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Representing Dimensions
An international standard called Système International d'Unites (SI), breaks every quantity down into a combination of the dimensions mass, length (or position), time, charge, temperature, intensity
, and angle. To be reasonably general, our system would have to be able to represent seven or more fundamental dimensions. It also needs the ability to represent composite dimensions that, like force
, are built through multiplication or division of the fundamental ones.
In general, a composite dimension is the product of powers of fundamental dimensions. If we were going to represent these powers for manipulation at runtime, we could use an array of seven ints, with
each position in the array holding the power of a different fundamental dimension:
typedef int dimension[7]; // m l t ...
dimension const mass = {1, 0, 0, 0, 0, 0, 0};
dimension const length = {0, 1, 0, 0, 0, 0, 0};
dimension const time = {0, 0, 1, 0, 0, 0, 0};
In that representation, force would be:
dimension const force = {1, 1, -2, 0, 0, 0, 0};
that is, mlt^-2. However, if we want to get dimensions into the type system, these arrays won't do the trick: they're all the same type! Instead, we need types that themselves represent sequences of
numbers, so that two masses have the same type and a mass is a different type from a length.
Fortunately, the MPL provides us with a collection of type sequences. For example, we can build a sequence of the built-in signed integral types this way:
#include <boost/mpl/vector.hpp>
typedef boost::mpl::vector<
signed char, short, int, long> signed_types;
How can we use a type sequence to represent numbers? Just as numerical metafunctions pass and return wrapper types having a nested ::value, so numerical sequences are really sequences of wrapper
types (another example of polymorphism). To make this sort of thing easier, MPL supplies the int_<N> class template, which presents its integral argument as a nested ::value:
#include <boost/mpl/int.hpp>
namespace mpl = boost::mpl; // namespace alias
static int const five = mpl::int_<5>::value;
In fact, the library contains a whole suite of integral constant wrappers such as long_ and bool_, each one wrapping a different type of integral constant within a class template.
Now we can build our fundamental dimensions:
typedef mpl::vector<
mpl::int_<1>, mpl::int_<0>, mpl::int_<0>, mpl::int_<0>
, mpl::int_<0>, mpl::int_<0>, mpl::int_<0>
> mass;
typedef mpl::vector<
mpl::int_<0>, mpl::int_<1>, mpl::int_<0>, mpl::int_<0>
, mpl::int_<0>, mpl::int_<0>, mpl::int_<0>
> length;
Whew! That's going to get tiring pretty quickly. Worse, it's hard to read and verify: The essential information, the powers of each fundamental dimension, is buried in repetitive syntactic "noise."
Accordingly, MPL supplies integral sequence wrappers that allow us to write:
#include <boost/mpl/vector_c.hpp>
typedef mpl::vector_c<int,1,0,0,0,0,0,0> mass;
typedef mpl::vector_c<int,0,1,0,0,0,0,0> length; // or position
typedef mpl::vector_c<int,0,0,1,0,0,0,0> time;
typedef mpl::vector_c<int,0,0,0,1,0,0,0> charge;
typedef mpl::vector_c<int,0,0,0,0,1,0,0> temperature;
typedef mpl::vector_c<int,0,0,0,0,0,1,0> intensity;
typedef mpl::vector_c<int,0,0,0,0,0,0,1> angle;
Even though they have different types, you can think of these mpl::vector_c specializations as being equivalent to the more verbose versions above that use mpl::vector.
If we want, we can also define a few composite dimensions:
// base dimension: m l t ...
typedef mpl::vector_c<int,0,1,-1,0,0,0,0> velocity; // l/t
typedef mpl::vector_c<int,0,1,-2,0,0,0,0> acceleration; // l/(t^2)
typedef mpl::vector_c<int,1,1,-1,0,0,0,0> momentum; // ml/t
typedef mpl::vector_c<int,1,1,-2,0,0,0,0> force; // ml/(t^2)
And, incidentally, the dimensions of scalars (like pi) can be described as:
typedef mpl::vector_c<int,0,0,0,0,0,0,0> scalar;
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mainly covers the subjects of beam design, axial forces and combined bending and axial forces,
wood structural panels, diaphragms, and shearwalls as discussed in Chapters 6, 7, 8, 9, and 10
of the textbook as well as relevant material in the 2018 NDS, NDS Supplement and 2015
SDPWS. Should you need to consult the IBC at some point, the 2018 Edition should be used.
Free access to 2018 IBC online version at https://codes.iccsafe.org/content/IBC2018P4. Refer
to sources of where you obtain any information needed to work these problems and show all
your work. Work all problems in ASD.
Problems 6 and 7 = 2 points each
Problems 2, 3, 4, and 8 = 3 points each
Problem 1 = 4 points each
Problem 5 = 5 points
Bonus problem = 3 points each
1. Given:
2. Given:
The beam in the figure below has the compression side of the member
supported laterally at the ends and the quarter points. The span length
L=24 ft. The member is glulam 3-1/8 x 16-1/2 20F-V7 DF. The load is a
combination of D+L. Cand Care all equal to 1.0.
The maximum allowable bending moment in ft-kips.
The corresponding allowable load, P, in kips.
Floor joists supporting a wood structural panel floor deck are spaced at
24 inches on center. The panels are oriented with their strong direction
perpendicular to the supporting joists and are continuous over two or
more spans. Underlayment, %-in thick, is to be applied over the
sheathing. The floor dead load is 10 psf and the floor live load is 50 psf.
The total load deflection is to be less than or equal to L/360. Using the
allowable span and load tables in the 2018 IBC:
Find: a.
Is Table 2304.8(3) or 2304.8(5) applicable in designing the
wood structural panel deck?
What minimum span rating and thickness is required for
wood structural panel sheathing?
Do the continuous edges require support?
Fig: 1
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Comparing And Ordering Rational Numbers 6th Grade Worksheet 2024 - NumbersWorksheets.com
Comparing And Ordering Rational Numbers 6th Grade Worksheet
Comparing And Ordering Rational Numbers 6th Grade Worksheet – A Reasonable Numbers Worksheet might help your child become a little more acquainted with the principles associated with this rate of
integers. In this particular worksheet, individuals are able to resolve 12 diverse issues related to reasonable expression. They are going to figure out how to increase two or more amounts, group
them in sets, and determine their products and services. They are going to also exercise simplifying reasonable expression. After they have learned these methods, this worksheet might be a valuable
resource for continuing their research. Comparing And Ordering Rational Numbers 6th Grade Worksheet.
Rational Figures certainly are a ratio of integers
There are two forms of amounts: irrational and rational. Logical phone numbers are understood to be entire numbers, whilst irrational numbers tend not to recurring, and have an limitless quantity of
digits. Irrational phone numbers are no-absolutely no, low-terminating decimals, and rectangular origins that are not excellent squares. They are often used in math applications, even though these
types of numbers are not used often in everyday life.
To define a realistic amount, you need to realize what a realistic variety is. An integer can be a entire quantity, as well as a logical amount is actually a rate of two integers. The proportion of
two integers is definitely the variety on the top divided up through the variety on the bottom. For example, if two integers are two and five, this would be an integer. However, there are also many
floating point numbers, such as pi, which cannot be expressed as a fraction.
They may be created into a small percentage
A realistic amount includes a numerator and denominator that are not absolutely no. Consequently they can be depicted like a small fraction. Along with their integer numerators and denominators,
rational phone numbers can also have a unfavorable value. The adverse importance ought to be positioned left of and its complete value is its range from absolutely nothing. To streamline this
example, we shall claim that .0333333 can be a portion which can be written like a 1/3.
As well as adverse integers, a realistic number may also be produced in to a small percentage. As an example, /18,572 can be a reasonable number, while -1/ is not really. Any small percentage
consisting of integers is reasonable, so long as the denominator fails to include a and may be composed as being an integer. Likewise, a decimal that leads to a point can be another logical number.
They are sense
Even with their name, logical phone numbers don’t make much sensation. In mathematics, they are single entities with a unique size around the amount collection. This means that once we count one
thing, we can purchase the size and style by its rate to its authentic amount. This contains accurate even when there are actually unlimited rational numbers involving two certain amounts. If they
are ordered, in other words, numbers should make sense only. So, if you’re counting the length of an ant’s tail, a square root of pi is an integer.
In real life, if we want to know the length of a string of pearls, we can use a rational number. To get the duration of a pearl, by way of example, we might add up its width. Just one pearl weighs 15
kilograms, which is a reasonable variety. Additionally, a pound’s body weight means ten kilos. Therefore, we should be able to divide a lb by ten, without the need of concern yourself with the length
of just one pearl.
They can be conveyed being a decimal
If you’ve ever tried to convert a number to its decimal form, you’ve most likely seen a problem that involves a repeated fraction. A decimal amount can be composed being a a number of of two
integers, so four times 5 various is equivalent to eight. The same dilemma requires the recurring fraction 2/1, and either side must be separated by 99 to have the right answer. But how can you make
the transformation? Here are some cases.
A reasonable amount may also be designed in various forms, such as fractions as well as a decimal. A great way to signify a rational number within a decimal is always to divide it into its fractional
comparable. You can find 3 ways to break down a reasonable number, and every one of these approaches brings its decimal counterpart. One of these brilliant techniques is to separate it into its
fractional equal, and that’s what’s known as the terminating decimal.
Gallery of Comparing And Ordering Rational Numbers 6th Grade Worksheet
Comparing And Ordering Integers Worksheets 6th Grade Worksheets Master
Pin On Math Short Cuts
16 Best Images Of Adding Integers Worksheets 7th Grade With Answer Key
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John A. Pelesko
Professor and Chair
Mathematical Sciences
University of Delaware
Department of Mathematical Sciences
501A Ewing Hall
Newark, DE 19716
Email: pelesko@math.udel.edu
Phone: (302) 831-1467
Fax: (302) 831-4511
Website: http://www.math.udel.edu/~pelesko
• Ph.D. Mathematical Sciences, New Jersey Institute of Technology, 1997
BS Mathematics, University of Massachusetts, 1992
Research Overview:
Dr. Pelesko is an applied mathematician interested in the development and application of mathematical methods to the physical and biological sciences. He has worked on problems in the microwave
heating of ceramics, electron beam welding, diffusion in polymers, solidification thermomechanics, thermoelastic stability and shock dynamics. His current research is focused upon the mathematical
modeling of microelectromechanical systems (MEMS), nanoelectromechanical systems (NEMS) and self-assembly. He is interested in bio-mimetic devices, i.e., those that utilize nature’s mechanisms in
their function.
Dr. Pelesko has been at the University of Delaware since 2002. Prior to arriving at UD he was a faculty member at the Georgia Institute of Technology and a postdoc at the California Institute of
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Evidence of unspanned stochastic volatility in crude-oil market
The Academy of Economic Studies
The Faculty of Finance, Insurance, Banking and Stock Exchange Doctoral School of Finance and Banking (DOFIN)
Dissertation Paper
Evidence of the unspanned stochastic volatility in crude-oil market
Student: R
zvan Daniel Tudor
Supervisor Professor: PhD. Mois
The purpose of this dissertation paper is to conduct a comprehensive analysis of unspanned stochastic volatility in commodity markets with focus and empirical evidence on crude-oil market. Using
crude-oil futures and options on futures data from New York Mercantile Exchange (NYMEX) there are presented model-free results that strongly
suggest the presence of unspanned stochastic volatility in the crude-oil market. Sharp oil prices changes exert influence on macroeconomic activity in general and crude-oil industry in particular.
The importance of the results is that they show the extent to which volatility risk is spanned by the futures contracts. The extent to which crude-oil futures contracts trading span volatility will
indicate if options on futures are redundant securities or there is needed a mixed strategy combining both types of crude-oil market derivatives (futures and options) to fully hedge against
volatility risk.
1. Introduction
Over the last few years, the persistent sharp oil prices changes in both the spot and futures markets have represented perhaps the most striking challenge to the forecasting abilities of private and
public institutions worldwide. From the demand side increasing crude-oil prices led to new challenges in hedging against volatility risk.
While volatility is clearly stochastic, it is not clear to what extent volatility risk can be hedged by trading in the commodities themselves or, more generally, their associated futures contracts,
forward or swap contracts, in other words, the extent to which volatility is spanned.
Existing equilibrium models from commodity markets imply that volatility risk is largely spanned by the futures contracts. Mainly, they suggest that market volatility is embedded in inventories which
are the basis for futures price formation. Therefore by construction futures offer a high degree of volatility spanning.
The consequence of these models is that they imply that options on futures are redundant securities. In spite of this, the data provided form Bank of International Settlements – BIS – strongly
suggests that the market for commodity derivatives has exhibited phenomenal growth over the past few years. For exchange-traded commodity derivatives, the BIS estimates that the number of outstanding
contracts more than doubled from 12.4 million in June 2003 to 32.1 million in June 2006. For over-the-counter (OTC) commodity derivatives, the growth has been even stronger with the BIS estimating
that, over the same period, the notional value of outstanding contracts increased five-fold from USD 1.04 trillion to USD 6.39 trillion. Importantly, a large and increasing fraction of the commodity
derivatives are options (as opposed to futures, forwards and swaps). According to BIS statistics, options now constitute over one-third of the number of outstanding exchange-traded contracts and
almost two-thirds of the notional value of outstanding OTC contracts.
The purpose of this paper is to show that if, for a given commodity, volatility contains important unspanned components it cannot be fully hedged and risk-managed using only the underlying
instruments and options are not redundant securities.
The unspanned stochastic volatility evidence research is conducted in crude-oil market because it is by far the most liquid commodity derivatives market. The data for the analysis was provided by New
York Mercantile Exchange – NYMEX – and contains a large set of futures and options on futures contracts prices. Since volatility is not directly observable I will use, for different options
maturities, straddle returns and implied volatility of the at-the-money options straddles as proxies for the true volatility and I will show the extent to which futures contracts span volatility. If
volatility is completely spanned by trading in futures contracts then the equilibrium models for commodity markets are correct in assuming that commodities futures prices formation incorporates
market volatility. If shown on contrary, it means that options on futures are not redundant securities and their role is to extend the degree of hedging which futures contracts traditionally offer.
The reason for choosing these two volatility proxies is that straddle returns are not conditioned on a particular pricing model. Returns are obtained from daily options on futures market prices from
NYMEX. While using the implied volatility, though it might be more accurate, it involves using a pricing model.
Previously, this approach was used to evidence the unspanned stochastic volatility in fixed-income market, more specifically to show the extent to which trading of bonds span the term structure of
interest rates.
The dissertation paper is organized as follows. Section 2 contains a literature review of the models which treated the stochastic volatility in commodity and financial markets. Section 3 briefly
presents the crude-oil derivatives data used in this paper and the computational aspects behind data which was used as input for the model. In section 4 the paper contains the model used to evidence
of the unspanned stochastic volatility in crude-oil market. Section 5 presents model estimation and analysis. Finally, in Section 6 there are to be found the conclusion which can be drawn from this
paper. Section 7 contains the reference list and Section 8 the relevant additional information – Annexes - which are mentioned in the paper content.
2. Literature Review
The first equilibrium models from commodity markets implied that futures contracts provide insurance against price volatility, the level of inventories being negatively related to the required risk
premium of commodity futures. The starting point of these models was the traditional Theory of Storage originally proposed by Kaldor (1939). The theory provides a link between the term structure of
futures prices and the level of inventories of commodities. This link, also known as “cost of carry arbitrage,” predicts that in order to induce storage, futures prices and expected spot prices of
commodities have to rise sufficiently over time to compensate inventory holders for the costs associated with storage. Developments in this area were made by Deaton and Laroque (1992), Chambers and
Bailey (1996), Routledge, Seppi and Spatt (2000). Their models predict a link between the level of inventories and future spot price volatility. Inventories act as buffer stocks which can be used to
absorb shocks to demand and supply, thus dampening the impact on spot prices. Deaton and Laroque show that at low inventory levels, the risk of “stock-out” (exhaustion of inventories) increases and
expected future spot price volatility rises. In an extension of the Deaton and Laroque model which includes a futures market, RSS show how the shape of the futures curve reflects the state of
inventories and signals expectations about future spot price volatility. DL (1992) and RSS (2000) have explained the existence of a convenience yield as arising from the probability of a stock-out of
inventories. Because they study storage in a risk-neutral world, risk premiums are zero by construction, and futures prices simply reflect expectations about future spot prices.
Another reference model, the model in Litzenberger and Rabinowitz (1995) and Ng and Pirrong (1994), incorporates the embedded option in reserves of extractable resource commodities. Finally it has
similar implications. The relationship between volatility and the slope of the futures - Litzenberger and Rabinowitz (1995) showed it for crude oil, and Ng and Pirrong (1994), for metals - show that
the degree of backwardation is indeed positively related to volatility, implying that volatility does contain a component that is spanned by the futures contracts. However, whether volatility also
contains important unspanned components was not shown.
Other papers, which emphasize production/extraction and investment decisions for the formation of futures prices, include those of Casassus, Collin-Dufresne, and Routledge (2003), Kogan, Livdan, and
Yaron (2005) and Carlson, Khoker, and Titman (2006).
In their paper, Gordon, Hayashi and Rouwenhorst (2005), analyzed the fundamentals of commodity futures returns and predicted a link between the state of inventories, the shape of the futures curve,
and expected futures risk premiums. They showed that show that the convenience yield is a decreasing, non-linear relationship of inventories and also linked the current spot commodity price and the
current (nearest to maturity) futures price to the level of inventories, and empirically documented the nonlinear relationship predicted by the existence of the non-negativity constraint on
inventories. In particular, they showed that low inventory levels for a commodity are associated with an inverted (“backwardated”) term structure of futures prices, while high levels of inventories
are associated with an upward sloping futures curve (“contango”).
The existence of unspanned volatility factors was first evidenced in fixed income market. Collin-Dufresne and Goldstein (2002) and Heidari and Wu (2003) defined unspanned stochastic volatility as
being those factors driving Cap and Swaption implied volatilities that do not drive the term structure of interest rates. In other words they showed that trading in underlying bonds do not span the
term structure of interest rates. There are embedded factors in Cap and Swaption that bonds do not contain and make them more valuable in hedging against interest rates volatility risk That is, in
contrast to the predictions of standard short-rate models, bonds do not span the fixed income market.
Using Collin-Dufresne and Goldstein (2002) approach, Trolle and Schwartz (2006) extended the problem with existence of unspanned stochastic volatility to commodity markets. They developed a tractable
model for pricing commodity derivatives in the presence of unspanned stochastic volatility. The model features correlations between innovations to futures prices and volatility, quasi-analytical
prices of options on futures and futures curve dynamics in terms of a low-dimensional affine state vector. Their evidence was on crude-oil market due to its liquidness and showed that in the presence
of unspanned stochastic volatility factors options are not redundant securities. The model and the evidence could be extended as well on the other commodity markets.
Richter and Sorensen (2007) have a work in progress for a stochastic volatility model in the presence of unspanned volatility factors for the soybean market.
3. Overview of the Data
As mentioned before, crude-oil market is the most liquid commodity market. The data used in this paper was delivered by New York Mercantile Exchange - NYMEX. It contains large data set of futures and
options on futures prices with different maturities and strike prices. The futures data contains daily prices for futures contracts starting with January 1987 and ending with May 2008. Since options
on futures prices were available for research purposes only for June 2002 – December 2006 period, I chose to use the futures contracts prices for the same interval.
The NYMEX futures contract trades in units of 1,000 barrels, and the delivery point is Cushing, Oklahoma, which is also accessible to the international spot markets via pipelines. The contract
provides for delivery of several grades of domestic and internationally traded foreign crude, and serves the diverse needs of the physical market. The NYMEX symbol for light-sweet crude-oil is CL.
Crude oil futures are listed nine years forward using the following listing schedule: consecutive months are listed for the current year and the next five years; in addition, the June and December
contract months are listed beyond the sixth year. Additional months will be added on an annual basis after the December contract expires, so that an additional June and December contract would be
added nine years forward, and the consecutive months in the sixth calendar year will be filled in. The futures expire on the third business day prior to the 25th calendar day of the month proceeding
the delivery month. If the 25th calendar day of the month is a non-business day, expiration is on the third non-business day prior to the non-business day proceeding the 25th calendar day.
For my purpose I extracted from various maturities only futures contracts with time-to-maturity 1 Month, 3 Months, 6 Months, 9 Months and 1 Year. The reason is that crude-oil spot prices established
on spot markets are not available. Mainly, spot prices are settled on one to one transactions between partners based on current market conditions. Therefore the 1 Month time-to-maturity futures
contract serves as a proxy for the crude-
oil spot prices. The 1 Year time-to-maturity futures prices are continuous during the June 2002 – December 2006 sample so I chose not to include them in the analysis. The quarterly maturities
correspond to the traditional hedging strategy of a crude-oil refining based company. Being given the optimal refining capacity usually the company engages in a rolling futures contract with
quarterly maturities providing the company with the necessary crude oil amount at a certain price which can be used for financial forecasts.
While futures offer price protection by allowing the holder of a futures contract to lock in a price level, a major advantage of options is that the holder of an options contract is afforded price
protection, but still has the ability to participate in favorable market moves. Because the buyer of an options contract has the options contract but not the market moves against a position, and a
trader holds on to this option, the maximum cost is the price he has already paid for the option.
On the other hand, if the market moves in favor of a position, the virtually unlimited profit potential to the buyer of an options contract is parallel to a futures position, net of the premium paid
for the options contract. Therefore, protection from unfavorable market moves is achieved at a known cost, without giving up the ability to participate in favorable market moves. Options on futures
contracts expire three business days prior to the expiration of the underlying futures.
For the research I chose crude-oil calendar spread options on futures. The reason is that calendar spread options are the most traded crude-oil options derivatives on NYMEX and thus the results of my
study will be more representative. Also they imply delivery of the underlying asset as opposed to other derivatives which are only settled in cash, for example European Style options – NYMEX symbol
LO. Their NYMEX trading symbol for calendar spread options is WA. The contract is simply an options contract on the price differential between two delivery dates for the same commodity. The price
spread between contract months can be extremely volatile because the energy markets are more sensitive to weather and news than any other market. A widening of the month-to-month price relationships
can expose market participants to severe price risk which could adversely affect the effectiveness of a hedge or the value of inventory. The calendar spread options can allow market participants who
hedge their risk to also take advantage of favorable market moves.
For the corresponding three maturities of the futures prices I extracted corresponding calendar spread straddles. One straddle consists of a call and a put option with the same strike. More, I chose
the at-the-money straddles since they are sensitive to market volatility (“Vegas” peak for the at-the-money straddles). At-the-money property of an option means that the option has the strike price
equal or near to spot price.
This computation is helpful to extract from the whole options and futures sample the data I need for the evidence of unspanned stochastic volatility.
4. The Model
If equilibrium models are correct and changes in crude oil prices are spanned by the futures contracts then in order to hedge against volatility risk one may construct a portfolio of futures
contracts for this purpose.
One simple alternative to evidence if the conclusion of these models is correct is to simply regress changes in volatility from crude oil markets on futures contracts prices and see if they fully
explain volatility changes. But volatility in crude oil market as well as in other commodity and financial markets is stochastic and not directly observable. Therefore I will use two reasonable
proxies for the true and unobservable volatility – at-the-money calendar spread straddles prices and at-at-the-money calendar spread straddles implied volatility.
A straddle consists of a call and a put option on the same underlying with the same strike. When purchasing a straddle the investor expects the market to spike in either direction. This is the case
of long at-the-money straddle strategy which will be used throughout this paper as opposite to short at-the-money straddle which is used when market is expected to be quiet (expecting minor changes
in volatility). Therefore we can say that by purchasing a straddle the investor trades volatility. Straddle profits are unlimited in either direction while losses are limited to the premium paid for
both options which form the straddle.
The reason for selecting straddles as volatility proxies is the straddle Greeks indicators. The price of a near-ATM straddle has low sensitivity to variations in the price
of the underlying futures contract (since “deltas” are close to zero for ATM straddles) but high sensitivity to variations in volatility (since “Vegas” peak for ATM straddles).
Delta (Δ) and Vega (ν ) for at-the-money straddles:
⎪⎩ ⎪ ⎨ ⎧ − − = ∂ ∂ = Δ [−] − Put d e Call d e S V qT qT ), ( * ), ( * 1 2 φ φ τ φ σ ν V =S*e−qT (d[1]) ∂ ∂
= , for both Call and Put options.
The indicators were derived from the Black-Scholes option pricing formula where:
V – Value of the option; S – Stock price;
q – Annual dividend yield;
τ - Time to maturity (T-t);
σ - Volatility; r – Risk free rate;
Φ(d1) - The probability of exercise under the equivalent exponential martingale probability measure and the equivalent martingale:
∞ − − ∞ − − = =
x y x y dy e dy e x
π π φ 2 2 ) ( 2 2 2 2 τ σ τ σ /2)* ( ) / ln(
[+] [−] [+]
2 =
S K r q y
To avoid the non-stationary problem with straddle and futures prices, which is common to almost all asset prices I will use for further analysis straddle returns and futures returns.
Straddle returns are computed as follows:
⎪ ⎩ ⎪ ⎨ ⎧ < + + + − − + + > + − − = else K S S K K S K S
r i call put i call put
put call i put call i i straddle , 0 ) ( ), ( ) ( ), ( , π π π π π π π π Where: i
S - The spot price of the underlying commodity. For calendar spread options type
the underlying commodity spot price is the price differential from current market price and the futures price of the futures contract which at the maturity of the option.
K – The strike price; put
call π
π , - Call and Put option prices;
The futures contracts returns are simply computed this way:
jMonth i i
i Spot ice FuturesContract ice
futures_ [,] = Pr − Pr [,]
Where: i
SpotPr - The crude-oil spot price. In absence of a transparent spot market the
spot price is computed as the price of the futures contract with shortest time to maturity, the contract with expiration the following month.
jMonth i
ice tract
FuturesCon Pr [,] - The today observed market price of the futures
contract with expiration in “j” months. As mentioned before j = 3, 6, 9 Months.
There are three alternatives to evidence the presence of unspanned stochastic volatility in crude oil market:
- Investigate how much of the variation in the prices of derivatives highly exposed to stochastic volatility (so-called “straddles”) can be explained by variation in the underlying futures prices;
- Investigate how much of the variation in implied volatilities (which is related to expectations under the risk-neutral measure of future volatility) can be explained by variation in the underlying
futures prices;
- Investigate how much of the variation in realized volatility, estimated from high
frequency data, can be explained by variation in the underlying futures prices.
- Investigate how much of the volatility of the variance swaps can be explained by variation in the underlying futures prices.
Unfortunately high-frequency data is available only for calendar spread options. Also variance swaps are quite illiquid in the market therefore I will use only the first two approaches for evidence.
For the approach which requires the use of straddle implied volatility, this is computed as the average of straddle component put and call options implied volatilities. The put and call implied
volatilities are obtained from put and call formulas of the Black-Scholes model.
Briefly, the formulas for call and put options derived from the Black-Scholes partial differential equation are:
0 2 1 2 2 2 2 [−] [=] ∂ ∂ + ∂ ∂ + ∂ ∂ rV S V rS S V S t V [σ] (Black-Scholes PDE) ) ( ) ( ) , (S T S d[1] Ke d[2] C = φ − −rTφ ) ( ) ( ) , ( ( ) [2] [1] d S d Ke T S P = −r T−tφ − − φ − Where: τ σ τ
σ /2)* ( ) / ln( 2 1 + − + = S K r q d T d T r K S d σ τ σ σ − = − + = 2 [1] 2 * ) 2 / ( ) / ln(
Φ is the cumulative distribution function. Φ(d1) and Φ(d2) are the probabilities of exercise under the equivalent exponential risk neutral measure and the equivalent risk neutral probability measure,
Being given the call and put option prices, the underlying futures price, the current and maturity date, the strike price and the risk free rate, the implied volatility is computed using the
Newton-Raphson method. It is a root-finding algorithm that uses the first few terms of the Taylor series of a function f(x)in the vicinity of a suspected root. Newton's method is also known as
Newton's iteration.
The Taylor series of f(x)about the point x=x[0] +εis given by: ... ) ( '' 2 1 ) ( ' ) ( ) ( 2 0 0 0 0 +ε = f x + f x ε + f x ε + x f
Keeping terms only to first order:
ε ε) ( ) '( ) (x[0] f x[0] f x[0]
f + ≈ +
This expression can be used to estimate the amount of offset ε needed to land closer to the root starting from an initial guessx[0]. Setting f(x[0] +ε)=0and solving the
above equation for ε ≡ε[0]gives:
) ( ' ) ( 0 0 0 x f x f − = ε
This is the first-order adjustment to the root's position. By letting x[1] = x[0] +ε,
calculating a new ε[1], and so on, the process can be repeated until it converges to a fixed point (which is precisely a root) using:
) ( ' ) ( 0 0 0 x f x f − = ε
Therefore with a good initial choice of the root's position, the algorithm can be applied iteratively to obtain: ) ( ' ) ( 1 n n n n x f x f x x [+] = −
Many commodity markets as well as financial markets are characterized by a high degree of collinearity between returns. In order to extract the most uncorrelated sources of variation in a
multivariate system I will use principal components analysis (PCA) for the futures returns.
Mainly, principal components analysis objective is to:
- Reduce dimensionality by taking into account only the most relevant principal components from the whole data set;
- Avoid near multicollinearity issues for the returns and use in further analysis only uncorrelated components. Also, these components characterize the data and are useful for drawing conclusions;
Mathematical background:
The data input to principal component analysis must be stationary. Principal component analysis is based on eigenvalues and eigenvector analysis of V =X’XT, the k x k symmetric matrix of correlations
between the variables in X. Each principal component is a linear combination of these columns, where the weights are chosen in such way that:
- the first principal component explains the greatest amount of the total variation in X, the second component explains the greatest amount of the remaining variation, and so on;
- the principal components are uncorrelated to each other; Denoting by W the k x k matrix of eigenvectors of V. Thus:
Where Λ is the k x k diagonal matrix of eigenvalues of V. Then we order the columns of W according to the size of corresponding eigenvalue. Thus if W =(wij) for i,j=1,…,k then the m-th column of W,
denoted wm =(w[1]m,...,wkm), is the k x 1 eigenvector corresponding to the eigenvalue λmand the column labeling has been chosen so that λ[1] >λ[2] >...>λ[k].
Therefore the m-th principal component of the system is defined by: k km m m m w X w X w X P = [1] [1] + [2] [2] +...+
Where Xidenotes the i-th column of X, the standardized historical input data on the i-th variable in the system. In matrix notation the above definition becomes:
m m Xw
P =
Each principal component is a time series of the transformed X variables, and the full T x m matrix of principal components, which has Pmas its m-th column, may be written as:
The procedure leads to uncorrelated components because: P’P=W’X’XW=TW’WΛ
W is an orthogonal matrix, which means ' [=] −1 W
W and so P’P=TΛ. Since this is
a diagonal matrix the columns of P are uncorrelated, and the variance of the m-th
principal component is λ[m](sum of eigenvalues). However, the sum of the eigenvalues is k, the number of variables in the system. Therefore, the proportion of variation explained by the first n
principal components together is:
k n i i/ 1
= λ
Because of the choice of labeling in W the principal components have been ordered so that P[1]belongs to the first and largest eigenvalue λ[1], P[2]belongs to the first and largest eigenvalue λ[2],
and so on. In a highly correlated system the first eigenvalue will be much larger than the others, so the first principal component alone will explain a large part of variation.
Since ' [=] −1 W
W , is equivalent to X=PW’, that is:
k ik i i i w P w P w P X = [1] [1]+ [2] [2] +...+
Thus each vector of the data input may be written as a linear combination of the principal components.
To sum up principal component analysis it is a way of identifying patterns in data, and expressing the data in such a way as to highlight their similarities and differences. Since patterns in data
can be hard to find in data of high dimension, where the luxury of graphical representation is not available, principal components analysis is a powerful tool to achieve this.
The principal components analysis will be illustrated in the next section on the highly correlated crude-oil futures prices returns.
In order to evidence the presence of unspanned stochastic volatility in crude-oil market I will use the first two approaches mentioned above: investigate how much of straddle returns and straddle
implied volatilities variation can be explained by the variation of the futures returns.
The evidence procedure consists of three steps:
a) The first step is principal components analysis of the correlation matrix of daily futures returns. We retain all the principal components identified in the analysis in an attempt to not exclude
from evidence any source of variation embedded in a component, though that component might have minor significance.
b) For each futures contract “i” I regress the daily closest to the at-the-money straddle returns on the futures returns principal components.
For each futures contract “i” I regress the daily straddle implied volatilities on futures returns principal components. The daily straddle implied volatility is related to the average expected
(under the risk-neutral measure) volatility of the underlying futures contract over the life of the option.
Since in commodity and financial markets the returns dependency is rarely linear I will introduce in the regression equation the squared principal components also, in an attempt to take into account
non-linearities between straddle returns and implied volatilities and futures returns. Therefore if I take into account just one principal component the regression equation to catch non-linearity
will be:
ε β β α + + + = 2 2 1x x y
The coefficient for the squared principal component will be important (as long as it is significant) just for the sign indicating the convexity or concavity of the dependency. Another aspect is
taking into account the cross-products dependencies of the straddle returns and implied volatility. These dependencies reflect changes in the marginal effect of one explanatory variable given others.
Considering straddle returns and first two principal components the transformation can be written as:
ε β β β β β α + + + + + + = x w x w xw y [1] [2] [3] 2 [4] 2 [5]
By rewriting the equation above as:
ε β β β β β α + + + + + + =( 2) ( [1] [3] [5] ) 4 2w w x x w y
We can interpret the intercept as a function of w and the slope of x as changing
with w and x.
To sum up, taking into consideration both squared components and cross-product between components the regression equations for both approaches may be written as:
Straddle returns regression:
t i i i i i i i i i i i i x x x x x x x x x x x x y =α +β 1 [1] +β 2 [2] +β 3 [3] +β 4 [1]2 +β 5 [2]2 +β 6 [3]2 +β 7 [1] [2] +β 8 [1] [3] +β 9 [2] [3] +ε Implied volatility regression:
t i i i i i i i i i i i i [x] [x] [x] [x] [x] [x] [x] [x] [x] [x] [x] [x] z = α + β 1 [1] + β 2 [2] + β 3 [3] + β 4 [1]2 + β 5 [2]2 + β 6 [3]2 + β 7 [1] [2] + β 8 [1] [3] + β 9 [2] [3] + ε Where: 3 ,
2 , 1 ,i=
xi - The principal components of the futures return data. They numbered according to they variation explanatory power from principal component with highest eigenvalue to the principal component with
smallest eigenvalue.
y - The straddle returns at “i” maturity. In my case i = 3, 6, 9 Months.
z - The implied volatility of the straddles at “i” maturity.
Both regressions will indicate the extent to which volatility is spanned by the futures contracts.
c) Finally, I will analyze the principal components of the time series of residuals from the straddle return regressions and the implied volatility regressions. The principal components of the
residuals are, by construction, independent of those of the futures returns. If there is unspanned stochastic volatility in the data, there should be at least one significant explanatory principal
component for the variation due to unspanned factors. If the residuals are simply due to noisy data, there should not be one principal component with high explanatory power among residuals.
5. The Model Estimation and Analysis
Commodity futures prices are characterized by some important properties:
- Commodity futures prices are often “backwardated" in that they decline with time to delivery,
- Spot and futures prices are mean reverting;
- Commodity prices are strongly heteroscedastic and price volatility is correlated with the degree of backwardation;
- Unlike financial assets, many commodities have pronounced seasonality in both price levels and volatilities.
Being given S(t) the time-t crude-oil spot price and F(t, T ) [P(t, T )] the time-t price of a crude-oil futures contract [zero-Coupon bond] with maturity T - t. The futures contract is backwardated
if S(t) - P(t, T )F(t, T ) > 0 and strongly backwardated if S(t) - F(t, T ) > 0.
For our futures date the results confirm the above “backwardation” property:
Backwardation type vs.
Maturity of the Futures Contract
3 Months 6 Months 9 Months Backwardation Degree(%) 45.6 52.7 55.3
Strongly Backwardation Degree(%) 94.3 95.4 96.2
Table 1 – The simple and strong “backwardation” degree
As time to maturity increases so does the backwardation degree. If we take a look on the futures prices graphical representation for various maturities we see that clearly the market was in contango,
although market expectations derived from the prices of futures contracts for the same maturities were bearish.
The strongly heteroscedasticity property of commodities, which can be translated as the property of futures prices to have time dependant functions for mean (μ(t)) and variance ( 2( )
σ ) poses a serious problem for out further econometric estimations.
In order to test the validity of property I will use Augmented Dickey-Fuller (ADF) test for the 3 Months futures prices. The ADF test is a unit root test which is carried out by estimating the
following equation:
t p t t t t t y y y y = α + β + γ + δ Δ + + δ Δ + ε Δ [−][1] [1] [−][1] ... [1] [−]
This is the most restrictive form of the test, which includes the intercept ( )α and the trend (β). The null hypothesis is that the coefficient of the level variable (γ ) is 0, which means the series
is non-stationary, or less than 0 otherwise. I carried out the test for the futures prices with 3 Month maturity in levels using only the intercept. The results were:
Null Hypothesis: FUTURES_3M has a unit root Exogenous: Constant
Lag Length: 0 (Automatic based on SIC, MAXLAG=22)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -1.200488 0.6763
Test critical values: 1% level -3.435876
5% level -2.863868
10% level -2.568060
*MacKinnon (1996) one-sided p-values.
Table 2 – The ADF test results for 3 Months Futures Prices
The value of the ADF test is larger than the critical values for all levels of confidence meaning that we cannot reject the null hypothesis of the futures prices series
being non-stationary. Therefore I will further use futures returns instead of futures prices. Futures
returns are computed as shown in 4th section as:
jMonth i i
i Spot ice FuturesContract ice
Building the futures returns time series offer the advantage of stationary. Indeed if we carry out once again the ADF test for the 3 Months futures returns, the value of the ADF test will reject the
null hypothesis.
Null Hypothesis: FCR_3M has a unit root Exogenous: Constant, Linear Trend
Lag Length: 0 (Automatic based on SIC, MAXLAG=22)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -6.498157 0.0000
Test critical values: 1% level -3.966124
5% level -3.413762
10% level -3.128951
*MacKinnon (1996) one-sided p-values.
Table 3 – The ADF test results for 3 Months Futures Returns
The ADF unit root tests for the 6 Months and 9 Months futures returns may be found in the Annex section of this paper (Table 13, 14, 15).
The graphical representation of the crude-oil futures returns for the chosen maturities show that in crude-oil market futures returns are highly correlated.
This highly correlation suggests that for example the 9 Months futures returns are not only influenced by the crude-oil spot price but also by intermediate maturities futures returns. Next I will
present the futures returns correlation matrix. The correlation coefficients are closed to 1 indicating as well a high correlation degree.
FCR_3M FCR_6M FCR_9M
FCR_3M 1.000000 0.984578 0.965624
FCR_6M 0.984578 1.000000 0.995120
FCR_9M 0.965624 0.995120 1.000000
Table 4 – Futures returns correlation matrix
If we examine the first column of the model we see that the correlation degree tends to decrease with maturity though very slightly.
Since crude-oil futures returns are explanatory variables in my classical linear regression model, the highly correlated returns pose the problem of near multicollinearity. In this case, it is not
possible to estimate all of the “betas” from the model. In the presence of multicollinearity, it will be hard to obtain small standard errors.
Therefore I will use futures returns principal components analysis as a solution for the near multicollinearity problem.
The starting point in identifying the futures returns principal components is the futures returns correlation matrix. Bellow there is presented the eigenvalues and eigenvectors of the futures returns
correlation matrix. The eigenvectors are ordered after the corresponding eigenvalue, starting with the highest.
Date: 07/06/08 Time: 17:16
Sample (adjusted): 6/10/2002 10/20/2006 Included observations: 1140 after adjustments Correlation of FCR_3M FCR_6M FCR_9M
Comp 1 Comp 2 Comp 3
Eigenvalue 2.963599 0.035471 0.000930
Variance Prop. 0.987866 0.011824 0.000310
Variable Vector 1 Vector 2 Vector 3
FCR_3M -0.574725 -0.769887 0.277426
FCR_6M -0.580496 0.144590 -0.801323
FCR_9M -0.576815 0.621585 0.530016
Table 5 – The eigenvalues and eigenvectors of futures returns correlation matrix The first principal component, which will be further denoted asPC[1], has the highest eigenvalue, which is responsible
for explaining 98.76% (λ[1]/k, where k is the matrix dimension, in my case 3) of the variation of the future returns. If we look at corresponding eigenvector weights they are quite similar due to
strong correlation between futures returns.
The significance of the first principal component corresponding eigenvector weights is that an upward shift in the first principal component induces a downward parallel shift of the futures returns
curve. For this reason first principal component is called the trend component.
As shown in the theoretical section of my paper starting from the eigenvectors we can get to the original data applying the following formula:
k ik i i i w P w P w P X = [1] [1] + [2] [2] +...+
The graph above shows by comparison the 3M futures return curve after inducing an upward shift in the first component. The downward parallel shift is explainable due to negative and similar weights
of the eigenvector.
The second principal component, which will be further denoted asPC[2], explains only 1.18% of futures returns variation. The weights are increasing from “-” to “+”. Thus an upward movement of the
second principal component induces a change in slope of the futures returns, where short maturities move down and long maturities move up. The second principal component significance is that 1.18% of
the total futures return variation is attributed to changes in slope.
Graph 4– 3M futures return curve reaction to PC[2]upward shift.
The third principal component, which will be further denoted asPC[3], explains
only 0.03% of the futures returns variation. The weights are positive for the short term returns, negative for the medium term returns and positive for the long term returns. Therefore we can say
that the third component influences the convexity of the returns curve. The significance of the third principal component is that 0.03% of the total variation is due to changes in convexity.
Graph 4– 3M futures return curve reaction to PC[3]upward shift.
Given the variance explained by each principal component I may choose to drop the third component and use only the first two components in the regression, since they cumulated explain 99.97% of the
futures returns variation. I chose not to drop it since I want to see if changes in convexity have significance in explaining volatility in crude-oil markets as well.
As I previously mentioned the main purpose when using principal components analysis was to eliminate the strong correlation among futures returns. Indeed if we check the correlation matrix of
principal components we see that correlation indices are close to 0 indicating that we managed to extract patterns from original data which move independently.
PC1 PC2 PC3
PC1 1.00000000000000 0.00000000000000 -0.00000000000002 PC2 0.00000000000000 1.00000000000000 -0.00000000000011 PC3 -0.00000000000002 -0.00000000000011 1.00000000000000
Table 6 – Principal components correlation matrix
I decided to include squared principal components and principal components cross-product in an attempt to take into account possible non-linearity between volatility
proxies and futures returns. Though, this may lead as well to near multicollinearity issue. In Annex part of this paper you may find the correlation matrix of principal components (Table 16). The
correlation indices are not high, the highest values are for correlations between squared principal components of the first two components and cross-product between them (0.336674).
Next, I will compute the time series of volatility proxies I chose. The first volatility proxy is straddle returns for the same maturity as futures returns (3 Months, 6 Months and 9 Months). Straddle
returns were computed using:
⎪ ⎩ ⎪ ⎨ ⎧ < + + + − − + + > + − − = else K S S K K S K S
r [i] [call] [put] [i] [call] [put]
put call i put call i i straddle , 0 ) ( ), ( ) ( ), ( , π π π π π π π π
The period of the sample from which I extracted the straddles was 10/6/2002 – 12/14/2006. When I built the straddles I looked mainly for at-the-money straddles (straddles with strike price near or
equal to spot price). The daily frequency of the data was not so high, therefore there were days when just only straddle may be computed from the put and call options available. I decided to take it
into account for further evidence imposing though the condition that strike price divided by the underlying asset spot price to be in the interval (0.75; 1.25). Where straddle could not be computed
due to lack of data or values outside the (0.75; 1.25) interval I used for the missing daily straddle information the previous available straddle returns value.
The 3Months straddle returns series is represented in the bellow graphic.
Further, I will carry out the ADF unit root test to see if we can work with the level series or there is needed at least on difference in order to obtain a stationary series. The output of the ADF
test (carried out with both intercept and trend included) is:
Null Hypothesis: WA_3M has a unit root Exogenous: Constant, Linear Trend
Lag Length: 2 (Automatic based on SIC, MAXLAG=22)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -7.405778 0.0000
Test critical values: 1% level -3.966139
5% level -3.413769
10% level -3.128955
*MacKinnon (1996) one-sided p-values.
Table 7 – Calendar spread straddle returns ADF test result
The value of the ADF test is lower than test critical values. Therefore the null hypothesis can be rejected leading to conclusion that calendar spread options straddle returns for the mention period
are stationary.
In Annex part of this paper there are shown unit root tests results carried out for straddle returns for 6 Months and 9 Months maturities (Table 17, 18).
The second volatility proxy is the straddle implied volatility. As mentioned, straddle implied volatility is computed as the average of Call and Put options which form the straddle implied
2 IV IV IV Put Call Straddle = +
The implied volatility is derived from Black-Scholes formulas for Call and Put options using the options market prices and the risk free rate of the US T-Bills with 3 Months and 6 Months maturities.
For the 9 Months maturity the risk free rate was not available, therefore I computed it as the average of 6 Months and 1 Year risk free rate.
Graph 6– 3M straddle implied volatility curve
The layout of 3M of straddle implied volatility ADF unit root test shows the series is stationary allowing use it as volatility proxy for our unspanned stochastic volatility research.
Null Hypothesis: WA_IV_3M has a unit root Exogenous: Constant, Linear Trend
Lag Length: 4 (Automatic based on SIC, MAXLAG=22)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -5.262452 0.0001
Test critical values: 1% level -3.966153
5% level -3.413776
10% level -3.128960
*MacKinnon (1996) one-sided p-values.
Table 8 – Calendar spread 3M straddle returns ADF test result
Next, I will regress calendar spread (NYMEX symbol WA) straddle returns on principal components of the futures returns, squared principal components of the returns and cross-products between
components. Since 2
R is the square of the correlation
coefficient between the values of the dependant variables and corresponding fitted values from the regression model. Using straddle returns as a volatility proxy the 2
indicate the extent to which volatility is spanned by trading in the futures contracts, which information is emphasized by the principal components used as regressors. However, there are some issues
around the 2
R as goodness of fit measure.
- If we change the order of the regressors the value will change;
- 2
R will never fall if we add extra regressors ;
Therefore, I will rely on Adjusted 2
R as goodness of fit measure since it takes into
account the loss of degrees of freedom associated with adding extra variable (squared principal components and cross-products between them).
2 R = [⎢⎣]⎡ − [⎥⎦]⎤ − − − 1(1 ) 1 2 R k T T
Where, k is the number of degrees of freedom.
The layout of the 3M straddle returns regression on futures returns principal components is presented bellow.
Dependent Variable: WA_3M Method: Least Squares Date: 06/30/08 Time: 21:04
Sample (adjusted): 6/10/2002 10/20/2006 Included observations: 1140 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 0.425578 0.013792 30.85750 0.0000 PC1 0.173028 0.005772 29.97657 0.0000 PC2 0.275921 0.057700 4.782031 0.0000 PC3 2.162054 0.385089 5.614420 0.0000 PC1*PC1 0.057807 0.002972 19.45140 0.0000 PC2*PC2
0.583068 0.196789 2.962914 0.0031 PC3*PC3 -1.675284 2.113742 -0.792568 0.4282 PC1*PC2 0.311658 0.028326 11.00240 0.0000 PC1*PC3 0.643695 0.181391 3.548663 0.0004 PC2*PC3 -1.974297 1.457376 -1.354693
R-squared 0.612262 Mean dependent var 0.616018
S.E. of regression 0.313793 Akaike info criterion 0.528565
Sum squared resid 111.2664 Schwarz criterion 0.572764
Log likelihood -291.2818 F-statistic 198.2596
Durbin-Watson stat 0.685380 Prob(F-statistic) 0.000000
Table 9 – Calendar spread 3M straddle returns regression on principal components of futures returns output
We notice that the coefficients for squared third principal component and product between second and third component are not significant. Though, the third principal component (the convexity
influence on variance) explained only 0.03% of the total futures returns variation. Lack of significance for the coefficient does not influence the results.
We see that straddle returns have a non-linear dependency on the futures returns trend component. The values of the coefficient for PC[1]and PC21are both positive which means the straddle returns
dependency on trend component takes the shape of an increasing convex function. Since PC[1]is responsible for explaining 98.76% of the futures return variation we might say that an upward movement in
PC[1]will lead to a parallel downward shift of the straddle returns. The slope (PC[2]) coefficient is significant as well.
An upward movement in (PC[2])will make straddle returns to decrease for short maturities
and increase for long maturities. This change has also a degree of convexity (PC22coefficient is significant), but since PC[2]is responsible only with 1.18% explanation for the whole futures returns
variation the convexity is slight. Also the marginal influence of the components is significant – trend component change on slope component change and trend component change on convexity component
The regression both 2
R and R2are low 0.61 and 0.60, which indicates that trading
in futures contracts do not span much of crude-oil prices volatility embedded in our volatility proxy – straddle returns. For commodity and financial markets high 2
R and 2
R should exceed 0.85, whereas R2and R2 bellow 0.7 indicate that volatility risk cannot
One problem which may appear is the residuals autocorrelation. A key assumption for Ordinary Least Squares method is that residuals have the following property: j i u ui, j)=0,∀ ≠ cov(
Since by uiwe denote the residuals of the regression estimation the property assumes that covariance between errors over time are 0. Having autocorrelation among residuals it is not a problem by
itself. But since it is a key assumption of OLS if it is violated it means that estimated coefficients may not be significant. In the Annex part of this paper there are presented the test performed
to evidence and eliminate residuals autocorrelation (Table 19,20).
If we examine the squared residuals correlogram we see that we have partial autocorrelation among squared residual at lag 1 and 2. We try to model the residuals in order to get rid of the partial
autocorrelation by introducing two MA (Moving Average) terms – MA (1) and MA (2). Running the regression with second order MA terms leads to a different regression output. We eliminate residuals
autocorrelation (Durbin-Watson test is close to 2). The 2
R and R2increase (0.76 and 0.75) but the most important fact is
the significance of the main coefficients remains unchanged - PC[1],PC[1]-squared, 2
PC .
Later on I will use regression residuals to evidence the presence of unspanned stochastic volatility in crude-oil market. Therefore I will retain the residuals form the original regression.
Running the regressions for the 6 Months and 9 Months maturities exhibits same low values for 2
R and R2for 6 Months regression – 0.64 and 0.63 – whereas for 9
Months the results are even lower – 0.24 and 0.23. The results indicate that most volatility risk cannot be hedged by trading in the futures contracts.
Futures Returns (Principal Components) Regression Output
R^2 Adjusted
S.E. From
Regression Sum of the Squared Residuals
3M Straddle
Returns 0.612262 0.609173 0.313793 111.2664
6M Straddle
Returns 0.639721 0.636852 0.478931 259.1941
Straddle Returns (Volatility Proxy)
9M Straddle
Returns 0.239385 0.233327 1.041255 1225.16
Table 10 – 2
R and R2from straddle return regressions
We notice that the explanatory power of the futures contracts decreases with maturity. The standard errors from regression as well as the sum of the squared residuals increases with maturity meaning
that the gap between y−yˆ(actual versus fitted of
straddle returns) increases while time-to-maturity increases.
Now we want to investigate how much of the variation in straddle implied volatilities (which is related to expectations under the risk-neutral measure of future volatility) can be explained by
variation in the underlying futures prices.
I used the same approach as in straddle returns regressions. The coefficient significance it is quite similar. There is the same partial autocorrelation problem for the squared residuals. Introducing
MA terms in the regression equation does not change the significance of the estimated coefficients therefore we may assume that the regression estimation was successful.
In straddle implied volatility case the capacity of futures contracts variation to span crude-oil market volatility is even lower which confirms what the conclusion from straddle returns regression
that one cannot hedge much against volatility risk using only trading of futures contracts.
Implied volatility regressions output as well as the procedure for eliminating partial autocorrelation among residuals are shown in the Annex part of this paper (Table 21-25). I will retain the
implied volatility residuals as well for further evidence.
Futures Returns (Principal Components) Regression Output R^2 Adjusted R^2 S.E. From Regression
Sum of the Squared Residuals 3M Straddle Implied Volatility 0.179701 0.173167 0.447045 491.46 6M Straddle Implied Volatility 0.064359 0.056907 0.850868 818.0939 Straddle Implied Volatility
(Volatility Proxy) 9M Straddle Implied Volatility 0.055037 0.04751 1.424951 2294.449 Table 11 – 2
R and R2from straddle implied volatility regressions
The first remark is that the highest explanatory power is for the shortest maturity (3M) but still very low. Straddle implied volatilities 2
R and R2evidence that there is a
very low extent in which futures returns can be used to hedge against volatility. Also the explanatory power of the futures contracts decreases with maturity. The standard errors from regression as
well as the sum of the squared residuals increases with maturity meaning that the gap between y−yˆ(actual versus fitted of straddle returns) increases
while time-to-maturity increases.
There are both advantages and disadvantage for using these approaches to evidence the presence of unspanned stochastic volatility.
Straddle returns:
“+” Straddle returns are not conditioned on a particular pricing model. They are computed based on NYMEX observed call and put premiums, corresponding strike prices. The only assumption in straddle
computation is the choice of the shortest time-to-maturity futures contract as a proxy for the crude-oil spot price.
“-“ Straddles have high gammas ( 2[2]
S V ∂ ∂ = T S d σ ϕ( [1])
, where V is the option premium – Call or Put). Gamma shows how much will vary the value of the option at high changes in crude-oil spot price. It indicates the convexity of the option value. Since
straddles are built to hedge against significant changes in crude-oil prices they are subject
to high gammas. The assumed significant variant spot price is used in computation of both straddle returns and futures returns. As shown in futures returns principal components analysis and in the
significance of estimated coefficients from the straddle returns regression straddle returns are convex in futures returns. Though, even if volatility is completely unspanned by the futures contracts
the presence of squared principal components (measuring convexity of the dependencies) may not lead to results close to 0. This may be one of the explanations for higher 2
R and R2in straddle returns regressions
than in straddle implied volatility regressions. Straddle implied volatilities:
“+” If volatility is completely unspanned by futures contracts result will be 0 or closed to 0. If we look at the results this is the case for 9M straddle implied volatility regression.
“-“ The results for straddle implied volatilities are conditioned on the accuracy of the pricing model we use. In our case we conditioned on Black-Scholes model.
The third and final step in my evidence is analyzing of the residuals from the regression. I retained the three sets of residuals from each regression type. Next, I will extract the principal
components out of each set of regression residuals. If there is unspanned stochastic volatility in the data, there should be large common variation in the residuals. Using the principal components
analysis properties this should lead us to a first principal component which embeds most of the variation from the residuals. If the residuals are simply due to noisy data, there should not be common
variation in the residuals.
The output of principal components analysis for the two data sets containing regression residuals are presented in the Annex part of this paper (Table 26, 27). Bellow can be found the synthesis of
the analysis.
Principal Components Analysis
Common Variation Among Residuals
PC1 explanatory power for the
variance among residuals (%)
PC2 explanatory power for the variance among
residuals (%)
explanatory power for the variance among
residuals (%)
Straddle Returns
Residuals 76.69 16.86 6.43
Straddle Implied
Volatility Residuals 77.64 16.62 5.72
Table 12 – explanatory power of the first three principal components of the regression residuals
For the straddle return regressions, the first principal component explains 76.69% of the variation in the residuals across maturities, while for the implied volatility regressions, it explains
77.64%. The main property of principal component analysis is that it identifies patterns in data. The strong explanatory power of the first component evidence the presence of large common variation
in the residuals, which strongly indicates that low 2
R and R2 from the regressions are primarily due to an unspanned
stochastic volatility factor rather than noisy data.
One potential weakness of above procedure is that it assumes the estimated coefficients are constant over the 1140 observation length sample. In reality this is not the case, they are time varying.
To compensate this I will split the entire sample in four “windows” of 285 observations each. I will repeat the procedure for these rolling windows and see if the new results are consistent with
previously illustrated unspanned stochastic volatility evidence. The aggregated results are displayed in the Annex part of this paper (Table 28, 29).
Briefly, the rolling window results display the same low 2
R and ,R2meaning that
futures variance has low explanatory power on straddle returns– the volatility proxy - even if we split the sample. 2
R and R2are higher for 6 Months maturity than for 3
Months but sensible lower for the 9 Month maturity. The sum of the squared residuals increases with maturity.
Analyzing the principal components of the residuals of the rolling window straddle return regression we notice that first component explanatory power ranges from
49.5% to 92%. This suggest as well the presence of large common variation in the residuals, the signal that low 2
R and R2 are due to an unspanned stochastic volatility
factor rather than noisy data.
6. Conclusions
In this dissertation paper I presented evidence of the unspanned stochastic volatility in crude-oil market. The results are important since they contradict the general commodity equilibrium models
derived mainly from Kaldor’s (1939) Theory of Storage, models applied to crude-oil market as well, which suggest that crude-oil market spot prices volatility is determined by the levels of
inventories. On the other hand these models suggest the inventories levels are the basis for futures prices formation.
If we rely on these approaches it will mean that trading in futures contracts will be enough to protect against volatility risk. Though, the data obtained from BIS – Bank of International Settlements
– states that the number of options on futures derivatives is highly increasing for crude-oil market.
Secondly, there are oil refining companies who still use hedging strategies based on entering a rolling futures contract with different maturities to protect against volatility risk. In my example I
simulated one of these strategies with a rolling futures contract with quarterly maturities. The results obtained from the evidence procedure suggest that there is at least one unspanned stochastic
volatility factor which cannot be hedged. The low results for 2
R and R2 clearly show the low extent to which futures contracts hedge
against volatility risk. Therefore, the rolling futures contract strategy is not of much help. The results are important because they do not rely on a particular pricing model – the case of straddle
returns. The implied volatility regression results, though they are model dependant, emphasize the straddle returns regression result.
Further direction in this area will mean to extend the evidence procedure taking into account high frequency data as Andersen and Benzoni (2005) did for fixed income market.
There could be interesting to investigate the unspanned stochastic volatility in other commodity markets less liquid where the futures contracts trading covers the most part of the transactions. For
example metals commodities markets.
7. Bibliography:
Alexander, C. (2001): “Market Models. A Guide to Financial Analysis”, John Wiley & Sons Ltd.
Andersen, T. G. and L. Benzoni (2005): “Can bonds hedge volatility risk in the U.S. treasury market? A specification test for affine term structure models,” Working paper, Kellogg School of
Management, Northwestern University
Brooks, C. (2002): “Introductory Econometric for Finance”, Cambridge University Press
Carlson, M. , Khoker, Z. , Titman, S. (2006): “Equilibrium Exhaustible Resource Price Dynamics”, NBER Working Paper 12000.
Cassassus, J. and P. Collin-Dufresne (2005): “Stochastic convenience yield implied from commodity futures and interest rates,” Journal of Finance, 60:2283–2331.
Deaton, A. and G. Laroque (1992): “On the behaviour of commodity prices,” Review of Economic Studies, 59:1–23.
Deaton, A. and G. Laroque (1996): “Competitive storage and commodity price dynamics,” Journal of Political Economy, 104:896–923.
Elekdag, S., Lalonde, R., Laxton, D., Muir, D. and Pesenti, P. (2008): “Oil prices movements and the global economy: A model-based assessment”, NBER Working Paper 13792.
Gibson, R. and E. S. Schwartz (1990): “Stochastic convenience yield and the pricing of oil contingent claims,” Journal of Finance, 45:959–976.
Heidari, M. and L. Wu (2003): “Are interest rate derivatives spanned by the term structure of interest rates?,” Journal of Fixed Income, 13:75–86.
Kogan, L., D. Livdan, and A. Yaron (2005): “Futures prices in a production economy with investment constraints,” Working paper, NBER # 11509.
Litzenberger, R. H. and N. Rabinowitz (1995): “Backwardation in oil futures markets: Theory and empirical evidence,” Journal of Finance, 50:1517–1545.
Miltersen, K. (2003): “Commodity price modeling that matches current observables: A new approach,” Quantitative Finance, 3:51–58.
Newey, W. and K. West (1987): “A simple, positive semi-definit, heteroscedasticity and autocorrelation consistent covariance matrix,” Econometrica, 55:703–708.
Nielsen, M. J. and E. S. Schwartz (2004): “Theory of Storage and the Pricing of Commodity Claims,” Review of Derivatives Research, 7:5–24.
Richter, M. and C. Sørensen (2002): “Stochastic volatility and seasonality in commodity futures and options: The case of soybeans,” Working paper, Copenhagen Business School.
Routledge, B. R., D. J. Seppi, and C. S. Spatt (2000): “Equilibrium forward curves for commodities,” Journal of Finance, 55:1297–1338.
Schwartz, E. S. (1997): “The stochastic behavior of commodity prices: Implications for valuation and hedging,” Journal of Finance, 52:923–973.
Trolle, A. and E. Schwartz (2006): “A general stochastic volatility model for the pricing and forecasting of interest rate derivatives,” Working paper, UCLA and NBER # 12337.
8. Annex
Table 13 - ADF test for crude-oil spot futures prices Null Hypothesis: FUTURES_SPOT has a unit root Exogenous: Constant
Lag Length: 0 (Automatic based on SIC, MAXLAG=22)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -1.337612 0.6138
Test critical values: 1% level -3.435876
5% level -2.863868
10% level -2.568060
*MacKinnon (1996) one-sided p-values.
Table 14 – ADF test for crude-oil 6 Months futures returns Null Hypothesis: FCR_6M has a unit root
Exogenous: Constant, Linear Trend
Lag Length: 0 (Automatic based on SIC, MAXLAG=22)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -5.408393 0.0000
Test critical values: 1% level -3.966124
5% level -3.413762
10% level -3.128951
*MacKinnon (1996) one-sided p-values.
Null Hypothesis: FCR_9M has a unit root Exogenous: Constant, Linear Trend
Lag Length: 0 (Automatic based on SIC, MAXLAG=22)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -5.131246 0.0001
Test critical values: 1% level -3.966124
5% level -3.413762
10% level -3.128951
*MacKinnon (1996) one-sided p-values.
Table 16– Principal components, Squared principal components and Cross-products between principal components correlation matrix.
PC1 PC2 PC3 PC1^2 PC2^2 PC3^2 PC12 PC13 PC23 PC1 1.000000 PC2 1.96E-15 1.000000 PC3 0.00 0.00 1.000000 PC1^2 0.076823 -0.15 -0.14 1.000000 PC2^2 -0.22 -0.37 -0.01 0.207121 1.000000 PC3^2 -0.05
0.100960 0.480248 0.015384 0.096237 1.000000 PC12 -0.12 -0.36 0.125049 0.336674 0.314062 -0.02 1.000000 PC13 -0.17 0.083183 -0.37 0.080146 0.016955 -0.14 -0.12 1.000000 PC23 0.081019 0.002317
0.429687 -0.08 -0.25 0.627921 0.016908 -0.14 1.000000
Table 17 – ADF test for calendar spread 6 Months straddle returns Null Hypothesis: WA_6M has a unit root
Exogenous: Constant, Linear Trend
Lag Length: 0 (Automatic based on SIC, MAXLAG=22)
t-Statistic Prob.*
Test critical values: 1% level -3.966124
5% level -3.413762
10% level -3.128951
*MacKinnon (1996) one-sided p-values.
Table 18 – ADF test for calendar spread 9 Months straddle returns
Null Hypothesis: WA_RET_9M has a unit root Exogenous: Constant
Lag Length: 0 (Automatic based on SIC, MAXLAG=15)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -3.715733 0.0044
Test critical values: 1% level -3.454626
5% level -2.872121
10% level -2.572482
*MacKinnon (1996) one-sided p-values.
Table 19 – Partial autocorrelation among residuals of 3M straddle returns regression on principal components of the futures returns
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Central limit theorem
12. Central limit theorem#
Recommended reference: Wasserman [Was04], Sections 5.3–5.4.
The central limit theorem is a very important result in probability theory. It tell us that when we have \(n\) independent and identically distributed random variables, the distribution of their
average (up to suitable shifting and rescaling) approximates a normal distribution as \(n\to\infty\).
12.1. Sample mean#
Consider a sequence of independent random variables \(X_1, X_2, \ldots\) distributed according to the same distribution function (which can be discrete or continuous). We say \(X_1, X_2, \ldots\) are
independent and identically distributed (or i.i.d.).
We saw the notion of independence of two random variables in def-indep. For a precise definition of independence of an arbitrary number of random variables, see Section 2.9 in Wasserman [Was04].
The \(n\)-th sample mean of \(X_1,X_2,\ldots\) is
\[ \overline{X}_n = \frac{1}{n}\sum_{i=1}^n X_i. \]
Note that \(\overline{X}_n\) is itself again a random variable.
12.2. The law of large numbers#
As above, consider i.i.d. samples \(X_1, X_2, \ldots\), say with distribution function \(f\). We assume that \(f\) has finite mean \(\mu\). The law of large numbers says that the \(n\)-th sample
average is likely to be close to \(\mu\) for sufficiently large \(n\).
(Law of large numbers)
For every \(\epsilon>0\), the probability
\[ P(|\overline{X}_n-\mu|>\epsilon) \]
tends to \(0\) as \(n\to\infty\).
The above formulation is known as the weak law of large numbers; there are also stronger versions, but the differences are not important here.
Fig. 12.1 illustrates the law of large numbers for the average of the first \(n\) out of 1000 dice rolls.
Show code cell source Hide code cell source
import numpy as np
from matplotlib import pyplot
from myst_nb import glue
rng = np.random.default_rng()
N = 1000
n = np.arange(1, N + 1, 1)
X = rng.integers(1, 7, N)
Xbar = np.cumsum(X) / n
fig, ax = pyplot.subplots()
ax.plot((0, 1000), (3.5, 3.5))
ax.plot(n, Xbar)
glue("lln", fig)
If the mean of the distribution function \(f\) does not exist, then the law of large numbers is meaningless. This is illustrated for the Cauchy distribution (see Definition 11.1) in Fig. 12.2.
Show code cell source Hide code cell source
import numpy as np
from matplotlib import pyplot
from myst_nb import glue
rng = np.random.default_rng(seed=37)
N = 1000
n = np.arange(1, N + 1, 1)
X = rng.standard_cauchy(N)
Xbar = np.cumsum(X) / n
fig, ax = pyplot.subplots()
ax.plot((0, 1000), (0, 0))
ax.plot(n, Xbar)
glue("lln-cauchy", fig)
12.3. The central limit theorem#
Again, we consider a sequence of i.i.d. samples \(X_1, X_2, \ldots\) with distribution function \(f\). We assume \(f\) has finite mean \(\mu\) and variance \(\sigma^2\).
By the law of large numbers, the shifted sample mean
\[ \overline{X}_n-\mu = \frac{1}{n}(X_1+\cdots+X_n)-\mu \]
has expectation 0.
(Central limit theorem)
The distribution of the sequence of random variables
\[ Y_n = \frac{\sqrt{n}}{\sigma}(\overline{X}_n-\mu) \]
converges to the standard normal distribution as \(n\to\infty\).
Roughly speaking, we can write this as
\[ \overline{X}_n \approx \mu + \frac{\sigma}{\sqrt{n}}Z \]
where \(Z\) is a random variable following a standard normal distribution, or as
\[ \overline{X}_n \longrightarrow \mathcal{N}\biggl(\mu,\frac{\sigma}{\sqrt{n}}\biggr) \quad\text{as }n\to\infty, \]
where \(\mathcal{N}(\mu,\sigma)\) denotes the normal distribution with mean \(\mu\) and standard deviation \(\sigma\).
