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From Wikipedia

Recurrence relation

In mathematics, a recurrence relation is an equation that recursively defines a sequence: each term of the sequence is defined as a function of the preceding terms.

The term difference equationsometimes (and for the purposes of this article) refers to a specific type of recurrence relation. Note however that "difference equation" is frequently used to refer to any recurrence relation.

An example of a recurrence relation is the logistic map:

x_{n+1} = r x_n (1 - x_n) \,

Some simply defined recurrence relations can have very complex (chaotic) behaviours, and they are a part of the field of mathematics known as nonlinear analysis.

Solving a recurrence relation means obtaining a closed-form solution: a non-recursive function of n.

Example: Fibonacci numbers

The Fibonacci numbers are defined using the linear recurrence relation

F_n = F_{n-1}+F_{n-2} \,

with seed values:

F_0 = 0 \,
F_1 = 1 \,

Explicitly, recurrence yields the equations:

F_2 = F_1 + F_0 \,
F_3 = F_2 + F_1 \,
F_4 = F_3 + F_2 \,

etc.

We obtain the sequence of Fibonacci numbers which begins:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

It can be solved by methods described below yielding the closed form expression which involve powers of the two roots of the characteristic polynomial t2 = t + 1; the generating function of the sequence is the rational function

\frac{t}{1-t-t^2}.

Structure

Linear homogeneous recurrence relations with constant coefficients

An order d linear homogeneous recurrence relation with constant coefficients is an equation of the form:

a_n = c_1a_{n-1} + c_2a_{n-2}+\cdots+c_da_{n-d} \,

where the d coefficients ci(for all i) are constants.

More precisely, this is an infinite list of simultaneous linear equations, one for each n>d−1. A sequence which satisfies a relation of this form is called a linear recursive sequence or LRS. There are d degrees of freedom for LRS, the initial values a_0,\dots,a_{d-1} can be taken to be any values but then the linear recurrence determines the sequence uniquely.

The same coefficients yield the characteristic polynomial (also "auxiliary polynomial")

p(t)= t^d - c_1t^{d-1} - c_2t^{d-2}-\cdots-c_{d}\,

whose d roots play a crucial role in finding and understanding the sequences satisfying the recurrence. If the roots r1, r2, ... are all distinct, then the solution to the recurrence takes the form

a_n = k_1 r_1^n + k_2 r_2^n + \cdots + k_d r_d^n,

where the coefficients kiare determined in order to fit the initial conditions of the recurrence. When the same roots occur multiple times, the terms in this formula corresponding to the second and later occurrences of the same root are multiplied by increasing powers of n. For instance, if the characteristic polynomial can be factored as (x − r)3, with the same root r occurring three times, then the solution would take the form

a_n = k_1 r^n + k_2 n r^n + k_3 n^2 r^n.\,

Rational generating function

Linear recursive sequences are precisely the sequences whose generating function is a rational function: the denominator is the auxiliary polynomial (up to a transform), and the numerator is obtained from the seed values.

The simplest cases are periodic sequences, a_n = a_{n-d}, n\geq d, which have sequence a_0,a_1,\dots,a_{d-1},a_0,\dots and generating function a sum of geometric series:

\begin{align} & \frac{a_0 + a_1 x^1 + \cdots + a_{d-1}x^{d-1}}{1-x^d} \\[6pt] & = \left(a_0 + a_1 x^1 + \cdots + a_{d-1}x^{d-1}\right) \\[3pt] & {} \quad + \left(a_0 + a_1 x^1 + \cdots + a_{d-1}x^{d-1}\right)x^d \\[3pt] & {} \quad + \left(a_0 + a_1 x^1 + \cdots + a_{d-1}x^{d-1}\right)x^{2d} + \cdots. \end{align}

More generally, given the recurrence relation:

a_n = c_1a_{n-1} + c_2a_{n-2}+\cdots+c_da_{n-d} \,

with generating function

a_0 + a_1x^1 + a_2 x^2 + \cdots,

the series is annihilated at a_d and above by the polynomial:

