
Garbage collection is rarely used on embedded or real-time systems because of the usual need for very tight control over the use of limited resources. However, garbage collectors compatible with many limited environments have been developed.[29] The Microsoft .NET Micro Framework, .NET nanoFramework and Java Platform, Micro Edition are embedded software platforms that, like their larger cousins, include garbage collection. The Arduino language includes GC as well.
Compile-time use[edit]
Real-time systems[edit]
Incremental, concurrent, and real-time garbage collectors have been developed, for example by Henry Baker and by Henry Lieberman.[32][33][34]
Dynamic dead-code elimination
^ Zorn, Benjamin (1993-01-22). "The Measured Cost of Conservative Garbage Collection". Software: Practice and Experience. Department of Computer Science, University of Colorado Boulder. 23 (7): 733–756. CiteSeerX 10.1.1.14.1816. doi:10.1002/spe.4380230704. S2CID 16182444.
^ "Reference Counting Garbage Collection".
^ "RAII, Dynamic Objects, and Factories in C++, Roland Pibinger, 3 May 2005". 2005-04-17.
^ a b Yossi Levanoni; Erez Petrank (2006). "An on-the-fly reference-counting garbage collector for java". ACM Trans. Program. Lang. Syst. 28: 31–69. CiteSeerX 10.1.1.15.9106. doi:10.1145/1111596.1111597. S2CID 14777709.
^ Salagnac, G; et al. (2005-05-24). "Fast Escape Analysis for Region-based Memory Management". Electronic Notes in Theoretical Computer Science. 131: 99–110. doi:10.1016/j.entcs.2005.01.026.
^ "Altair 8800 Basic 4.1 Reference Manual" (PDF). The Vintage Technology Digital Archive. April 1977. p. 108. Retrieved 2021-06-29.
^ "I did some work to speed up string garbage collection under Applesoft..." Hacker News. Retrieved 2021-06-29.
^ Little, Gary B. (1985). Inside the Apple IIc. Bowie, Md.: Brady Communications Co. p. 82. ISBN 0-89303-564-5. Retrieved 2021-06-29.
^ "Fast Garbage Collection". Call-A.P.P.L.E.: 40–45. January 1981.
^ Worth, Don (1984). Beneath Apple Pro DOS (March 1985 printing). Chatsworth, Calif.: Quality Software. pp. 2–6. ISBN 0-912985-05-4. Retrieved 2021-06-29.
^ "Objective-C 2.0 Overview". Archived from the original on 2010-07-24.
^ Lieberman, Henry; Hewitt, Carl (1983). "A real-time garbage collector based on the lifetimes of objects". Communications of the ACM. 26 (6): 419–429. doi:10.1145/358141.358147. hdl:1721.1/6335. S2CID 14161480.
^ Baker, Henry G. (1978). "List processing in real time on a serial computer". Communications of the ACM. 21 (4): 280–294. doi:10.1145/359460.359470. hdl:1721.1/41976. S2CID 17661259. see also description
Wilson, Paul R. (1992). "Uniprocessor Garbage Collection Techniques". Proceedings of the International Workshop on Memory Management (IWMM 92). Lecture Notes in Computer Science. Springer-Verlag. 637: 1–42. CiteSeerX 10.1.1.47.2438. doi:10.1007/bfb0017182. ISBN 3-540-55940-X.
Wilson, Paul R.; Johnstone, Mark S.; Neely, Michael; Boles, David (1995). "Dynamic Storage Allocation: A Survey and Critical Review". Proceedings of the International Workshop on Memory Management (IWMM 95). Lecture Notes in Computer Science (1 ed.). 986: 1–116. CiteSeerX 10.1.1.47.275. doi:10.1007/3-540-60368-9_19. ISBN 978-3-540-60368-9.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Garbage_collection_(computer_science)&oldid=1084458229"
|
Section 4.1 - First-Order Partial Derivatives
Section 4.2 - Higher-Order Partial Derivatives
Section 4.3 - Chain Rule
Section 4.4 - Directional Derivative
Section 4.5 - Gradient Vector
Section 4.6 - Surface Normal and Tangent Plane
Section 4.7 - Approximations
Section 4.8 - Unconstrained Optimization
Section 4.9 - Constrained Optimization
Section 4.10 - Optimization on Closed Domains
Section 4.11 - Differentiability
Chapter 4 treats differentiation of functions of several variables. Derivatives of such functions are called partial derivatives, use the special notation
∂
d
, and are de facto what the diff command delivers. When Maple takes an ordinary derivative, it holds constant all variables except the differentiation variable. That is precisely the notion of a partial derivative. As with ordinary derivatives, there are higher-order partial derivatives. For nearly every function met in a first course in multivariate calculus, the order in which these higher-order partial derivatives are taken does not matter.
Thinking of a function such as
w=g\left(x,y,z\right)
w
represents a physical quantity such as temperature, the question "At what rate does the temperature vary in a specified direction?" leads to the notion of the directional derivative. The directional derivative can be expressed in terms of the gradient vector, a vector orthogonal to the level sets (either curves or surfaces, depending on the number of independent variables). The orthogonality of the gradient vector supports calculations of tangent planes for surfaces and tangent lines for curves.
Of course, differentiation is at the heart of techniques for finding extrema for functions of several variables. As in single-variable calculus, optimization problems can be either unconstrained, or constrained. For unconstrained optimization problems, Maple has the SecondDerivativeTest command for classifying the nature of any critical points found by differentiation techniques. Constrained optimization problems yield to the Lagrange multiplier technique, which Maple implements with its LagrangeMultipliers command.
Finally, the last section clarifies a fundamental difference between the differential calculus of one and several variables. In single-variable calculus, the derivative is defined first, and the differential is given in terms of the derivative. In multivariate calculus, the differential is defined first, and functions that have a differential are said to be differentiable. In either case, it remains true that functions that are differentiable are continuous.
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.