Show code cell source Hide code cell source
# Adapted from
# https://furnstahl.github.io/Physics-8820/notebooks/Basics/visualization_of_CLT.html
# by Dick Furnstahl (license: CC BY-NC 4.0)
from math import comb, factorial
import numpy as np
from matplotlib import pyplot
from myst_nb import glue
# Mean and standard deviation of our distribution
mu = 0.5
sigma = 1 / np.sqrt(12)
def bates_pdf(x, n):
# https://en.wikipedia.org/wiki/Bates_distribution
return (n / (2 * factorial(n - 1)) *
sum((-1)**k * comb(n, k) * (n*x - k)**(n-1) * np.sign(n*x - k) for k in range(n + 1)))
def normal_pdf(x, mu, sigma):
return (1/(np.sqrt(2*np.pi) * sigma) *
np.exp(-((x - mu)/sigma)**2 / 2))
def plot_ax(ax, n):
Plot the n-th sample mean and the limiting normal distribution.
sigma_tilde = sigma / np.sqrt(n)
x_min = mu - 4*sigma_tilde
x_max = mu + 4*sigma_tilde
ax.set_xlim(x_min, x_max)
# plot a normal pdf with the same mu and sigma
# divided by the sqrt of the sample size.
x_pts = np.linspace(x_min, x_max, 200)
y_pts = normal_pdf(x_pts, mu, sigma_tilde)
z_pts = bates_pdf(x_pts, n)
ax.plot(x_pts, y_pts, color='gray')
ax.plot(x_pts, z_pts)
sample_sizes = [1, 2, 3, 4, 8, 16]
# Plot a series of graphs to show the approach to a Gaussian.
fig, axes = pyplot.subplots(3, 2, figsize=(8, 8))
for ax, n in zip(axes.flatten(), sample_sizes):
plot_ax(ax, n)
glue("clt", fig)
Show code cell source Hide code cell source
from math import comb, factorial
import numpy as np
from matplotlib import pyplot
from myst_nb import glue
p = 0.7
# Mean and standard deviation of our distribution
mu = p
sigma = np.sqrt(p*(1 - p))
def binomial_pmf(k, n):
return comb(n, k) * p**k * (1 - p)**(n - k)
def normal_pdf(x, mu, sigma):
return (1/(np.sqrt(2*np.pi) * sigma) *
np.exp(-((x - mu)/sigma)**2 / 2))
def plot_ax(ax, n):
Plot a histogram on axis ax that shows the distribution of the
n-th sample mean. Add the limiting normal distribution.
sigma_tilde = sigma / np.sqrt(n)
x_min = mu - 4*sigma_tilde
x_max = mu + 4*sigma_tilde
ax.set_xlim(x_min, x_max)
# plot a normal pdf with the same mu and sigma
# divided by the sqrt of the sample size.
x_pts = np.linspace(x_min, x_max, 200)
y_pts = normal_pdf(x_pts, mu, sigma_tilde)
ax.plot(x_pts, y_pts, color='gray')
for k in range(n + 1):
ax.add_line(pyplot.Line2D((k/n, k/n), (0, n * binomial_pmf(k, n)),
sample_sizes = [1, 2, 4, 8, 16, 32]
# Plot a series of graphs to show the approach to a Gaussian.
fig, axes = pyplot.subplots(3, 2, figsize=(8, 8))
for ax, n in zip(axes.flatten(), sample_sizes):
plot_ax(ax, n)
glue("clt-bern", fig)
The law of large numbers and a fortiori the central limit theorem are ‘well-known’ results, but their applicability relies on properties of random variables that are not necessarily satisfied in all
cases. We already saw an example where the law of large numbers fails in Fig. 12.2. Likewise, one needs to be careful when applying the central limit theorem. Mathematical proofs help to clarify
under what conditions these results hold and why. Perhaps surprisingly, a way to prove the law of large numbers and the central limit theorem is through Fourier analysis, via the characteristic
function introduced in Section 10.9. We will not go into the details here.
12.4. Exercises#
Suppose a certain species of tree has average height of 20 metres with a standard deviation of 3 metres. A dendrologist measures 100 of these trees and determines their average height. What will the
mean and standard deviation of this average height be?
Suppose \(X\) and \(Y\) are two independent continuous random variables which follow the standard normal distribution \(\mathcal{N}(0,1)\). In whatever programming language you like, compute and plot
a histogram of \(n\) samples from \(X+Y\) for \(n = 100,1000,\) and \(10000\). What can you conclude about the random variable \(X+Y\)? Repeat the same exercise, but this time with \(X/Y\). What can
you conclude about the random variable \(X/Y\)? Think about the central limit theorem as you explain your reasoning.
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August 2016
August 2016
• 43 participants
• 112 discussions
Dear list, I experience exactly same issue explained here <https://mailman.ntg.nl/pipermail/ntg-context/2015/083790.html>, but apparently the link provided for Hans doesn't exist. I will appreciate
any help. Thanks.
Dear list, when I use a body font size other than the standard, issuing \vfill will move the content off the grid. MWE is below. How do I fix this? Cheers, Henri --- \setupbodyfont[9pt] \setuplayout
[grid=yes] \showgrid \starttext Hello \vfill World \stoptext
Hello, it seems my ConTeXt doesn't find its own 'cow.pdf'. The file is located in c:\Ctx-Beta\tex\texmf-context\tex\context\sample\common\cow.pdf when the whole ConTeXt is located in c:\Ctx-Beta\tex\
My resulting .pdf shows no image; and also there is no info in the .log where ConTeXt tried to find the image. How to tell ConTeXt where to look for "installed-in" images? Best regards, Lukas -- Ing.
Lukáš Procházka | mailto:LPr@pontex.cz Pontex s. r. o. | mailto:pontex@pontex.cz | http://www.pontex.cz | IDDS:nrpt3sn Bezová 1658 147 14 Praha 4 Tel: +420 241 096 751 (+420 720 951 172) Fax: +420
Dear list, I was trying to place float captions in the margin, which according to setup-en.pdf should work with location=outermargin, but it doesn't. The MWE below reproduces this on TL 2016 and on
http://live.contextgarden.net/. Please fix. Cheers, Henri --- \setupfloatcaption [figure] [location=outermargin] \starttext \placefigure [top] {The famous dutch cow!} {\externalfigure[cow]} \stoptext
Hi guys, i need some help. I have installed the linux-armhf version of ConTeXt on my android phone via android deploy, the installation was smooth but when i try to create any document, it compile
fine, but the resulting document seems to be corrupted. You can see the result in the attached file. Thanks in advance. MWE %%%%%% \starttext \input davis \stoptext %%%%%% ppa
Hello, I noticed a problem with standalone context and not with texlive 2016 2 Question 3 Question instead of 1 Question 2 Question Thanks fabrice ####################################################
######## \defineframed [FRAME] [frame=off, offset=0pt, location=low, width=\struttotal, background=color, backgroundcolor=darkred, foregroundcolor=white, forgroundstyle=bold] \defineprocessor
[ACPROCESSOR][command=\FRAME] \defineconversionset[ACCONVERSION][][ACPROCESSOR->n] \startsetups[table:initialize] \setupTABLE[start] [align={middle,lohi},width=0.12\textwidth,type=\tfx,offset=1ex] \
setupTABLE[column][first] [align={middle,right},width=0.52\textwidth] \setupTABLE[1][1][frame=off] \stopsetups \definelabel[TableRow][text=,numbercommand=\FRAME] \starttext \startmidaligned \bTABLE
[setups=table:initialize] \bTR \bTD \eTD \bTD a \eTD \bTD b \eTD \bTD c \eTD \bTD d \eTD \eTR \bTR \bTD \TableRow\ question \eTD \bTD \eTD \bTD \eTD \bTD \eTD \bTD \eTD \eTR \bTR \bTD \TableRow\
question \eTD \bTD \eTD \bTD \eTD \bTD \eTD \bTD \eTD \eTR \eTABLE \stopmidaligned \stoptext
Hello guys, am working with some basic maths here and the latest version of ConTexT. However, i think something has changed in ConTexT, because the following MWE, from Aditya, was working fine
before: \definefontfamily[mainface][rm][Minion Pro] \definefallbackfamily[mainface][math][Minion Pro][math:lowercaseitalic][force=yes] \definefallbackfamily[mainface][math][Minion Pro]
[math:digitsnormal][force=yes] \definefontfamily[mainface][math][TeX Gyre Pagella Math] \setupbodyfont[mainface] \starttext \startformula c^2 = a^2 + b^2 \stopformula \stoptext Now, the italic font
of Minion Pro is replaced for regular. Well, i have two questions: 1) How can i set the Minion Pro italic font for math? 2) Is there any way to set the italic font for numbers too in math? Thank you
Hi, This is based on a question on TeX.SE https://tex.stackexchange.com/questions/326653/context-wrapfigure-interacts… All marginrules except those of level 1 stop working after \placefigure[right].
Here is a minimal example: \useexternalfigure[ctanlion][http://www.ctan.org/lion/ctan_lion_350x350.png][width=4cm] \starttext \placefigure[here,none,right]{}{\externalfigure[ctanlion]} \input knuth \
startmarginrule[2] \input ward \stopmarginrule \startmarginrule[1] \input ward \stopmarginrule \startmarginrule[3] \input ward \stopmarginrule \stoptext The output is attached. Aditya
Dear list, when I typeset multi line equations using mathalignment and grid the descenders of the last line of equation run into the first line of text after that equation. The problem is
particularly bad with the Lucida fonts (I think the large operators are relatively larger than for Latin Modern). I could just increase the spacing after equation (commented out in the MWE below),
but that would introduce an imbalance between space before and space after for all equations without descenders. Furthermore the problem does not arise for single line equations with descenders.
Without grid there is also no problem. How do I fix mathalignment to prevent this? Cheers, Henri --- \setuplayout[grid=yes] \setupbodyfont[lucidaot,9pt] %\setupformulas[spaceafter=
{back,nowhite,2*line}] \showgrid \starttext \input ward \startformula \startmathalignment \NC \sum_{i,\alpha,\beta} c_{i,\alpha}^\dagger c_{i,\beta} \NR \NC \sum_{i,\alpha,\beta} c_{i,\alpha}^\dagger
c_{i,\beta} \NR \stopmathalignment \stopformula %\blank[2*line] \input ward \startformula \sum_{i,\alpha,\beta} c_{i,\alpha}^\dagger c_{i,\beta} \stopformula \input ward \startformula \text{no
descenders} \stopformula \input ward \stoptext
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Use of Which, Who, Whose, Where on GMAT | Experts' Global GMAT PrepUse of Which, Who, Whose, Where on GMAT | Experts' Global GMAT Prep
This short video explains the correct usage of which, who, whose, and where from the perspective of the GMAT sentence correction.
Use of Which, Who, Whose, Where
This short article will cover the usage of which, who, whose, and where from the perspective of GMAT sentence correction. Understanding the usage of these terms is very important for navigating
modifier-based sentence correction questions.
Usage of Which, Who, whose, and Where
These terms, when preceded by a comma, refer to the noun just before the comma. Let us illustrate this concept, through the following example:
Example 1 -
France would play against Brazil, which is a stronger team, in the finals.
In this sentence, the word "which" refers to the noun "Brazil" because it is directly before the comma that "which" follows.
Please go through the following examples to gain further clarity on the usages of these terms.
Example 2 -
Jack would play against John, who is a stronger contender, in the finals.
In this sentence, the word "who" refers to the noun John because "John" directly precedes the comma that "which" follows. The word "who" does not refer to the noun "Jack", in this sentence, as "Jack"
is far away from the comma. "Jack" is the subject of the sentence and the core meaning of the sentence is that "Jack" will play against "John" in the finals; the phrase "who is a stronger contender"
modifies "John", as "John" directly precedes it.
Example 3 -
Jack would play against John, whose record is 90% wins, in the finals.
In this sentence, the word "whose" refers to the noun "John" and conveys the meaning that the record belongs to John. Once again, the word "whose" does not refer to the noun "Jack", in this sentence,
as "Jack" is far away from the comma. "Jack" is the subject of the sentence and the core meaning of the sentence is that "Jack" will play against "John" in the finals; the phrase "whose record is 90%
wins" modifies "John", as "John" directly precedes it.
Example 4 -
The finals are in DC, where Jack made his debut.
In this sentence, the word "where" refers to the noun DC; in this case, the noun is a location. Of course, in this case, there is confusion regarding which noun the term refers to because "which" can
only be used to refer to a place. Nevertheless, please do keep in mind that which will also refer to the noun that directly precedes the comma that it follows.
Understanding the usage of which, who, whose, and where will go a long way towards helping you eliminate answer choices, in the GMAT sentence correction questions.
This article has deliberately been kept brief; for a more elaborate explanation, please refer to Experts' Global's Stage One Sentence Correction videos.
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Planar Graphs – Graphs and Networks – Mathigon
Graphs and NetworksPlanar Graphs
Here is another puzzle that is related to graph theory.
In a small village there are three houses and three utility plants that produce water, electricity and gas. We have to connect each of the houses to each of the utility plants, but due to the layout
of the village, the different pipes and cables are not allowed to cross.
Try to connect each of the houses to each of the utility companies below, without any of your lines intersecting:
Just like the Königsberg bridges before, you quickly discover that this problem is also impossible. It seems that some graphs can be drawn without overlapping edges – these are called planar graphs –
but others cannot.
The complete graph K5 is the smallest graph that is not planar. Any other graph that contains K5 as a subgraph in some way is also not planar. This includes K6, K7, and all larger complete graphs.
The graph in the three utilities puzzle is the bipartite graph K3,3. It turns out that any non-planar graph must either contain a K5 or a K3,3 (or a subdivision of these two graphs) as a subgraph.
This is called Kuratowski’s theorem.
This is a planar graph, but the ${n} vertices have been scrambled up. Rearrange the vertices so that none of the edges overlap.
Euler’s Formula
All planar graphs divide the plane they are drawn on into a number of areas, called faces.
11 Vertices + Faces
15 Vertices + Faces
25 Vertices + Faces
When comparing these numbers, you will notice that the number of edges is always than the number of faces plus the number of vertices. In other words, F + V = E + 1. This result is called Euler’s
equation and is named after the same mathematician who solved the Königsberg Bridges problem.
Unfortunately, there are infinitely many graphs, and we can’t check every one to see if Euler’s equation works. Instead, we can try to find a simple proof that works for any graph…
The simplest graph consists of a single vertex. We can easily check that Euler’s equation works.
Let us add a new vertex to our graph. We also have to add an edge, and Euler’s equation still works.
If we want to add a third vertex to the graph we have two possibilities. We could create a small triangle: this adds one vertex, one face and two edges, so Euler’s equation still works.
Instead we could simply extend the line by one: this adds one vertex and one edge, and Euler’s equation works.
Let’s keep going: if we now create a quadrilateral we add one vertex, two edges and one face. Euler’s equation still works.
Any (finite) graph can be constructed by starting with one vertex and adding more vertices one by one. We have shown that, whichever way we add new vertices, Euler’s equation is valid. Therefore, it
is valid for all graphs.
The process we have used is called mathematical induction. It is a very useful technique for proving results in infinitely many cases, simply by starting with the simplest case, and showing that the
result holds at every step when constructing more complex cases.
Many planar graphs look very similar to the nets of polyhedra, three-dimensional shapes with polygonal faces. If we think of polyhedra as made of elastic bands, we can imagine stretching them out
until they become flat, planar graphs:
This means that we can use Euler’s formula not only for planar graphs but also for all polyhedra – with one small difference. When transforming the polyhedra into graphs, one of the faces disappears:
the topmost face of the polyhedra becomes the “outside”; of the graphs.
In other words, if you count the number of edges, faces and vertices of any polyhedron, you will find that F + V = E + .
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Structural Formula - Complete Structural Formulas, Line Formulas, Three Dimensional Formulas - Condensed structural formulas
Structural Formula
Complete Structural Formulas, Line Formulas, Three Dimensional FormulasCondensed structural formulas
A structural formula is a chemical formula that gives you a more complete picture of a compound than its molecular formula can. While a molecular formula, such as H[2]O, shows the types of atoms in a
substance and the number of each kind of atom, a structural formula also gives information about how the atoms are connected together. Some complex types of structural formulas can even give you a
picture of how the atoms of the molecule are arranged in space. Structural formulas are most often used to represent molecular rather than ionic compounds.
There are several different ways to represent compounds in structural formulas, depending on how much detail needs to be shown about the molecule under consideration.
We will look at complete structural formulas, condensed formulas, line formulas, and three-dimensional formulas.
After you become familiar with the rules for writing complete structural formulas, you find yourself taking shortcuts and using condensed structural formulas. You still need to show the complete
molecule, but the inactive parts can be more sketchily shown. Thus the two formulas above look like this when written in condensed form:
Additional topics
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The Stacks project
Lemma 29.47.7. Let $X$ be a scheme.
1. The category of universal homeomorphisms $Y \to X$ has an initial object $X^{awn} \to X$.
2. Given $Y \to X$ in the category of (1) the resulting morphism $X^{awn} \to Y$ is an isomorphism if and only if $Y$ is absolutely weakly normal.
3. The category of universal homeomorphisms $Y \to X$ which induce ismomorphisms on residue fields has an initial object $X^{sn} \to X$.
4. Given $Y \to X$ in the category of (3) the resulting morphism $X^{sn} \to Y$ is an isomorphism if and only if $Y$ is seminormal.
For any morphism $h : X' \to X$ of schemes there are unique morphisms $h^{awn} : (X')^{awn} \to X^{awn}$ and $h^{sn} : (X')^{sn} \to X^{sn}$ compatible with $h$.
Comments (2)
Comment #4306 by correction_bot on
The proof says we'll only prove (1) and (2) but then talks about $X^{sn}$ which only plays a role in (3) and (4) (should be $X^{awn}$ instead).
Comment #4467 by Johan on
Thanks and fixed here.
There are also:
• 2 comment(s) on Section 29.47: Absolute weak normalization and seminormalization
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Theanets 0.7.3 documentation
class theanets.layers.recurrent.LSTM(size, inputs, name=None, activation='relu', **kwargs)¶
Long Short-Term Memory (LSTM) layer.
An LSTM layer is composed of a number of “cells” that are explicitly designed to store information for a certain period of time. Each cell’s stored value is “guarded” by three gates that permit
or deny modification of the cell’s value:
□ The “input” gate turns on when the input to the LSTM layer should influence the cell’s value.
□ The “output” gate turns on when the cell’s stored value should propagate to the next layer.
□ The “forget” gate turns on when the cell’s stored value should be reset.
The output \(h_t\) of the LSTM layer at time \(t\) is given as a function of the input \(x_t\) and the previous states of the layer \(h_{t-1}\) and the internal cell \(c_{t-1}\) by:
\[\begin{split}\begin{eqnarray} i_t &=& \sigma(x_t W_{xi} + h_{t-1} W_{hi} + c_{t-1} W_{ci} + b_i) \\ f_t &=& \sigma(x_t W_{xf} + h_{t-1} W_{hf} + c_{t-1} W_{cf} + b_f) \\ c_t &=& f_t c_{t-1} +
i_t \tanh(x_t W_{xc} + h_{t-1} W_{hc} + b_c) \\ o_t &=& \sigma(x_t W_{xo} + h_{t-1} W_{ho} + c_t W_{co} + b_o) \\ h_t &=& o_t \tanh(c_t) \end{eqnarray}\end{split}\]
where the \(W_{ab}\) are weight matrix parameters and the \(b_x\) are bias vectors. Equations (1), (2), and (4) give the activations for the three gates in the LSTM unit; these gates are
activated using the logistic sigmoid so that their activities are confined to the open interval (0, 1). The value of the cell is updated by equation (3) and is just the weighted sum of the
previous cell value and the new cell value, where the weights are given by the forget and input gate activations, respectively. The output of the unit is the cell value weighted by the activation
of the output gate.
The LSTM cell has become quite popular in recurrent neural network models. It works amazingly well across a wide variety of tasks and is relatively stable during training. The cost of this
performance comes in the form of large numbers of trainable parameters: Each gate as well as the cell receives input from the current input, the previous state of all cells in the LSTM layer, and
the previous output of the LSTM layer.
The implementation details for this layer come from the specification given on page 5 of [Gra13a].
□ b — vector of bias values for each hidden unit
□ ci — vector of peephole input weights
□ cf — vector of peephole forget weights
□ co — vector of peephole output weights
□ xh — matrix connecting inputs to four gates
□ hh — matrix connecting hiddens to four gates
□ out — the post-activation state of the layer
□ cell — the state of the hidden “cell”
[Hoc97] S. Hochreiter & J. Schmidhuber. (1997) “Long short-term memory.” Neural computation, 9(8), 1735-1780.
[Gra13a] (1, 2) A. Graves. (2013) “Generating Sequences with Recurrent Neural Networks.” http://arxiv.org/pdf/1308.0850v5.pdf
LSTM layers can be incorporated into classification models:
>>> cls = theanets.recurrent.Classifier((28, (100, 'lstm'), 10))
or regression models:
>>> reg = theanets.recurrent.Regressor((28, dict(size=100, form='lstm'), 10))
This layer’s parameters can be retrieved using find:
>>> bias = net.find('hid1', 'b')
>>> ci = net.find('hid1', 'ci')
__init__(size, inputs, name=None, activation='relu', **kwargs)¶
│__init__(size, inputs[, name, activation]) │ │
│add_bias(name, size[, mean, std]) │Helper method to create a new bias vector. │
│add_weights(name, nin, nout[, mean, std, ...])│Helper method to create a new weight matrix. │
│connect(inputs) │Create Theano variables representing the outputs of this layer. │
│find(key) │Get a shared variable for a parameter by name. │
│initial_state(name, batch_size) │Return an array of suitable for representing initial state. │
│log() │Log some information about this layer. │
│output_name([name]) │Return a fully-scoped name for the given layer output. │
│setup() │Set up the parameters and initial values for this layer. │
│to_spec() │Create a specification dictionary for this layer. │
│transform(inputs) │Transform the inputs for this layer into an output for the layer. │
│input_size│For networks with one input, get the input size. │
│num_params│Total number of learnable parameters in this layer. │
│params │A list of all parameters in this layer. │
Set up the parameters and initial values for this layer.
Transform the inputs for this layer into an output for the layer.
inputs : dict of theano expressions
Symbolic inputs to this layer, given as a dictionary mapping string names to Theano expressions. See base.Layer.connect().
outputs : dict of theano expressions
A map from string output names to Theano expressions for the outputs from this layer. This layer type generates a “cell” output that gives the value of each hidden cell in the
Returns: layer, and an “out” output that gives the actual gated output from the layer.
updates : list of update pairs
A sequence of updates to apply inside a theano function.
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Order Of Operations Worksheet 8Th Grade
Order Of Operations Worksheet 8Th Grade - 8th grade 7 units · 121 skills. We'll now cater to all the multiplication. Order of operations worksheets worksheets » order of operations worksheet
operations example; These pdf worksheets on order of operations are ideal for students in grade 4 through grade 8. The worksheets are available both in pdf and html formats (html is editable) and can
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operations (no parentheses; No exponents) on these very basic order of operations worksheets and task cards, expressions and equations have no parenthesis and no exponents. Learn where to start, and
walk through examples in this quick, free lesson for middle grades.
PEMDAS rule & Worksheets
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20++ Order Of Operations Worksheet 7Th Grade Worksheets Decoomo
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Related Post:
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A Student Just Proved Paradox-Free Time Travel Is Possible
In a new peer-reviewed paper, a senior honors undergraduate says he has mathematically proven the physical feasibility of a specific kind of time travel. The paper appears in Classical and Quantum
University of Queensland student Germain Tobar, who the university’s press release calls “prodigious,” worked with UQ physics professor Fabio Costa on this paper. In “Reversible dynamics with closed
time-like curves and freedom of choice,” Tobar and Costa say they’ve found a middle ground in mathematics that solves a major logical paradox in one model of time travel. Let’s dig in.
The math itself is complex, but it boils down to something fairly simple. Time travel discussion focuses on closed time-like curves (CTCs), something Albert Einstein first posited. And Tobar and
Costa say that as long as just two pieces of an entire scenario within a CTC are still in “causal order” when you leave, the rest is subject to local free will.
“Our results show that CTCs are not only compatible with determinism and with the local 'free choice' of operations, but also with a rich and diverse range of scenarios and dynamical processes,”
their paper concludes.
In a university statement, Costa illustrates the science with an analogy:
“Say you travelled in time, in an attempt to stop COVID-19's patient zero from being exposed to the virus. However if you stopped that individual from becoming infected, that would eliminate the
motivation for you to go back and stop the pandemic in the first place. This is a paradox, an inconsistency that often leads people to think that time travel cannot occur in our universe. [L]
ogically it's hard to accept because that would affect our freedom to make any arbitrary action. It would mean you can time travel, but you cannot do anything that would cause a paradox to
Some outcomes of this are grouped as the “butterfly effect,” which refers to unintended large consequences of small actions. But the real truth, in terms of the mathematical outcomes, is more like
another classic parable: the monkey’s paw. Be careful what you wish for, and be careful what you time travel for. Tobar explains in the statement:
“In the coronavirus patient zero example, you might try and stop patient zero from becoming infected, but in doing so you would catch the virus and become patient zero, or someone else would. No
matter what you did, the salient events would just recalibrate around you. Try as you might to create a paradox, the events will always adjust themselves, to avoid any inconsistency.”
While that sounds frustrating for the person trying to prevent a pandemic or kill Hitler, for mathematicians, it helps to smooth a fundamental speed bump in the way we think about time. It also fits
with recent quantum findings from Los Alamos, for example, and the way random walk mathematics behave in one and two dimensions.
At the very least, this research suggests that anyone eventually designing a way to meaningfully travel in time could do so and experiment without an underlying fear of ruining the world—at least not
right away.
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From Scholarpedia
Xiaofei He (2010), Scholarpedia, 5(8):9324. doi:10.4249/scholarpedia.9324 revision #91419 [link to/cite this article]
Laplacianfaces refer to an appearance-based approach to human face representation and recognition. The approach uses Locality Preserving Projection(LPP) to learn a locality preserving subspace which
seeks to capture the intrinsic geometry of the data and the local structure. When the projection is obtained, each face image in the image space is mapped to the low-dimensional face subspace, which
is characterized by a set of feature images, they are called Laplacianfaces. Specifically, Laplacianfaces are the optimal linear approximations to the eigenfunctions of the Laplace Beltrami operator
on the face manifold. The approach of using Laplacianfaces for recognition was developed by Xiaofei He et al. (Xiaofei He et al.2005). This is the first devoted work on face representation and
recognition which explicitly considers the manifold structure.
The motivation of Laplacianfaces is:
• Recently, a number of research efforts have shown that the face images possibly reside on a nonlinear submanifold and some nonlinear method have yield impressive performance on some benchmark
artificial data sets. However, for most of them, it is still unclear to evaluate the maps on novel test data points. So they might not be suitable for some computer vision tasks, such as face
recognition. On the other hand, low-dimensional representations through kernel-based techniques have been developed for face recognition and can discover the nonlinear structure of the face
images. But they are computationally expensive. The Laplacianfaces is proposed against this background.
• Laplacianfaces method aims to preserve the local structure of the image space. It considers the manifold structure which is modeled by an adjacency graph. In many real-world classification
problems, the local manifold structure is more important than the global Euclidean structure, especially when nearest-neighbor like classifiers are used for classification.
• LPP (the algorithm generates Laplacianfaces) shares some similar properties to LLE, such as a locality preserving character. Moreover, it is linear and defined everywhere, may be simply applied
to any new data point to locate it in the reduced representation space. In many real-world classification problems, the local manifold structure is more important than the global Euclidean
structure, especially when nearest-neighbor like classifiers are used for classification. LPP seems to have discriminating power although it is unsupervised.
So Laplacianfaces are expected to be a natural alternative to Eigenfaces in face representation and recognition.
Laplacianfaces generation
Before generating Laplacianfaces, original images are normalized (in scale and orientation) such that the two eyes are aligned at the same position. Then all resampled at the same pixel resolution.
Give M face images with the size of \(h \times w \ ,\) A face image can be represented as a high dimension vector of size \( D (D=h \times w)\) in image space and placed into the set: {\
(x_1,x_2,...,x_m\)}. Then we need to project the image set {\( x_i\)} into the PCA subspace by throwing away the smallest principle components as processing for noise reduction. For the sake of
simplicity, we still use \(x\) to denote the images in the PCA subspace in the following step. The transformation matrix of PCA is denoted by \(W_{pca}\ .\) Laplacianfaces are extracted out of the
image data space by means of Locality Preserving Projection (LPP) in the following manner:
1. Constructing the nearest-neighbor graph. Let G denote a graph with n nodes. The \(i\)th node corresponds to the face image \(x_i\ .\) Put an edge between nodes \(i\) and \(j\) if \(x_i\) and \
(x_j\) are "close". i.e., \(x_j\) is among k nearest neighbors of \(x_i\ ,\) or \(x_i\) is among k nearest neighbors of \(x_j\ .\) The constructed nearest-neighbor graph is an approximation of
the local manifold structure.
2. Choosing the weights. If node\(i\) and \(j\) are connected, put
\(S_{ij} = e^{ - \frac{{\left\| {x_i - x_j } \right\|^2 } } {t} }\)
where \(t\) is a suitable constant. Otherwise, put \(S_{ij}=0. \) The weight matrix \(S\) of graph \(G\) models the face manifold structure by preserving local structure.
3. Eigenmap. Computer the eigenvectors and eigenvalues for the generalized eigenvector problem:
\(XLX^T w = \lambda XDX^T w \)
where \(D\) is a diagonal matrix whose entries are column(or row, since \(S\) is symmetric) sums of \(S\ ,\) \(D_{ii} = \sum\nolimits_j {S_{ij} .} L=D-S\) is the Laplacian matrix. The \(ith\) row
of matrix \(X\) is \(x_i\ .\)
Let \(w_0, w_1, ..., w_{k-1}\) be the solution of the equation above, ordered according to their eigenvalues, \(0 \leqslant \lambda _0 \leqslant \lambda _1 \leqslant ... \leqslant \lambda _{k -
1}\ ,\) These eigenvalues are equal to or greater than zero because the matrices \(XLX^T\) and \(XDX^T\) are both symmetric and positive semidefinite. Thus, the embedding is as follow
\(x \to y = W^T x\)
\(W = W_{PCA} W_{LPP}\)
\(W_{LPP} = [w_0, w_1, ..., w_{k - 1} ] \)
where \(y\) is a k-dimensional vector. \(W\) is the transformation matrix. This linear mapping best preserves the manifold's estimated intrinsic geometry in a linear sense. The column vectors of
\(W\) are the so-called Laplacianfaces.
With its neighborhood preserving character, the Laplacianfaces seem to be able to capture the intrinsic face manifold structure to a larger extent. Figure 3 and Figure 4 shows examples that the face
images with various pose and expression of a person are mapped into two-dimensional subspace.
Locality Preserving Projection(LPP)
This section discusses the Locality Preserving Projection which is used to generate Laplacianfaces. The algorithm are linear projective maps that arise by solving a variational problem that optimally
preserves the neighborhood structure of the data set.
1. The linear dimensionality reduction problem
The generic problem of linear dimensionality reduction is: Given a set \( x_1, x_2, ..., x_m \) in \(R_l\) ( \(l \ll n\)), such that \(y_i\) "represents" \(x_I\ ,\) where \(y_i=A^T x_i\ .\) Our
method is of particular applicability in the special case where \(x_1, x_2, ..., x_m \in\mathcal {M}\) and \(\mathcal{M}\) is a nonlinear manifold embedded in \(R_n\ .\)
2. The algorithm
Locality Preserving Projection is a linear approximation of the nonlinear Laplacian Eigenmap. The algorithmic procedure is formally stated below:
1. Constructing the adjacency graph: Let \(G\) denote a graph with m nodes. We put an edge between nodes \(i\) and \(j\) if \(x_i\) and \(x_j\) are "close". There are two variations:
(a) \(\varepsilon\)-neighborhoods. [ parameter \( \varepsilon \in R \) ] nodes \(i\) and \(j\) are connected by an edge if \(\left\| {x_i - x_j } \right\|^2 < \varepsilon \) where the norm is
the usual Euclidean norm in \(R^n\ .\)
(b) k nearest neighbors. [ parameter \( k \in R \) ] nodes \(i\) and \(j\) are connected by an edge if \(i\) is among \(k\) nearest neighbors of \(j\) or \(j\) is among \(k\) nearest
neighbors of \(i\ .\)
2. Choosing the weights: Two variations for weighting the edges. \(W\) is sparse symmetric \(m \times m \) matrix with \(W_{ij}\) having the weight of the edge joining vertices \(i\) and \(j\ ,
\) and 0 if there is no such edge.
(a)Heat kernel. [parameter \(t \in R\)]. If nodes \(i\) and \(j\) are connected. put
\(W_{ij} = e^{ - \frac{{\left\| {x_i - x_j } \right\|^2 } } {t} }\)
(b) Simple-minded. [No parameter]. \(W_{ij}=1\) if and only of vertices \(i\) and \(j\) are connected by an edge.
3. Eigenmaps: Computer the eigenvectors and eigenvalues for the generalized eigenvector problem:
\(XLX^T a = \lambda XDX^T a \)
where\(D\) is a diagonal matrix whose entries are column(or row, since \(W\) is symmetric) sums of \(W\ ,\) \( D_{ii} = \sum\nolimits_j {W_{ji} } \) is the Laplacian matrix. The \(i_{th}\)
column of matrix \(X\) is \(x_i\ .\)
Let the column vectors \(a_0, ..., a_{l-1} \) be the solutions of the generalized eigenvector problem, ordered according to their eigenvalues, \( \lambda _0 \leqslant \lambda _1 \leqslant ...
\leqslant \lambda _{l - 1} \ .\) Thus the embedding is as follows:
\( x \to y = A^T x_i, \) \(A = (a_0 , a_1, ..., a_{l - 1} ) \)
where \(y_i\) is a \(l\)dimensional vector, and \(A\) is a \( n \times l \) matrix.
The relation between LPP and PCA
This section discusses the close relationship between LPP and PCA. It is worthwhile to point out that if the Laplacian matrix \(L\) is \(\frac{1} {n}I - \frac{1} {{n^2 } }ee^T \ ,\) the matrix \(XLX^
T\) is the data covariance matrix. where \(n\) is the number of data points, \(I\) is the identity matrix, and \(e\) is a column vector taking one at each entry. The Laplacian matrix here has the
effect of removing the sample mean from the sample vectors. In this case, the weight matrix \(S\) takes \(\frac{1} {{n^2 } } \) at each entry, i.e., \( S_{ij} = \frac{1} {{n^2 } }\ .\) \( D_{ii} = \
sum\nolimits_j {S_{ij} = } \frac{1} {n} \ .\) Hence, the Laplacian matrix is \( L = D - S = \frac{1} {n}I - \frac{1} {{n^2 } }ee^T \ .\) Let \(m\) denote the sample mean, i.e., \( m = \frac{1} {n}\
sum\nolimits_i {x_i } \ .\) The proof is follow: \[ XLX^T = \frac{1} {n}X(I - \frac{1} {n}ee^T )X^T \ :\] \[ = \frac{1} {n}XX^T - \frac{1} {{n^2 } }(Xe)(Xe)^T \ :\] \[ = \frac{1} {n}\sum\limits_i
{x_i x_i ^T - \frac{1} {{n_2 } }(nm)(nm)^T } \]
\[ = E[(x - m)(x - m)^T ] + 2mm^T - 2mm^T \ :\]
\[= E[(x - m)(x - m)^T ] \] where \(E[(x - m)(x - m)^T ] \) is just the covariance matrix of the data set. The above analysis shows that the weight matrix \(S\) play a key role in the LPP algorithm.
When we aim at preserving the global structure (PCA), we take \( \varepsilon \) (or \(k\)) to be infinity and choose the eigenvectors of
\(XLX^Tw = \lambda w \)
associated with the largest eigenvalues. Hence, the data points are projected along the directions of maximal variance. When we aim at preserving the local structure, we take \(\varepsilon \) to be
sufficiently small and choose the eigenvectors (of the matrix \(XLX^T\)) associated with the smallest eigenvalues. Hence, the data points are projected along the directions preserving locality. It is
important to note that, when \(\varepsilon \) (or \(k\)) is sufficiently small, the Laplacian matrix is no longer the data covariance matrix and, hence, the directions preserving locality are not the
directions of minimal variance. In fact, the directions preserving locality are those minimizing local variance.
The relation between LPP and LDA
This section presents a theoretical analysis of LPP and its connection to LDA. LDA seeks directions that efficient for discrimination. The projection is found by solving the generalized eigenvalue
\(S_B w = \lambda S_W w\)
where \(S_W\) and \(S_B\) are defined as:
\(S_W = \sum\limits_{i = 1}^l {(\sum\limits_{j = 1}^{n_i } {(x_j ^{(i)} - m^{(i)} )(x_j ^{(i)} - m^{(i)} )^T } )} \)
\( S_B = \sum\limits_{i = 1}^l {n_i (m^{(i)} - m)(m^{(i)} - m)^T } \)
suppose there \(l\) classes. The \(i\)th class contains \(n_i\) sample points. Let \(m^{(i)}\) denote the average vector of the \(x_{(i)}\) denote the random vector associated to the \(i\)th class
and \( {x_j ^{(i)} } \) denote the \(j\)th sample point in the \(i\)th class. We can rewrite the matrix \(S_W\) as: \[ S_W = \sum\limits_{i = 1}^l {(\sum\limits_{j = 1}^{n_i } {(x_j ^{(i)} - m^{(i)}
)(x_j ^{(i)} - m^{(i)} )^T } )} \ :\] \[ = \sum\limits_{i = 1}^l {(\sum\limits_{j = 1}^{n_i } {(x_j ^{(i)} (x_j ^{(i)} )^T - m^{(i)} (x_j ^{(i)} )^T - x_j ^{(i)} (m)^{(i)} } )^T + m^{(i)} (m^{(i)} )^
T ))} \ :\] \[ = \sum\limits_{i = 1}^l {(\sum\limits_{j = 1}^{n_i } {(x_j ^{(i)} (x_j ^{(i)} )^T - m^{(i)} (x_j ^{(i)} )^T - X_j ^{(i)} (m^{(i)} )^T } + m^{(i)} (m^{(i)} )^T ))} \ :\] \[ = \sum\
limits_{i = 1}^l {(\sum\limits_{j = 1}^{n_i } {x_j ^{(i)} (x_j ^{(i)} )^T } - n_i m^{(i)} (m^{(i)} )^T )} \ :\] \[ = \sum\limits_{i = 1}^l {(X_i X_i ^T - \frac{1} {{n_i } }(x_1 ^{(i)} + \cdot \cdot \
cdot + x_{n_i } ^{(i)} )(x_1 ^{(i)} + \cdot \cdot \cdot + x_{n_i } ^{(i)} )^T )} \ :\] \[ = \sum\limits_{i = 1}^l {(X_i X_i ^T - \frac{1} {{n_i } }X_i (e_i e_i ^T )X_i ^T )} \ :\] \[ = \sum\limits_{i
= 1}^l {(X_i L_i X_i ^T )} \]
where \(\sum\limits_{i = 1}^l {(X_i L_i X_i ^T )} \) is the data covariance matrix of the \(i\)th class and \( X_i = [x_1 ^{(i)} ,x_2 ^{(i)} , \cdot \cdot \cdot ,x_{n_i } ^{(i)} ] \) is a \( d \times
n_i \) matrix. \( L_i = I - \frac{1} {{n_i } }e_i e_i ^T \) is a \( n_i \times n_i \) matrix where \(I\) is the identity matrix and \(e_i=(1, 1, ..., 1)^T\) is an \(n_i\)-dimensional vector. To
further simplify the above equation, define\[ X = (x_i, x_2, \cdot \cdot \cdot, x_n ) \]
if \(x_i\) and \(x_j\) both belong to the \(k\)th class \(W_{ij}=\frac{1} {{n_k } }\ ,\) otherwise, \(W_{ij}=0\ .\) \( L = I - W \)
Thus, we get\[ S_W = XLX^T \]
we could regard the matrix \(W\) as the weight matrix of a graph with data points as its nodes. Specially, \(W_{ij}\) is the weight of the edge (\(x_i, x_j\)). \(W\) reflects the class relationships
of the data points. The matrix \(L\) is thus called graph Laplacian, which plays key role in LPP.
Similarly, we can computer the matrix \(S_B\) as follow: \[ S_B = \sum\limits_{i = 1}^l {n_i (m^{(i)} - m)(m^{(i)} - m)^T } \ :\] \[ = (\sum\limits_{i = 1}^l {n_i m^{(i)} (m^{(i)} )^T ) - m(\sum\
limits_{i = 1}^l {n_i (m^{(i)} )^T } )} - (\sum\limits_{i = 1}^l {n_i m^{(i)} } )m^T {\text{ + (}}\sum\limits_{i = 1}^l {n_i } {\text{)mm}}^{\text{T}} \ :\] \[ = (\sum\limits_{i = 1}^l {\frac{1}
{{n_i } } } (x_1 ^{(i)} + \cdots + x_{n_i } ^{(i)} )(x_1 ^{(i)} + \cdots + x_{n_i } ^{(i)} )^T ) - 2nmm^T + nmm^T \ :\] \[ = XWX^T - X(\frac{1} {n}ee^T )X^T \ :\] \[ = X(W - \frac{1} {n}ee^T )X^T \ :
\] \[ = X(W - I + I - \frac{1} {n}ee^T )X^T \ :\] \[ = - XLX^T + X(I - \frac{1} {n}ee^T )X^T \ :\] \[ = - XLX^T + C \]
where \( e = (1, 1, ..., 1)^T \) is a \(n\)-dimensional vector and \( C = X(I - \frac{1} {n}ee^T )^T \) is the data covariance matrix. Thus, the generalized eigenvector problem of LDA can be written
as follow: \[ S_B w = \lambda S_W w \] \[ \Rightarrow (C - XLX^T )w = \lambda XLX^T w \] \[ \Rightarrow Cw = (1 + \lambda )XLX^T w \] \[ \Rightarrow XLX^T w = \frac{1} {{1 + \lambda } }Cw \]
Thus, the projection of LDA can be obtained by solving the following generalized eigenvalue problem,
\(XLX^T w = \lambda Cw\)
The optimal projections correspond to the eigenvectors associated with the smallest eigenvalues. If the sample mean of the data set is zero, the covariance matrix is simply \(XX^T\) which is exactly
the matrix \(XDX^T\) in the LPP algorithm. The analysis above shows that LDA actually aims to preserve discriminating information and global geometrical structure. Moreover, LDA can be induced in the
LPP framework. However, LDA is supervised while LPP can be performed in either supervised or unsupervised manner.
Using Laplacianfaces in face representation and recognition
The locality preserving face subspace is spanned by a set of eigenvectors of \( \hat W_{lpp} = \{ w_0 ,w_1 ,...,w_{k - 1} \}\ .\) Each eigenvector can be displayed by an image. These images are
called Laplacianfaces. A face image can be mapped into the locality preserving subspace by using the Laplacianfaces.
When the Laplacianfaces are created, face recognition become a pattern classification task.
The recognition process has three steps:
• Calculate the Laplacianfaces from the training set of face images. \( \hat W_{lpp} = \{ w_0, w_1, ..., w_{k - 1} \}\ .\) each column vector of \(\hat W_{lpp}\) is a Laplacianfaces.
• A new face\(x_i\) is projected into the face space by \(y_i = \hat W^T _{lpp} x_i\ .\) Where \(\hat W_{lpp}\) is the set of eigenvectors. The vector \(y_i\) is the representation of the new face
in face space.
• To determine which face class \( x_i \) belongs to is find the minimum value of
\(d_k = \left\| {y_i - y_k } \right\|\)
where \(y_k\) is the vector representing the \(k\)th face class. The face \( x_i \) is considered as belonging to class \(k\) if the minimum \(d_k\) is smaller than some predefined threshold \(\
theta _d\ ;\) Otherwise, it is classified as unknown.
Different pattern classifiers, including nearest-neighbor, Bayesian, Support Vector Machine can be applied for face recognition.
• He, Xiaofe; Yan, Shuicheng; Niyogi, Partha and Zhang, HongJiang (2005). Face Recognition Using Laplacianfaces. IEEE Trans. Pattern Anal. Mach. Intell. 27(3): 328-340.
• He, Xiaofei and Niyogi, Partha (2003). Locality Preserving Projections. NIPS, 2003.
• P(1997). Eigenfaces vs. Fisherfaces: Recognition Using Class Specific Linear Projection. IEEE Transactions on Pattern Analysis and Machine Intelligence 19(7): 711-720: 1997.
• Belkin, Mikhail and Niyogi, Partha (2001). Laplacian Eigenmaps and Spectral Techniques for Embedding and Clustering. NIPS, 2001.
• Belkin, Mikhail and Niyogi, Partha (2002). Using Manifold Structure for Partially Labeled Classification. NIPS, 2002.
• M. , Brand (2002). Charting a Manifold. NIPS, 2002.
• Chung, F.R.K (1997). Spectral Graph Theory. Regional Conf.Series in Math, 1997.
• Tenenbaum, J.B.; de Silva, V. and Langford, J.C. (2000). A Global Geometric Framework for Nonlinear Dimensionality Reduction. Science, 2000.
• Y., Chang; Hu, C. and Turk, M. (2003). Manifold of Facial Expression. IEEE Int'l Workshop Analysis and Modeling of Faces and Gestures, 2003.
Internal references
• Sheng Zhang and Matthew Turk (2008) Eigenfaces. Scholarpedia, 3(9):4244.
See also
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Writing and Evaluating Expressions with Exponents | sofatutor.com
Writing and Evaluating Expressions with Exponents
Basics on the topic Writing and Evaluating Expressions with Exponents
Writing and Evaluating Expressions with Exponents
Exponents are a critical component of algebra and are essential for simplifying and solving mathematical expressions efficiently. They provide a shorthand way of expressing repeated multiplication
and significantly streamline the representation and calculation of large numbers. By understanding exponents, students can unlock new levels of mathematical proficiency and tackle more complex
problems with confidence.
An exponent is a small numeral, known as the power, placed to the upper right of a base number, indicating how many times the base number is to be multiplied by itself. The expression $5^{2}$, for
instance, means that the base, $5$, is used twice in multiplication: $5 \times 5$.
Writing Expressions with Exponents – Explanation
When it comes to writing expressions using exponential notation, it's crucial to understand the basic rules and how they can be applied to represent multiplication and division succinctly.
Expressions with exponents can include positive powers, negative powers, and even powers of powers.
Evaluating Expressions with Exponents – Example
Evaluating expressions with exponents requires careful adherence to the order of operations. By following theorder of operations, you ensure accurate calculations. Let's evaluate the expression $2^
{3} \times (2 + 3)$. First, calculate the exponent: $2^{3} = 8$. Then add the numbers inside the parentheses: $2 + 3 = 5$. Finally, multiply the results: $8 \times 5 = 40$.
Writing Expressions with Exponents – Application
Writing Expressions with Exponents – Summary
Key Learnings from this Text:
• Exponents simplify the representation of repeated multiplication.
• Positive exponents indicate the number of times a base is multiplied by itself.
• Negative exponents represent division or taking the reciprocal of the base.
• Writing and evaluating expressions correctly involves following the rules of exponents and order of operations.
Continue practicing writing and evaluating expressions with exponents to solidify your understanding of this fundamental math concept.
Writing Expressions with Exponents – Frequently Asked Questions
Transcript Writing and Evaluating Expressions with Exponents
Ever since she was a little girl, Leelee's wanted to be famous, but now that she’s a teenager, she realizes that she’s not especially good at anything. Poor Leelee, she realizes she may have an
impossible dream, but one day, by complete accident, she made the most amazing video. Really, it's amazing! Leelee thinks that this video is her ticket to stardom and wonders, how long will it take
until the video will go viral? Let’s help Leelee figure it out by using expressions with exponents. Leelee has a plan, as soon as her amazing video uploads, she'll send it to three friends and ask
each friend to send it to three friends, and then ask each of those friends to send it to three of their friends, and so on, and so on. The video is really short, so Leelee figures that it should
only take a minute for each of her three friends to watch the amazing video and then share it with three of their friends, and so on, and so on. Let’s write this information in a chart. After the
first minute, her three friends will have watched the video, so that makes three views. After they each share it with three friends, that'll be another nine views. And, when those friends each share
it with three more, that'll be another 27 views. Oh boy! By minute four, those 27 friends will each share with three of their friends, so that’s another 81 more views. Do you see a pattern? The
number of views grows exponentially, so we can write each of these expressions using a base and an exponent. Let's take a look: 3 is equal to 3^1. 3 times 3 is the same as 3 squared, which we all
know is 9, and 3 times 3 times 3 is equal to 3 cubed or 27, and so on, and so on. For each of these exponent expressions, the base, the number we multiply, remains the same. But the exponent, the
number of times we multiply the base with itself, increases by one each time. What’s really neat is you can use this pattern to write an expression to calculate how many new views there will be any
given minute. We can write this as 3 raised to the 'x' power, with ‘x’ representing the given minute. Leelee is impatient. She wants to be super famous ASAP. But what if she shares the video with
five people, and they each share it with five people, and so on, and so on. At minute 10, how many new views will there be? If we multiply 5 by itself 10 times, that's the same as 5 raised to the
10th power! Calculating this out, there will be 9,765,625 people watching her video at minute 10. WOW! Leelee is really psyched! The video has finally finished uploading! She's gonna watch it so that
she can be the very first viewer of this soon-to-be-famous video. Soooo cool! Just like Leelee predicted, the video went viral, and the fly is really famous, but Leelee - not so much.
Writing and Evaluating Expressions with Exponents exercise
Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the video Writing and Evaluating Expressions with Exponents.
• Show how to write the given situation as an exonential expression.
Each of the three friends recommends the video to another three friends.
Each indicates multiplication.
$a^n$ is a shorter way to write a multiplication: $a^n=a\times a\times ...\times a$, or $a$ multiplied by itself $n$ times.
If Leelee wants to know how many people are viewing her video in the fifth minute, she calculates $3^5=243$.
She plans to send it to three friends, and after a minute she wants each of those friends to send it to another three friends each, and so on. Let's see what Leelee´s results should be:
□ During the first minute, three people will view the video.
□ During the second minute, $3\times 3$ people will view the video, which simplifies to $9$ people total.
□ During the third minute, $3\times 3\times 3$ people will view the video, which simplifies to $27$ people total.
□ During the fourth minute, $3\times 3\times 3\times 3$ people will view the video, which simplifies to $81$ people total.
□ We can generalize this by saying that during the $x^{th}$ minute, $3^x$ people will view the video.
• Express the following problem as a mathematical expression.
We have that $5^3=5\cdot 5\cdot 5\cdot 5$, while $3^5=3\cdot 3\cdot 3\cdot 3\cdot 3$.
Five friends who should send it to five friends each, gives us $5\times 5=5^2=25$ people viewing the video.
You finally have to multiply $25$ by $5$ once again: $25\times 5=125$.
Check this result with each of the given alternatives.
First, Leelee sends the videos to five friends. Then all of those friends send the video to five friends each. This leads to $5\times 5=5^2=25$.
Then, all of these $25$ people send the video to five friends each. So we finally get $5\times 5\times 5=5^3=125$ people viewing the video in the end.
• Explain how to transform the word problems.
Here you see how a video is shared after $1$, $2$, $3$, and $4$ minutes, when it is shared with three people in the first minute, and those three people share it with three more people in the
second minute, and so on.
Keep the notation for exponents in mind:
□ $3=3^1$
□ $3\times 3=3^2$
□ $3\times 3\times 3=3^3$
□ $3\times 3\times 3\times3=3^4$
So after $3$ minutes, the video is shared with $3^3=27$ people.
In each of the given examples you have to find the base as well as the exponent of a power:
The video is shared with $4$ people. All of those share it with $4$ people each after one minute and so on. So after 5 minutes, we get $4\times 4\times 4...\times 4=4^5=1024$.
Leelee shares the video with $10$ people. All of those share it again with $10$ people each after one minute. And so on. Thus, after 4 minutes, we get $10\times 10\times 10...\times 10=10^4=
This time Leelee sends the video to $7$ people and asks them to send it out to $7$ people each after one minute. Then the video will be sent to $10$ people and so on. So after 6 minutes, we have
$7\times 7\times 7...\times 7=7^6=117649$.
• Find the right exponential expression.
The base of $a^n$ is $a$.
The exponent of $a^n$ is $n$.
For example, $4\times 4\times 4=4^3$.
New Years Messages
She sends the messages to $6$ friends. After one hour each of those friends sends a message to another $6$ friends. So it's $6\times 6$. After another hour all of those $6\times 6$ friends send a
message to $6$ friends each ... So we get $6^4$ messages sent after three hours.
A bacterium doubles after one period to $2=2^1$ bacteria. After another period there are $2^2$ bacteria. So we can see that $2^n$ there are bacteria after $n$ periods.
Giraffe Legs
First, determine the number of giraffes: $4$ giraffes in $4$ zoo exhibits leads to $4\times 4$ giraffes. Each of the giraffes has $4$ legs. So, in total we can count $4\times 4\times 4=4^3$ legs.
Mateo's Flyer
To promote his new taco bar, Mateo distributes flyers: first to $5$ people. All of those people distribute flyers to $5$ people each. After $4$ distributions $5^4$ flyers are given away.
• Label the base as well as the exponent.
The exponent is the number of times you multiply the base by itself.
In general a power is given by $a^n$, where $a$ stands is the base of the power.
Keep the meaning of the corresponding positions in mind.
In the example beside, $7$ is the base while $5$ is the exponent.
In general a power is given by $a^n$, where $a$ is called the base and $n$ is called the exponent.
You can read it as $a$ raised to the power of $n$.
• Determine the corresponding expression.
The volume of any rectangular prism is given by
width $\times$ length $\times $ height.
You can simplify as follows:
Just multiply the coefficients.
The volume of a cube with the length $4$ is given by $64$.
The volume of a cube can be determined by raising the side to the power of $3$. This leads to $a^3$.
The volume of a rectangular prism is given by width $\times$ length $\times $ height. Thus, we get $(3h)(4h)(h)=(3)(4)h^3=12h^3$.
The area of a square is given by the length squared: $s^2$.
More videos in this topic Writing and Evaluating Expressions and Formulas
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Tag: FEM
• Let’s calculate material tangent of hyperelastic material, taking neo-Hookean for instance. The elastic energy of neo-Hookean material is written as \psi^e=\frac{\mu}2tr(\bold{C})-\mu\ln J A
penalty method is applied to constrain volume unchanged \psi^b=\frac{\kappa}2(J^2-2\ln J-1) The total free energy density is therefore written into \psi=\psi^e+\psi^b The stress is obtained by
derivative P_{iA}=\frac{\partial\psi}{\partial F_{iA}}=\frac{\partial\psi^e}{\partial F_{iA}}+\frac{\partial\psi^b}{\partial F_{iA}} The elastic…
• F-bar method is a element technique used in linear elements to alleviate volumetric locking. This method replaces F of each integration point with F-bar. Index notation Displacement interpolation
u_i=N_{iK}d_k Deformation gradient is calculated as F_{iA}=\delta_{iA}+N_{iK,A}d_K=\delta_{iA}+B_{iKA}d_K where d_K is nodal displacement Total lagrangian shceme without traction and body force
is \delta U=\int P_{iA}\delta F_{iA}dV and nodal…
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Paired Folder - Action Taken Property
Could the PairedFolder object add a property to show whether a nonexistent folder action was taken. My use case is for deciding whether it is okay to link the paired tab. I only want to do that if
the requested pair path exists.
Wouldn't this be the job of FSUtil.Exists()?
My problem occurs when If non-existent: is set to Go up to first existing parent. For example, if C:/Test is paired with D:/Test and Apply setting to all sub-folders is turned on then opening C:/Test
/NoMatchInPair returns a valid pair of D:/Test (the parent). I only want to link folders that exist in both locations.
Hmm, yeah, I don't think you have any way of knowing what the original matched path was, as the object will return the parent path if that's where it ultimately ended up (when configured to do so).
My simple test, FWIW:
C:\ProgramData exists but D:\ProgramData doesn't.
The PairedFolder object for C:\ProgramData pairs it with D:\ (which makes sense; it's where the lister would go),
var fsu = DOpus.FSUtil;
var pair = fsu.GetFolderPair("C:\\ProgramData");
var otherPath = pair.path;
if (fsu.Exists(otherPath))
DOpus.Output("Exists: " + otherPath);
DOpus.Output("Doesn't exist:" + otherPath);
I haven't looked at the code yet, but maybe we can add a property that tells you the original matched path, before any existence checks and parenting was done, as well as a flag saying if the path
was changed. I've added that to the list.
13.9.8 adds the parent_level which will return >0 if the initial folder didn't exist.
1 Like
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Quantitative Strategy
Quantitative trading is a strategy which uses mathematical models and price movement theories to enter and exit a trade. The strategies are back-tested using the historical price data points and
refined to create the future entry and exit points. These techniques have been in existence even before the advent of computers, but then it was very time consuming and complex. Many common
investment theories like those of diversification have all evolved from quantitative studies.
The advantage of trading with quant strategies is that they suggest trades based on back-tested strategies which have offered good returns in the past. Also, there’s a proper discipline attached to
the strategy due to the model defined entry and exit points. Most strategies also provide the maximum profit or loss generated for any specific time frame, which helps define the risk in the
The disadvantages of these types of strategies are that there can be many unforeseen circumstances or events that can cause the entire strategy to collapse and create a much bigger loss/ risk than
the one foreseen earlier.
Pair trade strategy:
Pair trade strategy is one of the many strategies based on Mathematical Quant analysis. This strategy involves taking a long and a short position in two pairs respectively which have good fundamental
and mathematical correlation. Pair trading strategy is also known as ‘Market Neutral’ Strategy which means overall market direction doesn’t affect the trade as the trade is made of hedge positions,
i.e. one buy trade and one sell trade.
The pair trade strategy is a trade on the spread or the ratio of two products. The prerequisite for any pair trade strategy is the requirement of a couple of pairs which have very similar
fundamentals, for example, German DAX Index & France CAC index. These pairs also need to have a positive correlation in terms of movement statistically. Correlation measures the strength of
association between the two pairs and the direction of their relationship. Pairs which have a correlation of 0.8 and above are ideal for such kind of strategies.
Spread Vs Ratio Trade:
For understanding the strategy wealso need to understand the difference between the change in spread &ratio. Spread means the absolute difference between the rates of the two instruments.Ex.: If the
current price of US S&P 500 index is 2723 &Russel 2000 index is 1623, then the spread is 2723 – 1623 = 1100. When we buythis spread and it increases to 1200, we make a profit of 200 points.
The ratio for the same index is 1.68 if the ratio is bought, which means buy S$P 500 & sell Russel 2000 index. Now if the ratio increases to 1.8, considering the assumption that the price of Russel
2000 remains the same and the price of S&P 500 would increase to 2921, it would lead to a profit of 100 points approximately.
Another example of absolute spread based trade can be seen in the above chart. One could have initiated the buy Nasdaq 100 index and sell S&P 500 index trade at 3800 levels of spread and closed the
positions when the spread increased to 4000. One would need to do a proportionate number of units in this case which means 1 unit of Nasdaq 100 index for 1 unit of S&P index to execute this view.
In case of a ratio based trade, one needs to trade equal dollar amount of both indices when entering a trade and exiting the same.
Mean Reversion Strategy:
One of the popular ways of defining entry and exit points for a pair trade is the Mean reversion strategy. This strategy uses the moving average of the ratio of prices to determine when there is a
substantial deviation from the mean of the price ratio, and accordingly, one enters into a Buy and Sell trade in the respective instruments based on the expected reversal.
To understand the mean reversion we need to understand the concept of bell curve or normal distribution of price movement.
The above image portrays a normal distribution curve of a sample size. The law of normal distribution states that if we were to map all the price data in a chart, they would form a curve as the
above. The Greek letter σ represents the standard deviation or variation of the price from the mean. One can also see the probability of prices deviating to higher levels given in percentage terms.
Coming back to the case of the pair trades, we expect the spread/ ratio to go back to the mean whenever it deviates substantially from the mean spread/ ratio. One of the easier ways of measuring
deviation for mean reversion strategy is the Z-Score. Z score represents the point on the X-axis above which the price point is plotted.
The following is the back-tested performance on the pair of Nasdaq & S&P 500 for an exposure value of 50,000 dollars on each leg of the strategy. It takes entry and exit for a target of 2000 dollars
and stop-loss of 1500 dollars. The strategy uses 20-day average to calculate the entry and exit points.
│Year │2009│2010│2011│2012 │2013│2014│2015 │2016│2017│2018│
│Profit │2977│0 │2335│-2537│2010│2698│-3106│6220│4722│2566│
The maximum draw down on the same has been $3106.
There are many advantages of pair trades as it’s a direction neutral strategy, but one must maintain strict discipline in terms of the exposure taken, as high exposures can cause huge risk in terms
of any six-sigma event which is beyond the strategy’s purview. Diversification into multiple pairs for the pair trading strategy is also advisable.
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Data Structures Merge Sort - Tutoline offers free online tutorials and interview questions covering a wide range of technologies, including C, C++, HTML, CSS, JavaScript, SQL, Python, PHP, Engineering courses and more. Whether you're a beginner or a professional, find the tutorials you need to excel in your field."
Merge Sort is a popular sorting algorithm that falls under the category of divide and conquer algorithms. It is widely used due to its efficiency and ability to handle large datasets. In this
article, we will explore the concept of Merge Sort, its working principle, and provide examples to illustrate its implementation.
At its core, Merge Sort works by dividing the input array into smaller subarrays, sorting them individually, and then merging them back together to obtain the final sorted array. The algorithm
follows a recursive approach, repeatedly dividing the array until each subarray contains only one element. Then, it merges the subarrays in a sorted manner to obtain the final sorted array.
The key step in Merge Sort is the merging process. It takes two sorted subarrays and combines them into a single sorted subarray. To achieve this, the algorithm compares the elements from both
subarrays and selects the smaller element to be placed in the merged array. This process continues until all the elements from both subarrays are merged into the final sorted subarray.
One of the advantages of Merge Sort is its ability to handle large datasets efficiently. Since the algorithm divides the array into smaller subarrays, it can take advantage of parallel processing or
multi-threading to speed up the sorting process. This makes Merge Sort a suitable choice for sorting large datasets in parallel computing environments.
Another advantage of Merge Sort is its stability. A sorting algorithm is considered stable if it maintains the relative order of elements with equal keys. In other words, if two elements have the
same key, their relative order in the sorted array should be the same as their order in the original array. Merge Sort achieves this stability by always merging the subarrays in a way that preserves
the relative order of equal elements.
However, Merge Sort also has some drawbacks. One of the main drawbacks is its space complexity. The algorithm requires additional space to store the subarrays during the merging process. This can be
a concern when sorting very large arrays, as it may require a significant amount of additional memory.
In conclusion, Merge Sort is a powerful sorting algorithm that offers efficient and stable sorting of large datasets. Its divide and conquer approach, along with its ability to handle parallel
processing, makes it a popular choice in various applications. However, the additional space required for the merging process can be a drawback in certain scenarios. Nonetheless, Merge Sort remains a
valuable tool in the field of sorting algorithms.
Working Principle of Merge Sort
The Merge Sort algorithm follows a simple yet effective approach to sort a given list of elements. It breaks down the list into smaller subproblems, sorts them individually, and then merges them back
together to obtain the final sorted list.
The key steps involved in the Merge Sort algorithm are as follows:
1. Divide: The given list is divided into two halves until each sublist contains only one element. This is done recursively until the base case is reached.
2. Conquer: Each sublist is sorted individually using the Merge Sort algorithm.
3. Merge: The sorted sublists are merged back together to obtain the final sorted list. This is done by comparing the elements from each sublist and placing them in the correct order.
Let’s take a closer look at each step in the Merge Sort algorithm:
During the divide step, the given list is divided into two halves. This process is repeated recursively until each sublist contains only one element. The base case is reached when a sublist has only
one element, as a single element is already considered sorted.
For example, let’s say we have a list of 8 elements: [7, 3, 2, 6, 1, 4, 8, 5]. The divide step would first split the list into two halves: [7, 3, 2, 6] and [1, 4, 8, 5].
Next, each of these halves is further divided into smaller sublists until each sublist contains only one element. This is done recursively until the base case is reached.
Continuing with our example, the first half [7, 3, 2, 6] would be divided into [7, 3] and [2, 6]. Then, these sublists would be divided further until each sublist contains only one element.
Once the divide step is complete and each sublist contains only one element, the conquer step begins. In this step, each sublist is sorted individually using the Merge Sort algorithm.
For example, let’s consider the sublist [7, 3]. The conquer step would sort this sublist by recursively applying the divide and conquer steps. The sublist would be divided into [7] and [3], which are
already considered sorted. Then, these sublists would be merged back together to obtain the sorted sublist [3, 7].
This process is repeated for each sublist until all sublists are sorted individually.
After the conquer step, we have a set of sorted sublists. The final step is to merge these sublists back together to obtain the final sorted list. This is done by comparing the elements from each
sublist and placing them in the correct order.
For example, let’s consider the sorted sublists [2, 6] and [3, 7]. The merge step would compare the first elements from each sublist and place them in the correct order. In this case, 2 is smaller
than 3, so it would be placed first. The process is repeated until all elements from both sublists are merged together, resulting in the sorted sublist [2, 3, 6, 7].
This merging process is repeated for all sorted sublists until the final sorted list is obtained.
In summary, the Merge Sort algorithm follows a divide and conquer approach to sort a given list of elements. It divides the list into smaller subproblems, sorts them individually, and then merges
them back together to obtain the final sorted list.
Example of Merge Sort
Let’s consider an example to understand how Merge Sort works. Suppose we have an unsorted list of integers: [7, 2, 1, 6, 8, 5, 3, 4].
Step 1: Divide the list into smaller sublists:
• [7, 2, 1, 6] and [8, 5, 3, 4]
Step 2: Sort each sublist individually:
• [1, 2, 6, 7] and [3, 4, 5, 8]
Step 3: Merge the sorted sublists:
By following these steps, we successfully sorted the given list using the Merge Sort algorithm.
Merge Sort is a popular sorting algorithm that follows the divide-and-conquer approach. It works by dividing the unsorted list into smaller sublists, sorting them individually, and then merging them
back together to obtain the final sorted list. This algorithm is efficient for large data sets and has a time complexity of O(n log n).
The first step in Merge Sort is to divide the list into smaller sublists. This is done recursively until each sublist contains only one element. In our example, we divided the list [7, 2, 1, 6, 8, 5,
3, 4] into two sublists: [7, 2, 1, 6] and [8, 5, 3, 4].
Next, we sort each sublist individually. This is done by recursively applying the same divide-and-conquer approach. In our example, we sorted the sublists [7, 2, 1, 6] and [8, 5, 3, 4] to obtain [1,
2, 6, 7] and [3, 4, 5, 8] respectively.
Finally, we merge the sorted sublists back together to obtain the final sorted list. This is done by comparing the elements of the sublists and placing them in the correct order. In our example, we
merged [1, 2, 6, 7] and [3, 4, 5, 8] to obtain the sorted list [1, 2, 3, 4, 5, 6, 7, 8].
Merge Sort has several advantages over other sorting algorithms. It is a stable sorting algorithm, meaning that the relative order of equal elements is preserved. It is also efficient for large data
sets and has a worst-case time complexity of O(n log n). However, Merge Sort requires additional space to store the sublists during the sorting process.
In conclusion, Merge Sort is a powerful sorting algorithm that efficiently sorts large data sets. By dividing the list into smaller sublists, sorting them individually, and merging them back
together, Merge Sort achieves the final sorted list. Its divide-and-conquer approach and efficient time complexity make it a popular choice for sorting tasks.
Advantages of Merge Sort
Merge Sort offers several advantages over other sorting algorithms:
• Efficiency: Merge Sort has a time complexity of O(n log n), making it highly efficient for large datasets. This efficiency is achieved through its divide and conquer approach. Merge Sort divides
the dataset into smaller subproblems, sorts them individually, and then merges them back together in the correct order. This divide and conquer strategy reduces the number of comparisons and
swaps required, resulting in a faster sorting process.
• Stability: Merge Sort is a stable sorting algorithm, meaning that it preserves the relative order of equal elements. This can be useful in certain scenarios where maintaining the original order
of equal elements is important. For example, when sorting a list of employees by their salaries, if two employees have the same salary, Merge Sort will ensure that their original order is
• Divide and conquer: The divide and conquer approach of Merge Sort allows for easy parallelization, making it suitable for parallel processing. In parallel computing, tasks are divided into
smaller subtasks that can be executed simultaneously on multiple processors. Merge Sort’s divide and conquer strategy naturally lends itself to parallelization. The sorting of each subproblem can
be assigned to a separate processor, and then the sorted subproblems can be merged back together. This parallel processing can significantly reduce the overall sorting time, especially for large
• Adaptability: Merge Sort is an adaptable algorithm that can handle various data types and sizes. It can be easily modified to sort different types of data, such as integers, floating-point
numbers, strings, or even custom objects. Additionally, Merge Sort performs well on both small and large datasets. While it may not be the fastest sorting algorithm for small datasets, its
efficiency becomes more apparent as the dataset size increases.
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using GMP's algorithms in other projects
Torbjorn Granlund tg at swox.com
Fri Aug 29 09:06:39 CEST 2008
"Kevin Sopp" <baraclese at googlemail.com> writes:
I'm working on a public domain c++ multi precision integer library.
I'm specifically interested in the subquadratic algorithm used to
convert strings to large integers. My question is, is it generally
allowed to use GMP's algorithms? While this is probably true for the
ones described in your documentation, is it true as well for the ones
that exist only in code?
If you write your code with the GMP code in the next window, I think
your code might rightfully be seen as derivate work of GMP.
But if you figure out some basic algorithm by reading the code, then
reimplement the algorithm without referring to GMP code, your work is
more likely to be independent.
For the radix conversion algorithms, I think the best approach might
be to read the description in the manual.
(Note however that this list is about GMP, and not the right forum for
discussing other bignum libraries.)
More information about the gmp-discuss mailing list
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Poincaré Conjecture - SOUL OF MATHEMATICS
Poincaré Conjecture
If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the
other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking
either the rubber band or the doughnut. We say the surface of the apple is “simply connected,” but that the surface of the doughnut is not. Poincaré, almost a hundred years ago, knew that a two
dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere.