1- c_1x^1 - c_2 x^2 - \cdots - c_dx^d. \,

That is, multiplying the generating function by the polynomial yields

b_n = a_n - c_1 a_{n-1} - c_2 a_{n-2} - \cdots - c_d a_{n-d} \,

as the coefficient on x^n, which vanishes (by the recurrence relation) for n \geq d. Thus

(a_0 + a_1x^1 + a_2 x^2 + \cdots {} ) (1- c_1x^1 - c_2 x^2 - \cdots - c_dx^d) = (b_0 + b_1x^1 + b_2 x^2 + \cdots + b_{d-1} x^{d-1})

so dividing yields

a_0 + a_1x^1 + a_2 x^2 + \cdots =

\frac{b_0 + b_1x^1 + b_2 x^2 + \cdots + b_{d-1} x^{d-1}}{1- c_1x^1 - c_2 x^2 - \cdots - c_dx^d},

expressing the generating function as a rational function.

The denominator is x^d p\left(x^{-1}\right), a transform of the auxiliary polynomial (equivalently, reversing the order of coefficients); one could also use any multiple of this, but this normalization is chosen both because of the simple relation to the auxiliary polynomial, and so that b_0 = a_0.

Relationship to difference equations narrowly defined

Given an ordered sequence \left\{a_n\right\}_{n=1}^\infty of real numbers: the first difference \Delta(a_n)\, is defined as

\Delta(a_n) = a_{n+1} - a_n\,.

The second difference \Delta^2(a_n)\, is defined as

\Delta^2(a_n) = \Delta(a_{n+1}) - \Delta(a_n)\,,

which can be simplified to

\Delta^2(a_n) = a_{n+2} - 2a_{n+1} + a_n\,.

More generally: the kth difference of the sequence a_n\, is written as \Delta^k(a_n)\, is defined recursively as

\Delta^k(a_n) = \Delta^{k-1}(a_{n+1}) - \Delta^{k-1}(a_n)\,.

The more restrictive definition of difference equation is an equation composed of anand its kth differences. (A widely used broader definition treats "difference equation" as synonymous with "

Mean difference

The mean difference is a measure of statistical dispersion equal to the average absolute difference of two independent values drawn from a probability distribution. A related statistic is the relative mean difference, which is the mean difference divided by the arithmetic mean. An important relationship is that the relative mean difference is equal to twice the Gini coefficient, which is defined in terms of the Lorenz curve.

The mean difference is also known as the absolute mean difference and the Gini mean difference. The mean difference is sometimes denoted by Δ or as MD. Themean deviation is a different measure of dispersion.

Calculation

For a population of size n, with a sequence of values yi, i = 1 to n:

MD = \frac{1}{n(n-1)} \Sigma_{i=1}^n \Sigma_{j=1}^n | y_i - y_j | .

For a discrete probability functionf(y), where yi, i = 1 to n, are the values with nonzero probabilities:

MD = \Sigma_{i=1}^n \Sigma_{j=1}^n f(y_i) f(y_j) | y_i - y_j | .

For a probability density functionf(x):

MD = \int_{-\infty}^\infty \int_{-\infty}^\infty f(x)\,f(y)\,|x-y|\,dx\,dy .

For a cumulative distribution function F(x) with quantile function x(F):

MD = \int_0^1 \int_0^1 |x(F_1)-x(F_2)|\,dF_1\,dF_2 .

Relative mean difference

When the probability distribution has a finite and nonzero arithmetic mean, the relative mean difference, sometimes denoted by ∇ or RMD, is defined by

RMD = \frac{MD}{\mbox{arithmetic mean}}.

The relative mean difference quantifies the mean difference in comparison to the size of the mean and is a dimensionless quantity. The relative mean difference is equal to twice the Gini coefficient which is defined in terms of the Lorenz curve. This relationship gives complementary perspectives to both the relative mean difference and the Gini coefficient, including alternative ways of calculating their values.