Clay Mathematics Institute
If a compact three-dimensional manifold M^3 has the property that every simple closed curve within the manifold can be deformed continuously to a point, does it follow that M^3 is homeomorphic to the
sphere S^3?
Henri Poincare’ commented, with considerable foresight, “Mais cette question nous entraˆınerait trop loin”. Since then, the hypothesis that every simply connected closed 3-manifold is homeomorphic to
the 3-sphere has been known as the Poincare’ Conjecture. It has inspired topologists ever since, and attempts to prove it have led to many advances in our understanding of the topology of manifolds.
From the first, the apparently simple nature of this statement has led mathematicians to overreach. Four years earlier, in 1900, Poincare’ himself had been the first to err, stating a false theorem
that can be phrased as follows.
The fundamental group plays an important role in all dimensions even when it is trivial, and relations between generators of the fundamental group correspond to two-dimensional disks, mapped into the
manifold. In dimension 5 or greater, such disks can be put into general position so that they are disjoint from each other, with no self-intersections, but in dimension 3 or 4 it may not be possible
to avoid intersections, leading to serious difficulties. Stephen Smale announced a proof of the Poincare’ Conjecture in high dimensions in 1960. He was quickly followed by John Stallings, who used a
completely different method, and by Andrew Wallace, who had been working along lines quite similar to those of Smale.
Let M be an n-dimensional complete Riemannian manifold with the Riemannian metric g[ij] . The Levi-Civita connection is given by the Christoffel symbols
where g^ij is the inverse of g[ij] . The summation convention of summing over repeated indices is used here and throughout the book. The Riemannian curvature tensor is given by
We lower the index to the third position, so that
The curvature tensor R[ijkl] is anti-symmetric in the pairs i, j and k, l and symmetric in their interchange:
Also the first Bianchi identity holds
The Ricci tensor is the contraction
and the scalar curvature is
We denote the covariant derivative of a vector field v = v^j (∂/∂x^j)by
and of a 1-form by
These definitions extend uniquely to tensors so as to preserve the product rule and contractions. For the exchange of two covariant derivatives, we have
and similar formulas for more complicated tensors. The second Bianchi identity is given by
For any tensor T = T^i[jk] we define its length by
and we define its Laplacian by
the trace of the second iterated covariant derivatives. Similar definitions hold for more general tensors.
The Ricci flow of Hamilton is the evolution equation
for a family of Riemannian metrics g[ij] (t) on M. It is a nonlinear system of second order partial differential equations on metrics.
In August 2006, Grigory Perelman was offered the Fields Medal for “his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow”, but he
declined the award, stating: “I’m not interested in money or fame; I don’t want to be on display like an animal in a zoo.” On 22 December 2006, the scientific journal Science recognized Perelman’s
proof of the Poincaré conjecture as the scientific “Breakthrough of the Year”, the first such recognition in the area of mathematics.
If a compact three-dimensional manifold M3 has the property that every simple closed curve within the manifold can be deformed continuously to a point, does it follow that M3 is homeomorphic to the
sphere S3?
Rajarshi Dey
Publisher Name
Soul Of Mathematics
Publisher Logo
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Quadratic Inequalities Worksheet With Answers
Quadratic equation is a four-dimensional algebraic equation and solving it using quadratic equation worksheets is a great way to learn more about it. This subject deals with slopes, derivatives,
roots and volumes. It also uses complex numbers which are very useful in algebraic equations. The main goal of the quadratic equation is to find the area that a function cuts the x-axis. In addition
to this, it helps in finding solutions for functions on the x-axis, slopes and volumes.
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This free worksheet enables you to plot various functions that solve quadratic equations for you. These include the logarithm of the sinus, the sine wave formula and the exponential functions. These
worksheets were developed by Tony Buzan, and they are very easy to use and understand as well.
quadratic equations can be used for a lot of things. For example, you can plot the solutions for the functions x(i) = a(i) + exp(i) where a(i) is a real number between zero and infinity. It can also
plot the solutions for the quadratic equation ax2 bx c where bx c is a real number between zero and infinity. The other use of the quadratic inequality worksheet is when you want to solve the
quadratic equation using the least-squares method. You can add the function y(i) to the left of the x, and then solve for x using the function y(i) where i is the i-th root of the real number i. This
way you get the solutions for the x and y, where the y-axis shows the x-value of theta.
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The first thing you may notice when using a solving quadratic inequalities worksheet is that it is difficult at first to get the answers. However, once you get used to how it works, you will find
that it gives you very accurate results. And because it is so accurate, you should not worry if it gives you wrong answers – with this type of formula, any calculation is correct.
A good solution is to use the graphing inequalities worksheet. Instead of solving for x, you plot a line connecting the x-values of the quadratic equation. Using the kuta software, you can draw any
type of graphical representation for your results. Moreover, it makes it much easier to plot the solutions, since the kuta software has built in plotting capabilities.
Inequalities Problems Worksheet from quadratic inequalities worksheet with answers , source:topsimages.com
When using a solving quadratic equations worksheet, it is also important to note that the solutions are given in a format that can be understood by anyone. For example, if one of the x values is
invalid, then you simply erase that value from the equation, as it is not worth entering in the answer section. This makes the problem solving process easy and painless. Even better, the figures and
graphs are clear, and the answers are simple to read and understand. What more could you ask for?
There are a number of other features that make the solving quadratic equations worksheet a useful tool for any student or teacher. One of the best features is the availability of a back up worksheet,
which will come in handy should you need to check your calculations. Most graphing worksheets come with tutorials to help new users learn how to use the functions, and even include worksheets and
charts to give you a visual idea of what your results may look like. Many also come with support for common office software like Microsoft Excel. If you’re going to be using this for official
purposes, this is definitely something you want.
Worksheet Quadratic Equations solve Quadratic Equations by Peting from quadratic inequalities worksheet with answers , source:sblomberg.com
So, the conclusion might be that a quadratic equation with one side looking like the sum of the corresponding sides when graphed out can be solved using a quadratic equation worksheet. Of course, you
will probably want more than just an answer on your worksheet. It is always nice to get a sense of the answer so that you know if you’re really getting a big advantage. But, most students won’t
bother much with these worksheets anyway, as they’re basically unnecessary.
Free Worksheets Library Download and Print Worksheets from quadratic inequalities worksheet with answers , source:comprar-en-internet.net
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Solving linear equations and linear inequalities — Basic example from quadratic inequalities worksheet with answers , source:khanacademy.org
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Dynamic Light Scattering (DLS) - Definition & Terms (2024)
There are a number of sources of information that give a mathematical description of the terms used in light scattering. However, these will not usually give assistance in understanding their use in
the practical application of the technique. (Ref, 1,2,3,4,5,6)
The following list of terms gives a descriptive definition, with notes on their specific use in the context ofDynamic Light Scattering.
Z-Average size
The Z-Average size or Z-Average mean used in dynamic light scattering is a parameter also known as the cumulants mean. It is the primary and most stable parameter produced by the technique. The
Z-Average mean is the best value to report when used in a quality control setting as it is defined in ISO 13321 and more recently ISO 22412 which defines this mean as the 'harmonic intensity averaged
particle diameter'.
The Z-average size will only be comparable with the size measured by other techniques if the sample is monomodal (i.e. only one peak), spherical or near-spherical in shape, monodisperse (i.e. very
narrow width of distribution), and the sample is prepared in a suitable dispersant, as the Z-Average mean size can be sensitive to even small changes in the sample, e.g. the presence of a small
proportion of aggregates. It should be noted that the Z-average is a hydrodynamic parameter and is therefore only applicable to particles in a dispersion or molecules in solution.
Cumulants analysis
This is a simple method of analysing the autocorrelation function generated by a DLS experiment. The calculation is defined in ISO 13321 and ISO 22412. As it is a moments expansion it can produce a
number of values, however only the first two terms are used in practice, a mean value for the size (Z-Average), and a width parameter known as the Polydispersity Index (PdI). The Z-Average is an
Intensity-based calculated value and should never be confused with or directly compared to a Mass or Number mean value produced by other methods. The calculation is defined in the ISO standards, so
all systems that use this calculation as recommended should give comparable results if the same scattering angle is used.
Polydispersity Index
This index is a number calculated from a simple 2 parameter fit to the correlation data (the cumulants analysis). The Polydispersity Index is dimensionless and scaled such that values smaller than
0.05 are rarely seen other than with highly monodisperse standards. Values greater than 0.7 indicate that the sample has a very broad size distribution and is probably not suitable for the dynamic
light scattering (DLS) technique. The various size distribution algorithms work with data that falls between these two extremes. The calculations for these parameters are defined in the ISO standard
document 13321:1996 E and ISO 22412:2008.
In light scattering, the term polydispersity and % polydispersity are derived from the Polydispersity Index, a parameter calculated from a Cumulants analysis of the DLS-measured intensity
autocorrelation function. In the Cumulants analysis, a single particle size mode is assumed and a single exponential fit is applied to the autocorrelation function and the polydispersity describes
the width of the assumed Gaussian distribution. In terms of a protein analysis, a % polydispersity less than 20% indicates that the sample is monodisperse.
Diffusion Coefficient
Particles and molecules in suspension/solution undergo Brownian motion. This is the motion induced by the bombardment by solvent molecules that themselves are moving due to their thermal energy. If
the particles or molecules are illuminated with a laser, the intensity of the scattered light fluctuates at a rate that is dependent upon the size of the particles as smaller particles are "kicked"
further by the solvent molecules and move more rapidly. Analysis of these intensity fluctuations yields the velocity of the Brownian motion and hence the particle size using the Stokes-Einstein
relationship. The diffusion coefficient, therefore, defines this Brownian motion of the analyte or particle in that particular solvent environment. The translational diffusion coefficient will depend
not only on the size of the particle "core" but also on any surface structure, as well as the concentration and type of ions in the medium.
Hydrodynamic diameter
The hydrodynamic size measured by Dynamic Light Scattering (DLS) is defined as "the size of a hypothetical hard sphere that diffuses in the same fashion as that of the particle being measured". In
practice though, particles or macromolecules in solution are non-spherical, dynamic (tumbling), and solvated. Because of this, the diameter calculated from the diffusional properties of the particle
will be indicative of the apparent size of the dynamic hydrated/solvated particle. Hence the terminology, Hydrodynamic diameter. The hydrodynamic diameter, or Stokes diameter, therefore is that of a
sphere that has the same translational diffusion coefficient as the particle being measured, assuming a hydration layer surrounding the particle or molecule.
Correlation Curve - or correlation function
The measured data in a dynamic light scattering (DLS) experiment is the correlation curve which should be a smooth, single exponential decay function for a mono-size particle dispersion. Embodied
within the correlation curve is all of the information regarding the diffusion of particles within the sample being measured. By fitting the correlation curve to an exponential function, the
diffusion coefficient (D) can be calculated (D is proportional to the lifetime of the exponential decay). With the diffusion coefficient (D) now known, the hydrodynamic diameter can be calculated by
using a variation of the Stokes-Einstein equation. For a polydisperse sample this curve is a sum of exponential decays.
Y-Intercept or Intercept
In DLS the Y-Intercept, or more simply Intercept, refers to the intersection of the correlation curve on the y-axis of the correlogram. The y-intercept can be used to evaluate the signal-to-noise
ratio from a measured sample and thus is often used to judge data quality. It is usually scaled such that an ideal signal will give a value of 1, and a good system will give intercepts in excess of
0.6, and greater than 0.9 for the best systems.
Deconvolution or Deconvolution algorithm
An algorithm-based approach to resolving a mixture of exponentials derived from a polydisperse sample into a number of intensity values each associated with a discrete size band. The particle size
distribution from dynamic light scattering (DLS) is derived from a deconvolution of the measured intensity autocorrelation function of the sample. Generally, this is accomplished using a
non-negatively constrained least squares (NNLS) fitting algorithm, a common examples being CONTIN.
Count Rate or Photon Count Rate
In DLS this is simply the number of photons detected and is usually stated in a "per second" basis. This is useful for determining the sample quality, by monitoring its stability as a function of
time, and is also used to set instrument parameters such as the attenuator setting and sometimes analysis duration. The count rate needs to be above some minimum value in order to have enough signal
for analysis, however all detectors have a maximum count rate where the response remains linear, and if the count rate is not adjusted automatically, the manufacturer recommendations for adjusting
the count rate must be observed.
Intensity Distribution
The first order result from a DLS experiment is an intensity distribution of particle sizes. The intensity distribution is naturally weighted according to the scattering intensity of each particle
fraction or family. For biological materials or polymers the particle scattering intensity is proportional to the square of the molecular weight. As such, the intensity distribution can be somewhat
misleading, in that a small amount of aggregation/agglomeration or presence or a larger particle species can dominate the distribution. However this distribution can be used as a sensitive detector
for the presence of large material in the sample.
Volume Distribution
Although the fundamental size distribution generated by DLS is an intensity distribution, this can be converted, using Mie theory, to a volume distribution or a distribution describing the relative
proportion of multiple components in the sample based on their mass or volume rather than based on their scattering (Intensity.)
When transforming an intensity distribution to a volume/mass distribution, there are 4 assumptions that must be accepted.
• All particles are spherical
• All particles are homogeneous
• The optical properties of the particles are known, i.e. the real & imaginary components of the refractive index
• There is no error in the intensity distribution
An understanding of these assumptions is particularly important since the DLS technique itself produces distributions with inherent peak broadening, so there will always be some error in the
representation of the intensity distribution. As such, volume and number distributions derived from these intensity distribution are best used for comparative purposes, or for estimating the relative
proportions where there are multiple modes, or peaks, and should never be considered absolute. It is therefore good practice to report the size of the peak based on an intensity analysis and report
the relative percentages only (not size) from a Volume distribution analysis.
Adaptive Correlation
DLS instruments monitor the amount of light scattered by diffusive particles. The intensity of the scattered light is significantly influenced by the size of the particles. For isotropic scatterers,
for example, the intensity is proportional to the 6th power of the particle diameter. In the Zetasizer Nano range, 50% of the sub-runs with the highest count rates are discarded, in order to reduce
the impact of erratic data caused by sample contaminants (larger particles = higher count rates). The new Zetasizer Range uses a new statistical approach - each sub run is individually studied and
depending on how much they differ statistically from the other sub runs, they can be classified as steady-state or transient data.
Steady state data
Steady-state data sets describe particles that are consistently part of the measurement volume and hence are characteristic of the whole sample being analysed. The polydispersity index (PDI) of each
sub run is the key parameter for this data classification. The reasoning behind this approach is that PDI is particularly susceptible to the presence of larger populations but also to other effects
(noise) in the correlation function. With the aid of statistic models, the statistical relevance of PDI can be determined, and sub runs can be classified as representative of the sample or just as
transient events.
Transient data
Conversely, transient data are usually particles that are not representative of the detection volume or the bulk of the sample (i.e. aggregates, dust and other contaminants). The impact (or
frequency) of data labelled as transient events can be verified in the “Run Retention” parameter, which shows the percentage of runs that have been used for the steady-state analysis, and
consequently the percentage of runs that have been excluded. It is important to note that transient data is never deleted from the analysis, and it can be shown by itself, or how it would have
affected the original sample by displaying the unfiltered results. This way the analyst can monitor how adaptive correlation improves the results.
Another advantage of adaptive correlation is the ability to further reduce the time of analysis. It has been found that shorter sub-runs’ length produces more reliable results by limiting the overall
impact of transient events in the analysis. It has been shown in another application note [Adaptive Correlation: A new approach to produce the most reliable DLS data in less time] that 10x one-second
long sub-runs produce more repeatable results than ten-second long sub-runs. In the new Zetasizer the number of sub-runs’ and their length is determined until a point where adding more data would not
significantly improve the confidence in the result, thereby providing a final result with improved reproducibility.
Particles larger than 1/10 of the λlaser display an angular dependency on the intensity of light scattered. Furthermore, this effect becomes exponentially more significant with increasing particle
sizes, up to a point where the particles’ scattering is a complex function of maxima and minima depending on the detection angle.
With its considerable distortion towards forward angles of detection, it has been recommended that when searching for the presence of aggregates, the 13° detector should be used. In the Zetasizer
Nano software, there is a dual angle measurement feature, which allows for two individual measurements to be performed at backscattering and forward scattering angles of detection which allows for a
more complete picture to be collected. Nevertheless, analysts would be presented with two different results rather than just one plot representative of their whole sample. In the new Zetasizer Ultra,
instead of just two, there are three detectors positioned at different angles (back, side and forward) which can be used to obtain a single higher resolution result - Multi-Angle Dynamic Light
Scattering (MADLS^®).
DLS as a technique is known to have limitations in resolving different size populations within the same sample. MADLS uses the angular dependency of the scattered light to improve the resolution of
the technique by combining the information obtained at the different angles and giving a single, higher resolution size distribution. It is important to note that the range of concentrations that can
be used in this type of measurement is more limited compared to those of backscattering (NIBS®) analysis, as some effects usually present at forward and side scattering measurements may also be
detected (e.g. multiple scattering, number fluctuations, etc). MADLS results are primarily displayed as volume-weighted particle size distributions, but can also be converted to intensity
(back-scattered weighted) and number particle size distributions allowing for even more information to be extracted.
Particle concentration
The Zetasizer Ultra, by measuring the particle size and the angular dependent intensity of the scattered light - from which the buffer scattered intensity (background) is subtracted, can provide
information on the number particles per mL of solution. Moreover, if different populations are present in a sample, it can also produce a reliable particle concentration for each mode present, since
it uses the same principles of a MADLS measurement (higher resolution size determination). Particle concentration results can be reported as cumulative particle concentration plots, distributed
particle concentration or as a total particle concentration value. Analogous to a MADLS measurement, the range of concentrations that can be used is more limited than when performing a NIBS®
Even though particle concentration is displayed as a stand-alone measurement in the ZS XPLORER software, it is an extension of a multi-angle DLS measurement, and thus a MADLSresult is also obtained.
1. International Standard ISO13321 Methods for Determination of Particle Size Distribution Part 8: Photon Correlation Spectroscopy, International Organisation for Standardisation (ISO) 1996.
2. International Standard ISO22412 Particle Size Analysis - Dynamic Light Scattering, International Organisation for Standardisation (ISO) 2008.
3. Dahneke, B.E. (ed) Measurement of Suspended Particles by Quasi-elastic Light Scattering, Wiley, 1983.
4. Pecora, R. Dynamic Light Scattering: Applications of Photon Correlation Spectroscopy, Plenum Press, 1985.
5. Washington, C. Particle Size Analysis In Pharmaceutics And Other Industries: Theory And Practice, Ellis Horwood, England, 1992.
6. Johnson, C.S. Jr. and Gabriel, D.A. Laser Light Scattering, Dover Publications, Inc., New York 1981
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Real Madrid stadium requires seats for home and away end. The
seating project results in $1.2...
In: Finance
Real Madrid stadium requires seats for home and away end. The seating project results in $1.2...
Real Madrid stadium requires seats for home and away end. The seating project results in $1.2 MM/yr of annual savings. The seating project requires a fixed capital investment of $3.5 MM. The working
capital investment is taken as 15/85 of the fixed capital investment. The annual operating cost of the stadium seating is $0.5 MM/yr. Straight-line depreciation is calculated over 10 years (no
salvage value). The corporate income tax rate for the project is 35%. Assuming a discount rate of 15%.. Assume that the FCI and WCI are expended immediately at the outset of the project.
a. Determine the NPV after 10 years, the Discounted Cash Flow Payback Period and the Discounted Cash Flow Return on Investment
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Minimum Uncertainty Calculator - Calculator Wow
Minimum Uncertainty Calculator
In the realm of quantum mechanics and precision science, uncertainty plays a crucial role. The Minimum Uncertainty Calculator is designed to quantify the minimum possible uncertainty in a system, an
essential aspect for scientists and researchers. This tool helps in understanding the limits of measurement accuracy, which is vital for conducting experiments and interpreting results. By
calculating the minimum uncertainty, researchers can better grasp the constraints imposed by fundamental physical principles, leading to more precise and reliable scientific work.
The Minimum Uncertainty Calculator holds significant value for several reasons:
1. Precision in Measurements: It helps in determining the smallest uncertainty that can be achieved, which is crucial for experiments requiring high precision. Understanding this limit ensures that
measurements are as accurate as possible within the constraints of physical laws.
2. Compliance with Heisenberg’s Principle: The calculator is grounded in Heisenberg’s Uncertainty Principle, which states that certain pairs of physical properties cannot be simultaneously measured
with arbitrary precision. This principle is fundamental in quantum mechanics and the calculator provides a practical way to apply it.
3. Improving Experimental Design: By knowing the minimum uncertainty, scientists can design experiments that account for these limits, leading to better experimental setups and more reliable
4. Enhancing Data Interpretation: Accurate calculation of minimum uncertainty allows for more precise data interpretation. This is critical in fields such as quantum physics, where even small
uncertainties can impact the outcome of experiments.
5. Guiding Technological Advances: In technology and engineering, understanding the limits of uncertainty can guide the development of more advanced instruments and techniques, enhancing overall
performance and accuracy.
How to Use
Using the Minimum Uncertainty Calculator is straightforward:
1. Enter the Uncertainty in Position: Input the uncertainty in position (uxu_xux) into the calculator. This value represents the uncertainty associated with the position measurement in meters (m).
2. Calculate Minimum Uncertainty: Click the "Calculate Minimum Uncertainty" button. The calculator uses the formula:up=h4π⋅uxu_p = \frac{h}{4 \pi \cdot u_x}up=4π⋅uxhwhere:
□ upu_pup is the minimum uncertainty.
□ hhh is Planck's constant, approximately 6.626×10−346.626 \times 10^{-34}6.626×10−34 Joule-seconds.
□ uxu_xux is the uncertainty in position.
3. Review the Result: The result will display the minimum uncertainty, helping you understand the fundamental limits imposed on your measurements.
FAQs and Answers
1. What is minimum uncertainty?
Minimum uncertainty is the smallest possible uncertainty that can be achieved in a measurement, given the constraints imposed by fundamental physical principles like Heisenberg’s Uncertainty
2. Why is calculating minimum uncertainty important?
Calculating minimum uncertainty is crucial for ensuring that measurements and experiments are as precise as possible, considering the fundamental limits of measurement accuracy.
3. How do I use the Minimum Uncertainty Calculator?
Enter the uncertainty in position into the calculator and click "Calculate Minimum Uncertainty" to get the result.
4. What if my uncertainty in position is zero?
If the uncertainty in position is zero, it is theoretically impossible to calculate the minimum uncertainty because division by zero is undefined. Ensure that the input value is a positive number.
5. Can I use this calculator for any type of measurement?
This calculator is specifically designed for quantum mechanical measurements where Heisenberg’s Uncertainty Principle applies. It may not be applicable for other types of measurements.
6. What is Planck's constant?
Planck's constant (hhh) is a fundamental physical constant used in quantum mechanics, approximately 6.626×10−346.626 \times 10^{-34}6.626×10−34 Joule-seconds.
7. How precise is the result?
The result is typically displayed in scientific notation to handle very small values accurately.
8. Can this calculator handle different units?
The calculator works with meters for uncertainty in position and provides results in Joule-seconds per meter.
9. Is there a limit to the input values?
The input values should be positive numbers. Ensure the uncertainty in position is realistic and within practical measurement limits.
10. How often should I calculate minimum uncertainty?
Minimum uncertainty should be calculated whenever precision is crucial, such as in quantum mechanics experiments or high-precision measurements.
The Minimum Uncertainty Calculator is a vital tool for anyone involved in high-precision scientific work. By calculating the minimum uncertainty, researchers can gain insights into the fundamental
limits of measurement accuracy, leading to better experimental designs and more reliable results. Understanding and applying these principles ensures that scientific work adheres to the constraints
of physical laws, enhancing the quality and accuracy of research. Whether in quantum mechanics or advanced technological fields, this calculator is essential for achieving precision and excellence.
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How to Think Exponentially and Better Predict the Future
This is the third in a four-part series looking at the big ideas in Ray Kurzweil’s book The Singularity Is Near. Be sure to read the other articles:
“The future is widely misunderstood. Our forebears expected it to be pretty much like their present, which had been pretty much like their past.” –Ray Kurzweil, The Singularity Is Near
We humans aren’t great predictors of the future. For most of history, our experience has been “local and linear.” Not much change occurred generation to generation: We used the same tools, ate the
same meals, lived in the same general place.
Though the pace of technology is progressing exponentially, the default mode of our caveman brains is to think linearly.
As a result, we’ve developed an intuitive outlook of the future akin to how we approach a staircase—having climbed a number of steps, our prediction of what’s to come is simply steps followed by more
steps, with each day expected to be roughly like the last.
But, as Ray Kurzweil describes in The Singularity Is Near, the rapid growth of technology is actually accelerating progress across a host of domains. This has led to unexpected degrees of
technological and social change occurring not only between generations, but within them.
Against our intuition, today the future is unfolding not linearly but exponentially, making it challenging to predict just what will happen next and when. This is why the pace of technological
progress tends to surprise us, and we find ourselves in situations like this:
How do we prepare for a future tracking to exponential trends, if we aren’t accustomed to thinking this way? Let’s start with the basics of exponential growth.
What is exponential growth?
Unlike linear growth, which results from repeatedly adding a constant, exponential growth is the repeated multiplication of a constant. This is why linear growth produces a stable straight line over
time, but exponential growth skyrockets.
Here’s another way to think about it: imagine you are going to walk down a road taking steps a meter in length. You take 6 steps, and you’ve progressed six meters (1, 2, 3, 4, 5, 6). After 24 more
steps, you’re 30 meters from where you began. It’s easy to predict where 30 more steps will get you—that’s the simplicity of linear growth.
However, setting anatomy aside, imagine you could double the length of your stride. Now when you take six steps, you’ve actually progressed 32 meters (1, 2, 4, 8, 16, 32), which is significantly more
than the 6 meters you’d move with equal steps. Amazingly, by step number 30, doubling your stride will put you a billion meters from where you started, a distance equal to twenty-six trips around the
That’s the surprising, unintuitive power of exponential growth.
Exponential growth is deceptive, then explosive
What’s interesting about exponential growth is that when you double your stride, you progress the same distance with each step as all the previous steps combined. Before you hit a billion miles at
step 30, you’re at 500 million miles at step 29. That means that any of your previous steps look minuscule compared with the last few steps of explosive growth, and most of the growth happens over a
relatively short period of time.
Another example: let’s say you want to get to a certain location and you’re going to double your stride again to get there. Progress toward your destination appears distant at one percent of the way
there, but in fact, you’re only seven steps (or doublings) away—and much of all that progress happens in the last step.
The point is we often miss exponential trends in their early stages because the initial pace of exponential growth is deceptive—it begins slow and steady and is hard to differentiate from linear
growth. Hence, predictions based on the expectation of an exponential pace can seem improbable.
Ray Kurzweil gives this example: “When the human genome scan got underway in 1990 critics pointed out that given the speed with which the genome could then be scanned, it would take thousands of
years to finish the project. Yet the fifteen-year project was completed slightly ahead of schedule, with a first draft in 2003.”
Here’s a great visual of exponential growth’s deceptive then explosive nature in computers. See how most of the progress happens right at the end after years of doubling?
Image courtesy of Pawel Sisiak/AI Revolution.
Will exponential growth eventually end?
In practice, exponential trends do not last forever. However, some trends can continue for long periods, driven along by successive technological paradigms.
A broad exponential trend, computing for example, is made up of a series of consecutive S-shaped technological life cycles, or S-curves.
Each curve looks like the letter ‘S’ because of the three growth stages it represents—initial slow growth, explosive growth, and leveling off as the technology matures. These S-curves overlap, and
when one technology slows, a new one takes over and speeds up. With each new S-curve, the amount of time it takes to reach higher levels of performance is less.
Kurzweil lists five computing paradigms in the 20th century: electromechanical, relay, vacuum tubes, discrete transistors, and integrated circuits. When one technology exhausted its potential, the
next took over making more progress than its predecessors.
Planning for an exponential future
“[T]he future will be far more surprising than most people realize, because few observers have truly internalized the implications of the fact that the rate of change itself is accelerating.” Ray
Kurzweil, The Singularity Is Near
The rule of thumb here is: expect to be surprised, then plan accordingly.
For example, what might the next five years look like? One way to forecast them would be to look at the last five and extend this pace forward. By now, the problem with this thinking should be clear:
The pace itself is changing.
A better forecast would be to look at the last five and then reduce the time it will take to make a similar amount of progress in the next five. It’s more likely that what you think will happen in
the next five years will actually happen in the next three.
The practice of exponential thinking isn’t really about the ins and outs of how you plan—you know how to do that—it’s about better timing your plan (whatever it may be).
In fact, Kurzweil’s law of accelerating returns arose from very practical of origins.
“As an inventor in the 1970s, I came to realize that my inventions needed to make sense in terms of the enabling technologies and market forces that would exist when the inventions were introduced,
as that world would be a very different one from the one in which they were conceived,” Kurzweil wrote in the Singularity Is Near.
With a little practice, we can all make better plans by becoming consciously aware of our intuitive, linear expectations and adjusting them for an exponential future.
Why is learning to think exponentially valuable?
This isn’t just an interesting concept—our linear brains can get us into real trouble.
Thinking linearly causes businesses, governments, and individuals to get blindsided by factors that trend to exponential growth. Big firms get disrupted by new competition; governments struggle to
keep policy current; all of us worry our future is out of control.
Exponential thinking reduces some of this disruptive stress and reveals new opportunities. If we can better plan for the accelerating pace, we can ease the transition from one paradigm to the next,
and greet the future in stride.
To learn more about the exponential pace of technology and Ray Kurzweil’s predictions, read his 2001 essay “The Law of Accelerating Returns” and his book, The Singularity Is Near.
Image Credit: Shutterstock
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CSCI 1440/2440 Topics in Algorithmic Game Theory - Spring 2023
This course examines topics in game theory and mechanism design through the lens of computation. Like such a course in an economics department, the focus is the design and analysis of systems
utilized by self-interested agents. Students investigate how the potential for strategic agent behavior can/should influence system design, and the ramifications of conflicts of interest between
system designers and participating agents. Unlike a traditional economics course, however, emphasis on computational tractability is paramount, so that simplicity may trump other design desiderata.
Students will learn to analyze competing designs using the tools of theoretical computer science. They will also use artificial intelligence to build autonomous agents for computational advertising
markets, wireless spectrum auctions, automated negotiation, and prediction markets.
There are two primary learning outcomes intended for students taking this course. Students should be able to:
• Reason about the design of multiagent systems taking into account agents' incentives, a perspective borrowed from economists, as well as computational performance, the bread and butter of
computer scientists.
• Design and build effective autonomous agents for market domains that again incorporate both strategic and computational considerations.
Mathematical maturity and programming experience are necessary for success in this course. The following are specific areas in which basic expertise is assumed, along with suggested courses for
acquiring said expertise.
• Comfort with continuous mathematics: e.g., Math 0180, Math 0350, APMA 0350, or APMA 0360.
• Comfort with probability and statistics: e.g., CS 1450, APMA 1650, APMA 1655, or Math 1620.
• Comfort writing proofs: e.g., CS 22, CS 1010, CS 1550, or any 1000-level Math class.
• Comfort with programming: e.g., CS 4, CS 111, CS 15, CS 17, CS 19, or equivalent.
Some knowledge of Java and Python is assumed; neither language is taught.
For students wishing to enroll in the graduate section of this course, CSCI 2440, knowledge of Markov decision processes and linear programming is also assumed.
This course has no formal prerequisites. However, a sufficient level of mathematical maturity is required, as is knowledge of programming. For the most part, labs and homework assignments will be in
Python, while the final project will be in Java. All assignments are programmed in pairs. If you worry that your programming skills are not up to snuff, take this opportunity to pair with someone who
can help you get up to speed.
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Video Tutorials on the Internal Capital Adequacy Assessment Process for Interest Rate Risk - SAS Risk Data and Analytics
Basel II required that regulated institutions implement an “internal capital adequacy assessment process” (“ICAAP”) as part of the pillar 2 component of regulations promulgated by the Basel Committee
on Banking Supervision. Even five years later, many thoughtful financial institutions risk managers have concerns about whether the approach followed by their institution for ICAAP can be improved.
This blog summarizes some of the key areas in which common practice in interest rate risk management (and by logical extension the associated ICAAP) can be improved.
The recent low interest rate period in the United States, combined with more than two decades of low interest rates in Japan, has highlighted many of the concerns that thoughtful interest rate risk
managers have had with “common practice” interest rate risk analytics and risk systems:
1. Common practice interest rate risk (“ALM”) systems typically use a one factor term structure model (2 factor models are rare), but yield curve movements almost always show a twist in rates that is
inconsistent with one factor models.
2. Common practice interest rate risk (ALM) systems typically assume that interest rate volatility is either constant (as in the Ho and Lee and Merton term structure models, implying a parallel shift
in the yield curve) or declining with maturity (as in the Vasicek and extended Vasicek or Hull-White models), but this is inconsistent with historical data from the U.S. Treasury market, one of the
deepest fixed income markets in the world.
3. Common practice interest rate risk systems have difficulty dealing with the observed interest rate volatility, which shows a strong correlation between the level of short term interest rates and
short term volatility and a much weaker link between rate level and volatility level at longer maturities.
In a recent series of blogs, we have shown how multi-factor interest rate models based on the Heath-Jarrow-Morton “no arbitrage” restrictions can deal with these three issues very successfully:
van Deventer, Donald R. “Heath Jarrow and Morton Example One: Modeling Interest Rates with One Factor and Maturity-Dependent Volatility,” Kamakura blog, www.kamakuraco.com, March 2, 2012.
van Deventer, Donald R. “Heath Jarrow and Morton Example Two: Modeling Interest Rates with One Factor and Rate and Maturity-Dependent Volatility,” Kamakura blog, www.kamakuraco.com, March 6,
van Deventer, Donald R. “Heath Jarrow and Morton Example Three: Modeling Interest Rates with Two Factors and Rate and Maturity-Dependent Volatility,” Kamakura blog, www.kamakuraco.com, March 13,
van Deventer, Donald R. “Heath Jarrow and Morton Example Four: Modeling Interest Rates with Three Factors and Rate and Maturity-Dependent Volatility,” Kamakura blog, www.kamakuraco.com, March 28,
In this blog, we show the implications of a multi-factor model and realistic empirical interest rate volatility assumptions for ICAAP analysis. We use the 3 factor Heath-Jarrow-Morton model and the
interest rate volatility assumptions for those three factors outlined in the March 28 blog.
Stress Testing and ICAAP
One of the most important aspects of the internal capital adequacy assessment process is stress testing, a discipline with very long history in interest rate risk management. Over the last three
decades, both bankers and regulators have relied heavily on parallel shifts of the yield curve itself and ignored the implications of the yield curve shift for interest rate volatility. Ignoring the
impact of interest rate volatility shifts in the ICAAP process when yield curves move is a serious error. The data from the U.S. Treasury market from 1962 to 2011 shows that interest rate volatility
depends on the level of various points on the yield curve in a complex but intuitively attractive way. When these volatility shifts are correctly taken into account, one can see that the projected
levels of interest rates have a much wider potential dispersion than a similar projection from a lower starting yield curve level.
This narrated video from Kamakura shows the stress testing of the U.S. Treasury yield curve of March 31, 2011 in combination with the 3 U.S. Treasury risk factors identified in the March 28, 2012
blog. The volatility of these 3 risk factors generally (but not always) rises with the level of interest rates. The volatilities both rise and fall with the maturity of the forward rate being
analyzed. The result, as shown in this video, is a change in the level of future interest rate dispersion projected by a multi-factor interest rate model:
The video shows 1250 parallel shifts, in one basis point increments, of the Heath Jarrow Morton “bushy tree” for the one year U.S. Treasury spot rate based on the shifted initial yield curve and the
volatilities of the three driving risk factors that are appropriate at starting interest rate risk levels. A video of Monte Carlo scenarios using the same Heath Jarrow Morton no arbitrage
restrictions would give similar results.
Common practice ALM systems which do not have this multi-factor interest rate risk capability and the related ability to pick up multi-factor volatilities will produce incorrect risk assessments and
under-estimate capital needs for most typical interest rate risk positions seen in practice. This is well-known to sophisticated financial institutions regulators and it is likely that a firm using
such common practice techniques will be under increasingly tough regulatory scrutiny with respect to ICAAP. This is particularly true in the current interest rate environment in many countries,
where interest rates have little potential to fall and a much higher potential for sharp increases.
Benchmarking Interest Rate Forecasts for Realism Under ICAAP
Another important failure of common practice in interest rate risk management and ALM analytics involves the consistency of the interest rate forecast assumptions with both history and market
prices. As explained above and in many recent blogs, one factor term structure models assume away yield curve twists even though a twist is in fact what took place on 94% of the business days
studied for the U.S. Treasury market.
More generally, the ICAAP process should ensure that the projected volatility of rates and the collective set of interest rate paths that is assumed was, after the fact, realistic. The best way to
check the realism of interest rate assumptions is to apply them to historical data and measure whether the projected dispersion of interest rates is consistent with the actual paths of interest rates
that came about. The video below does exactly that. In doing this analysis, an interest rate assumption would be rejected at the desired level of statistical significance N (say the 99th or 95th
percentile) if actual rates are either too high or too low relative to the set of assumed interest rate paths. A more subtle interest rate forecasting error comes about if the actual movement of
rates is much less volatile (varying, say, between the 53rd and 47th percentile 100% of the time) than the set of alternative scenarios used for modeling.
The video below could be done either using Monte Carlo simulation or a bushy tree using the Heath Jarrow and Morton approach. For ease of exposition, we again take the bushy tree approach. We
assume that there are three factors driving interest rates and that these three factors have volatilities that vary with the level of rates. We use the actual volatilities that prevailed from 1962
to 2011 and hold them constant. We ask the question, “Are these assumptions consistent with the ultimate movement in rates that actually occurred?” We show that by comparing a projection of 1 year
U.S Treasury rates in a bushy tree with the actual 1 year U.S. Treasury rates that actually came about 1 year, 2 years and 3 years later. The actual rates are depicted by the heavy red line for
12,395 business days in this video:
The video shows that the volatility assumptions were generally accurate over the entire 50 year period with the exception of the extremely high interest rates in the late 1970s and early 1980s and
the extremely low interest rates that have prevailed in the aftermath of the 2006-2010 credit crisis. Given this, updating interest rate volatility assumptions frequently, rather than holding them
constant for 50 years as we have done here, is the obvious way to increase the accuracy of the ICAAP analytics.
Stepping Away from Common Practice to Best Practice
A wide array of financial institutions are committed to moving away from common practice to best practice in reaction to the large number of firms who failed using legacy risk systems in the credit
crisis. Kamakura advises clients on how to do this in a progressive, scalable step by step process that ultimately leads to a sophisticated multi-factor interest rate modeling environment for the
full range of enterprise risks. For information on this process, please contact us at info@kamakuraco.com.
Donald R. van Deventer
Kamakura Corporation
Honolulu, April 17, 2012
© Copyright 2012 by Donald R. van Deventer. All rights reserved.
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Ks3 math
ks3 math Related topics: equations involving rational algebraic expressions
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Author Message
tom369 Posted: Thursday 07th of Oct 10:29
Hi there I have almost taken the decision to hire a math tutor , because I've been having a lot of stress due to math homework lately . each time when I come home from school I waste all
my time with my algebra homework, and in the end I still seem to be getting the incorrect answers. However I'm also not completely sure whether a algebra tutor is worth it, since it's not
cheap , and who knows, maybe it's not even that good . Does anyone know anything about ks3 math that can help me? Or maybe some explanations regarding distance of points,percentages or
decimals? Any ideas will be valued.
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espinxh Posted: Friday 08th of Oct 15:30
I don’t think I know of any resource where you can get your calculations of ks3 math checked within hours. There however are a couple of websites which do offer help , but one has to wait
for at least 24 hours before getting any response.What I know for sure is that, this program called Algebrator, that I used during my college career was really good and I was quite happy
with it. It almost gives the type of results you need.
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Momepi Posted: Saturday 09th of Oct 18:27
I always use Algebrator to help me with my math assignments. I have tried several other online help tools but so far this is the best I have seen. I guess it is the detailed way of
explaining the solution to problems that makes the whole process appear so easy. It is indeed a very good piece of software and I can vouch for it.
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Roun Toke Posted: Monday 11th of Oct 08:21
I think I need to get a copy of this program right away. All I want to know is, where can I get it? Anyone?
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Ashe Posted: Tuesday 12th of Oct 11:07
Take a look at https://softmath.com/about-algebra-help.html. You can find out more information about it and purchase it too.
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Left cosets partition the parent group
Given a group \(G\) and a subgroup \(H\), the left cosets of \(H\) in \(G\) partition \(G\), in the sense that every element of \(g\) is in precisely one coset.
Firstly, every element is in a coset: since \(g \in gH\) for any \(g\). So we must show that no element is in more than one coset.
Suppose \(c\) is in both \(aH\) and \(bH\). Then we claim that \(aH = cH = bH\), so in fact the two cosets \(aH\) and \(bH\) were the same. Indeed, \(c \in aH\), so there is \(k \in H\) such that \(c
= ak\). Therefore \(cH = \{ ch : h \in H \} = \{ akh : h \in H \}\).
Exercise: \(\{ akh : h \in H \} = \{ ar : r \in H \}\).
Suppose \(akh\) is in the left-hand side. Then it is in the right-hand side immediately: letting \(r=kh\).
Conversely, suppose \(ar\) is in the right-hand side. Then we may write \(r = k k^{-1} r\), so \(a k k^{-1} r\) is in the right-hand side; but then \(k^{-1} r\) is in \(H\) so this is exactly an
object which lies in the left-hand side. <div><div>
But that is just \(aH\).
By repeating the reasoning with \(a\) and \(b\) interchanged, we have \(cH = bH\); this completes the proof.
Why is this interesting?
The fact that the left cosets partition the group means that we can, in some sense, “compress” the group \(G\) with respect to \(H\). If we are only interested in \(G\) “up to” \(H\), we can deal
with the partition rather than the individual elements, throwing away the information we’re not interested in.
This concept is most importantly used in defining the quotient group. To do this, the subgroup must be normal (proof). In this case, the collection of cosets itself inherits a group structure from
the parent group \(G\), and the structure of the quotient group can often tell us a lot about the parent group.
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template <class F, class T, class Tol>
std::pair<T, T>
bisect( // Unlimited iterations.
F f,
T min,
T max,
Tol tol);
template <class F, class T, class Tol>
std::pair<T, T>
bisect( // Limited iterations.
F f,
T min,
T max,
Tol tol,
std::uintmax_t& max_iter);
template <class F, class T, class Tol, class Policy>
std::pair<T, T>
bisect( // Specified policy.
F f,
T min,
T max,
Tol tol,
std::uintmax_t& max_iter,
const Policy&);
These functions locate the root using bisection.
bisect function arguments are:
A unary functor (or C++ lambda) which is the function f(x) whose root is to be found.
The left bracket of the interval known to contain the root.
The right bracket of the interval known to contain the root.
It is a precondition that min < max and f(min)*f(max) <= 0, the function raises an evaluation_error if these preconditions are violated. The action taken on error is controlled by the Policy
template argument: the default behavior is to throw a boost::math::evaluation_error. If the Policy is changed to not throw then it returns std::pair<T>(min, min).
A binary functor (or C++ lambda) that specifies the termination condition: the function will return the current brackets enclosing the root when tol(min, max) becomes true. See also predefined
termination functors.
The maximum number of invocations of f(x) to make while searching for the root. On exit, this is updated to the actual number of invocations performed.
The final Policy argument is optional and can be used to control the behaviour of the function: how it handles errors, what level of precision to use etc. Refer to the policy documentation for more
Returns: a pair of values r that bracket the root so that:
f(r.first) * f(r.second) <= 0
and either
tol(r.first, r.second) == true
where m is the initial value of max_iter passed to the function.
In other words, it's up to the caller to verify whether termination occurred as a result of exceeding max_iter function invocations (easily done by checking the updated value of max_iter when the
function returns), rather than because the termination condition tol was satisfied.
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Do Cities With Many Immigrants Have More Crime?
I played with stats from large cities in the US to see if anything correlates well with crime rates.
Warning...this is not rigorous. This is just me playing with data sets that were easy to google because I was curious and couldn't sleep. There are probably legit papers that provide much more
accurate answers.
First, let's check the title question. What do we get if we plot crime rate vs foreign-born population? Defining crime here as total number of murders, rapes, robberies, and aggravated assaults per
100,000 people, you get the following:
There's a slight negative slope. If you aren't familiar, the way you'd read that is that knowing nothing else about two cities, if one has a 10 point higher immigrant population percentage, you'd
expect to see 124 fewer violent crimes per 100,000 people in that city every year. The R^2 is only 0.12 so that's pretty low and implies that the relationship isn't very strong/there are other, major
An obvious question is how this differs if you consider only illegal immigrants. I couldn't find as much data for that, so here's the same plot for 61 cities considering only illegal immigrant
For completeness, here's the original plot for foreign-born population with just those 61 cities:
Same basic trend.
There are a lot of other factors that came to mind, so I decided to put them together and see if any factors were well-correlated with violent crime. I used total population, foreign-born population,
illegal immigrant population, GINI index, and black population stats for 34 cities. This is not very many and this is not rigorous analysis at all, but it's fun to run the numbers.
Based on that, the overall regression r^2 is 0.43. The only stat with a p value < 0.01 is 'black population %', and the coefficient for that stat is 18. The confidence interval for it is 8 to 27.
Like before, the crude interpretation is that if you know nothing else about two cities, if one has a 10 point higher black population percentage, you'd expect to see between 80 and 270 more violent
crimes per 100,000 people in that city every year.
Foreign-born population was the only other one that came close but the p value is 0.08 (coefficient was -18).
I used the following data sources. Note that when a city source was unavailable, I used the stat for that metro area so those are even less accurate.
Sample gaps in the data...
• very limited data set for black populations
• had to mix metro and city data
• data is not all from the exact same time periods
• few cities overall...I'd ideally have a couple hundred US cities
• lots of factors I didn't find data sets for in the limited time I looked...e.g., 'average age', 'average air quality', and 'average July heat index' came to mind as possible contributors
0 comments:
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[Solved] What mass of 95% pure CaCO3 will be required to neutralise 5
What mass of 95% pure CaCO[3] will be required to neutralise 50 mL of 0.5 MHCl solution according to the following reaction?
CaCO[3(s)] +2HCl[(aq)] → CaCl[2(aq)] + CO[2][(g)] + 2H[2]O[(l)]
[Calculate upto second place of decimal point]
This question was previously asked in
NEET 2022 Official Paper (Held On: 17 July, 2022)
View all NEET Papers >
Answer (Detailed Solution Below)
Option 3 : 1.32 g
RPMT Optics Test 1
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10 Questions 40 Marks 10 Mins
Mass percent - It is a concentration term used to determine the strength of the solution. It is defined as the total percentage of the mass of solute present in the given solution.In the given
reaction -
CaCO3(s) +2HCl(aq) → CaCl2(aq) + CO2(g) + 2H2O(l)
The number of moles of pure CaCO3 is half of the number of moles of HCl;
So, the number of moles of pure CaCO3 = \(\frac{1}{2}× moles\hspace{0.1cm} of HCl\)
Moles of HCl = molarity of HCl × volume of HCl in liters (∵ molarity = \(\frac{no. of moles }{1L of solution}\))
= 0.5 × \(\frac{50}{1000} \)
= 0.5 × 0.05 = 0.025
So, the number of moles of pure CaCO[3] is = \(\frac{1}{2}× 0.025\)
weight of pure CaCO3 is = number of moles of CaCO3 pure × molecular wt. of CaCO3
= 0.0125 × 100 (∵ mol wt. of CaCO3 is 100)
Percentage of purity of CaCO3 in sample = \(\frac{wt. \hspace{0.1cm} of\hspace{0.1cm} pure \hspace{0.1cm} CaCO3 }{total \hspace{0.1cm} weight\hspace{0.1cm} of\hspace{0.1cm} sample}\times100\)
given purity = 95%
So, 95% = \(\frac{1.25g }{total \hspace{0.1cm} weight\hspace{0.1cm} of\hspace{0.1cm} sample}\times100\)
total wt. of sample(impure) = \(\frac{1.25g }{95}\times100\) = 1.3157 ∼ 1.32 g
So, a 1.32 g sample of CaCO[3] is required to neutralize the given HCl.
Hence, the correct answer is option 3.
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Sample Chapter - Ryan Cleckner
Sample Chapter
This sample chapter from the Long Range Shooting Handbook is Chapter 9, Units of Measurement. In this chapter I introduce yards, maters, MOA, Mils, velocity, ballistic coefficient, and more! These
topics are explored greater (how to actually use them) in other sections of the book. Below, you’ll find just a preview. If you want to read the whole chapter, subscribe to my newsletter at the
bottom of the page. After doing so, you’ll be able to view the content of the entire chapter on this page and you’ll also get a pdf version emailed to you. Enjoy!
9 Units of Measurement
There are many measurements that we must take into consideration when shooting long range: distance to the target, size of the target, elevation compensation, windage compensation, barometric
pressure, temperature, and others. You need to get familiar with all of them, as we need to speak the same language.
9.1 Linear Measurements
Linear measurements are generally used to describe the distance to a target. However, they are also sometimes used to describe a target’s size for range estimation purposes.
9.1.1 Yards (yds)
A yard is an English/Standard unit of measurement and it equals exactly 3 feet (36 inches).
9.1.2 Meters (m)
A meter is a metric unit of measurement and it is the basic linear unit in the metric system. From this unit of measurement, prefixes are added to describe different lengths. For example, since the
metric prefix for 1000 is “Kilo,” 1000 meters is 1 Kilometer. Likewise, “Centi” is the metric prefix for 1/100th and therefore 100 Centimeters make up 1 meter.
9.1.3 Converting Between Yards and Meters
Notice how the number for the meters is smaller than the number for yards. This will always be the case and it is a good way to confirm your math, when you’re cold, tired and hungry.
(answers at end of chapter)
1. Which unit of measurement is longer, a yard or a meter?
2. When converting from yards to meters, will the number for meters be larger or smaller than the number for yards?
3. 900 yards equals approximately how many meters?
4. 420 meters equals approximately how many yards?
9.1.4 Linear Conversion Charts
The following charts (Figures 9.1-2 and 9.1-3) are used for converting linear measurements.
9.2 Angular Measurements
Angular measurements are used to describe linear size relative to distance. The most common uses are incremental adjustments to the bullet impact, estimating the distance of a known-size target,
“holding” for windage or elevation, and measuring accuracy by shot-group size.
The most important thing to understand about these measurements is that they are angular! For example, when we adjust our scopes, we move the reticle inside the scope which then forces us to move the
barrel of the rifle up, down, left, or right in order to get the reticle back on to the target. This difference between where the rifle’s barrel was pointed prior to an adjustment in windage or
elevation and where the barrel is pointed after the adjustment is a change in angle. This same angular adjustment translates into smaller changes in the bullet’s impact at closer distances and larger
changes at further distances.
To help you understand how an angular measurement translates into a different sizes at different distances, imagine holding two laser pointers next to each other and pointing them down range. If you
spread the two laser pointers apart at a certain angle, the lasers would gradually get further and further apart from each other as they went down range. For a certain angle, however, the rate at
which the dots spread apart is consistent. The dots will be twice as far apart at 200 yds – and ten times as far apart at 1000 yds – as they were at 100 yds. See Figure 9.2-1.
9.2.1 Minute of Angle (MOA)
In the term Minute of Angle, the word “minute” means 1/60th (for example, there are 60 minutes in 1 hour so 1 minute of time is 1/60th of 1 hour) and the word “angle” refers to one of the 360 degrees
in a circle. So, 1 Minute of Angle is 1/60th of a degree. See Figure 9.2-2.
If we spread two laser pointers apart 1 MOA (1/60th of a degree), the dots would be about 1 inch apart at 100 yards, about 2 inches apart at 200 yards, about 3 inches apart at the 300 yards and so
on. Simply stated, this means that 1 Minute of Angle is . . . (to keep reading, subscribe below!)
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Multiply Unit Fractions By Whole Numbers Worksheet 2024 - NumbersWorksheets.com
Multiply Unit Fractions By Whole Numbers Worksheet
Multiply Unit Fractions By Whole Numbers Worksheet – This multiplication worksheet concentrates on training individuals the way to emotionally increase complete amounts. Students can use custom made
grids to match particularly a single question. The worksheets also deal withfractions and decimals, and exponents. You can even find multiplication worksheets with a dispersed residence. These
worksheets really are a should-have for your personal mathematics school. They are often found in type to discover ways to mentally grow entire line and numbers them up. Multiply Unit Fractions By
Whole Numbers Worksheet.
Multiplication of total figures
You should consider purchasing a multiplication of whole numbers worksheet if you want to improve your child’s math skills. These worksheets will help you learn this standard idea. You are able to go
for 1 digit multipliers or two-digit and about three-digit multipliers. Abilities of 10 may also be an incredible alternative. These worksheets will help you to practice extended practice and
multiplication studying the phone numbers. Also, they are the best way to assist your youngster fully grasp the value of knowing the various kinds of total amounts.
Multiplication of fractions
Experiencing multiplication of fractions on a worksheet will help professors prepare and get ready classes effectively. Utilizing fractions worksheets will allow educators to swiftly determine
students’ understanding of fractions. Students can be questioned to finish the worksheet inside a a number of some time and then symbol their strategies to see in which they want further
instructions. Students can be helped by expression issues that relate maths to genuine-daily life scenarios. Some fractions worksheets consist of samples of comparing and contrasting numbers.
Multiplication of decimals
Whenever you multiply two decimal figures, be sure to team them vertically. The product must contain the same number of decimal places as the multiplicant if you want to multiply a decimal number
with a whole number. As an example, 01 x (11.2) x 2 will be equal to 01 x 2.33 x 11.2 except if the merchandise has decimal areas of below two. Then, the item is circular towards the closest total
Multiplication of exponents
A math concepts worksheet for Multiplication of exponents will allow you to training dividing and multiplying numbers with exponents. This worksheet may also supply issues that will need students to
grow two various exponents. By selecting the “All Positive” version, you will be able to view other versions of the worksheet. Apart from, you may also get into special directions on the worksheet on
its own. When you’re completed, you may click “Produce” along with the worksheet will be downloaded.
Section of exponents
The fundamental principle for department of exponents when multiplying numbers is to subtract the exponent inside the denominator from the exponent from the numerator. You can simply divide the
numbers using the same rule if the bases of the two numbers are not the same. By way of example, $23 divided by 4 will identical 27. This method is not always accurate, however. This procedure can
result in misunderstandings when multiplying numbers that happen to be too large or too small.
Linear capabilities
If you’ve ever rented a car, you’ve probably noticed that the cost was $320 x 10 days. So, the total rent would be $470. A linear function of this type has got the kind f(by), exactly where ‘x’ is
the amount of days and nights the car was booked. Furthermore, it offers the form f(by) = ax b, exactly where ‘b’ and ‘a’ are actual numbers.
Gallery of Multiply Unit Fractions By Whole Numbers Worksheet
Multiplying Whole Number With Fractions Worksheets EduMonitor
Free Multiplying Fractions With Whole Numbers Worksheets
Multiplying Fractions By A Whole Number Lessons Tes Teach
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Frequencies and amplitudes of high-degree solar oscillations
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.
Measurements of some of the properties of high-degree solar p- and f- mode oscillations are presented. Using high-resolution velocity images from Big Bear Solar Observatory, we have measured mode
frequencies, which provide information about the composition and internal structure of the Sun, and mode velocity amplitudes (corrected for the effects of atmospheric seeing), which tell us about the
oscillation excitation and damping mechanisms.
We present a new and more accurate table of the Sun's acoustic vibration frequencies, [...], as a function of radial order n and spherical harmonic degree l. These frequencies are averages over
azimuthal order m and approximate the normal mode frequencies of a nonrotating, spherically symmetric Sun near solar minimum. The frequencies presented here are for solar p- and f- modes with [...],
[...], and [...]. The uncertainties, [...] , in the frequencies are as low as 3.1 pHz. The theoretically expected f-mode frequencies are given by [...], where g is the gravitational acceleration at
the surface, [...] is the horizontal component of the wave vector, and [...] is the radius of the Sun. We find that the observed frequencies are significantly less than expected for l > 1000, for
which we have no explanation.
Observations of high-degree oscillations, which have very small spatial features, suffer from the effects of atmospheric image blurring and image motion (or "seeing"), thereby reducing the amplitudes
of their spatial-frequency components. In an attempt to correct the velocity amplitudes for these effects, we have simultaneously measured the atmospheric modulation transfer function (MTF) by
looking at the effects of seeing on the solar limb. We are able to correct the velocity amplitudes using the MTF out to [...]. We find that the frequency of the peak velocity power (as a function of
l) increases with l. We also find that the mode energy is approximately constant out to [...], at which point it begins to decrease. Mode energy is expected to be constant as a function of f if the
modes are excited by stochastic interactions with convective turbulence in the solar convection zone. Finally, we discuss the accuracy of the seeing correction and a test of the correction using the
1989 March 7 partial solar eclipse.
Item Type: Thesis (Dissertation (Ph.D.))
Degree Grantor: California Institute of Technology
Division: Physics, Mathematics and Astronomy
Major Option: Physics
Thesis Availability: Public (worldwide access)
Research Advisor(s): • Libbrecht, Kenneth George
Thesis Committee: • Unknown, Unknown
Defense Date: 2 November 1990
Record Number: CaltechETD:etd-04022004-155134
Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-04022004-155134
DOI: 10.7907/X6C6-R543
Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code: 1249
Collection: CaltechTHESIS
Deposited By: Imported from ETD-db
Deposited On: 02 Apr 2004
Last Modified: 21 Dec 2019 04:01
Thesis Files
PDF (Kaufman_jm_1991.pdf) - Final Version
See Usage Policy.
Repository Staff Only: item control page
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Open Circuit and Short Circuit Characteristics of Synchronous Machine - Electrical Concepts
Open Circuit and Short Circuit Characteristics of Synchronous Machine
Open Circuit Test and Short Circuit Test are performed on a Synchronous Machine to find out the parameters of Synchronous Machine and hence to have an idea of their performance. Open Circuit Test of
Synchronous Machine is also called No Load, Saturation or Magnetizing Characteristics for the reason which will be clear after going through the post.
For getting the Open Circuit Characteristics of Synchronous Machine, the alternator is first driven at its rated speed and the open terminal voltage i.e. voltage across the armature terminal is noted
by varying the field current. Thus Open Circuit Characteristic or OCC is basically the plot between the armature terminal voltage E[f] versus field current I[f] while keeping the speed of rotor at
rated value. It shall be noted that for OCC, the final value of E[f] shall be 125% of the rated voltage.
Figure below shows the connection diagram for performing the Open Circuit Test of Alternator.
As clear from the figure above, an Ammeter is connected in series with the field circuit to measure the field current and a Voltmeter is connected across the armature terminals to note down the
voltage generated. Figure (b) shows the plot between I[f] and E[f]. It can be seen from the graph that the relationship between the field current I[f]and no load generated voltage E[f] is linear up
to certain value of field current but as the the field current increases the relationship no longer remains linear. The linear part of the relationship is because, at small value of filed current the
whole mmf is required by the air gap to create magnetic flux but as the value of mmf exceeds some certain value, the iron parts get saturated and hence the relationship between the flux (No load
generated emf is proportional to flux) and field current no longer remain linear.
Next assume that if there were no saturation (assuming no iron part is present rather only air gap is present), the relationship between the field current and no load voltage would have been a
straight line and that is why the straight line ob in the figure is called Air Gap Line.
Thus we observe that because of saturation in iron parts of machine, the no load generated voltage E[f]does not increase in the same proportion as the increase in field current.
Short Circuit Test of Synchronous Machine:
For performing Short Circuit Test on an Alternator, the machine is driven at rated synchronous speed and the armature terminals are short circuited through an Ammeter as shown in figure below.
Now the field current If is gradually increased from zero until the armature short circuit current reaches its maximum safe value i.e. 125 to 150% of its rated current value. Readings of field
current If and short circuit current are noted and plotted.
If you see the above plot of Short Circuit Test, you notice that the short circuit characteristics of a synchronous machine is a straight line.
Why Short Circuit Characteristics of Synchronous Machine is Straight Line?
For short circuit test, as the armature terminals are shorted, therefore terminal voltage V[t] = 0. Therefore the air gap emf E[r] shall only be enough to provide the leakage impedance drop in the
armature i.e.
E[r] = I[a](R[a] + jX[al]) where X[al] = Armature Leakage Reactance
As we know that, for a Synchronous machine the value of X[al] is of the order of 0.1 to 0.2 per unit and Ra (Armature Resistance) is negligible thus we can write as
X[al] = 0.15 (Taking average value of 0.1 and 0.2)
R[a] = 0
then E[r] = I[a] (R[a] +jX[al]) = 0.15I[a]
Taking rated current of armature, I[a] = 1 pu
Therefore, E[r] = 0.15 pu
You must read Per Unit System in Electrical Engineering
Thus we observe that during short circuit test, the air gap generated emf Er is only 0.15 pu which mean that air gap flux must also be 0.15 pu. As the resultant air gap flux is only 0.15 of its rated
value under normal voltage condition, such a low value of air gap flux does not saturate the iron parts of synchronous machine and hence the short circuit characteristics is a straight line. It shall
also be noted here that, in case of short circuit test the armature mmf is almost entirely demagnetizing in nature which results in very low value of air gap flux.
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7-1 Ratio and Proportion Warm Up Lesson Presentation Lesson Quiz - ppt video online download
1 7-1 Ratio and Proportion Warm Up Lesson Presentation Lesson QuizHolt Geometry
2 A ratio compares two numbers by division.The ratio of two numbers a and b can be written as: a to b a:b , where b ≠ 0
3 Example 1: Writing RatiosWrite a ratio expressing the slope of l.
4 Check It Out! Example 1 Given that two points on m are C(–2, 3) and D(6, 5), write a ratio expressing the slope of m.
5 A ratio can involve more than two numbersA ratio can involve more than two numbers. For the rectangle, the ratio of the side lengths may be written as 3:7:3:7.
6 Example 2: Using Ratios The ratio of the side lengths of a triangle is 4:7:5, and its perimeter is 96 cm. What is the length of the shortest side?
7 Check It Out! Example 2 The ratio of the angle measures in a triangle is 1:6:13. What is the measure of each angle?
8 A proportion is an equation stating that two ratiosare equal. In the proportion , the values a and d are the extremes. The values b and c are the means. When the proportion is written as a:b = c:d,
the extremes are in the first and last positions. The means are in the two middle positions.
9 The product of the extremes ad and the product of the means bc are called the cross products.
16 Example 4: Using Properties of ProportionsGiven that 18c = 24d, find the ratio of d to c in simplest form.
17 Check It Out! Example 4 Given that 16s = 20t, find the ratio t:s in simplest form.
18 Example 5: Problem-Solving ApplicationMarta is making a scale drawing of her bedroom. Her rectangular room is 12 feet wide and 15 feet long. On the scale drawing, the width of her room is 5
inches. What is the length?
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Using the Fibonacci Extensions Using Visual Basic?
Using the Fibonacci Extensions in Visual Basic involves incorporating the mathematical concept of the Fibonacci sequence into a programming environment. The Fibonacci sequence is a series of numbers
where each number is the sum of the two preceding ones, starting with 0 and 1. In Visual Basic, these numbers can be used to calculate Fibonacci Extensions, which are commonly used in technical
analysis to predict potential future price levels in trading.
To implement Fibonacci Extensions in Visual Basic, you first need to calculate the Fibonacci numbers using a loop or recursion. Once you have the Fibonacci numbers, you can use them to calculate the
extension levels by multiplying them by certain ratios (typically 0.382, 0.618, 1.000, 1.382, 1.618, etc.). These extension levels can be used to identify potential support and resistance levels in
the price movements of a financial asset.
By utilizing the Fibonacci Extensions in Visual Basic, traders and analysts can make more informed decisions about when to enter or exit trades based on the predicted price levels. This can help
improve the accuracy of trading strategies and potentially increase profits in the financial markets.
How to combine Fibonacci Extensions with other technical indicators in Visual Basic?
To combine Fibonacci Extensions with other technical indicators in Visual Basic, you can create a custom function or subroutine that calculates the Fibonacci Extensions based on the highs and lows of
a stock price and then compare this data with other technical indicators. Here is an example of how you can do this:
1. Create a function to calculate Fibonacci Extensions:
1 Function FibonacciExtension(highs As List(Of Double), lows As List(Of Double)) As List(Of Double)
2 Dim fibExt As New List(Of Double)
4 ' Calculate Fibonacci levels based on highs and lows
5 ' Add the calculated Fibonacci levels to the list fibExt
7 Return fibExt
8 End Function
1. Create a function to calculate other technical indicators, such as Moving Average or Relative Strength Index (RSI):
1 Function MovingAverage(data As List(Of Double), period As Integer) As List(Of Double)
2 Dim ma As New List(Of Double)
4 ' Calculate Moving Average based on data and period
5 ' Add the calculated Moving Average values to the list ma
7 Return ma
8 End Function
1. Use these functions in your main program to combine Fibonacci Extensions with other technical indicators:
1 Sub Main()
2 Dim highs As New List(Of Double) From {100, 110, 120, 130, 140}
3 Dim lows As New List(Of Double) From {90, 100, 110, 120, 130}
4 Dim fibExt As List(Of Double) = FibonacciExtension(highs, lows)
6 Dim prices As New List(Of Double) From {105, 115, 125, 135, 145}
7 Dim ma As List(Of Double) = MovingAverage(prices, 5)
9 ' Compare Fibonacci Extensions with Moving Average values
10 For i As Integer = 0 To fibExt.Count - 1
11 If fibExt(i) > ma(i) Then
12 Console.WriteLine("Fibonacci Extension is higher than Moving Average")
13 Else
14 Console.WriteLine("Fibonacci Extension is lower than Moving Average")
15 End If
16 Next
17 End Sub
By following these steps, you can combine Fibonacci Extensions with other technical indicators in Visual Basic to analyze stock price movements more effectively.
What are some real-world examples of using Fibonacci Extensions in Visual Basic?
1. Stock Market Analysis: Fibonacci Extensions are commonly used in technical analysis to predict potential price targets in financial markets. Traders and analysts can use Visual Basic to create a
program that calculates Fibonacci Extensions based on historical price data and identifies potential price levels for buying or selling assets.
2. Project Management: Fibonacci Extensions can also be used in project management to estimate time and cost extensions for completing a project. By incorporating Fibonacci Extensions into a Visual
Basic program, project managers can more accurately predict project completion dates and budget requirements.
3. Game Development: Fibonacci Extensions can be utilized in game development to create dynamic and visually interesting animations. Visual Basic can be used to program animations that incorporate
Fibonacci sequences, such as spiral patterns, to add an extra level of depth and complexity to games.
4. Graphic Design: Fibonacci Extensions can be applied in graphic design to create visually appealing layouts and compositions. Visual Basic can be used to create tools that allow designers to
easily incorporate Fibonacci Extensions into their designs, ensuring aesthetic harmony and balance.
5. Architecture and Engineering: Fibonacci Extensions can be used in architecture and engineering to optimize structural designs and layouts. By implementing Fibonacci Extensions into Visual Basic
programs, architects and engineers can efficiently calculate and evaluate design alternatives, leading to more efficient and aesthetically pleasing structures.
What is the accuracy of Fibonacci Extensions predictions in Visual Basic?
The accuracy of Fibonacci Extensions predictions in Visual Basic (or any programming language, for that matter) can vary depending on a number of factors, including the quality of the code, the
accuracy of the input data, and the complexity of the analysis being performed. Generally speaking, Fibonacci Extensions can be a useful tool for predicting potential price levels in financial
markets, but they should be used in conjunction with other technical analysis tools and methods to improve their accuracy. It is important to note that no prediction method is 100% accurate, and
there is always a degree of uncertainty in forecasting future price movements in financial markets.
What are some common pitfalls to avoid when using Fibonacci Extensions in Visual Basic?
1. Incorrect calculation of Fibonacci numbers: Ensure you are accurately calculating Fibonacci numbers when using them as extensions in Visual Basic. Mistakes in this calculation can lead to
incorrect results.
2. Misinterpreting the extension values: Make sure you understand the significance of the Fibonacci extension values and how they are applied in your code. Misinterpreting these values can result in
errors or unexpected behavior.
3. Not validating input values: It is essential to validate the input values used in Fibonacci extension calculations to avoid potential errors or exceptions in your code.
4. Ignoring edge cases: Remember to consider edge cases such as zero and negative values when working with Fibonacci extensions. Failure to account for these cases can lead to incorrect results or
unexpected behavior.
5. Not properly handling overflow: Fibonacci numbers can grow rapidly, leading to potential overflow issues when calculating extensions. It is essential to handle these cases properly to prevent
errors in your code.
How to develop a Fibonacci Extensions trading strategy in Visual Basic?
To develop a Fibonacci Extensions trading strategy in Visual Basic, you can follow these steps:
1. Import necessary libraries: Start by importing the necessary libraries for handling data and mathematical calculations in Visual Basic.
2. Define Fibonacci Levels: Define the Fibonacci levels for your trading strategy. In Fibonacci Extensions, these levels are typically 0%, 100%, 161.8%, and 261.8%.
3. Fetch Data: Retrieve historical price data for the asset you want to trade. This data will be used to calculate Fibonacci Extensions.
4. Calculate Fibonacci Extensions: Use the historical price data to calculate Fibonacci Extension levels. You can do this by identifying swing highs and swing lows in the price data and applying the
Fibonacci ratio to them.
5. Trading Signal: Define your trading signal based on the Fibonacci Extension levels. For example, you may decide to buy when the price reaches the 161.8% level or sell when it reaches the 261.8%
6. Backtesting: Backtest your trading strategy with historical data to see how well it would have performed in the past.
7. Risk Management: Implement risk management techniques to protect your capital, such as setting stop-loss orders and position sizing.
8. Automation: If you want to automate your trading strategy, you can use Visual Basic to create a script that executes trades based on the signals generated by your Fibonacci Extensions strategy.
By following these steps, you can develop a Fibonacci Extensions trading strategy in Visual Basic and potentially improve your trading results.
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7th grade math 7 utahs course 2 workbooks
Algebra Tutorials!
7th grade math 7 utahs course 2 workbooks
Related topics:
Home ti-89 will not factor complex quadratic | fun seventh grade algebra lesson plans | simplifying square roots worksheet"" | games for dividing polynomials by
Rational Expressions monomials | exponents on calculator | c++ code to get factors of positive number | solving quadratic equations completing the square | convert square meters to
Graphs of Rational lineal metres
Solve Two-Step Equations
Multiply, Dividing; Author Message
Exponents; Square Roots;
and Solving Equations mesareve Posted: Thursday 28th of Dec 20:55
LinearEquations Hello Guys , I am urgently in need of assistance for getting through my mathematics exam that is approaching . I really do not want to opt
Solving a Quadratic for the assistance of private masters and web tutoring since they prove to be quite expensive . Could you recommend a perfect tutoring
Equation software that can support me with learning the basics of Intermediate algebra. Particularly, I need assistance on rational expressions and
Systems of Linear factoring polynomials.
Equations Introduction Registered:
Equations and 24.08.2005
Inequalities From: 42° 3' N 83°
Solving 2nd Degree 22' W
Review Solving Quadratic
System of Equations espinxh Posted: Friday 29th of Dec 20:42
Solving Equations & Well I do have a suggestion for you.Sometime back even I was stuck on questions relating to 7th grade math 7 utahs course 2 workbooks,
Inequalities that’s when my elder sister suggested that I should try Algebrator. It didn’t just solve all my queries, but it also explained those
Linear Equations answers in a very nice step-by-step manner. It’s hard to believe but one night I was actually crying because I would miss yet another
Functions Zeros, and assignment deadline, and a couple of days from that I was actually helping my classmates with their assignments as well. I know how strange
Applications Registered: it might sound, but really Algebrator helped me a lot.
Rational Expressions and 17.03.2002
Functions From: Norway
Linear equations in two
Lesson Plan for
Comparing and Ordering Homuck Posted: Sunday 31st of Dec 09:58
Rational Numbers I might be able to help if you can send more details regarding your problems. Alternatively you may also try Algebrator which is a great
LinearEquations piece of software that helps to solve math questions . It explains everything systematically and makes the topics seem very simple . I must
Solving Equations say that it is indeed worth every single penny.
Radicals and Rational
Exponents Registered:
Solving Linear Equations 05.07.2001
Systems of Linear From: Toronto,
Equations Ontario
Solving Exponential and
Logarithmic Equations
Solving Systems of
Linear Equations Jot Posted: Tuesday 02nd of Jan 09:30
DISTANCE,CIRCLES,AND I am a regular user of Algebrator. It not only helps me complete my assignments faster, the detailed explanations given makes understanding
QUADRATIC EQUATIONS the concepts easier. I strongly suggest using it to help improve problem solving skills.
Solving Quadratic
Quadratic and Rational Registered:
Inequalit 07.09.2001
Applications of Systems From: Ubik
of Linear Equations in
Two Variables
Systems of Linear
Test Description for
RATIONAL EX
Exponential and
Logarithmic Equations
Systems of Linear
Equations: Cramer's Rule
Introduction to Systems
of Linear Equations
Literal Equations &
Equations and
Inequalities with
Absolute Value
Rational Expressions
SOLVING LINEAR AND
Steepest Descent for
Solving Linear Equations
The Quadratic Equation
Linear equations in two
7th grade math 7 utahs course 2 workbooks
Related topics:
Home ti-89 will not factor complex quadratic | fun seventh grade algebra lesson plans | simplifying square roots worksheet"" | games for dividing polynomials by
Rational Expressions monomials | exponents on calculator | c++ code to get factors of positive number | solving quadratic equations completing the square | convert square meters to
Graphs of Rational lineal metres
Solve Two-Step Equations
Multiply, Dividing; Author Message
Exponents; Square Roots;
and Solving Equations mesareve Posted: Thursday 28th of Dec 20:55
LinearEquations Hello Guys , I am urgently in need of assistance for getting through my mathematics exam that is approaching . I really do not want to opt
Solving a Quadratic for the assistance of private masters and web tutoring since they prove to be quite expensive . Could you recommend a perfect tutoring
Equation software that can support me with learning the basics of Intermediate algebra. Particularly, I need assistance on rational expressions and
Systems of Linear factoring polynomials.
Equations Introduction Registered:
Equations and 24.08.2005
Inequalities From: 42° 3' N 83°
Solving 2nd Degree 22' W
Review Solving Quadratic
System of Equations espinxh Posted: Friday 29th of Dec 20:42
Solving Equations & Well I do have a suggestion for you.Sometime back even I was stuck on questions relating to 7th grade math 7 utahs course 2 workbooks,
Inequalities that’s when my elder sister suggested that I should try Algebrator. It didn’t just solve all my queries, but it also explained those
Linear Equations answers in a very nice step-by-step manner. It’s hard to believe but one night I was actually crying because I would miss yet another
Functions Zeros, and assignment deadline, and a couple of days from that I was actually helping my classmates with their assignments as well. I know how strange
Applications Registered: it might sound, but really Algebrator helped me a lot.
Rational Expressions and 17.03.2002
Functions From: Norway
Linear equations in two
Lesson Plan for
Comparing and Ordering Homuck Posted: Sunday 31st of Dec 09:58
Rational Numbers I might be able to help if you can send more details regarding your problems. Alternatively you may also try Algebrator which is a great
LinearEquations piece of software that helps to solve math questions . It explains everything systematically and makes the topics seem very simple . I must
Solving Equations say that it is indeed worth every single penny.
Radicals and Rational
Exponents Registered:
Solving Linear Equations 05.07.2001
Systems of Linear From: Toronto,
Equations Ontario
Solving Exponential and
Logarithmic Equations
Solving Systems of
Linear Equations Jot Posted: Tuesday 02nd of Jan 09:30
DISTANCE,CIRCLES,AND I am a regular user of Algebrator. It not only helps me complete my assignments faster, the detailed explanations given makes understanding
QUADRATIC EQUATIONS the concepts easier. I strongly suggest using it to help improve problem solving skills.
Solving Quadratic
Quadratic and Rational Registered:
Inequalit 07.09.2001
Applications of Systems From: Ubik
of Linear Equations in
Two Variables
Systems of Linear
Test Description for
RATIONAL EX
Exponential and
Logarithmic Equations
Systems of Linear
Equations: Cramer's Rule
Introduction to Systems
of Linear Equations
Literal Equations &
Equations and
Inequalities with
Absolute Value
Rational Expressions
SOLVING LINEAR AND
Steepest Descent for
Solving Linear Equations
The Quadratic Equation
Linear equations in two
Rational Expressions
Graphs of Rational
Solve Two-Step Equations
Multiply, Dividing;
Exponents; Square Roots;
and Solving Equations
Solving a Quadratic
Systems of Linear
Equations Introduction
Equations and
Solving 2nd Degree
Review Solving Quadratic
System of Equations
Solving Equations &
Linear Equations
Functions Zeros, and
Rational Expressions and
Linear equations in two
Lesson Plan for
Comparing and Ordering
Rational Numbers
Solving Equations
Radicals and Rational
Solving Linear Equations
Systems of Linear
Solving Exponential and
Logarithmic Equations
Solving Systems of
Linear Equations
Solving Quadratic
Quadratic and Rational
Applications of Systems
of Linear Equations in
Two Variables
Systems of Linear
Test Description for
RATIONAL EX
Exponential and
Logarithmic Equations
Systems of Linear
Equations: Cramer's Rule
Introduction to Systems
of Linear Equations
Literal Equations &
Equations and
Inequalities with
Absolute Value
Rational Expressions
SOLVING LINEAR AND
Steepest Descent for
Solving Linear Equations
The Quadratic Equation
Linear equations in two
7th grade math 7 utahs course 2 workbooks
Related topics:
ti-89 will not factor complex quadratic | fun seventh grade algebra lesson plans | simplifying square roots worksheet"" | games for dividing polynomials by monomials | exponents on
calculator | c++ code to get factors of positive number | solving quadratic equations completing the square | convert square meters to lineal metres
Author Message
mesareve Posted: Thursday 28th of Dec 20:55
Hello Guys , I am urgently in need of assistance for getting through my mathematics exam that is approaching . I really do not want to opt for the assistance of
private masters and web tutoring since they prove to be quite expensive . Could you recommend a perfect tutoring software that can support me with learning the basics
of Intermediate algebra. Particularly, I need assistance on rational expressions and factoring polynomials.
From: 42° 3' N 83°
22' W
espinxh Posted: Friday 29th of Dec 20:42
Well I do have a suggestion for you.Sometime back even I was stuck on questions relating to 7th grade math 7 utahs course 2 workbooks, that’s when my elder sister
suggested that I should try Algebrator. It didn’t just solve all my queries, but it also explained those answers in a very nice step-by-step manner. It’s hard to
believe but one night I was actually crying because I would miss yet another assignment deadline, and a couple of days from that I was actually helping my classmates
with their assignments as well. I know how strange it might sound, but really Algebrator helped me a lot.
From: Norway
Homuck Posted: Sunday 31st of Dec 09:58
I might be able to help if you can send more details regarding your problems. Alternatively you may also try Algebrator which is a great piece of software that helps
to solve math questions . It explains everything systematically and makes the topics seem very simple . I must say that it is indeed worth every single penny.
From: Toronto,
Jot Posted: Tuesday 02nd of Jan 09:30
I am a regular user of Algebrator. It not only helps me complete my assignments faster, the detailed explanations given makes understanding the concepts easier. I
strongly suggest using it to help improve problem solving skills.
From: Ubik
Author Message
mesareve Posted: Thursday 28th of Dec 20:55
Hello Guys , I am urgently in need of assistance for getting through my mathematics exam that is approaching . I really do not want to opt for the assistance of private masters
and web tutoring since they prove to be quite expensive . Could you recommend a perfect tutoring software that can support me with learning the basics of Intermediate algebra.
Particularly, I need assistance on rational expressions and factoring polynomials.
From: 42° 3' N 83°
22' W
espinxh Posted: Friday 29th of Dec 20:42
Well I do have a suggestion for you.Sometime back even I was stuck on questions relating to 7th grade math 7 utahs course 2 workbooks, that’s when my elder sister suggested that
I should try Algebrator. It didn’t just solve all my queries, but it also explained those answers in a very nice step-by-step manner. It’s hard to believe but one night I was
actually crying because I would miss yet another assignment deadline, and a couple of days from that I was actually helping my classmates with their assignments as well. I know
how strange it might sound, but really Algebrator helped me a lot.
From: Norway
Homuck Posted: Sunday 31st of Dec 09:58
I might be able to help if you can send more details regarding your problems. Alternatively you may also try Algebrator which is a great piece of software that helps to solve
math questions . It explains everything systematically and makes the topics seem very simple . I must say that it is indeed worth every single penny.
From: Toronto,
Jot Posted: Tuesday 02nd of Jan 09:30
I am a regular user of Algebrator. It not only helps me complete my assignments faster, the detailed explanations given makes understanding the concepts easier. I strongly
suggest using it to help improve problem solving skills.
From: Ubik
Posted: Thursday 28th of Dec 20:55
Hello Guys , I am urgently in need of assistance for getting through my mathematics exam that is approaching . I really do not want to opt for the assistance of private masters and web tutoring since
they prove to be quite expensive . Could you recommend a perfect tutoring software that can support me with learning the basics of Intermediate algebra. Particularly, I need assistance on rational
expressions and factoring polynomials.
Posted: Friday 29th of Dec 20:42
Well I do have a suggestion for you.Sometime back even I was stuck on questions relating to 7th grade math 7 utahs course 2 workbooks, that’s when my elder sister suggested that I should try
Algebrator. It didn’t just solve all my queries, but it also explained those answers in a very nice step-by-step manner. It’s hard to believe but one night I was actually crying because I would miss
yet another assignment deadline, and a couple of days from that I was actually helping my classmates with their assignments as well. I know how strange it might sound, but really Algebrator helped me
a lot.
Posted: Sunday 31st of Dec 09:58
I might be able to help if you can send more details regarding your problems. Alternatively you may also try Algebrator which is a great piece of software that helps to solve math questions . It
explains everything systematically and makes the topics seem very simple . I must say that it is indeed worth every single penny.
Posted: Tuesday 02nd of Jan 09:30
I am a regular user of Algebrator. It not only helps me complete my assignments faster, the detailed explanations given makes understanding the concepts easier. I strongly suggest using it to help
improve problem solving skills.
|
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6th European Congress of Mathematics
6th European Congress of Mathematics
On Monday, July 2, right after the final of the UEFA European Championship, the doors open to the
6th European Congress of Mathematics
in the beautiful historic city of Kraków in Poland. Since 1992, the European Mathematical Society (EMS) invites every four years mathematicians from all over the world to this important event.
Previous congresses have been held in Paris, Budapest, Barcelona, Stockholm and Amsterdam. This year, the congress is organized by colleagues from the Polish Mathematical Society and the Jagiellonian
University in Krakow, chaired by Prof. Stefan Jackowski (Warsaw). The Polish President, Mr. Bronislaw Komorowski has accepted the honorary patronage for the congress.
Close to 1.000 mathematicians are expected to participate in the congress that will take place over a whole week at the Auditorium Maximum of the Jagellionian University in the city center of Kraków.
They are looking forward to the opening ceremony on Monday morning with excitement for a very particular reason: A total of 12 prizes installed by the European Mathematical Society will be awarded by
EMS President Prof. Marta Sanz-Solé (Barcelona, Spain) to laureates selected by three prize committees. The monetary value of each prize is 5000 Euro. All prize winners will be invited to deliver
lectures at 6ECM.
Ten EMS prizes
10 EMS prizes will be awarded to young researchers not older than 35 years, of European nationality or working in Europe, in recognition of excellent contributions in mathematics. The prize winners
were selected by a committee of around 15 internationally recognized mathematicians covering a large variety of fields and chaired by Prof. Frances Kirwan (Oxford, UK). Funds for this prize have been
endowed by the Foundation Compositio Mathematica.
Previous prize winners have proved to continue their careers with high success. Several of them have won the most important distinction for young mathematicians, the Fields Medal, of which at most
four are awarded every four years by the International Mathematical Union. Congress participants may thus be able to attend a lecture by a future Fields Medal winner!
European research politicians should be concerned: Among the ten selected extremely talented young mathematicians, five have chosen to pursue their career in the United States!
List of Prize winners Simon Brendle
, 31 years old, received his PhD from Tübingen University in Germany under the supervision of Gerhard Huisken. He is now a Professor of mathematics at Stanford University, USA. An EMS-prize is
awarded to him
for his outstanding results on geometric partial differential equations and systems of elliptic, parabolic and hyperbolic types, which have led to breakthroughs in differential geometry including the
differentiable sphere theorem, the general convergence of Yamabe flow, the compactness property for solutions of the Yamabe equation, and the Min-Oo conjecture. The talk Emmanuel Breuillard
, 35 years old, graduated in mathematics and physics from Ecole Normale Superieure (Paris); then he pursued graduate studies in Cambridge (UK) and Yale (USA) where he obtained a PhD in 2004. He is
currently a professor of mathematics at Universite Paris-Sud, Orsay. He receives an EMS-prize
for his important and deep research in asymptotic group theory, in particular on the Tits alternative for linear groups and on the study of approximate subgroups, using a wealth of methods from very
different areas of mathematics, which has already made a long lasting impact on combinatorics, group theory, number theory and beyond. Alessio Figalli
, 28 years old, graduated in mathematics from the Scuola Normale Superiore of Pisa (2006) and he received a joint PhD from the Scuola Normale Superiore of Pisa and the Ecole Normale Supérieure of
Lyon (2007). Currently he is a professor at the University of Texas at Austin. An EMS-prize goes to him
for his outstanding contributions to the regularity theory of optimal transport maps, to quantitative geometric and functional inequalities and to partial solutions of the Mather and Mañé conjectures
in the theory of dynamical systems. The talk Adrian Ioana
, 31 years old, obtained a bachelor of Science from the University of Bucharest (2003) AND received his Ph.D. from UCLA in 2007 under the direction of Sorin Popa. Currently, he is an assistant
professor at the University of California at San Diego. An EMS prize is awarded to him
for his impressive and deep work in the field of operator algebras and their connections to ergodic theory and group theory, and in particular for solving several important open problems in
deformation and rigidity theory, among them a long standing conjecture of Connes concerning von Neumann algebras with no outer automorphisms. The talk Mathieu Lewin
, 34 years old, studied mathematics at the École Normale Supérieure (Cachan), before he went to the university of Paris–Dauphine where he got his PhD in 2004. He currently occupies a full-time CNRS
research position at the University of Cergy-Pontoise, close to Paris. He receives an EMS-prize
for his ground breaking work in rigorous aspects of quantum chemistry, mean field approximations to relativistic quantum field theory and statistical mechanics. The talk Ciprian Manolescu
, 33 years old, studied mathematics at Harvard University; he received his PhD in 2004 under the supervision of Peter B. Kronheimer. He worked for three years at Columbia University, and since 2008
he is an Associate Professor at UC in Los Angeles. An EMS-prize goes to him
for his deep and highly influential work on Floer theory, successfully combining techniques from gauge theory, symplectic geometry, algebraic topology, dynamical systems and algebraic geometry to
study low-dimensional manifolds, and in particular for his key role in the development of combinatorial Floer theory.
The talk
Grégory Miermont
received his education at Ecole Normale Supérieure in Paris during 1998–2002. He defended his PhD thesis, which was supervised by Jean Bertoin, in 2003. Since 2009 he is a professor at Université
Paris-Sud 11 (Orsay). During the academic year 2011–2012 he is on leave as a visiting professor at the University of British Columbia (Vancouver). An EMS prize is awarded to him
for his outstanding work on scaling limits of random structures such as trees and random planar maps, and his highly innovative insight in the treatment of random metrics. The talk Sophie Morel
, 32 years old, studied mathematics at the École Normale Supérieure in Paris, before earning her PhD at Université Paris-Sud, under the direction of Gerard Laumon. Since December 2009, she is a
professor at Harvard University. She receives an EMS-prize
for her deep and original work in arithmetic geometry and automorphic forms, in particular the study of Shimura varieties, bringing new and unexpected ideas to this field. Tom Sanders
studied mathematics in Cambridge; he received his PhD in 2007 under the supervision of William T. Gowers. Since October 2011, he is a Royal Society University Research Fellow at the University of
Oxford. An EMS-prize goes to him
for his fundamental results in additive combinatorics and harmonic analysis, which combine in a masterful way deep known techniques with the invention of new methods to achieve spectacular
applications. The talk Corinna Ulcigrai
, 32 years old, obtained her diploma in mathematics from the Scuola Normale Superiore in Pisa (2002) and defended her PhD in mathematics at Princeton University (2007), under the supervision of Ya.
G. Sinai. Since August 2007 she is a Lecturer and a RCUK Fellow at the University of Bristol. An EMS prize is awarded to her
for advancing our understanding of dynamical systems and the mathematical characterizations of chaos, and especially for solving a long-standing fundamental question on the mixing property for
locally Hamiltonian surface flows. Felix Klein Prize
The Felix Klein prize, endowed by the Institute for Industrial Mathematics in Kaiserslautern, will be awarded to a young scientist (normally under the age of 38) for using sophisticated methods to
give an outstanding solution, which meets with the complete satisfaction of industry, to a concrete and difficult industrial problem. The Prize Committee that selected the winner consisted of six
members, chaired by Prof. Wil H.A. Schilders from Eindhoven in the Netherlands.
Emmanuel Trélat
, 37 years old, obtained his PhD at the University of Bourgogne in 2000. Currently he is a full professor at the University Pierre et Marie Curie (Paris 6), France, and member of the Institut
Universitaire de France, since 2011. He receives the Felix Klein Prize
for combining truly impressive and beautiful contributions in fine fundamental mathematics to understand and solve new problems in control of PDE’s and ODE’s (continuous, discrete and mixed
problems), and above all for his studies on singular trajectories, with remarkable numerical methods and algorithms able to provide solutions to many industrial problems in real time, with
substantial impact especially in the area of astronautics. The talk Otto Neugebauer Prize
For the first time ever, the newly established Otto Neugebauer Prize in the History of Mathematics will be awarded for a specific highly influential article or book. The prize winner was selected by
a committee of five specialists in the history of mathematics, chaired by Prof. Jeremy Gray (Open University, UK). The funds for this prize have been offered by Springer-Verlag, one of the major
scientific publishing houses.
Jan P. Hogendijk
obtained his Ph.D. at Utrecht University in 1983 with a dissertation on an unpublished Arabic treatise on conic sections by Ibn al-Haytham (ca. 965-1041). He is now a full professor in History of
Mathematics at the Mathematics Department of Utrecht University. He is the first recipient of the Otto Neugebauer Prize
for having illuminated how Greek mathematics was absorbed in the medieval Arabic world, how mathematics developed in medieval Islam, and how it was eventually transmitted to Europe. The talk Photos
From the prize ceremony, and in particular, photos of all prize winners, will be publicly available around 12 am on the web pages
Translations to several European languages will be added later during the day.
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Odds Ratios Versus Relative Risk
Many great things have been written about the difference between Odds Ratios (OR) and Relative Risks (RR). Every medical student at some point has been taught the difference. Yet these statistical
terms are confused and misused every day in both the writing of and the interpretation of literature (which we’ll talk more about at the end if you get bored of the numbers). They are unfortunately
assumed to be the same, or basically the same, thing.
Odds are derived from the world of gambling, and are simply an expression of how many times an event is likely to occur compared to how many times the event is not likely to occur. If you roll a
dice, there are 6 possible outcomes. The odds of rolling a 3 is 1:5 (or 1 to 5) because there is only one 3 on the dice and there are five other options. This 1:5 could be expressed as 0.2.
The probability of rolling a 3 is 1/6 since there is one 3 among the 6 faces of the dice. This 1/6 is also expressed as 0.166.
So it should be apparent that odds and probability are not the same thing. Odds are not intuitive. When someone asks, “What are the odds?”, they usually mean, “What is the chance?” or the
probability. We use probability to talk about chance or risk. We use odds to gamble.
This gets more confusing when we compare odds or risks. We can compare two sets of odds with an Odds Ratio (OR) and we can compare two sets of risks or probabilities with the Relative Risk (RR).
Since odds are not intuitive, the odds ratio will also not be intuitive. Risk or probability is more intuitive, and in the same way relative risk seems to be more intuitive. The OR and RR are not the
same thing, yet we tend to substitute them for one another, as we will see.
An Odds Ratio (OR) then is simply the comparison of two odds, OR=Odds(A)/Odds(B).
The Relative Risk (RR) is simply the comparison of two risks or probabilities, RR=Probability(A)/Probability(B). This is made more clear when the term is referred to as the Risk Ratio.
Let’s look at this graphically. Below is an experiment where we are attempting to pick a blue card. In situation A, there are five cards, one of which is blue. In situation B, there are also five
cards, but now three are blue. The difference in the situations could be anything you like. Think of an intervention like a surgery or a medication that changes the number of blue cards; or think of
a risk factor that is different in the two groups. It doesn’t really matter. But there are two situations, A and B, and we are interested in picking the blue card:
So we see that odds makes sense when we say there is a 1 to 4 chance of picking a blue card in A versus a 3 to 2 chance of picking a blue card in B. Odds make less sense when we convert those to 0.25
odds and 1.5 odds.
Probability makes wonderful sense. There is a 20% chance of picking blue in A and a 60% chance of picking blue in B. We can all understand this. And we can relate probability back to odds if we think
about it. A 1 to 4 odds means there are five possible outcomes. If you are interested in the blue card, it happens once among those five outcomes, or 20% of the time.
Likewise, relative risk makes a lot of sense. The relative risk of picking a blue card in group A compared to group B is 1/3 or 0.33. We can understand that by looking at the picture. And the
relative risk of picking a blue card in group B compared to group A is 3. Put in other words, you are 3 times as likely to pick a blue card in B as you are in A. This makes perfect sense.
But the odds ratio makes no sense (to non-statisticians). The odd ratio of picking blue in A relative to B is 0.16. What!? Exactly. The odds ratio of picking blue in B relative to A is 6. Yet we
understand intuitively (and from the RR) that you are only 3 times as likely to pick blue in this situation, not 6 times as likely.
It should be obvious that you cannot substitute ORs for RRs in your thinking. You cannot say with an OR of 6 that you are 6 times as likely to pick a blue card in situation B. Let’s look at picking
red, just for fun, and see how the numbers change for the more likely event:
In this case, the numbers are even more exaggerated. We understand that you are twice as likely to pick red in A as in B; and this is borne out in the RR of 2. But the OR is even more misleading
since it is 6. In B, you are as half as likely to pick red as in A, or a RR of 0.5, but the OR for this event would be 0.166! So let’s never use OR to imply risk again.
It is true, however, that, in many cases, the OR is approximately (though never exactly) equal to the RR. This is true in rare events. But what does rare mean? Let’s look at another example.
If an event happens in a patient 1/1000 times with an intervention but 2/1000 times without an intervention, then the RR for the intervention group would be:
• RR=(1/1000)/(2/1000)=1/2=0.5
The OR would be:
• OR=(1:999)/(2:998)=0.4995
So in practical terms, these are the same number. How rare is rare? Let’s do the same math for events happening on the scale of hundreds and tens:
For the n per 100s:
• RR=(1/100)/(2/100)=1/2=0.5
• OR=(1:99)/(2:98)=0.495
For the n per 10s:
• RR=(1/10)/(2/10)=1/2=0.5
• OR=(1:9)/(2:8)=0.444
Still not much of a difference, but also not exactly true to use the OR result in the sentence, “You are x times as likely to experience the event if you have the intervention.”
If the OR < 1, it will always be smaller than the RR. Conversely, if the OR > 1, it will always be larger than the RR by some amount. To demonstrate a case where the OR > 1, let’s look at the
opposite of that last problem:
• RR=(2/10)/(1/10)=2
• OR=(2:8)/(1:9)=2.25
2 and 2.25 are still very close, but it is clearly not fair to say that something is 2.25 times as likely to happen in this experiment. The more common the problem, or the greater the effect, the
more misleading an OR might be. For example, let’s imagine that our study relates to obesity (a common problem). Pretend that if you eat below a certain number of calories per day, you have a 1/15
chance of being obese, but if you eat above a certain amount of calories per day, you have a 1/3 chance of being obese. What do these numbers look like:
• RR=(1/3)/(1/15)=5
• OR=(1:2)/(1:14)=7
So these numbers are different enough to be misleading (you are not 7 times as likely to be obese) but close enough that the average non-statistician (read: doctors, reporters, politicians, patients,
other researchers) will assume that 7 times as likely makes sense.
Okay, but does this stuff happen in real life? How often do smart researchers and smart editors make these type of mistakes? Unfortunately, very often (and this is an easy thing not to screw up
compared to some very advanced statistics and statistical concepts that are misused in the literature).
In 2001, Holcomb et al studied the misuse of the OR in OB/GYN literature published in Obstetrics and Gynecology and the American Journal of Obstetrics and Gynecology in the years 1998-1999. They
studied 107 articles and found the following results:
• 44% of the articles published ORs that were more than 20% different than the appropriate RRs, almost all magnifying the reported effect.
• In 26% of the articles, the ORs was explicitly interpreted by the authors as a RR without justification, meaning that the authors stated, “There is an X-fold increased risk…” Only one study did
so and explained to the readers the inherent assumptions they were making by using the OR as a RR.
• In one study on the familial occurrence of dystocia, the authors stated, “The risk is increased by more than 20-fold (odds ratio 24.0, 95% confidence interval 1.5 to 794.5) if one sister had
dystocia…” The baseline risk was 11% and the actual RR was 6.75. (Another issue for discussion later is the ridiculously wide confidence interval.)
Articles have been published in almost every field examining the inappropriate use of ORs in the literature, and the conclusions are striking. Grimes and Schulz (in an excellent article) have argued
that the use of ORs should be limited to case control studies and logistic regression, where they are necessary mathematically. Unfortunately, the incidence of publication of ORs is increasing and so
too are the false conclusions sometimes based on them. More thoughtful reading on the subject can be found here.
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Revision Notes for Class 9 Maths Chapter 15 Probability
NCERT Notes for Class 9 Maths Chapter 15 Probability
Class 9 Maths Chapter 15 Probability Notes
│Chapter Name │Probability Notes │
│Class │CBSE Class 9 │
│Textbook Name │Probability Class 9 │
│ │ • Notes for Class 9 │
│Related Readings│ • Notes for Class 9 Maths │
│ │ • Revision Notes for Probability │
Probability is the study of the uncertainty. The uncertainty of any doubtful situation is measured by means of Probability.
Uses of Probability
Probability is used in many fields like Mathematics, Physical Sciences, Commerce, Biological Sciences, Medical Sciences, Weather Forecasting, etc.
Basic terms related to Probability
1. Randomness
If we are doing an experiment and we don't know the next outcome of the experiment to occur then it is called a Random Experiment.
2. Trial
A trial is that action whose result is one or more outcomes. Like –
• Throw of a dice
• Toss of a coin
3. Independent trial
A trial will be independent if it does not affect the outcome of any other random trial. Like throwing a dice and tossing a coin are independent trials as they do not impact each other.
4. Event
While doing an experiment, an event will be the collection of some outcomes of that experiment.
Example: If we are throwing a dice then the possible outcome for even number will be three i.e. 2, 4, 6. So the event would consist of three outcomes.
Probability: An Experimental Approach
Experimental probability is the result of probability based on the actual experiments. It is also called the Empirical Probability.
In this probability, the results could be different, every time you do the same experiment. As the probability depends upon the number of trials and the number of times the required event happens.
If the total number of trials is ‘n’ then the probability of event D happening is
Examples: If a coin is tossed 100 times out of which 49 times we get head and 51 times we get tail.
a. Find the probability of getting head.
b. Find the probability of getting tail.
c. Check whether the sum of the two probabilities is equal to 1 or not.
a. Let the probability of getting head is P(H)
b. Let the probability of getting tail is P(T)
c. The sum of two probability is
= P(H) + P(T)
Impossible Events
While doing a test if an event is not possible to occur then its probability will be zero. This is known as an Impossible Event.
Example: You cannot throw a dice with number seven on it.
Sure or Certain Event
While doing a test if there is surety of an event to happen then it is said to be the sure probability. Here the probability is one.
Example 1: It is certain to draw a blue ball from a bag contain a blue ball only.
This shows that the probability of an event could be
0 ≤ P (E) ≤ 1
Example 2: There are 5 bags of seeds. If we select fifty seeds at random from each of 5 bags of seeds and sow them for germination. After 20 days, some of the seeds were germinated from each
collection and were recorded as follows:
│Bag │1 │2 │3 │4 │5 │
│No. of seeds germinated │40│48│42│39│41│
What is the probability of germination of
(i) more than 40 seeds in a bag?
(ii) 49 seeds in a bag?
(iii) more than 35 seeds in a bag?
(i) The number of bags in which more than 40 seeds germinated out of 50 seeds is 3.
P (germination of more than 40 seeds in a bag) = 3/5 = 0.6
(ii) The number of bags in which 49 seeds germinated = 0.
P (germination of 49 seeds in a bag) = 0/5 = 0.
(iii) The number of bags in which more than 35 seeds germinated = 5.
So, the required probability = 5/5 = 1.
Elementary Event
If there is only one possible outcome of an event to happen then it is called an Elementary Event.
If we add all the elementary events of an experiment then their sum will be 1.
The general form
P (H) + P (T) = 1
P (H) + P
P (H) – 1 = P
P (H) and P
Example: What is the probability of not hitting a six in a cricket match, if a batsman hits a boundary six times out of 30 balls he played?
Let D be the event of hitting a boundary.
So, the probability of not hitting the boundary will be
= 0.8
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C Program to Compute Charges for Sending Parcels as Per Their Weight
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C Program to Compute Charges for Sending Parcels as Per Their Weight
In this post, we will learn about how to create a program in C that will compute and print the charge (that has to be paid by the sender) to send a parcel as per its weight. That is, the program will
ask the sender (or user) to enter the weight of the parcel and will calculate and print the charge for sending that parcel (of the given weight).
The program is designed in the following manner:
A post office charges parcel senders according to the weight of the parcel. For each parcel having a weight of 2 kg or less, the charge is 32.50, and for each extra kg, there is an additional charge
of 10.50.
int main()
float weight, initCharge=32.50, perCharge=10.50, tempWeight, addCharge, totalCharge;
printf("Enter weight of the parcel (in Kg): ");
scanf("%f", &weight);
printf("Charge = %0.2f", initCharge);
tempWeight = weight-2;
addCharge = tempWeight * perCharge;
totalCharge = addCharge + initCharge;
printf("Charge = %0.2f", totalCharge);
return 0;
As the above program was written in the Code::Blocks IDE, here is the sample run after a successful build and run. This is the first snapshot:
The user must now enter the weight of the parcel to determine the total charge that must be paid. Here is the second snapshot of the same sample run:
If the weight of the parcel is less than or equal to 2, then there is only 32.50 that has to be paid by the sender. Let's take a look at another sample run:
Let's take another sample run, as given below. This time the user has entered the parcel weight as 1.5 kg:
Below are some of the main steps used in the above program:
• Take any two variables, say initCharge and perCharge. Set initCharge to 32.50 (charge for parcels weighing less than or equal to 2 kg) and perCharge to 10.50 (charge for each additional kg of
parcel weight).
• At runtime, get the weight of the parcel from the user.
• Check whether the given weight is less than or equal to 2 or not.
• If it is, then just print the value of initCharge as output that holds the charge value of a parcel with a weight less than or equal to 2 kg.
• Otherwise, if the weight of the parcel is greater than 2 kg, then follow the steps given below.
• Because the first 2 kg of parcel weight is 32.50,Therefore, we have to subtract the original weight by 2 and then compute the charge as per the given rate, which is $10.50 per extra kilogram. And
then add the calculated charge to 32.50; after this, you will get the final charge that has to be paid by the parcel sender to send the parcel.
• For example, if the user has entered the weight of the parcel as 10 kg, Because the value 10 is greater than 2, the program flow enters the else block.
• Inside the else block, weight-2 or 10-2 or 8 is initialized to tempWeight, then tempWeight * perCharge or 8 * 10.50 or 84 is initialized to addCharge, and then addCharge + initCharge or 84 +
32.50 or 116.5 is initialized to totalCharge
• Print the value of totalCharge as output, which will be the final charge (to be paid) to send the parcel.
« Previous Program Next Program »
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Overview of recent advances in stability of linear systems with time‐varying delays
Overview of recent advances in stability of linear systems with time-varying delays
First published: 01 January 2019
This study provides an overview and in-depth analysis of recent advances in stability of linear systems with time-varying delays. First, recent developments of a delay convex analysis approach, a
reciprocally convex approach and the construction of Lyapunov–Krasovskii functionals are reviewed insightfully. Second, in-depth analysis of the Bessel–Legendre inequality and some affine integral
inequalities is made, and recent stability results are also summarised, including stability criteria for three cases of a time-varying delay, where information on the bounds of the time-varying delay
and its derivative is totally known, partly known and completely unknown, respectively. Third, a number of stability criteria are developed for the above three cases of the time-varying delay by
employing canonical Bessel–Legendre inequalities, together with augmented Lyapunov–Krasovskii functionals. It is shown through numerical examples that these stability criteria outperform some
existing results. Finally, several challenging issues are pointed out to direct the near future research.
1 Introduction
A time-delay system is also called a system with after-effect or dead-time [1]. A defining feature of a time-delay system is that its future evolution is related not only to the current state but
also to the past state of the system. Time-delay systems are a particular class of infinite dimensional systems, which have complicated dynamic properties compared with delay-free systems. A large
number of practical systems encountered in areas, such as engineering, physics, biology, operation research and economics, can be modelled as time-delay systems [2, 3]. Therefore, time-delay systems
have attracted continuous interest of researchers in a wide range of fields in natural and social sciences, see, e.g. [4–9].
Stability of time-delay systems is a fundamental issue from both theoretical and practical points of view. Indeed, the presence of time-delays may be either beneficial or detrimental to stability of
a practical system. Time-delays are usually regarded as a factor of system destabilisation. However, practical engineering applications reveal that, for some dynamical systems, intentional
introduction of a specific time-delay may stabilise an unstable system [
]. Hence, one main concern about a time-delay system is to determine the
maximum delay interval
on which the system remains stable. For non-linear time-delay systems, it is quite challenging due to complicated dynamical properties. For linear time-delay systems, a lot of effort has been made on
delay-dependent stability analysis in the last two decades, see, e.g. [
]. In this paper, we focus on the following linear system with a time-varying delay described by
) is called a linear system with an
time-varying delay [
], which allows the system to be unstable at
]. For simplicity of presentation, in what follows, let
Recalling some existing results, there are two types of approaches to delay-dependent stability of linear time-delay systems: frequency domain approach and time domain approach. Frequency domain
approach-based stability criteria have been long in existence [1, 9]. For some recent developments in the frequency domain, we mention an integral quadratic constraint framework [21–23], which
describes the stability of a system in the frequency domain in terms of an integral constraint on the Fourier transform of the input/output signals [24]. In the time domain approach, the direct
Lyapunov method is a powerful tool for studying stability of linear time-delay systems [1, 9]. Specifically, there are complete Lyapunov functional methods and simple Lyapunov–Krasovskii functional
methods for estimating the maximum admissible delay upper bound that the system can tolerate and still maintain stability. Complete Lyapunov functional methods can provide necessary and sufficient
conditions on stability of linear systems with a constant time-delay [25–28]. Simple Lyapunov-Krasovskii functional methods only provide sufficient conditions on stability of linear time-delay
systems. Compared with stability criteria based on complete Lyapunov functional methods, although stability criteria based on simple Lyapunov-Krasovskii functional methods are more conservative, they
can be applied easily to control synthesis and filter design of linear time-delay systems [29].
A simple Lyapunov–Krasovskii functional method for estimating the maximum delay upper bound for linear time-delay systems is based on a proper simple Lyapunov–Krasovskii functional candidate, which
usually includes a double integral term as
The key to the stability criterion is how to deal with the quadratic integral term
1. Model transformation approach. The model transformation approach employs the Leibniz–Newton formula 1) to a system such that a cross-term
or the improved basic inequality [30], for
or the general basic inequality [31]: if
the cross-term offsets ’ the quadratic integral term 32], ‘parameterised first-order transformation’ [33], ‘second-order transformation’ [34], ‘neutral transformation’ [35] and ‘descriptor model
transformation’ [36, 37]. As pointed out in [38, 39], under the first-order transformation, or the parameterised first-order transformation, or the second-order transformation, the transformed
system is not equivalent to the original one due to the fact that additional eigenvalues are introduced into the transformed system. Under the neutral transformation, although no explicit
additional eigenvalue is introduced, some additional eigenvalue constraints for the stability of an appropriate operator should be satisfied [39]. The descriptor model transformation delivers
some larger delay upper bounds since the transformed system is equivalent to the original one.
2. Free-weighting matrix approach. Compared with model transformation approaches, a free-weighting matrix approach can provide an easier way to deal with the quadratic integral term 40, 41]
3. Integral inequality approach. An integral inequality approach directly provides an upper bound for the quadratic integral term 42–44]. By using the Leibniz–Newton formula, an integral inequality
In [
], Jensen integral inequality is introduced
for the first time
in the stability of time-delay systems. In [
], Jensen integral inequality is used to derive a different upper bound for
This inequality can be also derived from (
), where the free weighting matrix
is selected as
) or (
) can obtain the
delay upper bound
that based on the free-weighting matrix approach. Notice that
Thus, the relationship between the integral inequalities (
) and (
) can be disclosed by
which means that the integral inequality (
) provides a minimum upper bound for
). Therefore, the integral inequality (
) has been attractive in the stability analysis since it can derive some stability criteria without introducing any extra matrix variable, if compared with the integral inequality (
) and the free-weighting matrix approach.
In [
], a
bound for
convex delay analysis
method [
], the
reciprocally convex
approach [
] and a proper Lyapunov–Krasovskii functional. It is shown through a number of numerical examples that the obtained stability criterion can produce an admissible maximum upper bound closely
approaching to the system analytical value [
]. Boosted by the Wirtinger-based integral inequality, increasing attention is paid to integral inequality approaches, and a great number of results have been reported in the open literature, see,
e.g. [
In this paper, we provide an overview and in-depth analysis of integral inequality approaches to stability of linear systems with time-varying delays. First, an insightful overview is made on convex
delay analysis approaches, reciprocally convex approaches and the construction of Lyapunov–Krasovskii functionals. Second, in-depth analysis of Bessel–Legendre inequalities and some affine integral
inequalities is made, and recent stability results based on these inequalities are reviewed. Specifically, the refined allowable delay sets are discussed with insightful understanding. Third, we
develop a number of stability criteria based on a canonical Bessel–Legendre inequality recently reported, taking three cases of time-varying delay into account. Simulation results show that the
canonical Bessel–Legendre inequality plus an augmented Lyapunov–Krasovskii functional indeed can produce a larger delay upper bound than some existing methods. Finally, some challenging issues are
proposed for the near future research.
The remaining part of the paper is organised as follows. Section 2 gives an overview of recent advances in convex and reciprocally convex delay analysis approaches, as well as the construction of
Lyapunov–Krasovskii functionals. Recent integral inequalities and their applications to stability of linear systems with time-varying delay are reviewed in Section 3. A canonical Bessel–Legendre
inequality and its affine version, together with a proper augmented Lyapunov–Krasovskii functional is developed to derive some stability criteria for three cases of time-varying delays. Section 5
concludes this paper and proposes some challenging problems to be solved in the future research.
Notation: The notations in this paper are standard. N, where N is a positive integer; and the notation
2 Recent advances in convex analysis approaches, reciprocally convex approaches and the construction of Lyapunov–Krasovskii functionals
2.1 Convex delay analysis approach
The convex delay analysis approach provides an effective way to handle the time-varying delay
]. Suppose that a stability condition is given in the following form:
) is
to two boundary linear matrix inequalities (LMIs) as
Some other conditions on stability are of the following form [
) is that
]. However, if the constraint
satisfied, the above conclusion is not necessarily true. In this situation, two sufficient conditions are established in [
] and [
], which are given, respectively, as
Clearly, these two sufficient conditions described in (
) and (
) are independent. The relationship between them needs to be further investigated. It should be mentioned that Theorem 1 in [
] is
correct. In fact,
equivalent to
, Theorem 1].
2.2 Reciprocally convex delay analysis approach
In the stability analysis of the system (
), applying some integral inequalities usually yields a reciprocally convex combination on the time-varying delay
real vectors. A reciprocally convex inequality for
]: For any
the following inequality holds:
The reciprocally convex inequality (
) provides a lower bound for
) usually leads to a significant stability criterion in the sense of two aspects: (i) it requires less decision variables; and (ii) it can derive the same delay upper bound as the one using the
free-weighting matrix approach, which is verified through some numerical examples [
]. Nevertheless, the second aspect only holds for stability conditions based on Jensen's inequality. This is indeed not the case anymore when employing for instance the Wirtinger-based integral
inequality, which can be seen from [
] or the simulation results in the next section. Moreover, insight analysis of the reciprocally convex approach [
] is made in [
], which points out that the reciprocally convex inequality (
) can be interpreted as a discretised version of the Jensen's inequality.
Recently, an improved reciprocally convex inequality is proposed in [
] and developed in [
], which reads as
Compared with (
), the significance of (
) lies in two aspects: (i) the matrix constraint (
) is
from (
); and (ii) the improved reciprocally convex inequality (
) provides a larger lower bound than (
) for
] in detail.
By introducing more slack matrix variables, a general reciprocally convex inequality is proposed in [
]: For any
the following inequality holds:
The reciprocally convex inequality (
) is rather general: (i) Taking
) reduces to (
); (ii) Taking
) becomes (
). However, four slack matrix variables are introduced and two constraints are imposed on them, which lead to higher computation complexity of a stability criterion. Fortunately, following the idea
in [
], an improved reciprocally convex inequality for
]: For any
Clearly, just two slack matrix variables are introduced and the constraints in (
) are also removed from (
). Combining the improved reciprocally convex inequality (
) with the convex delay analysis approach, some less conservative stability criteria can be derived, which is verified through some numerical examples, for details see [
If setting
) immediately reduces to
Based on (
), an estimate of
], which is in a different form as
) can be referred to [
]. The link between (
) and (
) is revealed if one considers the following slack variables:
Several calculations allow indeed to recover (
) from (
In the sequel, it will be shown that the inequality (23) is a special case of the inequality obtained from the following lemma (see [74, Lemma 1]).
Apply (
) to each term in (
) to obtain
If one sets
) immediately reduces to (
From the analysis above, it is clear that the inequality (
) is the general form of those reciprocally convex inequalities (
), (
), (
), (
) and (
). Following the idea above, a general inequality for the reciprocally convex combination
is derived, which can be referred to [
, Lemma 4].
2.3 Construction of Lyapunov–Krasovskii functionals
A proper Lyapunov–Krasovskii functional is crucial for deriving less conservative stability criteria for time-delay systems. However, it is still challenging to construct an exact Lyapunov–Krasovskii
functional so that a necessary and sufficient stability condition can be derived for the system (
). In general, such a Lyapunov–Krasovskii functional is based on parameters which are solutions to partial differentiable equations, see, e.g. [
]. Hence, many researchers have turned to a simple Lyapunov–Krasovskii functional as
can be weakened by [
such that
). In the following, typically we mention the several kinds of Lyapunov–Krasovskii functionals.
2.3.1 Augmented Lyapunov–Krasovskii functionals
An augmented Lyapunov functional is introduced in [
]. A key feature of it is to augment some terms in (
) such that more information on the delayed states is exploited to derive a stability criterion. For example, in [
], the first two terms
) are augmented, respectively, by
The augmented Lyapunov–Krasovskii functional can make the system state and some delayed states coupled closely, which possibly enhances the feasibility of the related LMIs in a stability criterion.
Numerical examples show that such a stability criterion indeed can produce a larger upper bound
The purpose of the augmentation of a Lyapunov–Krasovskii functional is to help provide a tighter estimate on its derivative by introducing some new matrix variables as well as some new state-related
vectors. It is true that the estimate of the derivative of a Lyapunov–Krasovskii functional depends mainly on the treatment with some integral terms. However, such an estimate sometimes is not enough
for a less conservative stability criterion. In both [72, 80], it has been proven that the Wirtinger-based inequality can produce a tighter estimate on the derivative of a Lyapunov–Krasovskii
functional than Jensen integral inequality, but both the obtained stability criteria are of the same conservatism if the Lyapunov–Krasovskii functional is not augmented. Recent research [81, 82]
shows that using an augmented Lyapunov–Krasovskii functional plus the N -order Bessel–Legendre inequality indeed can yield nice stability criteria of less conservatism.
2.3.2 Lyapunov–Krasovskii functionals with multiple-integral terms
Another development on constructing a proper Lyapunov–Krasovskii functional is to introduce a triple-integral term as [
Following this idea, a quadruple-integral term is introduced to the augmented Lyapunov–Krasovskii functional, which is in a different form as [
More generally, multiple-integral terms are introduced as [
is a certain positive integer. Based on the augmented Lyapunov-Krasovskii functionals with multiple-integral terms, it is shown through numerical examples that the resulting delay-dependent stability
conditions for the system (
) are less conservative, see, e.g. [
]. However, the introduction of multiple-integral terms gives rise to some new integral terms to be estimated in the derivative of the Lyapunov–Krasovskii functional [
2.3.3 Lyapunov–Krasovskii functionals for linear systems with interval time-varying delays
For a system with an interval time-varying delay
Based on (
), a number of (augmented) Lyapunov-Krasovskii functionals are introduced by exploiting more delayed states such as
] and the references therein.
It should be mentioned that, if both lower and upper bounds of 61], where the Lyapunov matrix P is chosen as a convex combination
2.3.4 Lyapunov–Krasovskii functionals based on a delay-fractioning approach
Another kind of Lyapunov–Krasovskii functionals is based on the delay-fractioning approach [
]. The key idea is to introduce fractions
is a positive integer
] that, as the integer
becomes larger, the obtained stability criterion is less conservative. The idea of delay-fractioning is extensively used to construct various Lyapunov–Krasovskii functionals, see, e.g. [
3 Recent developments of integral inequality approaches to stability of linear systems with time-varying delays
In this section, we focus on the recent developments of integral inequality approaches to stability of the system (1). To begin with, we first give an overview of integral inequalities developed
3.1 Recent integral inequalities
Employing the Wirtinger inequality provides a larger lower bound than the well-used Jensen integral inequality for a non-negative integral term [47]. Soon after, by introducing a proper auxiliary
function, an auxiliary-function-based integral inequality is reported in [53]. Both of them are given in the following.
Lemma 2.For any constant matrix a and b with
1. Wirtinger-based integral inequality:
2. Auxiliary-function-based integral inequality:
Clearly, the auxiliary-function-based integral inequality (33) is an improvement over the Wirtinger-based integral inequality (32). A natural inspiration from (33) is to extend the inequality to a
general form, which is completed by introducing the Legendre polynomials, leading to the canonical Bessel–Legendre inequality [81, 82].
Lemma 3.Under the assumption in Lemma 2, the following inequality holds:
The canonical Bessel–Legendre inequality includes the Wirtinger-based integral inequality and the auxiliary-function-based integral inequality as its special cases. The underlying idea of
Bessel–Legendre inequality is to provide a generic and expandable integral inequality which is asymptotically (in the sense that
goes to infinity) not conservative, because of the Parseval's identity [
]. The proof of (
) relies on the expansion of the following non-negative quantity:
denoted by
Lemma 3 gives a canonical Bessel–Legendre inequality of two different forms (
) and (
), where
, Lemma 1] can be expressed on the basis of
] is equivalent to the above canonical Bessel–Legendre inequality if setting
Since 35) depends on the Legendre polynomial, the inequality (35) is not convenient for use. In [82], a useful form of the canonical integral inequality is developed for stability analysis of
time-delay systems, which is given in the following lemma.
Lemma 4.For an integer a and b with
Notice that k -integral of the vector 40) discloses an explicit relationship between
The other class of integral inequalities is called affine integral inequalities, or free-matrix-based integral inequalities, where the coefficient
) can be readily obtained based on the fact that the following inequality holds for any real matrix
with compatible dimensions
Thus, an affine canonical Bessel–Legendre inequality is given by
The affine versions of (32) and (33) can be found in [67, 97] or [98]. As pointed out in [97] and [58], the affine version and its corresponding integral inequality provide an equivalent lower bound
for the related integral term. It should be mentioned that those affine integral inequalities can be regarded as special cases of (46). For example, Lemma 1 in [98] is a special case of (46) with
3.2 Recent developments on stability of the system (1) using recent integral inequalities
Although the canonical Bessel-Legendre inequality in Lemma 3 provides a lower bound for the integral term as tight as possible if 47, 99, 100] and 52, 53, 64, 101]. It is proven in [80] that a
tighter bound of the integral term in the derivative of the Lyapunov–Krasovskii functional should not be responsible for deriving a less conservative stability criterion. Therefore, although the
integral inequality (33) provides a tighter bound than (32), it is not a trivial thing to derive a less conservative stability criterion using the inequality (33). The main difficulty is that the
vectors 34) are not easily handled in the stability analysis of the system (1). It is shown from [72] that the vectors 1) can be obtained using the integral inequality (33). The recent development on
this issue is briefly summarised as follows.
For simplicity of presentation, suppose that the time-varying delay
) belongs to one of three cases:
• Case 1:
• Case 2:
• Case 3:
In what follows, we consider the three cases.
3.2.1 Case 1
Since information on the upper and lower bounds of the time-varying delay and its time-derivative is available, in order to formulate some less conservative stability criteria, an augmented
Lyapunov–Krassovskii functional is introduced in [69] on the basis of the Lyapunov–Krassovskii functional in (28), where the first term 33). Combining with the extended reciprocally convex inequality
(18), some nice results are derived therein.
In [
], an augmented Lyapunov–Krasovskii functional is constructed as
) lies in two aspects: (a) The quadratic term
) is merged into the first two integral terms such that the vectors
) are included in the derivative of
), a less conservative stability criterion is presented in [
In [
], using the Bessel–Legendre inequality (
), an
-dependent stability criterion is established by choosing the following augmented Lyapunov–Krasovskii functional
Similar to [
], it is proven that the
-dependent stability criterion presented in [
] also forms a hierarchy, which means that its conservatism will be reduced if
is increased. On the other hand, an observation is that Lyapunov–Krasovskii functional
dependent on the Legendre polynomials
3.2.2 Case 2
Information on the lower bound of
], in which two augmented terms are given as
) yields four vectors
However, taking the derivative of the Lyapunov–Krasovskii functional just gives three vectors
), a stability criterion for the system (
) is obtained [
3.2.3 Case 3
The time-varying delay is only known to be continuous (possibly not differentiable), which implies that information on the derivative of the time-varying delay is unavailable. Thus, the above
Lyapunov–Krasovskii functionals in Cases 1 and 2 can be no longer used to produce the vectors similar to 52], where the quadratic 28) is augmented by 35) appear in the derivative of the
Lyapunov–Krasovskii functional, for details see [52, Theorem 1]. It should be pointed that stability of the system (1) with unknown information on the derivative of 53] using the
auxiliary-function-based integral inequality (33), but those vectors induced from (33) do not exist in the derivative of the chosen Lyapunov–Krasovskii functional. Thus, one can claim that [53,
Theorem 1] is of the same conservatism as that using the Wirtinger-based integral inequality (32) instead of (33).
From the above analysis, one can see that, it is still challenging to investigate the stability for the system (1) with time-varying delay based on the recent integral inequalities. When both the
time-varying delay and its derivative are bounded from above and from below, most existing stability criteria are based on the Wirtinger-based or the auxiliary-function-based integral inequalities or
the second-order Bessel–Legendre inequality. In the other cases where the information on the derivative of the time-varying delay is partly known or completely unknown, relatively few results on
stability of the system (1) are obtained, even using the second-order Bessel–Legendre inequality.
3.3 Refinement of allowable delay sets
Recently, another development on stability of the system (
) is the
of allowable delay sets, see [
]. To make it clear, suppose that a stability condition can be derived from the matrix inequality
on both
The set
) with four vertices as
A stability condition is thus readily obtained provided that
]. However, it is pointed out [
] that such a stability condition is conservative in some situation. In fact, if
], the vertices
Such an idea is extended to neural networks with time-varying delays [
]. It has been shown through simulation that, a stability criterion based on the new delay set
On the one hand, the above analysis just keeps an eye on the two vertices not be reached. Let us still consider the above delay function 55). Set
On the other hand, an important observation is that the allowable delay set
suitable for the description of the time-varying delay function. To reveal such a fact, we stick to
). After simple algebraic manipulations, it is found that
which means that the function
. Unfortunately, the set
cover the convex ellipsoid
) with
). Thus, the set
It is a good idea to refine the allowable delay set such that less conservative stability criteria can be obtained. However, based on the above analysis, one cannot claim that the set 57) is a
refinement of 55), one can build a polygon (such as the octagon with green dashed lines in Fig. 3) as small as possible to cover the ellipsoid, while for other different not necessarily reached by
4 Stability criteria based on the canonical Bessel–Legendre inequalities (40) and (46)
In this section, we develop some stability criteria using the canonical Bessel–Legendre inequalities (40) and (46), in order to show the effectiveness of canonical Bessel–Legendre inequalities, and
confirm some claims made in the previous sections as well.
4.1 Stability criteria for case 1
Under case 1, the time-varying delay
). In this situation, we choose the following augmented Lyapunov–Krasovskii functional:
The first augmented term in (59) is motivated from Lemma 4 such that the vectors in (60) induced from the integral inequality (40) appear in the derivative of the Lyapunov–Krasovskii functional 62].
It should be mentioned that the Lyapunov–Krasovskii functional 59) is different from the one in (51), which is dependent on the Legendre polynomials.
4.1.1 N -dependent stability criteria
Proposition 1.For constants N, the system (1) subject to
), and
th block-row matrix such that
Proof.First, we introduce a vector as 59) along with the trajectory of the system (1) yields
Now, we estimate the integral term in (
) using the integral inequality (
). Apply the integral inequality (
) to obtain
), and
), one has
). Substituting (
) into (
) yields
Note that
) and (
) are satisfied, by the Schur complement, one has
Thus, from (
), there exists a scalar
) is asymptotically stable for
Instead of the integral inequality (40), one can also use its affine version (46) to derive another N -dependent stability criterion by slightly modifying the Lyapunov–Krasovskii functional (59),
where the term
Proposition 2.For constants 1) subject to
and the other notations are the same as those in Proposition 1.
Remark 1.Propositions 1 and 2 deliver two N -dependent stability criteria for the system (1) subject to (47), thanks to the canonical integral inequality (40). The number of required decision
variables can be calculated as 76] or [81].
4.1.2 Hierarchy of LMI stability criteria
In [81], it is proven that the stability criterion in terms of LMIs forms a hierarchy. In the following, it is shown that such a hierarchical characteristic is also hidden in the LMIs of Propositions
1 and 2. Based on Proposition 1, one has
Proposition 3.For the system (1) subject to (47), one has that
), (
) and (
), respectively.
Proof.Without loss generality, suppose that
). Let
In fact, denote
After some algebraic manipulations, one has
Similar to Proposition 3, one can prove that the LMIs in Proposition 2 also form a hierarchy. For given scalars
4.2 Stability criteria for case 2
In the case where the time-delay
), it is challenging to establish an
-dependent stability criterion using Corollary 4 since one cannot exploit the convex property to cope with the derivative of
) if
Proposition 4.For constants 1) subject to (48) is asymptotically stable if there exist real matrices
) and (
) with
Proof.Taking the time-derivative of 74) yields
), and
Then for any real matrix
From (
), (
) and (
), one obtains
). Apply the integral inequality (
) with
), one has
Substituting (
) into (
) yields
). If the LMIs in (
) and (
) are satisfied,
) subject to (
) is asymptotically stable. □
Similar to Proposition 2, if using the affine integral inequality (46) instead of (40), we have the following result.
Proposition 5.For constants 1) subject to (48) is asymptotically stable if there exist real matrices
and the other notations are defined in Proposition 4.
Remark 2.Propositions 4 and 5 provide two stability criteria for the system (1) subject to (48). Compared with [64, Theorem 1], the main difference lies in that Propositions 4 and 5 are derived based
on such a condition as linear matrix-valued function on linear matrix-valued function contributes to the introduction of the vectors 81). However, in the proof of [64, Theorem 1], quadratic function
on 13) only gives a sufficient condition such that
Remark 3.The purpose of introducing the vectors 81) is to absorb linear on 1) in Case 2. The number of decision variables required is
4.3 Stability criteria for case 3
Under case 3, the time-varying delay
). Using the integral inequality (
) and its affine version (
), similar to the proof of Propositions 4 and 5, we have the following two stability criteria.
Proposition 6.For a constant 1) subject to (49) is asymptotically stable if there exist real matrices
1. The LMIs in (75) and (76) are satisfied, where
2. The LMIs in (86) are satisfied, where
The other notations are defined in Proposition 4.
Proof.The proof can be completed by the following the proof of Proposition 4.□
Remark 4.In Case 3, Proposition 6 presents two stability criteria for the system (1). By using the second-order Bessel–Legendre inequality, a stability criterion for the system (1) with (49) is also
reported in [52, Theorem 1]. The main difference between them lies in the chosen Lyapunov–Krasovskii functional. In Proposition 6, an augmented vector 88), but not in [52, Theorem 1]. As a result,
taking the derivative of the augmented term yields
That is, the vectors
) are coupled by
, which enhances the feasibility of the stability conditions in Proposition 6.
Remark 5.The number of decision variables required in Proposition 6 is 52, Theorem 2].
Remark 6.It should be pointed out that, the proposed results in this section can be easily extended to a linear system with an interval time-varying delay
4.4 Illustrative examples
In this section, we compared the above stability criteria with some existing ones through two numerical examples.
Example 1.Consider the system (1), where
The time-varying delay
Example 1 is well used to calculate the admissible maximum upper bound (AMUB)
Case 1: 47) with 54). For different values of 1 lists the obtained AMUBs of 102, Theorem 1], Zhang et al. [72, Proposition 1], Zhang et al. [69, Theorem 2], Lee et al. [100, Theorem 1], Zeng et al. [
67, Corollary 1], Seuret and Gouaisbaut [81, Theorem 8 with 21], the quadratic separation approach [104] and Propositions 1 and 2 with 1, one can see that
Method 0 0.1 0.5 0.8
[21] 6.117 4.714 2.280 1.608
[104] 6.117 4.794 2.682 1.957
[69] 6.165 4.714 2.608 2.375
[67] 6.059 4.788 3.055 2.615
[102] 6.0593 4.71 2.48 2.30
[100] 6.0593 4.8313 3.1487 2.7135
[72] 6.168 4.910 3.233 2.789
[81] 6.1725 5.01 3.19 2.70
Proposition 1 ( 6.0593 4.8344 3.1422 2.7131
Proposition 1 ( 6.1689 4.9192 3.1978 2.7656
Proposition 1 ( 6.1725 4.9203 3.2164 2.7875
Proposition 1 ( 6.1725 4.9246 3.2230 2.7900
Proposition 2 ( 6.0593 4.8377 3.1521 2.7278
Proposition 2 ( 6.1689 4.9217 3.2211 2.7920
Proposition 2 ( 6.1725 4.9239 3.2405 2.8159
Proposition 2 ( 6.1725 4.9297 3.2527 2.8230
• • Propositions 1 and 2 with 67, 69, 72, 100, 102], the IQC approach [21] and the quadratic separation approach [104]. Even for 69, Theorem 2], [100, Theorem 1], [67, Corollary 1], the IQC
approach [21] and the quadratic separation approach [104].
• • For 81, Theorem 8 with 81, Theorem 8].
• • For the same N, Proposition 2 delivers a larger upper bound 46) can derive a larger upper bound 40) and the improved reciprocally convex inequality (21).
Case 2: The time-varying delay 48). In order to show the effectiveness of Propositions 4 and 5, the AMUBs of 2 for different values of 64, Theorem 1] and [79, Theorem 1], while Propositions 4 and 5
require more decision variables than [79, Theorem 1] (64, Theorem 1] (46) can result in a larger upper bound 40) plus the improved reciprocally convex inequality (21).
Example 2.Consider the system (1) subject to (49), where
The time delay
Method 0 0.1 0.5 0.8 1
[79] 6.059 4.704 2.420 2.113 2.113
[64] 6.168 4.733 2.429 2.183 2.182
Proposition 4 6.168 4.800 2.533 2.231 2.231
Proposition 5 6.168 4.800 2.558 2.269 2.263
This example is taken to illustrate the validity of Proposition 6.
For comparison, we calculate the upper bound of 47, Theorem 7], [53, Theorem 1], [52, Theorem 2] and Proposition 6, the obtained results and the required number of decision variables are listed in
Table 3, from which one can see that Proposition 6 outperforms those methods in [47, 52, 53]. Moreover, it is clear that using the affine integral inequality (46) can yield a larger upper bound 40)
though more decision variables are required.
Method Number of decision variables
[47] 1.59
[53] 1.64
[52] 2.39
Proposition 6-(i) 2.39
Proposition 6-(ii) 2.53
In summary, through two well-used numerical examples, it is shown that, the obtained stability criteria in this paper are more effective than some existing ones in deriving a larger upper bound for a
linear system with a time-varying delay.
As a counterpart of integral inequalities, finite-sum inequalities for stability analysis of discrete-time systems with time-varying delays also have gained much attention. A large number of
finite-sum inequalities and stability criteria have been reported in the published literature, see, [105–114]. Since discrete-time systems with time-varying delays are not the focus of the paper,
stability criteria based on finite-sum inequalities developed recently are not mentioned in the paper.
5 Conclusion and some challenging issues
An overview and in-depth analysis of recent advances in stability analysis of time-delay systems has been provided, including recent developments of integral inequalities, convex delay analysis
approaches, reciprocally convex approaches and augmented Lyapunov–Krasovskii functionals. Then, some existing stability conditions have been reviewed by taking into consideration three cases of
time-varying delay, where information on the upper and lower bounds of the delay-derivative are totally known, partly known and completely unknown. Furthermore, a number of stability criteria have
been developed by employing the recent canonical Bessel–Legendre integral inequalities and an augmented Lyapunov–Krasovskii functional. When information on the lower and upper bounds of both
However, although there has been significant progress in stability analysis of time-delay systems, the following issues are still challenging.
• • If the positive integer N approaches to infinity, the canonical N -order Bessel–Legendre inequality can provide an accurate estimate on the integral term. Thus, using such an integral
inequality, it is possible to derive a necessary and sufficient condition on stability for linear systems with time-varying delays, which is interesting but challenging. Moreover, extending the
canonical N -order Bessel–Legendre inequality to multi-dimensional systems like 2D systems with time-varying delays is also an interesting topic [115, 116].
• • For the system (1) subject to (48) or (49), no N -dependent stability criteria are derived using the integral inequality (40) due to the vectors 40) to the system (1) possibly yields such a
stability condition as
• • In the proof of Proposition 4, four extra vectors linearly on the time-varying delay
• • The integral inequality (40) is established based on a sequence of orthogonal polynomials. Is it possible to formulate some integral inequality based on a sequence of non-orthogonal polynomials
such that the scalars
• • Simulation in this paper shows that the canonical Bessel-Legendre inequality approach can yield some nice results on stability. However, how to apply it to deal with control problems of a
number of practical systems, such as networked control systems [117, 118], event-triggered control systems [119–121], vibration control systems [122–124], formation control systems [125] and
multi-agent systems [126–129], deserves much effort of researchers.
6 Acknowledgment
This work are supported in part by the Australian Research Council Discovery Project under Grant DP160103567.
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Inductive with negative types?
In Haskell, I can define a data type like
data LC = App LC LC | Abs (LC -> LC)
This isn't possible with Inductive, as far as I know. What's the right way to model this in Coq?
There are lots of ways. One way is to switch to data LC = App LC LC | Abs LC | Var int, with the last constructor representing a de Bruijn index. Another way is to use parametric higher-order
abstract syntax.
Can you tell me more about the parametric higher-order abstract syntax?
Basically, it is data LC = App LC LC | Abs (V -> LC) | Var V for an arbitrary type V (hence the parametricity). But you should google it, as it is hardly something that can be explained in a few
The point of the Haskell data type is that it leans on the native notion of function and substitution rather than defining it explicitly. Chlipala's "Parametric Higher-Order Abstract Syntax for
Mechanized Semantics" and the earlier Despeyroux, Felty and Hirschowitz' "Higher order abstract syntax in Coq" both avoid negative occurrences rather than build a non-inductive type:
The first obstacle is that negative occurrences of the type being defined are not allowed in inductive definitions. If L is the type of terms of the language being defined, the usual way to
express the higher-order syntax of an abstraction operator such as function abstraction in our example is to introduce a constant such as Lam and assign it the type (L -> L) -> L. That is, Lam
takes one argument of functional type. Thus function abstraction in the object-language is expressed using λ-abstraction in the meta-language. As a result, bound variables in the object-language
are identified with bound variables in the meta-language. In inductive types in Coq, negative occurrences such as the first occurrence of L in the above type are disallowed. As in [3], we get
around this problem by introducing a separate type var for variables and giving Lam the type (var -> L) -> L. We must then add a constructor for injecting variables into terms of L. Thus, in our
restricted form of higher-order syntax, we still define function abstraction using λ-abstraction in Coq and it is still the case that a-convertible terms in our object-language map to
α-convertible terms in Coq, but we cannot directly define object-level substitution using Coq's β-reduction. Instead we define substitution as an inductive predicate. Its definition is simple and
similar to the one found in [12].
Is there no way to define object-level substitution using Coq's β?
I think this is the latest incarnation of Felty et al.'s regular HOAS: https://www.site.uottawa.ca/~afelty/HybridCoq/
No, it is not possible. Your original LC type is trivially inconsistent, by cardinality. But, (P)HOAS comes quite close to functions and substitution. Instead of writing Abs (\x -> App x x), you
write Abs (\x -> App (Var x) (Var x)).