Properties

The mean difference is invariant to translations and negation, and varies proportionally to positive scaling. That is to say, if X is a random variable and c is a constant:

  • MD(X + c) = MD(X),
  • MD(-X) = MD(X), and
  • MD(cX) = |c| MD(X).

The relative mean difference is invariant to positive scaling, commutes with negation, and varies under translation in proportion to the ratio of the original and translated arithmetic means. That is to say, if X is a random variable and c is a constant:

  • RMD(X + c) = RMD(X) · mean(X)/(mean(X) + c) = RMD(X) / (1+c / mean(X)) for c≠ -mean(X),
  • RMD(-X) = −RMD(X), and
  • RMD(cX) = RMD(X) for c> 0.

If a random variable has a positive mean, then its relative mean difference will always be greater than or equal to zero. If, additionally, the random variable can only take on values that are greater than or equal to zero, then its relative mean difference will be less than 2.

Compared to standard deviation

Both the standard deviation and the mean difference measure dispersion—how spread out are the values of a population or the probabilities of a distribution. The mean difference is not defined in terms of a specific measure of central tendency, whereas the standard deviation is defined in terms of the deviation from the arithmetic mean. Because the standard deviation squares its differences, it tends to give more weight to larger differences and less weight to smaller differences compared to the mean difference. When the arithmetic mean is finite, the mean difference will also be finite, even when the standard deviation is infinite. See the examples for some specific comparisons. The recently introduced distance standard deviation plays similar role than the mean difference but the distance standard deviation works with centered distances. See also E-statistics.

Sample estimators

For a random sample S from a random variable X, consisting of n values yi, the statistic

MD(S) = \frac{\sum_{i=1}^n \sum_{j=1}^n | y_i - y_j |}{n(n-1)}

is a consistent and unbiasedestimator of MD(X). The statistic:

RMD(S) = \frac{\sum_{i=1}^n \sum_{j=1}^n | y_i - y_j |}{(n-1)\sum_{i=1}^n y_i}

is a consistentestimator of RMD(X), but is not, in general, unbiased.

Confidence intervals for RMD(X) can be calculated using bootstrap sampling techniques.

There does not exist, in general, an unbiased estimator for RMD(X), in part because of the difficulty of finding an unbiased estimation for multiplying by the inverse of the mean. For example, even where the sample is known to be taken from a random variable X(p) for an unknown p, and X(p) - 1 has the Bernoulli distribution, so that Pr(X(p) = 1) = 1 âˆ’ p and , then

RMD(X(p)) = 2p(1 âˆ’ p)/(1 + p).

But the expected value of any estimator R(S) of RMD(X(p)) will be of the form:


From Encyclopedia

Einstein's Special Theory of Relativity EINSTEIN'S SPECIAL THEORY OF RELATIVITY

The German Jewish theoretical physicist Albert Einstein formulated the special theory of relativity in 1905. Relativity is that area of physics that has to do with how observers in motion with respect to the phenomenon observed can account for their observations given that two different frames of reference (that of the observer and that of what is observed) are involved. Einstein labeled his 1905 theory "special" because it dealt with a limited range of phenomena, namely uniform linear motion at constant but high velocities. The consequences of special relativity became cornerstones of twentieth-century physics and displaced some of the central tenets of Newtonian physics that had been pillars of scientific thought for two centuries. First, Einstein showed that time, space, and matter are interdependent, as expressed in the famous formula e - mc2 , where e is energy, m is mass, and c is the speed of light. The mass of material objects is determined by their energy; if they give off energy, their mass decreases. Mass increases with velocity and since the velocity of light is so great, a small mass traveling at the speed of light is equivalent to a vast amount of energy. (Atomic energy is an example of the special theory but was not based on it.) Second, time is not absolute: it depends on the circumstances of the ob-servers. Third, there are no privileged observers: what one person sees or measures may not be what another person measures, even if both think they are measuring the same phenomenon, especially if one is moving faster than the other. For example, according to the special theory, a person on a moving train and another on an adjacent embankment do not see the same light signal on the railroad station at the same time and thus cannot say that the two observations were simultaneous. Fourth, space is not absolute; it is only a conventional way of describing the relationships of objects. Fifth, as American physicist Albert Michelson had already demonstrated, the ether, an invisible substance that supposedly filled the entire universe and through which light waves were propagated, does not exist. The theory gave rise to all kinds of widely discussed paradoxes that delighted Einstein and his supporters and enraged physicists opposed to the theory. The most famous was the "twin paradox" of the space traveler. An astronaut travels through space at speeds approaching that of light for many years. When he returns to Earth, he finds his twin brother to be an old man while he has hardly aged at all. Einstein's theory explained that at very high speeds time slows down. There were few theoretical physicists in the United States when Einstein published his theory. Most physicists were experimentalists and were hostile to the theory that struck many of them as hopelessly abstract and counter to intuition and common sense. Moreover, they were committed to Maxwell's principles of electrodynamics, which purported space to be filled with aether. The first Americans to comment on relativity were Gilbert N. Lewis, professor of physical chemistry at MIT, and his student, Richard C. Tolman. In a paper published in 1909 they argued that Einstein's theory was both practical and based on empirical evidence. Other American scientists, with considerably less understanding, ridiculed special relativity, attacking it as metaphysical and unrelated to observation. Many were bothered by the counterintuitive nature of the theory and asserted that if a scientific theory were true, it would be, almost by definition, comprehensible to everyone. Tolman, on the other hand, always responded to such doubts by reasserting his conviction that Einstein's hypotheses could be tested by experiment. Both Einstein's early supporters and detractors in the United States, therefore, appealed to American scientific traditions of practicality and experimental verification. In 1914 Einstein broadened his concepts to include nonlinear motion in a general theory of relativity. This introduced his famous theory of curved space. When certain predictions contained in the theory were later verified (as when observations during the solar eclipse of 1919 confirmed that light rays emitted by stars bent when passing through the gravitational field of the sun), many scientists who previously had been skeptical now accepted the special theory as well. Einstein won the Nobel Prize in 1921, not for relativity—still considered too controversial—but for another of his discoveries of 1905, the photoelectric effect. The first book in English on special relativity was published by mathematician Robert D. Carmichael of Indiana University in 1912. At the same time, physicist Percy W. Bridgman began to work out the role of relativity in the philosophy of modern physics. But, perhaps because of the difficulty of the theory and its lingering aura of controversy, coverage of the theory in textbooks was sparse. Prior to World War II most American textbooks simply presented special relativity as a theory suggested by the Michelson-Morley experiment, which was understood as having proven that the speed of light was invariant. Such discussions were wrong on two counts: first, Michelson-Morley measured the speed of light but made no claims regarding its invariance; second, Einstein was in all likelihood unaware of that famous experiment when he conceived his theory. In 1906, four years after he had founded the Edison Portland Cement Company at Stewartsville, New Jersey, Thomas Edison announced that his solution to the problem of the housing shortage and inner city slums was the reinforced concrete house, which, if he had his way, could be built in a week. Edison thought he might be able to pour an entire house in one operation. Houses could then roll off a production line and be sold at a low price. Such houses could be built almost entirely of fire-resistant materials, thus also saving the cost of fire insurance. Edison identified bentonite clay as a substance that had the binding and stability requirements suitable to such monumental concrete structures and determined that more than one structure would have to be built on a single site in order to save on construction costs. All architectural and decorative features, from staircases to exterior flourishes, were included in the mold, and colors were added to the concrete mix to avoid any separate painting of the structure. After years of hoopla Edison had failed to produce a single house. Finally two houses were cast in Montclair, New Jersey (they remain standing today), but Edison decided that the process was too grandiose and complex. He then designed a smaller, two-bedroom model with a front porch, weighing 250,000 pounds, some 200,000 less than the Montclair prototypes. This house was built in South Orange, New Jersey, in 1910. Edison's ideas for the utility of concrete extended even to house-hold furnishings. He boasted that a line of furniture made from a kind of foamy concrete would cost half as much as wood and outlast the marriages of their buyers. In 1911 he actually molded some prototypes, including cabinets, a bathtub, and even a piano! Edison's cement business was to have some great achievements, including the building of Yankee Stadium, and several successful Edison houses were built in Union, New Jersey, in 1917, but the idea never caught on. In spite of the longevity of those houses that were built, Edison, at least in the area of design, was a prophet without honor. Michael Peterson, "Thomas Edison's Concrete Houses," The American Heritage of Invention and Technology, 11 (Winter 1996): 50-56. Thomas F. Glick, ed., The Comparative Reception of Relativity (Dordrecht, Netherlands: D. Reidel, 1987).