Ah! I didn't know that Coq's types weren't just computable sets, all of which necessarily have countable cardinality. OK, I can see now why that wouldn't work. Thanks.
@Karl Palmskog Thanks, I'll have a look.
They're countable from the outside :)
Last updated: Oct 13 2024 at 01:02 UTC
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Problem J
Heck S. Quincy (or Hex as his friends call him) is in charge of recycling efforts in the Quad Cities within the greater Tri-state area as well as the Twin Cities in the Lone Star State. One of the
programs he oversees is the placement of large recycling bins at various locations in the cities. Transporting these bins is very expensive so he tries to keep them at any given location for as long
as possible, emptying them once a week. He’s willing to keep a bin at a given location as long as it is full at the end of each week.
Hex has very reliable estimates of the amount of recyclable materials (in cubic meters) that he can expect each week at each location. Given this information he would like to know when to install the
recycling bin of an appropriate size to maximize the amount of material recycled. He will keep the bin at that location up to (but not including) the week when they don’t expect it to be filled to
For example, suppose Hex has the following estimates for the next seven weeks: $2\ 5\ 7\ 3\ 5\ 10\ 2$. Hex has several options for placing bins, including:
• A capacity-$2$ bin from week $1$ until week $7$, recycling $14\textrm{m}^3$
• A capacity-$5$ bin from week $2$ until week $3$, recycling $10\textrm{m}^3$
• A capacity-$3$ bin from week $2$ until week $6$, recycling $15\textrm{m}^3$ (this is the maximum possible)
Hex would not place a capacity-$5$ bin from week $2$ until week $6$, since it would not be filled in week $4$.
Input starts with a line containing a single positive integer $n$ ($n \leq 100\, 000$) indicating the number of weeks which Hex has estimates for. Weeks are numbered $1$ to $n$. Following this are
$n$ non-negative integers listing, in order, the amount of recycling expected for each of the $n$ weeks. These values may be over multiple lines.
Output three integers $s$ $e$ $r$ where $s$ and $e$ are the start and end weeks for when to place the bin to maximize the total recycling and $r$ is that maximum amount. If there are two or more time
periods that lead to the same maximum amount, output the one with the smallest $s$ value.
Sample Input 1 Sample Output 1
Sample Input 2 Sample Output 2
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Open-Ended Math Tasks Part Three: Grouping, Grading, and Creating
So by now, you understand the reasons that these crazy, open-ended math tasks are so beneficial (even if you were a bit reluctant at first, like me!). You know why you’re using them and you’ve got
your routines in place (If not, check out part 1 & part 2 here). And honestly, that is definitely enough to get started, but experience quickly showed me there are some hacks you can use to make
things run even more smoothly. Knowing how to group your students, how to create your own math tasks, and how to grade students’ work are things you will definitely need to consider. The exact
methods will vary from teacher to teacher, but I have some tried-and-true suggestions to help you as you begin!
Through some trial and error, I have discovered I like to have students work on open ended math tasks in groups of 2 or 3. Any larger than that, and there tends to be at least one student sitting
quietly, and any smaller and you lose the discussion and collaboration piece. As with everything. feel free to experiment until you find the size that best meets your students’ needs.
I always have my groups pre-planned to eliminate any arguing or hurt feelings in the classroom. There are lots of approaches to grouping your students, and each has its positives and negatives. Your
groups will likely change frequently, and should be created with your desired outcome in mind.
Ability-grouping can be used very effectively with open-ended math tasks, although maybe not in the traditional method. You can start by considering each student’s test scores, anecdotal evidence,
and other data you have about their math capabilities to sort them into high, medium, or low groups. (Obviously, it goes without saying that these groups are kept confidential.) From here, you can
select different ways to form your open-ended math task groups. Sometimes you might want to provide some built-in scaffolding by grouping a high, medium, and low student together. Other times, you
might want to allow for students to challenge each other, and group two higher students together. I have even randomly grouped students at times. I look at the specific task and consider what my
goals are when deciding the best approach. I also reflect on how previous groups have worked together, and use that to inform my future decisions.
Since I had my groups pre-determined, it did not take much time at all to set them up in class. I just called the students’ names for each group and they would come forward with a pencil. I’d hand
them one paper for the group, and they would go find a comfy spot to work. This routine was not hard to learn, and took minutes once they had it down pat. Having a routine is critical because you do
not want to waste valuable time arranging groups when they could be working!
Creating Open-Ended Math Tasks:
When I first started using open-ended math tasks, I was able to find some through sites like TeachersPayTeachers. These were great resources, but sometimes I would get frustrated that I couldn’t
find tasks that directly related with the topic we were working on. I quickly realized, though, that I could make some small changes to the traditional word problems in the math book to make them
Keeping the key components of open-ended math tasks in mind, I realized there were a few things I could do with traditional word problems to make them more open-ended. After all, these problems
generally had many “knowns” (or numbers that we are given) and one “unknown” (number we are trying to find out). It couldn’t take too much work to change them around to fit my needs.
1. Turn the question around to make it open-ended. Instead of asking for a specific unknown, you can ask how you would figure it out. This is simple, but may require some guidance to show students
that you expect them to walk you through the steps in their answer.
2. Ask for explanations. This might be the easiest approach, as you can simply ask students to “Solve this problem in two ways”, or present another student’s work and have them decide if they agree
with the answer and the approach. I like this type of problem, but keep in mind it works best when you have a less-intensive task since you are requiring them to solve it more than once.
3. Remove the known. Quite honestly, this is my preferred method and the first one I felt comfortable using. By eliminating one or more of the given numbers, you can instantly make the problem
open-ended and provide multiple points of entry.
4. Swap the “known” with the “unknown”. Many of the word problems in our math program followed a rather prescribed format. By simply turning the “known” into an “unknown” (and sometimes providing a
value for the “unknown”), I could change the problems to be more open-ended and still give some numbers to my class (which the often found comforting).
5. Change the unknown. Sometimes, you just need to change the unknown, or even provide a follow-up question that asks them to solve for a second unknown.
Having these different strategies for converting traditional problems into open-ended math tasks really allowed me to have a much larger pool to pull from than I had before. I was also able to always
find a task that aligned precisely with the topic we were working on. Almost any word problem can be changed into a math task using one (or two) of these methods.
Grading and Tracking
With all of the changing groups and the fact that there is no “right” answer to this type of work, I had no clue how I was going to grade (or even keep track of) these problems at first. I also
wasn’t comfortable not giving a grade since this was fairly labor intensive for my students. So, eventually I came up with this grading scheme. It is a great place to start, and then you can
obviously make any necessary tweaks to make it work for you and your class!
Participation Grades: After realizing that not every group was able to complete the entire task every week (even when they had been diligently working on it), I decided that a participation grade was
the way to go. After all, my goal was to build confidence in solving problems using new strategies. If students were getting bad grades because these they had to switch gears or revise some work
halfway through and were unable to finish, my grading would be counter to my intent!
Tracking: Once I decided to assign participation grades, I felt like a weight had been lifted. But then I realized I still needed a way to keep track of these grades…and I was stressed again! This
is where these grading pages became a life-saver.
They are simple to use, and kept my data nice and organized in one place. That was great for calculating grades, and also because I could see which students might need more support at a glance.
The basics of this sheet are fairly standard. I put the dates across the top, and the names alphabetically down the side. (I loved this particular chart because I could fit 3 months’ worth of weekly
task grades on one sheet of paper!) The bottom row gave me a spot to record the standard (or topic) that we worked on each week.
From there, you can decide how you want to use it! You can get as detailed as you like, or keep it fairly open. At the most basic level, you can mark a check for students who are working. I like to
combine that system with a “C” for students who complete the task, and “NC” for students who weren’t able to finish. I don’t use the “C” or “NC” in grading, but it helps me see if there are some
students who may need some guidance managing their time, or help me arrange the next week’s’ groups. You could create your own system to show different levels of participation as well. I have seen
teachers use “check-plus”, “check”, and “check-minus to rate participation. You can fine-tune this scale to meet your needs, obviously. However, this is a simple and effective way to keep data about
what is happening in your class during this time!
Open-ended math tasks hold such potential to get our students thinking at that higher level. They do take a bit of work to get going in your classroom, but they pay dividends if you can do it! I have
seen growth in independence and confidence in all of my students, and especially in some of the ones who had traditionally struggled with math! So, despite my initial reservations, open-ended math
tasks will always have a place in my math program.
If you are ready to begin, you might want to check out the Create-abilities Open-Ended Math Tasks Webinar that has a wealth of information and an eBook full of resources.
Do you have any good tricks for pairing or grouping students in your classroom?
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Unusual Rules in LSAT Logic Games: Part II - PowerScore Test Prep
This is Part II of our blog series covering the emerging trend of making you labor over strangely-worded or confusing rules in LSAT Logic Games. You can read the first part of the blog here. It’s
worth repeating that none these rules can be described as ambiguous, i.e. none of them are open to multiple interpretations. Indeed, each rule entails a concrete and definitive outcome.
In the second part of this blog series, we will analyze a conditional rule frequently encountered in Linear Games. This rule is particularly prone to misinterpretation:
The Victorian painting must be restored later in the week than the Renaissance painting or later in the week than the Modernist painting, but not both.
Notice that the rule has two parts.
Part One
The Victorian painting must be restored later in the week than the Renaissance painting OR later in the week than the Modernist painting.
The first part of the rule has the classic form of an “either/or” statement, outlining two possible sequences—R > V or M > V—one of which must always occur. So, if the first sequence does not occur,
i.e. if the Victorian painting is NOT restored later in the week than the Renaissance painting, then it must be restored later in the week than the Modernist painting:
By the contrapositive, if the Victorian painting is NOT restored later in the week than the Modernist painting, then it must be restored later in the week than the Renaissance painting:
Part Two
The Victorian painting CANNOT be restored later in the week than both the Renaissance painting and the Modernist painting
According to the second half of the rule, the two sequences outlined above CANNOT occur simultaneously. So, if the first sequence does occur, i.e. if the Victorian painting is restored later in the
week than the Renaissance painting, then it CANNOT be restored later in the week than the Modernist painting:
By the contrapositive, if the Victorian painting is restored later in the week than the Modernist painting, then it CANNOT be restored later in the week than the Renaissance painting:
When combined, the two parts of this rule produce the following conditional relationship between the M > V and R > V sequences:
Assuming that no two paintings can be restored simultaneously, we can simplify the diagram by eliminating any negated terms:
Thus, we can create two separate, exhaustive, and mutually exclusive sequencing chains, one of which will always govern the order in which the three paintings are being restored:
This is a much simpler representation of the original rule, showing us an important inference that you may not have seen otherwise: the Victorian painting can never be the first or the last painting
restored that week.
After you practice with this type of rule a few times, you can probably jump straight into creating the two sequencing chains without diagramming the original conditional statements. While there is
always the danger of misinterpretation, as long as you understand the precise conditional relationship between each two pairs of sequencing rules you should not forfeit accuracy for the sake of
speed. Indeed, superior conceptual understanding does not automatically entail more time spent analyzing the rule in question: on the contrary, over time, this understanding will enable you to
formulate ever more quickly the concrete and definitive outcome of the rules governing the composition of variables in your game.
Jarrett Ezekiel Reeves
Couldn’t the rule above also allow for this scenario: V> M/R?? Wherein the Victorian painting is before BOTH the Renaissance painting and the modernist painting?
Nikki Siclunov
Thanks for your question. The rule does not allow for a V> M/R scenario, because it clearly states that V must be restored LATER than either M or R. So, V cannot precede both.
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Index Feature
1 Multiple peak flag
2 Envelope length
3 Envelope area
4 Envelope mean
5 Envelope variance
6 Envelope skewness
7 Envelope kurtosis
8 Area in frequency domain
9 Mean in frequency domain
10 Variance in frequency domain
11 Skewness in frequency domain
12 Kurtosis in frequency domain
13 Ratio of peak to distance
14 Ratio of peak to CFAR threshold
15 Slope of second envelop
16 Ratio of mean to distance
17 Ratio of variance to distance
18 Ratio of skewness to distance
19 Ratio of kurtosis to distance
20 Ratio of peak to CFAR threshold with distance
21 First peak value in frequency domain
22 First peak index in frequency domain
23 Second peak value in frequency domain
24 Second peak index in frequency domain
25 Ratio of first peak value to distance in frequency domain
26 Ratio of second peak value in frequency domain
27 Ratio of first peak index to distance in frequency domain
28 Ratio of second peak value to distance in frequency domain
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Cost-benefit analysis of electric highways
1 Introduction
It should be noted that even the calculation "to the name" applies to a specific stretch, all cost estimates and calculations are done on a flat-rate basis. The calculation is therefore valid in
principle for any stretch with the same volume of heavy traffic.
2 Starting points for the calculation
The minimum value for heavy traffic (ÅTD) on the Södertälje-Helsingborg route is 2600. In the socioeconomic calculation, this value is used as an estimate of through traffic. No traffic estimation is
performed, but the flow is assumed to be constant over the calculation period.
Thus, approximately 530 million heavy vehicle kilometers are driven annually between Södertälje and Helsingborg.
The calculation assumes that a share of traffic is transferred from diesel to electric power. How much this share would be if the distance was really provided with the necessary infrastructure
obviously depends on business-related aspects, which have not been investigated here. The basic assumption is that 10 percent of the heavy traffic is transferred to electric power (hybrid vehicles).
53 million vehicle kilometers would then be transferred to electrical operation. Provided that the choice between electric and diesel operation is voluntary and without regulation, the proportion
will be Traffic that is really passed over to be determined by business-related aspects, which are not investigated here.
We assume that an average vehicle for long-haul transport rolls 600 km per day (219 000 km / year). There would then be needed about 240 electrified long haulers to perform the estimated traffic
2.1 Discount rate
. . .
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Return-on-Investment (ROI) Calculator
Accurate ROI Calculator
This ROI calculator (return-on-investment) calculates an annualized ROR (rate-of-return) using exact dates. This financial calculator allow you to compare the result of different investments. More
© 2022, Pine Grove Software, LLC
$ : mm/dd/yyyy
Original Size
Click to make smaller (-) or larger (+).
As a side benefit of this calculator's date accuracy, you can also use it to do date math calculations. That is, it will find the date that is "X" days from the start date or given two dates, it will
calculate the number of days between them.
Calendar Tip: When using the calendar, click on the month at the top to list the months, then, if needed, click on the year at the top to list years. Click to pick a year, pick a month, and pick a
day. Naturally, you can scroll through the months and days too. Or you can click on "Today" to quickly select the current date.
If you prefer not using a calendar, single click on a date or use the [Tab] key (or [Shift][Tab]) to select a date. Then, as mentioned, type 8 digits only - no need to type the date part separators.
Also, because the date is selected, you do not need to clear the prior date before typing. If your selected date format equals mm/dd/yyyy, then for March 15, 2023, type 03152023.
Recent: Your desired ROR
At some point, a user might need to know what they should pay for an investment to achieve a desired return-on-investment. Or what they need to sell it for if they have already entered into the
With the most recent update, this calculator can now perform either calculation. All the user need do is provide the goal ROI (and click "Calc" to update). The calculator calculates the adjustment
amount required for both the initial investment and the final value. It also calculates the absolute amount for both.
To double-check the accuracy of the results, copy and paste the value into the appropriate location and recalculate. The ROI should now equal your goal ROI (plus or minus a minimal rounding amount).
And now for an essential word about ROI/ROR financial calculators.
Because two different calculators may use different equations, don't compare the results from one ROI calculator for one investment with results from another calculator for a different investment.
Always use the same calculator to compare two different investments.
What is ROI?
ROI or return-on-investment is the annualized percentage gained or lost on an investment (ROR, or rate-of-return is the same calculation).
Enter the "Amount Invested" and the date the investment was made ("Start Date"). Enter the total "Amount Returned" and the end date.
You can change the dates by changing the number of days. Enter a negative number of days to adjust the "Start Date". Or as you change a date the "Number of Days" will update.
The results include the percentage gained or loss on the investment as well as the annualized gain or loss also expressed as a percent. The annualized return can be used to compare one investment
with another investment.
Example: If you bought $25,000 worth of your favorite stock on January 2nd 2024 and sold it for $33,000 on June 7th 2025, you would gain $8,000 which is 32%. The annualized gain is 21.4%.
Now, let's say you made a second investment on January 2nd, 2024. This time, it was for $10,000, and you sold it for $11,000 on March 1st, 2024 (a leap year). The gain is only $1,000 or 10%. However,
the annualized gain is 82.3%. Ignoring risk (which can be very dangerous), one would generally consider the latter investment to be better than the former.
58 Comments on “Roi Calculator”
Join the conversation. Tell me what you think.
I want to invest 6000000 USD for a lab. The estimated net income is 50000nUSD/month for the first year, 100000 USD/month for the second year, 200000 USD/month for the third year and will be
around this amount/month for the next 5 years. Which formula is needed to see whether this is a profittable investment and how can I play around with time and amount of money invested to make
sure the investment is profitable
When using the calcualtor for the following data, I receive an answer of 144% yet, when I calculate using ((SalePrice/PurchasePrice)/PurchasePrice)/NumDays*365 I get 91.6643% which seems
Purchase Price 123/25
Sale Price 129.75
Days in trade 21
Where did I go wrong, or is my understanding of the calculator’s purpose wrong?
The ROI is an annualized rate of return. This means the calculator assumes you will get the same results from your investment for an entire year AND that the funds are left invested for the
The equation you are using does not allow for the reinvestment of the gain.
Here’s what I mean (with some rounding for simplicity). Using your example, the gain is $6.50 or a gross return of 5.3% over the 21 days. There are about 17.4 investment periods of 21 days in
the course of a year. 17.4 periods * 5.3% gain = 91.6%. Thus this result assumes that the $6.50 profit is withdrawn from the investment at the end of 21 days.
I am investing $325,537 (equipment) that will provide savings of $64,800 per year.
The equipment is expected to last for 20 years. This means for all 20 years my total savings will be $1,296,000.
1. Do I have to enter $1,296,000 in the “amount returned” section?
2. Will the date be a range of 20 years?
3. Will the “amount invested” be $325,537?
Since you, in essence, have cash flows and since money changes value over time, I think you should use this internal rate of return calculator. IRR is an ROI calculation that allows for cash
I would enter -325,537 as the initial investment (thus negative) and then you can use the copy feature to enter 64,800 (positive) for each year for 20 years. For the final value at the end of
the 20 years, you might also want to enter the amount you think you’ll be able to sell the equipment for since that would also represent a return.
If you have questions, you can ask them at the bottom of the above-linked page.
When using the option to calculate selling a put call option I get a different result then using other roi calculators (ex. https://www.calculator.net/roi-calculator.html)
If I sold the put tying up $2100 collected $13 after expiration (considering it expire worthless) I keep the $13.
Amount Invested (PV)?: $2100.00
Amount Returned (FV)?: $2113.00
Days (-9,999 < # 1969)?: Aug 3, 2021
End Date (year < 2100)?: Aug 6, 2021
Hi, so what’s your question? You made $13 in 3 days on a $2,100 investment. The result is an annualized rate of return of about 111%.
It seems the other calculator is not taking into account reinvestment of the gain. An annualized rate of return should assume (to my way of thinking) that the gain is reinvested (or left in
the investment). Using your example, the next investment amount would be $2113 (reinvesting the gain), and if you do this for a year, you’ll have a return of about 111%. And the next
investment will be for $2126.?? and some change for the slightly higher gain for the second trade. That is the gain the 2nd time will not be $13, but $13 and some cents on the $2113
But, on a more important note, an ROI calculation is used to compare different investments, and if you are using it for that purpose, it’s important to use the same calculator (and you can
see why).
One more thing, I believe I must have misunderstood what you meant.
You said you sold a put call option. I took that to mean you were short, that is, that you sold it first. But now I think you must have meant you bought either an option and then sold it. In
that case, the amount invested, I guess, would be $2,100 and the amount returned would be just the $13.
Your post is a little unclear, particularly since you didn’t even ask a question.
Comments, suggestions & questions welcomed...
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James Clerk Maxwell: the father of light
The lonesome grave lies in a tiny crumbling stone church beside Loch Ken in southwest Scotland. You might think some otherwise undistinguished local lord lies here – until you see the name on the
granite headstone: James Clerk Maxwell.
This is the final resting place of a giant of physics, the man who discovered that light is a wave created by the mutual push of magnetic and electric impulses. His discovery 150 years ago opened the
door to the modern era of wireless communication.
Ask physicists to rank their heroes, and Maxwell is in the top three, standing a shade below Newton and Einstein. But when it comes to being celebrated by the public, somehow Maxwell got left behind.
Einstein’s image is well known and Newton’s pilgrims regularly flock to his tomb at Westminster. But few of us would recognise Maxwell’s face or know of the forgotten grave in the crumbling kirk.
It is a pity, because Maxwell was one of the most likeable men in the annals of science. How can you not like a man who sends a heartfelt letter of condolence on the death of a friend’s dog? A man
who patiently nursed his dying father, and later his wife, and who regularly gave up his time to volunteer at the new “Working Men’s Colleges” for tradesmen? It seems that everyone who knew him
thought of him as kind and generous, albeit a little eccentric. He was “one of the best men who ever lived”, according to his childhood friend and biographer, Lewis Campbell.
Born in Edinburgh, Maxwell grew up on his family estate at Glenlair, not far from the church where he is buried. He became laird of Glenlair when he was only 24, on the death of his father with whom
he shared a close bond – he was an only child and his mother died when he was eight.
His interest in physics was a natural extension of his fascination with how things work, and his love of nature. Maxwell was also a gifted mathematician. At 14 he developed a new method of
constructing some unusual geometric curves; his father brought it to the attention of the Edinburgh Royal Society which declared “the simplicity and elegance of the method” worthy of publication in
their proceedings. In his early 20s, Maxwell used Newton’s laws to show mathematically that Saturn’s rings are not solid, as they appear through a telescope, but are made of many smaller bodies. His
paper won him a prize from Cambridge and more than a century later in the 1980s, Voyager proved him right.
Maxwell also loved language: he was a poet and a lucid science writer. His feeling for mathematics and for language came together in his unique approach to the most important scientific problem of
his time: understanding electromagnetism.
We don’t often think about the importance of language in making scientific theories. Yet language clearly shapes our perception of the world. Place English speakers with their single word for snow
amid the Inuit of Alaska, and they’ll be at a loss to describe details of the landscape. The Inuit by contrast have dozens of words to describe snow in all its forms.
Language can also express a prejudice or lock in an attitude. For instance, do we describe a wilderness as beautiful or threatening?
If everyday language can be so subjective, then what about scientific language? Science’s guiding principle is objectivity: scientists must be able to agree upon the results of a given experiment,
regardless of their personal beliefs. At the same time science stands at the cusp between reality and language – between what is really out there, and what we are able to describe. Maxwell’s genius
lay in recognising that problem and painstakingly searching for the right mathematical language to minimise it. But we’ll get to that shortly.
First, let’s set the scene. What did physics look like when young Maxwell entered the picture?
Most auspiciously he was born in 1831, the same year that the self-educated English physicist Michael Faraday made an astounding discovery. Faraday slid a magnet through a coiled wire, and presto, an
electric current began flowing through it – no batteries required. Ten years earlier, Danish physicist, Hans Øersted had discovered the opposite: by switching on an electric current, he found a
nearby magnetic compass needle jumped, as if the changing electric current were itself a magnet.
This mysterious interaction between electricity and magnetism was called “electromagnetism”.
No-one knew how this force was actually transmitted between wires and magnets, but most physicists assumed it acted instantaneously, without any intermediary mechanism – the same way gravity,
magnetism and static electricity seemed to act. Apples immediately fall to the ground. Iron nails placed near a magnet are immediately pulled towards it, and two electric charges immediately attract
or repel each other.
This kind of remote, instantaneous process was dubbed “action at a distance”. But by giving it such a name, physicists were playing with words: it was just an intuitive idea yet it seemed to be
confirmed by mathematics – particularly Newton’s inverse square law (which says the force of gravity between two objects decreases according to the square of the distance between them). Using
Newton’s laws you could work out the paths of planets without worrying about how the force of gravity travelled from the Sun to the Earth, for instance. So if the maths worked without the need to
consider how gravity moved through space, then maybe gravity really did act at a distance?
This is how most of Newton’s disciples saw it – although Newton himself did not. As it happened, action at a distance was a headache for Newton: in the late 1600s many scholars dismissed his whole
theory because they couldn’t imagine how gravity could possibly act instantaneously across a vast chasm of nothingness, no intermediary required.
Nevertheless, Newton’s laws worked. As well as describing planetary motion, they accurately predicted phenomena such as the date of the return of Halley’s comet and the existence of Neptune (deduced
from distortions in the orbit of Uranus). In time, action at a distance was accepted as self-evident. This idea became even more entrenched when, in 1785 – nearly a century after Newton formulated
his theory – French physicist Charles- Augustin de Coulomb showed that the electric force between two charged particles obeyed the same kind of inverse square law as gravity.
With the discovery of electromagnetism in the 19th century, the picture became much more complicated. Most physicists – the so-called Newtonians – assumed action at a distance still applied.
But Faraday dissented. The co-discoverer of electromagnetism thought the electromagnetic force must be communicated step by step through space, just as a breeze blowing through a farmer’s field moves
every stalk in turn. Indeed he used the term “field” to describe the space around magnets and currents, and he imagined the field contained lines of force radiating from the electric and magnetic
sources. He didn’t believe gravity acted remotely either, and knew that Newton had never thought so: just because the maths seemed to imply action at a distance, that didn’t mean the concept was
real. “Newton was no Newtonian,” Faraday once quipped.
But the 19th-century Newtonians held sway. Their maths worked brilliantly for gravity, as well as for static electric and magnetic forces; they saw no reason not to apply it to electromagnetism too.
As George Airy, Britain’s Astronomer Royal summed it up: “I declare that I can hardly imagine anyone … to hesitate an instant in the choice between the simple and precise [Newtonian] action, on the
one hand, and anything so vague as lines of force on the other …”
* * *
This was the intellectual backdrop against which Maxwell, newly graduated from Cambridge, began developing a theory to explain electromagnetism. He carefully considered both sides of the debate. But
something about Faraday’s lines of force resonated with him. As a two-year-old at Glenlair, he had been amazed that when he pulled a rope in one room, a bell rang in another, as if by magic. Then he
discovered the holes in the walls where the bell-wires came through, and he dragged his father through the house, enthusiastically pointing them out.
Now, all these years later, he recalled those bell-wires. They deepened his conviction that just as a bell needed a wire, so electromagnetic effects must act through the agency of some sort of field.
Faraday’s lack of formal education meant he hadn’t been able to present his field concept in mathematical language. This is why Airy called the idea vague, and why few mainstream physicists paid it
much attention. Maxwell believed that if he could find the right mathematical language to describe Faraday’s meticulous measurements of the forces surrounding electromagnetic objects, then perhaps
the Newtonians might reconsider their objections.
For Maxwell, language held the key to unlocking the true nature of electromagnetism. He felt physicists’ choice of language was partly influenced by their style of thinking – by whether they tended
to think primarily in mathematical terms or with the help of concrete images. Each style had its advantages and disadvantages, which Maxwell summed up later in a speech to the British Association for
the Advancement of Science. The “natural mathematicians” are “quick to appreciate the significance of mathematical relationships”, he said. But the problem in physics was that such a thinker was
often “indifferent” as to whether or not “quantities actually exist in nature which fulfil this relationship”. (Maxwell might also have been thinking here about those who used mathematics to justify
a “magical” concept such as action at a distance.)
On the other hand, some scientists need concrete imagery to flesh out their equations. Such thinkers, Maxwell said, “are not content unless they can project their whole physical energies into the
scene which they conjure up. They learn at what rate the planets rush through space, and they experience a delightful feeling of exhilaration. They calculate the forces with which the heavenly bodies
pull at one another, and they feel their own muscles straining with the effort. To such men, [concepts such as] momentum, energy, and mass are not mere abstract expressions of the results of
scientific enquiry. They are words of power, which stir their souls like memories of childhood.”
Maxwell employed both types of thinking.
Maxwell was clear: analogies were useful as scaffolds in erecting a theory, but should not be taken for reality.
To explore Faraday’s field idea, Maxwell began by searching for analogies, “words of power” that conjure up concrete physical images. In his first electrical paper, he showed how Faraday’s imaginary
lines of force around magnets and electric charges could be modelled using the analogy of streamlines, such as you see in eddies in a flowing river. The mathematics of fluid flow had been pioneered
in the 18th century, and Maxwell adapted these equations so they fitted the meticulous data Faraday had collected measuring the forces in the space around magnets and current-carrying wires.
Maxwell sent a copy of his streamlines paper to Faraday. Now 65, and depressed at the mainstream rejection of his idea, Faraday was overjoyed to hear from this unknown 25-year-old. He wrote to
Maxwell that he had “never communicated with one of your mode and habit of thinking”. He also asked if Maxwell could express his work in “common language” as well as in mathematical “hieroglyphics”
so that he could understand it?
Maxwell spent the next five years developing mathematical descriptions of various mechanical models that helped him imagine how a field could transmit changing electromagnetic forces. Yet he knew
that such models did not qualify as a true theory of electromagnetism. It was about language again: by imagining Faraday’s field to be like a fluid – or like heat, or acting via mechanical cogs and
flywheels – he was making assumptions for which he had no actual evidence.
Maxwell was clear in his writing about this: models and analogies are useful as scaffolds in erecting a theory, but they should not be mistaken for reality. So, with the mathematical insights he’d
gained from his models now in hand, he dismantled his scaffolds, and started erecting his theory from scratch. His goal was to build a mathematical theory using only established physical principles,
and the data gleaned from the papers of electromagnetic experimentalists such as Faraday. It would take him three more years.
Finally, by 1865, Maxwell was able to describe all that was known about electromagnetism in a set of “partial differential equations”. That in itself was a remarkable feat. Then he combined his
equations and carried out one more mathematical operation.
And something extraordinary happened …
He found himself looking at the mathematical description of a transverse wave – the sort that travels along a plucked string.
With growing excitement, Maxwell realised his purely electromagnetic wave had exactly the same signature as a light wave – the same form, the same speed. The mathematical coincidence was too
delicious to ignore. In his 1865 paper, he announced with understated triumph: “We have strong reason to conclude that light itself (including radiant heat, and other radiation if any) is
electromagnetic …”
In one fell swoop, Maxwell’s equations seemed to resolve two of the major conundrums in physics: how electromagnetism is transmitted through space and the nature of light.
As Einstein said later of this discovery: “Imagine Maxwell’s feelings … at this thrilling moment! To few men in the world has such an experience been vouchsafed.”
If Maxwell was right, then Faraday was vindicated: electromagnetism did not act instantaneously at a distance, but through a field. And fluctuations in this field were propagated as waves. Just as a
wave rippling along a string can vibrate with a range of frequencies, so too electromagnetic waves had different frequencies, some of which we perceive as light.
The rippling light wave was created by the mutual nudging of electric and magnetic fields. It was not so different to the way a Mexican wave travels across a stadium: one row of fans stand up and sit
down, and trigger the next row to do the same. But in this case, the stadium is populated by electric and magnetic fields, each nudging the other on.
Maxwell hadn’t assumed anything about the nature of light.
But did the waves that Maxwell described mathematically actually exist? To prove it, someone would have to generate an electromagnetic wave from electrical and magnetic impulses. German physicist
Heinrich Hertz took up the challenge.
He rigged up an electric circuit that sent sparks jumping back and forth between a spark gap made of two brass knobs. According to Maxwell the changing electric current would generate an
electromagnetic wave.
To detect it, Hertz set up a receiver a few metres away – a loop of wire with a spark gap but no source of electricity. Then he started generating sparks from his oscillator. Lo and behold, across
the room, another series of sparks began oscillating in the receiver!
To prove that these sparks had been generated from energy carried by an electromagnetic wave, Hertz aimed his oscillator at a metal screen. If the oscillator really did produce waves, then the screen
would reflect the incoming waves, and a reflected wave would combine with an incoming one to form a series of “nodes” where the two waves cancel each other out.
This is what happens when the ripples from two pebbles dropped in a pond combine. By moving his apparatus, Hertz did indeed detect the neutral spots that must be the nodes; measuring the distance
between adjacent nodes, he found the wavelength of his radiation. The year was 1887, more than two decades after Maxwell’s theory was published. Hertz had produced the first deliberately engineered
radio waves. They had the same speed as light but a different frequency and wavelength, so they are part of what is now called the electromagnetic spectrum.
Sadly, Maxwell did not live to see Hertz’s confirmation of his theory. Nor did he live to see the widespread consequences of his mathematical field analysis, so that today we speak not only of
electromagnetic fields, but also of gravitational and quantum fields.
A hundred and fifty years ago, though, Maxwell’s theory was so controversial that many of his peers refused to teach it. Critics such as Maxwell’s friend, Lord Kelvin, thought mathematics should
describe tangible facts, analogies and models. They did not think that mathematical language itself might reveal new knowledge about the physical world.
Today, Einstein’s E=mc^2 is the best known example of the power of mathematical language to reveal hidden truths. Einstein hadn’t set out to prove that energy and matter were essentially the same –
but there it was in his equation!
Similarly, Maxwell hadn’t assumed anything about the nature of light – but there, hidden in his electromagnetic equations were mathematical waves travelling at the speed of light.
They were not ordinary waves travelling through a medium such as water or air, but purely mathematical waves of changing electric and magnetic intensity. In abandoning his earlier concrete models,
Maxwell instinctively seemed to know that in the unseen realms – the “hidden, dimmer regions where thought weds fact”, as he put it – the closest we may come to perceiving physical reality is to
imagine it mathematically.
The shy poetic Maxwell may not have achieved the celebrity status of Einstein, but in 2015 – the Year of Light – he deserves to be celebrated. You could do worse than don one of the T-shirts
occasionally worn by students at university physics departments. They read: “And God said … Maxwell’s equations … and there was light.”
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[Solved] What universal set (s) would you propose for each of t... | Filo
What universal set (s) would you propose for each of the following:
The set of isosceles triangles.
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The set of all triangles in a plane
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What universal set (s) would you propose for each of the following:
Question Text The set of isosceles triangles.
Updated On May 6, 2023
Topic Sets
Subject Mathematics
Class Class 11
Answer Type Text solution:1 Video solution: 3
Upvotes 381
Avg. Video Duration 7 min
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Research Grants 16/25053-8 - Dinâmica complexa, Sistemas hamiltonianos - BV FAPESP
Grant number: 16/25053-8
Support Opportunities: Research Projects - Thematic Grants
Duration: August 01, 2017 - January 31, 2024
Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
Principal Investigator: André Salles de Carvalho
Grantee: André Salles de Carvalho
Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
Pesquisadores principais: Albert Meads Fisher ; Clodoaldo Grotta Ragazzo ; Edson de Faria ; Edson Vargas ; Fábio Armando Tal ; Pedro Antonio Santoro Salomão ; Rodrigo Bissacot Proença ; Salvador
Addas Zanata
Associated researchers: Claudio Gorodski ; Fábio Armando Tal ; Luciana Luna Anna Lomonaco ; Ricardo dos Santos Freire Júnior ; Sinai Robins ; Sylvain Philippe Pierre Bonnot ; Yoshiharu Kohayakawa
19/16278-4 - Exploring universality in 1-D systems, AP.R SPRINT
18/06267-2 - Invariant measures in weakly hyperbolic dynamics, AV.EXT
Associated grant(s): 17/26645-9 - Thermodynamic formalism and KMS states on Countable Markov shifts, AV.BR
17/50139-6 - Rigidity in mildly smooth 1-D systems, AP.R SPRINT
17/13160-7 - Continuity of entropy and classification of partially hyperbolic systems with one-dimensional central bundle, AV.EXT
23/16187-4 - Random compositions of $T^2$ homeomorphisms and rotation sets, BP.PD
Associated scholarship 23/18381-2 - The 8-vertex model, the toric code and quantum information, BP.IC
(s): 23/07381-1 - A classic geometry view of Teichmüller theory and variations on the Gromov-Lawson-Thurston conjecture, BP.PD
+ associated scholarships - associated scholarships
This project is a continuation of two previous thematic projects supported by FAPESP with numbers 2006/03829-2 and 2011/16265-8. The present group includes researchers working on dynamical systems
and low-dimensional geometry and has senior as well as and young researchers, including recent hires. The areas covered by the project are: dynamics in dimension 2: dynamics of homeomorphisms and
diffeomorphisms of the torus; topological dynamics on surfaces; Hénon maps; Teichmüller Theory and its connections with dynamics and geometry in low dimensions; endomorphisms of the interval,
critical circle maps, renormalization and parameter space; hamiltonian dynamics; pseudo-holomorphic curves and symplectic dynamics; complex dynamics in dimensions 1 and 2; continuous and
differentiable ergodic theory of finite and infinite measures; thermodynamic formalism and ergodic optimization. The purpose of this proposal is to continue to the work we have been doing and also
aims to expand the activities of the group that has grown and includes new researchers and new areas of research. (AU)
Articles published in Agência FAPESP Newsletter about the research grant:
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Articles published in other media outlets ( ):
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VEICULO: TITULO (DATA)
VEICULO: TITULO (DATA)
Articles published in Agência FAPESP Newsletter about the research grant:
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Articles published in other media outlets ( ):
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VEICULO: TITULO (DATA)
VEICULO: TITULO (DATA)
(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
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Angle Sum Property of a Triangle: Theorem, Examples and Proof (2024)
Angle Sum Property of a Triangle is the special property of a triangle that is used to find the value of an unknown angle in the triangle. It is the most widely used property of a triangle and
according to this property, “Sum of All the Angles of a Triangle is equal to 180º.”
Angle Sum Property of a Triangle is applicable to any of the triangles whether it is a right, acute, obtuse angle triangle or any other type of triangle. So, let’s learn about this fundamental
property of a triangle i.e., “Angle Sum Property “.
Table of Content
• What is the Angle Sum Property?
• Angle Sum Property Formula
• Proof of Angle Sum Property
• Exterior Angle Property of a Triangle Theorem
• Angle Sum Property of Triangle Facts
• Solved Example
• FAQs
What is the Angle Sum Property?
For a closed polygon, the sum of all the interior angles is dependent on the sides of the polygon. In a triangle sum of all the interior angles is equal to 180 degrees. The image added below shows
the triangle sum property in various triangles.
This property holds true for all types of triangles such as acute, right, and obtuse-angled triangles, or any other triangle such as equilateral, isosceles, and scalene triangles. This property is
very useful in finding the unknown angle of the triangle if two angles of the triangle are given.
Angle Sum Property Formula
The angle sum property formula used for any polygon is given by the expression,
Sum of Interior Angle = (n − 2) × 180°
where ‘n’ is the number of sides of the polygon.
According to this property, the sum of the interior angles of the polygon depends on how many triangles are formed inside the polygon, i.e. for 1 triangle the sum of interior angles is 1×180° for two
triangles inside the polygon the sum of interior angles is 2×180° similarly for a polygon of ‘n’ sides, (n – 2) triangles are formed inside it.
Example: Find the sum of the interior angles for the pentagon.
Pentagon has 5 sides.
So, n = 5
Thus, n – 2 = 5 – 2 = 3 triangles are formed.
Sum of Interior Angle = (n − 2) × 180°
⇒ Sum of Interior Angle = (5 − 2) × 180°
⇒ Sum of Interior Angle = 3 × 180° = 540°
Proof of Angle Sum Property
Theorem 1: The angle sum property of a triangle states that the sum of interior angles of a triangle is 180°.
The sum of all the angles of a triangle is equal to 180°. This theorem can be proved by the below-shown figure.
Follow the steps given below to prove the angle sum property in the triangle.
Step 1: Draw a line parallel to any given side of a triangle let’s make a line AB parallel to side RQ of the triangle.
Step 2: We know that sum of all the angles in a straight line is 180°. So, ∠APR + ∠RPQ + ∠BPQ = 180°
Step 3: In the given figure as we can see that side AB is parallel to RQ and RP, and QP act as a transversal. So we can see that angle ∠APR = ∠PRQ and ∠BPQ = ∠PQR by the property of alternate
interior angles we have studied above.
From step 2 and step 3,
∠PRQ + ∠RPQ + ∠PQR = 180° [Hence Prooved]
Example: In the given triangle PQR if the given is ∠PQR = 30°, ∠QRP = 70°then find the unknown ∠RPQ
As we know that, sum of all the angle of triangle is 180°
∠PQR + ∠QRP + ∠RPQ = 180°
⇒ 30° + 70° + ∠RPQ = 180°
⇒ 100° + ∠RPQ = 180°
⇒ ∠RPQ = 180° – 100°
⇒ ∠RPQ = 80°
Exterior Angle Property of a Triangle Theorem
Theorem 2: If any side of a triangle is extended, then the exterior angle so formed is the sum of the two opposite interior angles of the triangle.
As we have proved the sum of all the interior angles of a triangle is 180° (∠ACB + ∠ABC + ∠BAC = 180°) and we can also see in figure, that ∠ACB + ∠ACD = 180° due to the straight line. By the
above two equations, we can conclude that
∠ACD = 180° – ∠ACB
⇒ ∠ACD = 180° – (180° – ∠ABC – ∠CAB)
⇒ ∠ACD = ∠ABC + ∠CAB
Hence proved that If any side of a triangle is extended, then the exterior angle so formed is the sum of the two opposite interior angles of the triangle.
Example: In the triangle ABC, ∠BAC = 60° and ∠ABC = 70° then find the measure of angle ∠ACB.
The solution to this problem can be approached in two ways:
Method 1: By angle sum property of a triangle we know ∠ACB + ∠ABC + ∠BAC = 180°
So therefore ∠ACB = 180° – ∠ABC – ∠BAC
⇒ ∠ACB = 180° – 70° – 60°
⇒ ∠ACB = 50°
And ∠ACB and ∠ACD are linear pair of angles,
⇒ ∠ACB + ∠ACD = 180°
⇒ ∠ACD = 180° – ∠ACB = 180° – 50° = 130°
Method 2: By exterior angle sum property of a triangle, we know that ∠ACD = ∠BAC + ∠ABC
∠ACD = 70° + 60°
⇒ ∠ACD = 130°
⇒ ∠ACB = 180° – ∠ACD
⇒ ∠ACB = 180° – 130°
⇒ ∠ACB = 50°
Read More about Exterior Angle Theorem.
Angle Sum Property of Triangle Facts
Various interesting facts related to the angle sum property of the triangles are,
• Angle sum property theorem holds true for all the triangles.
• Sum of the all the exterior angles of the triangle is 360 degrees.
• In a triangle sum of any two sides is always greater than equal to the third side.
• A rectangle and square can be divided into two congruent triangles by their diagonal.
Also, Check
□ Area of a Triangle
□ Area of Isosceles Triangle
Solved Example on Angle Sum Property of a Triangle
Example 1: It is given that a transversal line cuts a pair of parallel lines and the ∠1: ∠2 = 4: 5 as shown in figure 9. Find the measure of the ∠3?
As we are given that the given pair of a line are parallel so we can see that ∠1 and ∠2 are consecutive interior angles and we have already studied that consecutive interior angles are
Therefore let us assume the measure of ∠1 as ‘4x’ therefore ∠2 would be ‘5x’
Given, ∠1 : ∠2 = 4 : 5.
∠1 + ∠2 = 180°
⇒ 4x + 5x = 180°
⇒ 9x = 180°
⇒ x = 20°
Therefore ∠1 = 4x = 4 × 20° = 80° and ∠2 = 5x = 5 × 20° = 100°.
As we can clearly see in the figure that ∠3 and ∠2 are alternate interior angles so ∠3 = ∠2
∠3 = 100°.
Example 2: As shown in Figure below angle APQ=120° and angle QRB=110°. Find the measure of the angle PQR given that the line AP is parallel to line RB.
As we are given that line AP is parallel to line RB
We know that the line perpendicular to one would surely be perpendicular to the other. So let us make a line perpendicular to both the parallel line as shown in the picture.
Now as we can clearly see that
∠APM + ∠MPQ = 120° and as PM is perpendicular to line AP so ∠APM = 90° therefore,
⇒ ∠MPQ = 120° – 90° = 30°.
Similarly, we can see that ∠ORB = 90° as OR is perpendicular to line RB therefore,
∠QRO = 110° – 90° = 20°.
Line OR is parallel to line QN and MP therefore,
∠PQN = ∠MPQ as they both are alternate interior angles. Similarly,
⇒ ∠NQR = ∠ORQ
Thus, ∠PQR = ∠PQN + ∠NQR
⇒ ∠PQR = 30° + 20°
⇒ ∠PQR = 50°
FAQs on Angle Sum Property
Define Angle Sum Property of a Triangle.
Angle Sum Property of a triangle states that the sum of all the interior angles of a triangle is equal to 180°. For example, In a triangle PQR, ∠P + ∠Q + ∠R = 180°.
What is the Angle Sum Property of a Polygon?
The angle sum property of a Polygon states that for any polygon with side ‘n’ the sum of all its interior angle is given by,
Sum of all the interior angles of a polygon (side n) = (n-2) × 180°
What is the use of the angle sum property?
The angle sum property of a triangle is used to find the unknown angle of a triangle when two angles are given.
Who discovered the angle sum property of a triangle?
The proof for triangle sum property was first published by, Bernhard Friedrich Thibaut in the second edition of his Grundriss der Reinen Mathematik
What is the Angle Sum Property of a Hexagon?
Angle sum property of a hexagon, states that the sum of all the interior angles of a hexagon is 720°.
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Multiple Gaussian targets - The track-on-jam problem
When a radar with amplitude comparison monopulse arithmetic encounters signals from multiple Gaussian sources it will 'point' to the centroid of the incident radiation. The probability density
function (pdf) of the monopulse ratio when N independent samples of difference and sum signals are processed in a maximum likelihood receiver is derived. For finite jam-to-noise ratio the estimate
has a bias which is independent of N. The variance in the estimate does however depend upon N. Central moments of order less than or equal to 2N - 2 exist and are given by a simple formula. Plots of
the pdf and its bias and variance for various jam-to-noise ratios, locations of the centroid with respect to the boresight direction, and number of samples processed are presented in the accompanying
IEEE Transactions on Aerospace Electronic Systems
Pub Date:
November 1977
□ Jamming;
□ Maximum Likelihood Estimates;
□ Monopulse Radar;
□ Probability Density Functions;
□ Radar Detection;
□ Signal To Noise Ratios;
□ Boresights;
□ Electronic Countermeasures;
□ Normal Density Functions;
□ Signal Processing;
□ Variance (Statistics);
□ Communications and Radar
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homework 4w
\documentclass[11pt]{article} \usepackage[margin=1in]{geometry} \usepackage{fancyhdr} \usepackage{amssymb} \pagestyle{fancy} \lhead{Homework 4W} \chead{Geoffrey Bostany} \rhead{Recitation number 304}
\begin{document} \begin{enumerate} \item $\frac{4!}{2!}* {11 \choose 3}$ \\ if we are operating under the constraints that everone gets a fruit and a distinct person gets the mango and the pair, we
can satisfy both of these constraints by finding the permutation of 2 apples, a mango, and a pear. this is 4! divided by 2! for the duplicate apples. Then, we can now order the remaining 8 apples
freely between four people. So its ${8+4-1 \choose 8}$ and we get the answer by multiplying those two together. \item ${100 \choose 4}$ \\ We can think about this problem in terms of sticks and
crosses. We have 4 sticks to divide our x's up and then 197 crosses. In order to satisfy the constraint that all x's must be positive, we can assume that at least one cross must go in between each
stick. That leaves us with 192 sticks left. Now, in order to satisfy the constraint that all x's must be odd, we can group the crosses in pairs of 2 so that we only add them two at a time. Since
there is already 1 cross in each group, no matter how many pairs of x's we add, we can write it as $2k+1$ which we know is always odd. Thus we divide the 192 remaining crosses by 2 to get 96 crosses
and 4 sticks. Thus the answer is ${96+5-1 \choose 4}$ \item \begin{enumerate} \item \item \end{enumerate} \item \item \item Let $x$ be any arbitrary but particular element of $(P(A) \cap P(B))$ \\ $x
\in [P(A) \cap P(B)]$ \\ $x\in P(A) \wedge x\in P(B)$ \\ $x\subseteq A \wedge x \subseteq B$ \\ $x \subseteq A \cap B$ \\ $x\in P(A \cap B)$ \\ since we know that $x\in [P(A) \cap P(B)]$ then $P(A \
cap B) = P(A) \cap P(B)$ \item \begin{enumerate} \item let $n$ be any arbitrary but particular integer such that it is divisible by 3 thus \\ $n^3-n=3k$ for some integer $k$ \\ in the case that $3|n$
we know that $3|n^3-n$ because $3|n \to 3|n^3 \wedge 3|n \to 3|n^3-n$ \\ so to prove original statement, we must satisfy the case where $3 \nmid n$ \\ so given our original statement $n^3-n=3k$ for
some integer k, we can reason that $n(n^2-1)=3k$ \\ and since $3 \nmid n$ then $3 \mid (n^2-1)$ thus \\ $n^2-1=3k$ thus\\ $(n-1)(n+1)=3k$ \\ NOW, since $3 \nmid n \to [3|(n+1) \vee 3|(n-1)]$ because
3 divides every third number. Thus $3|n^3-n$ for all integers $n$. \item let $x$ be any arbitrary but particular real number such that \\ $2x^2-4x+3>0$\\ $2x^x-4x+2>-1$ \\ $2(x^2-2x+1)>-1$ \\ $2(x-1)
(x-1)>-1$ \\ $(x-1)^2>-\frac{1}{2}$ \\ for any integer $k$, $k^2$ will never be negative so if $k=(x-1)$, then $k^2$ will never be negative \\ thus $(x-1)^2>-\frac{1}{2}$ thus proving the original
statement. \end{enumerate} \end{enumerate} \end{document}
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The EDGE-CALIFA Survey: Spatially Resolved 13 CO(1–0) Observations and Variations in 12 CO(1–0)/ 13 CO(1–0) in Nearby Galaxies on Kiloparsec Scales (Journal Article) | NSF PAGESAbstractAbstractAbstractAbstractAbstractAbstract
We measure the thermal electron energization in 1D and 2D particle-in-cell simulations of quasi-perpendicular, low-beta (β[p]= 0.25) collisionless ion–electron shocks with mass ratiom[i]/m[e]= 200,
fast Mach number$Mms=1$–4, and upstream magnetic field angleθ[Bn]= 55°–85° from the shock normal$nˆ$. It is known that shock electron heating is described by an ambipolar,B-parallel electric
potential jump, Δϕ[∥], that scales roughly linearly with the electron temperature jump. Our simulations have$Δϕ∥/(0.5miush2)∼0.1$–0.2 in units of the pre-shock ions’ bulk kinetic energy, in agreement
with prior measurements and simulations. Different ways to measureϕ[∥], including the use of de Hoffmann–Teller frame fields, agree to tens-of-percent accuracy. Neglecting off-diagonal electron
pressure tensor terms can lead to a systematic underestimate ofϕ[∥]in our low-β[p]shocks. We further focus on twoθ[Bn]= 65° shocks: a$Ms=4$($MA=1.8$) case with a long, 30d[i]precursor of whistler
waves along$nˆ$, and a$Ms=7$($MA=3.2$) case with a shorter, 5d[i]precursor of whistlers oblique to both$nˆ$andB;d[i]is the ion skin depth. Within the precursors,ϕ[∥]has a secular rise toward the
shock along multiple whistler wavelengths and also has localized spikes within magnetic troughs. In a 1D simulation of the$Ms=4$,θ[Bn]= 65° case,ϕ[∥]shows a weak dependence on the electron
plasma-to-cyclotron frequency ratioω[pe]/Ω[ce], andϕ[∥]decreases by a factor of 2 asm[i]/m[e]is raised to the true proton–electron value of 1836.
more » « less
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Miscellany № 59: the percent sign
A post from Shady Characters
Miscellany № 59: the percent sign
Reproduced from David Eugene Smith’s 1908 Rara Arithmetica, the last line of this manuscript contains the abbreviation pcº, with the ‘c’ pulled out into a long horizontal stroke. (Image from
A few weeks back, Nina Stössinger asked on Twitter:
Isn’t it odd that the percent sign looks like “0/0” rather than, say, “/100” or “/00”?
This, it turns out, is a very good question. Like Nina, I had assumed that the percent sign was shaped so as to invoke the idea of a vulgar fraction, with a tiny zero aligned on either side of a
solidus (⁄), or fraction slash. That said, something about those zeroes had always nagged at me. Specifically, as you divide any non-zero quantity by a smaller and smaller number the result tends
ever closer to infinity (or rather, ±∞ as appropriate), until finally, when dividing by zero itself, you reach a mathematical singularity where the result cannot be computed — a numerical black hole
of exotic properties and mind-bending implications. Throw in another zero as the numerator and you have a thoroughly nonsensical fraction. Though this is all terribly exciting from a philosophical
point of view, it is not an especially useful situation to be in when trying to communicate the simple concept of division into hundredths. Either the ‘%’ had stumbled, blinking, from some secret
garden of esoteric mathematics and into the real world, or there was more to the story. And so there was.
Writing in 1908, David Eugene Smith, later to be president of the Mathematical Association of America,1 reported on a peculiar find he had made in an Italian manuscript written sometime during the
early part of fifteenth century. (Smith was cataloguing the mathematical holdings of one George Arthur Plimpton, a publisher and philanthropist who had amassed a huge library of ancient books.) What
had caught Smith’s eye was an oddly attenuated abbreviation comprising a ‘p’, an elongated ‘c’, and a superscript ‘o’, or ‘^o’, balanced upon the extended upper terminal of the ‘c’, as seen at top.
From its context, Smith deduced that pc^o was a stand-in for the words per cento, or “per hundred”, more often abbreviated to per 100, p cento, or p 100.2 It was the first step towards a distinct
percent sign — and, counterintuitively, it had precisely nothing to do with the digit zero.
A percent sign written in 1684, as reproduced in David Eugene Smith’s History of Mathematics. (Image courtesy of archive.org.)
Smith picked up the trail with his weighty two-volume History of Mathematics, published in 1923,3 wherein he printed an image of the percent sign caught midway between pc^o and ‘%’. Taken from an
Italian manuscript of 1684, as seen above, by now the word per had collapsed into the tortuous but common scribal abbreviation seen here while the ‘c’ had morphed into a closed circle surmounted by a
short horizontal stroke. The imperturbable ‘^o’ sat atop it. All that remained was for the vestigial per to vanish and for the horizontal stroke to assume its familiar diagonal orientation — a change
that occurred sometime during the nineteenth century — and the evolution of the percent sign was complete.
Since then the ‘%’ has gone from strength to strength, and today we revel in a whole family of “per ————” signs, with ‘%’ joined by ‘‰’ (“per mille”, or per thousand) and ‘‱’ (per ten thousand). All
very logical, on the face of it, and all based on a fundamental misunderstanding of how the percent sign came to be. Nina and I can comfort ourselves that we are not the first people, and likely will
not be the last, to have made the same mistake.
Fite, W Benjamin.
“[Obituary]: David Eugene Smith”
The American Mathematical Monthly
52, no. 5 (1945): 237-238.
Smith, David Eugene. Rara Arithmetica; A Catalogue of the Arithmetics Written before the Year MDCI, With Description of Those in the Library of George Arthur Plimpton, of New York. Boston: Ginn &
company, 1908.
Smith, David Eugene. History of Mathematics. Boston; New York: Ginn & company, 1923.
20 comments on “Miscellany № 59: the percent sign”
1. Comment posted by on
Dividing by zero doesn’t result in infinity, but is simply undefined.
1. Comment posted by on
Hi Bernd — thanks for the catch! My maths isn’t what it used to be.
2. Comment posted by on
Correction: Zero divided by zero is undefined. A positive number divided by zero is infinity. A negative number divided by zero is negative infinity.
3. Comment posted by on
Correcting my correction, this is bollocks. Carry on.
4. Comment posted by on
Thanks for the honesty! Even so, your mention of positive and negative numbers reminded me to amend the text to include them.
2. Comment posted by on
It’s not entirely wrong. See the Wikipedia article on “division by zero” for examples of systems in which division by zero is defined. For most practical purposes, however, division by zero is
indeed undefined.
3. Comment posted by on
Anything divided by nothing is everything. No, not useful in practice, but conceptually it’s quite important—undefined seems poor design. Mathematical notation is like a chair, we can make it
ergonomic or uncomfortable.
1. Comment posted by on
But everything is still not enough, surely?
2. Comment posted by on
It’s undefined because every definition they considered resulted in the ability to write mathematical proofs with results analogous to 1 = 2.
It may be uncomfortable, but, like quantum theory, it’s our fault for having trouble accepting the nature of reality.
3. Comment posted by on
Actually, something divided by nothing is NOT everything, and here’s why:
6 divided by 2 is 3, because 3 multiplied by 2 is 6.
What would you multiply 0 by to get 6? 0 multiplied by infinity is still 0.
4. Comment posted by on
If that were the case then nothing would exist. We agree a line is made of consecutive dots. A dot is zero by zero in size. So it takes an infinite number of dots to make any length line.
This tells us that zero multiplied by different infinities gives you your lengths. eg 0 x ∞(6) = 6, 0 x ∞(9) = 9. I know there is no nomenclature for this but that is the actuality. The
‘real’ number system that we use does not fully equal reality.
5. Comment posted by on
Not quite. That doesn’t work because infinity is not a number. (Because anything times infinity is equal to infinity. If you could multiply by infinity or use it to cancel terms, you could
prove that 1 = 2)
You can’t multiply infinity by zero because it’s an algebraic “irresistible force meets immovable object” situation.
6. Comment posted by on
Actually it is correct in the real world. That’s why I differentiated between the man made ‘real’ number system and the real world. The ‘real’ number system does not have infinity but it does
not mean that it doesn’t exist. The problem with the 1=2 proof is that as soon as it starts multiplying by zero (a-b) where a = b then this allows (a+b) to equal b by the 4th line. The only
instance where this can occur is when a and b are both 0. Any other numbers (eg. 3) make the 4th line errornous (3+3)/0 = 3/0. The proof relies on a and b being the same number and by the 4th
line this can only be true when a and b = 0 in reality. 6/0 = ∞(6) and 3/0 = ∞(3) which are not equal. One infinity does not equal another. (It’s like ‘unreal’ numbers which are no more
unreal than ‘real’ numbers except per our man made interpretation of them).
7. Comment posted by on
No, infinitiy is not a number, be it real, imaginary, or irrational. Infinity only exists in limit theorum where it is used as a stand in for a “there is *no* number that bounds this limit”.
You therefore can not perform simple algebra on it, such as division or multiplication. Compare this with other non real numbers, like the square root of negative one, or i, on which you
*can* perform algebra.
8. Comment posted by on
It’s not a number in the sense of representing a quantity. Infinity does however exist. There is no nomenclature for using in our algebra system. Doesn’t mean that there couldn’t be; as I’ve
shown with my examples. If you choose to use alternatives you could do algebra with it. It is only rigid adherence that prevents it. Don’t ever mistake scientific and mathematical adherence
with reality. Yes, you can do algebra with unreal numbers. The reason is that they are just as real as ‘real’ numbers. We mistake positive squares as somehow being different to negative
squares. If you swap negative along one of the axis then the rotated quadrants then become the unreal ones. The mistake is that one length in one direction is somehow thought to be the same
as the same length in a different direction. They are not. Multiplying 8m west x 8m north actually gives you 64mw.mn. It’s fine to represent it as 64m² but we should not forget the fact the
the metres are not identical to each other. They point in different (perpendicular) directions. It’s important to remember this when dealing with unreal numbers so as to not see them as
4. Comment posted by on
You say:
> by now the ‘p’ had vanished entirely
Actually, the ‘p’ has not vanished, it’s still there. The strange “gliph” before the “o/o” is the standard abbreviatura for “per” in those centuries:
See Cappelli, Dizionario di Abbreviature Latine ed Italiane, p. 257 (you’ll need to scroll to the right to see the most common form of “per”, which is exactly the one depicted in your image.
The image therefore should be read “guadagnare 22 [per] [cto]”: not only the “p” is still there but the whole “per” is there, even though it’s abbreviated
1. Comment posted by on
Hi Francesco — you’re right! Thanks for the comment. And thank you for reminding me about the “per” sign.
5. Comment posted by on
Somewhere back when ah were nobbut a lad, one of my maths teachers alluded to this. He said that the bottom “o” was a mis-shapen “c”, but that the stroke was the “per”. I was never entirely happy
with that because it didn’t explain the top “o”, but at least he was correct about the part of it! Which leads to another question – did the use of the stroke for “per” come from misunderstanding
the stroke in per cent, or does it have another origin?
1. Comment posted by on
Hi Jeremy — thanks for the comment! Do you mean the use of ‘/’ in division operations? That’s a good question. I know that the obelus, or ‘÷’ comes from an ancient Greek editing symbol used
to mark spurious text, but I don’t know when it was joined by ‘/’.
6. Comment posted by on
I have to wonder if this is related to the “per” sign, which is very widely used in the 19th century and earlier, and yet has basically disappeared completely. It’s in unicode (⅌), but is so new
in unicode that it isn’t supported on my computer, and I just get a blank there between the parenthesis.
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AIPS HELP file
AIPS HELP file for UVMOD in 31DEC24
As of Sun Nov 3 15:18:02 2024
UVMOD: Task which inserts a model into uv data
INNAME Input UV file name (name)
INCLASS Input UV file name (class)
INSEQ 0.0 9999.0 Input UV file name (seq. #)
INDISK 0.0 9.0 Input UV file disk unit #
SRCNAME Source name
QUAL -10.0 Calibrator qualifier -1=>all
CALCODE Calibrator code ' '=>all
STOKES Stokes of output
TIMERANG Time range to use
SELBAND Bandwidth to select (kHz)
SELFREQ Frequency to select (MHz)
FREQID Freq. ID to select.
SUBARRAY 0.0 1000.0 Sub-array, 0=>all
BIF Low IF number to do
EIF Highest IF number to do
BCHAN 0.0 First channel included
ECHAN 0.0 last channel included
DOCALIB -1.0 101.0 > 0 calibrate data & weights
> 99 do NOT calibrate weights
GAINUSE CL (or SN) table to apply
DOPOL -1.0 10.0 If >0.5 correct polarization.
PDVER PD table to apply (DOPOL>0)
BLVER BL table to apply.
FLAGVER Flag table version
DOBAND -1.0 10.0 If >0.5 apply bandpass cal.
Method used depends on value
of DOBAND (see HELP file).
BPVER Bandpass table version
SMOOTH Smoothing function. See
HELP SMOOTH for details.
DOACOR Include autocorrelations?
OUTNAME Output UV file name (name)
OUTCLASS Output UV file name (class)
OUTSEQ -1.0 9999.0 Output UV file name (seq. #)
OUTDISK 0.0 9.0 Output UV file disk unit #
NGAUSS 0.0 9999.0 Number of sources
CTYPE 0.0 5.0 Component type: 0 point, 1
Gaussian, others see help
FMAX Peak I value of sources
FPOS (X,Y) position in asec
FWIDTH (Major,Minor,PA) in
INLIST List of sources up to 9999
ZEROSP Zero spacing flux for
(1) ignored, (2) Q, (3) U,
and (4) V in Jy.
FLUX Noise level in Jy/Weight or
Jy if FACTOR=0.
DPARM Spectral indices to use
FACTOR Multiplication factor.
WTUV If >0 then weights are
reset to a value of 1.
DOHIST > 0 => list sources in
history file
FQCENTER >= 0 -> center frequency axis
BADDISK Disks to avoid for scratch
Type: Task
Use: Modification of existing UV data by the addition of models.
INNAME.....Input image name (name). Standard defaults.
INCLASS....Input image name (class). Standard defaults.
INSEQ......Input image name (seq. #). 0 => highest.
INDISK.....Disk drive # of input image. 0 => any.
SRCNAME....Source name to be modified. Must specify if input is
a multi-source data set, otherwise all sources are
QUAL.......Qualifier of source to be copied. -1 => all.
CALCODE....Calibrator code of sources to copy. ' '=> all.
STOKES.....Specifies which STOKES parameters are written in the
output data set: ' ' => 'FULL'
'I','Q','U','V', 'IV', 'IQU', 'IQUV'
'RR','LL', 'RL', 'LR', 'RRLL', 'RLLR', 'RLRL'
'VV','HH', 'VH', 'HV', 'VVHH', 'VHHV', 'VHVH'
'HALF', 'CROS', and 'FULL' have sensible interpretations
depending on the Stokes present in the data. The last in
each of the 3 rows above == 'FULL'. Note that many
combinations of polarizations in the input and values
above are not supported.
TIMERANG...Time range of the data to be copied. In order: Start day,
hour, min. sec, end day, hour, min. sec. Days relative to
ref. date.
SELBAND....Bandwidth of data to be selected. If more than one IF is
present SELBAND is the width of the first IF required.
Units = kHz. For data which contain multiple
bandwidths/frequencies the task will insist that some form
of selection be made by frequency or bandwidth.
SELFREQ....Frequency of data to be selected. If more than one IF is
present SELFREQ is the frequency of the first IF required.
Units = MHz.
FREQID.....Frequency identifier to select (you may determine which is
applicable from the OPTYPE='SCAN' listing produced by
LISTR). If either SELBAND or SELFREQ are set, their values
override that of FREQID. However, setting SELBAND and
SELFREQ may result in an ambiguity. In that case, the task
will request that you use FREQID.
SUBARRAY...Sub-array number to copy. 0=>all.
BIF........First IF to include. 0 -> 1.
EIF........Last IF to include. 0 -> max.
BCHAN......First channel to copy. 0=>all.
ECHAN......Highest channel to copy. 0=>all higher than BCHAN
DOCALIB....If true (>0), calibrate the data using information in the
specified Cal (CL) table for multi-source or SN table for
single-source data. Also calibrate the weights unless
DOCALIB > 99 (use this for old non-physical weights).
GAINUSE....version number of the CL table to apply to multi-source
files or the SN table for single source files.
0 => highest.
DOPOL......If > 0 then correct data for instrumental polarization as
represented in the AN or PD table. This correction is
only useful if PCAL has been run or feed polarization
parameters have been otherwise obtained. See HELP DOPOL
for available correction modes: 1 is normal, 2 and 3 are
for VLBI. 1-3 use a PD table if available; 6, 7, 8 are
the same but use the AN (continuum solution) even if a PD
table is present.
PDVER......PD table to apply if PCAL was run with SPECTRAL true and
0 < DOPOL < 6. <= 0 => highest.
BLVER......Version number of the baseline based calibration (BL) table
to apply. <0 => apply no BL table, 0 => highest.
FLAGVER....specifies the version of the flagging table to be applied.
0 => highest numbered table.
<0 => no flagging to be applied.
DOBAND.....If true (>0) then correct the data for the shape of the
antenna bandpasses using the BP table specified by BPVER.
The correction has five modes:
(a) if DOBAND=1 all entries for an antenna in the table
are averaged together before correcting the data.
(b) if DOBAND=2 the entry nearest in time (including
solution weights) is used to correct the data.
(c) if DOBAND=3 the table entries are interpolated in
time (using solution weights) and the data are then
(d) if DOBAND=4 the entry nearest in time (ignoring
solution weights) is used to correct the data.
(e) if DOBAND=5 the table entries are interpolated in
time (ignoring solution weights) and the data are then
IMAGR uses DOBAND as the nearest integer; 0.1 is therefore
BPVER......Specifies the version of the BP table to be applied
0 => highest numbered table.
<0 => no bandpass correction to be applied.
SMOOTH.....Specifies the type of spectral smoothing to be applied to
a uv database . The default is not to apply any smoothing.
The elements of SMOOTH are as follows:
SMOOTH(1) = type of smoothing to apply: 0 => no smoothing
To smooth before applying bandpass calibration
1 => Hanning, 2 => Gaussian, 3 => Boxcar, 4 => Sinc
To smooth after applying bandpass calibration
5 => Hanning, 6 => Gaussian, 7 => Boxcar, 8 => Sinc
SMOOTH(2) = the "diameter" of the function, i.e. width
between first nulls of Hanning triangle and sinc
function, FWHM of Gaussian, width of Boxcar. Defaults
(if < 0.1) are 4, 2, 2 and 3 channels for SMOOTH(1) =
1 - 4 and 5 - 8, resp.
SMOOTH(3) = the diameter over which the convolving
function has value - in channels. Defaults: 1,3,1,4
times SMOOTH(2) used when input SMOOTH(3) < net
DOACOR.....> 0 => include autocorrelations as well as cross
correlation data.
OUTNAME....Output image name (name). Standard defaults.
OUTCLASS...Output image name (class). Standard defaults.
OUTSEQ.....Output image name (seq. #). 0 => highest unique
OUTDISK....Disk drive number of output image. 0 =>
highest number with sufficient space.
NGAUSS.....Number of sources 1 - 4 use CTYPE, FMAX, FPOS, and
FWIDTH; > 4, use INLIST file. Limit 9999.
Note, if INLIST not blank and NGAUSS > 1, the INLIST file
will be used BUT the value of NGAUSS will limit how many
components are read from INLIST. Set it large when you
use INLIST if you want the full contents of INLIST.
CTYPE......Type of each source: 1 Gaussian, 2 solid disk, 3 solid
rectangle, 4 optically thin sphere, 5 exponential
otherwise point.
FMAX.......I polarization max of each source
FPOS.......Offset value of the X,Y centroid of the sources in
arcsec. Positive values mean increasing R.A. and DEC.
These are offsets in the coordinates with R.A. scaled by
cos(DEC) and are applied with the W term taken into
account (as of 1997-04-16). See Explain at the end.
FWIDTH.....Model major and minor axis in arcsec and PA in degrees.
Full width or full width to half maximum (GAUS).
INLIST.....Set this blank unless you want an input components list.
Text file containing one line per source, giving
I, DX, DY, Maj, Min, PA, type, spix, Dspix, Q, U, V
blank separated free format and trailing zeros may be
omitted. Used if NGAUSS > 1 and INLIST not blanks. The
resulting NGAUS will be min (NGAUSS, #lines in INLIST).
Limit 9999. Note that I, Q, U, V and the spectral
indexes are assumed to apply at the header (reference)
frequency. Any line in INLIST beginning with a $ or a #
is taken as a comment and ignored. Negative values for I
are allowed, but I=0 lines are ignored.
ZEROSP.....Zero spacing flux of (2) Q, (3) U and (4) V in Jy used
with the first source only and only if INLIST blank or
NGAUSS=1. For more polarization in models, use the
INLIST option. Be sure to put in the zeros (or other
values) for the parameters that precede the Q,U,V values
on each line in INLIST.
FLUX.......Noise level to be added (in Jy. per Weight or Jy if
FACTOR=0). This means, when FACTOR is not zero, that
the noise added is FLUX/sqrt(weight) so that FLUX scales
the data rms, assumed to be 1/sqrt(weight).
DPARM......(1-5) Spectral index to use for components 1,2,3,4,noise
(6-10) Spectral index curvature for comp 1,2,3,4,noise
A non-blank INLIST gives these if NGAUS > 1. DPARM(5)
and (10) still used for noise. See explain at the end.
Curvature numbers are based on base 10 logarithms with a
reference frequency of the *** header frequency ***.
FACTOR.....Factor by which original data are multiplied before they
are added to the model. If FACTOR = 0 then only the model
will be left.
WTUV.......If WTUV > 0 then all Weights will be set to a
value of 1 in the output file.
DOHIST.....List sources in history file if NGAUS <= 4 and/or DOHIST
> 0.
FQCENTER...> 0 => Change frequency axis reference pixel to
Nchan / 2 + 1
else => do not change reference pixel
BADDISK....The disk numbers to avoid for scratch files (sorting
tables mostly).
UVMOD: Task which modifies UVDATA by scaling the existing data,
and adding a specified model (See also IMMOD).
DOCUMENTATION: Eric R. Nelson NRAO/VLA/UNM
DATE OF DOCUMENTATION: 14 JUNE 1983
RELATED PROGRAMS: IMMOD, UVMAP, APCLN, COMB
UVMOD modifies an already existing UV data file by the addition of
one of several model types. The original data may be scaled by a
multiplicative factor, including negative values and zero, before they
are added to the model. Random noise may also be added to the UV data.
The program could be useful in investigating the affects of CLEAN on a
specific geometry, or for removing models from data, i.e. planetary
The six available models to choose from are 0) point source, 1)
Gaussian, 2) solid disk 3) solid rectangle, 4) optically thin sphere,
and 5) exponential. These models are first Fourier transformed and
then added to the UV data. The resulting functions are:
0) Point -> A constant visibility amplitude is added to the
data. The GWIDTH adverbs have no affect on this
model. Used with CTYPE <= 0 or > 5.
1) Gaussian -> The function EXP(-3.559707*R**2) is added to
the UV data. The function R is given by:
R = Sqrt(UU**2 + VV**2) where
UU = BMAJ*(V*COS(BPA)+U*SIN(BPA))
VV = BMIN*(U*COS(BPA)-V*SIN(BPA))
2) Disk -> The function J1(R)/R is added to the UV data,
where J1 is the Bessel function of order 1 and R
is the same as above.
3) Rectangle -> The function SINC(UU)*SINC(VV) is added to
the UV data, where SINC(X) = SIN(X)/X. UU and
VV are defined as above for the Gaussian.
4) Sphere -> The function (SIN(A)/A - COS(A)) / (A*A) is
added to the UV data where
A = BMAJ * Sqrt (U*U + V*V)
A = max (A, 2 pi / 100)
The GWIDTH adverbs have no affect on this model.
5) Exponential -> The function
2 Pi / (1 + a * a * R * R) ** 3/2
is added to the UV data where R is defined in 2
above and a is Pi/ln(2).
Note that all functions are scaled by the total flux and a complex
vector representing the phase of the model before being added to the
scaled input visibility data. The spectral index is applied to make
the peak flux
log(F/F0) = spix * log(nu/nu0) + Dspix * log^2 (nu/nu0)
where F0 is the model peak flux at the header reference frequency
nu0 (BEFORE any application of FQCENTER). log functions are base 10.
BMAJ, BMIN, BPA :
In this discussion, BMAJ = GWIDTH(1,i) for the i'th component,
BMIN = GWIDTH(2,i) for the i'th component, and BPA = GWIDTH(3,i) for
the i'th component, The dimensions of the resulting functions are
determined by BMAJ, BMIN and BPA (position angle). For the Gaussian
the first two values are the FWHM of the two axis. For the Disk and
the Rectangle, the first two values are the absolute dimensions of the
two available axis. If BMAJ and BMIN are both zero then all the
models reduce to the point model.
This adverb, array indices 2-4, allows you to add polarization to
the first model component only.
FACTOR :
The FACTOR term allows one to add a scaled version of the
original data to the model. FACTOR is simply multiplied by the
original data which is then added to the model. If FACTOR = 0, then
only the model will remain in the final UV data base.
Coordinate considerations:
FPOS is translated to an RA, Dec following the formula (assuming
no rotations):
Dec = Dec0 + FPOS(2)
RA = RA0 + FPOS(1) / cos (Dec0)
These are then turned into l,m,n for phase = ul + vm + wn as (for -SIN
l = cos (Dec) * sin (Ra-Ra0)
m = sin (Dec) * cos (Dec0) -
cos (Dec) * sin (Dec0) * cos (Ra-Ra0)
n = sin (Dec) * sin (Dec0) +
cos (Dec) * cos (Dec0) * cos (Ra-Ra0)
Suitable formulae are used for -NCP geometry as well.
Spectral index parameters are entered as:
x = log10 (freq / 1 GHz)
F = F0 * exp ((Spix + Dspix * x) * x)
where F is the flux and F0 is the flux at 1.0 GHz. The model fluxes
at the reference frequency are converted to fluxes at 1 GHz.
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Getting Started with lumopt - Python API
Inverse design using lumopt can be run from the CAD script editor, the command line or any python IDE. In the first sub-section, we will briefly describe how to import the lumopt and lumapi modules;
although an experienced python user can likely skip this part. As a starting point, it is recommended to run the AppGallery examples and use these as templates for your own project. It may be helpful
following along with these files as you read this page or you may simply reference this page when running the examples later. In the project Init section, we outline the project inputs and necessary
simulation objects that should be included. Important lumopt specific considerations are highlighted; however, a valid simulation set-up is imperative, so convergence testing should be considered a
pre-requisite. Next important lumopt set-up classes, that should be updated to reflect your specifications are documented. Finally, a description of the scipy optimizer and lumopt optimization
classes are presented. Shape and topology optimization primarily differ in how they handle the optimizable geometry which is the subject of the next page in this series.
Running from the CAD script editor has the advantage that it requires no set-up and uses a Python 3 distro that ships with the Lumerical installer, so there is no need to install a separate python.
This method automatically configures the workspace to find the lumopt and lumapi modules and would be the preferred method for users with little experience in python. Using your own version of python
and running from an IDE or the command line may be preferable for more experienced users. To do this one simply needs to import lumopt and lumapi which will require specifying the correct path.
Either pass an explicit path using importlibutil, or updating the search path permanently appending the PythonPath object. Advanced user working with numerous libraries might want to create a
Lumerical virtual environment. For more information on these methods and os specific paths, see Session management - Python API.
Lumerical ships with a version of Python 3, including lumapi and lumopt modules, already installed. To run any of our examples 'out of the box' simply run the scripts from the script file editor in
the CAD.
Project Init
Base Sim
The base simulation needs to be defined using one of the following options
• Predefined simulation file - Initialize a python variable that specifies the path to the base file. Example Grating coupler.
base_sim = os.path.join(os.path.dirname(__file__), 'grating_base.fsp')
• An lsf set-up script - Create a python variable using the load_from_lsf function. Example Waveguide crossing.
from lumopt.utilities.load_lumerical_scripts import load_from_lsf
crossing_base = load_from_lsf('varFDTD_crossing.lsf')
• Callable python code that does the set-up using the API - This can be a function defined in the same file or an imported function. Example Y-branch.
sys.path.append(os.path.dirname(__file__)) #Add current directory to Python path
from varFDTD_y_branch import y_branch_init_ #Import y_branch_init function from file
y_branch_base = y_branch_init_ #New handle for function
Each method produces a file which the optimizer updates and runs. Since the resulting project files should be equivalent; the method each user employs is a matter of preference or convenience.
Required Objects
In the varFDTD, and FDTD base simulation it is also necessary to provide input/output geometry and define the following simulation objects that are used by lumopt.
Figure 1: Required simulation inputs
• A mode source - Typically named source, but can be set in the Optimization class.
• A mesh overide region covering the optimization volume - This is a static object, and the pixel/voxel size should be uniform.
• A frequency monitor over the optimization volume - Should be named opt_field, these field values are important for the adjoint method.
• A frequency monitor over the output waveguides - Used to calculate the FOM. The name of this monitor is passed to the ModeMatch class.
The mode source should have a large enough span and the modes should be compared to expectations see FDE convergence. This is used as the forward source. A mesh override is placed over the
optimization region to ensure that a fine uniform grid covers this space, and the opt_field monitor is used to extract the fields in this region. The FOM monitor should be aligned to the interface of
a mesh cell to avoid interpolation errors; therefore, it is a good idea to have a mesh override co-located with the FOM monitor. In the adjoint simulation, the adjoint source will take the place of
the FOM monitor. Passing the name of the FOM monitor, to modematch class, allows multiple FOM monitors to be defined in the same base file which is helpful for SuperOptimization.
Set-up Classes
Two important lumopt classes that should be updated with your parameters are wavelengths and modematch. These are used to define the spectrum and mode number(or polarization) of the FOM respectively.
It should be noted here that the FOM we accept is power coupling of guided modes. Other figures of merit, such as optimizing the target phase or power to a specified grating order are not supported.
To compute the broadband figure of merit we take the mean of the target minus the error using the p-norm.
$$F=\left(\frac{1}{\lambda_{2}-\lambda_{1}} \int_{\lambda_{1}}^{\lambda_{2}}\left|T_{0}(\lambda)\right|^{p} d \lambda\right)^{1 / p}-\left(\frac{1}{\lambda_{2}-\lambda_{1}} \int_{\lambda_{1}}^{\
lambda_{2}}\left|T(\lambda)-T_{0}(\lambda)\right|^{p} d \lambda\right)^{1 / p}$$
• \( T_{0} \) is the target_T_fwd
• \( \lambda_{1} \text{ and } \lambda_{2} \) are the lower and upper limits of the wavelength points
• \( T \) is the actual mode expansion power transmission
• \( p \) is the value of the generalized p-norm
Defines the simulation bandwidth, and wavelength resolution. Defining the target FOM spectrum is done in modematch.
from lumopt.utilities.wavelengths import Wavelengths
class Wavelengths(start,
: start: float
Shortest wavelength [m]
: stop: float
Longest wavelength [m]
: points: int
The number of points, uniformly spaced including the endpoints.
wavelengths = Wavelengths(start = 1260e-9, stop = 1360e-9, points = 11)
This class is used to define target mode, propagation direction, and specify the broadband power coupling.
from lumopt.figures_of_merit.modematch import ModeMatch
class ModeMatch(monitor_name,
: monitor_name: str
Name of the FOM monitor in the file.
: mode_number : str or int
Used to specify the mode.
If the varFDTD solver is used:
• ‘fundamental mode’
• int - user select mode number
If the FDTD solver is used:
• 'fundamental mode'
• 'fundamental TE mode'
• 'fundamental TM mode'
• int - user select mode number
: direction : str
The direction is determined by the FDTD coordinates; for mode traveling in the positive direction the direction is forward.
: multi_freq_source: boolean, optional
Should only be enabled by advanced users. See frequency Frequency dependent mode profile for more info. Default = False
: target_T_fwd: float or function
A function which will take the number of Wavelengths points and return values [0,1]. Usually passed as a lambda function or a single float value for single wavelength FOM. To specify a more advanced
spectrum one can define a function, it may be helpful to use, numpy windows as a template.
: norm_p: float
Is the generalized p-norm used in the FOM calculation. The p-norm, with \( p \geq 1 \) allows the user to increase the weight of the error. Since \( p=1 \) provides a lower bound on this function, a
higher p-number will increase the weight of the error term. Default p =1.0
: target_fom: float
A target value for the figure of merit. This will change the behavior of the printing and plotting only. If this is enabled, by setting a value other than 0.0, the distance of the current FOM is
given. Default = 0.0
class ModeMatch(monitor_name = 'fom', mode_number = 3, direction = 'Backward', target_T_fwd = lambda wl: np.ones(wl.size), norm_p = 1)
Optimization Classes
Here we describe the generic ScipyOptimizer wrapper, and lumopt Optimization class which is used to encapsulate the project.
This is a wrapper for the generic and powerful SciPy optimization package.
from lumopt.optimizers.generic_optimizers import ScipyOptimizers
Class ScipyOptimizers(max_iter,
: max_iter: int
Maximum number of iterations; each iteration can make multiple figure of merit and gradient evaluations. Default = 100
: method: str
Chosen minimization algorithm; experimenting with this option should only be done by advanced users. Default = ‘L-BFGS-B'
: scaling_factor: none, float, np.array
None, scalar or a vector the same length as the optimization parameters. This is used to scale the optimization parameters. As of 2021R1.1, the default behavior in shape optimization is to
automatically map the parameters the range [0,1] within the optimization routines; which was always the case in topology. The bounds, defined in the geometry class, or eps_min/eps_max are used for
this. Default = None
: pgtol: float
The iteration will stop when \( \max( |\text{proj }g_i | \text{ i = 1, ..., n} ) <= pgtol| \) where \( g_i \) is the i-th component of the projected gradient. Default = 1.0e-5
: ftol: float
The iteration stops when \( \left(( f^k - f^{k+1}) / \max (| f^k |\text{ , }|f^{k+1}|\text{ , }1 ) \right) <=ftol \). Default = 1.0e-5
: scale_initial_gradient_to: float
Enforces a rescaling of the gradient to change the optimization parameters by at least this much; the default value of zero disables automatic scaling. Default = 0.0
: penalty_fun: function, optional
Penalty function to be added to the figure of merit; it must be a function that takes a vector with the optimization parameters and returns a single value. Advanced feature. Default = None
:penalty_jac: function, optional
The gradient of the penalty function; must be a function that takes a vector with the optimization parameters and returns a vector of the same length. If a penalty_fun is included with no
penalty_jac, lumopt will approximate the derivative. Advanced feature. Default = None
optimizer = ScipyOptimizers(max_iter = 200,
method = 'L-BFGS-B',
scaling_factor = 1.0,
pgtol = 1.0e-5,
ftol = 1.0e-5,
scale_initial_gradient_to = 0.0,
penalty_fun = penalty_fun,
penalty_jac = None)
Encapuslates and orchestrates all of the optimization pieces, and routines. Calling the opt.run method will perform the optimization.
from lumopt.optimization import Optimization
class Optimization(base_script,
: base_script: callable, or str
Base simulation - See project init.
• Python function in the workspace
• String that points to base file
• Variable that loads from lsf script
: wavelengths: float or class Wavelengths
Provides the optimization bandwidth. Float value for single wavelength optimization and Wavelengths class provides a broadband spectral range for all simulations.
: fom: class ModeMatch
The figure of merit FOM, see ModeMatch
: geometry: Lumopt geometry class
This defines the optimizable geometry, see Optimizeable Geometry
: optimizer: class ScipyOptimizers
See ScipyOptimizer for more information.
: hide_fdtd_cad: bool
Flag to run FDTD CAD in the background. Default = False
: use_deps: bool
Flag to use the numerical derivatives calculated directly from FDTD. Default = True
: plot_history: bool
Plot the history of all parameters (and gradients). Default = True
: store_all_simulations: bool
Indicates if the project file for each iteration should be stored or not. Default = True
: save_global_index: bool
Flag to save the results from a global index monitor to file after each iteration (used for visualization purposes). Default = False
: label: str, optional
If the optimization is part of a super-optimization, this string is used for the legend of the corresponding FOM plot. Default = None
: source_name: str, optional
Name of the source object in the simulation project. Default = "source"
opt_2d = Optimization(base_script = base_sim_2d,
wavelengths = wavelengths,
fom = fom,
geometry = geometry,
optimizer = optimizer,
use_var_fdtd = False,
hide_fdtd_cad = True,
use_deps = True,
plot_history = True,
store_all_simulations = True,
save_global_index = False,
label = None)
Note(Advanced): For 2020R2 we exposed, a prototype user-requested debugging function that allows the user to perform checks of the gradient calculation. This is a method of the optimization class and
can be called as follows.
opt.check_gradient(intitial_guess, dx=1e-3)
Where initial_guess is a numpy array that specifies the optimization parameters at which the gradient should be checked. The scalar parameter dx is used for the central finite difference
Has two limitations; one the check performs an unnecessary adjoint simulation. Two the check_gradient is only available for regular optimizations and not superoptimizations of multiple FOMs.
The + operator has been overloaded in the optimization class so it is trivial to simultaneously optimize:
• Different output waveguides and\or wavelength bands CWDM
• Ensure balanced TE\TM performance or suppress one over the other
• Robust manufacturing by simultaneously optimizing underetch\overtech\nominal variations.
• Etc ...
Simply adds the various Optimization classes together to create a new SuperOptimization object. Then you will call run on this object to co-optimize the various optimization definitions. Each FOM
calculation requires 1 optimization object so the number of simulations at each iteration will be \( N_{sim} = 2\times N_{FOM} \).
opt = opt_TE + opt_TM
opt.run(working_dir = working_dir)
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On Bernoulli Numbers
Number Theory
On Bernoulli Numbers
My Journey of Rediscovering Some of Mathematics’ Most Beautiful Numbers
I love patterns and take great joy in searching for them. During my 11th grade, we came across the question:
Our professor suggested that there is no general formula for the summation in the numerator and we ended up solving the problem using integration.
However, this left me on a search for a general formula for the sum of powers of the first n natural numbers.
This section describes my research approach to the above question
I approached the question by finding the formula for known values of k and thereby try and generalise the results for any value of k
Let us express the sum as
For k = 0,
For k = 1,
For k = 2,
For k = 3,
For k = 4,
By this step, I felt a pattern emerging for consecutive values of S. It goes like this:
I use the approximate symbol here to point out a specific modification to the integral.
Post-integration, in case we run out of n¹ and n² in the equation, we just add them along multiplied by a constant. For example:
In such cases, we add factors for n¹ and n²
As we can calculate the values of the sum for various n, we find out the values of k_1 and k_2. Here, we get k_1= 1/30 and k_2 = 0. Therefore, we have
Using this trick, I could calculate the formulae as far as k = 21. Some of them are:
Note: I checked S9 from Bernoulli’s Ars Conjectandi and found it to have an error in the n² term.
Now, what caught my eye were the numbers popping up as a factor of n every time we “modify” and complete our integral. It was like magic. What were these numbers?
One peculiar feature was that these numbers originated only for even power summations, such as k = 2, 4, 6, 8, etc.
The “magical” numbers for some values of k:
I made a note of these numbers and paused my research at this point. As time passed by, I entered university and started reading on various topics in mathematics. During one such session, you should
have seen the excitement on my face when I stumbled upon these exact numbers at the Reimann Zeta Function.
Zeta Function
Our summation was concerned about adding powers of natural numbers. The Zeta function is about adding the powers of reciprocals of natural numbers.
Bernoulli numbers are represented by B followed by a subscript 2n as Bernoulli numbers on 2n+1 places have a value of zero.
The Zeta function in its expanded form contains Bernoulli numbers, similar to the summation in the previous section.
We finally have
Bernoulli numbers appear in various series expansions, a crucial one among them being
At one point I was obsessed with nothing but this question alone. It was the only thing on my mind. And now I had a solution right before me, given by one of mathematics’ greatest more than 300 years
ago. I had mixed feelings at that time — on one hand, it was this great feeling of rediscovering some of the most crucial numbers in all of mathematics. On the other, accepting the fact that my
research was not new after all. Probably, this was what a part of life is about. Learning, discovering, rediscovering, and overall accepting the truth.
This is my story on how a single question led me on a path to rediscover some breathtaking numbers teaching me a valuable lesson about life along the way.
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Types of Functions in SQL - Webeduclick.com
Types of Functions in SQL
Functions in SQL:
SQL supports functions that used to manipulate data. There are two types of functions in SQL:
• Single Row Functions
• Multiple Row Functions
1. Single Row Functions:
It works with a single row at a time. A single-row function returns a result for every row of the table, on which a query is made. There are three types of single-row SQL functions:
i. Character Functions:
Those are SQL functions that accept a character input and returns character and/or numeric values.
INITCAP (string): These function capitalizes the first character of each word in the string.
LOWER (string): The LOWER function converts all the characters in the string to lowercase letters.
UPPER (string): The UPPER function converts all the characters in the string to uppercase letters.
CONCAT (string1, string2): The CONCAT function returns string1 appended by string2.
SOUNDEX (string): The SOUNDEX function returns a phonetic representation of each word and allows you to compare words that are spelt differently but sound alike.
INSTR(): The INSTR() function is used to find out where the particular pattern of characters occur in the given string.
LENGTH (string): The LENGTH returns the length of its character argument.
LTRIM() and RTRIM(): The LTRIM() and RTRIM() functions take at least one and at most two arguments. The first argument is a character string. The optional second element is either a character or
string. These trim functions will remove characters in the first argument string up to this second-argument character.
The second argument is “blank” by default. That is if you provide only one argument, characters up to the blank space are removed.
The RTRIM() function removes characters from the right of the given string.
SUBSTR (string, M, N): This function returns a substring, N character long, from the string, starting from position M. If the number of characters, N is not specified, the string is extracted from
position M to end. Note that the blank spaces are also counted.
LPAD(): The LPAD() function is of the following form:
[sql]LPAD(string1, n[, string2])
The first argument is the character string to be operated on. The second is the number of characters to pad it with and the optional third argument is the character to pad it with. The third argument
defaults to a blank.
ii. Number Functions:
Those functions accept numeric values and after performing the required operation, return numeric values.
ABS(N): -This function returns the absolute value without any -ve sign of the column or value passed.
CEIL(N): This function is used to find the smallest integer greater than or equal to N.
FLOOR(N): This function is used to find the largest integer less than or equal to N. Note that N can also be a column name.
MOD(M, N): The MOD(M, N) function returns the remainder of M divided by N. If N=0, the function will return M.
POWER(M, N): The POWER function returns M raised to the power N. Note that N must be an integer.
ROUND(): The ROUND() function returns a number rounded as required. It takes two integer arguments. The first argument is the number to be rounded off and the second argument specifies the number of
places to which the number should be rounded off.
TRUNC(): These function returns a number with specified number of digits to be truncated.
iii. Date Functions:
Those functions operate on data of the type Date.
SYSDATE: It is used as a column in queries to retrieve the current date and time.
ADD_MONTHS(D, N): This function adds N months to the date D. The result is shown as DATE type.
Here n can be a positive or negative value. If N is positive, the new date will be N months more than the previous value. If it is -N, then the new date will be less than the previous value.
MONTHS_BETWEEN(D1, D2): It returns the number of months between the two dates, D1 and D2. If D1 is later than D2, the result is positive, if D1 is earlier than D2, the result is negative.
TO_CHAR(D, ‘DAY’): It converts the date D to character format. It will give the corresponding name of the weekday.
LAST_DAY(): It is used to return the last date of the given month.
NEXT_DAY(): This function returns the date of a specified day in the next week.
2. Multiple Row Functions:
It works with data of multiple rows at a time and returns aggregated or total values.
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How Large are the Egyptian Pyramids?
The Pyramids of Egypt are one of the wonders of the world. They are ancient architectural structures that were built in Ancient Egypt. Few know that the number of pyramids found so far are not just 4
or 5 but over 100.
In the Valley of Kings, there are more than 138 pyramids, which serve as the tombs of the pharaohs and their wives. All of the pyramids were built during the Old Kingdom and Middle Kingdom periods.
The oldest Egyptian pyramids were discovered in Saqqara, northwest of Memphis. Among these is the Pyramid of Djoser, built during the 3rd Dynasty, which is thought to be the oldest Egyptian pyramid.
It and the mortuary complex around it were built by Imhotep.
The Pyramid of Djoser impresses with its vast dimensions. It has a rectangular base 125 x 115 m (358 ft × 410 ft) and stands about 60 m (197 ft) tall.
The most famous pyramids in the world are located in the Giza necropolis on the outskirts of Cairo. These are the Pyramid of Khafre, the Pyramid of Khufu and the Pyramid of Menkaure.
The largest of the 3 is the Pyramid of Khufu. According to archaeologists, it was built about a century after the Pyramid of Djoser but surpasses it in every way possible.
The Pyramid of Khufu has a square base of 756 ft (230.4 m) per side and height from base to tip of 455 ft (137.3 m). At the time of its construction, archaeologists say it was 481 ft (147.3 m) tall
but at some point its top was broken, leaving a flat area there now.
The 2nd largest pyramid is the Pyramid of Khafre. Its base is 706 ft (215.5 m) per side and it stands at 448 ft (136.4 m) tall, its original height once being 471 ft (143.5 m). Upon its completion it
was decorated with pink granite, which is no longer present today.
The smallest of the 3 Pyramids of Giza is the Pyramid of Menkaure, barely about 213 ft (61 m) tall and 343 ft (104.6) at the base. Its volume is just 1/10th that of the Pyramid of Khufu.
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On Mean Estimation for General Norms with Statistical Queries
On Mean Estimation for General Norms with Statistical Queries
Proceedings of the Thirty-Second Conference on Learning Theory, PMLR 99:2158-2172, 2019.
We study the problem of mean estimation for high-dimensional distributions given access to a statistical query oracle. For a normed space $X = (\mathbb{R}^d, \|\cdot\|_X)$ and a distribution
supported on vectors $x \in \mathbb{R}^d$ with $\|x\|_{X} \leq 1$, the task is to output an estimate $\hat{\mu} \in \mathbb{R}^d$ which is $\varepsilon$-close in the distance induced by $\|\cdot\|_X$
to the true mean of the distribution. We obtain sharp upper and lower bounds for the statistical query complexity of this problem when the the underlying norm is \emph{symmetric} as well as for
Schatten-$p$ norms, answering two questions raised by Feldman, Guzmán, and Vempala (SODA 2017).
Cite this Paper
Related Material
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Getting mean value from specific cells in large data frame
Calculating Mean Values from Specific Cells in Large DataFrames: A Practical Guide
Working with large dataframes in Python, particularly when you need to calculate mean values from specific cells, can be a computationally intensive task. This article aims to provide a clear and
efficient approach to tackling this challenge.
Let's consider a scenario where you have a large dataframe named df with multiple columns and rows. You need to calculate the mean of values in column 'value' for rows where the 'condition' column
meets a certain criteria.
Here's a basic example:
import pandas as pd
import numpy as np
# Example data frame
data = {'condition': ['Yes', 'No', 'Yes', 'Yes', 'No'],
'value': [10, 20, 30, 40, 50]}
df = pd.DataFrame(data)
# Incorrect approach:
mean_value = df[df['condition'] == 'Yes']['value'].mean()
In this code, you might think that df[df['condition'] == 'Yes']['value'].mean() directly calculates the mean for the rows where 'condition' is 'Yes'. However, this approach becomes inefficient and
potentially memory-intensive for large dataframes. Let's discuss why and how to overcome these challenges.
The Problem with Direct Indexing:
• Memory Overhead: Directly selecting rows and columns can create a copy of the DataFrame, leading to increased memory consumption especially with large datasets.
• Performance Bottleneck: Accessing specific cells in a loop can be slow, especially with massive dataframes.
Efficient Solution: Using NumPy's Vectorized Operations
NumPy's vectorized operations excel at working with arrays efficiently. Instead of looping through rows, we can leverage these operations for faster calculations. Here's how:
1. Boolean Indexing: Create a boolean array that identifies rows where the condition is met.
2. Direct Calculation: Use the boolean array to directly access the relevant values in the 'value' column and calculate the mean.
# Efficient solution:
condition_mask = df['condition'] == 'Yes'
mean_value = np.mean(df['value'][condition_mask])
• condition_mask is a boolean array that identifies rows where 'condition' is 'Yes'.
• df['value'][condition_mask] directly extracts values from the 'value' column only for the rows where condition_mask is True.
• np.mean() efficiently calculates the mean of these selected values.
Additional Considerations
• Large Datasets: For extremely large datasets, consider using dask or vaex libraries, which can efficiently handle data that doesn't fit in memory. These libraries offer parallel processing and
distributed computing capabilities.
• Optimization Techniques: Profiling your code and using techniques like vectorization and optimized functions can further improve performance.
By leveraging the power of NumPy's vectorized operations, we can significantly improve the efficiency of calculating mean values from specific cells in large dataframes. This approach reduces memory
overhead and speeds up computations, making it a preferred method for data analysis tasks.
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Infinity is not the problem
An article published in May in Quanta Magazine had the following remark as its lead:
A surprising new proof is helping to connect the mathematics of infinity to the physical world.
My first thought was that the mathematics of infinity is already connected to the physical world. But Natalie Wolchover’s opening few paragraphs were inviting:
With a surprising new proof, two young mathematicians have found a bridge across the finite-infinite divide, helping at the same time to map this strange boundary.
The boundary does not pass between some huge finite number and the next, infinitely large one. Rather, it separates two kinds of mathematical statements: “finitistic” ones, which can be proved
without invoking the concept of infinity, and “infinitistic” ones, which rest on the assumption — not evident in nature — that infinite objects exist.
Mapping and understanding this division is “at the heart of mathematical logic,” said Theodore Slaman, a professor of mathematics at the University of California, Berkeley. This endeavor leads
directly to questions of mathematical objectivity, the meaning of infinity and the relationship between mathematics and physical reality.
It is becoming increasingly clear to me that harmonizing the finite and the infinite has been an almost ever-present human enterprise, at least as old as the earliest mythical descriptions of the
worlds we expected to find beyond the boundaries of the day-to-day, worlds that were below us or above us, but not confined, not finite. I have always been provoked by the fact that mathematics found
greater precision with the use of the notion of infinity, particularly in the more concept-driven mathematics of the 19th century, in real analysis and complex analysis. Understanding infinities
within these conceptual systems cleared productive paths in the imagination. These systems of thought are at the root of modern physical theories. Infinite dimensional spaces extend geometry and
allow topology. And finding the infinite perimeters of fractals certainly provides some reconciliation of the infinite and the finite, with the added benefit of ushering in new science.
Within mathematics, the questionable divide between the infinite and the finite seems to be most significant to mathematical logic. Wolchover’s article addresses work related to Ramsey theory, a
mathematical study of order in combinatorial mathematics, a branch of mathematics concerned with countable, discrete structures. It is the relationship of a Ramsey theorem to a system of logic whose
starting assumptions may or may not include infinity that sets the stage for its bridging potential. While the theorem in question is a statement about infinite objects, it has been found to be
reducible to the finite, being equivalent in strength to a system of logic that does not rely on infinity.
Wolchover published another piece about disputes among mathematicians about the nature of infinity that was reproduced in Scientific American in December 2013. The dispute reported on here has to do
with a choice between two systems of axioms.
According to the researchers, choosing between the candidates boils down to a question about the purpose of logical axioms and the nature of mathematics itself. Are axioms supposed to be the
grains of truth that yield the most pristine mathematical universe? … Or is the point to find the most fruitful seeds of mathematical discovery…
Grains of truth or seeds of discovery, this is a fairly interesting and, I would add, unexpected choice for mathematics to have to make. The dispute in its entirety says something intriguing about
us, not just about mathematics. The complexity of the questions surrounding the value and integrity of infinity, together with the history of infinite notions is well worth exploring, and I hope to
do more.
“I’m convinced that paying more attention to how we participate in building our reality will clarify quite a lot.” Spoken like a therapist I am happy to say. I always thought that mathematicians were
the universe’s therapist. Before you and I reach into, or better, before infinity reaches into us, I would like to reconnect in the flesh. Why not? Vince Migliore
Does the qualitative reside within the finite?
Could the perfection of the sphere be an example?
Is pi an example?
I’ve been playing with it within an examination
of the Planck scale: http://81018.com/the-three/
Might all the dimensionless constants be
the bridge between the finite and infinite?
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There are seven papers available to download on this site.
'Why Money Trickles Up' - This is the full text which includes detailed explanation, justification and background material. This document has 250 pages of text.
'Why Money Trickles Up - Bullet Points' - This document gives an overview of the main points of the modelling and theory of Why Money Trickles Up in a brief format.
'Why Money Trickles Up - Wealth & Income Distributions' - This is a condensed version of the first part of the full document. This includes the main details of the modelling and core mathematics. It
also gives the full explanations of wealth and income distributions. This document has 45 pages of text.
'Why Money Trickles Up - Companies, Commodities and Macroeconomics' - This is a condensed version of the second part of the full document. This covers the structure and results of the models for
company size distributions and for dynamic pricing in commodities and economies as a whole. This document has 37 pages of text.
'The Bowley Ratio' - This is a brief extract from the second part of the full document. It gives a full explanation of the constant ratio of labour share in national income. This document has six
pages of text.
'Pricing, Liquidity & the Control of Dynamics Systems in Finance & Economics' - Is a discussion of practical issues that arise from treating economics as a dynamic system. This document has 29 pages
of text.
'Wealth, Income, Earnings & the Statistical Mechanics of Flow Systems' - discusses economics as an example of an out of equilibrium thermodynamic system that may be exactly soluble. This document has
25 pages of text.
Abstract - Why Money Trickles Up
This paper combines ideas from classical economics and modern finance with Lotka-Volterra models, and also the general Lotka-Volterra models of Levy & Solomon to provide straightforward explanations
of a number of economic phenomena.
Using a simple and realistic economic formulation, the distributions of both wealth and income are fully explained. Both the power tail and the log-normal like body are fully captured. It is of note
that the full distribution, including the power law tail, is created via the use of absolutely identical agents. It is further demonstrated that a simple scheme of compulsory saving could eliminate
poverty at little cost to the taxpayer. Such a scheme is discussed in detail and shown to be practical.
Using similar simple techniques, a second model of corporate earnings is constructed that produces a power law distribution of company size by capitalisation.
A third model is produced to model the prices of commodities such as copper. Including a delay to capital installation; normal for capital intensive industries, produces the typical cycle of
short-term spikes and collapses seen in commodity prices.
The fourth model combines ideas from the first three models to produce a simple Lotka-Volterra macroeconomic model. This basic model generates endogenous boom and bust business cycles of the sort
described by Minsky and Austrian economists.
From this model an exact formula for the Bowley ratio; the ratio of returns to labour to total returns, is derived. This formula is also derived trivially algebraically. This derivation is extended
to a model including debt, and it suggests that excessive debt can be economically dangerous and also directly increases income inequality.
Other models are proposed with financial and non-financial sectors and also two economies trading with each other. There is a brief discussion of the role of the state and monetary systems in such
The second part of the paper discusses the various background theoretical ideas on which the models are built.
This includes a discussion of the mathematics of chaotic systems, statistical mechanical systems, and systems in a dynamic equilibrium of maximum entropy production.
There is discussion of the concept of intrinsic value, and why it holds despite the apparent substantial changes of prices in real life economies. In particular there are discussions of the roles of
liquidity and parallels in the fields of market-microstructure and post-Keynesian pricing theory.
To download 'Why Money Trickles Up', please click the link below:
(nb - this is a large file; 11Mb, 250 pages of text, 90 figures.)
click here to download ymtu - full paper
To download 'YMTU - Bullet Points' click on the link below:
(2.3Mb, 35 pages)
click here to download file 'bullet points'
Abstract - Wealth & Income Distributions
This paper combines ideas from classical economics and modern finance with the general Lotka-Volterra models of Levy & Solomon to provide straightforward explanations of wealth and income
distributions. Using a simple and realistic economic formulation, the distributions of both wealth and income are fully explained. Both the power tail and the log-normal like body are fully captured.
It is of note that the full distribution, including the power law tail, is created via the use of absolutely identical agents. It is further demonstrated that a simple scheme of compulsory saving
could eliminate poverty at little cost to the taxpayer.
To download 'Wealth & Income Distributions', please click the link below:
(2.5Mb, 45 pages of text, 36 figures.)
click here to download wealth & income distributions
Abstract - Companies, Commodities and Macroeconomics
This paper combines ideas from classical economics and modern finance with Lotka-Volterra models, and also the general Lotka-Volterra models of Levy & Solomon to provide straightforward explanations
of a number of economic phenomena.
Using a simple and realistic economic formulation, a model of corporate earnings is constructed that produces a power law distribution of company size by capitalisation. A second model is produced to
model the prices of commodities such as copper. Including a delay to capital installation; normal for capital intensive industries, produces the typical cycle of short-term spikes and collapses seen
in commodity prices.
The third model combines previous ideas to produce a simple Lotka-Volterra macroeconomic model. This basic model generates endogenous boom and bust business cycles of the sort described by Minsky and
Austrian economists. From this model an exact formula for the Bowley ratio; the ratio of returns to labour to total returns, is derived. This formula is also derived trivially algebraically. This
derivation is extended to a model including debt, and it suggests that excessive debt can be economically dangerous and also directly increases income inequality.
To download 'Companies, Commodities & Macroeconomics', please click the link below:
(1.6Mb, 37 pages of text, 20 figures.)
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Abstract - The Bowley Ratio
The paper gives a simple algebraic description, and background justification, for the Bowley Ratio, the relative returns to labour and capital, in a simple economy.
To download 'The Bowley Ratio', please click the link below:
(6 pages of text, no figures.)
click here to download the bowley ratio
Abstract - Pricing, liquidity and the control of dynamic systems in finance and economics
The paper discusses various practical consequences of treating economics and finance as an inherently dynamic and chaotic system. On the theoretical side this looks at the general applicability of
the market-making pricing approach to economics in general. The paper also discuses the consequences of the endogenous creation of liquidity and the role of liquidity as a state variable. On the
practical side, proposals are made for reducing chaotic behaviour in both housing markets and stock markets.
To download 'Dynamic Systems', please click the link below (0.5Mb, 29 pages text, 8 figures.):
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Abstract - Wealth, Income, Earnings and the Statistical Mechanics of Flow Systems
This paper looks at empirical data from economics regarding wealth, earnings and income, alongside a flow model for an economy based on the general Lotka-Volterra models of Levy & Solomon. The data
and modelling suggest that a simple economic system might provide a tractable model for giving an exact statistical mechanical solution for an 'out of equilibrium' flow model. This might also include
an exact mathematical definition of a 'dissipative structure' derived from maximum entropy considerations. This paper is primarily a qualitative discussion of how such a mathematical proof might be
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Invasion of the Math Snatchers
{Originally published at the Independence Institute in January 2004}
“Math is hard, let’s go shopping!”
When Mattel released a talking Barbie who offered that bit of teenage wisdom, public reaction was so furious they pulled her off the shelves. Mattel is still trying to recover from the PR disaster.
I assume they fired the guy who came up with that little gem. Not that it mattered much.
I have every confidence he’s enjoying a new career, designing math programs for American public schools. What else can I think about programs that encourage children to “shop” for the correct way to
multiply? That ask kids what “color” they think math is, like it’s some sort of lip gloss? It’d be funny, if it weren’t so tragic.
It’s tragic because, in a modern global economy, mathematical literacy is essential. The most important product humanity produces in the 21st century is information. Working with information requires
intellectual discipline and the ability to think abstractly. That’s what math is all about.
Unfortunately, other countries do a much better job of teaching math than we do, with potentially serious consequences. Why shouldn’t American firms contract out high-tech jobs to engineers from
overseas, if that makes them more competitive?
Do you know any immigrants at your school? Ask Asian or European families what they think about math classes. Chances are their children placed into the most advanced math the district has to offer,
yet are still having a very easy time.
My own experience as a teacher bears this out. I am proud to be on the faculty at one of the most selective colleges in America. My students are America’s best and brightest.
And yet, when I went to Russia on sabbatical, I couldn’t believe how good my students were at math. After two weeks of class, I had to redo all my lesson plans. I wound up covering more material in
more detail than I had thought possible. It was a great experience, but a sobering indictment of American education.
Fortunately, what American students lack in fundamentals they make up in initiative and creativity. It’s a constant struggle to get Russian students to ‘think outside the box,” while my American
classes are always abuzz with interesting ideas. Fix the math problems, and American students will do great things.
So how do we do that?
First we have to undo two decades’ worth of damage done by faddish mathematical programs. Here’s how you can tell if your school has one:
Your school emphasizes children “discovering” or “constructing” their own techniques for arithmetic. This is nice in theory, but most children lack the intellectual curiosity and focus to discover
even basic arithmetic rules.
Besides, it took humanity a couple of millennia to develop the math we have now. Asking a roomful of 4th graders to start from scratch is an idea only an education professor could’ve come up with.
Your school de-emphasizes drills. “Boring” facts like multiplication tables and algebra formulas are no fun to teach, but they’re an essential part of developing mathematical fluency. If your child’s
teacher doesn’t pay much attention to drills or thinks math facts aren’t important, be on the alert.
Your school encourages extensive, early calculator use.
Calculators are appropriate once mathematical fluency has been gained. But they’re crippling if introduced too soon, particularly in the early grades. There is a big difference between a child who
knows *why* six times seven is fortytwo, and a child who merely pushes “6 X 7 =” on a calculator.
Fortunately, all is not lost. There are some terrific mathematics programs out there, ones that are both rigorous and fun. They’re ready and available to replace the silliness we have now, if only
parents will demand them.
But it won’t be easy. We’ll have to do our part. We must support teachers who set high standards. We must support schools that hold students accountable. We must understand that self-esteem in
mathematics is earned, not given. It comes from getting the right answer.
These and other “back to basics” ideas fly in the face of the modern educational establishment. They are in direct contradiction to incentives parents, teachers, and administrators face on a daily
basis. Trying to solve this problem will be very, very hard.
But so what? Math is hard. Let’s go to work.
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CLE Math Placement
I am switching my daughter from MUS Beta(finished the book) to CLE math. She hasn't had any multiplication at all. We are working on knowing our addition and subtraction facts better before moving
ahead. So my question is where do I start her. She will be in third, but I noticed that their multiplication starts in book 205. Help? Do we need to start at the beginning at 200 or go ahead and
start at 205.
If you liked the program will you tell me why?
Does anyone buy the addition and subtraction flashcards for $15.00?
Edited by homeschoolmom4
You need to start with the placement test. Period. Please do not worry about the levels. The numbers are only put on the Light Units for brick and mortar schools. The one thing I have learned is that
CLE's math spans several 'grade' levels in one level of their math course. For example my daughter just finished the 400 level series in math. She wanted to go back to BJU math(her perogative not
mine) and I bought their 6th grade math. Here only to find out that the 400 series of CLE math has covered most of what BJU has in their 6th grade math program(minus the last three chapters).So
really CLE's math for 400 covers what you would find in a 4th, 5th andn 6th grade math program elsewhere all in one year.
Other math programs only begin with concepts that prepare them for multiplication for usually the 4th grade year.
I really do like the CLE math program. I think it is an excellent math program and even though my oldest doesn't care for the math program this was her best year with math by far. I wish I could
convince her otherwise that CLE really is a better choice for her. I will definitely be using CLE math for my 9 and 6yr old this year.
The addition and subtraction cards for that level are highly recommended because they are numbered in a way that it goes with the math program. Otherwise your going to be trying to spend a lot of
time trying to find the right cards for the right lessons. Even though I already have flash cards here I will be purchasing them to make life just a little bit easier. I believe when you reach the
400 level you can use any flash cards you would like.
I definately found that the CLE math is very much ahead of many of the other programs. My mathy entering-6th grader is using 500 this year. There is a good deal of review and drill built in for those
who need it, and if your DD simply needs help with multiplication and division I'd suggestion picking up a few of the "math rocks" CDs and DVDs and playing them over the summer under she gets the
basic facts memorized. You can sit her down and show her that multiplication is really just a faster way to skip-count (at least at the early-elementary level.)
If after taking the placement test, she does well EXCEPT on the multiplication & division parts, you can have her work through the Math 301 Lightunit, it's a review of everything taught in the 200
level, and you can order their extra drill pages if you think she needs them. You can also just print out worksheets from the MUS site for her to work on multiplication and division.
When you cook, have her multiply/divide your recipes. This is especially good to do when baking cookies, because she can eat the results fo her math. ;)
Guest lahmeh
Best of luck to you with CLE! It's the ONLY thing we have stuck with for math, LA and reading. I hope you enjoy it as much as we are! :001_smile:
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Capital budgeting is the process by which long-term fixed assets are evaluated and possibly selected or rejected for investment purposes. The purpose of capital budgeting is to evaluate po - NursingEssays
Capital budgeting is the process by which long-term fixed assets are evaluated and possibly selected or rejected for investment purposes. The purpose of capital budgeting is to evaluate potential
projects for possible investment by the firm.
Address one of the following prompts in a brief but thorough manner.
• What are the various methods for evaluating possible capital projects, in terms of their possible benefits to the firm? Describe the benefits and/or shortcomings of each.
• What is the NPV profile and what are its uses?
respond to 2 peer response
Net present value (NPV) can be evaluated by adding the present discount values of the incomes while subtracting the discounted present costs through the useful lifetime of the system. The economic
significance and the reliability of a metric depend on its compatibility with the Net Present Value (NPV). Traditionally, a metric is said to be NPV-consistent if it is coherent with NPV in signaling
value creation (Marchioni, & Magni, 2018). NPV is the actual value of the capital and RCs of a device over its lifespan. NPV is used as a primary economic measure for the evaluation of an energy
system. The discrepancy between the actual value of the profits and the expenses resulting from an investment is the net present value of the system (Edwin et al., 2019; Edwin & Sekhar, 2014b, 2016)
(Kumar et al., 2020). The NPV method allows a company to evaluate a possible investment’s probability of gains or losses by incorporating the time value of money into its calculation. The NVP is one
of the simplest ways to determine a possible investment’s value to a firm. The NPV shows how much a firm’s current value, and thus stockholders’ wealth, will increase if a capital budgeting project
is purchased. If the net benefit computed on a present value basis—that is, NPV—is positive, then the asset (project) is considered an acceptable investment. In other words, to determine whether a
project is acceptable using the NPV technique, we apply the following decision rule: NPV Decision Rule: A project is acceptable if NPV > $0 (Besley & Brigham, pg. 200, 2021). The NPV profile uses a
project’s NPV and required rates of return to create a graph. NPV is considered a theoretically reliable measure of economic profitability. NPV profile is constructed as a part of the overall NPV
analysis of capital budgeting. It uses different discount rates to display how a change in discount rate impacts the net present value (NPV) of a potential opportunity. The projects with positive NPV
profiles are expected to increase the firm’s wealth and are considered good candidates to invest in (Javed, 2024).
Besley, S., & Brigham, E. (2021). CFIN (7th ed.). Cengage Limited.
Javed, R. (2024, April 9). Net present value (NPV) profile. Accounting for Management.
Kumar, L., Mamun, M. a. A., & Hasanuzzaman, M. (2020). Energy economics. In Elsevier
eBooks (pp. 167–178). https://doi.org/10.1016/b978-0-12-814645-3.00007-9
Marchioni, A., & Magni, C. A. (2018). Investment decisions and sensitivity analysis: NPV-
consistency of rates of return. European Journal of Operational Research, 268(1),
361–372. https://doi.org/10.1016/j.ejor.2018.01.007
Capital budgeting is a critical process for businesses seeking to make long-term investment decisions regarding fixed assets. The evaluation of potential projects plays a crucial role in determining
the success and growth of a firm. There are various methods available to assess the benefits of potential capital projects, each with its own set of advantages and disadvantages.
One common method used in capital budgeting is the Net Present Value (NPV) analysis. NPV calculates the present value of expected cash flows from a project, discounted at a predetermined rate of
return. The benefit of NPV is that it provides a clear measure of the profitability of a project, considering the time value of money. However, NPV relies heavily on accurate cash flow estimations
and the chosen discount rate, which can introduce subjectivity into the analysis.
Another popular method is the Internal Rate of Return (IRR), which computes the rate of return generated by a project's cash flows. The advantage of IRR is that it is easy to interpret and compare
against the cost of capital. However, IRR can be misleading when comparing mutually exclusive projects or when cash flows change sign multiple times.
Payback Period is a simple method that calculates the time it takes for a project to recoup its initial investment. The benefit of Payback Period is its ease of understanding and application.
However, it fails to consider the time value of money and the project's entire cash flow stream, leading to potentially flawed investment decisions.
The Profitability Index (PI) is a ratio that compares the present value of cash inflows to the initial investment. PI offers a useful way to rank projects based on their return per unit of
investment. Nonetheless, PI does not provide an absolute measure of profitability and may lead to inconsistent rankings when used in conjunction with other methods.
Lastly, the Accounting Rate of Return (ARR) measures the profitability of a project based on accounting income. ARR is easy to calculate and understand, making it a popular choice for non-financial
managers. However, ARR ignores the time value of money and does not consider cash flows beyond the payback period, potentially leading to suboptimal decisions.
In conclusion, capital budgeting methods offer various ways to assess potential capital projects, each with its own set of benefits and shortcomings. It is essential for firms to consider multiple
evaluation techniques to make informed investment decisions and mitigate risks associated with capital expenditure. By understanding the nuances of each method, businesses can effectively allocate
resources and maximize shareholder value.
1. Spyrou, Spyros P., et al. "Capital Budgeting Practices: A Survey in the Greek Business Environment." Procedia Economics and Finance, vol. 5, 2013, pp. 696-705.
2. Brealey, Richard A., and Stewart C. Myers. Principles of Corporate Finance. McGraw-Hill Education, 2017.
3. Pike, Richard, and Bill Neale. Corporate Finance and Investment: Decisions & Strategies. Pearson Education, 2009.
4. Ross, Stephen A., et al. Fundamentals of Corporate Finance. McGraw-Hill Education, 2016.
5. Copeland, Thomas E., et al. "Real Options: A Practitioner's Guide." The McKinsey Quarterly, no. 4, 2000, pp. 21-30.
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alculate the
How to Calculate the Z Score: A Comprehensive Guide
In the world of statistics, it's essential to know how to compare different data points and understand how far they deviate from the average. One useful tool for this purpose is the Z score. In this
comprehensive guide, we will delve into the concept of the Z score, explain why it is important, and provide a step-by-step guide on how to calculate it. By the end of this blog post, you will have a
solid understanding of Z scores, their applications, and how to interpret them. If you quickly need to calculate a z score or a z score probability please check out:
Definition of Z score
The Z score, also known as the standard score, is a measure that expresses the number of standard deviations a data point is away from the mean (average) of a dataset. It is a dimensionless quantity
that enables us to compare data points from different distributions or scales.
To better understand the Z score, let's break down its components:
1. Standard deviation: This is a measure of the spread or dispersion of a dataset. It tells us how much the individual data points deviate from the mean, on average. The greater the standard
deviation, the more dispersed the data points are from the mean.
2. Mean: The mean, or average, is the sum of all data points divided by the number of data points. It represents the central value of a dataset.
The Z score essentially calculates
how many standard deviations
away a particular data point is from the mean of the dataset. This is useful for understanding how unusual or typical a data point is within the context of the dataset.
For example, consider the following dataset of ages of a group of people: {20, 25, 30, 35, 40, 45, 50}. The mean age is 35, and the standard deviation is approximately 10.07. To calculate the Z score
for the age of 45, we would use the formula:
Z score = (Data point - Mean) / Standard deviationZ score = (45 - 35) / 10.07 ≈ 0.99
In this case, the Z score of 0.99 indicates that the age of 45 is approximately one standard deviation above the mean age of the group. This suggests that the age of 45 is relatively common within
the dataset, as it is close to the average value.
On the other hand, if we calculate the Z score for the age of 20:
Z score = (20 - 35) / 10.07 ≈ -1.49
The negative Z score of -1.49 indicates that the age of 20 is approximately 1.5 standard deviations below the mean age of the group. This suggests that the age of 20 is less common within the
dataset, as it is farther away from the average value.
Importance of Z score in statistics
Z scores are widely used in statistics because they provide a standardized way of comparing data points, regardless of the original scale. They are particularly helpful in identifying outliers,
standardizing data for analysis, and comparing data from different sources.
Applications of Z score
1. Comparing data from different scales: Z scores allow us to compare data points from different distributions or scales, such as test scores or financial data, by standardizing the data.
2. Identifying outliers: Z scores can help determine whether a data point is an outlier, as unusually high or low Z scores suggest that the data point significantly deviates from the mean.
3. Standardizing data for analysis: Z scores are often used in statistical analyses, such as regression and hypothesis testing, to ensure that the data is on a common scale.
Prerequisites for calculating Z score
Understanding the normal distribution
1. Definition and characteristics: The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution characterized by its bell-shaped curve. It is symmetric
around the mean, with the majority of data points concentrated near the mean and fewer data points as we move away from the mean.
2. The bell curve: The bell curve is a visual representation of the normal distribution. It is called a bell curve because of its bell-like shape, with the peak corresponding to the mean, and the
tails extending towards infinity in both directions.
Familiarity with basic statistical conceptsTo calculate Z scores, you need to have a basic understanding of the following statistical concepts:
1. Mean: The mean, or average, is the sum of all data points divided by the number of data points. It is a measure of central tendency that provides an idea of where the center of the dataset lies.
2. Standard deviation: The standard deviation is a measure of the dispersion or spread of a dataset. It indicates how much the individual data points deviate from the mean on average.
3. Variance: Variance is the square of the standard deviation. It represents the average of the squared differences between each data point and the mean.
Step-by-step guide to calculating Z score
To calculate the Z score, follow these steps:
Step 1: Identify the data point
First, you need to identify the data point for which you want to
calculate the Z score
. This data point could be a test score, a financial figure, or any other value you wish to compare with the rest of the dataset.
Step 2: Calculate the mean of the dataset
calculate the mean (average) of the dataset
by adding up all the data points and dividing the sum by the number of data points.
Step 3: Calculate the standard deviation of the dataset
calculate the standard deviation
of the dataset, follow these steps:
1. Subtract the mean from each data point and square the result.
1. Add up all the squared differences obtained in the previous step.
2. Divide the sum of the squared differences by the number of data points (or by one less than the number of data points if using a sample rather than a population).
3. Take the square root of the result to obtain the standard deviation.
Step 4: Compute the Z score using the formula
Now that you have the mean and standard deviation, you can calculate the Z score using the following formula:
Z score = (Data point - Mean) / Standard deviation
The Z score formula expresses the distance between the data point and the mean in terms of standard deviations. A positive Z score indicates that the data point is above the mean, while a negative Z
score indicates that it is below the mean. A Z score of zero means that the data point is equal to the mean.
Example with detailed calculations:
Suppose we have the following test scores for a group of students: 72, 84, 90, 66, 78, 96, and 82. We want to calculate the Z score for the test score of 90. First, we calculate the mean and standard
deviation of the dataset:
Mean = (72 + 84 + 90 + 66 + 78 + 96 + 82) / 7 = 668 / 7 = 95.43 (rounded to two decimal places)
Standard deviation:
a. Subtract the mean from each data point and square the result:
(72 - 95.43)^2 = 547.90
(84 - 95.43)^2 = 131.04
(90 - 95.43)^2 = 29.58
(66 - 95.43)^2 = 863.04
(78 - 95.43)^2 = 303.80
(96 - 95.43)^2 = 0.32
(82 - 95.43)^2 = 181.04
b. Add up all the squared differences: 547.90 + 131.04 + 29.58 + 863.04 + 303.80 + 0.32 + 181.04 = 2,056.72
c. Divide the sum of squared differences by the number of data points (or by one less than the number of data points if using a sample): 2,056.72 / 7 = 293.82
d. Take the square root of the result: √293.82 = 17.14 (rounded to two decimal places)
Finally, we can calculate the Z score for the test score of 90:
Z score = (90 - 95.43) / 17.14 = -0.32 (rounded to two decimal places)
This means that the test score of 90 is 0.32 standard deviations below the mean.
Interpretation of Z scores
What a positive, negative, or zero Z score indicates
positive Z score indicates
that the data point is above the mean, a
negative Z score
indicates that it is below the mean, and a zero Z score means that the data point is equal to the mean.
Z score as a measure of distance from the mean
The Z score tells us how far a data point is from the mean in terms of standard deviations. A Z score of 1, for example, indicates that the data point is one standard deviation above the mean, while
a Z score of -2 indicates that it is two standard deviations below the mean.
Standard deviations and percentiles
1. One, two, and three standard deviation rules: In a normal distribution, approximately 68% of the data points fall within one standard deviation of the mean, 95% within two standard deviations,
and 99.7% within three standard deviations. These rules help us determine the probability of a data point falling within a specific range in relation to the mean.
2. Using Z score to determine percentiles: The Z score can be used to find the percentile rank of a data point in a dataset. Percentile rank represents the percentage of data points that fall below
a given data point. To find the percentile rank, you can use a Z score table or an online calculator that provides the area under the curve to the left of the Z score.
Applications of Z scores in real life
Comparing test scores and student performance:
Z scores are often used to compare students' test scores from different schools or standardized tests, making it easier to evaluate student performance and identify areas where improvement is
Financial data analysis and risk management:
Z scores can help analysts and investors
compare financial data
from different companies, industries, or time periods, enabling them to make more informed decisions. They can also be used to assess the risk associated with investments, such as the likelihood of a
stock price falling within a certain range.
Quality control in manufacturing:
In manufacturing, Z scores can be used to monitor product quality and identify any deviations from the established standards. By detecting outliers, manufacturers can take corrective action to
ensure that products meet the desired quality standards.
Limitations of Z scores
Applicability only for normally distributed data: Z scores are most accurate when the data follows a normal distribution. If the data is not normally distributed, the Z score may not accurately
represent the position of a data point within the dataset.
Inaccuracy when dealing with small sample sizes: Z scores can be less accurate when dealing with small sample sizes, as the standard deviation may not be a reliable measure of dispersion.
Potential misuse in data manipulation: While Z scores can be helpful in identifying outliers and standardizing data for analysis, they can also be misused to manipulate data or draw inaccurate
How to Calculate the Z Score using a TI-89 Calculator
The TI-89 calculator is a powerful and versatile tool that can help you quickly and easily calculate Z scores. In this section, we will explain how to calculate the Z score using a TI-89 calculator,
which can save you time and effort when working with statistical data.
Step-by-step guide to calculating Z score on a TI-89 calculator:
Input your dataset:
Before you can calculate a Z score on your TI-89 calculator, you need to input your dataset. Follow these steps:
a. Turn on the calculator and press the 'APPS' button. b. Scroll down the list of available applications and select '6: Data/Matrix Editor'. c. Choose '3: New' to create a new dataset. d. In the
'Type' field, select 'Data' and provide a suitable name for your dataset in the 'Var' field. Press 'Enter' to create the dataset. e. Now, input your data points into the 'c1' column. You can navigate
using the arrow keys and input values by typing the numbers and pressing 'Enter'.
Calculate the mean and standard deviation:
a. Press the '2nd' button, followed by the '5' button to access the 'List' menu. b. Scroll down and select '3: mean(' and then select the dataset you created earlier by pressing '2nd', 'VAR-LINK',
and choosing the dataset. Close the parenthesis and press 'Enter' to calculate the mean. c. To calculate the standard deviation, repeat the process, but select '7: stdDev(' from the 'List' menu
instead. Enter the dataset name, close the parenthesis, and press 'Enter'.
Take note of the mean and standard deviation values, as you will need them to calculate the Z score.
Calculate the Z score:
To calculate the Z score for a specific data point, use the following formula:
Z score = (Data point - Mean) / Standard deviation
a. Press the 'Home' button to return to the main screen. b. Input the data point you want to
calculate the Z score
for, followed by the subtraction operator '-'. c. Input the mean value you calculated earlier and press 'Enter' to calculate the difference between the data point and the mean. d. Divide the result
by the standard deviation value you calculated earlier, using the division operator '/'. e. Press 'Enter' to calculate the Z score for the data point.
How to Calculate a Z Score in Excel and Google Sheets
Calculating the Z score in Excel or Google Sheets is a straightforward process that can help you quickly analyze and compare data points within a dataset. We will guide you through the steps to
calculate a Z score in both Excel and Google Sheets.Step-by-step guide to calculating Z score in Excel and Google Sheets:
1. Input your dataset:
Create a new spreadsheet in either Excel or Google Sheets, and input your dataset into a single column or row. For our example, we will input the dataset into column A, starting from cell A1.
1. Calculate the mean:
To calculate the mean of your dataset, use the AVERAGE function. In an empty cell, type the following formula:=AVERAGE(A1:A[n])Replace 'A[n]' with the last cell in the range containing your dataset.
Press 'Enter' to calculate the mean. For example, if your dataset ends at cell A7, the formula would be:=AVERAGE(A1:A7)
1. Calculate the standard deviation:
To calculate the standard deviation of your dataset, use the STDEV.S (for a sample) or STDEV.P (for a population) function in Excel, and the STDEV (for a sample) or STDEVP (for a population) function
in Google Sheets. In an empty cell, type the following formula:=STDEV.S(A1:A[n]) (Excel) or =STDEV(A1:A[n]) (Google Sheets)Replace 'A[n]' with the last cell in the range containing your dataset, and
press 'Enter' to calculate the standard deviation.
1. Calculate the Z score:
To calculate the Z score for a specific data point, use the following formula:Z score = (Data point - Mean) / Standard deviationIn an empty cell adjacent to the data point you want to calculate the Z
score for, type the following formula:=(A1 - Mean_Cell) / Stdev_CellReplace 'A1' with the cell containing the data point, 'Mean_Cell' with the cell containing the mean value, and 'Stdev_Cell' with
the cell containing the standard deviation value. Press 'Enter' to calculate the Z score for the data point.
1. Calculate Z scores for all data points (optional):
If you want to calculate Z scores for all data points in your dataset, you can copy the formula from the previous step and paste it into the adjacent cells. Be sure to use absolute cell references
(by adding dollar signs before the column and row identifiers) for the mean and standard deviation cells to avoid errors when copying the formula. For example:=(A1 - $B$1) / $B$2Now you have
successfully calculated the Z score for your data points using Excel or Google Sheets!Bottom of Form
How to Calculate a Z Score in R and Python
Calculating the Z score in R and Python is an essential skill for anyone working with statistical data in these popular programming languages. Here we will guide you through the steps to calculate a
Z score in both
Calculating Z Score in R:
1. Input your dataset:
Create a vector in R containing your dataset. For example, if you have the dataset {20, 25, 30, 35, 40, 45, 50}, you can create a vector as follows:
RCopy code
data <- c(20, 25, 30, 35, 40, 45, 50)
1. Calculate the mean and standard deviation:
Use the
functions in R to calculate the mean and standard deviation of your dataset:
RCopy code
mean_data <- mean(data) sd_data <- sd(data)
1. Calculate the Z score:
To calculate the Z score for a specific data point, use the following formula:
Z score = (Data point - Mean) / Standard deviation
For example, to calculate the Z score for the data point 45, use the following code:
RCopy code
data_point <- 45 z_score <- (data_point - mean_data) / sd_data
Calculating Z Score in Python:
1. Import the required libraries:
To calculate the Z score in Python, you will need the NumPy library for calculating the mean and standard deviation. Install the library if you haven't already, and then import it:
pythonCopy code
import numpy as np
1. Input your dataset:
Create a list or NumPy array containing your dataset. For example, if you have the dataset {20, 25, 30, 35, 40, 45, 50}, you can create a NumPy array as follows:
pythonCopy code
data = np.array([20, 25, 30, 35, 40, 45, 50])
1. Calculate the mean and standard deviation:
Use the
functions in Python to calculate the mean and standard deviation of your dataset:
pythonCopy code
mean_data = np.mean(data) sd_data = np.std(data)
1. Calculate the Z score:
To calculate the Z score for a specific data point, use the following formula:
Z score = (Data point - Mean) / Standard deviation
For example, to calculate the Z score for the data point 45, use the following code:
pythonCopy code
data_point = 45 z_score = (data_point - mean_data) / sd_data
Z Score Calculation Examples
Example 1:
A teacher gives a math test to her students, and the scores are as follows: {55, 65, 75, 85, 95}. Calculate the Z score for a student who scored 75 on the test.
Step 1: Calculate the mean and standard deviation.
Mean = (55 + 65 + 75 + 85 + 95) / 5 = 375 / 5 = 75 Standard deviation = √[((55-75)^2 + (65-75)^2 + (75-75)^2 + (85-75)^2 + (95-75)^2) / 5] ≈ 14.14
Step 2: Calculate the Z score.
Z score = (Data point - Mean) / Standard deviation Z score = (75 - 75) / 14.14 ≈ 0
Answer: The Z score for a student who scored 75 on the test is 0. This means that the student's score is exactly at the mean of the dataset.
Explanation: The Z score of 0 indicates that the student's score is not deviating from the mean. Since the mean is the central value of the dataset, this suggests that the student's score is
relatively common within the dataset.
Example 2:
A company measures the time (in minutes) that its employees take to complete a specific task: {15, 20, 22, 24, 30}. Calculate the Z score for an employee who took 15 minutes to complete the task.
Step 1: Calculate the mean and standard deviation.
Mean = (15 + 20 + 22 + 24 + 30) / 5 = 111 / 5 = 22.2 Standard deviation = √[((15-22.2)^2 + (20-22.2)^2 + (22-22.2)^2 + (24-22.2)^2 + (30-22.2)^2) / 5] ≈ 5.09
Step 2: Calculate the Z score.
Z score = (Data point - Mean) / Standard deviation Z score = (15 - 22.2) / 5.09 ≈ -1.42
Answer: The Z score for an employee who took 15 minutes to complete the task is approximately -1.42.
Explanation: The negative Z score of -1.42 indicates that the employee's completion time is approximately 1.42 standard deviations below the mean completion time. This suggests that the employee's
completion time is relatively fast compared to the other employees in the dataset.
Example 3:
The weights (in kilograms) of a group of people are as follows: {60, 65, 70, 75, 80}. Calculate the Z score for a person who weighs 80 kg.
Step 1: Calculate the mean and standard deviation.
Mean = (60 + 65 + 70 + 75 + 80) / 5 = 350 / 5 = 70 Standard deviation = √[((60-70)^2 + (65-70)^2 + (70-70)^2 + (75-70)^2 + (80-70)^2) / 5] ≈ 7.07
Step 2: Calculate the Z score.
Z score = (Data point - Mean) / Standard deviation Z score = (80 - 70) / 7.07 ≈ 1.41
Answer: The Z score for a person who weighs 80 kg is approximately 1.41.
Explanation: The Z score of 1.41 indicates that the person's weight is approximately 1.41 standard deviations above the mean weight of the dataset. This suggests that the person's weight is
relatively high compared to the other individuals in the dataset.
Example 4:
A dataset represents the ages of a group of people: {25, 30, 35, 40, 45}. Calculate the Z score for a person who is 25 years old.
Step 1: Calculate the mean and standard deviation.
Mean = (25 + 30 + 35 + 40 + 45) / 5 = 175 / 5 = 35 Standard deviation = √[((25-35)^2 + (30-35)^2 + (35-35)^2 + (40-35)^2 + (45-35)^2) / 5] ≈ 7.07
Step 2: Calculate the Z score.
Z score = (Data point - Mean) / Standard deviation Z score = (25 - 35) / 7.07 ≈ -1.41
Answer: The Z score for a person who is 25 years old is approximately -1.41.
Explanation: The negative Z score of -1.41 indicates that the person's age is approximately 1.41 standard deviations below the mean age of the dataset. This suggests that the person's age is
relatively low compared to the other individuals in the dataset.
By calculating Z scores for specific data points, you can easily compare and analyze values within a dataset. This statistical measure can be a helpful tool in various fields, including education,
business, and research.
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The Stacks project
Lemma 85.8.1. In Situation 85.3.3 and with notation as above there is a complex
\[ \ldots \to g_{2!}\mathbf{Z} \to g_{1!}\mathbf{Z} \to g_{0!}\mathbf{Z} \]
of abelian sheaves on $\mathcal{C}_{total}$ which forms a resolution of the constant sheaf with value $\mathbf{Z}$ on $\mathcal{C}_{total}$.