From Yahoo Answers

Question:

Answers:If serious and formal, we should refer to the definitions accepted by ISO(1993) and the other related terms as uncertainty and precision. Accuracy describes how closely to the truth something is. The accuracy of the measurement refers to how close the measured value is to the true or accepted (reference) value. Classical way of expressing accuracy is the error of measurement: (X) = X(M) - X(S) X(M) measured value X(S) - true (correct) value (a problem: if not known so called conventionally true (reference) value). Various influences acting together with measured value result in difference between measured and true value of the measured quantity (systematic errors; random errors). The accuracy is related only to the systematic ones. Therefore, there are 3 ways to improve the ACCURACY: - by choosing the proper instrument or equipment (calibrated most closely to the reference value) - by elimination of the possible sources of systematic errors (if they are known and controllable) - by correcting the result with the value of all known systematic errors (if they are estimated). The accuracy, has no direct connections with the repeatability, the uncertainty and the precision of the used instrument. It makes little sense to quote values of high precision (low random error) beyond the expected accuracy of the measurement (the method). Without stating the estimated accuracy, such a reading cannot be used in serious computations. It sounds as joke, but it is true: "If you have only one watch, you always know exactly what time it is. If you have two watches, you are never quite sure..., so refer to GMT." Good luck.

Question:And why would the absolute humidity be more than the relative humidity in summer?

Answers:Absolute humidity is the amount of water vapor in the air, often expressed in grams per liter. Relative humidity is the amount of water vapor in the air *expressed as a percentage of the maximum amount possible at a given temperature.* Since warm air can hold more water vapor than cold air, if the absolute humidity remains constant, then the relative humidity will decrease as the temperature increases. Since absolute and relative humidity are simply different ways of describing the same amount of water vapor in the air, one cannot be larger than the other regardless of temperature. The numerical value for the absolute humidity may be greater than the numerical relative-humidity percentage in hot weather, but because the units of measurement are different this doesn't mean that one is "larger" than the other.