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Elementary Functions Tanh[z] Integration Indefinite integration Involving functions of the direct function and hyperbolic functions Involving powers of the direct function and hyperbolic
Involving power of cosh
Involving cosh^u(b z) tanh^nu(c z)
Elementary Functions Tanh[z] Integration Indefinite integration Involving functions of the direct function and hyperbolic functions Involving powers of the direct function and hyperbolic functions
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What is the temperature at which the motion of particles theoretically ceases? | HIX Tutor
What is the temperature at which the motion of particles theoretically ceases?
Answer 1
I think that you will have to tell us that one........
Google #"temperatura assoluta............"# or #"the Kelvin Scale"#. You might already have come across this scale.
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Answer 2
The temperature at which the motion of particles theoretically ceases is ${0}^{0} A$ or absolute zero, Absolute ${0}^{0}$ is the definition of ${0}^{0}$ on the Kelvin and Rankine temperature scales.
Absolute zero computes to #-459.67^0F#, which is also #-273.15^0C.#
At this temperature, both the enthalpy (heat content) and entropy (state of randomness or disorder) approach zero. Effectively, the molecules of the gas are slowing down towards being motionless.
Absolute zero also describes a gas reaching a temperature from which no more heat can be removed. Experiments have shown that molecules continue to vibrate at absolute zero.
There are more thoughts on this topic here: https://tutor.hix.ai
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Answer 3
$\text{0 K}$, kelvin. But that is not strictly true, as all molecules vibrate at $\text{0 K}$ at their zero-point energy, $\frac{1}{2} h \nu$.
ATOMIC MOTION AT 0 K
In regards to atoms, yes, the motion of particles will stop at $\text{0 K}$, or $- \text{273.15"^@ "C}$, because average atomic kinetic energy (which for atoms is entirely translational) depends on
the temperature and any intermolecular forces present.
When we consider non-interacting atoms in the classical limit, the equipartition theorem gives for the average per-particle kinetic energy:
#<< K_(avg,"trans") >> -= K_(avg,"trans")/N = 3/2 k_BT#, in $\text{J/particle}$
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Estimate Position and Orientation of a Ground Vehicle
This example shows how to estimate the position and orientation of ground vehicles by fusing data from an inertial measurement unit (IMU) and a global positioning system (GPS) receiver.
Simulation Setup
Set the sampling rates. In a typical system, the accelerometer and gyroscope in the IMU run at relatively high sample rates. The complexity of processing data from those sensors in the fusion
algorithm is relatively low. Conversely, the GPS runs at a relatively low sample rate and the complexity associated with processing it is high. In this fusion algorithm the GPS samples are processed
at a low rate, and the accelerometer and gyroscope samples are processed together at the same high rate.
To simulate this configuration, the IMU (accelerometer and gyroscope) is sampled at 100 Hz, and the GPS is sampled at 10 Hz.
imuFs = 100;
gpsFs = 10;
% Define where on the Earth this simulation takes place using latitude,
% longitude, and altitude (LLA) coordinates.
localOrigin = [42.2825 -71.343 53.0352];
% Validate that the |gpsFs| divides |imuFs|. This allows the sensor sample
% rates to be simulated using a nested for loop without complex sample rate
% matching.
imuSamplesPerGPS = (imuFs/gpsFs);
assert(imuSamplesPerGPS == fix(imuSamplesPerGPS), ...
'GPS sampling rate must be an integer factor of IMU sampling rate.');
Fusion Filter
Create the filter to fuse IMU + GPS measurements. The fusion filter uses an extended Kalman filter to track orientation (as a quaternion), position, velocity, and sensor biases.
The insfilterNonholonomic object has two main methods: predict and fusegps. The predict method takes the accelerometer and gyroscope samples from the IMU as input. Call the predict method each time
the accelerometer and gyroscope are sampled. This method predicts the states forward one time step based on the accelerometer and gyroscope. The error covariance of the extended Kalman filter is
updated in this step.
The fusegps method takes the GPS samples as input. This method updates the filter states based on the GPS sample by computing a Kalman gain that weights the various sensor inputs according to their
uncertainty. An error covariance is also updated in this step, this time using the Kalman gain as well.
The insfilterNonholonomic object has two main properties: IMUSampleRate and DecimationFactor. The ground vehicle has two velocity constraints that assume it does not bounce off the ground or slide on
the ground. These constraints are applied using the extended Kalman filter update equations. These updates are applied to the filter states at a rate of IMUSampleRate/DecimationFactor Hz.
gndFusion = insfilterNonholonomic('ReferenceFrame', 'ENU', ...
'IMUSampleRate', imuFs, ...
'ReferenceLocation', localOrigin, ...
'DecimationFactor', 2);
Create Ground Vehicle Trajectory
The waypointTrajectory object calculates pose based on specified sampling rate, waypoints, times of arrival, and orientation. Specify the parameters of a circular trajectory for the ground vehicle.
% Trajectory parameters
r = 8.42; % (m)
speed = 2.50; % (m/s)
center = [0, 0]; % (m)
initialYaw = 90; % (degrees)
numRevs = 2;
% Define angles theta and corresponding times of arrival t.
revTime = 2*pi*r / speed;
theta = (0:pi/2:2*pi*numRevs).';
t = linspace(0, revTime*numRevs, numel(theta)).';
% Define position.
x = r .* cos(theta) + center(1);
y = r .* sin(theta) + center(2);
z = zeros(size(x));
position = [x, y, z];
% Define orientation.
yaw = theta + deg2rad(initialYaw);
yaw = mod(yaw, 2*pi);
pitch = zeros(size(yaw));
roll = zeros(size(yaw));
orientation = quaternion([yaw, pitch, roll], 'euler', ...
'ZYX', 'frame');
% Generate trajectory.
groundTruth = waypointTrajectory('SampleRate', imuFs, ...
'Waypoints', position, ...
'TimeOfArrival', t, ...
'Orientation', orientation);
% Initialize the random number generator used to simulate sensor noise.
GPS Receiver
Set up the GPS at the specified sample rate and reference location. The other parameters control the nature of the noise in the output signal.
gps = gpsSensor('UpdateRate', gpsFs, 'ReferenceFrame', 'ENU');
gps.ReferenceLocation = localOrigin;
gps.DecayFactor = 0.5; % Random walk noise parameter
gps.HorizontalPositionAccuracy = 1.0;
gps.VerticalPositionAccuracy = 1.0;
gps.VelocityAccuracy = 0.1;
IMU Sensors
Typically, ground vehicles use a 6-axis IMU sensor for pose estimation. To model an IMU sensor, define an IMU sensor model containing an accelerometer and gyroscope. In a real-world application, the
two sensors could come from a single integrated circuit or separate ones. The property values set here are typical for low-cost MEMS sensors.
imu = imuSensor('accel-gyro', ...
'ReferenceFrame', 'ENU', 'SampleRate', imuFs);
% Accelerometer
imu.Accelerometer.MeasurementRange = 19.6133;
imu.Accelerometer.Resolution = 0.0023928;
imu.Accelerometer.NoiseDensity = 0.0012356;
% Gyroscope
imu.Gyroscope.MeasurementRange = deg2rad(250);
imu.Gyroscope.Resolution = deg2rad(0.0625);
imu.Gyroscope.NoiseDensity = deg2rad(0.025);
Initialize the States of the insfilterNonholonomic
The states are:
States Units Index
Orientation (quaternion parts) 1:4
Gyroscope Bias (XYZ) rad/s 5:7
Position (NED) m 8:10
Velocity (NED) m/s 11:13
Accelerometer Bias (XYZ) m/s^2 14:16
Ground truth is used to help initialize the filter states, so the filter converges to good answers quickly.
% Get the initial ground truth pose from the first sample of the trajectory
% and release the ground truth trajectory to ensure the first sample is not
% skipped during simulation.
[initialPos, initialAtt, initialVel] = groundTruth();
% Initialize the states of the filter
gndFusion.State(1:4) = compact(initialAtt).';
gndFusion.State(5:7) = imu.Gyroscope.ConstantBias;
gndFusion.State(8:10) = initialPos.';
gndFusion.State(11:13) = initialVel.';
gndFusion.State(14:16) = imu.Accelerometer.ConstantBias;
Initialize the Variances of the insfilterNonholonomic
The measurement noises describe how much noise is corrupting the GPS reading based on the gpsSensor parameters and how much uncertainty is in the vehicle dynamic model.
The process noises describe how well the filter equations describe the state evolution. Process noises are determined empirically using parameter sweeping to jointly optimize position and orientation
estimates from the filter.
% Measurement noises
Rvel = gps.VelocityAccuracy.^2;
Rpos = gps.HorizontalPositionAccuracy.^2;
% The dynamic model of the ground vehicle for this filter assumes there is
% no side slip or skid during movement. This means that the velocity is
% constrained to only the forward body axis. The other two velocity axis
% readings are corrected with a zero measurement weighted by the
% |ZeroVelocityConstraintNoise| parameter.
gndFusion.ZeroVelocityConstraintNoise = 1e-2;
% Process noises
gndFusion.GyroscopeNoise = 4e-6;
gndFusion.GyroscopeBiasNoise = 4e-14;
gndFusion.AccelerometerNoise = 4.8e-2;
gndFusion.AccelerometerBiasNoise = 4e-14;
% Initial error covariance
gndFusion.StateCovariance = 1e-9*eye(16);
Initialize Scopes
The HelperScrollingPlotter scope enables plotting of variables over time. It is used here to track errors in pose. The HelperPoseViewer scope allows 3-D visualization of the filter estimate and
ground truth pose. The scopes can slow the simulation. To disable a scope, set the corresponding logical variable to false.
useErrScope = true; % Turn on the streaming error plot
usePoseView = true; % Turn on the 3D pose viewer
if useErrScope
errscope = HelperScrollingPlotter( ...
'NumInputs', 4, ...
'TimeSpan', 10, ...
'SampleRate', imuFs, ...
'YLabel', {'degrees', ...
'meters', ...
'meters', ...
'meters'}, ...
'Title', {'Quaternion Distance', ...
'Position X Error', ...
'Position Y Error', ...
'Position Z Error'}, ...
'YLimits', ...
[-1, 1
-1, 1
-1, 1
-1, 1]);
if usePoseView
viewer = HelperPoseViewer( ...
'XPositionLimits', [-15, 15], ...
'YPositionLimits', [-15, 15], ...
'ZPositionLimits', [-5, 5], ...
'ReferenceFrame', 'ENU');
Simulation Loop
The main simulation loop is a while loop with a nested for loop. The while loop executes at the gpsFs, which is the GPS measurement rate. The nested for loop executes at the imuFs, which is the IMU
sample rate. The scopes are updated at the IMU sample rate.
totalSimTime = 30; % seconds
% Log data for final metric computation.
numGPSSamples = floor(min(t(end), totalSimTime) * gpsFs);
numSamples = numGPSSamples*imuSamplesPerGPS;
truePosition = zeros(numSamples,3);
trueOrientation = quaternion.zeros(numSamples,1);
estPosition = zeros(numSamples,3);
estOrientation = quaternion.zeros(numSamples,1);
idx = 0;
for sampleIdx = 1:numGPSSamples
% Predict loop at IMU update frequency.
for i = 1:imuSamplesPerGPS
if ~isDone(groundTruth)
idx = idx + 1;
% Simulate the IMU data from the current pose.
[truePosition(idx,:), trueOrientation(idx,:), ...
trueVel, trueAcc, trueAngVel] = groundTruth();
[accelData, gyroData] = imu(trueAcc, trueAngVel, ...
% Use the predict method to estimate the filter state based
% on the accelData and gyroData arrays.
predict(gndFusion, accelData, gyroData);
% Log the estimated orientation and position.
[estPosition(idx,:), estOrientation(idx,:)] = pose(gndFusion);
% Compute the errors and plot.
if useErrScope
orientErr = rad2deg( ...
dist(estOrientation(idx,:), trueOrientation(idx,:)));
posErr = estPosition(idx,:) - truePosition(idx,:);
errscope(orientErr, posErr(1), posErr(2), posErr(3));
% Update the pose viewer.
if usePoseView
viewer(estPosition(idx,:), estOrientation(idx,:), ...
truePosition(idx,:), trueOrientation(idx,:));
if ~isDone(groundTruth)
% This next step happens at the GPS sample rate.
% Simulate the GPS output based on the current pose.
[lla, gpsVel] = gps(truePosition(idx,:), trueVel);
% Update the filter states based on the GPS data.
fusegps(gndFusion, lla, Rpos, gpsVel, Rvel);
Error Metric Computation
Position and orientation were logged throughout the simulation. Now compute an end-to-end root mean squared error for both position and orientation.
posd = estPosition - truePosition;
% For orientation, quaternion distance is a much better alternative to
% subtracting Euler angles, which have discontinuities. The quaternion
% distance can be computed with the |dist| function, which gives the
% angular difference in orientation in radians. Convert to degrees for
% display in the command window.
quatd = rad2deg(dist(estOrientation, trueOrientation));
% Display RMS errors in the command window.
fprintf('\n\nEnd-to-End Simulation Position RMS Error\n');
End-to-End Simulation Position RMS Error
msep = sqrt(mean(posd.^2));
fprintf('\tX: %.2f , Y: %.2f, Z: %.2f (meters)\n\n', msep(1), ...
msep(2), msep(3));
X: 1.16 , Y: 0.98, Z: 0.03 (meters)
fprintf('End-to-End Quaternion Distance RMS Error (degrees) \n');
End-to-End Quaternion Distance RMS Error (degrees)
fprintf('\t%.2f (degrees)\n\n', sqrt(mean(quatd.^2)));
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Review Objectives for Physical Chemistry
Physical Chemistry Review Topics
Both CHEM 323 and CHEM 325
•S.I. units and dimensional analysis (“Appendix 1” in Atkins and de Paula Physical Chemistry 8^th Ed.)
•Be able to take derivatives (ordinary and partial) and integrals of simple polynomials, trigonometric functions (e. g., sin and cos), and exponential functions. In CHEM 323 there’s more emphasis
on derivatives and differentials. In CHEM 325 there’s more emphasis on integrals, series and matrix manipulations. Click here for a brief calculus review, or see “Appendix 2” in Atkins and de
Paula Physical Chemistry 8^th Ed. Some important things to know are
>>Euler chain rule
>>Product rule for derivatives and/or differentials
• Preparing graphs of data with spreadsheets and by hand.
Both CHEM 324 and CHEM 326
• Maintenance of laboratory notebooks
•Standard laboratory techniques including
>Proper use of volumetric glassware
>Use and care of spectrometers and analytical balances
>Solution preparation
•Proper chemical waste disposal
•Preparing graphs of data with spreadsheets and by hand
•Propagation of error analysis
•Writing skills are assumed to be carried from writing courses (e.g., grammar, spelling, logical flow)
Topics for CHEM 323
•Basic stoichiometry
•Nomenclature of simple organics, binary salts and complex ions
>Enthalpy, Hess’s Law and calorimetry
>Bond dissociation energies
>Entropy, Gibb’s energy
>Basic concepts and calculations
>Applications to
‡Acid-base reactions
‡Precipitation reactions
‡Coordination Chemistry
•Molecular basis for properties of matter
>Intermolecular interactions
>Gas Laws
>Phase changes (calorimetry of and source of)
>Phase diagrams (pure substances and binary mixtures)
>Colligative properties
Topics for CHEM 325
•Classical Mechanics (“Appendix 3” in Atkins and de Paula Physical Chemistry 8^th Ed.)
•Model of atom
>Historical development of quantum mechanics
‡Photoelectric effect
‡de Broglie
>Quantum mechanical model of the hydrogen atom
>Elemental properties: trends and explanations
•Bonding models
>Lewis structures
>VSEPR (predict shape and polarity using)
>Valence Bond Theory
>MO Theory
•Solid state and Materials
>Unit cell definition
>Types of unit cells
>Bragg equation
>Born-Haber cycle (lattice energy)
•Nomenclature of simple organics, binary salts and complex ions
•IR and NMR spectroscopy
Return to the Main Review Page
Last Update: August 12, 2008
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Alessandra Capanna reviews Le Corbusier for the Nexus Network Journal vol 2 no 1 January 2000
Willy Boesiger and Hans Girsberger . Le Corbusier 1910-65 (Basel: Birkhauser, 1999). To order this book, click here!
Reviewed by Alessandra Capanna
ne can conceive no better beginning to sketch the creative genius of Le Corbusier than the following sentences take from the laudatio accompanying the honorary doctorate degree from the faculty of
jurisprudence of Cambridge University in June 1959:
...He holds philosophic views on his art: he believes with Pythagoras that number, and with Plato that geometry, underlies the harmony of the universe and the beauty of objects, and with Cicero that
utility is the mother of dignity. He is also akin to Leonardo, in that he observes the principles of the engineer while applying to them the eye of a painter and sculptor, and for those who are
seeking the famous 'Divine Proportion', has proposed the standard he called 'Modulor', based on the stature of man, or to be exact, a six-foot Englishman...
The recent republication of this book by Birkhauser, the only volume of Le Corbusier's complete works, is an excellent vehicle for the divulgation and understanding of the mathematical spirit that
runs through Le Corbusier's "patient research", more evident in some instances, more hidden in others.
Most readers will already be familiar with the master from La Chaux-de-Fonds' study on the Modulor. Some pages in this book are dedicated to this "range of dimensions which makes the bad difficult
and the good easy" (Albert Einstein). To complete the chapter, these pages are followed by other that examine the pictorial work, the sculptures and the woven wall hangings created by Le Corbusier.
These are intended only to complete his architectural realizations, but more generally because his creative genius sank its roots in the ancient union of art and mathematics. This is an ideal and
practical correspondence that is completely synthesized in the rude outlines in low-relief of the human figure that was engraved in the concrete of the Unités d'habitation.
The present volume follows in large measure the impagination of the eight-volume edition, but the chronological catalog is subordinated to a subdivision in chapters that single out the great
Corbusian themes: private homes, large-scale constructions, museums, sacred architecture and urban design. It should be considered, in any case, a basic text for the educational formation of
architects. In terms of didactics, the study of Corbusier's works is an effective justification for the insistence on method in the practice of composition, even though this is somewhat less
cultivated today. For those who are interested in the mathematic structures and in the geometric architectonics, this is a useful basic tool that above all makes it possible to compare the theory
with the application. Beyond serving to verify the congruence of the method right through to the construction of a project, this is an exercise that allows us to understand that the objective of Le
Corbusier in the identification and use of harmonic proportions was to show that he was conscious that an insistence on the initiatory character, on the magico-ritual aspect, of the golden number did
not seem to be coherent with the scientific aspect of it, that which permitted the elaboration of a geometric grid in order to establish dimensional norms for each prefabricated habitable unit.
This book is also a useful instrument for approaching even those projects that are less well-known, but are equally rich in the correct scientific applications of complex geometries, in large part
conceptually derived from the studies undertaken for the Modulor. However, from this point of departure are derived a series of developments that are conceptually engaging not only as ulterior
evolutions of mathematically structured compositions, but also from a typological point of view, as in the spiral and hyperbolic geometry.
The spiral is the figurative matrix for the museum based on continuous growth, and even if the first design ideas relative to this way of organizing a course of exhibit spaces can be traced to a
period that precedes the studies for the Modulor, the intimate relationship between the laws that govern the growth of the logarithmic spiral, to which the form of the museum refers, and the value of
the golden number, is well-known. The evolution of this compostional theme is examined in the beginning of the chapter on museums, where the first projects may be analyzed in an uninterrupted
sequence, up to the realization of Ahmedabad and Tokyo.
Hyperbolic geometry, which in Le Corbusier's work has been studied in relation to those works defined as "sonorous architecture", or rather as the response to technical-functional demands, led to the
realization of volumes with original aesthetic characteristics. Because, as Le Corbusier affirms in the chapter dedicated to the mathematics in his book, The Modulor:
Mathematics is the majestic edifice imagined by man in order to comprehend the universe. There one encounters the absolute and infinite, what may be grasped and what may not be grasped. There
walls are erected in front of which one may pass and repass without results; sometimes a door is found; one opens it and enters, and is in other places, there where the gods are found, there
where lie the keys of the great systems. These door are those of miracles. Passing through one of these doors, the operating force is no longer man, but the contact with the universe. And before
him occur and develop the fabulous series of limitless combinations. He is in the land of numbers.
Great Buildings Online: Le Corbusier
Walk-through tour of Notre Dame du Haut, Ronchamp
Skewarch.com: Le Corbusier
Le Corbusier Database
Hermann Kühn, Le Corbusier 1887 - 1965 Der Modulor (in German, with good links)
Hermann Kühn, Le Corbusier, The Architect and His Works
Alessandra Capanna is an Italian architect living and working in Rome. She has taken her degree in Architecture at University of Rome 'La Sapienza', from which she also received her PHD, discussing a
thesis entitled "Strutture Matematiche della Composizione", concerning the logical paradigms in music and in architecture. She is the author of Le Corbusier. Padiglione Philips, Bruxelles, on the
correspondance between hyperbolic paraboloid geometry and technical and acoustic needs, and its final and aesthetics consequences. She has published articles on mathematical principles both in music
and in architecture such as "Una struttura matematica della composizione", remarking the idea of self-similarity in composition; "Musica e Architettura. Tra ispirazione e metodo", about three
architectures by Steven Holl, Peter Cook and Daniel Libeskind; and "Iannis Xenakis. Combinazioni compositive senza limiti", taken from a lecture given at 'Dipartimento di Progettazione Architettonica
e Urbana', University of
Rome. She is presenting a paper on Le Corbusier's Philips Pavillion at Nexus 2000.
Copyright ©2000 Kim Williams Books
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Linear Inequalities | Brilliant Math & Science Wiki
An inequality, as the name suggests, is a relationship between two quantities that are unequal.
One property of real numbers is that they have order. This order allows us to compare numbers and decide if they are equal to each other or one is greater or less than the other.
It is easiest to understand inequalities in the context of a number line (see above). This shows us that the numbers are ordered in a particular way. Those to the left are "less than" those to the
To show the inequality of numbers we use a symbolic notation:
• Less than: The \(<\) sign stands for "less than." So, \( 4 < 10 \) is true. Further, \( x<10 \) means \(x\) can be any number less than \(10.\)
• Greater than: The \(>\) sign stands for "greater than". So, \( 11 > 10 \) is true. Further, \( x>11 \) means \(x\) can be any number greater than \(11.\)
• Or equal to: Sometimes we want to show an inequality that is not strictly greater or less than. We use the same symbol, but with an underline, to show that the number might also be equal to the
value we are comparing to. So \( 4 \geq 4 \) is true, and so is \( 5 \geq 4 \). So, \( x \leq 3 \) means that \( x \) can be any number less than or equal to \(3.\)
How many positive integers are there that are \(<7?\)
Looking at the above number line, we see that the numbers that are less than \(7\) and positive are found in between \(0\) and \(7\).
So the numbers are \(1, 2, 3, 4, 5\) and \(6\), which make up a total of \(6\). \(_\square\)
What inequality operation should be used in place of the question mark below?
\[61\ ?\ 59\]
If we look at the extended number line, we would see that the number \(61\) is two steps ahead of \(59\). It means both "\(61\) is greater than \(59\)" and "\(59\) is less than \(61.\)"
We saw the appropriate sign to show one is greater than the other is "\(>\)".
So the answer is \(61>59\), which is true. \(_\square\)
How many positive integers are there that are less than \(100\) and are multiples of \(11?\)
We list all the multiples of \(11\) that are found between \(0\) and \(100\):
\[ 11\times1&=11\\ 11\times2&=22\\ 11\times 3&=33\\ 11\times 4& =44\\ 11\times 5& = 55\\ 11\times 6& = 66\\ 11\times 7 &=77\\ 11\times 8 &= 88\\ 11\times 9&=99. \]
Since \(11\times 10 =110\) and \(110\) is ten steps ahead of \(100\), we omit it and end there.
So the numbers are \(11, 22, 33, 44, 55, 66, 77, 88, 99\), giving a total of \(9\) such numbers. \(_\square\)
How many integers \(x\) satisfy \(13<x<21?\)
If we split the above inequality, we get two inequalities \(13<x\) and \(x<21\).
The first inequality \(13<x\) means \(13\) is less than \(x\), which also means \(x\) is greater than \(13\). So the numbers are found to the right of \(13\) on the number line.
The second inequality \(x<21\) means \(x\) is less than \(21\). So the numbers are found to the left of \(21\) on the number line.
Combining the two inequalities, we get a list of numbers that are to the right of \(13\) and a list of numbers that are to the left of \(21\), which means the numbers located between \(13\) and \
The numbers are \(14, 15, 16, 17 ,18 , 19, 20\), which gives a total of \(7\) numbers. \(_\square\)
Intervals on the Number Line
In the wiki Representation on the Real Line, we saw that real numbers can be visually represented on a number line, with each point on the line corresponding to a real number and each real number
corresponding to a point on the line. How can we use the number line to represent sets of real numbers?
When we draw regions of the number line, we use the following conventions:
• Solid dots and colored lines denote points that are included in the region.
• Open (unfilled) dots denote points that are not included in the region.
Let's consider an example.
Find an inequality representing the above region in the number line.
Since the circle on the real number \(-3\) is an open circle, \(-3\) is not included in the region. This shows the region includes all real numbers strictly greater than \(-3\), i.e., all real
numbers \(x\) satisfying the inequality \(x > -3\). \(_\square\)
Now, we will consider the same region of the number line and demonstrate that there may be more than one possible inequality to represent the same region.
Which of the following inequalities is represented in the above number line?
\[\begin{array} &&~(a)\ x+8\le 2x &&~(b)\ 2x-3>x+4 &&~(c)\ -x+3<4x+18 &&~(d)\ 3x+9\le -2x-1\end{array}\]
The solutions to the above inequalities are
\[\begin{array} &&(a)\ x+8\le 2x &\implies x\ge 8 \\ \\ &(b)\ 2x-3>x+4 &\implies x>7 \\ \\ &(c)\ -x+3<4x+18 &\implies x>-3\\ \\ &(d)\ 3x+9\le -2x-1 &\implies x\le -2. \end{array}\]
Therefore, the answer is \((c).\) \( _\square \)
This example demonstrates that it may be necessary to perform algebraic manipulation to an inequality before being able to determine if the inequality represents a given region in the number line.
Possible algebraic manipulations include
• add or subtract a constant to both sides;
• multiply both sides by a positive number;
• multiply both sides by a negative number and switch the inequality sign.
Which of the following number lines represents the inequality
\[\\ 2x-11\le -(x+2)?\]
We have
\[ 2x-11 &\le -(x+2) \\ 2x-11 &\le -x-2 \\ 3x &\le 9\\ x &\le 3. \]
Therefore, the answer is \((b).\) \( _\square \)
Which of the following inequalities is represented in the above number line?
\[\begin{array} &&~(a)\ \frac{x}{100}<\frac{x}{5}+\frac{19}{50} &&~(b)\ \frac{1}{5}-\frac{2x}{5}>\frac{x}{5}-1 &&~(c)\ \frac{x-1}{2}<\frac{x+3}{4} &&~(d)\ \frac{3x+1}{4}<\frac{x}{3}-1\end{array}
The solutions to the above inequalities are
\[\begin{array} &&(a)\ \frac{x}{100}<\frac{x}{5}+\frac{19}{50} &\implies \frac{19x}{100}>-\frac{38}{100} &\implies x>-2 \\ \\ &(b)\ \frac{1}{5}-\frac{2x}{5}>\frac{x}{5}-1 &\implies \frac{3x}{5}<\
frac{6}{5} &\implies x<2 \\ \\ &(c)\ \frac{x-1}{2}<\frac{x+3}{4} &\implies \frac{2x-2}{4}<\frac{x+3}{4} &\implies x<5\\ \\ &(d)\ \frac{3x+1}{4}<\frac{x}{3}-1 &\implies \frac{9x+3}{12}<\frac
{4x-12}{12} &\implies x<-3 . \end{array}\]
Therefore, the answer is \((c).\) \( _\square \)
Which of the above number lines represents the solution of the following system of linear inequalities?
\[\begin{cases} 2x-1 &\le &x+5\\x+1&>&0\end{cases}\]
The first inequality gives \(x\le 6.\) The second inequality gives \(x>-1.\) Thus, the solution is \(-1<x\le 6,\) implying that the appropriate representation of the number line is \((a).\) \(_\
Finding Solutions to Linear Inequalities
Linear inequalities are relationships that hold true between two different components. They are usually composed of a \(<, >, \leq,\) or \(\geq\) symbol.
The notation \(x < y\) means that \(x\) is less than \(y\).
The notation \(x \leq y\) means that \(x\) is less or equal to \(y\).
The notation \(x>y\) means that \(x\) is greater than \(y\).
The notation \(x \geq y\) means that \(x\) is greater or equal to \(y\).
Linear inequalities with one variable can be solved by algebraically manipulating the inequality so that the variable remains on one side and the numerical values on the other. Once this is done, we
obtain a relationship that expresses the solution of the inequality.
Linear inequalities can also be solved by graphing and thinking of them visually.
What is the smallest integer that satisfies \(x > 4?\)
Looking at the equality, we observe that it states all real numbers greater than four. But since we are looking for the smallest integer value, we notice that the smallest integer greater than \
(4\) is five. \(_\square\)
What is the smallest integer that satisfies \(x \geq 4?\)
Looking at the equality, we observe that it states all real numbers greater than or equal to four. Since these numbers are basically all numbers starting from \(4\) to \(\infty\), including \(4
\), we know that \(4\) is the smallest integer that satisfies the inequality. \(_\square\)
Use algebra to find the values of \(x\) for which \(6x + 4 < 2 + x.\)
We first gather like terms by adding \( -x -4 \) to both sides of the inequality and then divide both sides by \(5\) to obtain
\[ -x - 4 + 6x + 4 &< 2 + x - x - 4\\\\ 5x &< -2\\\\ x &< \frac{-2}{5} . \]
This means that all values of \(x\) less than \(\frac{-2}{5}\) are solutions to the relationship above.
In short, we can write the solution as \(\left(-\infty,\frac{-2}{5}\right),\) which means all real numbers between but not including negative infinity and \(\frac{-2}{5}\). This is essentially a
shorter way of rewriting the statement above. \(_\square\)
Show graphically that the solution obtained for the above example is true.
From our previous equation we know that \(6x + 4 < 2 + x\) reduces to \(x < -\frac{2}{5} \).
On this graph, we first plotted the line \(x = -\frac{2}{5},\) and then shaded the entire region to the left of the line. The shaded area is called the bounded region, and any point within this
region satisfies the inequality \(x < -\frac{2}{5}.\) Notice also that the line representing the region's boundary is a dashed line; this means that values along the line \(x = -\frac{2}{5}\) are
not included in the solution set of the inequality. \(_\square\)
One-step Linear Inequalities
Solving linear inequalities in one variable is the same as solving linear equations. Given an inequality, what we should do is to isolate the variable on one side.
For example, if we are given the inequality
leave the steps involved for now:
\[ x-3+3&>9+3\\ x+0&>12\\ x&>12. \]
Both \(x-3>9\) and \(x>12\) have the same solution because performing the basic arithmetic operations \(+,-,\times,\div\) to both sides of the inequality doesn't change the inequality. This is known
as an equivalent inequality. Thus all numbers greater than \(12\) satisfy the above inequality.
We have seen that transforming inequalities into an equivalent inequality will lead us to the solution. To change an inequality into an equivalent inequality, we use the four basic arithmetic
operations, but what operation to use is entirely dependent on the type of question, so we must keep our eyes sharp to identify what operation to use.
Addition to form equivalence inequality
When using addition, we rely on the fact that any number plus zero is the number itself \((x+0=x)\). So, for example, if the inequality is
to create an equivalence inequality, we have to isolate \(x\). We can do that by making the left term equal to \(x+0,\) which is basically the number itself. So we can add \(+8\) to make it zero:
\[ x-8+8&<1\\ x-0&<1\\ x&<1. \]
OK, we have isolated \( x\), but we have altered the inequality by simply adding \(8\). To correct this, we can add another \(8\) to the right side of the inequality to counter the change we made on
the left side:
\[ x-8+8&<1\\ x-0&<1+8\\ x&<9. \]
So, all numbers less than \(9\) satisfy the inequality.
Subtraction to from equivalence inequality
Suppose we are given the inequality \(x+3>7.\)
If we add \(3\) to both sides, \(x+3+3>7+3\) or \(x+6>10,\) which is no more helpful than the original. But if we subtract \(3\) from both sides, we can isolate \(x\):
\[ x+3&<7\\ x+3-3&<7-3\\ x&<4. \]
So, all numbers less than \(4\) satisfy the inequality.
Multiplication to form equivalence inequality
When using multiplication, we rely on the property that any number multiplied by \(1\) except for \(0\) is the number itself \((m\times1=m, m\neq0)\).
Given the inequality
\[ \frac { 1 }{ 2 }x <7,\]
let's see how we can isolate \(x\). We can see that if we multiply \(\frac{1}{2}\) by \(2\), we get one. So we have
\[ \left( \frac { 1 }{ 2 } x \right) \times 2 &<7\times 2\\ 1\times x&<14\\ x&<14. \]
Another thing not to forget is that when multiplying both sides of the inequality by a negative number, the inequality sign changes. For example, in the inequality \(-\frac{1}{3}x<5\), to isolate \(x
\) we multiply both sides of the inequality by \(-3\):
\[ -\left( \frac { 1 }{ 3 } x \right) \times (-3)&<5\times (-3)\\ 1\times x&>-15\\ x&>-15. \]
So, all numbers greater than \(-15\) satisfy the inequality.
Division to form equivalence inequality
Division relies on the same principle as multiplication since division is the reciprocal of multiplication: \(m\div n= m \times \frac{1}{n}\).
Consider the inequality \(6x<12\).
If we wanted to isolate \(x\), we can divide the left term by \(6\). Since any number divided by itself except for \(0\) is equal to \(1\), it follows that \(\frac{6}{6}=1\), implying
\[ 6x&<12\\ 6x\div 6&<12\div 6\\ 6x\times \frac { 1 }{ 6 } &<12\times \frac { 1 }{ 6 } \\ 1\times x&<2\\ x&<2. \]
So, all numbers less than \(2\) satisfy the inequality.
Just like what we did with multiplication, dividing both sides of an inequality with a negative number changes the sign of the inequality.
Let's look at some more difficult questions to strengthen our understanding.
Solve \(6x-3<9\).
This looks like we have to do both addition and division to isolate \(x\).
Step 1:
\[ 6x-3&<9\\ 6x-3+3&<9+3\\ 6x&<12. \]
Step 2:
\[ 6x\times \frac { 1 }{ 6 } &<12\times \frac { 1 }{ 6 } \\ x&<2. \ _\square \]
Note: We didn't necessarily have to do the addition process before the division process, but it is usually easier to do addition and subtraction process before multiplication and division
Given that \(x\) is an element of the set \( \left\{ -2,-1,-7,-4,-5,0 \right\} \), how many values of \(x\) satisfy the inequality \(-6x+13<37?\)
Instead of plugging in all the numbers and checking, let's isolate \(x\) to form equivalence inequality.
Step 1: Remove \(13 \) from the left-hand side:
\[ -6x+13-13&<37-13\\ -6x&<24. \]
Step 2: To remove \(-6\), divide both sides by \(-6\) and change the sign of the inequality:
\[ (-6x)\times \left( -\frac { 1 }{ 6 } \right) &<24\times \left( -\frac { 1 }{ 6 } \right) \\ x&>-4. \]
Since \(x\) is greater than \(-4\), the numbers \(-2,-1,0\) satisfy the inequality. Thus, a total \(3\) values satisfy the inequality. \(_\square\)
Solve \(\frac { 2 }{ 3 } x-\frac { 1 }{ 2 } >\frac { 3 }{ 2 }.\)
Step 1: Remove \(\frac{1}{2} \) from the left-hand side:
\[ \frac { 2 }{ 3 } x-\frac { 1 }{ 2 } &>\frac { 3 }{ 2 } \\ \frac { 2 }{ 3 } x-\frac { 1 }{ 2 } +\frac { 1 }{ 2 } &>\frac { 3 }{ 2 } +\frac { 1 }{ 2 } \\ \frac { 2 }{ 3 } x&>2. \]
Step 2: Here we could remove the \(2\) and \(3\) in \(\frac{2}{3}\) separately by dividing by \(2\) and multiplying by \(3\). But if we realize that the reciprocal of a fraction is \(1\), i.e. \
(\frac{m}{n}\times \frac{n}{m}=1\), we can multiply it by its reciprocal:
\[ \left( \frac { 2 }{ 3 } x\times \frac { 3 }{ 2 } \right) &>2 \times \left( \frac { 3 }{ 2 } \right) \\ 1\times x&>3\\ x&>3. \]
Thus, all numbers greater than \(3\) satisfy the inequality. \(_\square\)
Two-sided Linear Inequalities
Solve \(2x+3<x<3x+16.\)
The first inequality gives
\[ 2x+3 &<x \\ x&<-3. \qquad (1) \]
The second inequality gives
\[ x &<3x+16 \\ -16 &<2x \\ -8 &<x. \qquad (2) \]
Combining \((1)\) and \((2)\) gives \(-8<x<-3. \ _\square \)
Solve \(3x-8<2x<5x-33.\)
The first inequality gives
\[ 3x-8 &<2x \\ x&<8. \qquad (1) \]
The second inequality gives
\[ 2x &<5x-33 \\ 33 &<3x \\ 11 &<x. \qquad (2) \]
Since there is no value of \(x\) that satisfies both \((1)\) and \((2),\) there is no solution. \(_\square \)
If the solution to the following inequalities is all the non-negative numbers, what is the value of \(a?\)
\[x-9 \le 3x+1 \le 7x-a\]
The first inequality gives
\[ x-9 &\le 3x+1 \\ -10 &\le 2x \\ x &\ge -5. \qquad (1) \]
The second inequality gives
\[ 3x+1 &\le 7x-a \\ a+1 &\le 4x \\ x &\ge \frac{a+1}{4}. \qquad (2) \]
As the result of combining \((1)\) and \((2),\) we should have the required solution to this problem: \(x\ge 0. \qquad (3)\)
Since \((1)\) contains \((3),\) it must be true that \((2)\) is equivalent to \((3):\)
\[ x \ge \frac{a+1}{4} \Leftrightarrow x\ge 0.\]
This implies
\[a+1=0 \implies a=-1. \ _\square\]
Multi-step Linear Inequalities
To solve inequalities involving the expression \(mx+b\), we need to consider the properties of inequalities.
Properties of Inequalities:
1. The sense of inequality is unchanged if the same real number is added to both sides.
2. The sense of inequality is unchanged if both sides are multiplied by the same positive real number.
3. The sense of inequality is reversed if both sides are multiplied by the same negative real number.
4. If \(a>b\) and \(c>d ,\) then \(a+c> b+d.\)
5. \(a>b>0\) and \(c>d>0,\) then \(ac>bd.\)
Solve the solution set of \(3-4x \leq 2x +9.\)
By Property \(1\) above, \(-6x \leq 6.\)
Then by Property \(3\) above, \(x \geq -1. \ _\square\)
Linear Inequalities - Multi-step - Intermediate
Linear Inequalities - Problem Solving
When solving a problem (as opposed to merely using a given formula), it is generally good to proceed along the following lines:
1. First, you have to understand the problem.
2. After understanding, make a plan.
3. Carry out the plan.
4. Look back on your work. How could it be better?
For more details on each of these, see How to Solve Problems on Brilliant.
Prove that for all positive reals \(x\), the following inequality is always true:
\[ x + \frac1x \geq 2. \]
Because the square of a real number is always non-zero, we have \(\Big(\sqrt x - \frac1{\sqrt x}\Big)^2 \geq 0.\) Perform some algebraic manipulation, and you will get the desired inequality. \(_
First Second They are equal Not enough information
If \(A\) is not equal to \(B\), which of the following is greater?
\[\text{ First: } \frac{ A } {A-B} \quad \quad \text{ Second: } \frac{ B}{A-B} \]
Let \(a,b,c\) be positive reals. Also, let \(k\) be the largest possible real such that
\[\dfrac{a}{1}+\dfrac{b}{1}+\dfrac{c}{1}+\dfrac{a+b}{1}+\dfrac{b+c}{1}+\dfrac{c+a}{1}\le \dfrac{a+b+c}{k}.\]
If \(k\) can be expressed as \(\frac{p}{q}\) for relatively prime positive integers \(p\) and \(q\), then what is \(p+q?\)
• Inequality of Denominators: the harder version of this inequality
Linear Inequalities Word Problems
If that doughnut weighs heavier than the flip-flop by 100 grams, but that flip-flop is lighter than the dry cement by 1000 kg. Which is heavier, the doughnut or the dry cement? And by how much?
To be continued
Triangle > Rectangle > Circle Circle > Rectangle > Triangle Rectangle > Triangle > Circle Circle > Triangle > Rectangle Rectangle > Circle > Triangle Triangle > Circle > Rectangle
The diagram shows how a mobile will be balanced when left to hang.
What are the relative weights of these shapes?
Rectangle > Triangle > Circle Circle > Triangle > Rectangle Circle > Rectangle > Triangle Triangle > Circle > Rectangle Triangle > Rectangle > Circle Rectangle > Circle > Triangle
The diagram shows how a mobile will be balanced when left to hang, and the rods are all tilted to the maximum degree.
Assuming that the fulcrum is at the center of each rod, what are the relative weights of these shapes?
Linear Inequalities - Word Problems - Intermediate
A classroom has \(n\times n\) number of tables. If the teacher removes the least number of squares such that there is still a perfect square number of tables, then there will be 4 students who
don't have a sit. How many extra tables are there if the teacher did not remove the tables?
\( (n-1)^2 = m + 4 \) or something. To be continued.
A school bus carries 40 students, of which 20 are boys and 20 are girls.
At the first stop, 2 boys and 3 girls exit the bus.
At the second stop, 11 students exit the bus.
What is the fewest number of boys that must exit to ensure that more girls than boys have exited the bus?
\[ 0.33 < \dfrac {m}{n} < \dfrac {1}{3}\]
Find the smallest positive integer \(n\) such that there exists an integer \(m\) for which the inequality above is fulfilled.
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Mastering 3D Plotting in Python: Essential Methods and Examples - Adventures in Machine Learning
Using Python for 3D Plotting
Using Python for 3D plotting is a powerful tool that can assist engineers, scientists, and data analysts to create 3D models of complex systems. This revolutionary method of creating 3D models is
comparatively easy and simple to learn.
In this article, we will cover the essential methods required for plotting in 3D and walk through an example of creating a Solenoid using .plot3D() in Python.
1. Essential Libraries for 3D Plotting
To plot in 3D, Python relies on certain libraries, such as NumPy, SciPy, and matplotlib. These libraries allow users to create and manipulate 3D plots of various types.
2. Understanding the Mechanics of 3D Plotting
To create a 3D plot of a system, one must first understand some of the fundamental methods and principles of plotting in 3D. NumPy’s linspace() method is an excellent way to create a linear array of
numbers within a defined range.
This method is a potent tool as it allows for finer quality control, especially when plotting 3D models. The mgrid method, which is a multidimensional version of linspace, returns an N-dimensional
meshgrid where one can evaluate an N-dimensional function at N points.
The resulting arrays can then be used to create 3D plots. Together with the numpy.linspace() function, they are invaluable tools for creating 3D plots of various types.
3. Plotting a 3D Model using .plot3D() method
Now that we understand the mechanics of creating 3D plots let us delve into a practical example of creating a Solenoid. A Solenoid is an electromechanical device whose primary function is to produce
electromagnetism when an electric current flows through it.
We will make use of Numpy and Matplotlib modules for this practical example:
# Importing required libraries
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
The next step is to create the meshgrid that will be used to plot the 3D coordinates of the solenoid. This involves setting the X, Y, and Z-axis boundaries, as shown below:
# Create meshgrid for 3D plot
x = np.linspace(-2, 2, 1000)
y = np.linspace(-2, 2, 1000)
z = np.linspace(-5, 5, 1000)
X, Y, Z = np.meshgrid(x, y, z)
After creating the meshgrid, we can move on to the next step, which involves creating the actual 3D Solenoid.
Here, we will make use of NumPy’s sin and cos mathematical functions to manipulate the values of X, Y, and Z to create the Solenoid’s 3D structure.
# Create 3D Solenoid using numpy sin and cos
R = 1
a = 1
b = 1
theta = np.linspace(0, 2*np.pi, 1000)
x = np.outer((R + a * np.cos(theta)), np.cos(2*np.pi*Z/b))
y = np.outer((R + a * np.cos(theta)), np.sin(2*np.pi*Z/b))
z = np.outer(a * np.sin(theta), np.ones_like(Z))
Finally, we can plot the Solenoid using the plot3D method provided by the mplot3d module.
This 3D plot is fully interactive, and one can pan, zoom, and rotate the image using mouse clicks.
# Plot the created Solenoid
fig = plt.figure('Solenoid')
ax = fig.add_subplot(111, projection='3d')
ax.plot_surface(x, y, z, color='c')
By executing this script in a Python environment, we can successfully create a 3D Solenoid plotted on the X, Y, and Z-axis.
In conclusion, creating 3D plots in Python can be complex, but with the right modules such as Matplotlib, Numpy, and mplot3d, it becomes a straightforward process. The key is to have an in-depth
understanding of the plotting methods and mathematical functions required to create different types of 3D plots.
In this article, we have walked through the two primary methods required for plotting in 3D, namely numpy.linspace() and numpy.mgrid. Additionally, we provided a step-by-step example of how to create
a 3D Solenoid in Python using Matplotlib, Numpy, and mplot3d.
4. Plotting a 3D Model using .scatter3D() method
The scatter3D() method in Python’s mplot3d module is used to plot 3D datasets as scattered points. This method is useful when one needs to visualize 3D data points in a broad range and is suitable
for creating simple 3D models.
Let us walk through an example of creating a Scattered Dotted Solenoid in a 3D Graph using Python:
# Importing required libraries
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
# Create meshgrid for 3D plot
x = np.linspace(-2, 2, 1000)
y = np.linspace(-2, 2, 1000)
z = np.linspace(-5, 5, 1000)
X, Y, Z = np.meshgrid(x, y, z)
# Create 3D Solenoid using numpy sin and cos
R = 1
a = 1
b = 1
theta = np.linspace(0, 2*np.pi, 1000)
x = np.outer((R + a * np.cos(theta)), np.cos(2*np.pi*Z/b))
y = np.outer((R + a * np.cos(theta)), np.sin(2*np.pi*Z/b))
z = np.outer(a * np.sin(theta), np.ones_like(Z))
# Create Scattered Dotted Solenoid
x_coords = x.ravel()
y_coords = y.ravel()
z_coords = z.ravel()
c = np.random.random(len(x_coords))
fig = plt.figure('Scattered Dotted Solenoid')
ax = fig.add_subplot(111, projection='3d')
ax.scatter3D(x_coords, y_coords, z_coords, c=c, cmap='hsv', s=1)
In the above code, we use the scatter3D() method to create a Scattered Dotted Solenoid using the x, y, and z coordinates we defined earlier. Additionally, we generate a random color array and assign
it to the “c” parameter in scatter3D() so that each point has a different color.
Finally, by adjusting the “s” parameter, we control the size of the dots, and we set the “cmap” parameter to ‘hsv’ to obtain a more vibrant look.
5. Plotting a Surface from a List of Tuples
Apart from meshgrid function, a list of tuples can also be used to create surfaces in Python. To plot the surface, one will need to pass the x, y, and z datapoints as separate lists of tuples.
Let us look at an example of plotting the surface for some tuples in Python:
# Importing required libraries
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
# Creating List of Tuples
data = [(1, 1, 2), (2, 1, 4), (3, 1, 7), (1, 2, 3), (2, 2, 5), (3, 2, 8), (1, 3, 4), (2, 3, 6), (3, 3, 9)]
# Unpacking Data to separate coordinate arrays
x, y, z = zip(*data)
# Reshaping coordinate arrays
x = np.asarray(x).reshape((3,3))
y = np.asarray(y).reshape((3,3))
z = np.asarray(z).reshape((3,3))
# Creating surface plot
fig = plt.figure('Surface Plot')
ax = fig.add_subplot(111, projection='3d')
ax.plot_surface(x, y, z)
In the above script, we use the zip() method to unpack our data list, containing tuples of (x, y, z), into separate lists. We then reshape these lists into 2D NumPy arrays that can be used for
plotting surfaces.
Finally, we plot the surface using the plot_surface() method provided by the mplot3d module.
6. Plotting a 3D Model using .plot_surface() method
Python’s mplot3d module provides an assortment of methods for plotting 3D models. The plot_surface() method is a powerful tool that can create 3D models of various shapes and sizes.
Here, we will explore how to use this method to plot points on the surface of a sphere. To generate points on the surface of a sphere, one must first define a set of spherical coordinates.
Typically, these coordinates are represented as a tuple with three values: the radius, inclination angle, and azimuth angle. The radius determines the size of the sphere, while the inclination and
azimuth angles are used to position the point on the sphere.
Using these spherical coordinates, we can define the location of the points that make up the surface of the sphere. Let us look at an example of how to plot points on the surface of a sphere in
# Importing required libraries
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
# Define Spherical Coordinates
theta = np.linspace(0, np.pi, 100)
phi = np.linspace(0, 2*np.pi, 100)
theta, phi = np.meshgrid(theta, phi)
# Convert Spherical to Cartesian Coordinates
x = np.sin(theta) * np.cos(phi)
y = np.sin(theta) * np.sin(phi)
z = np.cos(theta)
# Create Surface Plot
fig = plt.figure('Sphere Surface')
ax = fig.add_subplot(111, projection='3d')
ax.plot_surface(x, y, z)
In the above script, we define the spherical coordinates for the points on the surface of the sphere. We then use NumPy’s meshgrid() function to create a mesh of these points.
Next, we convert these spherical coordinates to Cartesian coordinates using the standard conversion formulas. Finally, we plot these points as a surface using the plot_surface() method provided by
the mplot3d module.
In summary, we have covered the essential methods required for plotting 3D models in Python. We initially discussed numpy.linespace() and numpy.mgrid, which are both valuable tools for creating
linear arrays of numbers for finer quality control in 3D plotting.
We then covered the plot3D() method, which is useful for visualizing simple 3D models. The scatter3D() method is a powerful tool for visualizing 3D datasets as scattered points.
Furthermore, we discussed plotting surfaces from a list of tuples and using the plot_surface() method to create surfaces of different shapes and sizes. In the article, we presented practical
examples, such as creating a solenoid and plotting points on the surface of a sphere.
These examples demonstrated how to use the methods covered in this article to create and visualize different 3D models. By understanding these methods and applying them to different use-cases, one
can create various types of 3D plots that can assist professionals in different fields to visualize their data and gain insights into complex systems.
In conclusion, Python’s 3D plotting libraries provide a wealth of tools that can be used to create and manipulate 3D models. By implementing the methods outlined in this article and using the
examples provided, users can develop the skills required to create 3D visualizations of their data in Python.
This skill set can be used in numerous applications, such as data analysis, scientific visualization, computer graphics, and more. In conclusion, this article explored the different methods required
for plotting 3D models in Python, such as linspace(), mgrid, plot3D(), scatter3D(), and plot_surface().
We have walked through several practical examples that demonstrated how to utilize these methods, including creating a Solenoid, a Scattered Dotted Solenoid, plotting a surface from a list of tuples,
and defining and plotting the points on the surface of a sphere. These examples highlight the flexibility and versatility of Python’s 3D plotting libraries, making it an essential toolset for
engineers, scientists, and data analysts.
The takeaways from this article are that by understanding the methods and applying them to different applications, one can create complex 3D models swiftly and efficiently in Python, offering
valuable insights into the underlying data and enhancing the user’s decision-making capabilities.
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Lagrange polynomial calculator
This online calculator builds Lagrange polynomial for a given set of points, shows a step-by-step solution and plots Lagrange polynomial as well as its basis polynomials on a chart. Also, it can
interpolate additional points, if given
This content is licensed under Creative Commons Attribution/Share-Alike License 3.0 (Unported). That means you may freely redistribute or modify this content under the same license conditions and
must attribute the original author by placing a hyperlink from your site to this work https://planetcalc.com/8692/. Also, please do not modify any references to the original work (if any) contained
in this content.
I wrote this calculator to be able to verify solutions for Lagrange's interpolation problems. In these problems you are often asked to interpolate the value of the unknown function corresponding to a
certain x value, using Lagrange's interpolation formula from the given set of data, that is, a set of points x, f(x).
The calculator below can assist with the following:
1. It finds the final Lagrange polynomial formula for a given data set.
2. It shows step-by-step formula derivation.
3. It interpolates the unknown function by computing the value of the Lagrange polynomial at the given x values (points of interpolation)
4. It plots the data set, interpolated points, Lagrange polynomial and its basis polynomials on the chart.
First, enter the data points, one point per line, in the form x f(x), separated by spaces. If you want to interpolate the function by the Lagrange polynomial, enter the points of interpolation into
the next field, just x values, separated by spaces.
By default, the calculator shows the final formula and interpolated points. If you want to see a step-by-step solution for the polynomial formula, turn on the "Show Step-By-Step Solution" option. The
chart at the bottom shows the Lagrange polynomial, as well as its basis polynomials. These can be turned off.
You can also find some theory about the Lagrange polynomial below the calculator.
Digits after the decimal point: 2
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Lagrange Polynomial
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Lagrange polynomial
Let's suppose we have a set of data points for the unknown function, where no two x are the same:
$(x_{0},y_{0}),\ldots ,(x_{j},y_{j}),\ldots ,(x_{k},y_{k})$
Let's construct the following polynomial (called the Lagrange polynomial):
$L(x):=\sum _{j=0}^{k}y_{j}\ell _{j}(x)$
where $\ell _{j}(x)$ is Lagrange basis polynomial
$\ell _{j}(x):=\prod _{\begin{smallmatrix}0\leq m\leq k\\meq j\end{smallmatrix}}{\frac {x-x_{m}}{x_{j}-x_{m}}}={\frac {(x-x_{0})}{(x_{j}-x_{0})}}\cdots {\frac {(x-x_{j-1})}{(x_{j}-x_{j-1})}}{\frac
{(x-x_{j+1})}{(x_{j}-x_{j+1})}}\cdots {\frac {(x-x_{k})}{(x_{j}-x_{k})}}$
If you look at the formula of the basis polynomial for any j, you can find that for all points i not equal to j the basis polynomial for j is zero, and in point j the basis polynomial for j is one.
That is,
$y_{j}\ell _{j}(x_{j})=y_{j} \cdot 1=y_{j}$
$L(x_{j})=y_{j}+0+0+\dots +0=y_{j}$
which means that the Lagrange polynomial interpolates the function exactly.
Note that Lagrange's interpolation formula is susceptible to Runge's phenomenon. It is a problem of oscillation at the edges of an interval when using polynomials of high degree over a set of
equidistant interpolation points. It is important to keep in mind, because it means that going to higher degrees (i.e. having more data points in the set) does not always improve the accuracy of the
However, also note that unlike some other interpolation formulas, Lagrange’s formula does not require that the values of x should be equidistant. It is used in some techniques to mitigate the
problem, like the change of interpolation points using Chebyshev nodes.
Similar calculators
PLANETCALC, Lagrange polynomial calculator
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Simulation of the controlled movement based on the complexity principle for an automatic underwater vehicle
The paper deals with the mathematical modeling of the controlled motion of an automatic underwater vehicle under conditions of inaccuracy and uncertainty of information support. Methodological and
theoretical approaches based on the application of the principle of complexity and fuzzy logic are proposed.
1. Introduction
One of the most effective and frequently used technical means for the development and study of the oceans are automatic underwater vehicles (AUV) [1]. Their main advantages are: the ability of
independent spatial maneuvering, the ability to simultaneously perform a wide range of underwater work, a high level of automation of standard work operations, mobility and autonomy. The solution of
the tasks is ensured by AUV with the help of various subsystems combined into a single control system. Within the framework of this system, the dynamic modes of AUV operation is supported by the
information control complex and a high-performance computing environment.
The motion control system is an important element of the information and control complex of the underwater vehicle. This system implements one purposeful spatial maneuvering, and also provides a
given mode of movement. During designing a motion control system, the accepted mathematical model of the AUV as an object of dynamics and control has the main influence on the achievement of control
objectives. Accounting for the properties of all elements of the information and control system allows us to conclude that the underwater vehicle is a “complex” control object [2]. Modeling of
complex objects is a time consuming and expensive process. In practice, simplified models are used. In the process of models simplifying, the inaccuracies and uncertainties of the information used to
create the information and control complex increase. As a result, the software for the underwater vehicle based on which control signals are generated contains incorrect information. Practical
implementation of the methods of accounting for “complexity” in the mathematical description of the underwater vehicle as a control object in the design of a control system will improve the quality
and accuracy of achieving of the control goal – implementing the required maneuvering and ensuring of its predetermined movement. To do this, it is rational to apply new approaches, formalisms and
methods of modern control theory, focused on the application of the complexity principle [3] and the concept of soft computing [4], including the theory of fuzzy sets, artificial neural networks and
genetic algorithms.
2. The principle of complexity in mathematical modeling of the motion of an automatic underwater vehicle
A variety of options for the practical use of various types of underwater vehicles is based on existing technical support for the process of achieving of the required system-wide operating goals.
Managed spatial movement of the AUV is implemented in accordance with the desired motion mode, during which the specified types of trajectories are realized.
The consequence of this is the presence of elements of information support of such non-factors as inaccuracy and uncertainty. The practical implementation of this approach to modeling the AUV shows
that the incompleteness of the mathematical description of the control object and supporting the controlled dynamic information support process affects the characteristics of the motion control
The elements of information support, formalized by using fuzzy sets include AUV parameters, its equations of motion, a quantitative description of the inaccuracy and uncertainty of information
elements, and other information depending on the specific type of AUV and features of its operation. The object parameter values and the laws of their changes are considered unknown, but there is
some information about the preference of certain values of their elements, which allows us to determine some convex set $P$. Then the mathematical description of the AUV has the form of differential
$\frac{dx}{dt}\in f\left(x,u,P,t\right),\mathrm{}\mathrm{}\mathrm{}\mathrm{}t\in \left[{t}_{0},{t}_{N}\right],\mathrm{}\mathrm{}\mathrm{}\mathrm{}x\in X\subset {R}^{{n}_{x}},$$x\left({t}_{0}\right)\
in {X}_{0}\subset X,\mathrm{}\mathrm{}\mathrm{}\mathrm{}u\in U\subset {R}^{{n}_{u}},\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}p\in P\subset {R}^{{n}_{p}},$
where $x$ is the state vector of the AUV; $u$ – control vector; $p$ – AUV parameter vector; $f\left(\right)$ – vector function; $t$ – time; $X$, $U$, $P$ – known compact convex sub-sets of the
corresponding spaces; $\left[{t}_{0},{t}_{N}\right]$ – the time interval at which the controlled process of the movement of the AUV is simulated.
The output of the AUV model based on Eq. (1) is characterized by the observation vector $z$, $\mathrm{d}\mathrm{i}\mathrm{m}z={n}_{z}$, which belongs to the set $Z$, $z\in Z$, called the observation
The observation model is a functional interrelation of elements with $z\in Z$ and vector $x\in X$ in the form $z=h\left(x,t\right)$, where $h\left(x,t\right)$ is a known vector function.
When constructing a mathematical description of the AUV, a certain initial set of models is formed, from which the preferred option is selected. When forming of model set, it includes only those
models that meet the stated goal of modeling.
The selected models are combined into the initial set $A=\left\{a\right\}$, on which the possibility of element comparing among themselves for the analysis of preference can also be determined. Each
$a\in A$ is assigned to the purpose of modeling $aima$. The relation ${O}_{cm}$ denoting the purpose of the simulation is a binary equivalence relation, ${a}_{1}{O}_{cm}{a}_{2}$. For some elements $
{a}_{1}$, ${a}_{2}\in A$, a partial order ratio ${a}_{1}\sigma {a}_{2}$ can also be specified. The set of all model s$A$ that have a common goal of modeling with the equivalence relations defined on
this set is called the target model space (TMS) of the controlled AUV. Then the TMS has a tuple $TMS=⟨A,\left\{O\right\},\left\{\sigma \right\}⟩$, provided that ${O}_{cm}=aima$, ${O}_{cm}\in \left\{O
\right\}$ and set $\left\{O\right\}$ is closed.
In addition, with the TMS a variety of attributes of models (VAM), included in this space is used. All elements of VAM are reduced to the terms of the abstract finite alphabet $\sigma$. From the
symbols of this alphabet, words and word combinations are composed using a stitching operation. Then, the description of VAM corresponds to a universal set of words $U=\left\{u\right\}$ expressing
all the properties of the model $a$.
Part of the properties $a\in A$, called non-specific, is established by analyzing the model $a$itself without involving other elements. The set of non-singular words along with the operation of
stitching is called the model appearance space (MAS). Special call properties, the presence of which can be established only by comparing of the model $a$ with the elements of a certain subset $A\
mathrm{"}\subseteq A$. The set of such words and the operation of stitching is called the criterial space of models (CSM).
The MAS and CSM spaces for a specific TMS are formed on the basis of the display of the entire input information set ${U}^{in}$ of information elements available at the modeling stage of the AUV
controlled information set to characterize the set of information intended for the formation of the TMS and model selection $g:{U}^{in}\to U$.
In modeling, an equivalence ${O}_{complex}$ is introduced. ${O}_{complex}$ characterizes the complexity of models which correspond to specified conditions and requirements. This allows you to create
a family $\stackrel{-}{A}=\left\{{A}^{c}\right\}$ that is a cover of the TMS and represents a decomposition $A$ in complexity. The complexity functional is a mapping $s:A\to D$ where a subset $D$
characterizes a quantitative estimate of proximity $a$ to ${a}^{0}$. The model ${a}^{0}\in {\stackrel{-}{A}}^{0}$ has minimal complexity. For $a\in A$ functional complexity is indicated $s\left(a/{a}
^{0}\right)$. In mathematical modeling, taking into account the complexity principle, the problem of multicriteriality can arise. This necessitates the use of equivalence on lexicographic complexity.
The principle of complexity is formulated as follows: in a given main ideal ${J}_{i}$ of the element ${\stackrel{-}{A}}^{i}$ of the decomposition of the TMS taking into account complexity of $\
stackrel{-}{A}$ it is necessary to find an element $a$with the required property ${u}^{\mathrm{*}}\in U$ defined in the CSM. With the help of the complexity functional, the complexity principle is
written as ${u}^{\mathrm{*}}=\left[s\left(a/{a}^{0}\right)\le {\epsilon }^{\mathrm{*}}\right]$, $a\in {J}_{D}$, where ${J}_{D}$ is the principal ideal of the decomposition of the space generated by
the level sets of the functional $\sigma \left(a\right)$; ${\epsilon }^{\mathrm{*}}$ – specified level of complexity.
The set $\left\{{s}_{\epsilon }\right\}$ of representatives of the level sets of the complexity functional is taken as the scale of complexity $\left\{\left\{a,s\left(a/{a}^{0}\right)\le \epsilon \
right\}\right\}$, $\epsilon \in \left(0,\mathrm{\infty }\right)$.
Taking into account inaccuracy and uncertainty $U$, it is proposed to determine the elements of the TMS on its basis in the class of fuzzy models, in which the input words correspond to the desired
control, and the output – to the states of the dynamic system whose behavior is described by the model. In this case, a generalized description of a fuzzy model can be represented as a sequence of
fuzzy operators that correspond to fuzzy production rules based on a fuzzy implication operation. Such model representation can be written as:
$\begin{array}{l}c:\mathrm{}\mathrm{}\text{if}\text{}{z}_{1}\left(\tau \right)\in {\stackrel{~}{Z}}_{1}^{c},{z}_{2}\left(\tau \right)\in {\stackrel{~}{Z}}_{2}^{c},\dots ,\mathrm{}\mathrm{}\mathrm{}
{z}_{{n}_{z}}\left(\tau \right)\in {\stackrel{~}{Z}}_{{n}_{z}}^{c},\\ \text{then}\mathrm{}\mathrm{}\left\{\begin{array}{l}x\left(\tau +1\right)\in {\stackrel{~}{x}}^{c}\left(\tau +1\right)={\stackrel
{~}{f}}^{c}\left(x\left(\tau \right),\mathrm{}\stackrel{~}{p},u\left(\tau \right),\tau \right),\mathrm{}\mathrm{}\mathrm{}\mathrm{}c=\overline{1,{C}_{a}},\\ z\left(\tau \right)=h\left(x\left(\tau \
right),\tau \right),\end{array}\right\\end{array}$
where ${\stackrel{~}{Z}}_{1}^{c},{\stackrel{~}{Z}}_{2}^{c},\dots ,{\stackrel{~}{Z}}_{{n}_{z}}^{c}$ – fuzzy sets belonging $Z$, with given membership functions ${\mu }_{{\stackrel{~}{Z}}_{i}^{c}}$, $i
=\overline{1,{n}_{z}}$; ${\stackrel{~}{x}}^{c}\left(\tau \right)$ – a fuzzy state vector defined by the production rule with a number $c$; ${\stackrel{~}{f}}^{c}\left(x\left(\tau \right),\stackrel{~}
{p},u\left(\tau \right),\tau \right)$ – fuzzy display of the consequent part of the product, which characterizes the local dynamics of the AUV; ${C}_{a}$ – number of rules. A clear output Eq. (2) is
calculated in accordance with the selected defuzzification method.
Thus, the mathematical description of the AUV requires a theoretical solution of the complex of problems, the structure of the relationship between them is illustrated in Fig.1.
Fig. 1. The structure of the mathematical description of the AUV on the basis of the complexity principle
The elements of the structure reveal the following sequence for solving of the research problem:
• formation of the input information space based on the AUV modeling;
• design of an input information set;
• description of fuzzy information elements that form the project information space, combining VAM and CSM;
• selection of the purpose of modeling and equivalence relations;
• construction of a mathematical description of the TMS;
• determination of the equivalence relation by complexity, partitioning of the TMS into related classes and factor sets by complexity, choice of scale and complexity functional;
• decomposition of the TMS, VAM and CSM with respect to equivalence in complexity, the definition of the main ideals for the elements of the decomposition of the TMS;
• formalization of the complexity principle;
• model selection from the corresponding decompositions of the TMS;
On the basis of the chosen model, the tasks of analyzing of the AUV dynamics are solved.
The approach requires the specification of the complexity principle and its practical application using the basic elements of “soft” calculations for mathematical modeling of the controlled motion of
the AUV.
3. Mathematical description of the input information space
To construct a TMS, VAM, CSM and substantiate the choice of the complexity principle, the input information space ${U}^{in}$ should combine, along with the stitching operation ${u}^{in}$, elements of
information support, with the help of which the controlled motion of a dynamic object is described.
Traditional dynamic models of the AUV should be focused on the formation of a TMS, the elements of which model the behavior of the control object solving a set of management tasks. For this, a
mathematical description based on the theory of the dynamics of a rigid body moving in a viscous fluid is usually used.
The spatial orientation of the object is described by Euler angles: yaw $\phi$, pitch $\psi$ and roll $\theta$. To describe the kinematic parameters, linear $V={\left[\begin{array}{lll}{V}_{x}& {V}_
{y}& {V}_{z}\end{array}\right]}^{T}$ and angular $\mathrm{\Omega }={\left[\begin{array}{lll}{\omega }_{x}& {\omega }_{y}& {\omega }_{z}\end{array}\right]}^{T}$ velocity vectors are introduced. The
position vector $e$ is defined as ${e}^{T}=\left[\begin{array}{ll}r& \chi \end{array}\right]$, where ${r}^{T}=\left[\begin{array}{lll}{x}_{g}& {y}_{g}& {z}_{g}\end{array}\right]$ is the vector of the
coordinates of the AUV pole; ${\chi }^{T}=\left[\begin{array}{lll}\theta & \phi & \psi \end{array}\right]$ – vector of Euler angles. The velocity vector can be written as ${q}^{T}=\left[\begin{array}
{ll}V& \mathrm{\Omega }\end{array}\right]$.
The controlled motion of the AUV is described by a system of differential equations, which in the vector-matrix form is:
$\frac{d}{dt}\left[\begin{array}{l}q\\ e\end{array}\right]=\left[\begin{array}{cc}{\left({M}_{T}+{M}_{Z}\right)}^{-1}\left(-{C}_{T}\left(q\right)q-{C}_{Z}\left(q\right)q-D\left(q\right)q-g\left(q\
right)+{T}_{y}\right)& {0}_{6×6}\\ \left[\begin{array}{ll}{B}_{V}^{-1}& {0}_{3×3}\\ {0}_{3×3}& {B}_{\omega }^{-1}\end{array}\right]& {0}_{6×6}\end{array}\right]\cdot \left[\begin{array}{l}q\\ e\end
where ${M}_{T}$ is the inertia matrix of AUV as a solid; ${C}_{T}\left(q\right)$ – matrix of Coriolis and centrifugal forces of a solid; ${M}_{Z}$ – matrix of added masses; ${C}_{Z}\left(q\right)$ –
hydrodynamic matrix, similar to the matrix of Coriolis and centrifugal forces; $D\left(q\right)$ – matrix of forces and moments of viscous friction; $g\left(q\right)$ – the vector of forces and
moments caused by gravity and buoyancy; ${B}_{V}$ and ${B}_{\omega }$ – kinematic matrices describing the relative rotations of the corresponding coordinate systems; ${T}_{y}$ – the vector of forces
and moments created by the controls of the AUV.