Question:In my book, The Huntress, I have a character that I need help describing. His name is Caleb, and he's a shape-shifter that looks sixteen or seventeen but is actually a few centuries old. He is kind and considerate, and his true form is a jet black cat (American Shorthair) with green eyes. He's in love with Sasha, an evil shape-shifter, but also falls for a human girl, Isabelle, who he must protect. Caleb thought Sasha was dead for years, and he hated himself, thinking he had caused her death, so during the course of the book he feels hurt and cheated by her. I can't decide on which description of him to use: P.S. The story is written from Sasha's point of view. Description One: From my vantage point in the tree, I could see the scene in the clearing perfectly. Isabelle, the human I was instructed to kill, was sitting warming her hands by a campfire. She was pretty, but that was all I saw, because in the next second my eyes flashed to the second figure. All the breath whooshed out of my lungs and my heart skipped a beat. Caleb. It was Caleb. He looked exactly the smae as when I had last seen him, all those decades ago. Same curly black hair that he was constantly brushing out of his eyes. Same pale skin, pink lips and perfect features. And the same green eyes with very little white, and catlike slits for pupils. The eyes that were filled with all the compassion and kindness of Caleb's nature, the eyes that had held my gaze for years and years. He'd taken on a more active lifestyle since I left; he was still thin, but now his arms and chest had obvious definition, visible even through the trademark white shirt and black jacket. He still moved with the same feline grace, completely aware of his surrondings and any dangers they might pose. But there was a deep guilt in his eyes now, a flaming anger at himself that wouldn't go away no matter how much I stared. And I had a horrible feeling that I'd put that in his eyes. My heart twisted, and tears came to my eyes. "Oh, Caleb..." I choked. Description Two: From my vantage point in the tree, I could see the scene in the clearing perfectly. Isabelle, the human I was instructed to kill, was sitting warming her hands by a campfire. She was pretty, but that was all I saw, because in the next second my eyes flashed to the second figure. All the breath whooshed out of my lungs and my heart skipped a beat. Caleb. It was Caleb. He looked exactly the smae as when I had last seen him, all those decades ago. Same straight black hair that covered almost half his face and he was constantly brushing aside to see better. Same pale skin, pink lips and perfect features. And the same green eyes with very little white, and catlike slits for pupils. The eyes that were filled with all hidden the compassion and kindness of Caleb's nature, the eyes that had held my gaze for years and years. The eyes that told me, that for all his bad-boy good looks, he was gentle underneath. He'd taken on a more active lifestyle since I left; he was still thin, but now his arms and chest had obvious definition, visible even through the trademark white shirt and black jacket. He still moved with the same feline grace, completely aware of his surrondings and any dangers they might pose. But there was a brand new, deep guilt in his eyes now, a flaming anger at himself that wouldn't go away no matter how much I stared. And I had a horrible feeling that I'd put that in his eyes. My heart twisted, and tears came to my eyes. "Oh, Caleb..." I choked. So, which description of Caleb should I use in my story?

Answers:In my opinion you should use description two because most of the time people see that bad boy look and think trouble and this helps them see a different side. I also think that in description two you should change "a brand new, deep guilt" to "a deep guilt" because this guilt is supposed to be decades old not a brand new thing that just happened. It sounds like a great book. Good Luck!

Question:For example, 14 to 18 to me is slightly too big a gap. However, in 6 years, it would be 20 to 24 which to me, isn't so bad. I explained this to someone saying that if the ratio difference is lower (ie. 7:9 to 5:6), it's acceptable. We then argued that this is a retarded way of describing it. What do you guys think? I reckon it makes perfect sense. And before you say it, yes they are as smart as me. Promise.

Answers:Yes, in terms of general consensus, the ratio of ages in a relationship is a bit more representative than the straight difference. For example, I am 24 and suppose I were dating a 30 year old. (I'm actually dating a 40 year old, but the math made it suddenly seem inapprorpiate!) The ratio is 24 / 30 = 0.8, but the age difference is 6 years. Now, suppose the older person were only 20. Then, using the ratio, the younger person would be 16, which is marginal. I mean, its legality varies by jurisdiction, but few people would call a 20 year old a sicko for being interested in a 16 year old. Now, look at the age difference; the younger person would be only 14, which is much more clearly unacceptable. In general, we accept bigger age differences between partners who are themselves older, and a ratio is a good way to capture that.

From Youtube

Elephants Relatives? :Even though we may describe some of our family members as being as big as elephants, the hyrax is actually the elephants closest living relative. It is believed the giant hyracoid family evolved in different ways millions of years ago. Some evolved smaller and became the hyrax family, while others started the elephant family. Hyraxes, like elephants have four toes on each front foot and three on each back foot as well as other similar traits. Today they are broken down into 4 main species. The Southern and Western Hyrax, and then the Yellow-Spotted Rock and Cape Hyrax. They are listed as a vulnerable species due to their forest habitat being cut down for its wood. Some species are very social like elephants and can live in colonies up to of 50 members. Cape hyraxes often make large piles of dung and urine that eventually turn into hyraceum which has been used by locals as a medicine for treating epilepsy, convulsions, among other disorders. They are also hunted for their skin and as a food source. Some groups that protect the land that is home to wildlife like the hyrax: African Wildlife Foundation www.awf.org Conservation International http Wildlife Direct wildlifedirect.org Thanks for watching, and be sure to subscribe to my videos, which help me raise funds for wildlife conservation. -Brian *dedicated to Vinney Hyrax