For a quantitative assessment of inaccurate and uncertain parameters it is proposed to use triangular $LR$ numbers.
In this regard, the vector of parameters $\stackrel{~}{p}$ is represented by a tuple $⟨{p}^{0},{p}^{I}⟩$, where ${p}^{0}$ is a vector of nominal values of parameters, a ${p}^{I}$ is an interval
vector${p}^{I}=\left[\begin{array}{llll}{p}_{1}& {p}_{2}& \dots & {p}_{{n}_{p}}\end{array}{\right]}^{T}$, ${p}_{i}=\left[{p\mathrm{"}}_{i},{p\mathrm{"}\mathrm{"}}_{i}\right]$ or ${\stackrel{~}{p}}_
{i}=⟨{p\mathrm{"}}_{i},{p}_{i}^{0},{p\mathrm{"}\mathrm{"}}_{i}⟩$, $i=\overline{1,{n}_{p}}$, where ${\stackrel{~}{p}}_{i}$ is a fuzzy number; ${p\mathrm{"}}_{i}$, ${p\mathrm{"}\mathrm{"}}_{i}$ –
respectively, the upper and lower limits of the interval ${p}_{i}$; ${p}_{i}^{0}$ – nominal value of the parameter.
The mathematical description of the controlled motion of the AUV in the form of taking into account the intervaliness ${p}^{I}$ allows us to form the inclusion and the equation.
The method of forming of the set of possible phase trajectories in the state space $X$ based on a series of computational experiments on model Eq. (1).
Each interval ${p}_{i}$ is approximated by a finite set of points ${S}_{{p}_{i}}={\left\{{p}_{{i}_{j}}\right\}}_{j=1}^{{M}_{{p}_{i}}}$ that are elements of a set ${S}_{P}$. The number of elements $
{S}_{P}$ is equal to $Q=\prod _{i=1}^{{n}_{p}}{M}_{{s}_{i}}$. The series $Q$ of computational experiments consisting in solving the Cauchy problem for $p\in {S}_{p}$ forms $Q$ set of phase
trajectories ${x}^{w}\left(t\right)\text{,}$$w=\overline{1,Q}\text{.}$ The values ${x}^{l}\left(t\right)$ in moments ${t}_{\tau }^{}$ determine the information sets ${G}_{x\left({t}_{\tau }^{}\
right)}={\left\{{x}^{w}\left({t}_{\tau }\right)\right\}}_{w=1}^{Q}{n}_{x}$. The input information set ${G}_{x}={\left\{{\left\{{x}^{w}\left({t}_{\tau }\right)\right\}}_{\tau =0}^{N}\right\}}_{w=1}^
{Q}$ is proposed to be formed by approximating ${G}_{x\left({t}_{\tau }^{}\right)}$ with ellipsoids, which have the form:
${\mathrm{Э}}_{{t}_{\tau }^{}}\left({\pi }_{{t}_{\tau }^{}},{D}_{{t}_{\tau }^{}}\right)=\left\{x:\left({D}_{{t}_{\tau }^{}}^{-1}\left(x-{\pi }_{{t}_{\tau }^{}}\right),\left(x-{\pi }_{{t}_{\tau }^{}}\
right)\right)\le 1\right\},$
where ${D}_{{t}_{\tau }^{}}$ is a positive-definite symmetric matrix with size ${n}_{x}×{n}_{x}$; ${\pi }_{{t}_{\tau }^{}}$ – vector of coordinates of the center of the ellipsoid of ${n}_{x}$
The definition of the matrix ${D}_{{t}_{\tau }^{}}$ and the vector ${\pi }_{{t}_{\tau }^{}}$ is made from the condition of minimizing the volume of the ellipsoid ${\mathrm{Э}}_{{t}_{\tau }^{}}$. For
this, the optimization problem is solved using nonlinear programming methods:
$\underset{{q}_{{t}_{\tau }^{}},{D}_{{t}_{\tau }^{}}}{\mathrm{m}\mathrm{i}\mathrm{n}}\mathrm{t}\mathrm{r}{D}_{{t}_{\tau }^{}},\left({D}_{{t}_{\tau }^{}}^{-1}\left(x\left({t}_{\tau }^{}\right)-{\pi }_
{{t}_{\tau }^{}}\right),\left(x\left({t}_{\tau }^{}\right)-{\pi }_{{t}_{\tau }^{}}\right)\right)\le 1,$
where $\left(·,·\right)$ denotes the scalar product of vectors.
The study of the time variation of the characteristic dimensions of the approximating ellipsoids Eq. (6) is performed using the criterion:
${J}_{1}=\mathrm{t}\mathrm{r}\mathrm{\Theta },\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{\Theta }=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left\{{\gamma }_{1},{\gamma }_{2},\dots ,{\gamma }
_{{n}_{x}}\right\},\mathrm{}\mathrm{}\mathrm{}\mathrm{}{\gamma }_{i}=\frac{{{d}_{{t}_{\tau }^{}}}_{ii}}{{\stackrel{^}{v}}_{i}^{2}},\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}i=\overline{1,{n}_{x}},
\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}V=\left[\begin{array}{llll}{v}_{1}& {v}_{2}& \dots & {v}_{{n}_{x}}\end{array}{\right]}^{T},$
where ${{d}_{{t}_{\tau }^{}}}_{ii}$ are the diagonal elements ${D}_{{t}_{\tau }^{}}$; $V$ – the normalizing vector of dimension ${n}_{x}$.
The obtained results demonstrate the possibility of forming an input information space sufficient for fuzzy model constructing. Mapping of the input space to fuzzy VAM and CSM to design the TMS,
select the initial model from it and analyze the stability of the AUV motion on its basis.
A level set $E$ is introduced, which includes vectors $\epsilon$, $\mathrm{d}\mathrm{i}\mathrm{m}\epsilon =0.5{n}_{x}\left({n}_{x}+1\right)$ – sets of matrix elements $D\left(\tau \right)$, an
ellipsoid ${\mathrm{Э}}_{\tau }\left({x}^{\mathrm{*}}\left(\tau \right),D\left(\tau \right)\right)$, characterizing the “tube” in space $X$ with the central axis in the form ${G}_{{x}^{\mathrm{*}}}=
{\left\{{x}^{\mathrm{*}}\left({t}_{\tau }\right)\right\}}_{\tau =0}^{N}$.
The set of admissible domains $D=\left\{{D}^{i}\right\}$ in the set $E$ is partially ordered by nesting. The unit is the whole set $E$. Chains of nested sets ${D}^{i}\le D$ form the main ideals ${J}_
{{D}^{i}}$ in the set $D$. Full ordering of set $E$ and set $D$ can be implemented lexicographically for the alphabet $\sigma$.
The quality functional $\sigma$ is reduced to satisfying the constraints on the model output, i.e. for all $x\left({t}_{\tau }\right)\in {G}_{x}$, $\tau =\overline{0,N}$, the following condition is
$x\left(\tau \right)\in {\mathrm{Э}}_{\tau }\left({x}^{\mathrm{*}}\left(\tau \right),{D}^{add}\right),\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{D}^{add}=diag{\epsilon }^{add}=diag\left\{{d}_{1}^
{add}\dots {d}_{{n}_{x}}^{add}\right\}.$
To solve the problem, the principle of complexity is applied in the form ${\sigma }_{\mathrm{*}}=\left[s\left({G}_{x}|{G}_{{x}^{\mathrm{*}}}\right)\le {\epsilon }^{\mathrm{*}}\right]$, ${G}_{x}\in
The set $\left\{{\sigma }_{\epsilon }\right\}$ of representatives of the level sets of the complexity functional $s$ is taken as the scale of complexity.
The result of the mapping $g$ is determined by the TMS, which combines the algorithmic description of the fuzzy model Eq. (2). Output variables ${z}_{i}$, $i=\overline{1,{n}_{z}}$, are treated as
linguistic variables.
To estimate the measure of proximity of fuzzy models, it is proposed to use a fuzzy relationship, reflecting the degree of confidence that the models under consideration have adequate properties. For
this, the concept of fuzzy measure of proximity of models $\stackrel{~}{E}=r\left[\stackrel{~}{x}\left(\tau \right),\underset{_}{\stackrel{~}{x}}\left(\tau \right)\right]$ is introduced. Adequacy
assessment is performed by the value of the index of the function of belonging ${\mu }_{\stackrel{~}{E}}$. The function based on the well-known information sets ${G}_{x}$ and ${G}_{u}$ is formed.
A measure of the proximity of the outputs of the differential inclusion and fuzzy models and at the time $\tau$ is ${r}_{\tau }\left[\stackrel{~}{x}\left(\tau \right),\underset{_}{\stackrel{~}{x}}\
left(\tau \right)\right]=‖\stackrel{~}{x}\left(\tau \right)-\underset{_}{\stackrel{~}{x}}\left(\tau \right)‖$.
According to the principle of generalization [5], L. A. Zadeh the membership function of a fuzzy proximity measure is calculated by the formula:
${\mu }_{\stackrel{~}{E}}\left(y\right)=\mathrm{s}\mathrm{u}\mathrm{p}\left\{\underset{\tau =\overline{0,N}}{\wedge }{\mu }_{\stackrel{~}{E}},y=r\left(x,\underset{_}{x}\right)\right\}.$
On the basis of the differential inclusion Eq. (2), a computational experiment is organized to simulate the dynamics of the AUV with fuzzy elements of the parameter vector $p$. The simulation result
for the underwater vehicle “AFALINA” [1] is shown in Fig. 2.
Thus, the theoretical and methodological issues of the synthesis of the algorithmic description of mathematical models of the controlled motion of the AUV based on the complexity principle are
considered. The mapping of the input information space to the spaces of the TMS, VAM and CSM, sufficient to select a model that meets the requirements for complexity, is given.
Fig. 2Results of modeling fuzzy dynamics of underwater vehicle “AFALINA”
4. Conclusions
A feature of the practical application of modern complex AUV is the increasing requirements for the quality of their purposeful functioning under conditions of objectively increasing the level of
uncertainty, like a priori information about the system and its operating conditions used in the design, and information collected by the system about its current state and environment direct
performance of underwater work. The recognized direction of solving the problem of compensating the influence of inaccuracies and uncertainties of the information used on the dynamic capabilities of
the AUV is to improve the methods of its mathematical modeling as a control object. This assumes the use of effective information technologies developed in the theory of artificial intelligence, and
intended for use in inaccurate and uncertain information support. Therefore, the actual scientific and technical problem is the development of mathematical modeling methods that take into account the
modern principles of describing the properties of information used in modeling and the implementation-oriented results of its practical application in the next-generation computing systems.
• Siek Yu L., Smolnikov A. V., Yakovleva M. V. Controlling an Underwater Robot Based on Fuzzy Logic: a Monograph. SPbGMTU, St. Petersburg, 2008, p. 185.
• Siek Yu L., Soe Min Lwin Simulation of the controlled movement of a marine dynamic object based on the complexity principle. Proceedings of the XVI All-Russian Scientific Conference
“Telematics-2009”, 2009.
• Solodovnikov V. V., Tumarkin V. I. The Theory of Complexity and Design of Control Systems. Science, Moscow, 1990, p. 68.
• Nechaev Yu I. Fuzzy knowledge system for estimation of ship seaworthiness in onboard real time intelligence systems. Proceedings of 16th International Conference on Hydrodynamics in Ship Design,
Poland, 2005.
• Zadeh L. A. A Theory of Approximate Reasoning. Machine Intelligence, Vol. 9, Elsevier, New York, 1979, p. 149-194.
• Sokolov S., Zhilenkov A., Chernyi S., Nyrkov A., Mamunts D. Dynamics models of synchronized piecewise linear discrete chaotic systems of high order. Symmetry, Vol. 11, Issue 2, 2019, p. 236.
About this article
Mathematical models in engineering
automatic underwater vehicle
complexity principle
Copyright © 2019 Yuri Siek, et al.
This is an open access article distributed under the
Creative Commons Attribution License
, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Value of Perfect Information
sammyd22 Registered Posts: 207 Dedicated contributor 🦉
In my recent exam paper there was a 5mark question regarding the value of perfect information similar to this:
Project1 - Project2 - Project3
Probability - Weather:
.3 Good $600 $400 $200
.5 Moderate $200 $150 $100
.2 Poor ($200) $50 ($400)
What is the value that would be paid for perfect information?
I approached the question as follows:
(.3 x $600 + .5 x $200 + .2 x $50) = 180 + 100 + 10 = 290 Therefore a business could afford to pay anything upto this aqmount for perfect info.
As there is a vast amount of knowledge on this forum would anyone be kind enough to advise on whether i approached this correctly?
• Wouldn't it be a different value per project?
Question lacks info, but I'm assuming the probability is a weighting of importance?
• Use your probabilities to find the expected value in the normal way.
Then do the calculations on the basis of knowing what will happen i.e if good weather then probability x that outcome, if moderate prob x outcome and if bad that, then add them up.
The difference in expected value between the higher value (when you know which project to pick) and the lower value (based on the normal expected value calculation) is the value of perfect
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The wonder-full world of one-dimensional bosons
Hanns-Christoph Nägerl
University of Innsbruck
Laser-cooled bosons at ultralow temperatures, with strong interactions and under strong low-dimensional confinement, have myriad surprises in store. They don’t condense, instead fermionize, but
remain superfluid, though at the slightest of perturbations they may crystallize. In our recent experiments with strongly confined Cs atoms, for which we can tune the contact interactions from zero
to infinite, we have explored the 2D-to-1D crossover regime [1], implemented a very precise thermometer with which we have found cooling upon dimensional reduction [2], investigated into impurity
transport through a fermionized Bose gas, analyzed the short-time dynamics in a quantum Newton cradle experiment, and found evidence for many-body dynamical localization in a 1D quantum kicked-rotor
setting [3]. In particular, we have seen that uniformly moving impurities develop anyonic properties, with anyons being quasi particles that interpolate between bosons and fermions. On the side,
after 20 years of Bose-Einstein condensation of Cs atoms, we have condensed Cs atoms in a state other than the absolute ground state [4], and even that experiment has some surprise for us in store.
[1] Observation of the 2D-1D dimensional crossover in strongly interacting ultracold bosons,
Y. Guo et al., Nature Physics (2024), arXiv:2308.00411 (2023)
[2] Anomalous cooling of bosons by dimensional reduction,
Y. Guo et al., Science Advances 10, 6 (2024), arXiv:2308.04144 (2023)
[3] Observation of many-body dynamical localization,
Y. Guo et al., arXiv:2312.13880 (2023)
[4] Bose-Einstein condensation of non-ground-state caesium atoms,
M. Horvath et al., Nature Communications 15, 3739 (2024), arXiv:2310.12025 (2023)
Speaker's Brief Introduction：
Hanns-Christoph Nägerl studied physics in Göttingen, San Diego, and Innsbruck and completed his doctoral thesis in 1998 on the subject of “Ion Strings for Quantum Computation” with Prof. R. Blatt. As
a postdoctoral researcher he work at the California Institute of Technology (Caltech) with Prof. J. Kimble on single ultracold neutral atoms for quantum information purposes. In 2000 he joined Prof.
R. Grimm in Innsbruck to set up an ultracold-atom group and to work on atomic and molecular quantum gases and matter waves. He received his “habilitation” in 2006, with this became associate
professor, and then advanced to full professor for experimental physics at the University of Innsbruck in 2011. Professor Nägerl has been awarded numerous prizes, e.g. a START prize grant (2003), the
Rudolf-Kaiser prize (2010), an ERC-Consolidator prize grant (2011), the prestigious Wittgenstein award (2017), and an ERC-Advanced prize grant (2018). Presently, he is the director of the Institute
for Experimental Physics of the University of Innsbruck. His scientific interests center on quantum many-body physics and quantum simulation on the basis of ultracold atoms and molecules.
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How To Get Correlation Coefficient In Power BI
This content explains Pearson's correlation coefficient, the most widely used correlation coefficient, particularly in linear regression and measuring the relationship between two variables.
Interpretation of correlation coefficient values and provides step-by-step instructions.
There are many kinds of correlation coefficients, but Pearson’s correlation coefficient is the most popular. It is used in linear regression. It is also used to measure the relationship between two
variables. The value of a correlation coefficient is always between -1 to 1.
• 1 indicates a strong positive relationship
• -1 indicated a strong negative relationship
• 0 indicates no relationship between two values
The below graph images will help you to understand the positive, negative, and no correlation.
How to calculate the correlation coefficient?
To find the correlation coefficient you need to add a quick measure (see https://www.c-sharpcorner.com/article/what-is-quick-measure-in-power-bi/). First, we need to import the table so that we can
add a quick measure to it.
I am going to import data from the SQL server (see here).
Go to the home tab click on the get data drop down and then click on ‘sql server’.
A new window will be opened, provide your server name and database name and connect your SQL server by clicking on ‘OK’.
Once you connect your server a navigator window will be opened. Select your table and click on load. Your table will be loaded.
Loaded table will be shown at the right side of the tool.
Now go to the home tab and click on ‘Quick Measure’. A new quick-measure window will be opened.
Click on the drop-down menu of ‘Select a calculation’ and go to ‘Mathematical Operations’ and click on ‘Correlation coefficient’.
Calculate the correlation coefficient between two values over the category. You have to provide three data fields:
1. Category: Category to find the correlation over
2. Measure X: The first measure to find the correlation between
3. Measure: The second measure to find the correlation between
Click on ‘OK’ to calculate the correlation coefficient. After clicking on ‘Ok’ a new measure ‘OrderQty and UnitPrice correlation for ProductID’ will be created in the table. The background
calculation will be shown while you click on this newly added quick measure
OrderQty and UnitPrice correlation for ProductID =
VAR __CORRELATION_TABLE = VALUES('Sales SalesOrderDetail'[ProductID])
VAR __COUNT =
SUM('Sales SalesOrderDetail'[OrderQty])
* SUM('Sales SalesOrderDetail'[UnitPrice])
VAR __SUM_X =
CALCULATE(SUM('Sales SalesOrderDetail'[OrderQty]))
VAR __SUM_Y =
CALCULATE(SUM('Sales SalesOrderDetail'[UnitPrice]))
VAR __SUM_XY =
SUM('Sales SalesOrderDetail'[OrderQty])
* SUM('Sales SalesOrderDetail'[UnitPrice]) * 1.
VAR __SUM_X2 =
CALCULATE(SUM('Sales SalesOrderDetail'[OrderQty]) ^ 2)
VAR __SUM_Y2 =
CALCULATE(SUM('Sales SalesOrderDetail'[UnitPrice]) ^ 2)
__COUNT * __SUM_XY - __SUM_X * __SUM_Y * 1.,
(__COUNT * __SUM_X2 - __SUM_X ^ 2)
* (__COUNT * __SUM_Y2 - __SUM_Y ^ 2)
To check the value of this correlation coefficient, select a ‘Card’ visual from the visualization panel and select this newly added measure. You can see that the value of the correlation coefficient
lies between -1 to 1. See the below image,
This is how you can calculate the correlation coefficient between two values. I hope you understand how to find the correlation coefficient. This is also very useful for linear regression. I will
write a separate article on linear regression. So, stay tuned. Thanks for reading and have a great day.
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ENROLLMENT OPENS Jan. 16, 2022 – STREAM: Algebraic Thinking K-5 (Feb. 6 – Feb. 27, 2022)
This 3-week course is designed to address K-5 algebraic thinking standards. Participants in this module will examine learning progressions in Algebraic Thinking with a focus on foundational ideas
that promote algebraic understanding. Specific goals include the following:
• Recognize that addition, subtraction, multiplication, and division operate under the same properties in algebra as they do in arithmetic
• Learn variables are tools that are used to describe mathematical ideas in a variety of ways
• Develop strategies for describing relationships among quantities to determine equivalence or inequalities
15 Renewal Units
Enrollment opens – Jan. 16, 2022
Course runs Feb. 6 – Feb. 27, 2022
Visit STREAM: Algebraic Thinking (K-5) on the Teacher Learning Hub to enroll
ENROLLMENT OPENS Oct. 10, 2021 – STREAM: Algebraic Thinking K-5 (Oct. 31 – Nov. 21, 2021)
This 3-week course is designed to address K-5 algebraic thinking standards. Participants in this module will examine learning progressions in Algebraic Thinking with a focus on foundational ideas
that promote algebraic understanding. Specific goals include the following:
• Recognize that addition, subtraction, multiplication, and division operate under the same properties in algebra as they do in arithmetic
• Learn variables are tools that are used to describe mathematical ideas in a variety of ways
• Develop strategies for describing relationships among quantities to determine equivalence or inequalities
15 Renewal Units
Enrollment opens – October 10, 2021
Course runs Oct. 31 – Nov. 21, 2021
Visit STREAM: Algebraic Thinking (K-5) on the Teacher Learning Hub to enroll
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what is the probability of the destruction of the bridge if only 5 bo
Last Activity: 8 Years ago
for 2 bombs-
2^2=4 possible cases..-(H H),(H M),(M H),(M M)
2 Hits(H) req. for destruction...hence 1/4..
similarly, for 3 bombs-
2^3=8 possible cases..i.e-i.e-(H H H),(H H M),(H M H),(M H H),(H M M),(M H M),(M M H),(M M M)..All cases with more than 2 hits are favourable...hence 4/8..
similarly for other cases...for more number of bombs combination can be done to get the values quickly...
P.S-probability for hit or miss are the 1/2 so no need considering them
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Rick Lyons ●
January 10, 2019
●5 comments
Below is a little microprocessor history. Perhaps some of the ol' timers here will recognize a few of these integrated circuits. I have a special place in my heart for the Intel 8080 chip.
Image copied, without permission, from the now defunct Creative Computing magazine, Vol. 11, No. 6, June 1985.
Christian Yost ●
January 8, 2019
Last post we motivated the idea of viewing the classic phase vocoder as a Markov process. This was due to the fact that the input signal’s features are unknown to the computer, and the phase
advancement for the next synthesis frame is entirely dependent on the phase advancement of the current frame. We will dive a bit deeper into this idea, and flesh out some details which we left
untouched last week. This includes the effect our discrete Fourier transform has on the...
Christian Yost ●
January 8, 2019
Hello! This is my first post on dsprelated.com. I have a blog that I run on my website, http://www.christianyostdsp.com. In order to engage with the larger DSP community, I'd like to occasionally
post my more engineering heavy writing here and get your thoughts.
Today we will look at the phase vocoder from a different angle by bringing some probability into the discussion. This is the first part in a short series. Future posts will expand further upon the
The Discrete Fourier Transform (DFT) operates on a finite length time sequence to compute its spectrum. For a continuous signal like a sinewave, you need to capture a segment of the signal in order
to perform the DFT. Usually, you also need to apply a window function to the captured signal before taking the DFT [1 - 3]. There are many different window functions and each produces a different
approximation of the spectrum. In this post, we’ll present Matlab code that...
Steve Maslen ●
November 22, 2018
This article will look at a design approach for feedback controllers featuring low-latency "irrational" characteristics to enable the creation of physical components such as transmission lines. Some
thought will also be given as to the capabilities of the currently utilized Intel Cyclone V, the new Cyclone 10 GX and the upcoming Xilinx Versal floating-point FPGAs/ACAPs.
Fig 1. Making a Transmission Line, with the Circuit Emulator
Lyons Zhang ●
November 8, 2018
For any B-DMC $W$, the channels $\{W_N^{(i)}\}$ polarize in the sense that, for any fixed $\delta \in (0, 1)$, as $N$ goes to infinity through powers of two, the fraction of indices $i \in \{1, \
dots, N\}$ for which $I(W_N^{(i)}) \in (1 − \delta, 1]$ goes to $I(W)$ and the fraction for which $I(W_N^{(i)}) \in [0, \delta)$ goes to $1−I(W)^{[1]}$.
Mrs. Gerber’s Lemma
Mrs. Gerber’s Lemma provides a lower bound on the entropy of the modulo-$2$ sum of two binary random...
Lyons Zhang ●
October 19, 2018
Channel Combining
Channel combining is a step that combines copies of a given B-DMC $W$ in a recursive manner to produce a vector channel $W_N : {\cal X}^N \to {\cal Y}^N$, where $N$ can be any power of two, $N=2^n, n
The notation $u_1^N$ as shorthand for denoting a row vector $(u_1, \dots , u_N)$.
The vector channel $W_N$ is the virtual channel between the input sequence $u_1^N$ to a linear encoder and the output sequence $y^N_1$ of $N$...
The Google Summer of Code 2018 is now in its final stages, and I’d like to take a moment to look back at what goals were accomplished, what remains to be completed and what I have learnt.
The project overview was discussed in the previous blog posts. However this post serves as a guide to anyone who wishes to learn about the project or carry it forward. Hence I will go over the
project details again.
Project overview
The project “Digital Filter Blocks in MyHDL and PyFDA integration" aims...
This was my first time at Sensors Expo and my second time in Silicon Valley and I must say I had a great time.
Before I share with you what I find to be, by far, my best 'highlights' video yet for a conference/trade show, let me try to entertain you with a few anecdotes from this trip. If you are not
interested by my stories or maybe don't have the extra minutes needed to read them, please feel free to skip to the end of this blog post to watch the...
This post provides a Matlab function that designs linear-phase FIR sinx/x correctors. It includes a table of fixed-point sinx/x corrector coefficients for different DAC frequency ranges.
A sinx/x corrector is a digital (or analog) filter used to compensate for the sinx/x roll-off inherent in the digital to analog conversion process. In DSP math, we treat the digital signal applied
to the DAC is a sequence of impulses. These are converted by the DAC into contiguous pulses...
If you've read about the Goertzel algorithm, you know it's typically presented as an efficient way to compute an individual kth bin result of an N-point discrete Fourier transform (DFT). The
integer-valued frequency index k is in the range of zero to N-1 and the standard block diagram for the Goertzel algorithm is shown in Figure 1. For example, if you want to efficiently compute just
the 17th DFT bin result (output sample X17) of a 64-point DFT you set integer frequency index k = 17 and N =...
This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by deriving exact formulas for the phase and amplitude of a non-integer frequency real tone in a
DFT. The linearity of the Fourier Transform is exploited to reframe the problem as the equivalent of finding a set of coordinates in a specific vector space. The found coordinates are then used to
calculate the phase and amplitude of the pure real tone in the DFT. This article...
This final article in the series will look at -ve latency DSP and how it can be used to cancel the unwanted delays in sampled-data systems due to such factors as Nyquist filtering, ADC acquisition,
DSP/FPGA algorithm computation time, DAC reconstruction and circuit propagation delays.
Some applications demand zero-latency or zero unwanted latency signal processing. Negative latency DSP may sound like the stuff of science fiction or broken physics but the arrangement as...
Abstract: Dispersive linear systems with negative group delay have caused much confusion in the past. Some claim that they violate causality, others that they are the cause of superluminal tunneling.
Can we really receive messages before they are sent? This article aims at pouring oil in the fire and causing yet more confusion :-).
In this article we reproduce the results of a physical experiment...
Rick Lyons ●
November 7, 2007
●4 comments
Most of us are familiar with the process of flipping the spectrum (spectral inversion) of a real signal by multiplying that signal's time samples by (-1)n. In that process the center of spectral
rotation is fs/4, where fs is the signal's sample rate in Hz. In this blog we discuss a different kind of spectral flipping process.
Consider the situation where we need to flip the X(f) spectrum in Figure 1(a) to obtain the desired Y(f) spectrum shown in Figure 1(b). Notice that the center of...
Glue between Octave and NGSPICE for discrete- and continuous time cosimulation (download) Keywords: Octave, SPICE, Simulink
Many DSP problems have close ties with the analog world. For example, a switched-mode audio power amplifier uses a digital control loop to open and close power transistors driving an analog filter.
There are commercial tools for digital-analog cosimulation: Simulink comes to mind, and mainstream EDA vendors support VHDL-AMS or Verilog-A in their...
eywords: Quantization noise; noise shaping
A brief introduction to noise shaping, with firm resolve not to miss the forest for the trees. We may still stumble over some assorted roots. Matlab example code is included.
Fig. 1 shows a digital signal that is reduced to a lower bit width, for example a 16 bit signal being sent to a 12 bit digital-to-analog converter. Rounding to the nearest output value is obviously
the best that can be done to minimize the error of each...
Mark Newman ●
November 11, 2024
The Fourier Transform is a powerful tool, used in many technologies, from audio processing to wireless communication. However, calculating the FT can be computationally expensive. The Cooley-Tukey
Fast Fourier Transform (FFT) algorithm provides a significant speedup. It exploits the repetitive nature of calculations within the Discrete Fourier Transform (DFT), the mathematical foundation of
the FT. By recognizing patterns in the DFT calculations and reusing intermediate results, the FFT vastly reduces the number of operations required. In this series of articles, we will look at how the
Cooley-Tukey FFT algorithm works.
This blog discusses a little-known filter characteristic that enables real- and complex-coefficient tapped-delay line FIR filters to exhibit linear phase behavior. That is, this blog answers the
What is the constraint on real- and complex-valued FIR filters that guarantee linear phase behavior in the frequency domain?
I'll declare two things to convince you to continue reading.
Declaration# 1: "That the coefficients must be symmetrical" is not a correct
It seems to be fairly common knowledge, even among practicing professionals, that the efficiency of propagation of wireless signals is frequency dependent. Generally it is believed that lower
frequencies are desirable since pathloss effects will be less than they would be at higher frequencies. As evidence of this, the Friis Transmission Equation[i] is often cited, the general form of
which is usually written as:
Pr = Pt Gt Gr ( λ / 4πd )2 (1)
where the...
We are delighted to announce the launch of the very first new Related site in 15 years! The new site will be dedicated to the trendy and quickly growing field of Machine Learning and will be called
- drum roll please - MLRelated.com.
We think MLRelated fits perfectly well within the “Related” family, with:
• the fast growth of TinyML, which is a topic of great interest to the EmbeddedRelated community
• the use of Machine/Deep Learning in Signal Processing applications, which is of...
This blog explains why, in the process of time-domain interpolation (sample rate increase), zero stuffing a time sequence with zero-valued samples produces an increased-length time sequence whose
spectrum contains replications of the original time sequence's spectrum.
The traditional way to interpolate (sample rate increase) an x(n) time domain sequence is shown in Figure 1.
Figure 1
The '↑ L' operation in Figure 1 means to...
Markus Nentwig ●
August 18, 2012
• Octaveforge / Matlab design script. Download: here
• weighted numerical optimization of Laplace-domain transfer function
• linear-phase design, optimizes vector error (magnitude and phase)
• design process calculates and corrects group delay internally
• includes sinc() response of the sample-and-hold stage in the ADC
• optionally includes multiplierless FIR filter
Problem Figure 1: Typical FIR-DAC-analog lowpass line-up
Digital-to-analog conversion connects digital...
I've recently encountered a digital filter design application that astonished me with its design flexibility, capability, and ease of use. The software is called the "ASN Filter Designer." After
experimenting with a demo version of this filter design software I was so impressed that I simply had publicize it to the subscribers here on dsprelated.com.
What I Liked About the ASN Filter Designer
With typical filter design software packages the user enters numerical values for the...
Sometimes you may need to phase-lock a numerically controlled oscillator (NCO) to an external clock that is not related to the system clocks of your ASIC or FPGA. This situation is shown in Figure
1. Assuming your system has an analog-to-digital converter (ADC) available, you can sync to the external clock using the scheme shown in Figure 2. This time-domain PLL model is similar to the one
presented in Part 1 of this series on digital PLL’s [1]. In that PLL, we...
David ●
October 27, 2010
●1 comment
UPDATE: Added graphs and code to explain the frequency division of the branches
The focus of this article is to briefly explain an implementation of this transform and several filter bank forms. Theoretical information about DWT can be found elsewhere.
First of all, a 'quick and dirty' simplified explanation of the differences between DFT and DWT:
The DWT (Discrete Wavelet Transform), simply put, is an operation that receives a signal as an input (a vector of data) and...
Mike ●
January 5, 2016
●3 comments
Fixed point fractional representation always gives me a headache because I screw it up the first time I try to implement an algorithm. The difference between integer operations and fractional
operations is in the overflow. If the representation fits in the fixed point result, you can not tell the difference between fixed point integer and fixed point fractions. When integers overflow,
they lose data off the most significant bits. When fractions overflow, they lose data off...
Stephane Boucher ●
September 20, 2020
It is Sunday night as I write this blog post with a few days to go before the virtual doors of the very first DSP Online Conference open..
It all started with a post in the DSPRelated forum about three months ago. We had just had a blast running the 2020 Embedded Online Conference and we thought it could be fun to organize a smaller
event dedicated to the DSP community. So my goal with the post in the forum was to see if...
A cookbook recipe for segmented y=f(x) 3rd-order polynomial interpolation based on arbitrary input data. Includes Octave/Matlab design script and Verilog implementation example. Keywords: Spline,
interpolation, function modeling, fixed point approximation, data fitting, Matlab, RTL, Verilog
Splines describe a smooth function with a small number of parameters. They are well-known for example from vector drawing programs, or to define a "natural" movement path through given...
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Identity matrix - (Physical Sciences Math Tools) - Vocab, Definition, Explanations | Fiveable
Identity matrix
from class:
Physical Sciences Math Tools
An identity matrix is a square matrix in which all the elements of the principal diagonal are ones, and all other elements are zeros. This matrix acts as a multiplicative identity in linear algebra,
meaning that when any matrix is multiplied by the identity matrix, the result is the original matrix. It plays a crucial role in solving eigenvalue problems and in determining characteristics of
linear transformations.
congrats on reading the definition of identity matrix. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. The identity matrix is denoted as $$I_n$$, where $$n$$ indicates the size of the matrix (number of rows and columns).
2. For any square matrix $$A$$ of size $$n \times n$$, multiplying it by the identity matrix $$I_n$$ gives back the original matrix: $$A \cdot I_n = A$$.
3. The identity matrix serves as the equivalent of the number 1 in matrix multiplication, allowing for simplification in algebraic manipulations.
4. In eigenvalue problems, the identity matrix is used to formulate the characteristic equation by subtracting $$\lambda I$$ from the original matrix and calculating the determinant.
5. The presence of an identity matrix is essential for defining invertible matrices; a square matrix is invertible if it can be multiplied by another matrix to yield the identity matrix.
Review Questions
• How does an identity matrix facilitate the process of solving eigenvalue problems?
□ An identity matrix allows for the formulation of the characteristic equation in eigenvalue problems. By subtracting $$\lambda I$$ from a given square matrix $$A$$, we create a new matrix
whose determinant can be set to zero to find the eigenvalues. This step is crucial because it simplifies finding eigenvalues, ultimately leading to solving for eigenvectors as well.
• Explain how multiplying any square matrix by an identity matrix affects its properties and provide an example.
□ Multiplying any square matrix by an identity matrix retains all properties of that original matrix, effectively demonstrating that the identity acts like '1' in traditional arithmetic. For
example, if we have a 2x2 matrix $$A = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}$$ and multiply it by the 2x2 identity matrix $$I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$$, we get
back $$A$$: $$A \cdot I_2 = A$$. This property is fundamental in ensuring that transformations remain consistent.
• Evaluate the implications of having an identity matrix within a system of linear equations and how it relates to invertible matrices.
□ In a system of linear equations, the presence of an identity matrix indicates that there exists an inverse for a given coefficient matrix, meaning the system can be solved uniquely. If we can
express our system in terms of $$AX = B$$ and manipulate it to yield an identity matrix through row operations or inverse calculations, we establish that solutions are attainable. This also
highlights that for a square coefficient matrix to be invertible, it must be possible to manipulate it into an identity form through elementary row operations or multiplication with its
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BPU11 CONGRESS
Mrs Violeta Stanković (Faculty of Physics, University of Belgrade, Serbia)
The research topic is calculation of the drift velocity of electron transport in $N_2$ gas under the influence of an external crossed electromagnetic field, E x B. A Monte Carlo simulation code has
been used to obtain non-equilibrium electron energy distribution function within one oscillation of external crossed fields. In simulation, E is aligned with the z-axis, while B is parallel with
In order to test our simulation code validity under the condition of crossed RF electric and RF magnetic fields, we compared drift velocity components of electron transport in Reid’s model gas with
the available literature data. The results shows the transversal drift velocity (Vx, in E×B direction) obtained by our simulation and obtained by Petrovic et al., with their Monte Carlo code. The
calculation was performed for the frequency of 50 MHz, reduced electric field of 14 Td while the reduced magnetic field value was 500 Hx. One can clearly see the excellent matching of the compared
velocities which proves the validity of our code.
According to the results of that comparison, we have been encouraged to research and calculate the drift velocity components of electron transport in real $N_2$ gas. These results have been obtained
under the condition of reduced electric field, E/N, of 100 Td, frequency of 100 MHz and reduced magnetic field, B/N, of 1000 Hx.
1. Z. LJ. Petrović, Z.M. Raspopović, S. Dujko, T. Makabe, Applied Surface Science, 192 (2002) 1-25
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What our customers say...
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Keep up the good work Algebrator staff! Thanks!
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You know, for a step-by-step algebra solution teaching software program, I recommend Algebrator to every student, parent, tutor, teacher, and board member I can!
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"Polytope algebra" by Gaiane Panina
"Polytope algebra" by Gaiane Panina
We will gradually build up a beautiful algebraic object based on purely geometric objects — the polytope (graded) algebra developed by Peter McMulen. That is, we will introduce addition,
multiplication, and even such crazy things as exponent and logarithm for polytopes. This construction has helped to prove the f-vector problem (I’ll try to hint how) and is directly related to the
Chow rings of algebraic toric varieties (this is beyond our course). Surprisingly, there are no prerequisites. However, it is nice if you know what an abelian (that is, commutative) group, a ring,
and the (graded) algebra of polynomials are.
There are currently no items in this folder.
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National population projections
• The population of the UK is projected to increase by 3.0 million (4.5%) in the first 10 years of the projections, from an estimated 66.4 million in mid 2018 to 69.4 million in mid 2028.
• England’s population is projected to grow more quickly than the other UK nations: 5.0% between mid 2018 and mid 2028, compared with 3.7% for Northern Ireland, 1.8% for Scotland and 0.6% for
• Over the next 10 years, 27% of UK population growth is projected to result from more births than deaths, with 73% resulting from net international migration; although net migration falls during
this period, the number of deaths rises as those born in the baby boom after World War Two reach older ages.
• The UK population is projected to pass 70 million by mid 2031, reaching 72.4 million by 25 years into the projection (mid 2043).
• There will be an increasing number of older people; the proportion aged 85 years and over is projected to almost double over the next 25 years.
• The UK population growth rate is slower than in the 2016-based projections; the projected population is 0.4 million less in mid 2028 and 0.9 million less in mid 2043.
National population projections do not attempt to predict the impact of political circumstances such as Brexit.
Statistician’s comment
“The UK population is projected to grow by 3 million people by 2028. This assumes migration will have a greater impact on the size of the population than the combination of births and deaths.
Although migration declines at first and the number of births is stable, the number of deaths is projected to grow as those born in the baby boom after World War Two reach older ages.
“The population is increasingly ageing and this trend will continue. However, because of the expected rise in the State Pension age to 67 years, it is projected that slightly fewer than one in five
people will be of pensionable age in 2028, a similar proportion to today.”
Andrew Nash, Population Projections Unit, Office for National Statistics.
Follow the ONS Centre for Ageing and Demography on Twitter @RichPereira_ONS
Nôl i'r tabl cynnwys
Figure 1: UK population projected to rise to 69.4 million by mid 2028 and to 72.4 million by mid 2043
UK population estimates, mid 1993 to mid 2018, and projections to mid 2043
Source: Office for National Statistics – National population projections
Download this chart Figure 1: UK population projected to rise to 69.4 million by mid 2028 and to 72.4 million by mid 2043
Image .csv .xls
The UK population, which was 66.4 million in mid 2018, is projected to rise to 69.4 million over the decade to mid 2028. It is then projected to pass 70 million by mid 2031 and reach 72.4 million by
25 years into the projection (mid 2043).
The total projected increase in the UK population over the next 25 years is less than that over the past 25 years (Figure 1). Between mid 1993 and mid 2018, the population grew by 9.0 million
(15.1%); between mid 2018 and mid 2043, it is projected to grow by another 6.0 million (9.0%).
Table 1: Estimated and projected population of the
UK and constituent countries, mid 2018 to mid 2043
UK 66.4 68.1 69.4 70.5 71.4 72.4
England 56.0 57.6 58.8 59.8 60.8 61.7
Wales 3.1 3.2 3.2 3.1 3.1 3.1
Scotland 5.4 5.5 5.5 5.6 5.6 5.6
Northern Ireland 1.9 1.9 2.0 2.0 2.0 2.0
Download this table Table 1: Estimated and projected population of the UK and constituent countries, mid 2018 to mid 2043
.xls .csv
Focusing on the 10 years between mid 2018 and mid 2028 (Table 1), the total projected growth for the UK population is 3.0 million, or 4.5%. This represents an average annual growth rate of 0.4%.
Projected growth varies substantially between the four countries of the UK: England’s population is projected to grow 5.0% over this period; for Northern Ireland, the figure is 3.7%; while for
Scotland and Wales, the figures are 1.8% and 0.6% respectively.
Over the full 25 years between mid 2018 and mid 2043, England is projected to have the largest increase in population, at 10.3%. The other UK countries with a projected increase in population are
Northern Ireland at 5.7% and Scotland at 2.5%. Wales shows a projected population decline of 0.9%.
Nôl i'r tabl cynnwys
3. Births, deaths and migration
During the 10 years between mid 2018 and mid 2028, the projections for the UK as a whole suggest:
• 7.2 million people will be born
• 6.4 million people will die
• 5.4 million people will immigrate long term to the UK
• 3.3 million people will emigrate long term from the UK
This means that of the 3.0 million increase in total population, 0.8 million (27%) is projected to result from the higher number of births than deaths and 2.2 million (73%) is projected to result
directly from net international migration.
Over the full 25-year period between mid 2018 and mid 2043, the proportion of growth resulting from the balance of births and deaths is projected to be lower, at 16%, and that from net international
migration is projected to be higher, at 84%.
As Figure 2 shows, projected net international migration declines at first and then is constant from the year ending mid 2025. However, there is a steady increase in the number of deaths as people
born in the baby boom generations after World War Two and in the 1960s reach older ages. This means that although net migration is constant, it represents an increasing proportion of the projected
Figure 2: Over time, births and deaths reach similar levels so net international migration causes most growth
Projected births, deaths and net migration, UK, years ending mid 2019 to mid 2043
Source: Office for National Statistics – National population projections
Download this chart Figure 2: Over time, births and deaths reach similar levels so net international migration causes most growth
Image .csv .xls
Accounting for the indirect impact of international migration
As well as the direct impact, international migration has an indirect impact on the population as it changes the number of births and, to a lesser extent in the shorter term, the number of deaths.
For example, births to, and deaths of, people who had migrated to the UK; or births to, and deaths of, people who emigrated from the UK (and who would have given birth, or died, in the UK had they
not emigrated).
Once the indirect effect is included, international migration accounts for 79% of the projected UK population growth over the 10 years between mid 2018 and mid 2028. Over the 25 years between mid
2018 and mid 2043, the projected population would fall slightly if there were no migration. Because migrants are concentrated at young adult ages, the impact of migration on the projected number of
women of childbearing age is especially important over this period.
International migration to and from the UK before the projection base year of 2018 will also influence future population growth, in the sense that past migrants and their descendants will contribute
to the projected numbers of births and deaths. This aspect is complex, so our calculations of the indirect effect only take into account migration after mid 2018.
Section 5 provides details of our long-term assumptions for fertility, mortality and migration and how these assumptions have changed since the 2016-based projections.
Nôl i'r tabl cynnwys
4. Changing age structure
Figure 3: There is a growing number of older people in the UK
Age structure of the UK population, mid 2018 and mid 2043
Source: Office for National Statistics – National population projections
Download this image Figure 3: There is a growing number of older people in the UK
.png (73.8 kB) .xlsx (22.4 kB)
The population pyramid in Figure 3 compares the age structure of the population in mid 2018 with the projected age structure in mid 2043.
In mid 2018, there are more females than males at older ages, reflecting their higher life expectancy. The spike at age 71 years reflects the baby boom after World War Two and the wider area peaking
at age 53 years reflects the baby boom of the 1960s. The narrowing in the teenage years corresponds with the low birth rates around the turn of the millennium.
By mid 2043, all these features are still present in the pyramid, with the peaks and troughs now located 25 years higher up the age scale. The changes to the numbers at each age are constantly
evolving as a result of births, deaths, migration and everyone getting older. We will now consider each life stage.
More people at older ages
In mid 2043, there are projected to be many more people at older ages. This partly reflects the 1960s baby boomers now being aged around 80 years but also general increases in life expectancy. In mid
2018, there were 1.6 million people aged 85 years and over; by mid 2043, this is projected to nearly double to 3.0 million.
Variation in people of working ages
For people of working ages, some age groups have slightly more people, some slightly fewer. This is substantially affected by the number of people at each age in mid 2018, with international
migration also having the greatest impact at these ages.
Fewer young children and more adolescents
There are fewer young children in mid 2043 but more in their mid teens; this is influenced by our assumed fertility rates in the 2020s and 2030s being lower than those around 2010 but higher than
those around 2001 when UK fertility was at a record low.
Figure 4 shows the changing age structure by life stage: children, working age and pensionable age. By mid 2028, the number of children (those aged from 0 to 15 years) reduces slightly but taking
into account the planned increases in State Pension age (SPA) to 67 years old for both sexes, the number of those of pensionable age increases slightly. The number of people of working age has the
largest growth.
Continuing to mid 2043, Figure 4 shows that the numbers of children and people of working age are projected to remain around the mid-2028 levels, but the number of those at pensionable age increases
substantially. Over the full period from mid 2018 to mid 2043, the number of people of pensionable age increases by 3.6 million (30%).
Figure 4: The number of people of pensionable age is projected to grow the most
UK population by life stage, mid 2018, mid 2028 and mid 2043
Source: Office for National Statistics – National population projections
1. Children are defined as those aged 0 to 15 years.
2. Working age and pensionable age populations are based on State Pension age (SPA) for the stated year. Under current legislation, the SPA in mid 2028 and mid 2043 will be 67 years old for both
Download this chart Figure 4: The number of people of pensionable age is projected to grow the most
Image .csv .xls
The numbers of people in each life stage are important when considering dependency ratios, which inform government financial planning. A common measure is the old-age-dependency ratio (OADR), which
is the number of people of pensionable age for every 1,000 people of working age. The OADR is projected to decline from 295 in mid 2018 to 290 in mid 2028, then rise to 360 by mid 2043.
Interactive population pyramids
You can explore in more detail how the UK population is projected to evolve over time in our interactive population pyramids. As well as our principal projection, on which all the analysis in this
bulletin is based, they also include a range of variant projections. National population projections, variant projections: 2018-based has more information on these.
Figure 5: Use our interactive population pyramids to explore our projections
Nôl i'r tabl cynnwys
5. Changes since the 2016-based projections
The 2018-based projections differ from the previous set, the 2016-based projections. This is partly because they are based on the population estimate from mid 2018 rather than mid 2016, as well as
the latest data on births, deaths and migration. We have also updated our assumptions about the future.
Net international migration
We have assumed higher net international migration. This is because we have retained the approach of basing our assumption on the average levels of migration over the past 25 years. Average annual
net long-term international migration over the 25 years between mid 1993 and mid 2018 was 190,000. This compares with the average of 165,000 between mid 1991 and mid 2016, which we set as the
assumption in the 2016-based projections.
We have assumed that women will have fewer children. This reflects the recent fall in total fertility rates, which has continued in the two years since we published the 2016-based projections.
Life expectancy
Life expectancy increases less than in the 2016-based projections. This is a consequence of the continued limited growth in life expectancy over the last two years.
Table 2: Summary of changes to long-term assumptions in UK projections, 2016-based and 2018-based
2016-based 2018-based
Net annual long-term international migration (year ending mid 2025 onwards) +165,000 +190,000
Long-term average number of children per woman 1.84 1.78
Life expectancy at birth, males, 2043 (years) 83.6 82.6
Life expectancy at birth, females, 2043 (years) 86.4 85.5
Download this table Table 2: Summary of changes to long-term assumptions in UK projections, 2016-based and 2018-based
.xls .csv
The mid-2018 UK population estimate was 30,000 lower than projected in the 2016-based projections, meaning a slightly lower starting point. Also, the changes indicated in Table 2 combine to reduce
future population growth. In consequence, comparing the 2016-based projections with the 2018-based projections:
• the projected UK population in mid 2028 was 69.8 million; this has been reduced to 69.4 million
• the projected UK population in mid 2043 was 73.3 million; this has been reduced to 72.4 million
• the UK population was projected to pass 70 million by mid 2029; it is now projected to do so by mid 2031
• the old-age-dependency ratio (OADR) in mid 2043 was projected to be 372; this has been reduced to 360
More information on the 2018-based assumptions and how we set them is available in National population projections, how the assumptions are set: 2018-based.
Nôl i'r tabl cynnwys
6. Comparisons with other countries
The EU statistical office, Eurostat, publishes population projections for the current members of the EU. They are based on 2018 and use different methods from those of the Office for National
Statistics (ONS). Eurostat projects that the UK population at the start of 2040 will be 75.3 million. This is substantially higher than our (2018-based) mid-2040 projection of 71.8 million.
Eurostat’s projections suggest the total population of the current EU members will increase by 2% between 2018 and 2040, varying between 40% growth for Luxembourg and 18% decline for Lithuania.
On that basis, the UK’s projected growth of 14% between 2018 and 2040 is much higher than the EU average. It is also the highest growth rate among the four largest nations in the EU: over the same
period, France’s population is projected to grow by 6% and Germany’s is projected to see a slight increase of 1%, while Italy’s population is projected to decline by 5%.
The UN also produces population projections. Their methods are different again, projecting a UK population of 73.0 million in mid 2043, which is 0.6 million higher than our projection. This is an 8%
increase on mid 2020, compared with a projected world population increase of 20% over the same period.
The UN’s projections for the world’s three most populous nations, also for the period mid 2020 to mid 2043, see an increase of 17% for India and a 12% increase for the US, while China’s population is
projected to see a very slight decline. At opposite ends of the scale, Niger’s population is projected to grow by 121%, while the populations of Lithuania, Latvia and Bulgaria are projected to
decline by 18%.
These comparisons demonstrate that projected population growth or decline varies considerably across the globe. It also shows that different methods can lead to substantially different results.
Nôl i'r tabl cynnwys
7. National population projections data
National population projections dataset
Datasets | Released 21 October 2019
You can use our table of contents tool to navigate through this release. The tool contains links to our full range of data and documentation. It lists all the datasets available (over 200) and allows
you to filter by variable and geography. You can also access methodological information and all related background information associated with the 2018-based national population projections (NPPs).
Nôl i'r tabl cynnwys
Long-term assumptions
The 2018-based national principal projections are based on a set of long-term assumptions considered to best reflect recent patterns of future fertility, mortality and net migration. The assumptions
• average UK completed family size will reach 1.78 children per woman by 2043, increasing to close to 1.79 later in the projection
• by 2043, the annual improvement in UK mortality rates will be 1.2% for most ages for both males and females
• from the year ending mid 2025 onwards, average annual net international migration to the UK will be plus 190,000
Life expectancies
Life expectancies at birth are period expectations of life; this is the average number of years that a newborn baby could expect to live if the mortality rates at the time of their birth stayed
constant through their lives. For example, life expectancy in the year between mid 2042 and mid 2043 reflects that projected for the start of 2043. It does not account for the continuing decline in
mortality rates projected after that point.
Old-age-dependency ratio (OADR)
The number of people of pensionable age for every 1,000 people of working age.
Population projections
Population projections provide statistics on potential future population levels of the UK and its constituent countries by age and sex. They are based on assumptions of future levels of births,
deaths and migration.
Total fertility rate
The total fertility rate (TFR) represents the average number of children born per woman if women experienced the age-specific fertility rates (ASFRs) of the year in question throughout their
childbearing lives.
Variant projections
Variant projections are based on alternative assumptions of fertility, mortality and migration compared with the principal projection. These provide an indication of uncertainty but do not represent
upper or lower limits of future demographic behaviour.
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The 2018-based national population projections (NPPs) provide statistics on potential future population levels of the UK and its constituent countries by age and sex. We base them on the estimated
population at 30 June 2018, using an internationally accepted methodology that accounts for the impact over time of the latest births, deaths and migration flows. This release supersedes the
2016-based projections.
Our principal projection is based on assumptions considered to best reflect recent patterns of fertility, life expectancy and migration. It is not possible to know how these patterns may change in
future so, to reflect this uncertainty, we also produce a number of variant population projections, based on alternative future scenarios.
To create our projections, we also use a set of demographic long-term assumptions for fertility, mortality and migration. We derive the assumptions through extrapolation of past trends and by
consideration of expert views.
We produce the variant projections using the same method but with alternative assumptions of future levels of fertility, mortality and migration. An overview of our decision-making process and
further detail on our methods is included in National population projections, how the assumptions are set: 2018-based.
More information on the quality and methodology of the NPPs, including the accuracy of the release and how the outputs meet users’ needs, is available in the Quality and Methodology Information (QMI)
A general background and methodology report on the NPPs is also available. This provides more detailed information on the methodology used to produce the projections.
Proposed timing of next projections
We will publish the 2018-based subnational projections for England, which break the NPPs in this publication down to local authority and health authority level, on 24 March 2020. We will then publish
the 2018-based household projections for England, which also go down to local authority level, in late spring or early summer 2020.
We usually publish population projections every two years. However, we are currently proposing not to produce 2020-based projections, which would theoretically be published in autumn 2021 for the
national and spring 2022 for the subnational projections. This is because the first 2021 Census results are also expected in spring 2022; we therefore propose that the next round of projections will
be based on 2021, enabling them to use the updated base population that the 2021 Census results will offer. This approach would also apply to our household projections.
At this stage, this is not a definitive policy and we cannot be certain of exact timings. Factors that will affect our plans include how different the 2021 Census results are from the current
population estimates and our evaluation of the causes of any differences. However, we aim to produce NPPs using a mid-2021 population base by around the end of 2022.
We would welcome any feedback on this proposed approach. In addition, please note that updates on this will be communicated in our quarterly Migration and Population Statistics Newsletter. To sign up
to this, please contact us at pop.info@ons.gov.uk.
Transformation of population statistics
It is our mission to provide the best insights on population and migration using a range of new and existing data sources to meet the needs of our users. Our ambition is to deliver a fully
transformed system by 2023, making regular improvements to our statistics along the way as more administrative data become available. We will rigorously quality assure new methods and share the
impact of any changes made. The Transformation of the population and migration statistics system: overview gives more information on this work. The resulting improvements will also be incorporated
into future editions of population projections.
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10. Strengths and limitations
The Office for National Statistics’ (ONS’) national population projections (NPPs) are used both within and outside of government as the definitive set of NPPs. We produce them on a consistent basis
for the constituent countries of the UK using the internationally accepted cohort component methodology. Examples of their uses include informing fiscal projections, identifying future demand for
health and education services, and estimating the future cost of state pensions.
We base the projections on the latest mid-year population estimates for each UK country and the latest births, deaths and migration data. The projections are not forecasts and so will differ from
actual future outcomes to a greater or lesser extent.
There is already a margin of error in the underlying data, for example, estimates of the current population and past migration flows. In addition, our assumptions about the future cannot be certain
as patterns of births, deaths and migration are always liable to change and can be influenced by many factors. In most cases, each set of projections is superseded when the next scheduled release is
published. However, should there be cause to revise a specific set of projections – for example, because of an error in production – the policy on revisions is outlined in the Quality and Methodology
Information (QMI) report.
Two factors that may affect future population are political and economic changes, but it is not possible to know in advance what impact these will have. On that basis, the projections do not attempt
to predict the impact of the UK leaving the EU. However, the projections of people of State Pension age (SPA) do reflect future changes under existing legislation.
This bulletin focuses on the first 25 years of the projections, up to mid 2043. The data files include projections going forward 100 years, up to mid 2118. However, such long-term projections are
inevitably very uncertain as much may change over that timescale.
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Manylion cyswllt ar gyfer y Bwletin ystadegol
Andrew Nash
Ffôn: +44 (0) 1329 44 4661
Efallai y bydd hefyd gennych ddiddordeb yn:
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Superconvergence of collocation methods for a class of weakly singular Volterra integral equations
Diogo, Teresa; Lima, Pedro M.
Journal of Computational and Applied Mathematics, 218 (2008), 307-316
We discuss the application of spline collocation methods to a certain class of weakly singular Volterra integral equations. It will be shown that, by a special choice of the collocation parameters,
superconvergence properties can be obtained if the exact solution satisfies certain conditions. This is in contrast with the theory of collocation methods for Abel type equations. Several numerical
examples are given which illustrate the theoretical results.
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In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that
has a similar property of symmetry.
Paraboloid of revolution
Every plane section of a paraboloid by a plane parallel to the axis of symmetry is a parabola. The paraboloid is hyperbolic if every other plane section is either a hyperbola, or two crossing lines
(in the case of a section by a tangent plane). The paraboloid is elliptic if every other nonempty plane section is either an ellipse, or a single point (in the case of a section by a tangent plane).
A paraboloid is either elliptic or hyperbolic.
Equivalently, a paraboloid may be defined as a quadric surface that is not a cylinder, and has an implicit equation whose part of degree two may be factored over the complex numbers into two
different linear factors. The paraboloid is hyperbolic if the factors are real; elliptic if the factors are complex conjugate.
An elliptic paraboloid is shaped like an oval cup and has a maximum or minimum point when its axis is vertical. In a suitable coordinate system with three axes x, y, and z, it can be represented by
the equation^[1] ${\displaystyle z={\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}.}$ where a and b are constants that dictate the level of curvature in the xz and yz planes respectively. In this
position, the elliptic paraboloid opens upward.
Hyperbolic paraboloid
A hyperbolic paraboloid (not to be confused with a hyperboloid) is a doubly ruled surface shaped like a saddle. In a suitable coordinate system, a hyperbolic paraboloid can be represented by the
equation^[2]^[3] ${\displaystyle z={\frac {y^{2}}{b^{2}}}-{\frac {x^{2}}{a^{2}}}.}$ In this position, the hyperbolic paraboloid opens downward along the x-axis and upward along the y-axis (that is,
the parabola in the plane x = 0 opens upward and the parabola in the plane y = 0 opens downward).
Any paraboloid (elliptic or hyperbolic) is a translation surface, as it can be generated by a moving parabola directed by a second parabola.
Properties and applications
Elliptic paraboloid
Polygon mesh of a circular paraboloid
Circular paraboloid
In a suitable Cartesian coordinate system, an elliptic paraboloid has the equation ${\displaystyle z={\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}.}$
If a = b, an elliptic paraboloid is a circular paraboloid or paraboloid of revolution. It is a surface of revolution obtained by revolving a parabola around its axis.
A circular paraboloid contains circles. This is also true in the general case (see Circular section).
From the point of view of projective geometry, an elliptic paraboloid is an ellipsoid that is tangent to the plane at infinity.
Plane sections
The plane sections of an elliptic paraboloid can be:
• a parabola, if the plane is parallel to the axis,
• a point, if the plane is a tangent plane.
• an ellipse or empty, otherwise.
Parabolic reflector
On the axis of a circular paraboloid, there is a point called the focus (or focal point), such that, if the paraboloid is a mirror, light (or other waves) from a point source at the focus is
reflected into a parallel beam, parallel to the axis of the paraboloid. This also works the other way around: a parallel beam of light that is parallel to the axis of the paraboloid is concentrated
at the focal point. For a proof, see Parabola § Proof of the reflective property.
Therefore, the shape of a circular paraboloid is widely used in astronomy for parabolic reflectors and parabolic antennas.
The surface of a rotating liquid is also a circular paraboloid. This is used in liquid-mirror telescopes and in making solid telescope mirrors (see rotating furnace).
Parallel rays coming into a circular paraboloidal mirror are reflected to the focal point, F, or vice versa
Parabolic reflector
Rotating water in a glass
Hyperbolic paraboloid
A hyperbolic paraboloid with lines contained in it
Pringles fried snacks are in the shape of a hyperbolic paraboloid.
The hyperbolic paraboloid is a doubly ruled surface: it contains two families of mutually skew lines. The lines in each family are parallel to a common plane, but not to each other. Hence the
hyperbolic paraboloid is a conoid.
These properties characterize hyperbolic paraboloids and are used in one of the oldest definitions of hyperbolic paraboloids: a hyperbolic paraboloid is a surface that may be generated by a moving
line that is parallel to a fixed plane and crosses two fixed skew lines.
This property makes it simple to manufacture a hyperbolic paraboloid from a variety of materials and for a variety of purposes, from concrete roofs to snack foods. In particular, Pringles fried
snacks resemble a truncated hyperbolic paraboloid.^[4]
A hyperbolic paraboloid is a saddle surface, as its Gauss curvature is negative at every point. Therefore, although it is a ruled surface, it is not developable.
From the point of view of projective geometry, a hyperbolic paraboloid is one-sheet hyperboloid that is tangent to the plane at infinity.
A hyperbolic paraboloid of equation ${\displaystyle z=axy}$ or ${\displaystyle z={\tfrac {a}{2}}(x^{2}-y^{2})}$ (this is the same up to a rotation of axes) may be called a rectangular hyperbolic
paraboloid, by analogy with rectangular hyperbolas.
Plane sections
A hyperbolic paraboloid with hyperbolas and parabolas
A plane section of a hyperbolic paraboloid with equation ${\displaystyle z={\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}}$ can be
• a line, if the plane is parallel to the z-axis, and has an equation of the form ${\displaystyle bx\pm ay+b=0}$ ,
• a parabola, if the plane is parallel to the z-axis, and the section is not a line,
• a pair of intersecting lines, if the plane is a tangent plane,
• a hyperbola, otherwise.
STL hyperbolic paraboloid model
Examples in architecture
Saddle roofs are often hyperbolic paraboloids as they are easily constructed from straight sections of material. Some examples:
• Philips Pavilion Expo '58, Brussels (1958)
• IIT Delhi - Dogra Hall Roof
• St. Mary's Cathedral, Tokyo, Japan (1964)
• St Richard's Church, Ham, in Ham, London, England (1966)
• Cathedral of Saint Mary of the Assumption, San Francisco, California, US (1971)
• Saddledome in Calgary, Alberta, Canada (1983)
• Scandinavium in Gothenburg, Sweden (1971)
• L'Oceanogràfic in Valencia, Spain (2003)
• London Velopark, England (2011)
• Waterworld Leisure & Activity Centre, Wrexham, Wales (1970)
• Markham Moor Service Station roof, A1(southbound), Nottinghamshire, England
• Cafe "Kometa", Sokol district, Moscow, Russia (1960). Architect V.Volodin, engineer N.Drozdov. Demolished.
Warszawa Ochota railway station
, an example of a hyperbolic paraboloid structure
Surface illustrating a hyperbolic paraboloid
Restaurante Los Manantiales, Xochimilco, Mexico
Hyperbolic paraboloid thin-shell roofs at
, Valencia, Spain (taken 2019)
Markham Moor Service Station roof, Nottinghamshire (2009 photo)
Cylinder between pencils of elliptic and hyperbolic paraboloids
elliptic paraboloid, parabolic cylinder, hyperbolic paraboloid
The pencil of elliptic paraboloids ${\displaystyle z=x^{2}+{\frac {y^{2}}{b^{2}}},\ b>0,}$ and the pencil of hyperbolic paraboloids ${\displaystyle z=x^{2}-{\frac {y^{2}}{b^{2}}},\ b>0,}$ approach
the same surface ${\displaystyle z=x^{2}}$ for ${\displaystyle b\rightarrow \infty }$ , which is a parabolic cylinder (see image).
The elliptic paraboloid, parametrized simply as ${\displaystyle {\vec {\sigma }}(u,v)=\left(u,v,{\frac {u^{2}}{a^{2}}}+{\frac {v^{2}}{b^{2}}}\right)}$ has Gaussian curvature ${\displaystyle K(u,v)={\
frac {4}{a^{2}b^{2}\left(1+{\frac {4u^{2}}{a^{4}}}+{\frac {4v^{2}}{b^{4}}}\right)^{2}}}}$ and mean curvature ${\displaystyle H(u,v)={\frac {a^{2}+b^{2}+{\frac {4u^{2}}{a^{2}}}+{\frac {4v^{2}}{b^
{2}}}}{a^{2}b^{2}{\sqrt {\left(1+{\frac {4u^{2}}{a^{4}}}+{\frac {4v^{2}}{b^{4}}}\right)^{3}}}}}}$ which are both always positive, have their maximum at the origin, become smaller as a point on the
surface moves further away from the origin, and tend asymptotically to zero as the said point moves infinitely away from the origin.
The hyperbolic paraboloid,^[2] when parametrized as ${\displaystyle {\vec {\sigma }}(u,v)=\left(u,v,{\frac {u^{2}}{a^{2}}}-{\frac {v^{2}}{b^{2}}}\right)}$ has Gaussian curvature ${\displaystyle K
(u,v)={\frac {-4}{a^{2}b^{2}\left(1+{\frac {4u^{2}}{a^{4}}}+{\frac {4v^{2}}{b^{4}}}\right)^{2}}}}$ and mean curvature ${\displaystyle H(u,v)={\frac {-a^{2}+b^{2}-{\frac {4u^{2}}{a^{2}}}+{\frac {4v^
{2}}{b^{2}}}}{a^{2}b^{2}{\sqrt {\left(1+{\frac {4u^{2}}{a^{4}}}+{\frac {4v^{2}}{b^{4}}}\right)^{3}}}}}.}$
Geometric representation of multiplication table
If the hyperbolic paraboloid ${\displaystyle z={\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}}$ is rotated by an angle of π/4 in the +z direction (according to the right hand rule), the result is
the surface ${\displaystyle z=\left({\frac {x^{2}+y^{2}}{2}}\right)\left({\frac {1}{a^{2}}}-{\frac {1}{b^{2}}}\right)+xy\left({\frac {1}{a^{2}}}+{\frac {1}{b^{2}}}\right)}$ and if a = b then this
simplifies to ${\displaystyle z={\frac {2xy}{a^{2}}}.}$ Finally, letting a = √2, we see that the hyperbolic paraboloid ${\displaystyle z={\frac {x^{2}-y^{2}}{2}}.}$ is congruent to the surface ${\
displaystyle z=xy}$ which can be thought of as the geometric representation (a three-dimensional nomograph, as it were) of a multiplication table.
The two paraboloidal R^2 → R functions ${\displaystyle z_{1}(x,y)={\frac {x^{2}-y^{2}}{2}}}$ and ${\displaystyle z_{2}(x,y)=xy}$ are harmonic conjugates, and together form the analytic function ${\
displaystyle f(z)={\frac {z^{2}}{2}}=f(x+yi)=z_{1}(x,y)+iz_{2}(x,y)}$ which is the analytic continuation of the R → R parabolic function f(x) = x^2/2.
Dimensions of a paraboloidal dish
The dimensions of a symmetrical paraboloidal dish are related by the equation ${\displaystyle 4FD=R^{2},}$ where F is the focal length, D is the depth of the dish (measured along the axis of symmetry
from the vertex to the plane of the rim), and R is the radius of the rim. They must all be in the same unit of length. If two of these three lengths are known, this equation can be used to calculate
the third.
A more complex calculation is needed to find the diameter of the dish measured along its surface. This is sometimes called the "linear diameter", and equals the diameter of a flat, circular sheet of
material, usually metal, which is the right size to be cut and bent to make the dish. Two intermediate results are useful in the calculation: P = 2F (or the equivalent: P = R^2/2D) and Q = √P^2 + R
^2, where F, D, and R are defined as above. The diameter of the dish, measured along the surface, is then given by ${\displaystyle {\frac {RQ}{P}}+P\ln \left({\frac {R+Q}{P}}\right),}$ where ln x
means the natural logarithm of x, i.e. its logarithm to base e.
The volume of the dish, the amount of liquid it could hold if the rim were horizontal and the vertex at the bottom (e.g. the capacity of a paraboloidal wok), is given by ${\displaystyle {\frac {\pi }
{2}}R^{2}D,}$ where the symbols are defined as above. This can be compared with the formulae for the volumes of a cylinder (πR^2D), a hemisphere (2π/3R^2D, where D = R), and a cone (π/3R^2D). πR^
2 is the aperture area of the dish, the area enclosed by the rim, which is proportional to the amount of sunlight a reflector dish can intercept. The surface area of a parabolic dish can be found
using the area formula for a surface of revolution which gives ${\displaystyle A={\frac {\pi R\left({\sqrt {(R^{2}+4D^{2})^{3}}}-R^{3}\right)}{6D^{2}}}.}$
See also
External links
• Media related to Paraboloid at Wikimedia Commons
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